Digitalization of design for support structures in laser powder bed fusion of metals 303122955X, 9783031229558

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Light Engineering für die Praxis

Katharina Bartsch

Digitalization of design for support structures in laser powder bed fusion of metals Edited by Claus Emmelmann

123

Digitalization of design for support structures in laser powder bed fusion of metals

Vom Promotionsausschuss der Technischen Universität Hamburg zur Erlangung des akademischen Grades Doktor-Ingenieurin (Dr.-Ing.)

genehmigte Dissertation

von Katharina Lisa Bartsch

aus Hamburg

2023

Betreuer: Prof. Dr.-Ing. Claus Emmelmann 1. Gutachter: Prof. Dr.-Ing. Claus Emmelmann 2. Gutachter: Prof. Dr.-Ing. Benedikt Kriegesmann Tag der mündlichen Prüfung: 29. September 2022

Light Engineering für die Praxis Reihe herausgegeben von Claus Emmelmann, Hamburg, Deutschland

Technologie- und Wissenstransfer für die photonische Industrie ist der Inhalt dieser Buchreihe. Der Herausgeber leitet das Institut für Laser- und Anlagensystemtechnik an der Technischen Universität Hamburg sowie die Fraunhofer-Einrichtung für Additive Produktionstechnologien IAPT. Die Inhalte eröffnen den Lesern in der Forschung und in Unternehmen die Möglichkeit, innovative Produkte und Prozesse zu erkennen und so ihre Wettbewerbsfähigkeit nachhaltig zu stärken. Die Kenntnisse dienen der Weiterbildung von Ingenieuren und Multiplikatoren für die Produktentwicklung sowie die Produktionsund Lasertechnik, sie beinhalten die Entwicklung lasergestützter Produktionstechnologien und der Qualitätssicherung von Laserprozessen und Anlagen sowie Anleitungen für Beratungs- und Ausbildungsdienstleistungen für die Industrie.

Katharina Bartsch

Digitalization of design for support structures in laser powder bed fusion of metals

Katharina Bartsch iLAS Institut für Laser- und Anlagensystemtechnik TUHH Technische Universität Hamburg Hamburg, Germany

ISSN 2522-8447 ISSN 2522-8455 (electronic) Light Engineering für die Praxis ISBN 978-3-031-22955-8 ISBN 978-3-031-22956-5 (eBook) https://doi.org/10.1007/978-3-031-22956-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Abstract The digital transformation of the manufacturing sector is among the most important megatrends in the global value creation. Additive manufacturing, with laser powder bed fusion of metals presenting the most mature technology for the production of metal end-use products, is considered an essential technology for digital production and is increasingly applied in industry. The design freedom allows for highly complex parts optimized for performance and efficiency, while the nature of additive manufacturing – being based on a digital model of the part and requiring no tools – enables decentralized on-demand production as well as customization of parts without additional cost. However, several barriers towards the broad industrial application exist. Among the most crucial challenges are the associated cost and the required experience regarding the manufacturing process. A possible approach to reduce production cost and eradicate the need for user experience lies in the complete digitalization of the value creation process, including design and manufacturing. A digital, automated support structure design procedure addresses most of the process steps lacking. Support structures are essential to the successful manufacturing by laser powder bed fusion, but require design experience because the automation of their generation is limited currently. As they do not belong to the final product, though, they increase production cost without adding value to the product. In order to advance today’s additive manufacturing towards first-time-right production with no need of extensive user knowledge as well as to reduce support structure-induced production cost, this thesis digitalizes the support design process by developing an algorithm-based, automated process for part-specific support structure design. In the first step, process simulation determines the actual load cases of the support structures. Then, using the space colonization method from algorithmic botany, the support structures are generated, which have a tree-like topology. The algorithm includes specific design rules for the tree shape, which have been derived from a systematic topology optimization study. The support structure design procedure is validated experimentally by applying a benchmark approach specifically developed to compare support structures in terms of their technical and economic performance. The whole thesis is based on the titanium alloy Ti-6Al-4V, for which a complete model of the thermo-physical, optical, and mechanical material properties is presented. The validation demonstrates the procedure’s capability to design support structures tailored to a specific part without the need of user experience. This enables the adoption of laser powder bed fusion even in smaller companies, who cannot afford to build up experience over several years. Furthermore, by advancing the digitalization of the manufacturing process, the possibility of the digital transformation of manufacturing companies is eased. Although the support structure volume is significantly reduced, and therefore the production cost, no significant savings in support structure-induced cost is achieved at the

level of single-lot production due to the numerical efforts required. However, with the ongoing efforts to introduce laser powder bed fusion to serial production, significant cost savings could be pointed out at larger production scale, as the influence of the support structure design on the support structure cost is dimished. By this, the technology adoption barrier is lowered, contributing to the current efforts of achieving broad industrial application for fully digital production by additive manufacturing.

Table of content

1

Introduction

1

2

Digital production by additive manufacturing

5

2.1

2.2

2.3

2.4

2.5

2.6

Laser powder bed fusion of metals (PBF-LB/M) .............................................5 2.1.1 Technical process ................................................................................5 2.1.2 Digital transformation of (additive) manufacturing ............................9 Digitalization of part design by topology optimization..................................12 2.2.1 Topology optimization methods .......................................................14 2.2.2 Numerical challenges ........................................................................21 2.2.3 Physics addressed in topology optimization .....................................23 2.2.4 Solver algorithms ..............................................................................25 2.2.5 Topology optimization for additive manufacturing ..........................26 Digitalization of PBF-LB/M process .............................................................28 2.3.1 Process modeling ..............................................................................28 2.3.2 Material modeling .............................................................................34 Support structures in PBF-LB/M ...................................................................44 2.4.1 Integration of supports in the manufacturing process .......................44 2.4.2 Challenges in the application of supports .........................................48 Support structure optimization .......................................................................49 2.5.1 Support structure avoidance ..............................................................53 2.5.2 Optimization of available support structures ....................................56 2.5.3 Development of novel support structures..........................................60 2.5.4 General characteristics of optimization approaches ..........................64 2.5.5 Optimization goals ............................................................................65 2.5.6 Quantification of optimization success .............................................66 2.5.7 Automated support structure removal ...............................................69 Economic evaluation of additive manufacturing ...........................................71 2.6.1 Cost calculation .................................................................................73 2.6.2 Cost prediction ..................................................................................76

3

Research Hypothesis and Methodology

79

4

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

83

4.1

Thermo-physical properties of Ti-6Al-4V .....................................................84 4.1.1 𝛽-transus temperature .......................................................................85 4.1.2 Solidus temperature ..........................................................................90 4.1.3 Liquidus temperature ........................................................................90

VIII

Table of content

4.2 4.3

5

Support Structure Topology Optimization 5.1

5.2

6

6.2

6.3

7.2

145

Digital tree (support) structure generation ...................................................145 6.1.1 Academic approaches to tree support generation ............................145 6.1.2 Commercial implementations of tree support structures .................148 6.1.3 Algorithmic botany .........................................................................151 Tree support modeling procedure ................................................................156 6.2.1 Data import .....................................................................................158 6.2.2 Creation of interface points .............................................................159 6.2.3 Tree sampling..................................................................................162 6.2.4 Tree design ......................................................................................164 6.2.5 Data export ......................................................................................170 Validation of design rule implementation ....................................................170

Support Structure Performance Benchmark 7.1

125

Topology optimization setup .......................................................................125 5.1.1 Tree evaluation parameters .............................................................128 5.1.2 Experimental plan ...........................................................................129 5.1.3 Mesh convergence study .................................................................133 Results of systematic support structure topology optimization ....................133 5.2.1 General observations .......................................................................133 5.2.2 Details on relevant tree design parameters ......................................136

Support Structure Design 6.1

7

4.1.4 Evaporation temperature ...................................................................91 4.1.5 Martensite start temperature ..............................................................92 4.1.6 Density ..............................................................................................93 4.1.7 Specific heat capacity........................................................................96 4.1.8 Thermal conductivity ......................................................................101 4.1.9 Powder material properties .............................................................106 Optical properties of Ti-6Al-4V ..................................................................109 Mechanical properties of Ti-6Al-4V ............................................................110 4.3.1 Young’s modulus ............................................................................118 4.3.2 Yield strength ..................................................................................119 4.3.3 Ultimate tensile strength .................................................................121 4.3.4 Poisson’s ratio .................................................................................121 4.3.5 Coefficient of thermal expansion ....................................................123

173

Technical Performance ................................................................................173 7.1.1 Design of benchmark parts .............................................................176 7.1.2 Definition of measurement methods ...............................................178 Economic Performance ................................................................................179

Table of content 7.2.1 7.2.2 8

IX Cost model for support structures in PBF-LB/M ............................179 Procedure for quick cost evaluation during the benchmark procedure ........................................................................................186

Demonstration of algorithmic support structures 8.1

8.2

8.3

8.4

8.5

189

Process simulation .......................................................................................189 8.1.1 Simulation setup..............................................................................190 8.1.2 Results & discussion .......................................................................192 Support generation .......................................................................................195 8.2.1 Block & cone supports ....................................................................195 8.2.2 Algorithmic tree supports................................................................199 Specimen manufacturing and technical evaluation ......................................205 8.3.1 Data preparation ..............................................................................206 8.3.2 Manufacturing results .....................................................................207 8.3.3 Support removal ..............................................................................210 8.3.4 Dimensional accuracy .....................................................................212 Economic evaluation ....................................................................................213 8.4.1 Input parameters..............................................................................213 8.4.2 Results .............................................................................................215 Benchmark summary & conclusion .............................................................222

9

Conclusion

225

10

References

229

Appendix

283 Technical documentation of slice test specimen ..........................................283 Technical documentation of the benchmark parts ........................................285 Static input parameter of cost model ............................................................290 Experimental results of demonstration – block support ...............................293 Experimental results of demonstration – cone support ................................295 Experimental results of demonstration – tree support (𝑑𝑔 = 2 mm) ...........297 Experimental results of demonstration – tree support (𝑑𝑔 = 3 mm) ...........299

List of figures Figure 2.1: PBF-LB/M procedure from part design to finished product ............................6 Figure 2.2: Temperature gradient mechanism, cf. [30] ......................................................8 Figure 2.3: Staircase effect due to approximation via layers .............................................8 Figure 2.4: Types of structural optimization ....................................................................13 Figure 2.5: Overview of common topology optimization approaches .............................15 Figure 2.6: Geometry mapping approaches for level-set topology optimization (cf. [69]) .......................................................................................................21 Figure 2.7: Checkerboard pattern (left) and the corresponding sensitivity values (right), cf. [54] ....................................................................................................22 Figure 2.8: Scales and physics in PBF-LB/M modeling ..................................................30 Figure 2.9: Thermal cycle of a specific point in PBF-LB/M ...........................................37 Figure 2.10: 2D supports available in the data preparation software Materialise Magics 45 Figure 2.11: 3D supports available in the data preparation software Materialise Magics 46 Figure 2.12: Annual published studies on support optimization and their corresponding categories ...............................................................................................50 Figure 2.13: Parameters of a block support wall (based on [276]). In 3D space, walls are arranged in a rectangular grid. ...............................................................58 Figure 2.14: Schwartz diamond and Schoen gyroid.........................................................61 Figure 2.15: Benchmark part of Cheng & To [290] .........................................................67 Figure 2.16: Benchmark part of Jiang et al. [342] ...........................................................67 Figure 2.17: Technology adoption stages and corresponding cost evaluation goals ........71 Figure 2.18: PBF-LB/M cost model of Kranz (adapted from [373]) ...............................76 Figure 3.1: Methodology of algorithmic support application ..........................................80 Figure 3.2: Structure of the thesis ....................................................................................81 Figure 4.1: Specimen production and setup of DSC measurements ................................86 Figure 4.2: Heating sequence of DSC for 𝑇𝛽 ..................................................................88 Figure 4.3: Cooling sequence of DSC for 𝑇𝛽 ..................................................................89 Figure 4.4: Data for 𝑇𝛽 from literature and DSC ............................................................89 Figure 4.5: Literature data (experimental and computational studies) for 𝑇𝑠 ..................90 Figure 4.6: Literature data (experimental studies) for 𝑇𝑙 .................................................91 Figure 4.7: Literature data (computational studies) for 𝑇𝑒 ..............................................92 Figure 4.8: Literature data (experimental studies) for 𝑇𝛼′𝑆 .............................................93 Figure 4.9: Result of the fitting procedure for the modeling of the density .....................95 Figure 4.10: Experimental data from the literature for the specific heat capacity ...........96 Figure 4.11: Results of DSC for 𝑐𝑝 (heating sequence) ...................................................98 Figure 4.12: Result of the fitting procedure (1 st section) for the modeling of the specific heat capacity.........................................................................................100

XII

List of figures

Figure 4.13: Result of the fitting procedure (excl. phase transformation) for the modeling of the specific heat capacity .................................................................101 Figure 4.14: Experimental data from the literature for the thermal conductivity ...........102 Figure 4.15: Specimen production and setup of LFA ....................................................102 Figure 4.16: Experimental results of LFA – comparison of different base cylinders ....104 Figure 4.17: Experimental results of LFA for the comparison of 𝛼′- and 𝛼 + 𝛽microstructure ......................................................................................105 Figure 4.18: Result of the fitting procedure for the modeling of the thermal conductivity..........................................................................................106 Figure 4.19: Result of the thermal conductivity measurements of the powder sample ..108 Figure 4.20: Experimental data from the literature for the absorptivity (* corrected for radiation loss) .......................................................................................109 Figure 4.21: Experimental data from the literature for the absorptivity of powder .......110 Figure 4.22: Comparison of experimental data on the Young’s modulus with regard to the build orientation .............................................................................114 Figure 4.23: Comparison of experimental data on the Young’s modulus with regard to the year of publication ..........................................................................115 Figure 4.24: Comparison of experimental data on the yield strength with regard to the build orientation ...................................................................................116 Figure 4.25: Comparison of experimental data on the yield strength with regard to the surface condition ..................................................................................117 Figure 4.26: Comparison of experimental data on the yield strength with regard to the year of publication ...............................................................................117 Figure 4.27: Result of the fitting procedure for the modeling of the Young’s modulus 119 Figure 4.28: Result of the fitting procedure for the modeling of the yield strength .......120 Figure 4.29: Result of the fitting procedure for the modeling of the Poisson’s ratio .....122 Figure 4.30: Result of fitting procedure for the coefficient of thermal expansion (* data converted from 𝐶𝑉𝑇𝐸 to 𝐶𝑇𝐸) ............................................................124 Figure 5.1: Result export and evaluation procedure (for parameters shown in Section 5.1.1) ....................................................................................................127 Figure 5.2: Tree evaluation parameters ..........................................................................128 Figure 5.3: Design domain variation (reference value in bold letters) ...........................129 Figure 5.4: Topology optimization results for the rectangular design domain, at constant load of 200 MPa ..................................................................................134 Figure 5.5: Topology optimization results for varying part-support interface geometry, at constant load of 200 MPa ....................................................................135 Figure 5.6: Topology optimization results for increasing, linear load, reference design domain (𝑎Ω = 𝑏Ω = 50 mm) ..............................................................135 Figure 5.7: Trunk X-position deviation from load center ..............................................137 Figure 5.8: Crown width for linear load cases and 𝛼𝑡 = 0° ...........................................138

List of figures

XIII

Figure 5.9: Crown geometry results for 𝛼𝑡 = 0° ...........................................................139 Figure 5.10: Segment overhang angle of linear load (200 Mpa), square (𝑎Ω = 𝑏Ω = 50 mm) use case...................................................................................140 Figure 5.11: Segment angle to parent segment results ...................................................141 Figure 5.12: Segment length results ...............................................................................142 Figure 5.13: Segment width results ................................................................................143 Figure 5.14: Segment geometry for constant load (100 MPa), square (𝑎Ω = 𝑏Ω = 70 mm) use case...................................................................................144 Figure 6.1: Examples of periodic tree supports: Parametric CAD design [313], fractal tree [482] ..............................................................................................146 Figure 6.2: Examples of semi-periodic tree supports: Discretized design domain [485], cone constraint [486] ............................................................................147 Figure 6.3: Graphical representation of the Lindenmayer sequence shown in Equation (6.1). The letters a, b are associated with sharp tips; c,d represent lateral margins of lobes; k characterizes notches. ...........................................153 Figure 6.4: Space colonization algorithm (adapted from [508]) ....................................156 Figure 6.5: Tree generation procedure ...........................................................................157 Figure 6.6: Parameters of point grid at the bottom face of the design space’s bounding box (top view) ......................................................................................159 Figure 6.7: Projection scheme ........................................................................................161 Figure 6.8: Tree design procedure .................................................................................164 Figure 6.9: Data preparation steps for space colonization .............................................165 Figure 6.10: Situations encountered during the tree generation by space colonization .167 Figure 6.11: Joining of individual solutions for the left and right tree structure sides ...168 Figure 7.1: Procedure of the benchmark development...................................................173 Figure 7.2: Support cost model structure and corresponding cost types ........................180 Figure 8.1: Procedure of PBF-LB/M process simulation and result post-processing ....189 Figure 8.2: Part orientation on build platform (recoater movement along Y-axis) ........207 Figure 8.3: Failure of line support / part E .....................................................................208 Figure 8.4: Line of defect recoater blade and damaged parts.........................................208 Figure 8.5: Bending of tree support (2 mm) in part E ....................................................210 Figure 8.6: Overall support cost of respective configurations .......................................215 Figure 8.7: Contribution of the production phases to the overall support cost...............217 Figure 8.8: Design operator time variation ....................................................................218 Figure 8.9: Number of benchmark part sets in single build job .....................................219 Figure 8.10: Tree support (3 mm) production phase contribution for different number of benchmark part sets in single build job ................................................220 Figure 8.11: Support cost per benchmark set for serial production ...............................221

List of tables Table 2.1: Progress of digitalization in PBF-LB/M .........................................................10 Table 2.2: Chemical composition of Ti-6Al-4V by ASTM B348....................................35 Table 2.3: Overview of Ti-6Al-4V material properties to be modeled ............................38 Table 2.4: Ti-6Al-4V material modeling of thermo-physical and optical properties in PBF-LB/M process simulation ...............................................................38 Table 2.5: Ti-6Al-4V material modeling of mechanical properties in PBF-LB/M process simulation ...............................................................................................43 Table 2.6: Literature overview regarding support optimization approaches ....................50 Table 2.7: Overview of literature on PBF-LB/M cost modeling......................................73 Table 4.1: Overview of experimental works on thermo-physical properties of conventionally processed Ti-6Al-4V .....................................................84 Table 4.2: Powder and process parameters of specimens used in DSC experiments .......86 Table 4.3: Thermal sequence of DSC for 𝑇𝛽 determination ............................................87 Table 4.4: Material model of Ti-6Al-4V: Density (functions and temperature intervals for one complete cycle of heating and cooling) .....................................95 Table 4.5: Thermal sequence of DSC for 𝑐𝑝 determination ............................................97 Table 4.6: Material model of Ti-6Al-4V: Specific heat capacity (functions and temperature intervals for one complete cycle of heating and cooling) .101 Table 4.7: Laser pulse parameters for LFA ...................................................................103 Table 4.8: Material model of Ti-6Al-4V: Thermal conductivity (functions and temperature intervals for one complete cycle of heating and cooling) .106 Table 4.9: Overview of tensile tests on PBF-LB/M Ti-6Al-4V specimens at 𝑇𝑅 .........112 Table 4.10: Material model of Ti-6Al-4V: Young’s modulus (functions and temperature intervals for one complete cycle of heating and cooling) .....................119 Table 4.11: Material model of Ti-6Al-4V: Yield strength (functions and temperature intervals for one complete cycle of heating and cooling).....................120 Table 4.12: Material model of Ti-6Al-4V: Poisson’s ratio (functions and temperature intervals for one complete cycle of heating and cooling) .....................122 Table 4.13: Material model of Ti-6Al-4V: Coefficient of thermal expansion (functions and temperature intervals for one complete cycle of heating and cooling) ................................................................................................124 Table 5.1: Overview of decisions regarding the topology optimization setup ...............127 Table 5.2: Research questions targeted by the design domain variation ........................130 Table 5.3: Research questions targeted by the load variation ........................................132 Table 5.4: Technical specification of computing system for the topology optimization .........................................................................................133 Table 5.5: Summary of the main findings of the support topology optimization study .136

XVI

List of tables

Table 6.1: Overview of tree supports generated by data preparation software available in the market .............................................................................................149 Table 6.2: L-systems example: cell states and production rules ....................................152 Table 6.3: Comparison of topology optimization and tree generation results for the use case of 𝑎Ω = 100 mm and 𝑏Ω = 50 mm at constant 𝐹𝑡 = 200 MPa, with 𝛼𝑡 = 0° (top) and 𝛼𝑡 = 10° (bottom) ..........................................171 Table 6.4: Tree design parameter in validation ..............................................................172 Table 7.1: Defects due to insufficient support performance with regard to the support tasks .....................................................................................................174 Table 7.2: Requirement specification of the benchmark procedure ...............................175 Table 7.3: Overview of benchmark parts .......................................................................177 Table 8.1: Mesh parameters of process simulation applied to the benchmark parts (cf. Table 7.3) .............................................................................................191 Table 8.2: Computation times for PBF-LB/M process simulation and post-processing of the benchmark parts (cf. Table 7.3)......................................................191 Table 8.3: Overview of maximum stresses calculated by Simufact Additive ................193 Table 8.4: Overview of generated block and cone supports ..........................................196 Table 8.5: Indication of design domain (red) and part geometry (blue).........................201 Table 8.6: Tree design parameter in demonstration .......................................................202 Table 8.7: Overview of generated tree supports ............................................................203 Table 8.8: Process parameters for manufacturing ..........................................................206 Table 8.9: 1st step of benchmark – visual inspection .....................................................209 Table 8.10: 2nd step of benchmark – support removal....................................................210 Table 8.11: 3rd step of benchmark – dimensional accuracy ...........................................213 Table 8.12: Dynamic input parameter of cost model .....................................................214 Table 8.13: Support design time components ................................................................214

List of listings Listing 6.1: Projection of gridded points to the part geometry ......................................160 Listing 6.2: Accessibility check for interface points to ensure post-processing ............161 Listing 6.3: Sampling of support points to individual tree input data ............................163 Listing 6.4: Tree topology design by space colonization ...............................................166 Listing 6.5: 3D modeling of the branches’ cuboid surfaces ...........................................169 Listing 8.1: Processing of simulation results .................................................................191

List of abbreviations Abbreviation

Description

1D 2D 3D ABC AM ANFIS ASTM B2B bcc BESO CAD CNC CPU DDM DED DfAM DIN

One-dimensional Two-dimensional Three-dimensional Activity-based costing Additive manufacturing Adaptive neuro-fuzzy interference system American Society for Testing and Materials Business to business Body-centric cubic Bi-directional evolutionary structural optimization Computer aided design Computer numerical control Central processing unit Direct digital manufacturing Directed energy deposition Design for additive manufacturing Deutsches Institut für Normung (engl.: German Institute for Standardization) Differential scanning calorimetry Differential thermal analysis Electron beam Electrical discharge machining Europäische Norm (engl.: european standard) Evolutionary structural optimization Finite element Finite element method Genetic algorithm Globally convergent method of moving asymptotes Hexagonal closed packing Hot isotatic pressing Heaviside projection method Informed Lindenmayer system International Organization for Standardization Laser beam Laser flash analysis

DSC DTA EB EDM EN ESO FE FEM GA GCMMA hcp HIP HPM IL-system ISO LB LFA

XX Abbreviation LiDAR L-systems MBB beam MEX/P MMA MOLE MPE MTPS OC OL-System PBF-EB/M PBF-LB/M PBF-LB/P PSO RAMP SIMP SLA SME STL TGM VTM X-FEM

List of abbreviations Description Light detection and ranging data Lindenmayer systems Messerschmitt-Bölkow-Blohm beam Material extrusion of polymers Method of moving asymptotes Monotonicity based minimum length scale method Mechanical process equivalent Modified transient plane source Optimality criteria Informationless Lindenmayer system Electron beam powder bed fusion of metals Laser powder bed fusion of metals Laser powder bed fusion of polymers Particle swarm optimization Rational material with penalization Solid isotropic material with penalization Stereolithography Small- and medium-sized enterprise Standard tesselation language Temperature gradient mechanism Virtual temperature method eXtended finite element method

Nomenclature Symbol

Description

Unit

𝐴

Area of space

m2

𝐴𝐿𝐹𝐴

Amplification factor in LFA

−

𝐴𝑚

Manufacturing phase AM system area

m2

𝐴𝑜

Design phase office area

m2

𝐴𝑤

Post-processing phase wire eroding area

m2

𝐴𝑤𝑠

Post-processing phase workshop area

m2

𝐴Ω

Area of design domain in topology optimization

m2

𝐶𝐷

Support design phase costs

€

𝐶𝐷,𝑒

Design phase equipment cost

€

𝐶𝐷,𝑜

Design phase operator cost

€

𝐶𝐷,𝑟

Design phase room cost

€

𝐶𝑒

Equipment cost

€

𝐶𝑖

Capital investment of equipment

€

𝐶𝑖,𝑓

Manufacturing phase filter price

€

𝐶𝑖,𝐻𝑊

Design phase hardware capital investment

€

𝐶𝑖,𝑚

Manufacturing phase AM system capital investment

€

𝐶𝑖,𝑡

Post-processing phase tools capital investment

€

𝐶𝑖,𝑤

𝐶𝑀,𝑚

Post-processing phase wire eroding system capital invest- € ment Manufacturing phase cost € Manufacturing phase material cost €

𝐶𝑀,𝑝

Manufacturing phase production cost

€

𝐶𝑀,𝑝,𝑒

Manufacturing phase energy cost

€

𝐶𝑀,𝑝,𝑓

Manufacturing phase filter cost

€

𝐶𝑀,𝑝,𝑔

Manufacturing phase gas cost

€

𝐶𝑀,𝑝,𝑚

Manufacturing phase machine cost

€

𝐶𝑜

Operator cost

€

𝐶𝑃

Post-processing phase cost

€

𝐶𝑃,𝑟,𝑒

Post-processing removal equipment cost

€

𝐶𝑃,𝑟,𝑜

Post-processing removal operator cost

€

𝐶𝑃,𝑟,𝑟

Post-processing phase removal workshop cost

€

𝐶𝑃,𝑝,𝑤

Post-processing phase wire eroding system cost

€

𝐶𝑀

XXII

Nomenclature

Symbol

Description

Unit

𝐶𝑃,𝑤,𝑒

Post-processing phase wire eroding energy cost

€

𝐶𝑟

Room cost

€

𝐶𝑆

Support cost

𝐶𝐴𝑇𝐸

Coefficient of area thermal expansion

€ 1⁄K

𝐶𝐿𝑇𝐸

Coefficient of linear thermal expansion

1⁄K

𝐶𝑟𝑤𝑡.%

Element percentage by weight of chromium

−

𝐶𝑇𝐸

Coefficient of thermal expansion

𝐶𝑇𝐸𝑙

Coefficient of thermal expansion of liquid phase

1⁄K 1⁄K

𝐶𝑇𝐸𝛼′

Coefficient of thermal expansion of 𝛼 ′-phase

1⁄K

𝐶𝑇𝐸𝛽

Coefficient of thermal expansion of 𝛽-phase

1⁄K

𝐶𝑉𝑇𝐸

Coefficient of volume thermal expansion

𝑫

Effective thermal conductivity tensor

1⁄K W⁄(mK)

𝐷

Distance for placing new nodes

mm

𝐸

Young’s modulus

GPa

𝐸0

Young’s modulus of bulk material in SIMP

GPa

𝑬𝒅

GPa

𝐸𝑙

Design Young’s modulus global matrix in topology optimization Young’s modulus of the liquid phase

𝐸𝑀,𝑚

Manufacturing phase AM system energy consumption

kW

𝐸𝑚𝑒𝑑

Median of experimental data for Young’s modulus

GPa

𝐸𝑚𝑖𝑛

Minimum Young’s modulus in SIMP

GPa

𝐸𝑃,𝑤

kW

𝐸𝑣

Post-processing phase wire eroding system energy consumption Volume energy density

𝐸𝛼 ′

Young’s modulus of the 𝛼 ′-phase

GPa

𝐸𝛽

Young’s modulus of the 𝛽-phase

GPa

𝑭

Global force vector

N

𝐹

Objective function in topology optimization

𝐹𝐺

Weight load

N

𝐹𝑡

Tensile load

N

𝑭𝒕𝒉

Global thermal load vector

W⁄m2

𝒢

Graph of a tree

𝐺0

Volume constraint in topology optimization

𝐺𝑗

Optional constraints in topology optimization

𝐹𝑒𝑤𝑡.%

Element percentage by weight of iron

GPa

J/mm³

−

Nomenclature

XXIII

Symbol

Description

Unit

𝐼𝐿𝐹𝐴

Iris setting in LFA

𝑲

Global stiffness matrix

− N ⁄m

𝑲𝒕𝒉

Global thermal conductivity matrix

W⁄(mK)

𝑴 𝑀̇

Global mass matrix

kg

AM system melt rate

m3 /h

𝑀𝑜𝑤𝑡.%

Element percentage by weight of molybdenum

−

𝑁𝑏𝑤𝑡.%

Element percentage by weight of niobium

−

𝑃

Perimeter of a topology optimization result

m

𝑃1

Control point of design domain Beziér curve (coordinates) mm Control point of design domain Beziér curve (coordinates) mm

𝑃2 𝑃𝐿

Control point of design domain Beziér curve (coordinates) mm Laser power W

𝑃𝑚𝑎𝑥

Maximum perimeter length in topology optimization

m

𝑆 ̅̅̅ 𝑆𝐷

Set of auxin sources

mm

Mean difficulty of support removal rating

−

𝑆𝐷,𝐶,𝑖

Part C difficulty of support removal rating

−

𝑆𝐷,𝐶𝑇,𝑖

Part CT difficulty of support removal rating

−

𝑆𝐷,𝐸,𝑖

Part E difficulty of support removal rating

−

𝑆𝐷,𝑆,𝑖

Part S difficulty of support removal rating

−

𝑆𝐷,𝑆𝑇,𝑖 ̅̅̅ 𝑆𝑅

Part ST difficulty of support removal rating

−

Mean support residual rating

−

𝑆𝑅,𝐶,𝑖

Part C support residual rating

−

𝑆𝑅,𝐶𝑇,𝑖

Part CT support residual rating

−

𝑆𝑅,𝐸,𝑖

Part E support residual rating

−

𝑆𝑅,𝑆,𝑖

Part S support residual rating

−

𝑆𝑅,𝑆𝑇,𝑖

Part ST support residual rating

−

𝑇

Temperature

K

𝑻

Temperature field

K

𝑇𝑎𝑚𝑏

Ambient temperature

K

𝑇𝑒

Evaporation temperature

K

𝑇𝑙

Liquidus temperature

K

𝑇𝑅

Room temperature

K

𝑇𝑟𝑒𝑓

Reference temperature

K

𝑇𝑠

Solidus temperature

K

𝑇𝛼 ′ 𝑆

Martensite start temperature

K

𝑃3

XXIV

Nomenclature

Symbol

Description

Unit

𝑇𝛽

𝛽-transus temperature

K

𝑼

Global displacement vector

m

𝑼𝒕𝒉

Global temperature vector

K

𝑈𝑇𝑆

Ultimate yield strength

MPa

𝑈𝑇𝑆𝑚𝑒𝑑

Median of experimental data for ultimate tensile strength

MPa

𝑉

Speed function in level-set topology optimization

𝑉

Volume

m3

𝑉0

Initial volume

m3

𝑉𝐵𝐸

Volume of PBF-LB/M machine build envelope

m3

𝑉𝐵

Overall block support volume

mm3

𝑉𝐶

Overall cone support volume

mm3

𝑉𝐶,𝐵

Part C block support volume

mm3

𝑉𝐶,𝐶

Part C cone support volume

mm3

𝑉𝐶,𝑇2

Part C tree support (𝑑𝑔 = 2 mm) volume

mm3

𝑉𝐶,𝑇3

Part C tree support (𝑑𝑔 = 3 mm) volume

mm3

𝑉𝐶𝑇,𝐵

Part CT block support volume

mm3

𝑉𝐶𝑇,𝐶

Part CT cone support volume

mm3

𝑉𝐶𝑇,𝑇2

Part CT tree support (𝑑𝑔 = 2 mm) volume

mm3

𝑉𝐶𝑇,𝑇3

Part CT tree support (𝑑𝑔 = 3 mm) volume

mm3

𝑉𝐸,𝐵

Part E block support volume

mm3

𝑉𝐸,𝐶

Part E cone support volume

mm3

𝑉𝐸,𝑇2

Part E tree support (𝑑𝑔 = 2 mm) volume

mm3

𝑉𝐸,𝑇3

Part E tree support (𝑑𝑔 = 3 mm) volume

mm3

𝑉𝐿𝐹𝐴

Voltage applied in LFA

V

𝑉𝑀,𝑠

Support volume

m3

𝑉𝑆,𝐵

Part S block support volume

mm3

𝑉𝑆,𝐶

Part S cone support volume

mm3

𝑉𝑆,𝑇2

Part S tree support (𝑑𝑔 = 2 mm) volume

mm3

𝑉𝑆,𝑇3

Part S tree support (𝑑𝑔 = 3 mm) volume

mm3

𝑉𝑆𝑇,𝐵

Part ST block support volume

mm3

𝑉𝑆𝑇,𝐶

Part ST cone support volume

mm3

𝑉𝑆𝑇,𝑇2

Part ST tree support (𝑑𝑔 = 2 mm) volume

mm3

𝑉𝑆𝑇,𝑇3

Part ST tree support (𝑑𝑔 = 3 mm) volume

mm3

𝑉𝑇2

Overall tree support (𝑑𝑔 = 2 mm) volume

mm3

Nomenclature

XXV

Symbol

Description

Unit

𝑉𝑇3

Overall tree support (𝑑𝑔 = 3 mm) volume

mm3

𝑉𝑤𝑡.%

Element percentage by weight of vanadium

−

𝑾𝒔

Global matrix of current total strain energy

Nm

𝑊𝑠0

Initial total strain energy

Nm

𝑿

Point in level-set topology optimization

mm

𝑍𝑟𝑤.𝑡%

Element percentage by weight of zirconium

−

𝑎

Absorption

−

𝑎𝑝

Powder material’s effective absorption

−

𝑎Ω

Design domain height

mm

𝑏Ω

Design domain width

mm

𝑐

Contour level in level-set function

−

𝑐𝐷,𝑤,𝑔

Design phase operator gross wage

€/h

𝑐𝐷,𝑤,𝑖

Design phase operator incidental wage

€/h

𝑐𝑒

Energy price

€/kWh

𝑐𝑔

Manufacturing phase gas price

€/m3

𝑐𝑖𝑟

Inclusion rate in BESO

−

𝑐𝑀,𝑠𝑒𝑟

Manufacturing phase AM system service cost

€/a

𝑐𝑚𝑎𝑡

Material purchase price

€

𝑐𝑃,𝑠𝑒𝑟

Post-processing phase wire eroding system service cost

€/a

𝑐𝑃,𝑤,𝑔

Post-processing phase operator gross wage

€/h

𝑐𝑃,𝑤,𝑖

Post-processing phase operator incidental wage

€/h

𝑐𝑝

Specific heat capacity

J⁄(kgK)

𝑐𝑝,𝑙

Specific heat capacity of liquid phase

J⁄(kgK)

𝑐𝑝,𝑝

Powder material’s effective specific heat capacity

J⁄(kgK)

𝑐𝑝,𝛼′,𝑐𝑜𝑜𝑙

Specific heat capacity of 𝛼′-phase, cooling

J⁄(kgK)

𝑐𝑝,𝛼′,𝑐𝑜𝑜𝑙,𝑝ℎ𝑎𝑠𝑒 Specific heat capacity of 𝛼′-phase, 𝛽-transus enthalpy during cooling 𝑐𝑝,𝛼′,ℎ𝑒𝑎𝑡

Specific heat capacity of 𝛼′-phase, heating

𝑐𝑝,𝛼′,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 Specific heat capacity of 𝛼′-phase, 𝛽-transus enthalpy during heating

J⁄(kgK) J⁄(kgK) J⁄(kgK)

𝑐𝑝,𝛼′,𝑙𝑖𝑛

Specific heat capacity of 𝛼′-phase, linear model

J⁄(kgK)

𝑐𝑝,𝛽,𝑐𝑜𝑜𝑙

Specific heat capacity of 𝛽-phase, cooling

J⁄(kgK)

𝑐𝑝,𝛽,𝑐𝑜𝑜𝑙,𝑝ℎ𝑎𝑠𝑒

Specific heat capacity of 𝛽-phase, latent heat of fusion during cooling

J⁄(kgK)

𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡

Specific heat capacity of 𝛽-phase, heating

J⁄(kgK)

XXVI Symbol

Nomenclature Description

𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 Specific heat capacity of 𝛽-phase, latent heat of fusion during heating

Unit J⁄(kgK)

𝑐𝑝,𝛽,𝑙𝑖𝑛

Specific heat capacity of 𝛽-phase, linear model

J⁄(kgK)

𝑐𝑟,𝑜

Annual operational costs of rented room

(€/m2 )⁄a

𝑐𝑟,𝑜,𝑚

Annual manufacturing phase workshop operational costs

(€/m2 )⁄a

𝑐𝑟,𝑜,𝑜

Annual design phase office room operational costs

(€/m2 )⁄a

𝑐𝑟,𝑜,𝑤

𝑐𝑟,𝑟

Annual post-processing phase wire eroding workshop op- (€/m2 )⁄a erational costs Annual post-processing phase tooling workshop opera(€/m2 )⁄a tional costs Rejection rate in BESO − Annual rent of room (€/m2 )⁄a

𝑐𝑟,𝑟,𝑚

Annual manufacturing phase workshop rent

(€/m2 )⁄a

𝑐𝑟,𝑟,𝑜

Annual design phase office room rent

(€/m2 )⁄a

𝑐𝑟,𝑟,𝑤

Annual post-processing phase wire eroding workshop rent (€/m2 )⁄a

𝑐𝑟,𝑟,𝑤𝑠

Annual post-processing phase tooling workshop rent

(€/m2 )⁄a

𝑐𝑠𝑐𝑟𝑎𝑝

Scrap material sale price

€

𝑐𝑆𝑊

Annual software cost

€/a

𝑐𝑤,𝑔

Gross wage of employee

€/h

𝑐𝑤,𝑖

Incidental wage of employee

€/h

𝑑

Thermal diffusivity (usually denoted by 𝛼 or 𝑎)

cm2 ⁄s

𝑑𝑏

Birth distance of new vein nodes from auxin sources

mm

𝑑𝑐

Diameter of constant size cone support

mm

𝑑𝑐,𝑏

Diameter of cone supports at bottom

mm

𝑑𝑐,𝑖

Diameter of cone supports at part-support interface

mm

𝑑𝑓

Fragmentation distance of block support

mm

𝑑𝑔

Hatching distance for equispaced point grid

mm

𝑑𝑔𝑥

Hatching distance for point grid in X-direction

mm

𝑑𝑔𝑦

Hatching distance for point grid in Y-direction

mm

𝑑ℎ

AM system hatch distance

μm (m)

𝑑𝑘

Kill distance of vein nodes from auxin sources

mm

𝑑𝑜

Inward offset of equisized point grid

mm

𝑑𝑜𝑥

Inward offset of point grid in X-direction

mm

𝑑𝑜𝑦

Inward offset of point grid in Y-direction

mm

𝑐𝑟,𝑜,𝑤𝑠 𝑐𝑟𝑟

Nomenclature

XXVII

Symbol

Description

Unit

𝑑𝑃,𝑤

m

𝑑𝑊𝑒𝑒𝑘

Post-processing phase additional wire eroding distance due to supports Workdays per week

𝑑𝑌𝑒𝑎𝑟

Workdays per year

d/a

𝑒

Mathematical constant, ‘Euler’s number’

−

𝑒

Edges in tree graph

mm

𝑓𝑐

Correction factor for 𝑇𝛽 determination from DSC

−

̇ 𝑓𝑀,𝑔

AM system gas flow

m3 /h

𝑔

Heat generation rate

W ⁄s

𝑔

Gravitational acceleration

m⁄s 2

ℎ0

Initial mesh size

mm

ℎ𝑐

Tree crown height

mm

ℎ𝑚𝑎𝑥

Maximum element size in topolgy optimization

mm

ℎ𝑠

Support height

mm

ℎ𝑥𝑦,𝑏

Hatch distance of block support

mm

ℎ𝑥𝑦,𝑐

Hatch distance of cone support

mm

𝑘

Thermal conductivity

𝑘𝑙

Thermal conductivity of liquid phase

W⁄(mK) W⁄(mK)

𝑘𝑝

Powder material’s effective thermal conductivity

W⁄(mK)

′

d/week

𝑘𝛼 ′

Thermal conductivity of 𝛼 -phase

W⁄(mK)

𝑘𝛽

Thermal conductivity of 𝛽-phase

W⁄(mK)

𝑙𝑒𝑥

Bounding box edge length in X-direction

mm

𝑙𝑒𝑦

Bounding box edge length in Y-direction

mm

𝑙𝑡

Tree segment length

mm

𝑙𝑡𝑟

Trunk segment length

mm

𝑚

Mass

kg

𝑛𝑙

AM system number of lasers

−

𝑛𝑠

Number of segments after split

−

𝑛𝑡

Number of trees

−

𝑝

Pressure

hPa

𝑝

Penalization factor in SIMP and RAMP

−

𝑞

−

𝑟𝑎

Weighting factor in final topology optimization problem formulation Radius of attraction

𝑟ℎ𝑤

Ratio of tree crown height and width

−

mm

XXVIII

Nomenclature

Symbol

Description

Unit

𝑟ℎ𝑤,𝑚𝑒𝑎𝑛

Mean ratio of tree crown height and width

−

𝑟𝑙𝑤

Ratio of tree segment length and width

−

𝑟𝑡

Tree parent-to-child width ratio

−

𝑟̅𝑡

Mean tree parent-to-child width ratio

−

𝑟Ω

𝑡

Regulation factor for FE mesh in final topology optimiza- − tion problem formulation Auxin sources mm Pseudo-time in level-set topology optimization −

𝑡𝑎

Amortization period

a

𝑡𝑎,𝑓

Manufacturing phase filter lifespan

a

𝑡𝑎,𝐻𝑊

Design phase hardware amortization period

a

𝑡𝑎,𝑡

Post-processing phase tool amortization period

a

𝑡𝑎,𝑚

Manufacturing phase AM system amortization period

a

𝑡𝑎,𝑤

a

𝑡𝐷,𝐵

Post-processing phase wire eroding system amortization period Overall block support design time

𝑡𝐷,𝐶

Overall cone support design time

min

𝑡𝐷,𝐶,𝐵

Part C block support design time

min

𝑡𝐷,𝐶,𝐶

Part C cone support design time

min

𝑡𝐷,𝐶,𝑇2

Part C tree support (𝑑𝑔 = 2 mm) design time

min

𝑡𝐷,𝐶,𝑇3

Part C tree support (𝑑𝑔 = 3 mm) design time

min

𝑡𝐷,𝐶𝑇,𝐵

Part CT block support design time

min

𝑡𝐷,𝐶𝑇,𝐶

Part CT cone support design time

min

𝑡𝐷,𝐶𝑇,𝑇2

Part CT tree support (𝑑𝑔 = 2 mm) design time

min

𝑡𝐷,𝐶𝑇,𝑇3

Part CT tree support (𝑑𝑔 = 3 mm) design time

min

𝑡𝐷,𝐷

Overall design domain design time

min

𝑡𝐷,𝐷,𝐶

Part C design domain design time

min

𝑡𝐷,𝐷,𝐶𝑇

Part CT design domain design time

min

𝑡𝐷,𝐷,𝐸

Part E design domain design time

min

𝑡𝐷,𝐷,𝑆

Part S design domain design time

min

𝑡𝐷,𝐷,𝑆𝑇

Part ST design domain design time

min

𝑡𝐷,𝐸,𝐵

Part E block support design time

min

𝑡𝐷,𝐸,𝐶

Part E cone support design time

min

𝑡𝐷,𝐸,𝑇2

Part E tree support (𝑑𝑔 = 2 mm) design time

min

𝑠

min

Nomenclature

XXIX

Symbol

Description

Unit

𝑡𝐷,𝐸,𝑇3

Part E tree support (𝑑𝑔 = 3 mm) design time

min

𝑡𝐷,𝑒

Design phase equipment occupancy time

h

𝑡𝐷,𝑙

Design phase labor time

h

𝑡𝐷,𝑆,𝐵

Part S block support design time

min

𝑡𝐷,𝑆,𝐶

Part S cone support design time

min

𝑡𝐷,𝑆,𝑇2

Part S tree support (𝑑𝑔 = 2 mm) design time

min

𝑡𝐷,𝑆,𝑇3

Part S tree support (𝑑𝑔 = 3 mm) design time

min

𝑡𝐷,𝑆𝑇,𝐵

Part ST block support design time

min

𝑡𝐷,𝑆𝑇,𝐶

Part ST cone support design time

min

𝑡𝐷,𝑆𝑇,𝑇2

Part ST tree support (𝑑𝑔 = 2 mm) design time

min

𝑡𝐷,𝑆𝑇,𝑇3

Part ST tree support (𝑑𝑔 = 3 mm) design time

min

𝑡𝐷,𝑇2

Overall tree support (𝑑𝑔 = 2 mm) design time

min

𝑡𝐷,𝑇3

Overall tree support (𝑑𝑔 = 3 mm) design time

min

𝑡𝑙

Labor time

h

𝑡𝑀,𝑏

Build time of manufacturing phase

h

𝑡𝑀,𝑑𝑒𝑙

Manufacturing phase AM system delay time

h

𝑡𝑀,𝑝

Manufacturing phase AM system recoating time

h

𝑡𝑃,𝑙

Post-processing phase labor time

h

𝑡𝑃,𝑙,𝑟𝑒𝑚

Post-processing phase labor time for support removal

h

𝑡𝑃,𝑙,𝑟𝑒𝑠

Post-processing phase labor time for residual removal

h

𝑡𝑃,𝑟

Overall support removal time

min

𝑡𝑃,𝑟,𝐶,𝑖

Part C support removal time

min

𝑡𝑃,𝑟,𝐶𝑇,𝑖

Part CT support removal time

min

𝑡𝑃,𝑟,𝐸,𝑖

Part E support removal time

min

𝑡𝑃,𝑟,𝑆,𝑖

Part S support removal time

min

𝑡𝑃,𝑟,𝑆𝑇,𝑖

Part ST support removal time

min

𝑡𝑃,𝑤

Post-processing phase additional wire eroding time

h

𝑡𝑡

Tree thickness

mm

𝑡𝑢,𝑒

Equipment time in use

h

𝑡𝑢,𝑟

Room time in use

h

𝑡𝑊𝑒𝑒𝑘

Work hours per week

𝑡ℎ

Layer thickness

h/week μm (m)

𝒖

State field in topology optimization

𝑢𝐷,𝑒

Design phase utilization rate of equipment

h/a

XXX

Nomenclature

Symbol

Description

Unit

𝑢𝐷,𝑜

Design phase utilization rate of room

h/a

𝑢𝑒

Utilization rate of equipment

h/a

𝑢𝐻𝑊

Design phase hardware utilization rate

h/a

𝑢𝑀,𝑚

Manufacturing phase AM system utilization rate

h/a

𝑢𝑃,𝑡

Post-processing phase tool utilization rate

h/a

𝑢𝑃,𝑤

Post-processing phase wire eroding system utilization rate h/a

𝑢𝑃,𝑤𝑠

Post-processing phase workshop utilization rate

h/a

𝑢𝑟

Utilization rate of room

h/a

𝑣

Vein nodes

𝑣𝑃,𝑤

mm Volume of an individual element in topology optimization m3 Post-processing phase wire eroding velocity m/h

𝑣𝑠

AM system scan velocity

mm/s (m/h)

𝑤𝑐

Tree crown width

mm

𝑤𝑔

Gap width of block support fragmentation

mm

𝑤𝑡

Tree segment width

mm

𝑤𝑡,𝐶ℎ𝑖

Child tree segment width

mm

𝑤𝑡,𝑚𝑖𝑛

Minimum tree segment width

mm

𝑤𝑡,𝑃𝑎𝑟

Parent tree segment width

mm

𝑥𝑡

Tree root position in X-direction

mm

Γ

Boundary of a level-set function

Δ𝑑30

Diameter deviation (30 mm)

mm

Δ𝑑50

Diameter deviation (50 mm)

mm

Δ𝑒0°

mm

𝛥ℎ

Edge deviation (𝛼𝑡 = 0°) Manufacturing phase difference in height due to supports

Δ𝑤𝐶𝑇

Part CT wall deviation

mm

Δ𝑤𝑆𝑇

Part ST wall deviation

mm

Δ𝛼0° Ω

Angle deviation (𝛼𝑡 = 0°) Design domain

° m2 /m3

𝛼𝑐

Tree crown mean overhang angle

°

𝛼𝑡

Overhang angle

°

𝛽𝑡

Angle to parent segment

°

𝛽𝑡,1

Angle to parent segment for tree level 1

°

𝛽𝑡,2

Angle to parent segment for tree level 2

°

𝛽𝑡,34

Angle to parent segment for tree levels 3 and 4

°

𝑣𝑖

m

Nomenclature

XXXI

Symbol

Description

𝛾

𝜀𝑐𝑟

Allowed area fraction in final topology optimization prob- − lem formulation Inherent strain mm Creep strain mm

𝜀𝑒

Elastic strain

mm

𝜀𝑒𝐼 𝜀𝑒𝑆

Elastic strain of intermediate state

mm

Elastic strain of steady state

mm

𝜀𝑝

Plastic strain

mm

𝜀𝑝𝑐

Phase change strain

mm

𝜀𝑡ℎ

Thermal expansion/strain

mm

𝜀𝑡𝑜𝑡

Total strain

mm

𝜀𝑥𝑥

Normal strain in X-direction

mm

𝜀𝑦𝑦

Normal strain in Y-direction

mm

𝜀𝑧𝑧

Normal strain in Z-direction

mm

𝜈

Poisson’s ratio

−

𝜈𝑙

Poisson’s ratio of liquid phase

−

𝜈𝛼′,𝛽

Poisson’s ratio of both 𝛼′- and 𝛽-phase

−

𝜋

Mathematical constant

−

𝜌

Density

kg/m3

𝝆𝒅

Gloabl matrix of design variable in topology optimization − Design variable value for an individual element in topo− logy optimization Minimal value of design variable in topology optimization −

𝜀∗

𝜌𝑑,𝑖 𝜌𝑑,𝑚𝑖𝑛

Unit

−

𝜌𝑙

Material fraction of an element in fictitious material geometry mapping in topology optimization Density of liquid phase

𝜌𝑀,𝑠

Support material density

kg/m3

𝜌𝑝

Powder material’s effective density

kg/m3

𝜌𝛼 ′

Density of 𝛼 ′-phase

kg/m3

𝜌𝛽

Density of 𝛽-phase

kg/m3

𝜎̅

Stress constraint in topology optimization

MPa

𝜎𝑐

Compressive stress

MPa

𝜎𝑒𝑣𝑚

Von Mises-stress of an individual solid element in topol- MPa ogy optimization Approximated von Mises-stress of an individual void ele- MPa ment in topology optimization

𝜌𝑒

𝜎̃𝑒𝑣𝑚

kg/m3

XXXII

Nomenclature

Symbol

Description

Unit

𝜎𝑖

Stress of an element in topology optimization

MPa

𝑣𝑚 𝜎𝑚𝑎𝑥

Maximum von Mises-stress in topology optimization

MPa

𝜎𝑚𝑎𝑥,𝐶

Maximum stress in part C

MPa

𝜎𝑚𝑎𝑥,𝐶𝑇

Maximum stress in part CT

MPa

𝜎𝑚𝑎𝑥,𝑆

Maximum stress in part S

MPa

𝜎𝑚𝑎𝑥,𝑆𝑇

Maximum stress in part ST

MPa

𝜎𝑡

Tensile stress

MPa

𝜎𝑦

Yield strength

MPa

𝜎𝑦,𝑙

Yield strength of liquid phase

MPa

𝜎𝑦,𝑚𝑒𝑑

Median of experimental data for yield strength

MPa

𝜎𝑦,𝑠

Yield strength of supports

MPa

′

𝜎𝑦,𝛼′

Yield strength of 𝛼 -phase

MPa

𝜎𝑦,𝛽

Yield strength of 𝛽-phase

MPa

𝜑

Porosity of powder bed

−

𝜙

Level-set function in topology optimization

1 Introduction Companies operate in a macro-environment that is significantly changing because of large, transformative megatrends [1]. In the manufacturing industry, e.g. aviation, automotive, tooling, or medical, megatrends include individualization of products [2], increase of product efficiency to combat climate change and scarcity of resources [3], as well as the digital transformation [4], among others. Additionally to the global megatrends, which have been present for several years already, the Covid-19 pandemic has a tremendous effect on global business decisions: Aside from enforcing the already existing megatrend of digital transformation, sustainable and resilient supply chains are of high priority [5]. This development is enhanced by the growing ambition of many countries to reinstall domestic supply chains as a direct result from the Covid-19 pandemic issues in global transfer of goods. Additive manufacturing (AM) is considered a key technology to address those trends due to its characteristics and capabilities [6, 7]: First, AM provides large freedom in design, enabling the manufacturing of complex structures conventional manufacturing techniques are not able to realize. Second, AM does not require tools and is based on the 3D model of a part; therefore customization of products does not lead to an increase in production cost. The 3D model as base of the AM production process characterizes AM as a digital manufacturing technology, benefitting the overall digital transformation of companies involved in product development and production. It further enables decentralized, on-demand manufacturing, as only a digital model has to be exchanged. Last, AM reduces material waste significantly compared to subtractive manufacturing technologies. Consequently, AM market growth has been impressive with double-digit annual growth rates for most of the last two decades [8]. The worldwide AM market has been estimated at $ 9.8B in 2018 and forecast to reach $ 35.6B in 2024 [9]. As the Covid-19 pandemic affects those growth rates, the market forecast has been corrected to $ 24.9B by a report summarizing different forecasts [10]. Furthermore, AM is still mainly used for prototyping rather than end-use products in 2020 [11]. This is expected to change rapidly due to the sudden attention AM is receiving in the context of the Covid-19 pandemic [10]. This trend goes along with an increasing use of metal-based AM processes. Here, the laser powder bed fusion of metals (PBF-LB/M) represents the AM process with the widest application in industry [9, 12], with an estimated market share of 85 % in metal AM technologies [13]. Though the development of AM is impressing, its market size is far from broad industrial application: In 2023, the AM market is expected to be at 3 % of the subtractive machining market [14]. Thomas-Seale et al. [15] identified 18 barriers to the progression of AM for end-use products in 2018. The most impacting barriers were found to be the associated costs, education, software, material range, finishing, and validation. This is especially true for PBF-LB/M. Here, a high upfront investment is required to establish the complete process chain. PBF© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_1

2

Introduction

LB/M manufacturing systems are priced in the range of $ 80K − $ 1M, with industrial scale systems starting at appr. $ 400K.1 Even though many executive boards acknowledge the possible cost savings of a fully optimized AM product, the initial investment cost is often considered a business risk outweighting the savings [15]. When the financial risk is taken on, the investment cost – together with the material cost – directly translates to the part-related cost, making PBF-LB/M unfeasible for large-scale production compared to conventional manufacturing techniques. However, with the increasing competition in the AM system market [9] the pressure for competitive pricing grows, leading to more affordable PBF-LB/M systems. Furthermore, PBF-LB/M is a complex manufacturing process with many details regarding processing and materials [16, 17], requiring a significant amount of experience. Although many similarities to welding exist [17], the use of powder layers makes PBF-LB/M unique in many ways. Hence, considerable training of personnel is crucial to be able to achieve satisfactory results. This is also the case for part design: Design for Additive Manufacturing (DfAM) includes process-induced restrictions as well as optimization methods such as topology optimization or biomimetics. The general, digitalized design procedure differs significantly from the traditional computer-aided design (CAD) protocol [2, 18], so training of designers in DfAM is necessary to create value. Additionally, CAD software capable of supporting those novel design approaches is required. With an estimated market size of $ 460M in 2020 and a forecast of $ 3.7B in 2027 [19], the current restricted availability of appropriate software is expected to be opened within the next years, though. In conclusion, apart from issues of technical maturity of PBF-LB/M regarding e.g. surface finishing, the high investment, associated cost, and time to build the required experience are great barriers for entering the AM market to many companies, especially small and medium enterprises (SME). Consequently, to promote the industrial adoption of AM and PBF-LB/M in particular, the digitalization of PBF-LB/M by introducing intelligent and automated methods to decrease user experience requirements and cost related to the part are essential in addition to the already expanding manufacturing system and software markets. A key factor in achieving this goal for PBF-LB/M are support structures. Support structures are crucial to the successful manufacturing of complex, optimized parts. They do not belong to the final, functional part, though, and are removed after processing, hence not contributing to the value created by the AM part design. The design of support structures involves deep knowledge of the PBF-LB/M process and its physics [20, 21]. Because no fully automated software solution is available yet, the support structure design may induce one of the following risks: On the one hand, there is the technical risk of build

1

Estimations provided at https://www.aniwaa.com/buyers-guide/3d-printers/best-metal3d-printer/, accessed 19.02.2022. Prices may vary over time.

Introduction

3

failure due to insufficient layout and dimensioning. On the other hand, significantly increased production cost caused by the excessive use of support structures out of fear for build failure is possible, which results in increased material consumption, build time, and a decrease of surface quality at the part-support interface. Support structure design addresses the major adoption barriers of cost, software, education, and finishing, and is related to many process steps lacking digitalization. Therefore, this field presents a considerable opportunity to tackle the general challenge of increasing AM adoption in industry and further advancing the PBF-LB/M technology by digitalizing the support design process. In this thesis, the algorithmic, automated design of support structures to achieve first-time-right production and decrease the associated cost is investigated. With the method presented, users with little to no knowledge in PBF-LB/M are enabled to create a support structure design with sufficient performance at reasonable cost, lowering the AM adoption barrier for SMEs.

2 Digital production by additive manufacturing The production of end-use components by AM is often named Direct Digital Manufacturing (DDM) [22] due to AM creating products directly from a 3D model with no further tooling involved. Hence, AM is considered a key factor in the digital transformation of the manufacturing industry [23]. In the following section, the state of the art of the PBF-LB/M process is presented. The process is evaluated regarding its state of digitalization. Current achievements as well as challenges to the goal of digital production are discussed to highlight possible approaches to advance the digital transformation of (additive) manufacturing.

2.1

Laser powder bed fusion of metals (PBF-LB/M)

According to DIN EN ISO/ASTM 52900, AM is a manufacturing technology where an object is created by the successive addition of material, usually in a layer-wise manner. In the standard, AM processes are categorized depending on whether a secondary process such as sintering is required to produce a consolidated part. Furthermore, AM processes can be distinguished by material type and form, state of fusion, material application, source of fusion, and the basic principle or process category according to DIN EN ISO 17296-2. The basic part properties are determined by the material type and form, the material fusion mechanism, as well as the machine architecture. Today, the most prominent AM process in the production of metal end-use parts is the laser powder bed fusion of metals [14].

2.1.1 Technical process In the process category of powder bed fusion (PBF), powder is applied layer-by-layer and selectively melted by focused energy provided by either laser (-LB) or electron beam (EB). The complete process of creating a product by PBF-LB/M is visualized in Figure 2.1. First, a CAD model of the product needs to be created (step 1). The geometrical freedom of design provided by PBF-LB/M allows for the incorporation of digital design methodologies such as topology optimization to find the optimal part design [2, 8]. In step 2, the digital part model is prepared for the manufacturing machine file generation. Here, the parts to be manufactured simultaneously are placed on a virtual build platform and orientated with regard to the build direction, which is denoted by the (vertical) Z-axis. If required, support structures are added. Then the geometries are divided into layers of a defined thickness along the build direction – a process called slicing [8] – and the scan vectors of the laser beam are generated according to a pre-defined scan strategy and hatch distance, i.e. the distance between two parallel scan vectors. Together with other process parameters, the scan vectors are translated into machine code executable by the respective © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_2

6

Digital production by additive manufacturing

manufacturing machine. Important process parameters include (among others) the laser power, scan velocity, laser focus diameter, layer thickness, as well as the hatch distance. To ensure the manufacturability of the digital setup, both step 1 and 2 may include the use of process simulation. After the successful file generation, the files are transferred to the manufacturing machine (step 3). Additionally, the machine setup is prepared for the manufacturing process, including e.g. powder material supply and the evacuation or process gas flooding of the build chamber.

Figure 2.1: PBF-LB/M procedure from part design to finished product

The PBF-LB/M manufacturing process (step 4) is an iterative process of selectively scanning consecutive powder layers with one or multiple laser beams following the scan vectors generated in step 2 [24]. First, the powder material is supplied by the powder feed region (step 4a, e.g. container on the left in Figure 2.1) and deposited on the prior layer – or the build platform, if it is the first layer – by a coating element (step 4b), e.g. a blade or roller [25]. The term ‘recoater’ summarizes all different shapes. To ensure the deposition of a complete powder layer, there is more powder supplied than actually necessary. The remaining powder is collected in the overflow region (e.g. container on the right in Figure 2.1). After the powder deposition, the layer is selectively scanned by the laser beam (step

Laser powder bed fusion of metals (PBF-LB/M)

7

4c), which is positioned and shaped by a system of lenses and scanning mirrors or galvanometer [25]. Finally, the build platform is lowered by one layer thickness (step 4d) and the procedure starts over again until the manufacturing of the geometries is complete. Depending on the setup of the overflow region, lowering of its bottom may be required as well. Figure 2.1 depicts an example of a possible machine hardware setup, each system manufacturer has its own configuration and may have different models varying in terms of e.g. build envelope, build-up rate, or laser power [26]. After the manufacturing, post-processing is required. Together with the build platform, the parts are cleaned from the surrounding powder and removed from the build chamber (step 5). In step 6, the parts are detached from the build platform, using subtractive technologies such as milling or electrical discharge machining. Prior to step 6, depending on the material, a stress-relief heat treatment may be necessary to prevent distortion after the detachment due to the relaxation of residual stresses. Last, if support structures are present, they are removed, as they do not belong to the part itself (step 7). Although PBF-LB/M allows for great freedom in part design, there are still some restrictions due to the process characteristics. The powder material commonly applied in PBF-LB/M is defined by a particle size distribution range in the interval of 10 – 63 µm, and has a mainly spherical shape [27]. Because of the small size and high flowability of the powder [27], the powder bed is not able to support overhanging features with an angle between the down-facing surface and the build platform of less than 30° [28]. The critical angle is dependent on the material as well as the AM machine used and may vary significantly, with values up to 45° [18]. Consequently, support structures must be added if the overhang restriction is violated. This also prevents the free arrangement of the part geometries in the 3D build envelope as known from the polymer process (PBF-LB/P) [29] since all parts have to be connected to the build platform, either by their own geometry, support structures, or a combination of support structures and other parts built on a lower level. Another process characteristic to be considered in PBF-LB/M is the repeated heating and cooling including high thermal gradients [30] as well as heating and cooling rates [31], leading to non-uniform microstructures and relatively large residual stresses [32]. The critical temperature gradient mechanism (TGM) proposed by Mercelis & Kruth [30], which is able to clearly explain the origin of residual stresses in PBF-LB/M on the macroscale, links the residual stress to the thermal expansion and shrinkage when the material is heating or cooling, respectively. The thermal expansion (𝜀𝑡ℎ ) of the top layer during heating is restricted by the underlying material, i.e. the build platform or previously built layers, inducing compressive strain leading to plastic deformation (𝜀𝑝 , cf. Figure 2.2). During cooling, the plastically compressed top layer shrinks, adding tensile stress (𝜎𝑡 ) on top of the compressive stress (𝜎𝑐 ). The combination of tensile and compressive stress results in an upward bending of the layer. The residual stresses are the origin of various defects in

8

Digital production by additive manufacturing

PBF-LB/M parts [32–35]: If not constrained by support structures, the bending of the single layers can deform the whole part. Apart from the low dimensional accuracy, the deformation of a part during manufacturing may lead to collisions with the recoating unit, damaging it or even causing a complete build failure. Furthermore, deformation may occur after the detachment from the build platform since the part is no longer spatially restricted. Another defect caused by residual stresses are cracks, especially at geometrical transition areas such as the build platform/part or the part/support structure interface. Stress-relief heat treatment is a suitable counter-measure to residual stresses, but may not be able to eradicate them completely [36]. In-process approaches such as high pre-heating temperatures of the build chamber or re-scanning the current layer several times to achieve annealing effects are considered in the literature, but come at a high expense in terms of manufacturing time [37].

Figure 2.2: Temperature gradient mechanism, cf. [30]

Due to the layer-wise manufacturing, the part geometry is only approximated by layers with a defined thickness in build direction. The resulting geometrical error is called the staircase effect (cf. Figure 2.3) and diminishes the surface quality as the macroscopic roughness is increased [38]. This effect has to be taken into account when choosing the part orientation on the build platform to avoid excessive post-processing efforts for meeting respective requirements.

Figure 2.3: Staircase effect due to approximation via layers

Apart from the described process characteristics, there are many possibilities to control the PBF-LB/M process: Rehme [39] distinguished 157 influencing parameters, which indicates the enormous effort required to understand and master all influences present. To advance the industrialization of PBF-LB/M nevertheless, various studies [8, 18, 24–26, 40] identified the following main challenges: restrictions of DfAM, limited choice of materials, low productivity, and missing quality assurance methods.

Laser powder bed fusion of metals (PBF-LB/M)

9

While AM enhances the freedom of designers to explore new concepts regarding part functionality and shape, a change in design methodology is required as the AM design will be substantially different from design for conventional manufacturing methods, e.g. the aspect of design for assembly will slowly go out of focus as AM allows to combine several parts into a single one [8, 40]. To support the development and adaption of new design methodologies, the development of AM-oriented design tools including the AM design space and the possibilities of multifunctional design is important. Furthermore, CAD software has to be improved, as current software is based on the conventional design methodology of adding or subtracting basic shapes rather than following functional approaches such as topology optimization [8, 26]. Also, the design restrictions of the respective AM processes need to be assessed [18]. Currently, the selection of suitable and available materials for PBF-LB/M is strongly restricted in comparison to materials available in conventional manufacturing [25, 26]. To enable the industry to transform existing products or create new products, more materials have to be qualified for PBF-LB/M processing to meet specific requirements. Furthermore, the physical properties of PBF-LB/M parts need to be determined and publicly known, e.g. mechanical strength, fatigue strength, or corrosion resistance. In order to be competitive, it is necessary to increase the productivity of PBF-LB/M [25, 26]. This can be achieved by either increasing the processing speed in combination with the use of high-power lasers or by introducing the simultaneous operation of multiple lasers. These approaches pose a great challenge to the system manufacturers, as the corresponding hardware, software, and control have to be developed. Additionally, as the underlying physics of the process such as thermal conduction are time-dependent, there are physical limitations to the operating speed of a single laser. Smart use of post-processing technologies, e.g. hot isostatic pressing (HIP), may enable a higher productivity despite the parts produced by the PBF-LB/M process not being fully dense, as demonstrated by Herzog et al. [41]. Last, to be able to produce critical parts, quality assurance is crucial to the admission procedure, especially in metal-based AM processes [25, 26]. Fast-response sensors to apply real-time monitoring and prediction of defects, microstructure, and dimensional accuracy for a closed-loop control are essential to maintain quality standards. The acquired data also helps in tracing the origins of faulty parts to avoid systematic errors.

2.1.2 Digital transformation of (additive) manufacturing Business leaders [42], politics [23], and researchers [43] agree that the digital transformation of the manufacturing industry is a key factor to long-term success, and link the transformation process to AM, among other technologies such as artificial intelligence and the internet of things. Consequently, research and development activities in AM focus

10

Digital production by additive manufacturing

on the digitalization of the processes to provide the required tools for digital transformation. However, assessing the current state of AM regarding digital transformation is difficult. A major aspect here is the lack of a general, widely adopted definition of the terms “digitization”, “digitalization”, and “digital transformation”. Vial [44] identified 28 publications from various fields of expertise offering 23 unique definitions of digital transformation. While the details of the definitions differ, especially between the respective fields of expertise, they agree on the distinction of the terms on an abstract level: “Digital transformation” refers to the use of digital technologies to enable completely new business models. The preliminary stage of digital transformation is “digitalization”, which refers to the use of digital technologies to improve business processes, e.g. by automation or the use of software. At last, “digitization” denotes the sole conversion of physical to digital data [45]. Especially the terms “digitization” and “digitalization” are often used interchangeably, which may be due to the fact that the definitions available do not further specify what is considered a “digital technology” [44]. Table 2.1: Progress of digitalization in PBF-LB/M

Process step Design

PBF-LB/M

Postprocessing

Digital representation of process

Automation of process

1

●

◕

2

●

◑

3

○

◔

4

◑

●

5

○

◔

6

◑

◔

7

○

◔

Legend: ○ not available ◔ slightly available ◑ partially available ◕ mostly available ● fully available Though there are reports on how businesses adopt new business models by implementing e.g. digital platforms or marketplaces in connection with AM [43], the market situation sketched out in Section 1 indicates that full digital transformation of and by AM is hard to achieve yet, particularly in metal AM. This is partially due to the missing digitalization of the processes, which is the base of digital transformation. An overview of the digitalization

Laser powder bed fusion of metals (PBF-LB/M)

11

progress of the seven PBF-LB/M process steps (cf. Figure 2.1) is presented in Table 2.1. Here, the digitalization of the process itself, i.e. its mapping in software, as well as the automation of the process is rated. The design of the part (step 1) and supports as well as the generation of the machine code (step 2) is performed with the help of (several) software. In part design, CAD modeling is already the standard in contrast to physical technical drawings. Furthermore, methods such as generative design or topology optimization allow for mostly automated design procedures [46], which are increasingly available as the number of specialized software as well as corresponding modules implemented in commercial CAD software is on the rise [22]. The same observation is made for slicing software, where the number of opensource as well as commercial specialized software is increasing, and software packages such as Siemens NX (Siemens AG, Munich, Germany)2 or Altair Inspire (Altair Engineering, Inc., Troy, USA)3 start to include the complete process chain from part design to final machine code in a single software entity. The PBF-LB/M system preparation (step 3) is a highly manual, physical process. Only the transfer of the machine code file to the PBF-LB/M system as well as individual steps like the evacuation and flooding of the build chamber are automated to the extend that the user has to initiate these steps by clicking corresponding buttons in the control software. The actual PBF-LB/M manufacturing, though, is completely automated. However, the digital representation of the PBF-LB/M process is still in development: On the one hand, process simulation to determine the occuring thermal and mechanical loads is commercially available in both specialized and integrated software [22]. On the other hand, extended concepts such as the digital twin require the development of comprehensive frameworks and have not yet been implemented in practice [47]. The idea of the digital twin is to create a virtual model of the PBF-LB/M process, including components, sensor data, and physics. Depending on the interaction between the physical and virtual object, the framework may be classified as digital model (no interaction), digital shadow (one-way interaction, e.g. sensor data being integrated in the numerical modeling of the process as input), or digital twin (two-way interaction, e.g. sensor data being used to predict the possible formation of defects and automatic adjust of the process parameters by the virtual model to avoid this) [4]. The availability of process simulation is thus an important step toward the implementation of a digital twin, but is to be classified as digital shadow. Currently, published studies at the scale of a full digital twin present theoretical frameworks only [47].

2

https://www.plm.automation.siemens.com/global/de/products/manufacturing-planning/additive-manufacturing.html, accessed 19.02.2022 3 https://www.altair.com/inspire-print3d/, accessed 19.02.2022

12

Digital production by additive manufacturing

The post-processing phase (step 5 – 7) is the least digitalized production phase compared to the design and manufacturing, which is assumed to be caused by the high individualization regarding the procedure as heat treatment, detachment, support removal, and surface finish involves numerous technologies. Additionally, steps such as heat treatment and surface finish are optional and depend on the material used as well as the functional requirements of the product. Similar to the system preparation step, the unpacking of the build chamber as well as the cleaning of the parts (step 5) comprise of highly manual and physical labor, with single optionally automated procedures, e.g. the removal of the build chamber from the PBF-LB/M system if a complete build container is utilized. System manufacturers aim at increasing the automation of unpacking and cleaning, as this step is also part of a closed powder handling system. Examples are the EOS M400-4 system (EOS GmbH, München, Germany)4 and the Concept Laser X Line (GE Additive, Boston, USA) 5 . However, subsequent steps require different technical equipment, which is not integrated into a closed production system yet. Furthermore, these systems are at serial production scale coming at corresponding prices, and are thus not suited for most of the current PBF-LB/M use cases (cf. Section 1). Accordingly, the digital representation of the post-processing is sparse, too. The only exception are the heat treatment and detachment step, which are often included in process simulation to derive the release of residual stresses and possible deformation after the detachment from the build platform [22]. The least digitalized process step is the support removal, which is most commonly performed with manual tools, not even automated subtractive machining. In conclusion, the PBF-LB/M process with all its corresponding production phases has not yet achieved full digitalization, though effort is made in both industry and academia. Interestingly, the application of supports is a reoccurring topic in the process steps lacking digitalization, indicating a high potential for the advancement of the digitalization if addressed appropriately. In the following sections, the already mostly digitalized steps of part design and PBF-LB/M manufacturing are described in detail to give a full understanding of the current state of the art. Furthermore, supports, the corresponding challenges and current approaches to the optimization of support application are investigated.

2.2

Digitalization of part design by topology optimization

Since resources such as materials are limited and costly, the part design strives to arrange material such that the structure maintains given loads in the best possible way. To achieve this goal, numerical methods of structural optimization are applied, which focus on the

4

https://www.eos.info/de/additive-fertigung/3d-druck-metall/eos-metall-systeme/eos-m400-4, accessed 19.02.2022 5 https://www.ge.com/additive/additive-manufacturing/machines/dmlm-machines/x-line2000r, accessed 19.02.2022

Digitalization of part design by topology optimization

13

structure’s performance rather than e.g. the planned manufacturing technology. Especially in combination with AM, whose geometrical freedom allows for the realization of the optimized structures without having to change a significant portion of the design, structural optimization is gaining importance [48].

Figure 2.4: Types of structural optimization

In structural optimization, several approaches are distinguished with regard to the optimized variable [49, 50]: size, shape, and topology optimization. Figure 2.4 provides a visualization of the respective approaches, using a perforated beam as an example. Sizing optimization determines the values of geometrical parameters, e.g. the diameter of the holes, with respect to the applied load. Shape optimization, however, operates on the contours of the geometry. In the example of Figure 2.4, shape optimization adapts the circular shape of the holes in order to increase material efficiency. While sizing and shape optimization require a set topology, topology optimization aims at creating the most efficient part topology, e.g. the number, position, and general shape of the holes.

14

Digital production by additive manufacturing

A structure’s topology is defined as the spatial arrangement of structural members and joints or internal boundaries [51]. Following this definition, topology optimization describes the variation of the connectivity between the structural elements to enhance the structural performance efficiency, i.e. the weight-performance ratio. Considering discrete structures such as a set of trusses, varying the connectivity is realized by creating or eliminating trusses between joints. Additionally, joints may be generated or repositioned. Similarly, for continuous structures the variation of the connectivity means separating or joining structural domains as well as creating and reducing them. In this work, the topology optimization of continuous structures is focused. For a concise introduction to the optimization of discrete structures, the reader is kindly referred to the work of Achtziger [52].

2.2.1

Topology optimization methods

The general topology optimization problem aims at finding the material distribution within the design domain Ω that minimizes an objective function 𝐹, which is subject to a volume constraint 𝐺0 ≤ 0 as well as 𝑀 optional additional constraints 𝐺𝑗 ≤ 0, 𝑗 = 1, … , 𝑀 [53], which may arise from the real life applications of the part designs. The material distribution is represented by the design variable 𝜌𝑑 (𝒙). Currently applied topology optimization methods can be categorized with regard to the nature of the design variable. Figure 2.5 gives an overview of the different categories as well as the most common topology optimization methods. First, similar to the structure to be optimized, the design variable may be discrete or continuous. In the case of a discrete design variable, the general topology optimization problem is mathematically formulated as shown in Equation (2.1), adopting the notation of [53]. Here, 𝒖 denotes a state field satisfying a state equation. The discrete design variable takes either the value of 0 or 1, which is corresponding to void or solid material at any point in the design domain [53, 54]. The design domain is discretized by finite elements. Note that for computational reasons, 𝜌𝑑 is never exactly 0, but a very small number 𝜌𝑑,𝑚𝑖𝑛 , avoiding remeshing or renumbering of the finite element mesh.

min :

𝜌𝑑 𝜌𝑑

s.t.:

𝐹(𝒖(𝝆𝒅 ), 𝝆𝒅 ) = ∑ ∫ 𝑓(𝒖(𝜌𝑑,𝑖 ), 𝜌𝑑,𝑖 )𝑑𝑉 𝑖

Ω𝑖

𝐺0 (𝝆𝒅 ) = ∑ 𝑣𝑖 ∗ 𝜌𝑑,𝑖 − 𝑉0 ≤ 0

(2.1)

𝑖

𝐺𝑗 (𝒖(𝝆𝒅 ), 𝝆𝒅 ) ≤ 0, 𝑗 = 1, … , 𝑀 𝜌𝑑,𝑖 = 0 ∨ 1, 𝑖 = 1, … , 𝑁

}

The most common type of topology optimization approaches with discrete design variables is the evolutionary approach. Introduced by Xie & Steven [55] as a so-called hard-

Digitalization of part design by topology optimization

15

kill strategy, where elements with low material stress are progressively removed, the evolutionary structural optimization (ESO) was developed to a bi-directional scheme (BESO). This concept, first presented by Querin et al. [56], allows for the elimination as well as introduction of elements according to a certain stress threshold, see Equation (2.2). An element is removed if the von Mises-stress of the respective element 𝜎𝑒𝑣𝑚 falls below a 𝑣𝑚 prescribed critical or maximum von Mises-stress of the whole structure 𝜎𝑚𝑎𝑥 , which is combined with a rejection rate 𝑐𝑟𝑟 [54]. The same procedure is applied to the addition of elements, where the approximated von Mises-stress of a void element 𝜎̃𝑒𝑣𝑚 needs to exceed the critical stress controlled by an inclusion rate 𝑐𝑖𝑟 . 𝑣𝑚 𝜎𝑒𝑣𝑚 < 𝑐𝑟𝑟 ∗ 𝜎𝑚𝑎𝑥 𝑣𝑚 𝑣𝑚 𝜎̃𝑒 > 𝑐𝑖𝑟 ∗ 𝜎𝑚𝑎𝑥

→ element removal } → element addition

Figure 2.5: Overview of common topology optimization approaches

(2.2)

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Based on this concept, the BESO approach has been extended by applying gradient analysis and filtering techniques to stabilize the results [57–59] as the original BESO implementation suffers from numerical issues such as checkerboard creation or mesh dependency (cf. Section 2.2.2 for more details). Additionally, several approaches towards a softkill BESO method have been proposed by Hinton & Sienz [60], Rozvany & Querin [61], Zhu et al. [62], and Huang & Xie [63]. Here, instead of completely removing an element from the design domain, the core material property (e.g. the Young’s modulus for structural studies) is reduced to a very small value. This way the element is physically still existent, but does not influence the structural response. Two major challenges in BESO approaches are the lack of algorithmic convergence as well as the appropriate selection of the stopping criteria [53]. Ansola et al. [58, 59] stopped the algorithm when the desired volume fraction has been achieved. In [64], the stopping criterion considers the variation of the objective function over the last ten iterations and the average over the past five objective function values. The algorithm is stopped when the difference of both values is small enough. These procedures bear the risk of premature termination, as the objective function will increase until the desired volume fraction is reached by filling the design domain with elements, and will decrease afterwards as the position of the elements is adjusted to optimize the structure’s performance. The procedure of Ansola et al. will terminate before the element positioning is adjusted, and the approach of [64] may terminate early if the objective function values are equal directly before and after volume fraction fulfilment [53]. Contrary to the discrete design variable, the continuous design variable can take on any value between 0 and 1, yielding

min :

𝜌𝑑 𝜌𝑑

s.t.:

𝐹(𝒖(𝝆𝒅 ), 𝝆𝒅 ) = ∑ ∫ 𝑓(𝒖(𝜌𝑑,𝑖 ), 𝜌𝑑,𝑖 )𝑑𝑉 𝑖

Ω𝑖

(2.3)

𝐺0 (𝝆𝒅 ) = ∑ 𝑣𝑖 ∗ 𝜌𝑑,𝑖 − 𝑉0 ≤ 0 𝑖

𝐺𝑗 (𝒖(𝝆𝒅 ), 𝝆𝒅 ) ≤ 0, 𝑗 = 1, … , 𝑀 0 ≤ 𝜌𝑑,𝑖 ≤ 1, 𝑖 = 1, … , 𝑁

}

as problem formulation. The continuous topology optimization methods can be further differentiated with regard to derivatives the design variable operates on: On the one hand, similar to the discrete design variable methods, element- or nodal-based design variables are used, on the other hand, shape derivatives (e.g. contours) are used. Sigmund & Maute [53] named these Eulerian and Lagrangian approaches, respectively.

Digitalization of part design by topology optimization

17

Eulerian topology optimization approaches Within the Eulerian approaches, the density approach is the most popular. Introduced by Bendsøe [65], the so-called SIMP (Solid Isotropic Material with Penalization) approach was quickly picked up in research [66, 67] and is one of the most applied topology optimization approaches today [68, 69]. In SIMP, the relationship of the design variable, i.e. the density, and the material property is defined via the power-law, e.g. for structural problems: 𝑝

𝐸𝑑 (𝜌𝑑,𝑖 ) = 𝜌𝑑,𝑖 ∗ 𝐸0

(2.4)

Here, 𝑝 denotes the penalization parameter, whereas 𝐸0 is the Young’s Modulus of the solid bulk material. If 𝑝 > 1 is applied, intermediate values of the density are penalized, driving the result towards a discrete solution. Setting 𝑝 to a value too low or too high leads to either extensive grey-scale results or convergence towards local minima [53]. However, 𝑝 = 3 has been identified as providing almost discrete solutions while maintaining good convergence, and has been confirmed to ensure physical realizability of elements with intermediate density values by Bendsøe & Sigmund [70]. To ensure numerical stability, the formulation of Equation (2.4) has been extended to approximate 𝐸𝑑 (𝜌𝑑,𝑖 ) for elements representing void material rather than assign a value of 0 [71]. 𝑝

𝐸𝑑 (𝜌𝑑,𝑖 ) = 𝐸𝑚𝑖𝑛 − 𝜌𝑑,𝑖 ∗ (𝐸0 − 𝐸𝑚𝑖𝑛 ), 𝐸0 < 𝐸𝑚𝑖𝑛 < 0

(2.5)

Stolpe & Svanberg [71] further proposed an alternative interpolation scheme, the Rational Approximation of Material Properties (RAMP) method. As can be seen when comparing Equation (2.4) and Equation (2.6), the term approximating the effective material property does not include the power-law anymore. 𝐸𝑑 (𝜌𝑑,𝑖 ) =

𝜌𝑑,𝑖 ∗𝐸 1 + 𝑝 ∗ (1 − 𝜌𝑑,𝑖 ) 0

(2.6)

The RAMP method was created to ensure convergence to discrete solutions by defining a non-zero gradient for void elements, however the approach seems to have little influence on practical problems [53]. Another Eulerian method is the phase field approach. Its goal is the minimization of the functional 1 2 𝐹̅ (𝒖(𝝆𝒅 ), 𝝆𝒅 ) = ∫ ( ∗ 𝑤(𝜌𝑑,𝑖 ) + 𝜖 ∗ ‖∇𝜌𝑑,𝑖 ‖ ) 𝑑𝑉 + 𝜂 ∗ 𝐹 𝜖 Ω𝑖

(2.7)

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where the double well function is 𝑤(𝜌𝑑,𝑖 ) = 0 for 𝜌𝑑,𝑖 = 0 ∧ 1 [53]. Due to 𝑤(𝜌𝑑,𝑖 ) penalizing intermediate design variable values, the phase field approach operates directly on the design variable rather than associated material properties. The functional of Equation (2.7) may be minimized via the Cahn-Hilliard-Equation [72], involving the solution of a fourth order differential equation, though. Hence, researchers [73, 74] propose the direct minimization of 𝐹̅ , which becomes possible if a volume constraint is added. Compared to other topology optimization approaches, phase field approaches are commonly associated with slow convergence rates [53]. As this approach is not relevant within this thesis, readers interested in further details of the phase field approach and Equation (2.7)’s components are referred to [53]. Lagrangian topology optimization approaches To create precise optimal topologies from the discretized topology optimization results, the topology optimization procedure may be combined with a consecutive shape optimization step operating on the contours derived from the topology optimization [75– 77]. The result of the topology optimization is either manually [76, 77] or automatically [75] converted to CAD, and the created shapes are then optimized without changing the topology. Another approach to combine topology and shape optimization is presented by the so-called bubble method introduced by Eschenauer et al. [78]. Here, after standard shape optimization is carried out, a new hole (bubble) is inserted, with its position determined by a strain energy criterion. The shape of the hole is then adapted via shape optimization again. Iteratively creating new topologies, the optimal topology may be determined by a global objective function. A further development of the shape and topology optimization combination is presented by the Lagrangian approach of topological derivatives [53]. The core idea is to predict the influence of inserting an infinitesimal hole at any given point in the design domain, and use this as an orientation for the generation of new holes in the topology. Topological derivatives may either be used in the context of the bubble method, directly employed in element-based schemes [79–81], or integrated as part of a level set approach [82, 83], which will be explained in the next paragraph. However, the computation of topological derivatives involves complex mathematics requiring respective computational resources [84, 85]. Furthermore, while comparing different approaches to the implementation of topological derivatives, Bonnet & Guzina [86] point out that those are derived for infinitesimal holes, whereas in practice finite size holes are introduced to the numerical implementation. Therefore, it is not concluded yet whether the computed derivatives are useful or better than the application of simple criterions, e.g. the local strain energy density, for hole introduction [53]. Another major class of topology optimization techniques are the level-set methods [69]. Here, the interfaces between the material phases (solid and void) are described by the iso-

Digitalization of part design by topology optimization

19

contours of a level-set function 𝜙(𝑿) (cf. Equation (2.8)), usually the zero-level contour (𝑐 = 0) is chosen as interface. If the value of the level-set function at a point 𝑿 is greater than the interface contour value 𝑐, the point is part of the design domain Ω. The point is affiliated with the void domain 𝐷\Ω when the value is smaller than 𝑐. At equality, the 𝑿 directly represents the boundary. 𝜙(𝑿) > 𝑐 ⇔ 𝜙(𝑿) = 𝑐 ⇔ 𝜙(𝑿) < 𝑐 ⇔

𝑋∈Ω 𝑋∈Γ } 𝑋 ∈ (D\Ω)

(2.8)

Originally applied to model interface evolution in multi-phase flows [87, 88], Haber & Bendsøe [89] suggested the use in topology optimization. Various level-set methods developed simultaneously, however, the most popular approach is to incorporate shape sensitivity analysis [69, 83]. Core feature of the level-set methods is the update of 𝜙(𝑿) via the Hamilton-Jacobi equation: 𝜕𝜙 + 𝑉|∇𝜙| = 0 𝜕𝑡

(2.9)

In Equation (2.9), 𝑉 denotes the speed function, also called velocity field, advecting the level-set function, and 𝑡 is a pseudo-time representing the design’s evolution in the optimization process [53]. It is important to note that level-set methods operating the original Hamilton-Jacobi equation are able to change the topology by shape variation and merging of holes, but for the introduction of new holes the combination with topological derivatives is required [69], as demonstrated by e.g. [82, 83]. As an alternative to the update of 𝜙(𝑿) via the Hamilton-Jacobi equation, the parameters of the discretized levelset function may be considered as optimization variables of a parameter optimization problem [53]. While this approach enables the treatment of multiple constraints in a standardized fashion, it requires the explicit introduction of various regularization techniques to control the level-set function. To change the implicit representation of the boundary geometry by the level-set function to an explicit one in the discretized physical model used to assess the structural performance, three main approaches for geometry mapping have emerged [53, 69]: conforming discretization, immersed boundary techniques, and fictitious material approaches. The concepts are visualized in Figure 2.6. In conforming discretization, the solid material domain only is discretized by a conforming finite element (FE) mesh. The boundaries of the model are represented particularly well compared to other methods. However, the continuous re-meshing during each iteration of the optimization process induces additional computational efforts. Furthermore, challenges related to the meshing of arbitrary 3D geometries may arise. The immersed boundary technique was developed

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to avoid re-meshing. While the discretization of the design domain remains fixed, the changing boundary conditions are enforced locally. This is achieved by the eXtended Finite Element Method (X-FEM), which uses local enrichments of elements cut by the level-set contour (see e.g. [90, 91] for details on the implementation), or finite difference methods (cf. [92, 93]). Immersed boundary techniques produce a strictly binary interface in the FE model. The mapping procedure is reported to lead to ill-conditioning of the FE model in case of small element fractions, though [69, 93]. Furthermore, since specialized code is needed for the implementation of those techniques, the numerical setup can be time consuming and difficult [69]. Different from the two approaches already described, which are considered Lagrangian approaches, the fictitious material approaches are categorized as Eulerian geometry mapping method. Here, the geometry obtained by the level-set function is described by a density field 𝜌(𝑿) , similar to the density approaches [53, 69]. Elements within the boundary are assigned the full density (𝜌𝑑,𝑖 = 1), whereas elements outside are declared void (𝜌𝑑,𝑖 ≪ 1). If an element is intersected by the boundary, an intermediate density is determined. This may be realized following either a material fraction or direct approach. Both techniques are based on a Heaviside function describing the discrete density distribution. The material fraction approach assigns element-wise constant material fractions 0 < 𝜌𝑑,𝑖 ≤ 1 by considering the solid proportion of the intersected element, and is employed by e.g. Allaire et al. [83], de Ruiter & van Keulen [94], and van Dijk et al. [95]. The direct approach, however, defines the density field by directly mapping the levelset function by an approximate or smoothed Heaviside projection [96–98]. Here, the transition between void and solid regions is distributed over a few elements. The number of transitional elements depends on the sharpness of the Heaviside projection as well as the shape of the boundary. It is important to take care of the level-set function shape to avoid the creation of large grey-scale areas [98], e.g. by applying a distance restriction. The use of intermediate densities reduces the advantage of precise boundary representation associated with the level-set approach [53, 69]. Nevertheless, it is the most commonly applied geometry mapping procedure as it reduces computational efforts and is directly linked to the finite element description used for further evaluation. The use of the fictitious material approach demonstrates the possibility of interlacing Eulerian and Lagrangian topology optimization techniques. Especially the combination of level-set and density [99] or discrete [100] approaches is further developed today. This is due to the similarities in the various approaches, as Sigmund & Maute [53] point out, which also complicates the exact classification of novel adaptions. Apart from the topology optimization approaches described above, various other concepts are investigated to tackle the challenges provided by the established methods (cf. Section 2.2.2 for more details). Novel concepts include the use of the biologically inspired L-system methodology [101–104], cellular automata [105–107], and artificial intelligence

Digitalization of part design by topology optimization

21

[108–110]. The methods make use of the element-based design variable representation, which is interpreted as a graphical image consisting of a definite number of pixels. However, the detailed description of these approaches is out of this work’s scope, and the reader is kindly referred to the given references for more insights.

Figure 2.6: Geometry mapping approaches for level-set topology optimization (cf. [69])

Sigmund & Maute [53] estimate in 2013, that about 90 % of the applied continuous topology optimization approaches fall into the Eulerian scheme. Though the numbers recently started to shift in the literature, especially the density approaches are still very commonly employed in academics and industry [111]. Hence, the overall perception of [53] is still valid in the field of topology optimization for AM [48]. Therefore, Lagrangian approaches are not considered further in this work.

2.2.2

Numerical challenges

The Eulerian topology optimization approaches, no matter whether a continuous or discrete design variable is used, are prone to several numerical issues [54, 112]: checkerboard patterns, mesh-dependency, and local minima, among others. Checkerboard patterns The development of a checkerboard pattern, i.e. an alternating sequence of elements with a significant difference in the design variable value (see Figure 2.7), is a challenge especially in the discrete topology optimization approaches. The origin of this phenomenon was found to be bad numerical modeling of the checkerboard’s stiffness, which is artificially high and therefore preferred by the algorithm [113, 114]. To prevent checkerboard pattern development, high-order finite elements may be applied. In [113, 114], it has been demonstrated that 8- or 9-node elements are capable of suppressing almost any checkerboard pattern creation. However, the use of high-order finite elements results in a significant increase of computational time. Bendsøe et al. [115] introduced a patch technique to reduce the computational effort, but it was shown that this technique only partially removes checkerboard patterns.

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Figure 2.7: Checkerboard pattern (left) and the corresponding sensitivity values (right), cf. [54]

An alternative approach to prevent checkerboard pattern formation is the application of filtering techniques originating in image processing, as demonstrated by [73, 116, 117] for discrete topology optimization approaches. Here, a sensitivity field is applied to the finite element mesh. The sensitivity numbers are smoothed according to the number of solid elements and sensitivity values within a defined radius around the respective element (cf. Figure 2.7). While this approach itself is quite efficient [112], it is dependent on the finite element mesh characteristics. Mesh-dependency Eulerian topology optimization approaches’ results are highly dependent on the finite element mesh the design variable operates on. Ideally, a finer mesh should lead to the same optimal solution. However, topology optimization in general lacks solutions, because the introduction of holes will generally decrease the objective function when the structural volume is kept constant [112]. This is termed ‘nonexistence of solutions’. If such a numerical instability is present, a refinement of the mesh will create a more detailed solution, changing the topology significantly [54, 73]. Another issue are non-unique solutions. An illustrative example is a structure under uni-axial tension, where one sturdy bar will perform as well as several adjacent thin bars [112]. For bi-axial structural responses, Petersson [118] proved that the respective optimal solution is unique, though. The existence of non-unique solutions cannot be changed, but the nonexistence solution problem can be prevented by incorporating restriction methods. One approach is the perimeter control method [119, 120]. The perimeter of a structure is defined as the sum of the circumferences of all inner and outer boundaries. The scheme of the perimeter control method applies an upper limit to the perimeter length, effectively excluding microscopic perforations and driving the solution towards less detailed structures. This also presents a way of avoiding checkerboard patterns [54]. Regulation of the perimeter is realized by introducing penalized intermediate design variable values, which may be subjected to gradient-based algorithms, and restricting the total variation. A challenge remains in determining an appropriate value for the restriction [54, 119]. Another way of reducing mesh-dependency is the application of a constraint to the gradients of the design variable [112, 118, 121], which also effectively reduces the variation. Last, an extension of the

Digitalization of part design by topology optimization

23

checkerboard pattern filter suggested in [122], where the real sensitivity numbers are filtered after their determination, demonstrates similar results compared to the local gradient constraint implementation of Petersson & Sigmund [121]. The filter is comparatively simple to implement, but requires some additional computational time. Local minima Considering the variety in solutions available in the literature for e.g. the MBB beam, which research has somewhat agreed on as a benchmark example [53], it becomes obvious that one topology optimization problem has a considerable amount of local minima. Slight variations in the initial parameters, e.g. design domain geometry, constraint values, or the number of elements, may lead to significant changes in the result of ‘optimal design’ [112]. This is because most global optimization methods are unable to deal with problems the size of topology optimization. Continuation approaches are more likely to ensure solutions closer to a global optimum. The idea of continuation is to repeat the computation several times and gradually change controlling parameters such as penalization factors [120, 123, 124], filter parameters [122, 125], or weight functions [126]. That way, the respective topology optimization problem is expected to gradually converge to the aspired binary design of solid and void regions.

2.2.3

Physics addressed in topology optimization

The mathematical procedures of the various topology optimization approaches can be applied to different physics, including structural mechanics, heat transfer, fluid dynamics, and more. Furthermore, physics may be coupled [127]. In the context of supports in AM, the structural as well as thermal performance optimization is of interest. Hence, their applications to the continuous topology optimization problem formulation are described in the following. Structural mechanics In structural mechanics, a structure’s response to an external or internal mechanical load is investigated. The most common topology optimization problem is the compliance minimization problem. Equation (2.10) (see e.g. [128]) presents the mathematical formulation with the global displacement vector 𝑼, the global force vector 𝑭, and the global stiffness matrix 𝑲, describing the mechanical performance of a structure. Furthermore, the design variable is the relative material density, whose values depend on the topology optimization approach applied (discrete or continuous). Following Equation (2.1) and (2.3), respectively, the design domain is subject to a volume constraint.

24

Digital production by additive manufacturing 𝑁

min: 𝜌𝑑

s.t.:

𝑃 𝑐(𝝆𝒅 ) = 𝑼 𝑲𝑼 = ∑ 𝜌𝑑,𝑖 ∗ 𝒖𝑇𝑖 ∗ 𝒌0 ∗ 𝒖𝑖 𝑇

𝑖=1

𝐺0 (𝝆𝒅 ) = ∑ 𝑣𝑖 ∗ 𝜌𝑑,𝑖 − 𝑉0 ≤ 0

(2.10)

𝑖

𝑲𝑼 = 𝑭 0 < 𝜌𝑑,𝑖 ≤ 1, 𝑖 = 1, … , 𝑁

}

While the minimum compliance formulation restricts the overall mass of the structure and minimizes the occurring residual stresses, the minimum mass approach reverses the formulation. Here, the structure’s mass 𝑴 is the minimization target defined by the sum of the relative densities as well as the element volume, whereas a stress constraint 𝜎̅ is imposed (cf. e.g. [129, 130]): min: 𝜌𝑑

s.t.:

𝑴(𝝆𝒅 ) = ∑ 𝜌𝑑,𝑖 𝑉𝑖 𝑁

0 < 𝜎𝑖 ≤ 𝜎̅, 𝑖 = 1, … , 𝑁 𝑲𝑼 = 𝑭 0 < 𝜌𝑑,𝑖 ≤ 1, 𝑖 = 1, … , 𝑁}

(2.11)

Especially from the perspective of lightweight design, the minimum mass formulation – also called stress constraint problem – seems more suitable to achieve the intended goal of weight reduction. However, the stress constraint problem is associated with three major challenges [53]: stress singularities, the trivial solution of no structure at all for minimal stress, and the local nature of the stress constraint. The problem of occurring stress singularities has been partially solved by relaxation [131, 132]. The fact that no structure is the ideal solution for minimal stress is either ignored or dealt with by introducing an additional compliance objective forcing the algorithm to distribute material [130]. As the stress constraint is a local constraint operating on every element, there is a large number of variables, depending on the number of finite elements. These variables can be collected into a single constraint by aggregation functions such as the p-norm or the Kreisselmeier-Steinhauser functions [133–135]. In conclusion, the implementation and computational effort of the mass minimization problem is high in comparison to the compliance minimization problem. Furthermore, in the case of geometrically restricted use cases as usual in structural design, there is often no considerable difference between the results of both formulations [53]. Heat transfer Topology optimization is applied to conductive, convective, and conjugate heat transfer problems, and further often coupled with fluid dynamics [136]. In the context of support optimization, heat conduction is the heat transfer mechanism of interest. Objectives in heat

Digitalization of part design by topology optimization

25

conduction topology optimization are the thermal compliance as well as maximum temperature minimization [137]. The thermal compliance formulation is similar to the structural equivalent given in Equation (2.10) (see e.g. [138]): 𝑁

min: 𝜌𝑑

s.t.:

𝑃 𝑐(𝝆𝒅 ) = 𝑼𝑇𝒕𝒉 𝑲𝒕𝒉 𝑼𝒕𝒉 = ∑ 𝜌𝑑,𝑖 ∗ 𝒖𝑇𝑡ℎ,𝑖 ∗ 𝒌𝑡ℎ,0 ∗ 𝒖𝑡ℎ,𝑖 𝑖=1

𝐺0 (𝝆𝒅 ) = ∑ 𝑣𝑖 ∗ 𝜌𝑑,𝑖 − 𝑉0 ≤ 0

(2.12)

𝑖

𝑲𝒕𝒉 𝑼𝒕𝒉 = 𝑭𝒕𝒉 0 < 𝜌𝑑,𝑖 ≤ 1, 𝑖 = 1, … , 𝑁

}

Here, 𝑲𝒕𝒉 contains the effective thermal conductivity, 𝑼𝒕𝒉 denotes the temperature vector, and 𝑭𝒕𝒉 represents the thermal load vector. This formulation is based on the Fourier’s Law for static heat conduction, which can be directly employed as demonstrated by e.g. Yan et al. [139]: min:

𝑐(𝑻, 𝝆𝒅 )

s.t.:

𝐺0 (𝝆𝒅 ) = ∑ 𝑣𝑖 ∗ 𝜌𝑑,𝑖 − 𝑉0 ≤ 0

𝜌𝑑

(2.13)

𝑖

∇(𝑫(𝝆𝒅 )∇𝑻) + 𝑔 = 0 0 < 𝜌𝑑,𝑖 ≤ 1, 𝑖 = 1, … , 𝑁

}

𝑻 is the temperature field and 𝑫 represents the effective thermal conductivity tensor, while 𝑔 is the heat generation rate, which is independent from the material distribution. Again, the design variable is the relative material density, and a volume constraint is active. In the maximum temperature minimization, the objective function is simply the global maximum temperature [139, 140]. This scheme can be adapted to other measures, e.g. the mean temperature in the design domain. In summary, the thermal compliance and maximum temperature minimization allow for the minimization of the temperature gradient and peak temperature, respectively. Dbouk [136] notes in his review on heat transfer topology optimization that most studies consider steady-state heat transfer, whereas work on transient heat transfer is missing. While this would be more realistic regarding the application scenarios, it would require extensive computational effort, though.

2.2.4

Solver algorithms

To solve the problem formulated, optimizing algorithms are necessary. In the earlier applications of topology optimization, the Optimality Criteria (OC) method [141] has been broadly used due to its simplicity [128]. However, with increasing complexity of the topology optimization problems, the Method of Moving Asymptotes (MMA) developed by

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Svanberg [142] and later extended to the globally convergent MMA (GCMMA) [143] gained huge popularity due to its reliability and robustness [53]. Rojas-Labanda & Stolpe [144] performed a benchmark of various optimization solvers including OC, MMA, GCMMA, interior-point solvers IPOPT [145] and FMINCON (Mathworks Inc., documentation see [146]), as well as SNOPT [147], a solver based on the sequential quadratic programming methods. Comparing the performance of the solvers to MMA and GCMMA, they found the IPOPT solver to be the most efficient and reliable solver in terms of convergence, while being the only one outperforming (GC)MMA. A major drawback of the IPOPT solver observed in [147] is the very high computational time, though.

2.2.5

Topology optimization for additive manufacturing

While topology optimization is capable of creating efficient designs, traditional manufacturing methods struggle to realize the complex geometries. Additive Manufacturing presents itself as the missing link towards fully integrated optimal part design [148], as the geometrical freedom gained allows the manufacturing of the optimized designs without major changes to the topology [149]. Matching the generalized observation of Eulerian approaches being dominantly used in part design, the density approach is perceived to be the most popular topology optimization methodology in AM part design [148, 150]. However, despite AM simplifying the manufacturing of optimized parts, there are still some design restrictions to be paid attention to in topology optimization via manufacturing constraints [151]. These restrictions concern wall and gap sizes, overhanging features, curvatures, material accumulation, cavities, as well as bore holes and inner channels [8, 18, 28, 152–154]. The main restrictions with regard to the PBF-LB/M supports’ topology are compliance with the critical overhang angle, minimum & maximum feature size, minimal gap size, as well as the avoidance of closed cavities. All of these restrictions have been subject to research aiming at including them into topology optimization to avoid extensive changes during the remodeling of the results [155]. In the following, a brief overview of the corresponding approaches is given. For a detailed assessment, though, the reader is kindly referred to [155–157]. Critical overhang angle The most characteristic design restriction in PBF-LB/M is the critical overhang angle (cf. Section 2.1.1). To introduce this constraint into topology optimization, various approaches have been suggested in the literature, which may be categorized with regard to the overall procedure: approaches where the design variable values of the surrounding finite elements are taken into account, as well as procedures determining the overhang angle and applying a Heaviside Projection Scheme. In the element-based approaches, a gradient-based design variable constraint originating from topology optimization for casting may be used. The

Digitalization of part design by topology optimization

27

problem formulation is changed such that in a defined direction, e.g. the build direction in AM, the values of consecutive elements’ design variable cannot increase, but remain constant or decrease [158, 159]. This way only negative gradients of the design variable are allowed, and overhanging features as well as closed cavities are completely avoided. However, this constraint is over-restrictive for AM because overhangs can be manufactured up to the critical overhang angle. Further approaches to prevent the creation of critical overhanging features have already been described in Section 2.5.1. Minimum & maximum feature size The resolution of the PBF-LB/M process, i.e. the minimum wall or feature size, is governed by various laser and material parameters [18]. To enforce a minimum feature size in topology optimization, the perimeter control method (cf. Section 2.2.2) offers a simple tool. Perceiving the perimeter 𝑃 as the variation of the design variable within the design domain, it may be defined as follows and subjected to an upper limit 𝑃𝑚𝑎𝑥 [112]: 𝑃 = ∫ |∇𝝆𝑑 |2 𝑑Ω ≤ 𝑃𝑚𝑎𝑥

(2.14)

Ω

Equation (2.14) can be transformed into a formulation independent of the applied finite element mesh by scaling it using the maximum element size ℎ𝑚𝑎𝑥 .

𝑃=

ℎ0 ∗ ℎ𝑚𝑎𝑥 ∫|∇𝝆𝑑 |2 𝑑Ω ≤ 𝑃𝑚𝑎𝑥 𝐴Ω

(2.15)

Ω

The parameter ℎ0 stands for the initial mesh size, defining the resolution of the mesh, and 𝐴Ω indicates the area of the design domain for the two-dimensional use case. While this formulation is easily implemented into the topology optimization, it only indirectly affects the feature size. Other approaches include the Monotonicity Based Minimum Length Scale (MOLE) method [160], the Heaviside Projection Method (HPM) [161], and similar density filtering techniques [162, 163]. Furthermore, techniques employing geometric constraints have been proposed by several research groups [164–166]. Here, the structural skeleton is identified and used together with a defined value of the minimum feature size to locally thicken the structure where necessary. The techniques developed to control the minimum length scale may also be used for maximum length scale control, as material accumulation needs to be avoided [18, 28]. This is demonstrated by e.g. Carstensen & Guest [167] and Guest [168] for the projection scheme, as well as Lazarov & Wang [169] and Fernández et al. [156] for classical density filtering. Only slight reformulation is required.

28

Digital production by additive manufacturing

Minimum gap size Ensuring a minimum distance between two distinctive features to avoid the connection of those by agglomerated powder particles and enable the removal of the particles is critical in PBF-LB/M. This requirement poses a length scale control problem similar to the minimum feature size, where the techniques are applied to the void region rather than the solid region. Closed cavities In AM, closed cavities need to be avoided since the contained powder and possibly supports cannot be removed, skewing the part’s performance. Length scale control techniques such as the perimeter control method are capable to remove cavities to some extent, but are restricted to relatively small cavities originating from e.g. a developing checkerboard pattern. Liu et al. [155, 170] developed the Virtual Temperature Method (VTM), where a cavity is assumed to contain a heat source. Void material is assigned a high thermal conductivity, whereas solid regions receive low heat transfer capabilities. If the cavity is closed, solving the heat transfer problem will result in a high temperature at the boundary of the cavity, while an open cavity will be able to transfer the heat towards the edge of the design domain, which is subject to low a fixed temperature. The method is easy to implement, but was found to create convergence issues. Similar to the procedure of Liu et al., other algorithms to identify closed void regions may be introduced to topology optimization, as demonstrated in e.g. [171, 172]. Though each AM manufacturing constraint has received focused attention, a challenge is still present in the simultaneous application of those. Often the results are not directly manufacturable due to large regions of intermediate design variable values or one-node connected hinges [156]. Hence, the implementation of various constraints at the same time has to be handled with care.

2.3

Digitalization of PBF-LB/M process

As stated in Section 2.1.2, the current state of digitalization is characterized by the availability of PBF-LB/M process simulation. Within the modeling of the physical interaction of the laser with the material, a model (i.e. digital representation) of the material considered is required as well. These material models are also utilized in numerical methods for design, such as topology optimization or FEM.

2.3.1 Process modeling When modeling the PBF-LB/M process as it is described in Section 2.1.1, the major challenge lies within the different length scales [173]: The powder particle size distributions, layer thicknesses, and laser spot sizes are in the range of several micrometers [174], part

Digitalization of PBF-LB/M process

29

dimensions are in the centimeter regime, and as a result, scan path lengths are at several kilometers. Consequently, different modeling scales targeting different aspects of the physical phenomena involved in PBF-LB/M have developed. Various definitions of those scales are used. Here, the following definition is adopted, consisting of four distinguished modeling scales: micro-, meso-, macro-, and part-scale (cf. Figure 2.8). The micro-scale investigates the development of the microstructure in a material of choice. The meso-scale is characterized by the modeling of individual powder particles and their interactions either with themselves in powder application scenarios or with the energy introduced during manufacturing. Here, the moving laser heat source is applied and eventually, the melt pool dynamics are of interest. At the macro-scale, the individual powder particles are neglected by assuming the powder to be a homogeneous medium with corresponding material properties. Still modeling a moving laser heat source, individual scan vectors up to few complete layers are evaluated with regard to the effect of the energy to the macroscopic material. If only a single scan vector is investigated, the model may be considered meso-scale, too. Because of the difference in length scale between a single layer and a whole part, the part-scale modeling utilizes additional simplifications. First, the moving heat source may be substituted by a uniform heat input to a whole layer or a subset of it, which is significantly larger than the laser spot diameter. Depending on the dimensions of the subset, gradients in the current build plane can be acquired. The computational time increases with decreasing subset size, though, as the studies presented in [175, 176] demonstrate in their comparison of various heat source models. Second, several powder layers may be combined to one computational layer. This reduces the number of load steps applied and allows for coarser mesh element sizes since the layer thickness setting the smallest element edge. In addition to the length scale, models vary in the respective physics analyzed. While thermal processes are investigated mostly at the smaller length scales, macro- and part-scale modeling focus on mechanics, i.e. stress distributions and distortion. Here, purely mechanical approaches as well as physic-coupling thermo-mechanical models are developed. The meso-scale further employs fluid dynamics, and the micro-scale takes on nucleation and grain growth. The majority of publications dealing with PBF-LB/M process simulation concentrate on one modeling scale with the goal of answering a specific research question. However, the assumptions and simplifications of the respective scale only allow for a certain accuracy, as in reality the scales are based on each other. Hence, the development of multi-scale PBF-LB/M models is of high interest. Due to the high computational resources required, only few works are available, though. One major, holistic approach is being developed and researched at Lawrence Livermore National Laboratory (Livermore, USA) [177, 178]. With their custom FEM software, they are able to couple different modeling scales. The

30

Digital production by additive manufacturing

computing times are in the range of days, even though they do have a high power computation system available. Therefore, research regarding the increase of efficiency in the required algorithms is urgent. With the Lawrence Livermore National Laboratory pioneering in the field of multi-scale, researchers started to pick up the topic, recently. In their studies, Ao et al. [179] as well as Dezfoli et al. [180] presented approaches to couple the meso- and micro-scale. The temperature distribution of the scan path is used as input to the microstructure computation. That way, the characteristic grain orientation along the build direction is achieved in numerical simulation. Both [179, 180] employ cellular automata to realize the coupling. Ghanbani et al. [181] combined the meso- and macro-scale. Within a macro-scale 2D thermal model, used for computing very thin walls, the heat input of the individual layers is approximated by a meso-scale model calculating the respective thermal profile near the laser source. Li et al. [182–185] even realized the coupling of three scales, namely the meso-, macro-, and part-scale. At meso-scale, they determine the melt pool geometry and the corresponding temperature field. This is then applied to a thermo-mechanical model of a single layer. For a specific scan strategy, the temperature and residual stress fields are derived. Based on the results, the characteristic strains and stresses are calculated. Those strains and stresses are then applied to the part-scale model, which is purely mechanical.

Figure 2.8: Scales and physics in PBF-LB/M modeling

Different from the studies in [177, 178], where the detailed coupling is aimed for, usually methods for approximation to reduce the computational efforts are utilized [179–185]. Why this is necessary becomes evident when comparing different reports on resources and computing times. The Lawrence Livermore National Laboratory has a high-power computing system available. Their FEM code Diablo typically utilizes 32 to 256 central processing units (CPUs) [186], but is capable of scaling up significantly. In [186], investigating a bridge with the dimensions of 21 𝑥 9 𝑥 5 mm3 , they reported that for a layer height

Digitalization of PBF-LB/M process

31

of 500 μm, using 32 CPUs, a total time of 5.25 h is required. In a second run, they applied a layer thickness of 250 μm, resulting in a usage of 144 CPUs and a computing time of 50.8 h. Similar efforts are reported by Kundakcioglu et al. [187] (20 h for 52 s of actual manufacturing) and Luo et al. [188] (29,5 min. for a twin cantilever with dimensions of 10 𝑥 5 𝑥 1 mm3 ), who both employed a moving laser heat source in their thermal partscale models. In both [187, 188] standard desktop computers are used and no layer accumulation is applied. If all of these models were scaled to parts of industrial size, the computing times would increase accordingly. Having to wait days or even weeks for the simulation result is not feasible for application in industrial everyday business, though. Approaches to part-scale modeling distinguish in the investigated physics, the energy input modeling, and the layer accumulation. Some studies (e.g. [176, 187]) focus on the thermal phenomena, while the majority of publications investigates thermally induced stresses via thermo-mechanical modeling [35, 186, 189–193]. Other studies (e.g. [194, 195]) deal with mechanical loads only, at least at the part-scale. In terms of energy input modeling, moving heat sources [186, 187, 189–193] as well as the uniform heat input [35, 176] are commonly applied. The uniform heat input is usually related to a complete layer, smaller sections such as an exposure strip are investigated as well, though. In [196, 197], Yang et al. create intermediate models: First, they discretize the scan path with point sources and impose image fields of temperature distributions on them [196]. In [197], they simplify the lateral movement of the laser in a meandering hatch pattern by a line source. Regarding the layer thickness, most studies are in the range of a single layer (30 − 50 µm, no accumulation) up to 1 mm. Hodge et al. [198] accumulate 20 layers, which forces them to adjust their moving heat source in focus diameter and scan velocity. Peng et al. [199, 200] even combine 30 layers into one. That way, they decreased the time of computation to be shorter than an hour for the simple geometries investigated. However, the large element size allowing to do so struggles to accurately represent curved edges and surfaces, and hence the applicability to use cases of larger geometrical complexity is questionable. These examples demonstrate that the feature determining the mesh element size is changing from the layer thickness to the minimum part feature size when accumulating more and more layers. Part-scale modeling of PBF-LB/M is necessary to compute the stresses occurring during manufacturing for the identification of possible build failure risks. Therefore, the purely mechanical approach is of high interest since the information about the thermal history of the part is secondary. Here, the inherent strain method provides the only validated approach currently. The inherent strain method was originally invented by Ueda et al. [201] to predict the distortions in metal laser welding. In the welding process, which includes rapid melting

32

Digital production by additive manufacturing

and solidification of the material, the total strain 𝜀𝑡𝑜𝑡 consists of several components addressing different phenomena within the material during processing: plastic strain 𝜀𝑝 , elastic strain 𝜀𝑒 , thermal strain 𝜀𝑡ℎ , creep strain 𝜀𝑐𝑟 , and strain due to phase changes 𝜀𝑝𝑐 . 𝜀𝑡𝑜𝑡 = 𝜀𝑝 + 𝜀𝑒 + 𝜀𝑡ℎ + 𝜀𝑐𝑟 + 𝜀𝑝𝑐

(2.16)

The inherent strain denotes all inelastic strains, i.e. the residual strains in the stress-relieved state of the material, since they are permanently existent and cannot be relieved by post-processing techniques. Hence, the inherent strain 𝜀 ∗ is defined as the difference of the total and elastic strain at the mechanical equilibrium state of the weld region. 𝜀 ∗ = 𝜀𝑡𝑜𝑡 − 𝜀𝑒

(2.17)

In [201], Ueda et al. take into account the part’s cooled state after welding only. Here, no thermal strain is present, as the part temperature equals the ambient temperature. Furthermore, the creep and phase change strains are neglected, simplifying Equation (2.17) to 𝜀 ∗ = 𝜀𝑝

(2.18)

The predetermined inherent strain is applied to a static analysis, e.g. as coefficient of thermal expansion [202]. The inherent strain method was introduced to PBF-LB/M by Keller & Ploshikhin [194] and quickly adapted by other researchers [195, 203]. However, Bugatti & Semeraro [204] found that the original formulation of [201] is not suited to predict process-induced distortions due to the differences in the manufacturing processes. Because PBF-LB/M is characterized by many adjacent weld seams within one plane and a stacking of weld seam layers, the thermal history of a specific point includes several heating and cooling sequences. Additionally, weld track initial states are dependent on their previous layer’s state. This differs from conventional laser welding, where the individual weld seams do not influence each other. Consequently, research efforts have focused on including the 3D anisotropy of mechanical properties, the deposition process, as well as the effect of scan strategies within individual layers in the inherent strain method. Significant work has been done by the research group of Albert To [202, 205–207]. Based on the early works of [194, 195], they extend the formulation of Equation (2.18) to the contribution from thermal shrinkage combined with the layer-wise build. For any material point, its intermediate state is considered the beginning of elastic deformation accumulation. Then the elastic deformation is changed by thermal shrinking of the surrounding solid material. The change of elastic strain is interpreted as the conversion of thermal into mechanical strain, and is added to the original formulation. Here, 𝜀𝑒𝐼 denotes the elastic strain of the intermediate state, whereas the elastic strain of the steady state 𝜀𝑒𝑆 describes the part after cooling down.

Digitalization of PBF-LB/M process 𝜀 ∗ = 𝜀𝑝 + (𝜀𝑒𝐼 − 𝜀𝑒𝑆 )

33 (2.19)

A similar formulation is derived by Lu et al. [208], and is adopted by later works [209]. While in the described procedure the inherent strain is applied to the individual layers without accounting for the already existing part, Li & Anand [210] proposed a backwards interpolation scheme to incorporate results of earlier iterations. Their approach addresses the influence of the birth-death technique commonly used to model the layer-wise material addition. To add a new layer to the model, a new set of elements is usually initialized in a stress- and distortion-free state. However, as the new elements are connected to the elements of the previous layer, they are partially distorted in reality: The connected element nodes adjust to the previous layer, while the nodes not in touch with any existing element stay at the position of the initially designed mesh. The backward interpolation approach updates the new elements in terms of this distortion before the actual load is applied, coupling a new layer with the previous one. For large number of layers, no difference in the computed results is seen; Li & Anand found their approach to be more accurate when only a small number of layers is computed, though. To account for the anisotropy within a layer due to material properties and scan patterns, Keller & Ploshikhin [194] already introduced the inherent strain as tensor rather than a single value, which could be used to describe isotropic material behavior. Only the normal strains, i.e. the diagonal of the tensor, are considered. This simplification is justified by the layer thickness being significantly smaller than the layer dimensions, resulting in very limited amounts of shear stress [211]. Since many PBF-LB/M manufacturing systems rotate their scan pattern by 67° in-between layers to avoid identical scan vectors in different layers, the 𝜀 ∗ tensor is rotated accordingly in [194, 195, 203, 204, 206]. However, within one layer, 𝜀 ∗ is still constant. The most recent contribution of To’s research group aims at incorporating the scanning strategy to the inherent strain method [202]. They compute directional inherent strain vectors by applying the method of asymptotic homogenization to their 𝜀 ∗ determination results. That way they are able to fully incorporate different scanning strategies. They note that the results of the extended model are very similar to those of the original model for various scan strategies, though, raising concerns regarding the benefit of this method with higher computational demands. Prior to mechanical PBF-LB/M process simulation, the inherent strain tensor has to be determined to calibrate the model. This is done either numerically or experimentally. Numerical calibration includes the setup of a small thermal macro-scale model consisting of one or two layers. After full simulation of those layers, the normal strains 𝜀𝑥𝑥 , 𝜀𝑦𝑦 , and 𝜀𝑧𝑧 are evaluated. Setien et al. [212] present an experimental approach to deriving 𝜀 ∗ . They utilize twin-cantilever beams, which are manufactured in different orientations on the build platform (along X-, Y-, and 45°-direction). The support structures of the cantilevers

34

Digital production by additive manufacturing

are cut while the cantilevers remain on the build platform and the deflection of the cantilever beams are measured. From those measurements, the inherent strain tensor is calculated. That way, not only the process characteristics are incorporated, but the manufacturing system’s characteristics, too. This calibration method highlights a second drawback of the inherent strain method, though: A calibrated model is valid only for the calibration process parameter set and material. If any changes in the process parameters occur or a different material is to be investigated, the calibration has to be repeated. Depending on the use cases of the PBF-LB/M process simulation, this can lead to extensive efforts in calibration. Nevertheless, due to the computational efficiency and a lack of alternatives, the inherent strain method is broadly adopted by specialized commercial software such as Amphyon (Additive Works, Bremen, Germany), Simufact Additive (Simufact Engineering GmbH, Hamburg, Germany), Amp (General Electric Group, Boston, USA), and Ansys Additive Print (ANSYS Inc., Canonsburg, USA) [204, 207, 212–214].

2.3.2 Material modeling One of the most important alloys in PBF-LB/M is the titanium alloy Ti-6Al-4V, which will be considered in this thesis. Its low density in combination with high strength, high fracture toughness, as well as excellent corrosion resistance and biocompatibility [215– 217] promotes the application in various industries, making Ti-6Al-4V the most popular titanium alloy today taking up almost half of the titanium market [216, 218–220]. Originally developed for aerospace structural applications in 1954 [217], the aerospace industry is still dominating the demand of Ti-6Al-4V products [219, 220]. However, other industries apply Ti-6Al-4V with increasing interest due to the well-composed material properties. Especially the resistance to hazardous environments recommends the alloy to highperformance applications in medical [221–224] (e.g. implants) and chemical engineering [225, 226]. Ti-6Al-4V is a dual-phase (𝛼 + 𝛽) alloy, where aluminum as alloying element stabilizes the 𝛼-phase, extending the 𝛼-phase field to higher temperatures, and vanadium stabilizes the 𝛽-phase, shifting the 𝛽-phase field to lower temperatures [216, 217]. The chemical composition of Ti-6Al-4V is regulated by the ASTM B348 standard (cf. Table 2.2) for many industrial applications. Depending on the thermo-mechanical processing route, three distinctively different types of microstructure can be found [217]: fully lamellar structures, fully equiaxed structures, and bi-modal structures. Those types can be further classified as fine or coarse microstructures [216]. In PBF-LB/M, due to the layer-wise addition of material and high thermal gradients along the build direction, typically a lamellar microstructure with fine, elongated grains forms during processing [227].

Digitalization of PBF-LB/M process

35

Table 2.2: Chemical composition of Ti-6Al-4V by ASTM B348

Ti

Al

V

Fe

O

(balance)

5.5 − 6.75 %

3.5 − 4.5 %

< 0.4 %

< 0.2 %

C

N

H

Other

< 0.08 %

< 0.05 %

< 0.0125 %

< 0.4 %

The Ti-6Al-4V powder feedstock is mainly produced by gas atomization or plasma atomization [27, 228]. All atomization processes start with the melting of the feedstock alloy [229], which is either an ingot for gas atomization or a wire for plasma atomization. In gas atomization, the ingot is molten in a furnace. Then, the molten material is transferred to the atomization chamber through a nozzle. The liquid Ti-6Al-4V enters the atomization chamber from above and is free to fall to the bottom. High pressure gas jets, usually nitrogen or argon to reduce the risk of oxidation [229], are directed at the stream of molten material. There are two basic gas nozzle types [230]: The close-coupled nozzle is integrated in the entry nozzle, whereas the free-fall nozzle is a separate component positioned 50 − 150 mm below the entry nozzle. The gas streams atomize the powder particles, which solidify after leaving the gas stream and are collected at the bottom of the atomization chamber. In plasma atomization, plasma torches and gas jets melt and atomize the metal wire simultaneously [229]. The atomized droplets solidify before hitting the bottom of the atomization chamber. Plasma atomization is able to produce precisely spherical, smooth particles with a small number of so-called satellites, i.e. very small particles attached to the bigger particles’ surface, while gas atomization results in mostly spherical particles accompanied by some irregular particles [27]. However, due to the high cost of the feedstock wire [228], plasma atomization is characterized by higher cost and reduced productivity compared to gas atomization. Aside from the sphericity and number of satellites, PBF-LB/M powder is mainly categorized by the particle size distribution, which is referring to the particle diameter. Common distributions in PBF-LB/M are 15 − 45 µm, 20 − 53 µm, or 20 − 63 µm, with the lower value indicating the maximum particle diameter of 10 % and the upper number stating the maximum particle diameter of 90 % of all powder particles [27]. While Ti-6Al-4V exhibits excellent material properties compared to e.g. steel or aluminum alloys, there are some challenges to the broad application. The price6 of the powder feedstock at 200 − 300 €/kg is significantly higher than for other common alloys, such 6

Prices of powders are based on an online market research involving several online marketplaces conducted in March 2021.

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Digital production by additive manufacturing

as 1.4404 stainless steel (15 − 50 €/kg) or AlSi10Mg (50 − 100 €/kg). To decrease the cost associated with the powder material, the re-use of non-molten powder is a common practice [231]. Here, the powder is refurbished by sieving to remove agglomerated powder particles and optionally by mixing used and virgin powder. However, recycling results in a change of powder properties: On the one hand, the particle size distribution changes towards coarser particles [231, 232]. On the other hand, the chemical composition of the powder particles may change during the PBF-LB/M process [233, 234]. Regarding the PBF-LB/M processing of Ti-6Al-4V, a major challenge is the high energy input required, promoting the development of residual stresses [33]. Consequently, the application of supports to fixate the part is of high importance for Ti-6Al-4V processing. Additionally, heat treatment after manufacturing is often necessary to relax the residual stresses and prevent distortion. Furthermore, Ti-6Al-4V is highly reactive towards oxygen [27, 216–218]. An increase of the oxygen content stabilizes the 𝛼-phase and significantly affects the material properties, especially the mechanical properties. The exact oxygen content is rarely determined, though, because of the high cost and effort linked to the respective experimental techniques. To identify the material properties required in process simulation, the thermo-physical processes during PBF-LB/M need to be understood. Figure 2.9 illustrates the thermal cycle and its effect on Ti-6Al-4V at one specific point in the powder layer. In PBF-LB/M, the first powder layer is deposited directly on the build platform. Depending on the chosen process parameters and the available manufacturing system, the powder is either at room temperature or at any higher temperature induced by the pre-heating of the build platform. When the laser irradiation begins, the powder material is molten and may even be evaporated. To reduce the risk of enclosed gas pores in the finished part, evaporation is usually avoided, though. After the laser spot has passed the point in the powder bed observed, the molten material cools down to the ambient temperature of the surrounding powder particles (𝑇𝑎𝑚𝑏 ), which is limited by the build chamber temperature. During cooling, the material solidifies now being in the state of bulk material rather than powder. When one layer is finished, a new layer of powder material is deposited and the cycle is repeated. To ensure the complete bonding of adjacent layers, process parameters are chosen such that the depth of the melt pool exceeds the thickness of a single layer [235, 236]. As a result, a respective bulk layer will be reheated and even remolten several times as the process-induced heat is conducted via the bulk material towards the build platform. Modeling the temperature dependency of material properties is therefore of high importance. Throughout the thermal cycles, the microstructure of the material may change. Being a dual-phase (𝛼 + 𝛽)-alloy, Ti-6Al-4V transforms to a fully 𝛽-phase microstructure at elevated temperatures [216]. The associated temperature indicating the completion of the transformation process is called 𝛽-transus temperature (𝑇𝛽 ). In most conventional material

Digitalization of PBF-LB/M process

37

processing technologies, Ti-6Al-4V transforms back into the (𝛼 + 𝛽)-microstructure during cooling. In PBF-LB/M, though, the cooling rate is very high with values of over 106 K/s [237]. This leads to the development of a nearly complete martensitic (𝛼′) microstructure. The beginning of the 𝛼′-phase formation is marked by the martensite start temperature (𝑇𝛼′𝑆 ). When reheated at high heating rates, the 𝛼′-phase is directly transformed to 𝛽-phase again after 𝑇𝛽 is exceeded, while low heating rates result in a decomposition of the martensite into the original (𝛼 + 𝛽)-microstructure first. As each microstructural composition may exhibit its own respective material behavior, the different microstructures have to be taken into account individually in the material model of Ti-6Al-4V [216]. Additional to the temperature dependency and the different microstructural compositions, the solid material is present as powder and bulk material at the same time. It has to be considered that powder properties may differ significantly from the bulk material properties. While this is not always the case for the thermo-physical material properties, the mechanical properties of the powder are assumed close to zero (not equaling zero to avoid singularities during computation). Since the build chamber usually is not under pressure, the powder is free to move in the Z-direction, preventing the transmission of mechanical forces or deformation.

Figure 2.9: Thermal cycle of a specific point in PBF-LB/M

In Table 2.3, the individual material properties to be modeled for thermo-mechanical PBFLB/M process simulation at macro- or part-scale are summarized, including the properties involved in the thermo-mechanical processes and the phase transitions, as those influence the aforementioned properties significantly. Besides 𝑇𝛽 and 𝑇𝛼′ 𝑆 , which are specific for Ti-6Al-4V, the temperatures characterizing the melting interval are included: the liquidus temperature (𝑇𝑙 ), above which the material will be in its liquid state, as well as the solidus temperature (𝑇𝑠 ) indicating the solidification of the material during cooling. Also, it is

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Digital production by additive manufacturing

useful to know the evaporation temperature (𝑇𝑒 ) for interpretation, though 𝑇𝑒 may not necessarily be reached. Additional thermo-physical material properties required for modeling are the density (𝜌) of the material, which is also applied in purely mechanical computation, the specific heat capacity (𝑐𝑝 ) as the material’s ability to take in energy, as well as the thermal conductivity (𝑘) indicating the material’s ability to transmit heat. How much of the laser energy is actually coupled into the material, not reflected or transmitted, is denoted by the absorption (𝑎). Table 2.3: Overview of Ti-6Al-4V material properties to be modeled

Thermo-physical properties 𝛽-transus temperature

𝑇𝛽

[K]

Solidus temperature

𝑇𝑠

[K]

Liquidus temperature

𝑇𝑙

[K]

(Evaporation temperature)

(𝑇𝑒 )

[K]

Martensite start temperature

𝑇𝛼′𝑆

[K]

Density

𝜌

Specific heat capacity

𝑐𝑝

Thermal conductivity

𝑘

kg ] m3 J [ ] kg*K [

[

W ] m*K

Optical properties Absorption

𝑎

-

Young’s modulus

𝐸

[GPa]

Yield stress

𝜎𝑦

[MPa]

Poisson’s ratio

𝜈

[-]

𝐶𝑇𝐸

1 [ ] K

(𝑈𝑇𝑆)

([MPa])

Mechanical properties

Coefficient of thermal expansion (Ultimate tensile strength)

For the consideration of the mechanical material response, the Young’s modulus (𝐸) describes the stiffness in the elastic material behavior, while the yield strength (𝜎𝑦 ) labels the start of plastic deformation. The ultimate yield strength (𝑈𝑇𝑆) indicates the maximum stress bearable without material failure. It is not commonly applied in PBF-LB/M process simulation, but can help to interpret the derived stresses. The Poisson’s ratio (𝜈) describes

Digitalization of PBF-LB/M process

39

the material’s expansion or contraction perpendicular to the load direction. Last, the coefficient of thermal expansion (𝐶𝑇𝐸) indicates the change in volume of the material due to a temperature difference. The 𝐶𝑇𝐸 links the thermal and structural physics, enabling the calculation of stresses due to thermal loads. As the modeling of the PBF-LB/M process at different scales considers different physics, there may be different material properties involved. This is shown in Table 2.4 and Table 2.5, where an overview of computational studies modeling the PBF-LB/M process on various scales is given. All studies apply Ti-6Al-4V as material. The tables indicate the physics applied in the respective study (T – thermal, M – mechanical, TM – thermo-mechanical) as well as the modeling scale (MI – micro, ME – meso, MA – macro, PA – part). Furthermore, the material properties used and the manner of specification are given. Here, ‘E’ stands for a reference to published experimental work, whereas ‘S’ denotes references to other PBF-LB/M process simulations. The number prior to the letters gives the number of references for a specific material property. If no reference is given at all, it is marked with ‘x’. Where a material property is modeled as temperature dependent, ‘(T)’ is added.

Ref.

Physics

Scale

Table 2.4: Ti-6Al-4V material modeling of thermo-physical and optical properties in PBFLB/M process simulation

[238]

T

MI

[239]

T

ME

[240]

T

ME

[241]

T

ME 1S 1S

1S 1S

[242]

T

ME

1S

1E

1S

1E (T)

1E (T)

1E (T)

[243]

T

ME

1S

1S

1S X 3S (T) (T)

X (T)

X (T)

1S

[244]

T

ME

X

X

1S (T)

1S (T)

1S (T)

X

[245]

T

ME

1E

1E 1E

1E

1E (T)

1E (T)

Powder Material 𝑇𝛽

𝜌

𝑐𝑝

𝑘

𝑎

1S 1S

1S

1S (T)

1S (T)

1S

X

X (T)

X (T)

X (T)

X (T)

X (T)

X

X X X 1S 1S (T) (T) (T) (T)

1S (T)

1S (T)

1S

𝑇𝑠

𝑇𝑙

1S

X

𝑇𝑒 𝑇𝛼′𝑆

𝜌

𝑐𝑝

Bulk Material

𝑘

X

X

1S 1S 1S (T) (T) (T)

𝑎

X

Physics

Scale

Digital production by additive manufacturing

Ref.

40

[246]

T

ME

X

X

[247]

T

ME

1E

1E 1E

[248]

T

ME

1E

1E 1E

1E (T)

[249]

T

ME

X

X

X

[250]

T

ME

X

X

[251] TM ME

1E

1E

[252] TM ME

3ES 3ES

Powder Material 𝑇𝛽

𝑇𝑠

𝑇𝑙

𝑇𝑒 𝑇𝛼′𝑆

𝜌

𝑐𝑝

𝑘

𝑎

X 3ES (T) (T)

X

1E

MA

2E 2E

[254]

T

MA

X

[259] TM MA

X

X

T

PA

[186] TM PA X

1E (T)

1E (T)

1E (T)

X

1E (T)

1E

1E (T)

2E

2E

2E 1E (T) 2E (T)

1S (T)

1S (T)

1S (T)

1S (T)

1S 1S 2ES (T) (T) (T)

1S (T)

1S (T)

1S (T)

X (T)

X (T)

X (T)

X (T)

X (T)

2E (T)

2E (T)

2E (T) 1S

1S (T)

X

X

X (T)

X

X

X X (T) (T)

1E

2E (T)

2E (T) 1S (T)

X (T)

[37] TM MA [260]

1E

2E (T)

[174] TM MA

X

4E (T)

2E (T)

[255] TM MA

[257] TM MA

2E (T)

1E (T)

1E

1S

𝑎

3ES 3ES 3ES 3ES (T) (T) (T) 1E

1E

𝑘

1E

2ES (T)

2ES

𝑐𝑝

3ES 3ES (T) (T)

1E

T

[258] TM MA X

𝜌

X 1S 2ES 1E 1S (T) (T) (T) (T)

[253]

[256] TM MA

Bulk Material

1S (T) X (T)

1E

1S (T)

1S (T)

1S (T)

X (T)

X (T)

X (T)

X

Scale

Physics

Ref.

Digitalization of PBF-LB/M process

[191] TM PA

41

Powder Material 𝑇𝛽

𝑇𝑠

𝑇𝑙

𝑇𝑒 𝑇𝛼′𝑆

𝜌

𝑐𝑝

𝑘

1E 1E 1E (T) (T) (T)

Bulk Material

𝑎

𝜌

𝑐𝑝

𝑘

X

1E (T)

1E (T)

1E (T)

𝑎

Physics: T – thermal, M – mechanical, TM – thermo-mechanical Scale: MI – micro, ME – meso, MA – macro, PA – part Reference: X – no reference, E – experimental, S – simulation, number = number of references, (T) – temperature dependency modeled It can be concluded from Table 2.4 that only few PBF-LB/M process simulations consider the complete set of material properties named in Table 2.3: 𝑇𝛼′ 𝑆 is attributed once [258], 𝑇𝛽 thrice. For 𝑇𝑠 and 𝑇𝑙 it has to be noted that some studies do not differentiate those but give one temperature for the melting point rather than the melting interval [186, 239, 253, 254, 256, 258–260]. Here, the melting temperature has been assigned as 𝑇𝑙 . Even though some studies do not give explicit values for specific temperatures, they indirectly model them when considering temperature dependent material properties [174, 191, 255, 257]. A further observation concerns the modeling of respective powder properties, which is performed significantly less than the bulk material. While the explicit consideration of powder properties is not required at the meso-scale when the powder particles are geometrically represented, studies at meso-, macro-, and part-scale applying the homogeneous medium assumption need to include effective powder properties. Aside from the statement of which material properties are actually modeled in thermal PBF-LB/M process simulation, it is also important to assess the quality of the material model. Regarding the specific temperatures, often no reference is given for the stated value, e.g. 11 out of 22 values of 𝑇𝑙 provided are unreferenced. For the remaining thermophysical properties, this is also true to some extent. Furthermore, the number of references for a single material property or a set of material properties is low, with only a few studies naming more than one reference [246, 247, 252, 253, 256, 258]. As will be shown in the following sections, though, different experimental setups for the same material property can produce a wide range in values. This is due to the actual material configuration and the experimental procedure influencing the results, additionally to the respective measurement errors. The use of a single reference or dataset may therefore induce deviations in the result. Evaluating the type of references, it stands out that 12 out of 21 publications listed in Table 2.4, which give a reference for their material model, use other PBF-LB/M process simulations to do so. At the same time, 15 works employ experimental studies as material model reference. In [174, 242, 246, 260], both experimental and computational references are applied, depending on the material property. Saxena et al. [252],

42

Digital production by additive manufacturing

Fu & Guo [247], as well as Zhao et al. [256] even combine both reference types for a single material property. The broad application of other PBF-LB/M process simulations bears the risk of transmission errors. Sparse availability of experimental work on the thermo-physical properties of Ti-6Al-4V (cf. Section 4.1) makes it difficult, though, to rely fully on experimental values and requires extensive literature review. Table 2.5 shows a smaller number of mechanical PBF-LB/M process simulations compared to the thermal ones in Table 2.4For the consideration of the mechanical material response, the Young’s modulus (𝐸) describes the stiffness in the elastic material behavior, while the yield strength (𝜎𝑦 ) labels the start of plastic deformation. The ultimate yield strength (𝑈𝑇𝑆) indicates the maximum stress bearable without material failure. It is not commonly applied in PBF-LB/M process simulation, but can help to interpret the derived stresses. The Poisson’s ratio (𝜈) describes the material’s expansion or contraction perpendicular to the load direction. Last, the coefficient of thermal expansion (𝐶𝑇𝐸) indicates the change in volume of the material due to a temperature difference. The 𝐶𝑇𝐸 links the thermal and structural physics, enabling the calculation of stresses due to thermal loads. As the modeling of the PBF-LB/M process at different scales considers different physics, there may be different material properties involved. This is shown in Table 2.4 and Table 2.5, where an overview of computational studies modeling the PBF-LB/M process on various scales is given. All studies apply Ti-6Al-4V as material. The tables indicate the physics applied in the respective study (T – thermal, M – mechanical, TM – thermo-mechanical) as well as the modeling scale (MI – micro, ME – meso, MA – macro, PA – part). Furthermore, the material properties used and the manner of specification are given. Here, ‘E’ stands for a reference to published experimental work, whereas ‘S’ denotes references to other PBF-LB/M process simulations. The number prior to the letters gives the number of references for a specific material property. If no reference is given at all, it is marked with ‘x’. Where a material property is modeled as temperature dependent, ‘(T)’ is added. Table 2.4. The studies displayed mainly tend to the macro- or part-scale, whereas in the thermal modeling the meso-scale is focused. It is noted that most studies apply coupled physics. Only Rangaswamy et al. [261], Li et al. [262], Ahmad et al. [203], and Chen et al. [263] implement other methods such as the inherent strain method to derive internal stresses while not directly considering the thermal aspects of the PBF-LB/M process. Compared to Table 2.4, a more homogeneous use of material properties and references is observed. However, there is still a significant number of studies not referencing their material model. Furthermore, the broad involvement of computational references of Table 2.4, is matched, as well as the consideration of the temperature dependency (except for Ahmad et al. [203]).

Digitalization of PBF-LB/M process

43

Table 2.5: Ti-6Al-4V material modeling of mechanical properties in PBF-LB/M process simulation

Source

Physics

Scale

𝐸

𝜎𝑦

[261]

M

MI

X (T)

X (T)

X (T)

[264]

TM

ME

2ES (T)

2ES (T)

2ES (T)

[252]

TM

ME

1S (T)

1S (T)

1S (T)

[255]

TM

MA

2E (T)

2E (T)

2E (T)

[174]

TM

MA

1S (T)

1S (T)

1S (T)

[256]

TM

MA

X (T)

X (T)

X (T)

[257]

TM

MA

X (T)

[258]

TM

MA

1S (T)

[259]

TM

MA

1S (T)

[37]

TM

MA

[186]

TM

[191]

𝜈

𝐶𝑇𝐸

X (T) 1S (T)

1S (T)

1S (T)

1E (T)

1E (T)

2E (T)

1E (T)

PA

X (T)

X (T)

X

X (T)

TM

PA

X (T)

[262]

M

PA

1E (T)

1E (T)

1E (T)

[203]

M

PA

X

X

X

[263]

M

PA

X (T)

X (T)

X (T)

X (T) 1E (T) X (T)

Physics: T – thermal, M – mechanical, TM – thermo-mechanical Scale: MI – micro, ME – meso, MA – macro, PA – part Reference: X – no reference, E – experimental, S – simulation, number = number of references, (T) – temperature-dependency modeled

44 2.4

Digital production by additive manufacturing Support structures in PBF-LB/M

To overcome the challenges of overhanging features as well as residual stresses and resulting distortions, support structures (term shortened to ‘supports’ in the following) are applied in PBF-LB/M. Supports are auxiliary structures built together with the actual part, which fulfill three main tasks [265–270]: First, they support overhanging part features, preventing the molten material as well as the solidified part to drop. Second, supports dissipate process-induced heat from the part towards the build platform, reducing residual stresses by increasing the homogeneity of the parts temperature field. The third task of supports is to fixate the part in space, counter-acting distortions. Depending on the part geometry, supports are the key factor for successful manufacturing [267].

2.4.1 Integration of supports in the manufacturing process Considering supports is part of the complete design and manufacturing process. They need to be designed (step 2, Figure 2.1), manufactured (step 4), and finally removed as they do not belong to the actual part (step 7). For the design of supports, specialized software is required. While there is a variety of data preparation software capable of automatic support generation for the consumer AM market [271], most being free of charge, this is not the case for PBF-LB/M. Here, current innovation is driven from two sides: First, there is data preparation software, dominated by Materialise Magics (Materialise GmbH, Gilching, Germany). Second, providers of CAD software integrate new modules into their software to enable their users to complete the digital workflow of AM within a single product. Additionally, the AM software market is seeing an overwhelming amount of startups tending to the digital aspects of AM, including CAD design, data preparation, and production planning [272, 273]. Even though the respective software used for the support design may have slightly different settings, the general types of supports with regard to their geometrical layout are similar. They can be categorized with regard to their layout, which may be comprised of thin walls (2D supports) or solid volumes (3D supports). Different from the 3D supports, 2D supports are modeled without a volume in the software and machine code, gaining their physical volume by the laser beam diameter.

Support structures in PBF-LB/M

Figure 2.10: 2D supports available in the data preparation software Materialise Magics

45

46

Digital production by additive manufacturing

Figure 2.11: 3D supports available in the data preparation software Materialise Magics

Figure 2.10 displays the 2D supports available in Materialise Magics. The support type most frequently used is the block support [274]. It consists of a rectangular grid of thin walls, which optionally may be perforated to reduce material consumption and increase de-powdering capabilities [275]. Due to the grid layout, block supports are mainly applied to surfaces. Additionally, the high stability of the grid make block supports suitable for the support of features positioned in the upper region of the part. Furthermore, specialized supports exist to address various geometrical features [274, 276]: To support edges and other very small areas, the line and point support are used, respectively. To provide a minimum amount of stability, these support types are reinforced with small ribs along their main axis. This reinforcement leads to the cross section of the support being larger than

Support structures in PBF-LB/M

47

the area actually supported. The web support is adapted from the block support to improve the performance on circular surfaces. The circular walls are connected by transverse walls meeting in the center of the surface. The contour support neglects these transverse walls, but is able to follow the contours of the surface freely. Consequently, the contour support is not as stable. Last, the gusset support, which is applied at small overhangs, connects to the side surfaces of the part instead to the build platform. That way, little material is used, but the removal is complicated as the support is placed in a corner. The 3D supports available in Materialise Magics are presented in Figure 2.11. The cone support is applied if the 2D support’s strength is not sufficient and may be used for edges as well as surfaces [274]. The tree support is a variation of the cone support, where several cones (branches) are connected in one point and a single cone (stem) is reaching down to the build platform. This configuration reduces the material consumption compared to cone supports, but is also less stable. If very high stresses arise during the manufacturing process, volume supports may be used, which consist of a block of solid material. The effort for removal is very high, though, so the volume support is used only if all other support types are not applicable. The lattice support, which is designed by a grid of rods, represents a compromise between the stability of the 3D supports and the material efficiency of the 2D supports. The actual design requires high computational effort. The procedure to design supports is structured into two steps [265]: First, the surfaces that require support are identified. Commonly, supports are designed semi-automatically. Data preparation software is capable of identifying surfaces violating a defined restriction of the angle of overhanging features to the build platform. To be able to identify surface requiring support because of high residual stresses or heat accumulation, numerical process simulation has to be performed [20, 21]. Currently, many data preparation software packages do not provide this functionality, so additional software is needed. Furthermore, to ensure the removability of the supports after manufacturing, the tool accessibility has to be taken into consideration when determining the surfaces to be supported [277]. In the second step, the appropriate support layout und dimensions have to be chosen. Here, the respective overall performance of the applied configuration has to be known. To be able to link the assessment of surfaces in need of supports and the actual support design automatically, both support design and process simulation functionalities have to be coupled. In the Materialise Magics 25 version, this coupling is included as a beta test, i.e. not fully developed feature. After the PBF-LB/M process as well as an optional stress-relief heat treatment, the supports are usually detached from the part since they do not contribute to the actual product. Today, this step is done manually, using all kinds of saws, pliers, cutters, chisels or files [265, 277]. Alternatively, milling may be used for the support removal [265, 278]. Both methods do have their advantages: While the manual work does not require extensive machine programming as CNC (computer numerical control) milling does, the milling results in a smoother surface and may be combined with the post-processing

48

Digital production by additive manufacturing

of functional surface. The commonality of manual and CNC milling support removal is the extensive effort, though. As supports directly influence the manufacturing process, they also affect the final part properties. Nadammal et al. [269] investigated the influence of two different support configurations in PBF-LB/M of the superalloy Inconel 718. They found that the supports affected the microstructure and residual stress development on the surface, but the crystallographic texture in the core region of the samples remained independent. Varying grain sizes demonstrated the ability of supports to regulate cooling rates. Patterson et al. [270] found that the goal of supports to fixate overhanging features lead to increasing residual stresses, which were less uniform in nature. Their study suggests that supports may actually increase the probability of damage due to residual stresses, if the heat dissipation mechanism of the supports cannot prevent any tendency of distortion. Cao et al. [266] also investigated the influence of supports on the residual stresses, but contrary to the results of Patterson et al., they found the surface stresses to be more uniform than without supports. They further identified the compressive stresses to be dominating within supported specimens, while in non-supported specimens tensile and compressive stresses were alternating and unevenly distributed. In their investigation of the microstructure, the support structures altered the orientation of grain growth direction. Kajima et al. [268] applied the effect of supports on microstructure and residual stresses to the fatigue strength of PBFLB/M parts. Since the supports dissipated the heat more effectively than the powder bed surrounding the parts during manufacturing, the higher cooling rates resulted in a finer microstructure, enhancing the fatigue strength. Furthermore, they found the residual strain in the supported specimen to be lower than in non-supported specimen.

2.4.2 Challenges in the application of supports Nearly 80 % of build failures can be traced back to inadequate support design [279]. This observation on the one hand highlights the importance of supports to the manufacturing, and on the other hand demonstrates that there exist major challenges to be solved. When additive manufacturing was developed for rapid prototyping, supports were perceived as a minor inconvenience [265]. This is mostly due to the single item production and the other benefits of rapid prototyping overpowering the disadvantage of having to include additional structures. Furthermore, the requirements to the surface quality were low. With the technology maturing, the point of view regarding support changed, though. While supports are a necessity for many parts to be manufactured by PBF-LB/M, their application leads to increased cost throughout the whole manufacturing process [265, 274]. They have to be designed, increasing the time for the data preparation. During manufacturing, supports result in increased manufacturing time as well as energy and material consumption. After the manufacturing, the removal step requires personnel and tools.

Support structure optimization

49

Also, surfaces in contact with supports often need further finishing as the surface quality is worsened by residuals or sintered powder particles due to accumulated heat. Furthermore, the material of the supports cannot be directly recycled as unfused powder can. In conclusion, the goal of support application is to reduce the amount of supports to the necessary minimum while ensuring a sufficient support performance. Therefore, the design of the support structures as well as their layout has to consider various aspects: areas in need of support, support strength and thermal conduction, as well as the accessibility of the supports for removal. Currently, a fully automated support design incorporating all aspects is not commercially available. Therefore, the design of supports is heavily based on the experience of the designer. This entails major risks of either unnecessarily high cost because more supports than needed are applied or faulty parts with complete build failure as the worst-case scenario.

2.5

Support structure optimization

As the PBF-LB/M technology matures, there is an increasing interest in support optimization as the efforts to minimize cost intensify. Figure 2.12 indicates the number of publications explicitly referencing supports over time, visualizing an increase in publications during the last years. An overview of support optimization work is given in Table 2.6. Support optimization approaches can be differentiated into three main categories, each consisting of further subcategories that will be explained in detail in the following sections: 1 (a – d) Support avoidance or minimization without changing the support 2 (a – c) Optimization of the existing supports 3 (a – b) Development of new structures The three categories show an increasing degree of complexity to the optimization approach, which also becomes apparent in Figure 2.12. The first studies on support optimization dealt with approaches of category 1 and 2, trying to avoid supports or adjust their respective parameters, while later on studies aim at creating new structures (category 3) with the help of computational design methods. Interestingly, in 2018 there is a spike in category 1 approaches. This spike can be attributed to the increasing popularity of computational design methods such as topology optimization in combination with additive manufacturing, as formulations for design restrictions regarding support avoidance have been developed.

50

Digital production by additive manufacturing

Figure 2.12: Annual published studies on support optimization and their corresponding categories Table 2.6: Literature overview regarding support optimization approaches

Ref.

Cat.

Study type

Optimized tasks

Optimization goal

Validation

[274]

1a

E

-

↓ Volume

E

[280]

1a

C

Support

↓ Necessity

C

[281] [282]

1a 1a

C C

Support Support

↓ Necessity ↓ Necessity

C C

[283] [284]

1a 1a

Support Support

↓ Necessity ↓ Necessity

[285]

1a

Support

↓ Necessity

C C E C

[286]

1a 3b

C C E C E C

Support

↓ Necessity

C (E)

[287]

1a

C

-

↓ Volume

C

Own specimen design MBB beam, cantilever beam Cantilever beam MBB beam, cantilever beam MBB beam Cantilever beam MBB beam, Lbracket, bridge Bridge / cantilever beam, MBB beam Bracket

Support structure optimization

51

Ref.

Cat.

Study type

Optimized tasks

Optimization goal

Validation

[288]

1a 1b 1a 1b

C

Support

↓ Necessity

E

Bracket

C C

Support Support

E C

MBB beam Bracket

E

Support

E

[290]

1b 2a 1b

C

[291]

1b

C

Support Fixation Support

↓ ↑ ↑ ↓ ↑ ↓

[292]

C

[293]

1b 3a 1c

C

Own specimen design Own specimen design, bracket Own specimen design L-bracket, truss structure Dental bridge

[294]

1d

E

[275]

2a

E

[295]

2a

C

[175]

2a

C

[296]

2a

E

[297] [298]

2a 2a

E E

[299]

2a 3b

C E

Dissipation

[300] [301]

2a 2a

E C

Support Dissipation

[302]

2a

E

Support

[303]

2a

E

Dissipation

[289] [278] [276]

Necessity Removability Accessibility Volume Removability Volume

↓ Volume

C E C

Support

↓ Volume

C

Support Fixation Fixation

↓ Volume

E

↓ Necessity

E

Support Fixation Fixation Dissipation Fixation

↓ Volume ↑ Removability ↓ Volume

E

↑ Stiffness

C (E)

Support Fixation Fixation Support

↑ ↓ ↑ ↓ ↑ ↑ ↑

Removability Cost Stiffness Volume Removability Conduction Surface quality ↑ Removability ↑ Conduction

E

↓ Volume ↓ Cost ↑ Conduction

E

E E

Own specimen design Cantilever beam Own specimen design Own specimen design, same as [295] Rectangular plate Tensile specimen Cantilever beam

E

Own specimen design

E C

Cantilever beam Rectangular plate Rectangular plate Bridge + cantilever beam in one specimen

E

52

Digital production by additive manufacturing

Ref.

Cat.

Study type

Optimized tasks

Optimization goal

Validation

[304]

E

Support

E

Support

↓ Volume ↑ Removability ↑ Removability

E

[305]

2b 3a 2b

[306] [307] [308] [309]

2b 2c 2c 2c

E E E E

Fixation Support Support Support

↓ ↑ ↑ ↑

Volume Removability Removability Removability

E E E E

[310] [311] [312]

2c 2c 3a

E E C

↑ Removability ↑ Removability ↓ Volume

[313]

3a

E

Support Support Support Fixation -

E E C E E

[314]

3a

E

[315]

3a 3b 3a

C

Support Fixation Dissipation

E

-

C

Support

[318]

3a 3b 3a

C

Support

[319]

3a

C

Support

[320]

3b

C

Support Dissipation

↓ Volume ↑ Stiffness ↑ Conduction

C

[321]

3b

C

Support

C

[322]

3b

C

[323]

3b

C

Support Dissipation Fixation Support

↑ Removability ↓ Cost ↓ Volume

↓ Volume

C

[316] [317]

↑ Conduction

E

Rectangular plate Own specimen design not specified Bridge Tubular arch Stator segment, tensile specimen Spring Bridge Double cantilever beam Rectangular plate Own specimen design Cantilever beam

↓ Volume ↑ Removability ↓ Volume

E

Cantilever beam

C

Cantilever beam

↓ ↑ ↑ ↓

C

Bracket, turbine

C E

Bunny, armadillo, bird, stem, turbine MBB beam, Mshape, [321], chair, 3D beam, cantilever beam Cantilever beam (two-level) Double cantilever beam, tubular arc Hip implant

↓ Volume ↑ Removability ↓ Volume

Volume Removability Accessibility Volume

E

C

C

Support structure optimization

53

Ref.

Cat.

Study type

Optimized tasks

Optimization goal

Validation

[324]

3b

C

↓ Volume

C

[325] [326]

C C

↓ Cost ↓ Cost

C C

[327]

3b 3b 1b 3b

Support Dissipation Support Support

C

Support

C

[328]

3b

C

Support

↓ Cost ↑ Stiffness ↓ Volume

[329]

3b

C

Dissipation

↓ Volume

C E C

[330]

3b

C

Dissipation

↑ Conduction

C

[331] [332]

3b 3b

C

[333]

3b

C

↓ ↓ ↑ ↓

[334]

3b

C

(E) C E C E C

[335]

3b

C

Support Support Dissipation Support Fixation Support Dissipation Support

[336]

3b

C

Support

Volume Volume Removability Volume

↑ Conduction ↓ Volume ↑ Removability ↓ Volume

C E

Antenna support bracket Cantilever beam Cantilever beam Own specimen design Cantilever beam, tube Rectangular plate MBB beam, Lbracket Cantilever beam Dental implant Cantilever beam, bridge L-bracket Cantilever beam, letters Cantilever beam

Legend: C – computational E – experimental ↑ – Increase ↓ – Decrease

2.5.1 Support structure avoidance The most intuitive approach to reducing support-related cost is to avoid supports or at least reduce the necessary amount to the bare minimum, while not altering the supports themselves to decrease the required effort. In the literature, four ways to achieve this goal have been presented: 1a. Appropriate part design 1b. Optimized part orientation on build platform 1c. Support application based on numerical process simulation

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Digital production by additive manufacturing 1d. Appropriate process parameter choice

For support-free part design, respective design guidelines have to be derived. Besides the general work on PBF-LB/M design guidelines, see e.g. [28], Gralow et al. [274] investigated standard supports with regard to dimensional accuracy, material consumption, removability, tensile strength and resulting surface quality, and formulated design guidelines with respect to different use cases. Murphy et al. [287] designed a deep neural network to predict the amount of supports required, among other things. This prediction tool may be integrated into the design process to keep track of the potential support mass during design iterations. The most common approach to avoid supports is the application of an overhang constraint in the topology optimization design methodology. As AM allows for complex geometries to be produced, topology optimization as a method in the field of structural optimization is gaining popularity to create innovative, efficient part designs [281]. The inclusion of an overhang angle constraint to avoid overhanging features in need of support provides the opportunity to create self-supporting part designs. Brackett et al. [281] introduced this idea as an addition to the already existing minimum member thickness constraint to ensure compliance regarding the resolution of AM processes. Here, applying the BESO method, the overhanging edges of an iterations interim result are identified, linearly fitted, and the corresponding angle to the build platform is calculated. The overall violation of the critical overhang angle is quantified by a penalty function, which is then combined with the structural response to a single objective function. Leary et al. [284] used the SIMP method. After identifying the edges in need of support by computing the local gradients, they iteratively adapt the topology result of an iteration either by adding additional struts with noncritical overhang angles or by adding material to the respective edge such that the overall gradient complies with the angle constraint. In [282], a projection scheme is added to the RAMP method to prevent the formation of structural features violating the overhang rule rather than adjusting critical features. Various other approaches in metal and polymer AM have been published, addressing the most common topology optimization methods by either explicitly or implicitly constraining the overhang angle, e.g. [283, 285–289]. A more detailed description has been presented in Section 2.2.5. In the field of AM-related topology optimization constraints, only the overhang angle constraint directly addresses supports. In consequence, the support tasks of heat dissipation and part fixation are neglected, leading to limited application in PBF-LB/M. Together with the actual part design, the part orientation defines the amount of overhanging features violating the critical overhang angle. In 1994, Allen & Dutta [337] presented an algorithm to minimize the support faces on the part while simultaneously considering part stability on the build platform for the Stereolithography (SLA) process by simple approximation of the support geometry for a given set of part orientations and choosing the one with the minimal contact area to the part. Majhi et al. [338] proposed an algorithm

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to minimize the contact length between supports and part as well as the area of supports used in a 2D optimization problem. They calculate aforementioned parameters mathematically based on the CAD file. Until today, a huge variety of approaches has been published, especially in the field of polymer AM, and the latest research usually introduces part orientation into a framework with multiple goals. In the following, works referring to PBFLB/M are reviewed. Strano et al. [292] combined part orientation and novel cellular support structures in two consecutive steps to optimize supports. For the part orientation optimization, a set of possible part orientations is created by rotating the CAD file in 5° steps around each coordinate axis. Similar to [337], the corresponding support volume is calculated and the optimal orientation is chosen by the minimal volume. Calignano [276] presented a procedure also consisting of a part orientation optimization algorithm followed by a method automatically deciding on whether to apply line or block supports and designing them. Here, the CAD file is first checked regarding the presence of cylindrical surfaces and internal channels, and then corresponding procedures are defined to account for the staircase effect and surface roughness. The optimization is ended when a certain threshold of tolerable support volume is reached. In [291], Das et al. aimed at minimizing support volume as well as dimensional errors in the part by choosing an appropriate part orientation. Both minimization goals are combined into one multivariable minimization problem. Meanwhile, Chen & Frank [278] tend to the topic of ensuring the removability of supports by optimizing the part orientation. They defined a characteristic tool access volume for CNC milling, and mapped this tool access volume to the CAD file of the part. In the formulation of the optimization problem, minimal support contact area and inaccessible part faces are considered. Extending the work of [276], Salmi et al. [288] proposed a design methodology for PBF-LB/M components involving topology optimization, design for laser powder bed fusion rules, CAD modeling, as well as considerations regarding machining allowances and features for finishing. The part orientation is chosen to achieve minimal support volume and with respect to the powder recoating system in order to reduce the force applied by the recoater to the part. In the case study presented, the choice of part orientation is done manually by evaluating different orientations. Cheng & To [290] aimed at minimizing support volume as well as residual stresses by optimizing the part orientation. They introduced a voxel-based methodology to generate open-cell lattice structure supports and mesh the part, following [291], where the CAD file of the part is voxelized and supports also represented by voxels are generated via ray tracing. Cheng & To further expanded the framework to the calculation of residual stresses by the inherent strain method. Both optimization goals, minimal support volume and residual stresses, are combined into a multiobjective optimization model by linearly weighting the objectives. Similar to the angleconstraint topology optimization approaches, the part orientation optimization approaches consider overhanging features only in their support assessment, neglecting the remaining tasks.

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When the part design and orientation are fixed, it is the primary goal in the data preparation phase to apply supports as efficiently as possible. Kober et al. [293] proposed a simulationbased approach to position supports according to the predicted residual stresses. In a first step, the part is supported by identifying the local geometrical minima of the part. Here, cone supports with a constantly decreasing diameter towards the support-part interface are used since they can be approximated by a tripod of linear elements, reducing the size and complexity of the mesh for computation. Second, the MPE method is implemented to determine the residual stresses due to the manufacturing of the part with the initial support layout. In every iteration, the distortion is evaluated. If a defined threshold is exceeded, a new cone support is inserted. The simulation is started again from the first computational layer the new support influences the overall part from. The procedure is repeated until the whole part has been successfully computed and no change in the support layout occurs anymore. While the procedure demonstrates the importance of integrating process simulation, it is focused on the sole use of cone supports or similar support structures, leaving out the capabilities of other support structure types. Aside from the part design, the choice of material may enable support-less manufacturing via PBF-LB/M, as shown by Mumtaz et al. [294]. In their work, they transferred the principle of super-cooling polymers used in laser powder bed fusion of polymers (PBF-LB/P) preventing the need of supports to the metal-based process by the use of binary eutectic alloys. The chosen alloy typified by bismuth and zinc exhibits a melting point lower than the melting points of the pure elements. During manufacturing, a batch of bismuth and zinc powders is mixed in their un-alloyed eutectic proportions. The powder is deposited while the build platform is held at a temperature near the eutectic melting point of the alloy. This results in a slow solidification of the material after laser exposure, preventing the formation of stress concentrations. Though able to validate their approach experimentally, the avoidance of supports by picking a eutectic material does restrict the choice of material immensely and does not meet the needs of current industries applying the PBFLB/M process. In conclusion, the part design and data preparation provide powerful tools to reduce the application of supports to the necessity. However, as most AM products are part of an assembly group with other conventionally manufactured parts, it is often not possible to avoid supports completely due to the boundary conditions set by the assembly group. It is therefore critical to optimize the support structures themselves as well to minimize the additional cost induced by them.

2.5.2 Optimization of available support structures Commercial software for data preparation provides a variety of standard support structures, as shown in Section 2.4.1. Additionally, there is a great number of parameters to be individually set for each standard support type, geometrically as well as with regard to the

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manufacturing process. It is therefore obvious to optimize these if supports cannot be avoided. Three different approaches have been followed in research to do so: 2a. Optimizing the geometrical parameters of supports 2b. Adapting process parameters for supports 2c. Multi-material applications similar to those found in the material extrusion of polymers (MEX/P) process The most possibilities to adjust supports to a certain situation are given by the geometrical definition of the standard supports. Given its status as the most common support type, the block support has been focused. Calignano [276] investigated the influence of the hatch distance, i.e. the distance of parallel walls, tooth height and base interval, and the offset in Z-direction of the teeth into the part to strengthen the bonding between part and support. Furthermore, the application of wall perforation as well as wall fragmentation was evaluated. These relevant block support parameters are visualized in Figure 2.13. It was found that only the tooth height and hatch distance had a relative contribution to the performance criteria, which was defined as the signal-to-noise ratio of the Taguchi experimental design methodology. Künneke et al. [296] studied the hatch distance’s influence on support volume, removal time, surface roughness, and warpage. While the surface roughness reduces with decreasing hatch distance, an increase in hatch distance leads to less volume, easier removal, as well as only slightly increased warpage. In [297], Lindecke et al. also investigated the hatch distance, tooth top length, as well as the offset in Z-direction. Measuring the tensile strength of the supports, they observed a positive influence of the Z offset and the teeth top length on the tensile strength. A decrease in hatch distance also leads to increased tensile strength. Poyraz et al. [300] examined the effect of changes in hatch distance, top length, offset in Z-direction, as well as the fragmentation on the dimensional accuracy. In their study, it was observed that the hatch distance is the main characterizing parameter. Also attending to the dimensional accuracy as well as build time and removability, Sulaiman et al. [275] experimentally investigated the influence of hatch distance, perforation, and fragmentation. Their observations match the findings of [276, 296]. Furthermore, they determined that the changes in support parameter evaluated do not significantly change the build time. The same conclusions were drawn when repeating the experimental series with contour supports, though they found the contour support to be less effective regarding distortion prevention while easier to remove. Focusing on the teeth top length and hatch distance, Oter et al. [298] came to the same conclusion as [296] with a decreasing surface roughness if the hatch distance is increased. They also found a small tooth top length to be beneficial in reducing the surface roughness.

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Figure 2.13: Parameters of a block support wall (based on [276]). In 3D space, walls are arranged in a rectangular grid.

Aside from examining the influence of the block support parameters, some work has been done to create extensions to the base geometry in order to enhance a specific criterion. Aiming for reduced post-processing effort, Paggi et al. [299] developed contactless supports, where there is a small gap between part and support. That way, no support removal from the part is necessary and no residuals at the surface require further surface finishing. The appropriate choice of gap distance remains a critical issue, though, to prevent overheating or dross formation. Furthermore, the application of those contactless supports is unsuitable for setups where they are not able to prevent distortion solely by heat dissipation as no mechanical fixation is realized. Zhang et al. [303] added cuboids to the walls of the block support in order to increase its heat dissipation capability. They were able to show a positive effect on the deformation of their specimens, but also concluded that the teeth of the block support are more effective at resisting the forces due to residual stresses. Krol et al. first developed a method to numerically model the support performance of block, contour, and web supports [295], and later created an algorithmic fractal design of the block support [175] allowing for local stiffening of the support structure, which is based on the numerical assessment of the occurring residual stresses. Some work focusing on other types of supports in PBF-LB/M has been done by Song et al. [301], who developed a functionally graded cone support by varying the cone diameter or edge length, respectively, as the rectangular cross section was adopted to ease the numerical modeling of the support. With the help of a 1D numerical study, they demonstrated the effect of mass distribution to the global thermal history of an overhanging feature. Experimental validation is not presented. The parametric tree support is investigated by Zhang et al. [302] in terms of material savings compared to lattice supports as well as

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compressive strength. The tree supports exhibited higher yield strength, but at relatively big strut diameters. While there are many support parameters the designer is able to set individually, only a few seem to affect the support performance significantly, e.g. hatch distance and teeth geometry for block supports. Given that especially 2D supports already consist of thin features, the potential for support optimization via support parameter optimization is limited. A second way to optimize supports is the adaption of process parameters, as demonstrated by Morgan et al. [306]. They developed a set of process parameters for block and solid supports, respectively, tailored to reduce the support strength for easier removal, increasing the build rate at the same time. The procedure presented relies heavily on experimental work, though. Cloots et al. [304] reported a study, where the process parameters for a lattice support are varied in order to optimize the interface region between part and support. Here, the focus lies on defect-free, well-connected structures, while the remaining part of the support is manufactured with standard process parameters. The approach of utilizing the process parameters is taken a step further by Jhabvala et al. [305], who introduced the combination of continuous and pulsed modes of laser radiation for the production of the part and support, respectively. The use of the pulsed mode increases the manufacturing speed while leading to supports with a reduced strength, facilitating easy removal. All approaches dealing with process parameters or scan strategies require technical equipment to realize the new procedures, which poses a great challenge as there are close to no commercial manufacturing systems with an open infrastructure allowing for these changes. Similar to the dissolvable material used for supports in the MEX/P process, some work has been published towards the development of multi-material PBF-LB/M with explicit support material. In [307–310], a research group reports the development of dissolvable metal supports first for the directed energy deposition (DED) process, then for PBF-LB/M. Since PBF-LB/M is not directly capable of multi-material applications yet, the interface region between part und support is designed to consist of tightly spaced thin struts. In an electro-chemical etching process, the thin struts dissolve before the part surface is seriously damaged. In an extension of the general procedure, they sensitize the stainless steel samples in the annealing step prior to support removal. In consequence, the support is completely changed in its chemical composition at the part-support interface, while the chemical composition of the part is only changed on the surface. By using a different etching agent, only the new chemical composition is dissolved. The support is completely removed from the part, while the part etching is terminated when the original material is reached by the etching agent. They are able to apply the procedure to stainless steel [307], Inconel 718 [309], and Ti-6Al-4V [310]. This approach is a comparatively slow support removal process though, with etching times of 30 − 40 h. Wei et al. [311] made use of

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the multi-material PBF-LB/M system developed in earlier work [339], which combines the classic powder bed application with point-by-point nozzle ultrasonic powder delivery. After the usual layer of powder is applied, the powder is locally removed and a second powder material is deposited by the nozzle system. Wei et al. chose to supply silicon carbide to a 1.4404 stainless steel setup to create brittle supports for easy removal. While the study provides a proof of concept, various challenges such as material composition choice, process parameters, or the design of the transition zone between the two materials remain. In summary, the approaches to optimize existing support structures by adapting their geometrical features, developing specific process parameters, or applying multi-material procedures, may provide opportunities, which are limited, though. Especially regarding the last two topics, the requirement of novel equipment not commercially available yet restricts the further development of these approaches.

2.5.3 Development of novel support structures In order to create high-performance supports ensuring reliable manufacturing, the development of novel supports provides many opportunities to overcome the drawbacks of the support optimization approaches described in the prior sections. In general, developing new supports is based on either of these approaches: 3a. Unit-cell based supports 3b. Free topology supports The application of unit cell structures as supports promises a low weight while maintaining stability at the same time. Additionally, when designed to be an open unit cell, remaining powder can be recovered rather than be lost in the part detachment step [304]. Another benefit of unit cell structures are the usually small part-support interface regions, which facilitates easy support removal [319]. The design is focused on the respective unit cell, while the actual support is generated by repetitively placing the unit cell into the space the support is allowed to take up. Strano et al. [292] as well as Hussein et al. [316] investigated the ‘Schwartz’ and ‘Gyroid’ unit cells, which can be described by a combination of trigonometric functions (cf. Figure 2.14). They demonstrated the versatility of those unit cells, the implicit mathematical description easing the design effort, but also noted that the thin walls required to manufacture may be an on-going challenge. Cloots et al. [304] developed a strut-based unit cell with struts along the vertical edges as well as diagonal struts meeting in the unit cell center. They successfully applied them to their demonstrator parts, and point out the importance of the maximum horizontal edge length, i.e. the maximum distance between two supporting points. In [319], Vaissier et al. adopted the unit cell design of [304], added a vertical strut to the center, and extended the approach by an optimization step. Here, after the generation of an initial lattice support based on this unit cell, a genetic algorithm systematically removes struts to achieve the

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minimal cumulative strut length while respecting a sustainment constraint ensuring complete support of an overhanging surface. Compared to other common support structures created by commercial software, they were able to reduce the material consumption at the cost of additional computation time. Gan et al. [313] also designed strut-based unit structures, resembling trees with four branches. They investigated the design with the branches connected to the part as well as to the build platform, i.e. upside down. The tree design with the branches connected to the part were not successfully built due to heavy distortion. The thermal finite element (FE) analysis determined the cause to be the uneven supporting point distribution, which lead to local differences in the cooling rate. The upside-down tree exhibited good stability and minimal interface region, but was affected by manufacturing inaccuracies, too.

Figure 2.14: Schwartz diamond and Schoen gyroid

The unit cells in [292, 304, 313, 316, 319] are designed based on definite rules. Huang et al. [315] targeted the heat conduction capability of unit cells via topology optimization. After even distribution of the supporting points, basic unit cells for a rectangular, cylindrical, and arched design space were created for several values of the maximum volume fraction. They compared their design with the effective one of Gan et al. [313], and found their structures to result in less distortion of the overhangs manufactured. Cheng et al. [312] also utilized structural optimization in their work, but instead of optimizing the unit cell prior to the application to the part, the algorithm optimizes the lattice unit cell parameters to create a graded design. In consequence, unit cells bearing higher loads are assigned thicker struts. Though they successfully demonstrated the potential of the approach, the removal of the graded lattice support is a challenge as the unit cells are close to solid blocks in some areas, preventing manual support removal. Pellens et al. [317] reversed the approach of [312]: Instead of placing the unit cells first and then optimizing them, they conduct a standard topology optimization study and map the result onto a grid of strutbased unit cells, defining the strut diameters according to the material distribution.

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He et al. [314] applied the honeycomb structure aiming to overcome the problem of manufacturing thin struts while maintaining an open structure for powder removal. The honeycomb structure showed high mechanical strength and its ability to support large overhangs was demonstrated numerically. No evaluation of the removal of the honeycomb support was performed, though, reducing the reached optimization goal to lightweight design only. Different from the other unit cell approaches, which utilize rectangular unit cells, Vaidya & Anand [318] designed a unit cell support approach based on truncated octahedron and rhombic dodecahedron structures with perforated faces. The use of octahedral or dodecahedral unit cells allows for a denser packing of the unit cells, creating geometrical freedom regarding the arrangement of those. Experimental validation was not performed. Unit cell-based supports offer high stability at low volume and can be adapted to individual manufacturing situations. However, the application of these supports also comes with some disadvantages: While the material consumption is comparably low, the overall space occupied by the unit cells is significantly larger. This complicates the support of features with only narrow support space as well as the support of thin features, e.g. edges or thin walls. Furthermore, the low material consumption requires precise manufacturing of thin structures at the limit of the PBF-LB/M process resolution, carrying the risk of build failure. As unit cells are highly regular structures, the application to free-form part geometries may also pose a challenge in terms of interconnectivity. As additively manufactured parts are often characterized by highly individual geometries, it may seem obvious that the optimal support is also an individual structure. Vaissier et al. [319] demonstrated the individualization of an initial unit cell support by locally deleting non-loadbearing struts, resulting in tree-like supports. The geometry of a tree as support was also investigated by other researchers: Zhu et al. [336] proposed an algorithm to generate tree supports by discretizing the support design space, generating an initial tree structure by defining nodes such as connecting points or the root, and subsequent variation of the tree node positions by particle swarm optimization. They successfully validated their approach by comparison with tree supports generated by commercial software, decreasing material consumption and build time. Zhang et al. [332] employed the mathematical theory of L-systems to generate tree supports. Coupled with numerical process simulation, the tree supports are designed according to the temperature history during manufacturing. The computational cost of the process simulation challenges the broad application, though. In [331], Weber et al. presented a survey of the effects of a variation of the geometrical tree support parameters on build properties such as support and part strength, surface roughness, and manufacturing time. In order to create completely free support geometries, Leary et al. [328] established a voxel-based cellular automaton. They defined rules and boundary conditions operating on

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the design space discretized by voxels. That way, individual support topologies are generated at low computational cost. The approach suffers from the voxelized nature of the results, though, because angled features may have issues, as voxels are arranged edge-toedge with small connecting areas. A common way to design optimal structures is the method of topology optimization, as already discussed in Section 2.5.1. Gardan & Schneider [323] were among the first to apply topology optimization to supports by adding the support optimization step to a part optimization framework. Kuo et al. [321] applied the SIMP method to support overhanging features. They further included the requirement of minimal residuals after support removal, forcing the optimization to split the topology into many small struts in the designated support-part interface region. Zhou et al. [335] also tend to the design of easy-toremove supports for overhangs. In their SIMP optimization formulation, they integrated the filter of Langelaar [340] to ensure that the supports do not violate the critical angle restriction in PBF-LB/M themselves, and applied a local volume constraint proposed in [341] to achieve thin geometries in the interface region, similar to [335]. In [333], Zhang et al. focused on the support task of fixating the part in space. They implemented the SIMP method in combination with the inherent strain method to model residual stresses in the part and presented a framework capable of parallel computing to decrease the computational effort. At the same material usage, their supports achieved a decrease in distortion of over 70% compared to block supports. Giraldo-Londoño et al. [324] utilized a multimaterial topology optimization approach to optimize part and support geometry simultaneously. Topology optimization can be used to consider thermal as well as mechanical loads. Malekipour et al. [329] used the thermal formulation of the SIMP method to design supports for efficient heat dissipation from the part towards the build platform, and into the powder bed. Their result is a highly complex structure with many thin features, needing prove of manufacturability. In the works of Zhou et al. [334], the combination of the SIMP method and the filter of [340] lead to thermally conductive supports, as validated numerically. Wang & Qian [330] applied their own overhang angle constraint to the SIMP method, which were successfully demonstrated on common benchmark cases of topology optimization. While the studies presented above focus on the optimization of the supports only, Langelaar [325] proposed a holistic framework to optimize the topology of the part and supports simultaneously, aiming for a trade-off between design performance and support costs. He later extended the framework to optimize the part orientation first [326] and consider necessary features for post-processing via machining [327]. Allaire & Bogosel [320] also published their work on a framework capable of optimizing part and support topology simultaneously. Different from the other approaches employing topology optimization, the proposed framework combines the thermal and mechanical problem formulation for

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the supports. In [322], the framework is extended to perform part orientation optimization prior to the actual topology optimization, as done by Langelaar. Topology optimization is capable to generate truly individual supports for a given part at a given orientation, and can even be combined with the part design procedure. The utilization of topology optimization is accompanied by some challenges, though. The high computational cost leads to long support design times or limitations regarding the size of the design space of the supports. This may not be a critical factor in the context of mass production, where the design effort is only done once to define the production of years. For AM, whose benefits also lie in the comparatively low part costs at small production scale, this may hinder the establishment process in industry, so that the benefits in mass production will never be exploited. Furthermore, topology optimization is highly sensitive to its parameters and the initial guess of the result. Additionally, the result needs to be reintroduced to the CAD data, which is the base geometry format in AM. These circumstances require high expertise in terms of topology optimization to achieve optimal results reliably. The adoption of the topology optimization approach for support optimization in commercial data preparation software is therefore in question.

2.5.4 General characteristics of optimization approaches Comparing the three main categories of support optimization approaches, it becomes apparent that the category 1 (avoidance) and 3 (novel structures) approaches mostly rely on computational methods to achieve their goals, whereas category 2 (existing structures) studies use experimental procedures. In conclusion, 65 % of the studies listed in Table 2.6 follow computational approaches, highlighting the importance of computer-aided design. With regard to the support tasks considered in the optimization, 76 % of the studies include the task of supporting overhanging features, whereas heat dissipation (20 %) and part fixation (21 %) receive much less attention. Note that one publication can consider more than one support task. The discrepancy is due to two major reasons: On the one hand, there is more research on the overhang support task to build on, because in the polymerbased AM for rapid prototyping (most popular use case, cf. Section 1) heat and residual stresses have less influence than in metal AM. Aside from the design restrictions of the respective AM process, the general approaches are independent from the AM process. On the other hand, the support of overhanging features is a purely geometrical problem to solve; no information about the thermal history or internal stresses are required. As a result, the geometrical problem can be solved with less effort than it is the case for the other support tasks. In PBF-LB/M, the tasks of heat dissipation and part fixation are essential to successful manufacturing. To be able to address these additionally to the overhanging features, computational methods to predict the thermal history of a part during manufacturing as well

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as the resulting residual stresses are needed. The coupling of PBF-LB/M process simulation or other approximation approaches and support design is a key factor for efficient supports, as demonstrated in the computational works considering fixation [290, 293, 295, 312, 322, 333] and heat dissipation [295, 299, 301, 320, 322, 324, 330, 332, 334].

2.5.5 Optimization goals While there is a great variety in the approaches taken on, different optimization goals are pursued as well. Minimizing the cost due to supports being the overall target, the challenge of determining the exact cost leads to optimization approaches aiming for what is considered a cost driver rather than trying to quantify the overall cost. Additionally, specific properties of the supports or parts are addressed. It has to be noted that one publication can state more than one optimization goal. The main support property associated with cost is the volume of the support (considered by 50 % of the references), since the volume results in material consumption and extended production times as the volume has to be built alongside the part. The second important factor is the removal of the supports (31 % of all studies) because this is a mainly manual task today, requiring a human worker to be paid. With the goal of complete support avoidance by design (16 % of all studies) the three most important optimization goals cover the part design, part manufacturing, and post-processing stage. Only 10 % of the published studies directly address the support cost, and half of them belong to a series of consecutive publications of the same author. Nevertheless, no direct modeling of the support cost is performed: Künnecke et al. [296] use the support volume and removal time as cost indicator, while Zhang et al. [302] consider the volume only by calculating weight and scan time. Both [296] and [302] refrain from deriving cost values and stick to their cost indicators as references. Kuo et al. [321] as well as Langelaar [325– 327] incorporate the support cost into their topology optimization procedure. In [321], the additional travel time for support manufacturing (here of the extrusion head as MEX/P is considered as reference AM process, but the concept can be adapted to the additional scanning vectors of the laser) is introduced by weighting the distance of a support element to the part, such that a support element closer to the part is preferred by the algorithm. They further extended this approach to include the support material concentration, as highly concentrated supports require little travel distance within their region. Therefore, it may be beneficial to have a high concentration of support material at a greater distance to the part rather than having to cover a larger area closer to the part. Langelaar [325–327] considered two contributing factors, support material and support removal costs. The material costs are defined to be proportional to the support volume, whereas the removal costs are assumed to scale with the size of the interface region between part and support. Both aspects are implemented into the optimization problem formulation as constant factors related to the respective costs.

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Aside from optimization goals related to cost, several studies are targeting physical properties relevant to support or part performance. The capability of the support to dissipate heat (12 %) is either investigated by changing the parameters of established supports [299, 301, 303] or by the use of topology optimization [315, 320, 330, 334]. Furthermore, Allaire et al. [320] and Langelaar [327] aimed at increasing the stiffness of the part for postprocessing via machining with the help of supports. Paggi et al. [299] looked at enhancing the surface quality of a part, proposing a contactless support, i.e. there is no interface between support and part. That way, the removal of the support should not cause a reduction of the surface quality. This effect has not been observed, though, as the measured surface roughness did not change.

2.5.6 Quantification of optimization success Whereas the design of new supports is mainly generated computationally, the developed approach should be validated experimentally to ensure the applicability. Still, computational and experimental validation approaches are nearly tied in numbers for the studies of Table 2.6 with 34 and 33, respectively. About 12 % of the validations include both computational and experimental methods [284, 290, 312, 319, 328, 332, 333]. Linking the type of validation approach to the type of optimization approach, it becomes apparent that only 10 out of 40 studies employing computational methods for their support optimization perform an experimental validation. This is a great concern in terms of sustainable support optimization as FE analysis of either the support performance or the manufacturing process is not able to exactly mimic the real physical processes. One reason of missing experimental validation may be the non-existence of standardized benchmark procedures. Table 2.6 highlights the multitude of benchmark parts used in literature. The geometry used most is the so-called cantilever beam (34 % of all references), which consists of a base block and an overhang parallel to the build platform. The dimensions of the cantilevers used in the studies are not standardized, but chosen individually. Benchmark cases known in topology optimization are the Messerschmitt-Bölkow-Blohm (MBB) beam (13 % of all studies) and L-bracket (7 % of all studies). Various other bracket geometries have been applied also (7 % of all references); it is important to notice, though, that no bracket geometry has been applied twice. Nevertheless, brackets originating in aerospace or automotive industry represent common use cases for AM. 10 % of the studies in Table 2.6 use a simple rectangular plate, while 25 % use geometries not recognizable in their application. Further benchmark parts employed are listed in Table 2.6. In conclusion, there is a great variety in benchmark parts. Contrary to the polymer-based extrusion AM process, where 3D models such as the minotaur, gymnast, or rabbit have been established as standard use cases [342–344], no such geometry has been implemented in PBF-LB/M. In conclusion, neither specialized benchmark part designs nor commonly

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used other geometries to assess the performance of the optimization approaches are available to date.

Figure 2.15: Benchmark part of Cheng & To [290]

Cheng & To [290] take a first step towards a benchmark part by designing a combination of four different types of overhanging features (cf. Figure 2.15): a horizontal overhang, a concave as well as convex arrangement of two angled, plane surfaces, and a convex feature consisting of an upside-down square-based pyramid. The benchmark part is designed for the validation of their voxel-based overhang detection and support generation. Therefore, the tasks of heat dissipation and part fixation are not addressed. Jiang et al. [342] also identified the lack of standardized benchmarks in support optimization and are the first to publish a benchmark part to assess support performance, see Figure 2.16. Since they focused on the MEX/P process, similar to [290], again only the support of overhangs is considered. Their benchmark part includes different overhangs consisting of two angled surfaces with different angles, as well as concave and convex arcs of two different radii. Aside from the general functionality of the supports resulting in a successfully built part, they further evaluated the surface roughness of the part-support interface, the dimensional accuracy of the benchmark part, and the material consumption of the supports.

Figure 2.16: Benchmark part of Jiang et al. [342]

Although approaches towards a standardized benchmark procedure for support optimization are sparse, there are several studies aiming at the quantification of specific physical

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support characteristics. Lindecke et al. [297] developed a tensile specimen design, which is also employed in [274], where the support structure itself is located between two blocks used for clamping. During manufacturing, the support is strengthened by four cone supports, which are removed prior to tensile testing. They found a reduction of over 75 % in the block support tensile strength compared to the solid material. Furthermore, a tendency to break at the lower support-part interface was observed. In [274], Gralow et al. accompany the tensile specimen design by additional specimen design to evaluate material consumption, support removability, and surface quality. For material consumption, a defined cube with an edge length of 20 mm set off the build platform by a square-based pyramid is used. Support removability and surface quality after removal is tested by a plate with attached blocks for clamping. The cube design for material consumption evaluation is advanced in [345] to suit the lattice supports, which require more nominal space than block supports. Bobbio et al. [346] designed tensile specimen as well. Here, the geometry is a solid block, where lattice supports are built enclosed in the middle of the block. After manufacturing, a horizontal cut by wire eroding splits the block such that only the supports connect the two blocks. Different from [297], who included bore holes for clamping, the tensile specimen of [346] rely on wide grips and traction. Bobbio et al. observed the tensile strength of the different lattice supports to be in the range of 14 − 32 % of solid material. However, the strength of the part-support interface was found to be higher than 400 MPa in all cases, which was deemed sufficient to counter-act residual stresses. Weber et al. [347] designed tensile specimens in the style of the flat ASTM E8 standard. The entire mid-section consists of cone supports, while the solid regions for clamping were kept from the ASTM E8 design. The specimens were built without additional structures. They analyzed the inner and outer area of the specimens and concluded that while the inner solid region bears most of the load, the outer, not fully molten region still contributes to the overall load bearing. In [348], Leary et al. tend to the mechanical characterization of block support. Similar to [346], the tensile specimens include two solid blocks for clamping, which are connected by supports of varying height. Schmitt et al. [349] evaluated the shear and tensile strength. The shear strength is evaluated following DIN EN ISO 6789-1 by loading via a releasing torque wrench. The specimen design consists of a cylindrical shape with two solid sections connected by the supports, similar to the tensile test specimens. The tensile specimen design of Schmitt et al. is based on the round standard test geometry. In the mid-section, lattice supports are applied with a greater tensile specimen diameter. Furthermore, the supports are enclosed in a thin cylindrical wall. Depending on the process parameters in manufacturing, the lattice supports achieved 10 − 30 % of the solid material tensile strength. Different from the studies aiming at the quantification of effective mechanical properties of supports, Zeng et al. [350] investigate the effective thermal conductivity of supports. To do so, they set up a FE analysis of a representative volume element of the block support deriving the temperature distribution for a given thermal load. They compared the thermal

Support structure optimization

69

response of the support volume element with a parallel spring thermal element and found a significant similarity, thus proposing to model supports in PBF-LB/M process simulation via those elements. Still, experimental validation is to be conducted.

2.5.7 Automated support structure removal Aside from the additional resource consumption due to supports, the removal has been identified as one of the most critical challenges. The mostly manual work leads to high cost related to the required personnel, and is a risk in terms of the standards in post-processing essential for serial production. Therefore, efforts towards automated support removal have been made and reported in the literature. Because milling is often used for the surface finishing of parts with high requirements regarding the dimensional accuracy, the idea of using this subtractive process to remove the supports has received attention. Nelaturi et al. [351] proposed a procedure to automatically derive the path planing for multi-axis machining to remove supports. Their recursive algorithm identifies the supports accessible by the tool as well as the tool orientations the supports can be removed at. The least expensive tool path is then obtained by the so-called traveling salesman problem. That way, the supports are peeled away layer by layer, as supports in the middle of a full support block are only accessible if the surrounding supports are already removed. Their approach reveals high computational cost, which may be reduced by parallelization. Furthermore, the physics of the removal, e.g. the mechanics of the fracture process, are not considered yet. Cao et al. [277] experimentally investigated the machining of block and cone supports made of 1.4404 stainless steel. Measuring the Vickers micro-hardness, they found a zone of about 2.5 mm of gradual decrease starting from the part-support interface. A cutting depth of 0.4 mm was necessary to remove the supports completely, which was achieved by four consecutive cuts. For each cut, a new cutter was used. They also found the cutting forces to be high enough that they advised against the processing of thin walls to avoid damage to the part. Because they notice that cone supports tend to tilt during milling, their work is continued in a consecutive study [352]. Here, the space between the cone supports is filled with epoxy resin. Experimental investigations confirmed that not only the tilting of the cone supports is addressed this way, but also tool wear and damage to the part is significantly reduced. The topic of machining thin walls in combination with support removal is taken on by Hintze et al. [353]. They investigated different support geometries removed by circumferential milling and a cutting depth of 8 mm. It was noted that the feed force is significantly influenced by the type of the support, and the overall support volume fraction is stated to be the key factor. Furthermore, a high oscillation of the feed force in the milling of cone and block supports is observed because of the discontinuous contact between material and cutting edge. These oscillations can cause damage to the final work piece. Finally, they advise towards the

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combined machining and support removal for surfaces with strong requirements for surface roughness in order to reduce the overall machining time. In [354], the feed force as well as the resulting surface roughness are evaluated for several support designs. While the block supports were mainly cut, the cone support design showed no clear abrasion. All studies demonstrate that it is possible to remove supports via milling. They also point out the disadvantages of milling, though: On the one hand, cutting forces may prevent the processing of filigree parts. Furthermore, Cao et al. found that the collapse of the cone supports during milling resulted in plastic deformation of the part. On the other hand, the free-form geometries of AM parts require complex tool path planning to access every support. The feasibility of automated support removal by milling may be proven for the serial production of parts with high demands in terms of dimensional accuracy, but for complex parts at low production scale milling might not be the most economic choice. Möller et al. [355] extended the work of [353] and investigated continuous as well as impact shear separation, which resembles the manual support removal with chisel and hammer. They found a significantly smaller shear force in the impact separation of block supports compared to continuous shear separation, and are able to demonstrate the robot-guided process on straight as well as curved surfaces. However, applicability with regard to 3D part geometries remains to be proven, and further aspects such as tool wear and tool path generation need to be investigated. A way to automate support removal without having to consider the part geometry is chemical etching. The work of the group around Hildreth [307–310] has already been described in Section 2.5.2 since they changed the composition of the base material. Schmithuesen et al. [356, 357] investigated the etching of AlSi10Mg samples via sodium hydroxide (NaOH). The samples, a cantilever beam as well as a ring profile, were supported by perforated block supports and tree supports, respectively. They found that at elevated temperature of 80 °C and a solution concentration of 50 %, complete separation of part and supports can be achieved within a few minutes. The etching process changes the element fractions of the alloy, though. Furthermore, the setup is tailored to the AlSi10Mg alloy, whereas the adoption to other materials will require the identification of other suitable solutions. Additionally, as the process is not self-terminating as in [308], attention has to be paid to the etch duration. Chemical etching is a promising approach in the case of thin supports at the part support interface and parts with further surface finishing requirements. When the supports have to be designed to be sturdy because of high residual stresses, the time to completely dissolve the contact point also affects the remaining surface of the part. In addition, the question of whether the material choice is limited by this approach requires further investigation.

Economic evaluation of additive manufacturing 2.6

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Economic evaluation of additive manufacturing

To achieve broad industrial application of AM, this new technology has to be adopted by as many market participants as possible. The adoption of a novel technology is characterized by five consecutive stages [358, 359], namely awareness, interest, evaluation, trial, and adoption. Aside from technical innovation, production costs are a dominant factor throughout all of these stages: The impact of innovation on the costs is the most important aspect that an interested decision-maker analyzes before choosing a new technology [360]. First, to create interest in a decision-maker, the general cost advantages of the technology need to be communicated. During the evaluation stage, the specific financial benefit to that decision-maker is highlighted, and confirmed throughout the trial stage. Finally, when the technology is adopted, the on-going production requires constant monitoring to ensure profitability. Since the early adoption stages discuss future scenarios, cost prediction is important here. However, in the later stages the use cases are becoming more and more realistic, leading to a change in the need for cost modeling from prediction towards exact cost determination. This dynamic is visualized in Figure 2.17.

Figure 2.17: Technology adoption stages and corresponding cost evaluation goals

Consequently, research efforts have been directed towards the evaluation and prediction of the part production costs in AM. Kadir et al. [361] found that within the AM context, various cost classification techniques are utilized based on the overall point of view, namely the finance and accounting perspective, the manufacturing perspective, and the management perspective. Applying the finance and accounting perspective, a classical accounting technique also called the ‘method-based’ approach is commonly utilized. The method-based costing distinguishes quantitative and qualitative cost models, according to the way of model development: Quantitative models consist of parametric and analytical methods, while qualitative models include intuitive and analogical methods. Intuitive methods are based on subjective estimates made by an expert. The analogical methods extend the intuitive methods by including historical data for comparison. In the quantitative cost evaluation, historical data is collected and organized via statistical techniques. Analytical methods, however, consist of purely mathematical procedures aiming at the decomposition of a product into units, operations, or activities related to manufacturing.

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The manufacturing perspective commonly employs task-based methods, where the required tasks performed in product development and manufacturing are assessed individually. Kadir et al. differentiate design- and process-oriented cost models. Whereas the design-oriented models include involved scopes, e.g. part design or process planning, the process-oriented models cover the cost items relevant in production, which are often related to direct costs (e.g. material, energy) and indirect costs (e.g. administrative operations). These task-based methods are also known as activity-based costing (ABC) [362]. From the management perspective, cost models are build according to the investigated level, i.e. the process or system level. The process level addresses the costs related to the production, while the system level covers the product lifecycle including services, operation management, and supply chain. Capital, material, or administrative costs play an important role in this perspective. In conclusion, each of the discussed perspectives emphasizes different aspects of cost modeling. Table 2.7 gives an overview of cost models either explicitly addressing the PBF-LB/M process or compiled of a generic structure without a focus on a specific AM process. The cost models are categorized with regard to several criteria: First, the base scenario of the model is indicated, including buy (B) or make (M) scenarios. Second, the goal of the model is stated. Here, the model is designed either for cost prediction (P) or the detailed calculation (C) of costs after manufacturing. Furthermore, the considered stages of the product lifecycle, namely the product design, manufacturing, post-processing, or remaining lifecycle, are denoted to enable the classification of the models according to [361]. In terms of the considered scenario, all studies tend to the ‘make’ scenario, except for Baldinger et al. [363]. To evaluate the market, they gathered quotations from an online Business-to-Business (B2B) marketplace for PBF-LB/P and PBF-LB/M of various materials and performed regression analysis to identify a formula for cost prediction. However, the data on PBF-LB/M is sparse in comparison to PBF-LB/P, and therefore regression analysis is disregarded. While the cost calculation is essential in production optimization, the prediction of the product’s costs is an important tool at the design stage, as the costs are often included in the requirements to the part design. Furthermore, cost prediction is used to compare the economic suitability of different manufacturing processes for a given part. Due to both aspects being relevant to the broad application of PBF-LB/M, the research efforts are split between cost calculation and prediction with nine and twelve studies in Table 2.7, respectively. Calculating the exact costs of a part is possible by assessing detailed information during the different lifecycle stages. The challenge in the cost calculation is the identification of all cost items and the acquisition of the relevant data.

Economic evaluation of additive manufacturing

73

Table 2.7: Overview of literature on PBF-LB/M cost modeling

Modeled stages Ref

Type

Design

Manufacturing

Postprocessing

Lifecycle

[363]

B

P

x

[364]

M

P

x

[365]

M

P

x

[366]

M

P

[367]

M

P

[368]

M

P

[369]

M

P

x

x

[370]

M

P

x

x

[371]

M

P

x

x

[372]

M

P

x

[373]

M

C

x

[374]

M

C

x

[375]

M

C

x

[376]

M

C

x

[377]

M

C

x

[378]

M

C

x

[379]

M

C

x

[380]

M

C

x

x

x

[381]

M

C

x

x

x

[382]

M

C

x

x

[383]

M

C

x

x

[384]

M

C

x

x

x

x x

x

x

x x x

x

x

x

Legend: B/M - buy/make scenario C/P - calculative/predictive model

2.6.1 Cost calculation In 1998, Alexander et al. [384] introduced a generic cost model intending to provide a framework for process-specific cost evaluation. Their model is organized with respect to

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three phases of AM: pre-build preparation, manufacturing, and post-processing. They assume an appropriate 3D geometry is available. During pre-build preparation, the labor cost of the person positioning, scaling, and slicing the 3D geometry, as well as generating the supports via data preparation software is considered. Furthermore, the cost of the computer required for data preparation is included. The time necessary for the preparation of the manufacturing system and the respective cost is also assigned to this phase. The build phase costs consist of the material cost as well as the cost for running the manufacturing system during manufacturing. The material cost differentiates between part and support material, since the material may not always be the same depending on the applied AM process. The post-processing phase’s costs are based on the time to remove supports and finish the part surface, which is assumed to be done manually, leading to personnel cost. Additionally, the cost of operating required equipment is taken into account. In [382, 383], the respective authors also aimed at creating generic cost models. Schröder et al. [382] evaluated several calculative cost models and identified common characteristics. These characteristics represent the basis of the developed cost model, which is implemented into a cost calculation tool. Thomas & Gilbert [383] evaluated existing models for various AM processes, too, and further investigated inventory and supply chain costs to fully capture the effect of AM on production. They conclude that while there is an understanding of the AM process costs, the benefits due to e.g. part integration to reduce the number of assembly parts and steps are not represented adequately in cost modeling. To enable the assessment of these benefits, they suggest a cost representation incorporating the cost of inventory, raw material extraction and refining, trade of the finished product, as well as transportation cost throughout the supply chain. However, these figures are complicated to compile, requiring great effort to gather all information necessary for the application of the cost model. Parallel to the development of generic cost models, work has been directed towards the definition of PBF-LB/M specific cost models. In [381], Rickenbacher et al. built on the work of [384] by adapting it to PBF-LB/M. They extended the model to several parts with varying heights within one build job by allocating the build time in a layer-wise manner. Lindemann et al. [380] set up a highly detailed assessment of the machine costs per build, but did not define explicit functions for the calculation. In the end, the manufacturing costs are determined by the material costs and build costs, which consist of the fixed costs as well as the machine rate related to the build time. Baumers et al. [379] extended the model of Ruffo et al. [385], designed for the PBF-LB/P process, to the metal process. Here, they distinguished direct costs, i.e. material and energy costs, and indirect costs such as hardware and software investments or the involvement of employees. Similar to the framework of [384], Atzeni & Salmi [378] derived a cost model corresponding to a preparation, build, and post-processing phase. However, they considered only the labor cost for preparation, neglecting other cost items such as the hard- and software required. Post-processing includes manual work as well as the blank cost for external heat treatment. Väisto et al. [377]

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75

modified the approach of [379] to consider the exact part volume rather than approximating it with voxels. In [376], personnel costs are disregarded in order to compare different scenarios regarding the utilization of the build platform. Similar to [379], Grund [375] modified the model of Ruffo et al. [385, 386] for the application to PBF-LB/M. In her work, she considered direct costs (material and post-processing) as well as indirect costs due to the build time and the machine cost rate. Kopf et al. [374] derived a cost model to enable the cost-oriented design of PBF-LB/M manufacturing systems, mainly aiming to reduce activity durations such as machine setup or build time as well as minimizing investment costs related to the equipment. The cost model includes a build time calculation and considers the reusability of the powder feedstock. The most detailed calculative PBF-LB/M cost model is presented by Kranz [373], cf. Figure 2.18. The manufacturing cost of a PBF-LB/M part consists of material cost as well as production cost. The material cost comprises of the part material consumption, the support material consumption, as well as costs due to powder losses. Powder losses occur because of particles trapped in the part or supports, particles attached to the part surface, or agglomerated particles that are sieved out during the powder refurbishment. The production cost is structured by the respective production phases: preparation, building, postprocessing, and quality assurance. The preparation cost includes the labor cost of the person preparing the files and manufacturing system. The build process cost is split into machine cost (deprecation, required energy and space, service) as well as consumable cost (gas consumption, gas filter, other). The post-processing phase of PBF-LB/M consists of heat treatment, detachment from the build platform, support removal, and finishing. Each step represents a cost item. Kranz did not specify the quality assurance cost any further. In his definition of the respective cost items, Kranz followed the activity-based costing approach, as most of the cost models listed in Table 2.7 do.

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Figure 2.18: PBF-LB/M cost model of Kranz (adapted from [373])

2.6.2 Cost prediction While the calculative cost models provide functions of varying detail to describe the PBFLB/M process, demonstrating the reached understanding of the PBF-LB/M cost structure, the recent focus of literature shifted towards the goal of cost prediction. The challenge in

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77

cost prediction is the missing information on time intervals needed for the various activities involved. Allen [372] simply used the raw material cost to compare additive and subtractive manufacturing. In [371], Laureijs et al. investigated different manufacturing technologies for aerospace brackets. They did not calculate costs directly, though, but acquired quotations for various manufacturing scenarios. Then, the data is used to predict a range of costs and the main cost drivers are identified. Although [371] operates on service provider’s data, specific manufacturing parameters such as scan strategies and build orientation are evaluated. Therefore, this study is classified as considering a make-scenario. Mahadik & Masel [370] presented a generic cost estimation tool, which takes machine cost, material cost, labor cost, and post-processing cost into account. A primary and secondary user input is distinguished, where the primary information (AM process, machine type, material, part dimensions, labor rate, and post-processing) is required for the calculation, and the secondary information (specific times and parameters) may be provided for increased accuracy. Mahadik & Masel validated the tool by comparing calculated and actual cost. However, there is no indication of how the actual costs have been determined. Aside from the generic cost prediction approaches [370–372], there are various attempts to develop cost prediction methods tailored to PBF-LB/M [364–369, 381]. Lindemann et al. [369] first provided an approach toward lifecycle cost estimation, considering an aerospace scenario of supply chain logistics. The cost calculation model is based on previous work [380], but is not detailed any further. The design phase of a part is considered first by Yao et al. [368]. They developed a cost-driven design methodology, where the costs are linked with product family design. Starting from the reference design, for which the cost is calculated, possible changes in the reference cost due to the design of variants are integrated via cost increments defined as a function of a corresponding process setting variant factor. To be able to define such a factor, the model of the activity-based approach is analyzed regarding possibly changing activities and process settings when implementing a design variant. Furthermore, an Adaptive Neuro-Fuzzy Inference System (ANFIS)based expert system is trained to predict the activity’s time consumption, e.g. data preparation, as a function of the process setting adjustments due to the design variant. Schmidt [367] developed a cost estimation based on the material, part volume, and the part bounding box dimensions. Adopting the activity-based costing approach, the required time intervals for the various activities are estimated and assumed independent from the part. Only the build time is considered in more detail: Here, Schmidt evaluated the mean build rate as well as the mean powder application time of two different manufacturing systems equipped with two different materials. Using this information, the build time is calculated. Huang et al. [366] developed a cost prediction model and integrated their formulation into a topology optimization algorithm to enable cost consideration during part design optimization. Their cost model is based on the work of [371] and includes the part volume, build time, and laser power. For cost factors such as energy or labor, constant coefficients are

78 used. In [364], the model of Ruffo & Hague [386] is adopted by Griffiths et al. for application in cost-based build orientation algorithms. The indirect costs are again considered by a constant factor, whose value is not further specified nor explained. The build time is calculated by applying a linear regression model with the build height as well as the part and support volume as the input parameters. Rudolph & Emmelmann [365] presented a self-learning calculation implemented in a web-based platform for automated quoting. A calculative model is combined with the prediction of build height as well as manufacturing system capacity utilization, which is implemented in a self-learning system. The calculative model is purely based on the part dimensions, part volume, and corresponding build time. Comparing the calculative and predictive cost models, it becomes evident that the predictive models are less detailed. Furthermore, the databases of the predictive components are quite small, the largest amount of evaluated datasets being the one of Rudolph & Emmelmann with 25 build jobs. This is by far not sufficient for a reliable estimation of time-dependent cost items. Validation of the prediction is given in [365] only.

3 Research Hypothesis and Methodology AM is considered a key technology in digital production. Hence, the complete digitalization is a crucial step to enable the digital transformation of the manufacturing industry and its businesses. Today, part design and process modeling presents considerable progress in digitalization. However, other process steps are lacking digital representation as well as automation. Most of those steps in need of digitalization are related to the design and application of supports. Supports are essential, but auxiliary structures in PBF-LB/M that do not belong to the functional part. Therefore, the overall goal of support application is minimal support cost at sufficient performance. Today, support design is based on geometrical means, especially overhanging features, and user experience regarding stress development and heat flow during the manufacturing process. This often either leads to oversized supports or build failure. Additionally, the current support types are of regular shape, which does suit neither the load directions nor the free-form surfaces of AM parts. Furthermore, support removal is a manual, physical process often involving hand tools. This leads to low reproducability of the post-processing results, posing a considerable issue to quality assurance and management. Digitalization of the support design and removal has the potential to overcome these issues. Focusing on support design in this thesis, it is required to consider the load cases of the supports in support design. Furthermore, the support structure itself needs to be adaptable to the load directions. Most importantly, though, the support design procedure has to be automated to avoid user-induced errors. Hence, the following research hypothesis is investigated within this thesis:

By the digitalization of support design through an automated, load-based approach, it is possible to realize first-time-right production and decrease support cost.

Literature (see previous section) demonstrates the high potential of topology optimization to determine the optimal, digitalized support design for a given part-material configuration. However, artificial load cases have been applied only; a coupling with process simulation to determine realistic load cases has not been implemented yet. Preliminary studies exploring the fully digitalized procedure of computing the real structural and thermal load cases via PBF-LB/M process simulation and feeding these results into a topology optimization study as applied load case have been conducted, see [20, 21]. The studies as well as systematic investigation (see Section 5 for details) revealed that the optimal support

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_3

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Research Hypothesis and Methodology

structures have a tree-like topology for both structural and thermal loads. This finding allows substituting the topology optimization approach, which is sensitive to its parameters and requires a lot of experience in handling, with a generative, algorithm-based support design approach following the tree structure concept. Thus, there is a high potential of eliminating user actions through the digitalization of the support design process. Figure 3.1 summarizes the general methodology implemented and investigated in this thesis. Starting with a part geometry (CAD file), PBF-LB/M process simulation is performed to determine the load cases of the supports. The thesis is restricted to mechanical loads, i.e. residual stresses, as it is assumed that the support material required to withstand the occurring stresses is also sufficient for heat dissipation. The simulation results together with the part geometry are fed into a tree generation algorithm, which creates the tree supports. This concept of determining loads numerically and then using this information to algorithmically design a geometry provides a framework that is not limited to support design, but may be adopted in other design tasks addressing a base product with a large number of individual variations as well.

Figure 3.1: Methodology of algorithmic support application

To realize the algorithm-based support design, a three-step methodology is applied (see Figure 3.2): First, the design rules regarding the tree supports are developed via systematic topology optimization studies (cf. Section 5). Then, the actual algorithm for the support design is created and implemented as described in Section 6. Third, to enable the verification of the hypothesis, a benchmark methodology consisting of technical as well as economic assessment of supports is developed (Section 7). The experimental implementation of the benchmark is finally described in Section 8, where the novel supports of this thesis are tested against commonly used, well-known support types. Because all steps involving numerical or experimental tasks depend on the material used, a concise, but holistic material model is derived in a step preliminary to the support design methodology (Section 4). Here, the titanium alloy Ti-6Al-4V is chosen because of its popularity as well as its sensitivity regarding residual stresses.

Research Hypothesis and Methodology

Figure 3.2: Structure of the thesis

81

4 Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion The use of numerical methods requires a digital model of the utilized material. In Section 2.3.2, the following issues are identified in the current manner of Ti-6Al-4V material modeling in PBF-LB/M process simulation: 1.

In thermal modeling, caused by the variety in numerical approaches, no holistic material model including all thermo-physical material properties listed in Table 2.3 is available as different approaches require different material properties to be considered.

2.

A significant amount of computational studies does not provide references for their material model.

3.

Only a small number of references is given, neglecting the existing variation of experimental data.

4.

A considerable amount of referencing other PBF-LB/M process simulations is observed, inducing the risk of transmission errors.

5.

Experimental works used for reference rarely investigate PBF-LB/M specimens, and are therefore based on different microstructures of the material. Similarity with regard to specific material properties remains to be proven.

While current efforts in research to reduce the requirements regarding computational resources such that PBF-LB/M process simulation can be used by design engineers without access to computing centers target the algorithms or heat source modeling, each simplification comes at the cost of a decrease in accuracy of the result. At the same time, little attention is given to the other core component besides the heat source model, the material model, as demonstrated above. Whereas a poor material model may lead to significant errors in the temperature and stress calculation, a well-compiled material model is able to increase the accuracy of the calculation without any or little increased need for computational resources. In consequence, the development of a concise material model of Ti-6Al4V for PBF-LB/M is important. In the following sections, a model of all material properties listed in Table 2.3 is developed for use in the topology optimization, process simulation, and support design procedure. Because there is no complete material model published yet to build on, the model here is based on own and other’s experimental work. If the data available is considered too sparse, values stated in PBF-LB/M process simulations are used to extend the database. Due to the complex characteristics of the microstructure, the description of the material response may be complex, as well. Even though the knowledge in material science is constantly

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_4

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

improving, many processes and mechanisms are not fully understood today. This concerns the temperature dependency of the material properties caused by the changing microstructure as well as the influence of the respective alloyed elements and atmosphere. For example, the oxygen content in Ti-6Al-4V is known to significantly affect the 𝛽-transus temperature and the mechanical properties [27, 216, 217]. However, because of the difficulty of oxygen content determination, it is seldom reported in experimental studies. As a result of the complexity of the material behavior as well as the missing knowledge and data on all influencing factors, the material model developed in this section relies on stepwise linear functions to model the temperature dependency, if not constant. The oxygen influence is neglected due to scarce data.

4.1

Thermo-physical properties of Ti-6Al-4V

As a first step towards the modeling of the thermo-physical properties of Ti-6Al-4V, a literature review of experimental studies is conducted and summarized in Table 4.1. Interestingly, no published work on PBF-LB/M specimens has been identified; Table 4.1 therefore presents investigations of conventionally processed Ti-6Al-4V only. Table 4.1: Overview of experimental works on thermo-physical properties of conventionally processed Ti-6Al-4V

Source

𝑇𝛽

𝑇𝑠

𝑇𝑙

[31] [387]

𝑇𝑒

𝑇𝛼 ′ 𝑆

𝜌

𝑐𝑝

𝑘

𝑎

(x) (x)

(x)

[388]

x

[389]

x

[390]

x1

[391]

x

[392]

x

[393]

x

[394]

x

[395] [396]

x

x

x x

x

x

x

x

[397]

x

[398]

x

[216]

x

[217]

x

x

x x x

x

Thermo-physical properties of Ti-6Al-4V

85

[399]

x

[400]

x

[401]

x

[402]

x

[403] [404]

x

x

x

x

(x)

x x x

x

x

x

[407] [408]

x

x

[405] [406]

x

x x

x

x

x

1

data for powder material only (x) referenced data from different source As a consequence, the literature data is supplemented by own experimental work for material properties that are highly influenced by the microstructure (𝑇𝛽 , 𝑐𝑝 , 𝑘). The focus is on the investigation of the 𝛼′-microstructure resulting from PBF-LB/M and its influence on the aforementioned properties. Comparing to the usual (𝛼 + 𝛽)-microstructure of conventionally processed Ti-6Al-4V, conclusions are drawn regarding the adoption of those values in PBF-LB/Ti-6Al-4V modeling. In the following sections, the respective material properties are discussed and modeled either by a constant value (no temperature dependency) or piecewise linear function fitting the experimental data. As mentioned before, linear functions will not exactly represent most material properties due to numerous influencing factors such as the oxygen content. Because knowledge on those factors is missing and no reports in the literature are available, the modeling of those is neglected.

4.1.1

𝛽-transus temperature

In Table 4.1, a relatively high number of values is given for 𝑇𝛽 compared to other thermophysical properties. A wide range of data is reported, with values from 1160 K [398] to 1300 K [406]. The difference in the experimental results may be caused by the variance of chemical composition allowed by the standard specifications (ASTM B248, ASTM B348) with 5.5 − 6.75 wt.% aluminum and 3.5 − 4.5 wt.% vanadium. As already described, both alloying elements act as phase stabilizer (aluminum as 𝛼-stabilizer, vanadium as 𝛽-stabilizer), and therefore heavily influence the phase change taking place at 𝑇𝛽 . Also, the oxygen, nitrogen, and carbon content as 𝛼-stabilizers affect 𝑇𝛽 [27, 216, 217].

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Since the data in Table 4.1 is gathered on conventional specimens only and the microstructure of PBF-LB/M Ti-6Al-4V differs significantly, Differential Scanning Calorimetry (DSC) is performed on PBF-LB/M specimens for additional data. Experimental setup In the DSC approach based on the heat flux principle, the sample as well as a reference are subjected to a thermal sequence simultaneously in the same furnace [409]. The sample is contained in a crucible; the reference can consist of a sample with known material behavior, e.g. sapphire, or an empty crucible. Thermal sensors at the bottom of the crucibles measure the temperature history, and the difference between sample and reference is computed. Contrary to Differential Thermal Analysis (DTA), the temperature difference is converted to a heat flux difference by the instrument software. Table 4.2: Powder and process parameters of specimens used in DSC experiments

Parameter

Technical data

Manufacturing system

SLM500HL (SLM Solutions AG, Lübeck, Germany)

Process gas

Argon

Powder material provider

Tekna Advanced Materials, Inc., Sherbrooke, Canada

Powder particle size distribution

23 − 50 µm

Laser power

240 W

Scan velocity

1200 mm/s

Layer thickness

60 µm

Hatch distance

105 µm

Figure 4.1: Specimen production and setup of DSC measurements

Figure 4.1 depicts the specimen manufacturing as well as the DSC setup. For DSC, a cube with an edge length of 10 mm is manufactured via PBF-LB/M (step 1) applying the parameters displayed in Table 4.2. To fit the sample holder of the DSC equipment

Thermo-physical properties of Ti-6Al-4V

87

(TGA/DSC 2 HT 1600, Mettler-Toledo AG, Schwerzenbach, Switzerland), the cube is cut into smaller cubes by dividing it in three along each dimensional axis using wire electrical discharge machining (EDM) (step 2 & 3). The center cube, where the surface roughness influence is eliminated by the EDM cuts, is then subjected to the thermal cycle presented in Table 4.3 (step 4). Since the thermal response of 𝛼′-phase Ti-6Al-4V is not known yet, but expected to be similar to the (𝛼 + 𝛽)-phase alloy, the target temperature interval for phase transition is chosen to be 1073 − 1473 K to ensure not missing the point of phase transition on the one hand and completion of the transition process on the other hand. An additional heating step (step 2) prior to the relevant temperature section is included to adjust the specimen to the reduced heat rate. The heat rate of ±50 K⁄min is chosen according to Gadeev & Illarionov [410] in order to create a compromise between the fast heating required to evaluate high-alloyed materials and receiving a reasonable sensor signal during measurements. This heat rate is not able to reproduce the rapid heating in PBF-LB/M, which is in the order of 105 to 107 K⁄s [31, 411] and cannot be captured by current DSC or other suitable technology. As a result, the experimental measurements are not representative of the melting process of the powder layer. However, the heat conduction rate decreases the farther away from the zone of molten material. Therefore, the microstructural changes in the solidified bulk material, which is reheated several times during the manufacturing, are comparable to the DSC measurement setup. In order to prevent oxidation and mimic the manufacturing process, the measurements are conducted under argon atmosphere provided by a constant gas flow rate of 20 ml/min. Table 4.3: Thermal sequence of DSC for 𝑇𝛽 determination

Step

Temperature [K]

Heat rate [K⁄min]

Argon flow rate [ml⁄min]

1

298 − 973

100

20

2

973 − 1073

50

20

3

1073 − 1473

50

20

4

1473 − 1073

−50

20

5

1073 – 298

−100

20

Results of the DSC In Figure 4.2 and Figure 4.3, the result of the DSC is presented for the heating (step 1 to step 3) and the cooling (step 4 and step 5), respectively. To determine 𝑇𝛽 from the data, the procedure of Gadeev & Illarionov [410] is applied. They propose to utilize the

88

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

derivative maximum temperature, because they observed the low intensity of the thermal effect of metals compared to e.g. polymorphic transformation: 𝑇𝛽 = max(𝑑𝑇) − 𝑓𝑐 ± 1

(4.1)

They incorporated a correction factor 𝑓𝑐 , which is experimentally determined to 𝑓𝑐 = 15.25 for Ti-6Al-4V within their study. Equation (4.1) is applied to both heating and cooling results to calculate 𝑇𝛽 . The literature data as well as the own experimental values are displayed in Figure 4.4. The values for 𝑇𝛽 derived from DSC are in good agreement with the data obtained in other studies. The displayed range in values is due to the slight delay in the material’s response to the increasing temperature as well as the measurement signal delay. The experimental data accumulates around 𝑇𝛽 = 1268 K, though. Finally, to determine 𝑇𝛽 the median of both own experimental and literature data is calculated, yielding 𝑇𝛽 = 1268 K.

Figure 4.2: Heating sequence of DSC for 𝑇𝛽

Thermo-physical properties of Ti-6Al-4V

Figure 4.3: Cooling sequence of DSC for 𝑇𝛽

Figure 4.4: Data for 𝑇𝛽 from literature and DSC

89

90 4.1.2

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion Solidus temperature

In Table 4.1, only two references give values for 𝑇𝑠 , proposing either 𝑇𝑠 = 1873 K [396] or 𝑇𝑠 = 1878 K [403]. This data set is considered too small for reasonable fitting; therefore, it is extended by adding values used in published PBF-LB/Ti-6Al-4V process simulation [238, 240–252] as well as other related manufacturing processes, namely DED [412–416], PBF-EB/Ti-6Al-4V [417–419], and laser welding [420, 421]. The created data set is shown in Figure 4.5. Because the material will reach 𝑇𝑠 from the liquid state no matter the manufacturing technology, no further experimental work is conducted. A strong tendency towards 𝑇𝑠 = 1878 K is observed and affirmed by the median of the data set, which is taken as the final value.

Figure 4.5: Literature data (experimental and computational studies) for 𝑇𝑠

4.1.3

Liquidus temperature

Compared to 𝑇𝛽 , slightly less values are found in the literature for 𝑇𝑙 . The distribution of those is presented in Figure 4.6. A tendency to either 1923 K [396, 400, 408] or 1943 K [394, 398, 406] is noted. As the material will reach 𝑇𝑙 with a 𝛽-phase microstructure no matter the manufacturing technology, no further experimental work is conducted. The median of the data is considered, yielding 𝑇𝑙 = 1928 K [403].

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91

Figure 4.6: Literature data (experimental studies) for 𝑇𝑙

4.1.4

Evaporation temperature

No experimental data on 𝑇𝑒 has been found. For orientation, values applied in other PBFLB/Ti-6Al-4V and related manufacturing process simulations [186, 238, 239, 241, 245, 253, 254, 260, 414, 420–422] are investigated and displayed in Figure 4.7. With a minimum value of 3133 K [420] and a maximum value of 3613 K [238], 𝑇𝑒 data exhibits the largest range compared to other characteristic temperatures. In accordance with the procedure of determining a model value for a specific temperature of the previous sections, the median is considered, giving 𝑇𝑒 = 3533 K.

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Figure 4.7: Literature data (computational studies) for 𝑇𝑒

4.1.5

Martensite start temperature

To determine 𝑇𝛼′𝑆 , some experimental studies have been conducted as indicated by Table 4.1. Additionally, Neelakatan et al. [395] developed a formula for 𝑇𝛼′𝑆 prediction of 𝛽-Ti alloys based on their composition, which is expressed in terms of the elemental concentration. 𝑇𝛼′𝑆 = 1156 [K] − 150𝐹𝑒𝑤𝑡.% [K] − 96𝐶𝑟𝑤𝑡.% [K] − 49𝑀𝑜𝑤𝑡.% [K] − 37𝑉𝑤𝑡.% [K] − 17𝑁𝑏𝑤𝑡.% [K] − 7𝑍𝑟𝑤.𝑡% [K]

(4.2)

+ 15𝐴𝑙𝑤𝑡.% [K] For Ti-6Al-4V, only the elements Fe, V, and Al contribute to 𝑇α′ S with regard to Equation (4.2). To extend the data set on 𝑇𝛼′𝑆 , four chemical compositions, which have been experimentally measured, are applied to Equation (4.2): T-3.39Al-4.08V-0.19Fe [27], T-6.5Al4.06V-0.21Fe [27], T-6.27Al-4.1V-0.2Fe [27], and T-5.5Al-3.5V-0.03Fe [400]. In Figure 4.8, the resulting data is shown. The seemingly varying thickness of the bars is caused by contiguous values. The figure also points out the dependence of 𝑇𝛼′𝑆 on the cooling rate [401], which leads to the large range of values with a minimum value of 848 K [401] and a maximum value of 1104.5 K [395]. Because the cooling rate of PBF-LB/Ti-6Al-4V has only been estimated, the median of the data set is taken as model value, yielding 𝑇𝛼′𝑆 = 1073 K.

Thermo-physical properties of Ti-6Al-4V

93

Figure 4.8: Literature data (experimental studies) for 𝑇𝛼′𝑆

4.1.6

Density

The density of Ti-6Al-4V at room temperature is well known, because it is a value stated by the powder material providers in the range of 4410 kg⁄m3 [423] to 4430 kg⁄m3 [424]. The density at higher temperatures has not received much attention yet, as shown in Table 4.1 and Figure 4.9. This might be due to most computational studies keeping 𝜌 constant, since the material shrinkage or expansion is seldom modeled. In addition, measuring 𝜌 at higher temperatures remains challenging and is prone to errors. There is a possibility to create experimental data on the density indirectly, though: The coefficient of thermal expansion is linked to 𝜌 as a change in volume results in a change in density if mass conservation is ensured, and the 𝐶𝑇𝐸 is considerably easier to measure, e.g. by dilatometry (ASTM E228-95). To keep the material model consistent, the density is calculated from the 𝐶𝑇𝐸 model of Section 4.3.5. The term ‘coefficient of thermal expansion’ is commonly used to describe the coefficient of linear thermal expansion (𝐶𝐿𝑇𝐸), which states the material’s dimensional change in one direction. Extending the 𝐶𝐿𝑇𝐸 to other dimensions, the coefficient of area thermal expansion (𝐶𝐴𝑇𝐸) is created for the 2D case, as well as the coefficient of volume thermal expansion (𝐶𝑉𝑇𝐸) for considering three dimensions. The 𝐶𝑉𝑇𝐸 can be derived from the change in density [425]:

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

𝐶𝑉𝑇𝐸 =

1 𝑑𝑉 ∗ 𝑉 𝑑𝑇

(4.3)

The volume of the solid is denoted by 𝑉, and 𝑇 indicates the temperature in K. Because 𝑉 can be expressed by the mass 𝑚 and the density 𝜌, and mass conservation is assumed, Equation (4.3) can be written as 𝐶𝑉𝑇𝐸 =

1 𝑑𝜌 ∗ 𝜌 𝑑𝑇

(4.4)

The 𝐶𝑇𝐸 and 𝐶𝑉𝑇𝐸 are linearly related in accordance to the number of dimensions [425]: (4.5)

𝐶𝑉𝑇𝐸 = 3 ∗ 𝐶𝑇𝐸

To calculate 𝜌 from 𝐶𝑇𝐸, the model of Section 4.3.5 is transformed from 𝐶𝑇𝐸 to 𝐶𝑉𝑇𝐸 by applying Equation (4.5). In the next step, Equation (4.4) is solved for 𝜌 and evaluated stepwise with an initial value of 4420 kg⁄m3 at room temperature (293 K). That way, the following expression of 𝜌 is determined: 𝜌𝛼′ (𝑇) = −5.13 ∗ 10−5 [

kg 𝑇2 kg 𝑇 kg ] ∗ 2 − 0.09135 [ 3 ] ∗ + 4451 [ 3 ] 3 [K ] [K] m m m

kg 𝑇2 kg 𝑇 kg 𝜌𝛽 (𝑇) = −2.762 ∗ 10−6 [ 3 ] ∗ 2 − 0.1663 [ 3 ] ∗ + 4468 [ 3 ] [K ] [K] m m m

(4.6) (4.7)

As the heating rates are high in PBF-LB/Ti-6Al-4V, the material model does not include a detailed description of the so-called ‘mushy zone’, which denotes the temperature interval of 𝑇𝑠 < 𝑇 < 𝑇𝑙 , where solid and liquid phases co-exist during melting. Because of missing data on the 𝐶𝑇𝐸 of liquid Ti-6Al-4V, the data of Mills [400] and Schmon et al. [388] is linearly fitted. kg 𝑇 kg 𝜌𝑙 (𝑇) = −0.565 [ 3 ] ∗ + 5093 [ 3 ] [K] m m

(4.8)

The result of the fitting as well as the experimental data of Table 4.1 is shown in Figure 4.9. No significant change in density is noted at the phase transition to the 𝛽-microstructure. This observation is in good agreement with literature: The difference in density of Ti (hcp) and Ti (bcc) is Δ𝜌 ≪ 1% based on the volume of the respective unit cells using the lattice constants from Wood [426] and Levinger [427]. The calculation for pure Ti can be superimposed on the 𝛼/𝛼′- (hcp) and 𝛽- (bcc) microstructure of Ti-6Al-4V. For liquid Ti6Al-4V, Schmon et al. [388] found a significant drop in 𝜌 for temperatures above 2500 K. One obvious explanation for this drop is the material state transition from liquid to vapor. With 𝑇𝑒 = 3533 K determined in Section 4.1.4, there is a great difference in values, though. However, the value of 2500 K found by Schmon et al. is relatively close to the

Thermo-physical properties of Ti-6Al-4V

95

evaporation temperature of aluminum, which is around 2743 K [428]. It is concluded that the evaporation of the aluminum within the alloy is causing the drop in 𝜌. As mass conservation is assumed, the partial evaporation is not modelled. In Table 4.4, the complete material model for bulk Ti-6Al-4V covering one cycle of heating and cooling is summarized by listing the respective temperature intervals and corresponding functions. Table 4.4: Material model of Ti-6Al-4V: Density (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝜌𝛼′ (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝜌𝛽 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝜌𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝜌𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝜌𝛽 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝜌𝛼′ (𝑇)

Figure 4.9: Result of the fitting procedure for the modeling of the density

96 4.1.7

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion Specific heat capacity

The specific heat capacity describes the material’s ability to take in thermal energy, indicating the heating rate of the material as well as phase changes, and is highly dependent on the material’s temperature. Experimental studies on 𝑐𝑝 of conventionally processed Ti6Al-4V and their results are presented in Figure 4.10. A high variance in the values is noted, with a maximum difference of nearly 300 J⁄(kg*K) at around 1200 K. This variance is caused by some results [396, 398] exhibiting a peak at 𝑇𝛽 , indicating the microstructural phase transformation, while other results do not [400]. The reason for this lies within the measurement methods applied. The peak seen in 𝑐𝑝 is an artificial one due to the phase transformation from (𝛼 + 𝛽)- to 𝛽-microstructure. Here, the energy supplied to the material operates the phase transformation rather than increasing the material’s temperature, which can be interpreted as ‘storing energy’ and therefore as increasing specific heat capacity. If measurement methods which are based on an energy difference are applied, e.g. pulse heating [396, 398], it is not possible to distinguish between the actual 𝑐𝑝 and the phase transformation and the aforementioned interpretation is valid. This relationship between 𝑐𝑝 and phase transformation can be utilized in material modeling to model phase transformations and their enthalpies indirectly via the apparent (heat) capacity approach, which has been proven to be equal in efficiency compared to the more advanced heat integration method by Proell et al. [429]. In the apparent heat approach, the phase change enthalpy is included into the modeling of 𝑐𝑝 either as peak or valley, similar to the experimental data in Figure 4.10.

Figure 4.10: Experimental data from the literature for the specific heat capacity

Thermo-physical properties of Ti-6Al-4V

97

Figure 4.10 additionally demonstrates that phase transformations cause a discontinuity in the 𝑐𝑝 progression over temperature. At 𝑇𝛽 , a drop is observed, whereas 𝑐𝑝 increases at 𝑇𝑙 . To consistently model 𝑐𝑝 , these discontinuities need to be included. As a consequence, the PBF-LB/Ti-6Al-4V model is split into three sections corresponding to the 𝛼′-microstructure, 𝛽-microstructure, and liquid phase. Each section is represented by a linear function. Furthermore, additional terms for the phase transformations are provided to the first two sections. However, since the microstructure may influence 𝑐𝑝 and no data on 𝛼′-microstructure is available, own experimental work is conducted prior to the modeling to investigate possible differences between the (𝛼 + 𝛽)- and the 𝛼′-microstructure of Ti-6Al-4V. Experimental setup Again, DSC is used to obtain data on 𝑐𝑝 . A specimen of the manufactured batch described in Section 4.1.1 is subjected to the same thermal cycle (see Table 4.5) twice. Again, to prevent oxidation, argon is used as protective gas. The maximum temperature applied in this procedure significantly exceeds 𝑇𝛽 . Therefore, the specimen has a 𝛼′-microstructure in the first run, but a (𝛼 + 𝛽)-microstructure in the second run since the cooling rate in DSC is too low to result in a martensitic microstructure. The double measurement of the same specimen enables a direct comparison of the microstructures without any influences of the manufacturing process. Table 4.5: Thermal sequence of DSC for 𝑐𝑝 determination

Step

Temperature [K]

Heat rate [K⁄min]

Argon flow rate [ml⁄min]

1

293

(hold for 30 min)

20

2

293 − 1573

50

20

3

1573 − 298

−50

20

4

298

(hold for 5 min)

20

Results Figure 4.11 displays the measured signals of the respective experimental runs, limited to the heating sequence. Both curves of the corresponding microstructures are in parallel up until 1073 K. Then, they diverge during phase transformation, which is indicated by the drop in the signal as less heat is irradiated. This is also the reason to the figure’s emphasis on the given temperature interval rather than displaying the whole temperature range. The DSC signal is characterized by two maxima with a local minimum in-between. While the difference between the curves prior to the first maximum (1127 K for 𝛼′, 1131 K for 𝛼 + 𝛽 ) is at 5 mW, it increases to 9 mW at 1233 K (minimum of 𝛼 ′ ) and 20 mW at

98

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

1274 K (minimum of 𝛼 + 𝛽). At the second maximum (1323 K for both curves), the difference has reached 30 mW. While the first maximum and the minimum of the curves are shifted to each other, the second maximum denoting the end of the phase transformation is located at the same temperature for both microstructures. The offset of the first maximum and the minimum is due to the 𝛼′ as a meta-stable phase decomposing more easily than the (𝛼 + 𝛽)-composition. Up until 𝑇𝛽 , the difference between the 𝛼′- and (𝛼 + 𝛽)-measurements is ca. 5 % or lower in regard to the (𝛼 + 𝛽)-signal. It is therefore concluded, that both microstructures behave similar in terms of 𝑐𝑝 , and the experimental data displayed in Figure 4.10 can be used for the PBF-LB/Ti-6Al-4V material model.

Figure 4.11: Results of DSC for 𝑐𝑝 (heating sequence)

Modeling of the specific heat capacity The experimental data from the literature is fitted linearly for the three specified sections. Furthermore, the phase change is considered by applying a standard Gaussian distribution function, which is added to the respective linear function. By utilizing the Gaussian distribution, it is assumed that the phase change takes place in an even manner. Different from e.g. the density, due to the different temperature intervals and the involved phase transformation, the Gaussian terms have to be modeled for both heating and cooling. For the first section, the linear fit is determined to be

Thermo-physical properties of Ti-6Al-4V

99

J 𝑇 J ]∗ + 483 [ ] kg*K [K] kg*K

(4.9)

𝑐𝑝,𝛼′,𝑙𝑖𝑛 (𝑇) = 0.25 [

Because no specification of the 𝛽-transus enthalpy could be found, it is modeled by fitting the experimental peak values in Figure 4.10, also. The Gaussian standard distribution function is given in Equation (4.10).

𝑐𝑝,𝛼′,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 (𝑇) = 13,000 [

J √2𝜋 ]∗ ∗𝑒 kg*K 90

−0.5∗(

2 𝑇 −1160 [𝐾] ) 90

(4.10)

The overall result of the fitting with 𝑐𝑝,𝛼′,ℎ𝑒𝑎𝑡 (𝑇) = 𝑐𝑝,𝛼′,𝑙𝑖𝑛 (𝑇) + 𝑐𝑝,𝛼′,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 (𝑇) is displayed in Figure 4.12. For the second section of 𝑐𝑃 , the same procedure is applied. The actual material behavior is linearly fitted by J 𝑇 J 𝑐𝑝,𝛽,𝑙𝑖𝑛 (𝑇) = 0.14 [ ]∗ + 530 [ ] kg*K [K] kg*K

(4.11)

The phase transformation is then modeled with the help of the few values for the latent heat of fusion found in the literature, which are 286 kJ⁄kg [404], 290 kJ⁄kg [396], and 300 kJ⁄kg [394]. The value of Boivineau et al. [396] is chosen as it represents the median. For melting, the Gaussian distribution function is applied to the temperature interval of 𝑇𝑙 − 𝑇𝑠 = 50 K such that the area between the Gaussian distribution function and the linear material function equals the latent heat of fusion:

J √2𝜋 𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 (𝑇) = 41,650 [ ]∗ ∗𝑒 kg*K 9

2 𝑇 −1905 [𝐾] −0.5∗( ) 9

(4.12)

The overall specific heat capacity 𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡 (𝑇) is again defined by the sum of both terms 𝑐𝑝,𝛽,𝑙𝑖𝑛 (𝑇) and 𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 (𝑇). In the liquid state, Boivineau et al. [396], Kaschnitz et al. [399], and Mills [400] found 𝑐𝑝 to be constant. Therefore, the third section is described as constant value determined by the median of the experimental values from the literature. J 𝑐𝑝,𝑙 (𝑇) = 930 [ ] kg*K

(4.13)

The overall result of the fitting procedure is presented in Figure 4.13, excluding the terms for the phase transformation for better readability because the magnitude of the latent heat of fusion peak would prevent the distinction of the different features of the model. To meet the new phase transformation temperature intervals in the cooling phase, the respective functions have to be adjusted in terms of their absolute positioning while sustaining the values of the enthalpies:

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

𝑐𝑝,𝛼′,𝑐𝑜𝑜𝑙,𝑝ℎ𝑎𝑠𝑒 (𝑇) = 13,000 [

J √2𝜋 ]∗ ∗𝑒 kg*K 90

J √2𝜋 𝑐𝑝,𝛽,𝑐𝑜𝑜𝑙,𝑝ℎ𝑎𝑠𝑒 (𝑇) = 41,650 [ ]∗ ∗𝑒 kg*K 9

2 𝑇 −952 [𝐾] ) 90

(4.14)

2 𝑇 −1855 [𝐾] ) 9

(4.15)

−0.5∗(

−0.5∗(

Whereas the phase transformations of the heating phase are endothermic processes (Equation (4.16) and (4.17)), the cooling phase is characterized by exothermic behavior. In consequence, the difference of the two respective terms rather than the sum denotes the overall specific heat capacity (Equation (4.18) and (4.19)). 𝑐𝑝,𝛼′,ℎ𝑒𝑎𝑡 (𝑇) = 𝑐𝑝,𝛼′,𝑙𝑖𝑛 (𝑇) + 𝑐𝑝,𝛼′ ,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 (𝑇)

(4.16)

𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡 (𝑇) = 𝑐𝑝,𝛽,𝑙𝑖𝑛 (𝑇) + 𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡,𝑝ℎ𝑎𝑠𝑒 (𝑇)

(4.17)

𝑐𝑝,𝛼′,𝑐𝑜𝑜𝑙 (𝑇) = 𝑐𝑝,𝛼′,𝑙𝑖𝑛 (𝑇) − 𝑐𝑝,𝛼′ ,𝑐𝑜𝑜𝑙,𝑝ℎ𝑎𝑠𝑒 (𝑇)

(4.18)

𝑐𝑝,𝛽,𝑐𝑜𝑜𝑙 (𝑇) = 𝑐𝑝,𝛽,𝑙𝑖𝑛 (𝑇) − 𝑐𝑝,𝛽,𝑐𝑜𝑜𝑙,𝑝ℎ𝑎𝑠𝑒 (𝑇)

(4.19)

A summary of the 𝑐𝑝 model’s functions and corresponding temperature intervals is given in Table 4.6.

Figure 4.12: Result of the fitting procedure (1st section) for the modeling of the specific heat capacity

Thermo-physical properties of Ti-6Al-4V

101

Figure 4.13: Result of the fitting procedure (excl. phase transformation) for the modeling of the specific heat capacity Table 4.6: Material model of Ti-6Al-4V: Specific heat capacity (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝑐𝑝,𝛼′ ,ℎ𝑒𝑎𝑡 (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝑐𝑝,𝛽,ℎ𝑒𝑎𝑡 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝑐𝑝,𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝑐𝑝,𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝑐𝑝,𝛽,𝑐𝑜𝑜𝑙 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝑐𝑝,𝛼′,𝑐𝑜𝑜𝑙 (𝑇)

4.1.8

Thermal conductivity

In Figure 4.14, the experimental data reported in the references listed in Table 4.1 is presented. Two aspects are immediately noticed: First, the experimental data indicates a linear progression of 𝑘 with increasing temperature. Second, similar to 𝑐𝑝 , phase transformation results in discontinuities and changes in slope of the sections corresponding to the different microstructures are identified, highlighting the influence of the microstructure on 𝑘. Accordingly, it is experimentally investigated whether the 𝛼′- and (𝛼 + 𝛽)-microstructure exhibit similar behavior.

102

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Figure 4.14: Experimental data from the literature for the thermal conductivity

Experimental setup Laser Flash Analysis (LFA) is used for the determination of the thermal conductivity, following ASTM E 1461-01. Invented by Parker et al. [430], the LFA principle consists of a thermally insulated specimen with a thickness of a few millimeters, which is subjected to a high-intensity short-duration laser pulse. The laser radiation is absorbed by the specimen on one surface and the resulting temperature history of the opposite surface is measured. Based on the specimen thickness and time to a 50 % temperature decrease, the thermal diffusivity is calculated by the software.

Figure 4.15: Specimen production and setup of LFA

Thermo-physical properties of Ti-6Al-4V

103

An overview of the specimen production and the LFA setup is given in Figure 4.15. With the same manufacturing system (SLM500HL, SLM Solutions Group AG, Lübeck, Germany) and process parameters (cf. Table 4.2) as for the DSC specimens, two cylindrical specimens with a diameter of 12.7 mm are produced (step 1). To fit into the LFA equipment (Linseis LFA 1600, Linseis Messgeräte GmbH, Selb, Germany) consisting of a revolver sample holder, the cylinders are cut into 12 discs of a target thickness of 2 mm, following the LFA equipment provider’s suggestion (step 2). Up to six specimens can be evaluated in the same experimental run. Prior to the measurements, the discs are spraycoated with graphite to enhance the measured signal (step 3). Furthermore, after coating, the actual thicknesses of the specimens are measured with a caliper gauge. At first, a possible influence of the manufacturing is investigated by conducting measurements at room temperature with specimens from both cylinders. Because the sensor of the LFA is highly sensitive to the specimen’s and its own temperature, five consecutive measurements are taken for one specimen. The experiment is performed in vacuum with a pressure of 𝑝 < 2 ∗ 10−3 hPa. The parameters of the laser pulse are the iris sitting in front of the optic (𝐼𝐿𝐹𝐴 ), the amplification factor (𝐴𝐿𝐹𝐴 ), and the voltage (𝑉𝐿𝐹𝐴 ) of the pulse. The settings of those laser pulse parameters are given in Table 4.7. The parameter values at room temperature are determined after a reference run. Secondly, the temperature dependence of 𝑘 for both 𝛼′- and (𝛼 + 𝛽)-microstructure is investigated using specimens of the same cylinder. Similar to the experiment conducted to evaluate the microstructure’s influence on 𝑐𝑝 , the same specimens are subjected to the same thermal sequence twice, with the maximum temperature exceeding 𝑇𝛽 to change the microstructure of the specimens to 𝛼 + 𝛽 for the second thermal sequence. Table 4.7: Laser pulse parameters for LFA

𝑻 [K] 𝛼′

𝛼+𝛽

𝟐𝟗𝟑

𝟑𝟕𝟑

𝟒𝟕𝟑

𝟓𝟕𝟑

𝟕𝟕𝟑

𝟗𝟕𝟑

𝟏𝟏𝟕𝟑

𝟏𝟒𝟐𝟑

𝐼𝐿𝐹𝐴 [−]

2

2

2

2

2

4

4

8

𝐴𝐿𝐹𝐴 [−]

50

20

10

5

5

10

10

10

𝑉𝐿𝐹𝐴 [V]

500

300

250

240

200

240

220

400

𝐼𝐿𝐹𝐴 [−]

2

2

2

2

2

4

4

8

𝐴𝐿𝐹𝐴 [−]

50

50

20

10

2

5

2

5

𝑉𝐿𝐹𝐴 [V]

500

400

280

250

250

250

300

300

Results Figure 4.16 compares the values determined from specimens of two different cylinders at room temperature. Displayed on the Y-axis is the thermal diffusivity 𝑑 (usually denoted

104

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

by 𝛼 or 𝑎, changed here for better distinction from the absorptivity), which is the original parameter measured by LFA. Additionally, the mean values of 𝑑 at the respective temperatures as well as the linear fit of the data is indicated. The mean values are close to the linear fit and well within the standard deviation range of the experimental data. It is therefore concluded, that no significant difference between both cylinders is present and no manufacturing process influence has to be taken into account. Based on this observation, the investigation of the temperature dependence is performed with the remaining six discs of cylinder 1, and the results are presented in Figure 4.17. It is obvious that there is no significant difference between the 𝛼′ - and (𝛼 + 𝛽) microstructure. No tendency of the measured signal’s variance regarding the microstructures can be noted, and the linear fits of both data sets are nearly indistinguishable. Only at 1173 K an increase in the variance is visible. This is attributed to the furnace of the LFA equipment, which showed increasing trouble to keep the temperature constant with increasing furnace temperature. This inconsistency also affects the sensor in its performance. The values of the temperature level 𝑇 = 1423 K are not shown in Figure 4.17 as they do belong to the behavior of the 𝛽-microstructure. However, these values are included in Figure 4.18, which displays the overall fitting result.

Figure 4.16: Experimental results of LFA – comparison of different base cylinders

Thermo-physical properties of Ti-6Al-4V

105

Modeling of the thermal conductivity To be able to determine the thermal conductivity from the literature values as well as the own experimental data, the LFA results need to be converted from 𝑑 to 𝑘. This is done in accordance with ASTM E 1461-01 by applying the thermal diffusivity to the following equation: 𝑘(𝑇) = 𝑑(𝑇) ∗ 𝑐𝑝 (𝑇) ∗ 𝜌(𝑇)

(4.20)

The resulting data set is fitted by linear functions section-wise considering the three sections of corresponding microstructures and material states. For 𝑇𝑅 < 𝑇 ≤ 𝑇𝛽 , the thermal conductivity is determined as W 𝑇 W 𝑘𝛼′ (𝑇) = 0.012 [ ]∗ + 3.3 [ ] m*K [K] m*K

(4.21)

The following sections of 𝑇𝛽 < 𝑇 ≤ 𝑇𝑙 as well as 𝑇𝑙 < 𝑇 yield 𝑘𝛽 (𝑇) = 0.016 [

W 𝑇 W ]∗ −3 [ ] m*K [K] m*K

𝑘𝑙 (𝑇) = 0.0175 [

W 𝑇 W ]∗ − 4.5 [ ] [K] m*K m*K

(4.22) (4.23)

The overall result is shown in Figure 4.18. Furthermore, Table 4.8 summarizes the respective functions required for the modeling of the specific temperature intervals.

Figure 4.17: Experimental results of LFA for the comparison of 𝛼′- and (𝛼 + 𝛽)-microstructure

106

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Figure 4.18: Result of the fitting procedure for the modeling of the thermal conductivity Table 4.8: Material model of Ti-6Al-4V: Thermal conductivity (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝑘𝛼′ (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝑘𝛽 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝑘𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝑘𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝑘𝛽 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝑘𝛼′ (𝑇)

4.1.9

Powder material properties

Because PBF-LB/M is based on the melting of powder material rather than bulk material, the effective thermo-physical properties of the powder material have to be modeled as well. The density (𝜌𝑝 ) and specific heat capacity (𝑐𝑝,𝑝 ) can directly be related to the porosity 𝜑, as both material properties are independent of the material’s geometry. 𝜌𝑝 (𝑇) = (1 − 𝜑) ∗ 𝜌(𝑇)

(4.24)

𝑐𝑝,𝑝 (𝑇) = (1 − 𝜑) ∗ 𝑐𝑝 (𝑇)

(4.25)

Thermo-physical properties of Ti-6Al-4V

107

Technically, Equations (4.24) and (4.25) assume the build chamber to be evacuated, whereas in reality the build chamber is flooded with argon. The respective contributions of argon to 𝜌𝑝 and 𝑐𝑝,𝑝 are negligible compared to solid Ti-6Al-4V, though. For the thermal conductivity (𝑘𝑝 ), there is a strong influence of the geometry. The powder bed in PBF-LB/M is characterized by particles of a defined particle size distribution over a certain interval, e.g. 23 − 50 µm. The particles can be assumed spherical and smooth [27, 431]. The powder particles are randomly packed in the powder bed, as a non-defined application is created by the recoater. The whole system may be classified as multi-phase heterogeneous medium of a packed bed’s structure, with solid particles being dispersed in a continuous gas phase, touching each other, though [432]. The contact between the particles during the manufacturing process can be described as ideal point contacts or slightly elastically deformed contacts, depending on the powder application and the part already built. Although there is a long history in research regarding the thermal conductivity of packed beds (see e.g. De Beer et al. [433] for a comprehensive review), the case of the PBF-LB/M powder bed is still to be determined. In [432], Gusarov & Kovalev developed a model of 𝑘𝑝 based on the random packing of equally sized particles by modeling the discrete thermal resistance. Their calculations exhibit the same tendencies as the experimental data, but underestimate the absolute values. This difference is attributed to the nonsphericity of the powder particles used for the experiments as well as the size distribution in contrast to the assumed mono-size characteristic. Gu et al. [431] evaluated 𝑘𝑝 of Ti6Al-4V powder batches from several suppliers. They scanned the powder bed in the PBFLB/M manufacturing system until steady state conditions were met, and applied infrared imaging to capture changes in the temperature. No mention of the use of a process gas is made, although the processing of Ti-6Al-4V with process gas application is the standard due to the alloy’s tendency towards oxidation. In consequence, the data obtained in [431] cannot be included into the model. Nevertheless, Gu et al. were able to demonstrate the importance of the powder batch characteristics: While two of the three investigated powders resulted in a definition of 𝑘𝑝 ≈ 0.2 ∗ 𝑘, the third powder batch exhibited 𝑘𝑝 ≈ 0.6 ∗ 𝑘. The increase is explained by fine agglomerated powder particles observed in the Scanning Electron Microscopy (SEM) images. Wei et al. [434] also investigated 𝑘𝑝 by applying the transient hot wire method, and compared the values of different common PBFLB/M materials (Inconel 625 and 718, stainless steel 17-4 and 316L, Ti-6Al-4V). For Ti6Al-4V under ambient pressure, their experimental values range from 0.2 W⁄(m*K) at 300 K to 0.25 W⁄(m*K) at 450 K with an overall linear behavior. Because of the sparse data available in the literature, an own powder sample taken directly from a PBF-LB/M system is commissioned to the Thermal Analysis Lab (Fredericton, Canada). The powder sample is in the mixed state of fresh and recycled powder typical for industrial production. The thermal conductivity of the powder is examined using the C-Therm TCi Thermal Conductivity Analyzer (C-Therm Technologies, Inc., Fredericton,

108

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Canada). It operates on a Modified Transient Plane Source (MTPS) sensor in compliance with ASTM D7984-16. The MTPS technique is based on a central heating and sensing element [435]. A one-dimensional heat flow is applied to the sample, which is placed on the element, and the voltage drop in the element is measured, yielding the thermal effusivity of the sample. With the known density and specific heat capacity of the sample, the thermal conductivity can be calculated from the thermal effusivity.

Figure 4.19: Result of the thermal conductivity measurements of the powder sample

The powder is deposited in a high pressure liquids cell accessory. Prior to measurements, the cell is flushed with argon to recreate the environment during PBF-LB/M manufacturing. To compact the powder in the sample holder slightly for better contact to the heat source, a weight of 75 g is applied on top. For each value taken at a specific temperature, five consecutive measurements are conducted and the mean value is calculated. Figure 4.19 shows the results of the experimental measurements. A dip at 373 K is observed, which proved to be reproducible with other samples taken from the commissioned powder. This is explained by the powder characteristic, i.e. the fact that it consists of recycled powder and furthermore has been shipped from Germany to Canada. This possibly leads to water accumulation at the powder particle surface. During heating of the samples, the water evaporates around 373 K, and part of the energy provided to the sample is fostering the change of aggregate state instead of being conducted. As a result, the thermal conductivity seems lower at that point. Overall, a linear temperature dependency is found, which can be described as follows:

Optical properties of Ti-6Al-4V 𝑘𝑝 (𝑇) = 327.6 ∗ 10−6 [

4.2

W 𝑇 W ]∗ + 0.079 [ ] m*K [K] m*K

109 (4.26)

Optical properties of Ti-6Al-4V

The only relevant optical material property is the absorption of the laser radiation. The laser radiation used as heat input in PBF-LB/M is either reflected from, transmitted through or absorbed by the material. Only the absorped energy is contributing to the heating of the material. Thus, this material property needs to be modeled as well, although there is little experimental data available (cf. Table 4.1). However, some values have been determined experimentally, presented in Figure 4.20. The values range from 0.34 to 0.55. Keller et al. [397] as well as Munsch [392] both found a value of 0.4, but Keller et al. added a correction term to account for radiation losses during the measurements later on. Yang et al. [411] were able to proof that 𝑎 is independent from the material’s temperature for Ti-6Al-4V. Therefore, the model consists of the median of the experimental data, yielding 𝑎 = 0.4.

Figure 4.20: Experimental data from the literature for the absorptivity (* corrected for radiation loss)

The powder absorptivity (𝑎𝑝 ) is determined separately, as the small gaps between the powder particles act as ray traps where reflected rays may meet another particle surface and absorption, transmission, as well as reflection take place repeatedly. This mechanism leads to a significantly higher absorptivity compared to the bulk material of the same alloy,

110

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

which can also be seen when comparing the values for the bulk Ti-6Al-4V given in Figure 4.20 and the PBF-LB/M powder Ti-6Al-4V presented in Figure 4.21. The values determined for 𝑎𝑝 range from 0.65 to 0.74, whereas the bulk absorptivity has been determined to be 𝑎 = 0.4. Similar to the bulk absorptivity, 𝑎𝑝 was found to be independent of the temperature by Rubenchik et al. [390]. Therefore, the median of the literature data is considered for the material model, yielding 𝑎𝑝 = 0.7.

Figure 4.21: Experimental data from the literature for the absorptivity of powder

4.3

Mechanical properties of Ti-6Al-4V

For the mechanical material properties, a lot of research has been conducted as can be seen in Table 4.9. This attention is due to the high strength-to-weight ratio of Ti-6Al-4V, which leads to applications in industries characterized by high loads or long operating times, e.g. aerospace or medical engineering. Thus, the predictability of the additively manufactured Ti-6Al-4V’s mechanical properties is considered a key challenge [436]. Consequently, a lot of research has been conducted towards the mechanical rather than the thermo-physical properties, which is summarized in Table 4.9. Note that publications presenting their results as figures only have been excluded to evade possible errors due to reading values off figures. The remaining data set is deemed sufficient. Generally, experimental reports in the literature demonstrate a higher strength and, as a trade-off, a lower ductility of PBF-LB/Ti-6Al-4V in comparison to conventionally processed specimens [436]. This effect is attributed to the finer grains of the microstructure

Mechanical properties of Ti-6Al-4V

111

as well as the presence of defects. In terms of the required material properties, two major issues are identified: First, all data contained in Table 4.9 has been gathered by measurements at room temperature. No experimental work on high temperature properties could be found. Second, for PBF-LB/Ti-6Al-4V specimens no data on the Poisson’s ratio or 𝐶𝑇𝐸 is available. Even though the standard ASTM E8 describes a normed procedure to determine monotonic tensile properties, there are still several influencing factors to be considered in tensile testing. The main factor is the microstructure [437], especially the characteristic anisotropy caused by the fine lamellar grains oriented along the build direction, as observed by Mertens et al. [438], Simonelli et al. [439], and Qiu et al. [440]. The anisotropy is heavily influenced by the scan strategy as well as the build orientation of the specimens [441]. The difference in grain size either parallel or perpendicular to the build orientation, demonstrated by Thijs et al. [442], should influence 𝜎𝑦 according to the Hall-Petch law of grain boundary strengthening. However, this anisotropy is not always observed in experimental studies, as Edwards & Ramulu [443] point out. While the scan strategy is rarely mentioned in publications, the build orientation is specified with only little exceptions because of its already known importance to the mechanical properties. Therefore, the anisotropy in the microstructure and its influence on the tensile properties (𝐸, 𝜎𝑦 ) due to the build orientation is evaluated to ensure the data included in the material modeling procedure to be representative with respect to the considered system. The terminology to denote the build orientation exactly is given in ASTM/ISO 52921. Here, six different orientations of specimens are distinguished for mechanical testing. This standard has been developed throughout the last couple of years, though, and has not yet been widely adopted in scientific reports. Hence, Table 4.9 indicates only whether the main axis of the specimens has been oriented along the build direction (vertical orientation) or perpendicular to it (horizontal orientation). In addition to the anisotropy of the microstructure, the chemical composition affects the mechanical properties. Especially the oxidation of the highly reactive Ti-6Al-4V alloy has been identified to have a high impact on the material performance [27, 216, 217]. The exact chemical composition of specimens is seldom determined, though, because of the effort and cost linked to the required techniques, preventing a systematic evaluation of this aspect. Therefore, the influence of the oxygen content is not considered in the material model. Furthermore, post-processing methods such as heat treatments do have a major influence on the microstructure and thus on the mechanical properties. As the material model is meant for use in PBF-LB/M process simulation, any data on heat-treated specimens is excluded.

112

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Table 4.9: Overview of tensile tests on PBF-LB/M Ti-6Al-4V specimens at 𝑇𝑅

Ref.

as-built/ machined surface

horizontal/ vertical build orientation

[438]

as-built

[443]

𝜎𝑦

𝑈𝑇𝑆

horizontal

x

x

as-built

horizontal

x

x

[444]

as-built

horizontal

x

x

x

[445]

as-built

horizontal

x

x

x

[446]

as-built

vertical

x

x

[447]

as-built

vertical

x

x

[448]

as-built

vertical

[449]

as-built

vertical

[450]

as-built

vertical

[451]

as-built

vertical

[452]

as-built

horizontal, vertical

x

x

x

[453]

as-built

horizontal, vertical

x

x

x

[454]

as-built

horizontal, vertical

x

x

x

[455]

as-built

x

x

x

[431]

machined

horizontal

x

x

[456]

machined

horizontal

x

x

x

[457]

machined

horizontal

x

x

x

[458]

machined

horizontal

x

x

x

[459]

machined

vertical

[460]

machined

vertical

[461]

machined

vertical

[462]

machined

vertical

[463]

machined

vertical

[439]

machined

horizontal, vertical

x

x

x

[464]

machined

horizontal, vertical

x

x

x

𝐸

x x

x

x

x

x x

x x x

x

x

x

x

x

x x

𝜈

𝐶𝑇𝐸

Mechanical properties of Ti-6Al-4V

Ref.

as-built/ machined surface

horizontal/ vertical build orientation

[465]

machined

horizontal, vertical

[466]

machined

horizontal, vertical

[467]

machined

horizontal, vertical

[468]

machined

[469]

machined

113

𝐸

x1

x

𝜎𝑦

𝑈𝑇𝑆

x

x

x1

x1

x

x

x

x

x

x

1

𝐶𝑇𝐸

x1

[470]

machined

x

[471]

machined

x

x

x

[472]

as-built, machined

x

x

x

vertical

𝜈

1

data published as figure, raw data provided by authors upon request

Besides the microstructure characteristics, there are several other factors influencing the tensile test results. Various specimen geometries are used [473]: While the test coupons can be flat or cylindrical, the compact tensile specimens consist of a block with a notch in the middle. Then, Kasperovich & Hausmann [472] found that tensile specimens with machined surfaces exhibit greater strength than as-built specimens, whose surface roughness is higher compared to conventional or machined specimens due to the powder involved in the processing [437]. This observation is explained by concluding that the surface roughness acts as crack initiator, which has been noticed in fatigue testing as well [474]. To ensure the inclusion of representative data only, the influence of the surface condition on 𝜎𝑦 is evaluated for the works listed in Table 4.9. If hybrid conditions, i.e. partially machined specimens as produced by removing support structures from one side, are applied, the specimens are considered ‘machined’. If no post-processing is mentioned, the specimens are assumed to be investigated ‘as-built’. Because of the number of influencing factors in combination with a relatively high number of literature reports, no own experimental work is conducted as no substantial insights are expected to be gained. To determine the respective values of 𝐸 and 𝜎𝑦 at room temperature, the experimental data on PBF-LB/M tensile specimens is used. Furthermore, the 𝑈𝑇𝑆 is specified to assist in the interpretation of process simulation results. For the modeling of the temperature dependency, due to the unavailability of PBF-LB/M data, reports of conventionally processed Ti-6Al-4V as well as material models of published computational studies are evaluated to give an orientation for the material properties’ development

114

Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

with increasing temperature. Last, 𝜈 and 𝐶𝑇𝐸 are determined using the data obtained from other studies. Prior to modeling, the suspected influences of build orientation and surface condition on the tensile data are investigated. Additionally, since the innovation in the field of PBFLB/M manufacturing systems is high, the year of the publications is evaluated with regard to the tensile properties to rule out a possible influence of the manufacturing equipment and create a material model based on today’s state of the art.

Figure 4.22: Comparison of experimental data on the Young’s modulus with regard to the build orientation

The data for 𝐸 is presented in Figure 4.22 and Figure 4.23, respectively. Because the surface condition is relevant to crack initiation, which is not within the region of elastic material behavior, this parameter is not investigated for 𝐸. Figure 4.22 categorizes the data available with regard to the build orientation of the specimens. If a publication evaluates both build orientations, the corresponding data pairs are indicated by a connecting line. These data pairs are of special interest as the specimens have been manufactured under the same conditions, most times even in the same build job. Three out of five data pairs show a higher 𝐸 for the vertical orientation, whereas two data pairs demonstrate the opposite. The highest deviation in 𝐸 between the two orientations is found by Mower & Long [453] with a 5.6 % difference (108.8 GPa for horizontal, 114.9 GPa for vertical orientation). Taking the whole data set for 𝐸 into consideration, no clear difference between the horizontal and vertical build orientation is observed: The respective median of the Young’s modulus for the horizontal build orientation is 109 GPa, and 110 GPa for the

Mechanical properties of Ti-6Al-4V

115

vertical build orientation. Also, all data remains within the interval of 95 GPa < 𝐸 < 120 GPa with only few exceptions. In conclusion, the effect of the anisotropy known to be existent due to the oriented grain growth is not clearly identified within the present data set of tensile tests.

Figure 4.23: Comparison of experimental data on the Young’s modulus with regard to the year of publication

To investigate the effect of innovation in the field of PBF-LB/M, Figure 4.23 displays the experimental data of Table 4.9 with regard to the year of publication. The studies included were published between 2006 and 2020. A significant increase in publications from the year 2014 on is noticed and attributed to the increasing maturity of the PBF-LB/M process, leading to increased application in industry and therefore a high interest in material performance. Nevertheless, no significant influence of the publication year on 𝐸 is observed: Both the earliest reports [455, 458] exhibit a range in 𝐸 similar to the latest publications. Similar to 𝐸, the yield strength data is evaluated to identify possible influences to be considered, and presented in Figure 4.24, Figure 4.25, and Figure 4.26. Overall, the same conclusions are drawn for 𝜎𝑦 as for 𝐸. Yet again, the effect of anisotropy due to the build orientation (Figure 4.24) is not clearly identified, neither by the data pairs from the same publication nor the complete data set. Five data pairs display higher strength for the horizontal build orientation, whereas two data pairs show the opposite. The respective medians of 𝜎𝑦 do have a greater difference (3.3 %) compared to the ones of 𝐸 (0.9 %) with a median of 1100 MPa for the horizontal orientation, and 1064.8 MPa for the vertical build orientation; it is not considered significant, though. The maximum deviation within one

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

experimental study is shown by Vilaro et al. [454] with a yield strength of 1137 MPa (horizontal orientation) as well as 962 MPa (vertical orientation), resulting in a 18.1 % difference. Though this suggests a strong influence of the build orientation on 𝜎𝑦 , the data displayed in Figure 4.24 does not allow for a definitive conclusion.

Figure 4.24: Comparison of experimental data on the yield strength with regard to the build orientation

In Figure 4.25, the tensile data is compared with regard to the surface condition. Here, both as-built and machined specimens result in the same range of values with no outliers. For comparison, the median of the corresponding categories are 1095.5 MPa for the machined as well as 1027.3 MPa for the as-built specimens. Since the standard deviations are 100.9 MPa and 116.7 MPa, respectively, no significance is attributed. In conclusion, while a rough surface may provide the opportunity for crack initiation, no definite relationship between 𝜎𝑦 and the surface condition is determined with the given data set. Finally, the chronological development of the experimental results does not indicate any influence of the PBF-LB/M maturity on the yield strength, as can be seen in Figure 4.26.

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Figure 4.25: Comparison of experimental data on the yield strength with regard to the surface condition

Figure 4.26: Comparison of experimental data on the yield strength with regard to the year of publication

In summary, the data of the reports listed in Table 4.9 does not demonstrate a direct anisotropy in 𝐸 or 𝜎𝑦 related to either build orientation or surface condition, nor indicate a

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

chronological development of the mechanical properties in PBF-LB/M. For 𝜎𝑦 , there are significant differences in the related data pairs in terms of build orientation, but no definitive relationship can be specified. This indicates that while anisotropy may influence the mechanical performance, the effect is superimposed by other factors, e.g. residual stresses [473]. Since this material model is not meant to be tailored to a specific manufacturing system and process parameters, all references of Table 4.9 are included into the modeling procedure for the mechanical material properties. In consequence, the model does not take into account further microstructural properties such as grain boundaries or grain size.

4.3.1

Young’s modulus

To create the model of 𝐸, the different microstructural compositions of the bulk material (𝛼 ′, 𝛽, liquid) are considered individually as done in the model of the thermo-physical properties, too. In a first step, the value at room temperature 𝑇𝑅 = 293 K is calculated by the median of the experimental data in Table 4.9, yielding 𝐸𝑚𝑒𝑑 = 109.1 GPa. Following the approach of linear fitting, a second data point is required to create a linear function. However, as no experimental data on PBF-LB/Ti-6Al-4V specimens at any temperature other than 𝑇𝑅 is available, the material models in published computational works [37, 174, 186, 203, 235, 251, 252, 257, 263] are utilized as indicators of what the value of 𝐸 at 𝑇𝛽 as well as 𝑇𝑙 may be. Note that studies listed in Table 2.5, which give their material model as figures only, have been excluded. Because most material models are either not referenced at all or referenced to other simulation works, the data is not directly included into the fitting procedure. The result is presented in Figure 4.27. With the study of [251] being the only exception, the 𝐸 at 𝑇𝛽 is observed to have an assumed value of around 10 GPa, leading to the following linear function: 𝐸𝛼′ (𝑇) = −0.102 [GPa] ∗

𝑇 + 139 [GPa] [K]

(4.27)

In the liquid state of the material, the Young’s modulus is nonexistent. However, an assigned value of 𝐸𝑙 = 0 may result in singularities during computation. Therefore, 𝐸 is set to 10−9 GPa at 𝑇𝑙 to avoid possible singularities. Accordingly, the linear function for the temperature interval of 𝑇𝛽 < 𝑇 ≤ 𝑇𝑙 is determined to 𝐸𝛽 (𝑇) = −0.0145 [GPa] ∗

𝑇 + 28 [GPa] [K]

(4.28)

It has to be noted that Equation (4.28) had to be slightly adjusted due to the rounding errors of Equation (4.27) setting the function’s value to 9.664 GPa at 𝑇𝛽 rather than 10 GPa. At temperatures higher than 𝑇𝑙 , the Young’s modulus is assumed constant: 𝐸𝑙 (𝑇) = 10−9 [GPa]

(4.29)

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119

To provide a complete model including both heating and cooling of the material, Table 4.10 summarizes the temperature intervals and their respective functions. Table 4.10: Material model of Ti-6Al-4V: Young’s modulus (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝐸𝛼′ (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝐸𝛽 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝐸𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝐸𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝐸𝛽 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝐸𝛼′ (𝑇)

Figure 4.27: Result of the fitting procedure for the modeling of the Young’s modulus

4.3.2

Yield strength

A procedure similar to 𝐸 is applied for the modeling of 𝜎𝑦 . A median of 𝜎𝑦,𝑚𝑒𝑑 = 1094.5 MPa at 𝑇𝑅 is derived from the experimental data set of Table 4.9. In Figure 4.28, the experimental data as well as the material models of other PBF-LB/Ti-6Al-4V process

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

simulations are shown. Aside from [37, 251], the material models suggest a linear relationship between 𝜎𝑦 and the temperature. This tendency is supported by the experimental data on conventionally processed specimens [394, 475–479]. Most models agree on 𝜎𝑦 approaching a value of approximately 5 MPa at 𝑇𝛽 . Furthermore, similar to 𝐸, the yield strength for the liquid material is considered close to zero. Following this procedure, 𝜎𝑦 is described by Equations (4.30)-(4.32). 𝜎𝑦,𝛼′ (𝑇) = −1.117 [MPa] ∗

𝑇 + 1422.6 [MPa] [K]

𝜎𝑦,𝛽 (𝑇) = −9.46 ∗ 10−3 [MPa] ∗

𝑇 + 18.2 [MPa] [K]

𝜎𝑦,𝑙 (𝑇) = 10−6 [MPa]

(4.30) (4.31) (4.32)

Figure 4.28: Result of the fitting procedure for the modeling of the yield strength

Again, Equation (4.31) is slightly adjusted to meet the rounding errors of Equation (4.30), which yields 𝜎𝑦 = 6.244 MPa at 𝑇𝛽 . The result of the fitting procedure is displayed in Figure 4.28, together with the experimental data. For the complete model of the yield strength, the functions with respect to the corresponding temperature intervals for one full sequence of heating and cooling are given in Table 4.11.

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121

Table 4.11: Material model of Ti-6Al-4V: Yield strength (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝜎𝑦,𝛼′ (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝜎𝑦,𝛽 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝜎𝑦,𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝜎𝑦,𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝜎𝑦,𝛽 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝜎𝑦,𝛼′ (𝑇)

4.3.3

Ultimate tensile strength

Usually, the 𝑈𝑇𝑆 is not modeled in PBF-LB/M process simulation, because not only mesh deformation but mesh splitting would be required to depicture crack propagation. Nevertheless, it is important to be aware of the 𝑈𝑇𝑆 in order to be able to interpret computational results correctly. Hence, no full model of the 𝑈𝑇𝑆 is developed, but the median of the experimental data at 𝑇𝑅 is calculated as reference, yielding 𝑈𝑇𝑆𝑚𝑒𝑑 = 1202.5 MPa. The data of Sieniawski et al. [477] and Lütjering [479] suggests linear behavior with increasing temperature similar to 𝜎𝑦 .

4.3.4

Poisson’s ratio

For 𝜈, no experimental data of PBF-LB/Ti-6Al-4V specimens is available in Table 4.9. Furthermore, only a single reference [479] has been identified for the evaluation of conventionally processed Ti-6Al-4V. With no option of own experimental work due to missing equipment, data from PBF-LB/Ti-6Al-4V process simulations [37, 203, 257, 263] is incorporated into the fitting procedure (shown in Figure 4.29). As can be seen by the overlapping data points of [37, 263], both studies reference the same computational study of Rangaswamy et al. [261]. Here, no further reference or explanation of the data’s origin is given. The experimental data of Lütjering [479] is lower in regard of the actual values, but is characterized by a steeper slope in the development over increasing temperature. Above 𝑇𝛽 , no further change in the slope of the data is noticed. This may be caused by the lack of data. However, based on the data available, the 𝜈 of the solid material is determined. 𝜈𝛼′,𝛽 (𝑇) = 9 ∗ 10−5 ∗

𝑇 + 0.2655 [K]

(4.33)

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Evaluating Equation (4.33) at 𝑇𝑅 results in a value of 𝜈 = 0.29, which is well within the known range of 𝜈 for titanium alloys (0.25 − 0.4 [480]). When the material is in its liquid state, it expands and contracts isotropically, which leads to a constant value. (4.34)

𝜈𝑙 (𝑇) = 0.5

The complete sequence of the material model is summarized in Table 4.12, including the respective temperature intervals and corresponding functions. Table 4.12: Material model of Ti-6Al-4V: Poisson’s ratio (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝜈𝛼′,𝛽 (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝜈𝛼′,𝛽 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝜈𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝜈𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝜈𝛼′,𝛽 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝜈𝛼′,𝛽 (𝑇)

Figure 4.29: Result of the fitting procedure for the modeling of the Poisson’s ratio

Mechanical properties of Ti-6Al-4V 4.3.5

123

Coefficient of thermal expansion

The 𝐶𝑇𝐸 is of high importance in the thermo-mechanical process simulation as it couples the thermal and structural physics, enabling the calculation of heat strain and the corresponding residual stresses. Despite being a critical parameter, there is no data within the references for PBF-LB/Ti-6Al-4V found, though. In consequence, other experimental values [394, 476] as well as process simulation studies [37, 174, 203, 251, 252, 263] are used for fitting. The resulting functions are presented by Equations (4.35)-(4.37). 1 𝑇 1 𝐶𝑇𝐸𝛼′ (𝑇) = 4.197 ∗ 10−3 [ ] ∗ + 7.758 [ ] K [K] K

(4.35)

1 𝑇 1 𝐶𝑇𝐸𝛽 (𝑇) = 0.51 ∗ 10−3 [ ] ∗ + 12.427 [ ] K [K] K

(4.36)

1 𝐶𝑇𝐸𝑙 (𝑇) = 20.71 [ ] K

(4.37)

Because there is no data on liquid Ti-6Al-4V at temperatures significantly higher than 𝑇𝑙 , the 𝐶𝑇𝐸 is modeled as constant here. In PBF-LB/M, the liquid material is present only at the top layer, where no mechanical loads are transferred within the liquid material. Additionally, the liquid material is not subject to distortion. Therefore, the characteristic of 𝐶𝑇𝐸 in the liquid region is not of high interest, enabling the assumption of a constant value. Figure 4.30 displays the fitting result as well as the underlying data. It is noted that the data of Li [394] are experimental values from conventionally processed Ti-6Al-4V on the 𝐶𝑉𝑇𝐸 rather than 𝐶𝑇𝐸. Applying Equation (4.5), the values are converted to 𝐶𝑇𝐸 as plotted in Figure 4.30. The complete model of 𝐶𝑇𝐸 is given in Table 4.13 with respect to the corresponding microstructural differences present during PBF-LB/Ti-6Al-4V manufacturing.

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Material Model of Ti-6Al-4V Alloy in Laser Powder Bed Fusion

Figure 4.30: Result of fitting procedure for the coefficient of thermal expansion (* data converted from 𝐶𝑉𝑇𝐸 to 𝐶𝑇𝐸) Table 4.13: Material model of Ti-6Al-4V: Coefficient of thermal expansion (functions and temperature intervals for one complete cycle of heating and cooling)

Temperature

Microstructural phase

Function

𝑇𝑅 < 𝑇 ≤ 𝑇𝛽

𝛼′

𝐶𝑇𝐸𝛼′ (𝑇)

𝑇𝛽 < 𝑇 ≤ 𝑇𝑙

𝛽

𝐶𝑇𝐸𝛽 (𝑇)

𝑇𝑙 < 𝑇

liquid

𝐶𝑇𝐸𝑙 (𝑇)

𝑇𝑠 < 𝑇

liquid

𝐶𝑇𝐸𝑙 (𝑇)

𝑇𝛼′𝑆 < 𝑇 ≤ 𝑇𝑠

𝛽

𝐶𝑇𝐸𝛽 (𝑇)

𝑇𝑅 < 𝑇 ≤ 𝑇𝛼′𝑆

𝛼′

𝐶𝑇𝐸𝛼′ (𝑇)

5 Support Structure Topology Optimization The preliminary studies investigating the general suitability of topology optimization as digital support design method revealed that the results strongly resemble tree structures. Further studies presented in this section confirm this observation. Thus, a systematic investigation is conducted to derive design rules to be applied in the generative design of tree supports (cf. Section 6).

5.1

Topology optimization setup

Within this thesis, the overall support design goal is to minimize the support cost, represented by the support volume, while maintaining a defined structural performance. To achieve this goal, a systematic investigation of representative use cases is performed and the resulting structures are evaluated with regard to the tree parameters. Prior to the experimental design, general decisions regarding the topology optimization setup are made (see Table 5.1 for summary). As topology optimization approach, the density-based SIMP method is chosen. SIMP has been proven to lead to robust designs, and has been widely applied in industry and research. Furthermore, various techniques to combat known numerical issues as well as PBF-LB/M manufacturing constraints are available in the literature, offering an extensive toolbox to achieve the goal of design guideline determination. This toolbox is of special interest since topology optimization for AM is not the core topic of this thesis. The formulation of the effective design Young’s modulus 𝑬𝑑 is taken from Equation (2.4). In accordance with the literature, the penalization factor is set to 𝑝 = 3 (cf. Section 2.2.1). Preliminary runs of the topology optimization setup also used for the mesh convergence study revealed that no continuation, i.e. increase of 𝑝, is required to achieve satisfactory results. In terms of investigated physics and the corresponding optimization problem formulation, the study is focused on the structural mechanics rather than heat transfer or a combination of both. This decision is made to reduce the required computational effort, and is governed by the assumption that a support structure capable of withstanding the occurring loads will also be able to dissipate a significant amount of heat without having to tailor the support towards heat transfer. As the optimization goal is to minimize the supports’ mass at a prescribed structural load, the mass minimization problem formulation seems the intuitive choice. However, due to the unresolved issues pointed out in Section 2.2.3, the minimum compliance problem formulation is chosen over the minimum mass formulation. In order to avoid checkerboard patterns, the perimeter control method following the formulation of Equation (2.15) is added to the objective function. During preliminary runs of the topology optimization setup without the perimeter control only slight checkerboard © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_5

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Support Structure Topology Optimization

pattern occurrence was observed. Therefore, both objective functions are weighted with a weighting factor of 𝑞 = 0.1, which was found to sufficiently suppress any checkerboard pattern during preliminary runs without decreasing convergence significantly. No restrictions specific for PBF-LB/M are added in the first place to avoid interfering with the general observations regarding the optimal support design, as this study’s aim is the identification of general design rules rather than obtaining one highly optimized support design use case. The design variable is the relative density, and the design domain is subject to an area constraint, since the use cases are set up in the two-dimensional space. This is due to several reasons: First, in evaluation of the tree parameters, the trunk acts as reference feature. Hence, the investigation of a respective branch and the trunk reduces a 3D tree to a 2D situation. Furthermore, a 3D tree with a wide crown may be difficult to remove by the manual methods applied in today’s support removal. It may thus be beneficial to design 2D trees and arrange them similar to block support walls in parallel or in grids, such that the respective trees are easily broken off the part along the small edge. The upper limit on the area is defined by the design domain area 𝐴Ω and the user-defined allowed area fraction 𝛾. The final optimization problem is formulated as follows, with 𝑊𝑠0 denoting the initial and 𝑾𝑠 the current total strain energy:

𝜌𝑑

1−𝑞 ℎ0 ∗ ℎ𝑚𝑎𝑥 ∗ ∫ 𝑾𝑠 𝑑Ω + 𝑞 ∗ ∫ |∇𝝆𝑑 |2 𝑑Ω , with 𝑞 = 0.1 𝑊𝑠0 𝐴Ω

s.t.:

𝐺0 (𝝆𝒅 ) = ∫ 𝝆𝑑 𝑑Ω − 𝛾 ∗ 𝐴Ω ≤ 0

min:

Ω

Ω

(5.1)

Ω 𝑝 𝝆𝑑

𝐄d = ∗ 𝐸0 , with 𝑝 = 3 0 < 𝜌𝑑,𝑖 ≤ 1, 𝑖 = 1, … , 𝑁

}

For implementation of the optimization problem, the software COMSOL Multiphysics 5.2a (COMSOL AB, Stockholm, Sweden) is used. COMSOL Multiphysics is a mathematical modeling software with a focus on multi-physics modeling. While not specialized in topology optimization, it allows for the formulation of individual optimization problems and provides a wide range of solvers. It is therefore more flexible and efficient than the topology optimization modules built into commercial CAD software, and provides the same range of features a specialized software does. The design domains are discretized by a free triangular mesh. This type is well suited for structural computation. Additionally, the flexibility of the elements allows for a well discretization of the angled and curved sections of the use cases (see Section 5.1.1). For the

Topology optimization setup

127

optimization solver the MMA is chosen due to its robustness and reliability at reasonable computational effort (cf. Section 2.2.4). Table 5.1: Overview of decisions regarding the topology optimization setup

Item

Decision

Topology optimization approach

Solid isotropic material with penalization (SIMP)

Physic / Problem formulation

Structural mechanics / Minimum compliance

Additional implementations

Perimeter control method

Ambient temperature

293.15 K

Finite element mesh type

Free triangular mesh

Solver

Method of moving asymptotes (MMA)

Software

COMSOL Multiphysics 5.2a

The general procedure of computing is split into two consecutive steps: First, 𝑊𝑠0 is determined by performing an initial computation (‘Get Initial Values’ feature of COMSOL Multiphysics). In the second step, the appropriate volume fraction is determined by iteratively varying 𝛾 in intervals of 0.2 until the maximum von Mises-stress of the result is close to the yield strength of the material. Note that this is not done in a continuation manner; each computation is initiated by step no. 1. That way, differences in the number of continuation steps, which may influence the overall result, are avoided.

Figure 5.1: Result export and evaluation procedure (for parameters shown in Section 5.1.1)

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Support Structure Topology Optimization

The export of the results to enable the tree support parameter measurements consists of two steps. The contour of the topology optimization is exported to a text file. Here, a design density value of 0.8 is defined as contour level. Then the text file is imported into the same COMSOL study again as geometry. It is now possible to export the contour geometry as CAD file. Within a CAD software, SolidWorks 2017 (Daussalt Systèmes SOLIDWORKS Corp., Waltham, USA), the different evaluation parameters defined in the following section are measured. Figure 5.1 summarizes the individual stages.

5.1.1

Tree evaluation parameters

Parameters evaluated are either topology- or shape-related: First, the resulting volume fraction as well as the number of tree supports within the design domain are noted. Second, the tree shape is evaluated in detail to derive design constraints or rules to be included in the tree support design algorithm presented in Section 6. Figure 5.2 shows the tree parameters measured.

Figure 5.2: Tree evaluation parameters

Parameters directly defining the tree are investigated: The number of levels describes how often the trunk and branches are dividing, and the number of branches per split is attributed to the division of branches, too. In addition, the tree crown is analyzed in terms of its width (𝑤𝑐 ) and height (ℎ𝑐 ) or the ratio of both (𝑟ℎ𝑤 ), respectively. Then, the position of the tree root in X-direction (𝑥𝑡 ) is of interest. Furthermore, each trunk and branch (also named segment, when both trunk and branches are addressed) is evaluated. To do so, the tree skeleton is manually sketched over the imported CAD file of the tree contour. With the help of the skeleton structure, two different

Topology optimization setup

129

angles are identified for each segment: the overhang angle (𝛼𝑡 ) as well as the angle to the parent segment (𝛽𝑡 ), i.e. the previous element. Additionally, the length of each segment (𝑙𝑡 ) is directly taken from the skeleton. The thickness or width of the segments (𝑤𝑡 ) is measured at the middle point of the corresponding skeleton segment, as this is the thinnest and therefore most critical cross-section. If the graphical result of the topology optimization indicates the existence of a segment, but the contour does not include it, it is drawn from the result graphic and assigned a width of 𝑤𝑡 = 0.1 mm, which is considered the minimal possible feature size due to most laser spot diameters applied in PBF-LB/M being of this size.

5.1.2

Experimental plan

With the numerical setup of the topology optimization in place and the evaluation criteria identified, the experimental plan containing the investigated use cases is derived. The main parameters to be varied are the design domain geometry and the applied structural loads. The variations are accompanied with a corresponding research question to be answered by the respective variation (cf. Table 5.2 and Table 5.3). Design domain geometry Because geometrical characteristics of the tree supports will be evaluated (cf. Section 5.1.1), the design domain needs to consist of realistic geometries and dimensions. Therefore, a base geometry is defined and systematically varied as pictured in Figure 5.3.

Figure 5.3: Design domain variation (reference value in bold letters)

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The situation investigated is the support of an overhanging feature, because all support tasks apply to this situation. As base geometry a square with an edge length of 𝑎Ω = 𝑏Ω = 50 mm is defined. The lower edge represents the interface to the build platform, while the upper edge is the part-support interface. A fixed constraint is added to the edge connected to the build platform. The mechanical loads are applied to the part-support interface edge. The square defines an overhang angle of 𝛼𝑡 = 0°. Starting from this base design domain, the development of tree characteristics can be described with respect to the varied design domain parameters. That way, general tendencies in the tree design can be identified. The first design domain parameter is the square edge length, i.e. the size of the design domain. The second object of investigation is the overhang angle, which is increased in steps of 10° up to 𝛼𝑡 = 30°. The upper left corner is fixed, while the upper right is lifted up. Third, the aspect ratio (𝑏Ω : 𝑎Ω ) is varied by doubling the width (𝑏Ω ) and height (𝑎Ω ), respectively. Last, the overhang geometry is changed to an arch instead of a straight edge. The arch is defined by a quadratic Beziér curve with the control points 𝑃1 (0|𝑎Ω ) , 𝑃2 (

𝑏Ω 3 4

3

| 𝑎Ω ), and 𝑃3 (𝑏Ω | 𝑎Ω ). While the design domain size is only investigated for the 2

2

base square geometry, the overhang variations are applied to all aspect ratios. This is due to the fact that a topology optimization result is mesh-dependent rather than dimensiondependent, i.e. the volume fraction and topological layout of the linearly scaled design domain should remain constant with regard to the base design domain. Table 5.2: Research questions targeted by the design domain variation

Parameter

Research question

Expectation

Size of design domain

Will the dimensions of tree supports scale linearly, i.e. the volume fraction stay constant?

Yes

Overhang angle

Will the tree support crown follow the overhang edge, i.e. distort, or generate new levels of branching?

Distort

Aspect ratio 1:2

Is there a limit to the tree support crown height, i.e. will the trunk compensate for the additional height?

Yes

Aspect ratio 2:1

Is there a limit to the tree support crown width, i.e. will new tree supports be generated after the crown width exceeds a certain threshold?

Yes

Are tree supports evenly distributed over the design Even domain or are the tree parameter limits maxed out in distribution a specific sequence? Overhang geometry

Does the overhang geometry influence the tree support topology?

No

Topology optimization setup

131

In Table 5.2, the research questions corresponding to the individual design domain adaptions are formulated, and expectations towards the findings are stated. With an increasing overhang angle, the level of one edge is lifted while the other remains fixed, tilting the support-part interface. The tree support may react to this by either following the distortion by distorting itself, or by generating new branches to compensate for the additional height. As the latter will require more material than following the support-part interface, it is expected that the tree will also distort. The change of the aspect ratio targets the tree crown geometry, i.e. its height-to-width ratio. When increasing the height, it is expected that the height of the tree crown is limited and the tree’s trunk compensates most of the design domain height. It is assumed to be more efficient in material distribution to have one straight strut in the load’s direction than several, though thinner, struts that are fanning out. The 2:1 aspect ratio investigates the tree crown width. Here, it is expected that the angle of the branches are limiting the crown width as struts with an angle of over 45° to the load’s direction are undesirable from a mechanical point of view regarding load transmission. Hence, the tree crown will not widen limitlessly, but at some point, the topology will introduce a new tree. With regard to the generation of new tree supports, it is also of interest how these new tree supports are arranged within the design domain, i.e. whether all tree supports are evenly distributed over the design domain or the maximization of the tree supports is prioritized, and the remaining space is filled with comparatively narrow tree supports. It is expected that the tree supports will be evenly distributed, because a narrower tree is more suitable to transmit the load from the partsupport interface due to the branches being more aligned with the load transmission direction. Last, the overhang geometry is expected to have no influence on the tree support’s topology, because an arch may also be considered a concatenation of infinitesimal small straight edges. However, most AM parts include free-form surfaces, and therefore this assumption is checked. Structural loads Additional to the design domain geometry, several applied structural loads are examined. The first distinction is between tensile and compressive loads. The compressive load represents the weight of the material (powder and bulk) above the supports. To determine the maximum possible weight load, it is assumed that the complete build envelope of a PBF-LB/M system contains bulk Ti-6Al-4V. This assumption represents an extreme case to ensure the restuling structure’s integrity; realistic loads will be lower due to the mixture of powder and bulk material. The build envelope of the SLM500HL (500 mm 𝑥 280 mm 𝑥 365 mm) is used. The weight load is derived to be

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𝐹𝐺 = 𝜌 ∗ 𝑉𝐵𝐸 ∗ 𝑔 = 4420

kg m ∗ (500 mm ∗ 280 mm ∗ 365 mm) ∗ 9.81 2 3 m s

(5.2)

= 2216 N and the corresponding compressive load is determined by relating the weight load to the build platform’s area, yielding 𝐹𝐺 = 16,000 Pa. The tensile load is determined based on literature values. Lindecke et al. [297] as well as Bobbio et al. [346] performed tensile tests on Ti-6Al-4V block supports while varying geometric parameters. In [297], the yield strength of the supports was measured to be within the interval of 251 MPa ≤ 𝜎𝑦,𝑠 ≤ 283 MPa , whereas [346] present results of 176 MPa ≤ 𝜎𝑦,𝑠 ≤ 350 MPa. Although the experimental setups are quite different with regard to the specimen design, the measured yield strengths match. As block supports are often applied successfully in PBF-LB/M, the maximum yield strength of 𝜎𝑦,𝑠 = 350 MPa is considered a limit. To be able to observe the development of the tree supports with increasing load, tensile loads of 𝐹𝑡 = 100 MPa ∧ 200 MPa ∧ 300 MPa are investigated. The load is defined as constant load as well as linear load, which is increasing from 0 at the right corner to 𝐹𝑡 at the left corner. Table 5.3: Research questions targeted by the load variation

Parameter

Research question

Load type

Will the tree supports be different for tensile and compressive loads?

No

Load height

Will an increase of the load height orient the tree support branches towards the load direction?

Yes

Will an increase of the load height result in thicker tree support branches?

Yes

Does the tree support create a constant structure or adapt to changes in the load height throughout the part-support interface?

Adapt

Load distribution

Expectation

Again, a set of research questions regarding the development of the tree support structures under changing load conditions is formulated (see Table 5.3). The investigation of compressive and tensile loads is not expected to determine a significant difference in the support topology, because the load is on the same line of force, though in opposite direction. With an increase of the load height, it is expected that the tree branches translate into a structural more optimal shape and orientation. This is defined as orienting the branch middle axis along the line of force to minimize bending as well as increasing the branch diameter to reduce stresses. While the constant load distribution is expected to result in a

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symmetrical tree support, this is not the case for the linear load distribution. Here, the structure is assumed to adapt its parameters to the continuous load height increase, i.e. creating an asymmetrical topology.

5.1.3

Mesh convergence study

The computation of the topology optimization is performed on a regular desktop PC, whose technical specification is listed in Table 5.4. In order to determine an appropriate element size, a convergence study is performed. The mesh size is governed by ℎmax (following the notation of COMSOL Multiphysics), which is defined by ℎmax =

𝑎Ω + 𝑏Ω 𝑟Ω

(5.3)

The parameter 𝑟Ω describes the resolution of the mesh, i.e. is a regulation factor. This definition of ℎmax allows for the same refinement of the mesh no matter the design domain size and therefore ensures comparability of the results. For the convergence study, the quadratic design domain with an edge length of 70 mm is chosen as a compromise regarding the different design domain areas. The regulation factor 𝑟Ω is iteratively increased from 𝑟Ω = 80 in steps of 20 until no further changes in the topology is noticed. This is achieved for 𝑟Ω = 180. Table 5.4: Technical specification of computing system for the topology optimization

Item

Value

Operating system

Microsoft Windows 10 Pro (Version 10.0.19041 Build 19041)

Processor

Intel Core i7 – 6700 @ 3.40 GHz

Physical memory (RAM)

32 GB

5.2

Results of systematic support structure topology optimization

In this section, the results of the topology optimization study are presented. Answers to the research questions formulated in Section 5.1.2 are given, and the relevant tree parameters required for the algorithmic design procedure are evaluated.

5.2.1

General observations

The almost full factorial study of support topologies provides extensive insights regarding the ideal support structure and the influence of design domain as well as load. The first and most important observation is that all computed structures might be clearly classified as tree structure, confirming the hypothesis derived from the preliminary studies. This

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enables the algorithmic design of tree supports. In Figure 5.4 and Figure 5.5, selected results (constant load of 200 MPa) are shown to demonstrate the effect of design domain variation. The use cases with an edge length of 𝑏Ω = 50 mm develop a single tree structure, whereas the broader design domains lead to several tree structures. The general nature of the tree structures is very smilar, though.

Figure 5.4: Topology optimization results for the rectangular design domain, at constant load of 200 Mpa

First, the research questions defined in Section 5.1.2 are evaluated. All expectations have been met: The volume fraction stays constant with increasing size of Ω, and the increase of the overhang angle reveals that the tree support crown follows the part-support interface rather than building new segment levels (cf. Figure 5.5). Limits to the tree support crown have been found, i.e. the trunk compensates an increased 𝑎Ω at constant 𝑏Ω , whereas an increased 𝑏Ω results in new tree supports, which are evenly distributed throughout the design domain (cf. Figure 5.4). The comparison of the straight and arched overhang revealed no significant difference in the corresponding tree structure. Furthermore, no considerable variation of the tree support topology is observed for compressive and tensile loads. An increase of the load height creates tree supports with a slimmer crown (cf. Figure 5.6) due to the branches being oriented along the line of force, i.e. 𝛽𝑡 → 180°. In addition, the segment diameter is increased. Both reactions to the increased load height lead to less segment levels, since further splitting would contradict the tendencies described. Last, it is revealed that the tree supports adapt to a linear load distribution by applying the reaction to the load

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height in a continuous manner (cf. Figure 5.6). In conclusion, the achieved topology optimization results show the structural behavior expected from a theoretical point of view, validating the obtained results.

Figure 5.5: Topology optimization results for varying part-support interface geometry, at constant load of 200 MPa

Figure 5.6: Topology optimization results for increasing, linear load, reference design domain (𝑎Ω = 𝑏Ω = 50 mm)

A summary of the study’s main findings regarding the tree evaluation parameters is presented in Table 5.5. In the following section, the tree parameters relevant to the algorithm are evaluated in detail.

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Table 5.5: Summary of the main findings of the support topology optimization study

Parameter

Main findings

Volume fraction

𝛾

- Linear dependency on load height - Increase of 𝛾 with increasing overhang angle

Number of trees

𝑛𝑡

- Increase of design domain width (𝑏Ω ) leads to increased number of trees

Number of branches after split

𝑛𝑠

- Parent branches split in exactly 2 child branches

Trunk X-position

𝑥𝑡

- Identical with geometric center of load

Crown width

𝑤𝑐

- Even distribution of trees in design domain with full coverage of the part-support interface

Crown height

ℎ𝑐

- Development follows 𝑤𝑐

Crown geometry

𝑟ℎ𝑤

- Roughly constant

Branch overhang angle

𝛼𝑡

- Trees are mostly self-supporting - branch requiring support are located at the outer tree edges and inherit the need of support to the outer child branch (if any) - Increase of load height increases 𝛼𝑡

Branch angle to parent Branch

𝛽𝑡

- Decrease of 𝛽𝑡 with increasing branch level - Increase of variation with increasing branch level

Branch length

𝑙𝑡

- Trunk length dependent on design domain height (𝑎Ω ) - Linear increase of trunk length with increasing overhang angle (𝛼Ω ) - For other branches linear decrease of mean value and variance with increasing branch level

Branch width

𝑤𝑡

- Increase of load height increases 𝑤𝑡 - Decrease of mean value and variance with increasing branch level - Child 𝑤𝑡 add up to parent 𝑤𝑡

5.2.2

Details on relevant tree design parameters

For the algorithmic support design procedure described in Section 6, various tree parameters need to have limits or values attributed to control the tree support generation. Based on the general observations, these limits and values are identified from the topology optimization results.

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137

The volume fraction and number of trees will be indirectly considered by the choice of segment thicknesses as well as the clustering of the part-support interface points. However, it is important to note that every parent segment has only two child segments.

Figure 5.7: Trunk X-position deviation from load center

Figure 5.7 displays the difference of the measured trunk X-position to the load center of the rectangular design domain use cases with 𝛼𝑡 = 0° for illustration. The load type is noted, with ‘L’ for linear, ‘C’ for constant, and ‘G’ for gravitational load. If the topology optimization result consists of more than one tree structure, the mean value of the respective differences is taken into account. Deviations of up to ±2 mm are considered due to the manual determination of the tree skeletons and the associated errors. Most use cases fall into that range, which is marked by the red lines. Only 7 out of 28 presented use cases exhibit larger deviation. No correlation between these deviations and the use cases are apparent, the use cases include different load heights and types as well as geometrical variations. The same tendencies are observed in the remaining use cases with increasing overhang angle or arched overhang shape. Hence, it is concluded that the tree support’s trunk X-position should be located in line with the load center.

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The tree support crown is defined by its width as well as the crown geometry, i.e. the height-to-width ratio. It is observed that the tree supports are evenly distributed throughout the design domain width, if more than one tree structure is present. This is confirmed by the crown width, see Figure 5.8 for visualization. Here, the 𝑤𝑐 for use cases with 𝛼𝑡 = 0° and linear load is given. For a specific design domain geometry, the crown width is clustered around the same value, no matter the load height. Furthermore, with increasing design domain width, i.e. increasing number of tree structures, the crown width is decreasing. It is therefore concluded that a wide part-support interface needs to be divided into equally wide tree supports under a maximum crown width constraint, which is chosen as 𝑤𝑐,𝑚𝑎𝑥 = 45 mm based on the available data.

Figure 5.8: Crown width for linear load cases and 𝛼𝑡 = 0°

The tree support crown geometry gives insight on whether an increasing design domain height is compensated by the trunk of the crown. Figure 5.9 displays every use case with 𝛼𝑡 = 0°. The green line indicates a 1.0 ratio, which is equal to the crown’s mean overhang angle 𝛼𝑐 = 45°, whereas the red line marks the critical overhang angle of 𝛼𝑐 = 30° (ratio of 0.8). If a tree structure’s mark is in the area below the red line, its overall crown geometry is not self-supporting, pointing out a major risk of branches in need of support. The dotted lines mark 𝑟ℎ𝑤 in steps of 0.1. It becomes apparent immediately, that most tree

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139

structures form an overall self-supporting crown geometry. Hence, no repetition of the study with additional AM restrictions implemented is required. Only six use cases are in the critical zone. Generally, the ratio is observed to be 0.8 ≤ 𝑟ℎ𝑤 ≤ 1.2 with some exceptions, although the deviation from the upper limit is small. The mean value is 𝑟ℎ𝑤,𝑚𝑒𝑎𝑛 = 1.0028, thus 𝑟ℎ𝑤 = 1.0 is considered the ideal ratio for tree support design.

Figure 5.9: Crown geometry results for 𝛼𝑡 = 0°

The observation of a mainly self-supporting crown geometry translates to the overhang angle of the individual segments. The majority of tree segments is within the self-supporting regime. If segments with 𝛼𝑡 < 30° are present, they are located at the outer segment path. This is illustrated in Figure 5.10 as an example. Here, the overhang angle of consecutive segments building a segment path are connected with lines to visualize the tree structure. The outer paths indicate a linear decrease of 𝛼𝑡 , which may fall below the critical overhang angle. This is partly attributed to the algorithm striving to cover the complete part-support interface. Apart from the location of segments violating the critical overhang angle, it is noticed that an increase of the load height leads to an increase of the overhang angle, i.e. the segments are oriented towards the line of force.

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Figure 5.10: Segment overhang angle of linear load (200 Mpa), square (𝑎Ω = 𝑏Ω = 50 mm) use case

To control the branching during the tree support generation, 𝛽𝑡 is of high importance as it influences the load transmission capabilities of the tree structure. In Figure 5.11, 𝛽𝑡 is displayed for the individual tree structure levels of all use cases. The mean value of the respective use cases as well as the overall mean and standard deviation of a level are given. It is apparent that the mean values of the first, second, and third level decline linearly, whereas the fourth level does not follow this tendency. However, the fourth level also exhibits a significantly greater variation in the data. The cluster of segments with 𝛽𝑡 ≈ 160° is found in use cases with 𝛼𝑡 ≠ 0°, and is attributed to the algorithm’s desire to cover the part-support interface while following it through tree structure distortion. Based on the acquired data, limits to 𝛽𝑡 are determined for use in the tree support design. Following the linear development, segments of the first level are restricted to 𝛽𝑡,1 ≥ 160°, of the second level to 𝛽𝑡,2 ≥ 150°, and of the third and fourth level to 𝛽𝑡,34 ≥ 140°.

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141

Figure 5.11: Segment angle to parent segment results

Observations similar to those of the segment length are made with regard to the segment width (see Figure 5.13). With increasing tree structure level, the thickness of the segments decreases. In addition, the variance also decreases with increasing level. Therefore, when applying the segment widths during tree support design, a child segment’s width needs to be smaller than the parent’s one. It is further noted, that the parent segment width is approximately the sum of both child segment widths. The mean parent-to-childs width ratio (𝑟𝑡 = 𝑤𝑡,𝑃𝑎𝑟 ⁄∑ 𝑤𝑡,𝐶ℎ𝑖 ) has a value of 𝑟̅𝑡 = 0.94 for tree level 1, level 2 exhibits 𝑟̅𝑡 = 0.95, and level 3 is determined to 𝑟̅𝑡 = 0.89.

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Figure 5.12: Segment length results

The segment length, whose development over the tree structure levels is shown in Figure 5.12, reveals a linear decrease up to level 3. It is noted that the trunk length, i.e. level 0, does not follow any tendency but is characterized by a large variance. This is related to the fact that the crown geometry is quite constant, and the trunk compensates the remaining design domain height. For the tree support design, it is concluded that a parent segment is required to be longer than the child segment, except for the trunk. The general concept of the tree generation algorithm should account for this requirement without further intervention.

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143

Figure 5.13: Segment width results

During data evaluation, it is noted that the length-to-thickness ratio of the segments follows a roughly constant relationship. Figure 5.14 visualizes this observation exemplary, showing that on the one hand, the 𝑟𝑙𝑤 is clustered for the respective tree structure levels, and on the other hand the data follows a straight line with decreasing 𝑙𝑡 and 𝑤𝑡 , as described in the previous paragraphs. In conclusion, a constant 𝑟ℎ𝑤 can be derived to describe the segment geometry. This constant is dependent on the load height. Furthermore, the load type affects the ratio, as the linear load includes less stressed segments compared to the constant load type. For simplicity, the ratios for the different load heights are set to those acquired from the constant load, which will probably slightly oversize the generated tree supports. The overhang angle does not influence 𝑟𝑙𝑤 significantly. In conclusion, the length-to-thickness ratio of the segments is determined to 10, 𝐹𝑡 = 100 MPa 𝑟𝑙𝑤 = { 6, 𝐹𝑡 = 200 MPa 4, 𝐹𝑡 = 300 MPa

(5.4)

which is fitted to the following polynomial: 𝑟𝑙𝑤 (𝐹𝑡 ) = 0.1 ∗ 10−3 ∗ 𝐹𝑡2 − 0.07 ∗ 𝐹𝑡 + 16

(5.5)

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Figure 5.14: Segment geometry for constant load (100 MPa), square (𝑎Ω = 𝑏Ω = 70 mm) use case

6 Support Structure Design To generate tree supports adhering to the design rules of Section 5, a base algorithm is required, in which those rules can then be implemented. Approaches to the digital representation of tree structures are investigated and the most suitable approach is chosen. This base algorithm is then adapted and extended to meet the need of incorporating specific design rules. Last, the tree support design of the final design procedure is validated against some of the topology optimization results.

6.1

Digital tree (support) structure generation

Tree support structures in general are not a new phenomenon within AM, especially in the MEX/P process. However, the complexity of the tree support generation hinders the broad application, as will be demonstrated in Section 6.1.2. Hence, the focus of this literature review is on the techniques to generate tree supports with no restriction in terms of AM processes, considering both academic and commercial approaches.

6.1.1

Academic approaches to tree support generation

Approaches published in the academic literature may be distinguished with regard to the geometric layout of the tree supports: periodic, semi-periodic, and aperiodic tree supports. A periodic tree support is derived by iteratively applying simple design rules and taking advantage of symmetry. With more complex design rules operating on a discretized design domain, and the omission of symmetry, a semi-periodic tree support is created. These tree supports are capable of taking on individualized topologies, but are not completely free in their appearance due to the discrete procedure. Aperiodic tree supports overcome this issue, which is quite important given that AM parts are known for their complex surface geometries. However, the generation of aperiodic trees requires comparatively high effort. Periodic tree supports The most obvious way of creating a periodic tree structure is a parametric CAD design, as done by Gan & Wong [313], Zhang et al. [302], and Weber et al. [331]. Their designs consist of a tree with one level of branches. The trunk splits into several branches, making use of rotational symmetry where the trunk is the symmetry axis. A slightly more complex approach is the design of fractal trees [345, 481]. Fractal trees consist of several levels of branches, which are created iteratively using the same geometric rules. While Blunk et al. [345] designed 2D fractal trees and arrange them in a row, Díaz Lantada et al. [481] created 3D fractal trees. They further adjust the generated structures by eliminating unnecessary struts, but the resulting supports can still be classified as fractal tree as the periodicity is not violated. In [345, 481], the tree supports are created as discrete structures. Boyard et al. [482], however, first defined the crown of the tree support as a 3D solid geometry © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_6

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with a surface part-support interface rather than point-wise contact. In a second step, the tree support crown is hollowed in a fractal manner, resulting in a 3D fractal tree similar to the design of [345]. Examples of the periodic CAD as well as fractal tree supports are given in Figure 6.1. Zhang et al. [483] presented a completely different approach to supports in the MEX/P process for hollow parts based on the medial axis of a 3D model. Here, a base support structure is defined by the medial axis of the model. The medial axis is a skeletal representation and is defined as a set of centers of inscribed spheres, which touch the object boundary in at least two points [484]. Periodic tree supports similar to those of [302, 313, 331] are then applied to connect the base support to the model’s inner surface, which is represented by a hexagonal frame, by sustaining the corners of the hexagons.

Figure 6.1: Examples of periodic tree supports: Parametric CAD design [313], fractal tree [482]

Semi-periodic tree supports As an extension of the periodic design approach, semi-periodic tree supports make use of advanced geometrical or mathematical rules. They are therefore not governed by symmetry anymore. Two main approaches have emerged in the literature (see Figure 6.2): the use of discretized design domains and the cone constraint strategy. Vaidya and Anand [318] derived a tree support by considering a voxelized, i.e. discretized, design domain. The tree topology is built via the voxels, which are later assigned a defined unit cell geometry. In [336, 485], Zhu et al. demonstrated a tree support design approach where an algorithm operates on a design domain discretized by a uniform grid, connecting grid nodes to form the tree skeleton. These skeletons are later optimized by particle swarm optimization (PSO). While this procedure breaks up the orientation along the horizontal grid lines, the individual branches are still aligned with the vertical grid. Vaissier et al. [319] followed a different procedure by filling the design domain with a unit cell, which consist of a discrete strut arrangement, creating a lattice structure. During the discrete lattice optimization, single struts are removed from the design domain until the minimum volume objective is achieved. The resulting supports have a tree-like structure comparable

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147

to those of other studies. The use of a discretized design domain instead of fixed design rules allows for further design opportunities. However, the uniform discretization still restricts the tree support design, and creates challenges in the design of the part-support interface if the part consists of irregular, free-form surfaces.

Figure 6.2: Examples of semi-periodic tree supports: Discretized design domain [485], cone constraint [486]

To overcome the challenges of the discretized design domain approach, in 2014 both Schmidt & Umetani [487] as well as Vanek et al. [486] proposed a strategy based on a cone constraint. Here, the part-support interface is represented by points sampled on the part surface. A cone is defined by a maximum cone angle and attached to each point. The intersection of two cones determines the feasible space to put a point capable to support both interface points if connected via struts. This procedure may be applied iteratively, creating tree supports with multiple branch levels. Aperiodic tree supports Zhang et al. [344] adapted the cone constraint strategy by adjusting the support point determined by the cone intersection to coincide with the local barycenter of the interface points. They note, though, that the procedure generates new support points without taking into account the already created support and part above. This may lead to a mismatch of the barycenter and center of mass in the sub-regions, challenging the overall stability of the tree support. In [488], Tricard et al. used tree structures to create ribbed supports for sustaining hollow parts. Their procedure starts at the inner top surface of the part. Here, points in need of support are identified, and with an iterative process, 2D tree structures are created by propagating those points down. In between iterations the tree skeleton is straightened, which leads to a completely aperiodic tree support. The tree supports are pushed to connect to the surrounding part surface rather than propagating down to the build platform. Finally, the tree supports are filled in to create solid ribs.

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A completely new approach is taken on by Zhou et al. [489] and Zhang et al. [332]. They applied a method from the algorithmic botany, a discipline dealing with the mathematical description and modeling of plant growth and behavior, to generate tree supports, the socalled Lindenmayer (L)-systems. The L-system theory models the growth of plants by applying a pre-defined set of rules (cf. Section 6.1.3 for details). Zhou et al. combined the L-systems with an octree representation of the part geometry. They first generate a tree support via L-systems, then check for the connectivity to the part surface and prune accordingly. Zhang et al. integrated the L-system algorithm into a genetic algorithm (GA), which populates part-support interface points and tree root points to create various solutions. The solutions are evaluated and optimized by a multi-objective fitness function considering the minimization of the support volume and collisions of support structures, creating a pareto front. After the optimization run, feasible solutions of the pareto front are analyzed by process simulation to ensure sufficient support performance, and the best solution is chosen as result. The authors note, though, that the numerical evaluation of the tree support candidates poses a great challenge in terms of computational cost. Nevertheless, the L-system theory is deemed a suitable tool to create tree structures for technical applications, which allows for the creation of unique tree structures.

6.1.2

Commercial implementations of tree support structures

Tree supports are not only known in academia, but industry as well. To assess the current state of the art, a study is carried out to compare available implementations. Two test specimens are designed, consisting of a cantilever with an overhang angle of 0° and 30°, respectively. The exact dimensions are given in Appendix A.1. Both geometries are loaded into the respective software and tree supports are generated automatically. No manual setting of any parameters is done. In terms of software, the Materialise Magics (Materialise GmbH, Gilching, Germany) data preparation software is used. Since there is very few data preparation software for PBFLB/M, with none freely available, open-source software created for the MEX/P process is also investigated. Out of twelve so-called slicers, three offered the possibility to create tree supports: Ultimaker Cura (Ultimaker B.V., Utrecht, Netherlands), Autodesk Meshmixer (Autodesk, Inc., San Rafael, USA), and IceSL (Institut national de recherché en informatique et en automatique (Inria), Nancy, France). Note that IceSL is run by a research institution, not a company. The results of the study are presented in Table 6.1. The coloring of the supports is specific to the software and does not hold further relevant information.

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149

Table 6.1: Overview of tree supports generated by data preparation software available in the market

Overhang angle

Software 0°

Materialise Magics 22.0 (PBF-LB/M)

Ultimaker Cura 4.8.0 (MEX/P)

Autodesk Meshmixer 3.5.474 (MEX/P)

30°

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IceSL (MEX/P)

The automatic tree support generation result of the Materialise Magics slicer is very close to the standard cone support. Only at the very top, branches are built to connect to the part. The number of branches is variable, but greater than four. Interestingly, the overhang angle of the branches is comparatively small, which does not suit the compensation of mechanical loads in build direction. The simple topology can be altered through various functions. However, this needs to be done manually. Both Autodesk Meshmixer and IceSL create fractal trees with fixed branch angles and branching points. While Autodesk Meshmixer considers circular branch cross sections, IceSL uses thin walls for the 3D modeling of the trees. Furthermore, for the 0° overhang angle cantilever, the complete space below the overhang is used for support generation, whereas for the 30 ° overhang angle the trunks of the trees are positioned near the wall and areas at the free edge are reached via branching. The Ultimaker Cura algorithm behaves very differently for the two test geometries. For the 0° overhang angle, nearly the whole space below the overhang is used to create one large contiguous support structure. Tree-like features can be noticed; they are also of fractal nature. The structure consists of thin walls and hence is hollow. In the case of the 30° overhang angle, only a single tree with a trunk, one branch and a wall connected to the surface is created in the symmetry plane of the geometry. With the free edges on the side as well as at the top of the overhang, it is questionable whether this support fulfills all design restrictions of the MEX/P process. In summary, free-to-use data preparation software for MEX/P seems to rely on the concept of fractal trees, whereas the commercial software Materialise Magics, focused on PBFLB/M, presents an adapted, umbrella-like cone support. The software developers do not disclose the underlying concept or algorithm, though, except for the Autodesk Meshmixer software [487]. Therefore, this study is based on observations and may not grasp all aspects of the respective tree supports.

Digital tree (support) structure generation 6.1.3

151

Algorithmic botany

Algorithmic botany deals with the mathematical or algorithmic description of plant structures and growth. Tree modeling is relevant to a diverse set of fields, e.g. botany and forest inventory growth or movies and computer games [490]. While these techniques aim at creating trees as realistic as possible rather than trim them to technical applications, they provide a solid base of approaches, which may be adapted to the tree support application. There are two main categories of approaches: grammar- as well as environment-based. Grammar-based algorithms These procedures apply a defined, fixed set of rules that directly control the tree’s topology [491]. Thus, the techniques employed for tree support generation described in Section 6.1.1 qualify as grammar-based. A comparatively simple approach is provided by fractal trees. Here, a defined set of rules regarding branch orientation, diameter, and length are recursively applied. The rules may be global [492] (e.g. rotate each new branch by 10°) or referring to elements created in prior iterations [493] (e.g. length equals a specified fraction of the parent’s length). One of the most famous fractal objects are snowflakes with their individual, unique set of rules. Bejan & Lorente [494] apply the constructal law to generate tree structures for use in engineering, more specific heat conduction and fluid flow. The constructal law is formulated as follows: “For a flow system to persist in time (to survive) it must evolve in such a way that it provides easier and easier access to the currents that flow through it.” [495] Here, a specific sequence of inserting and adapting highly conductive blades is defined and coupled with FE analysis and a shape optimization step. Cellular automata, which operate on a voxelized space, have been used widely for spatial modeling of forests [490]. Each automaton can be defined as a triplet of a set of inputs, a set of states, and a nextstate function (rule) that is defined by the input-state pairs [496]. The set of inputs is defined as n-tuples of the states of a finite set of adjacent cells, also called ‘neighborhood’. With a small set of states and consequently small number of rules, cellular automata are capable of complex behavior. Cellular automata present the first technique capable of computing plant growth instead of fixed plant topologies [497]. The most common grammar-based tree modeling approach is the application of L-systems [490], developed by Lindenmayer [498]. L-systems are based on the concept of cellular automata and are indistinguishable in simple cases [499]. They use spatial units (cells), and the states and inputs are assumed discrete entities. L-systems are represented by symbols combined into a one-dimensional string. The cell array is grown by computing new cell states based on the previous states and inputs at certain time intervals. A change of the cell state may include the assignment of a new symbol or lead to cell division by substituting the current symbol by more than one new symbol. Eventually, cell death is also

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possible by changing the symbol to an empty string. The birth and death of cells is a distinctive feature of L-systems [499]. The rules defining how to substitute a cell state are called production rules, the current cell state is named ‘predecessor’, and the changed cell state is denoted ‘successor’. An example of the L-system procedure is given in the following [498]: Let there be five different cell states with respective production rules (see Table 6.2). Note that two production rules lead to cell division, and one production rule leaves the cell state as is. Table 6.2: L-systems example: cell states and production rules

Cell state symbol

Production rule predecessor → successor

𝑎

𝑎 → 𝑐𝑏𝑐

𝑏

𝑏 → 𝑑𝑎𝑑

𝑐

𝑐→𝑘

𝑑

𝑑→𝑎

𝑘

𝑘→𝑘

If 𝑎 is chosen as initial string, called axiom, the resulting sequence develops over a few iterations: (0) (1) (2) (3) (4) (5) (6) ⋮

𝑎 𝑐𝑏𝑐 𝑘𝑑𝑎𝑑𝑘 𝑘𝑎𝑐𝑏𝑐𝑎𝑘 𝑘𝑐𝑏𝑐𝑘𝑑𝑎𝑑𝑘𝑐𝑏𝑐𝑘 𝑘𝑘𝑑𝑎𝑑𝑘𝑘𝑎𝑐𝑏𝑐𝑎𝑘𝑘𝑑𝑎𝑑𝑘𝑘 𝑘𝑘𝑎𝑐𝑏𝑐𝑎𝑘𝑘𝑐𝑏𝑐𝑘𝑑𝑎𝑑𝑘𝑐𝑏𝑐𝑘𝑘𝑎𝑐𝑏𝑐𝑎𝑘𝑘 ⋮

(6.1)

The string array may be interpreted as geometrical shapes. In [498], to recreate the shape of a leaf, the symbols 𝑎 and 𝑏 are associated with sharp projecting tips, the symbols 𝑐 and 𝑑 are represented by lateral margins of lobes, and the symbol 𝑘 characterizes notches. Figure 6.3 displays the corresponding leaf shapes. Since the production rules do not include the termination of the sequence development, there would be infinite iterations unless some other termination criterion, such as a maximum number of iterations, is imposed. The example above describes an OL-system, also referred to as ‘zero-sided’ or ‘informationless L-system’ [498], where the production rules are a function of the present cell state only. However, the concept has been extended to IL-systems, i.e. systems with interaction among the cells. Here, production rules also account for the present cell state of the

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surrounding states. For each possible combination, production rules must be defined, increasing the required effort, but expressible complexity, too.

Figure 6.3: Graphical representation of the Lindenmayer sequence shown in Equation (6.1). The letters a, b are associated with sharp tips; c,d represent lateral margins of lobes; k characterizes notches.

To apply L-systems to the modeling of tree structures, the geometrical interpretation of Lsystems is of great importance. Earlier L-system approaches focus on the topology by considering a whole organism as an assembly of discrete units or modules [490]. Nevertheless, incorporating geometry, i.e. size and shape, extends the modeling power. Among the geometric interpretations of L-system strings proposed, the interpretation based on the notion of turtle geometry has turned out to be exceptionally useful [499]. The ‘turtle’ is a 3D cursor, whose movements are controlled by predefined L-system symbols acting as commands [500, 501]. The cursor moves in space, tracing the tree skeleton, and is capable of moving while or without drawing a line, turning and rolling left and right, bending up and down, marking the start and end of a branch, as well as setting line width and color. To create a graphical representation of the interpretation, the procedure of L-systems is extended by interpretation rules, which are applied immediately after the production rules. Different from the production rule application, interpretation rules change a predecessor only temporarily for graphical imaging. Environment-based algorithms Contrary to the grammar-based algorithms, environment-based algorithms and their rules do not control the tree structure, but interactions of tree components with the environment, e.g. point clouds from LiDAR (light detection and ranging) data applied in forest inventory [502]. The tree structure is partly given by the point data, though parts may have been hidden underneath other features. Therefore, a combination of point data and general knowledge about tree growth is required to reconstruct the tree structures.

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Xu & Mould [503] first proposed the generation of irregular tree structures by utilizing graphs. An irregular graph is built by creating a Poisson disc distribution of nodes (point data) within a designated volume. The nodes are connected with each other according to a policy defined in advance: One approach is to connect every two nodes that are within a certain distance from each other. While this procedure is generally effective, the appropriate choice of distance threshold is challenging, as it directly affects the procedure’s performance and result. An alternative is presented by the Yao graph, also known as Orthant Neighborhood Graph [504, 505]. Here, the space around a specific node is divided into sections, and the node is connected to the nearest neighbor within each section. This ensures the existence of edges (node connections) in every direction while simultaneously limiting the overall number of edges [503]. After the graph creation, the edges may be assigned random weights. Then, one node is chosen as root point whereas some nodes are marked as endpoints. Finally, the shortest (if no edge weights are assigned) or least-cost path from root to endpoints is determined to create the tree topology. A major challenge in graph-based approaches to tree modeling is the dependency on the graph structure, similar to the mesh dependency of topology optimization methods. Rodkaew et al. [506] propose a technique based on particle systems, which avoids this issue. Originally designed to model the growth of leaves, the goal of the algorithm is to create the leaf vein pattern, which can be adapted to tree structures. From botany, it is assumed that the leaf produces substances (auxins) during photosynthesis, which are then transported towards the plant via the leaf veins. The substances are represented by particles randomly distributed in a closed area, the leaf shape. The particles are iteratively moved towards a target point marking the attachment of the petiole, i.e. the leaf’s stem. If two particles get close, they are merged into the same path rather than creating individual paths for every particle. The direction of the particle movement is defined by the nearest neighbor and the target point. During each iteration, the particle is moved a predefined distance along its direction, and the new direction for the next iteration is computed. Last, depending of the amount of substance associated with the particles during the iterations, the diameter of the veins is assigned: the more substance, the thicker the vein. The particle system approach is further developed by Runions et al. [507]. They inverse the concept of the procedure by assuming that the veins grow towards the substances (auxins). Therefore, leaf venation patterns develop in a feedback process, in which auxin sources control the development of the venation pattern. The veins themselves further influence the placement of auxin sources. This means that after the leaf domain is populated with auxin sources, the vein structure is grown from the target point towards the auxins, which is also the origin of the method’s name: Space Colonization. For open venation, i.e. venation patterns with unconnected ends, the veins are represented by a tree graph 𝒢 = 〈𝑉, 𝐸〉, which is defined by vein nodes 𝑣 ∈ 𝑉. Adjacent nodes are connected by edges 𝑒 ∈ 𝐸 ⊂ 𝑉 × 𝑉. The procedure of [507] consists of three iterative steps:

Digital tree (support) structure generation 1.

Leaf growth

2.

Auxin source placement

3.

Vein node placement

155

Given an initial leaf shape at iteration 𝑖0 , the leaf model is enabled to determine the leaf shape at any iteration 𝑖1 > 𝑖0 by the leaf growth description. In their work, Runions et al. implement the growth descriptions for marginal, uniform, and non-uniform anisotropic growth. Next, with a given leaf shape, the auxin sources are placed. In 𝑖0 , the auxins are placed either randomly or controlled by the user. In any following iteration, auxin sources 𝑠 are assumed to emerge at positions that are farther away from the already existing sources as well as vein nodes than a specified threshold birth distance 𝑑𝑏 . An auxin source remains within the leaf domain until it is removed due to a vein reaching it, i.e. a vein node is closer than a threshold kill distance 𝑑𝑘 . The last step consists of the vein growth simulation by placing new vein nodes. It is assumed that each auxin source influences its nearest vein node, if it is within a defined radius of attraction 𝑟𝑎 . It is possible for a vein node to be influenced by several sources 𝑣 ∈ 𝑉, which is denoted a set of sources 𝑆(𝑣). If 𝑆(𝑣) ⊈ ∅, a new vein node 𝑣 ′ is generated and attached to the previous vein node. The node 𝑣 ′ is placed at a defined distance 𝐷 from 𝑣. The direction of placement is determined by the average of the normalized vectors from the vein node 𝑣 towards the auxin sources 𝑠 ∈ 𝑆(𝑣): 𝑣′ = 𝑣 + 𝐷 ∗

𝒏 , ‖𝒏‖

𝒏= ∑ 𝑠∈𝑆(𝑣)

𝑠−𝑣 ‖𝑠 − 𝑣‖

(6.2)

The described procedure to reproduce leaf venation is adapted by Runions et al. [508] for the modeling of trees. The leaf shape is replaced by a 3D tree crown envelope. Attraction points comparable to auxin sources are populated within the envelope. The procedure differs from [507] in skipping the leaf crown envelope growth. Furthermore, no new attraction points are created during one simulation run. The iterative process of tree skeleton generation terminates when all attraction points have been removed from the corresponding set or a user-defined number of iterations has been exceeded. The algorithm is illustrated in Figure 6.4. The displayed iteration (a) consists of a tree structure defined by four nodes (black) and five attraction points (blue). In the first step (b), the attraction points are associated with the respective nearest tree node, if there is any within 𝑟𝑎 . That way, the sets of attraction points 𝑆(𝑣) influencing the tree nodes are created. Then, the normalized vectors from the tree nodes to the attraction points are determined (c) and averaged for each tree node (d), if there are more than one. Those vectors denote the direction for the placement of new tree nodes, which is done in step (e) by adding new tree nodes at a defined distance from the parent tree node. Next, the attraction points are

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checked whether any tree nodes are within 𝑑𝑘 (f), and eliminated if so (g). The result of step (g) provides the initial state (a) of the next iteration.

Figure 6.4: Space colonization algorithm (adapted from [508])

The space colonization algorithm has been extended to adapt to obstacles [509, 510], and has been successfully applied to real-time tree generation in video games [511] or even the simulation of crowd formation [512], demonstrating the suitability in disciplines unrelated to tree structures. In [502], Favorskaya & Jain benchmark tree modeling approaches, including space colonization, irregular graph structures, and inverse procedural modeling. They found the space colonization to be the most effective algorithm, achieving 75 − 85 % of total similarity with the samples provided.

6.2

Tree support modeling procedure

Part designs in AM are often characterized by flexible, free-form geometries created by generative design methods. Therefore, the use of aperiodic tree structures as supports is required to ensure smooth transitions at the part-support interface. The techniques of the algorithmic botany have been proven to be capable of creating individual trees for various applications. Especially the fact that no discretization of space is required allows for highly individual and optimized structures. The most common approaches are the L-systems and

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157

the space colonization algorithm [510]. While both approaches have been proven to create realistic, but user-controlled trees at reasonable computational expense, the space colonization algorithm is chosen due to the following reasons: First, in space colonization, restrictions and design rules may be implemented directly rather than having to define new grammar rules, reducing the effort of implementation. Second, the growth of tree structures towards a part-support interface can be controlled via the attraction points. In summary, the space colonization algorithm provides a simple implementation of the algorithm itself, and allows for easy user-control. It also requires the slightest amount of discretization and is thus chosen as basic algorithm for the tree support generation.

Figure 6.5: Tree generation procedure

In Figure 6.5, the general procedure of the tree generation step of Figure 3.1 is shown. Circular components represent entities (geometries or files), whereas square components

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indicate functions or subroutines. First, the CAD data is imported, and then the support interface points are created. The third step samples those points into sets for individual trees and calculates the corresponding tree roots. Then the actual tree supports are designed. Last, the results are exported for use in data preparation. The respective functions and subroutines are explained in detail in the following sections. The tree generation procedure is implemented using the Grasshopper Plugin of the CAD software Rhino 6 (Robert McNeel & Associates, Seattle, USA). Grasshopper is a graphical algorithm editor developed for generative design, and is tightly linked to the Rhino CAD modeling tools. While Grasshopper does not require programming skills, it is possible to code own modules in the programming languages Visual Basic, C#, or Python. The combination of simple programmatic CAD modeling via the basic modules and the opportunity to create own procedures where necessary enables geometrical modeling of tree supports using algorithmic techniques different from common CAD modeling. That way, the tree generation is fully automated except for the import of the CAD and simulation result files as well as the export of the tree supports. Note that all modules developed for this thesis are written in Python, which is an arbitrary choice of the author based on the availability of extensive resources regarding the Rhino-specific functions. The Grasshopper procedure requires the part STL file, which includes both the part and design domain geometry, to be opened in Rhino. Similar to topology optimization, the design domain defines the volume tree supports are allowed to be created in and needs to be modeled by the user either directly in Rhino or any CAD software at hand. One requirement regarding the design domain modeling is that part and design domain need to share at least one surface to allow tree structures to grow to the part-support interface. By saving both part and design domain in a single STL file (standard tesselation file format, used for geometry exchange between different CAD softwares), both geometries are already correctly aligned when imported. If two separate STL files exist, the user may have to align the geometries manually.

6.2.1

Data import

In the first step, the required data is imported. For the CAD geometries of part and design domain, two containers for mesh data are provided, since the STL data format represents geometries through a triangular mesh. The two containers need to have the part mesh and design domain mesh assigned, respectively. This is done by right-clicking the container, use the ‘Set one mesh’ option and selecting the corresponding mesh in the Rhino application. After setting the meshes, making them available in Grasshopper, the meshes can be rotated. This may be necessary e.g. because different CAD software use different global coordinate systems, leading to orientation issues when importing STL data. By default, Rhino defines the Z-axis as the vertical axis, equal to the AM coordinate system defined by ISO/ASTM 52900. The axis and angle for the rotation has to be set by the user. After

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159

the optional orientation of the meshes, the meshes are translated to the origin of Rhino’s global coordinate system by evaluating the bounding box, i.e. the cuboid enclosing the meshes, and moving the nodes of the meshes. This repositioning is important to ensure that the volume of the meshes fit the nodes simulation results and superposition of the simulation results on the part mesh is possible. The simulation results (as text files) do not need a specific import routine, only the file paths to the text files have to be set in the corresponding modules: One for the nodes evaluated, one for the corresponding stresses. By default, the delimiter in the node coordinates text file is a comma.

6.2.2

Creation of interface points

The base procedure to create the support-part interface points, i.e. the tips of the tree supports, starts with a rectangular grid of points, which is projected onto the part geometry. The rectangular grid is derived from the bounding box of the design domain: The bottom of the bounding box defines the base plane of the grid. The edge lengths in X- and Ydirection (𝑙𝑒𝑥 ,𝑙𝑒𝑦 ) give the size of the grid in the respective direction. The distance between adjacent points, the so-called hatching, can be set by the user and may differ for X- and Y-direction (𝑑𝑔𝑥 , 𝑑𝑔𝑦 ). Furthermore, an inward offset (𝑑𝑜𝑥 , 𝑑𝑜𝑦 ) of the grid can be used to avoid interface points exactly on the edge of the design domain. The geometric parameters of the grid are shown in Figure 6.6.

Figure 6.6: Parameters of point grid at the bottom face of the design space’s bounding box (top view)

After the grid is created, the points are projected on the part geometry. This is done with a self-written module, see the pseudo-code in Listing 6.1 as well as Figure 6.7 for details. The input of the module includes the part and design domain geometry, as well as the grid points (coordinates). First, it is noted in a separate array if a point is positioned within the

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part geometry. This can occur in areas where the design domain (grey) is not cuboid due to the bounding box (white with dashed contour) used for grid creation. If a point seems to be inside the part geometry (white), it is double-checked with the design domain geometry. This is done because the function used to determine whether a point is inside a geometry considers the point inside if it is exactly on the outer surface. Therefore, if the point is inside both part and design domain geometry, it is exactly in the interface of both and considered outside the part in this module. Listing 6.1: Projection of gridded points to the part geometry

PROJECT POINTS ON PART Grid point coordinates Part geometry Design domain geometry Output: Array of points Input:

1 2 3 4 5 6 7 8 9 10

initialize [support points] global array for every point do check if point is within part geometry project points to part geometry find nearest and second nearest projection point if point is in part geometry then add second nearest projection point to [support points] global array else add nearest projection point to [support points] global array return [support points] global array

Then, the points are projected to the part geometry. The in-built function determines all intersection points of the geometry and a semi-finite line. In consequence, to project the grid points to the nearest surface, the nearest and second nearest projection points have to be noted. Depending on whether the grid point is inside the part geometry or not, the (second) nearest projection point is assigned as interface (support) point, see Figure 6.7. Here, the grey area displays the design domain. The dotted lines indicate the bounding box of the design domain. The (projected) support points are the output of the module. When the point projection is finished, all support points that are outside the design domain are deleted, since they cannot be reached by the tree structure. For further post-processing, all support points are subjected to the PBF-LB/M angle restriction. If the normal vector of the part surface at a specific support point does not violate the angle restriction, the support point is also removed from the array containing those.

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Figure 6.7: Projection scheme Listing 6.2: Accessibility check for interface points to ensure post-processing

ACCESSIBILITY CHECK FOR POINTS Support points at part interface Normal vectors of support points at part interface Ray direction vectors Part geometry Output: Array of accessible points Array of inaccessible points Input:

1 2 3 4 5 6 7 8 9 10

initialize [accessible points, inaccessible points] global arrays convert part geometry (Brep) to part geometry (Mesh) for every point do for every direction do create ray from point and direction if ray does not intersect part mesh then add point to [accessible points] global array continue outer for-loop add point to [inaccessible points] global array return [accessible points, inaccessible points] global arrays

Last, the accessibility of the interface points needs to be ensured for the removal of the tree supports. Similar to the projection, this accessibility check is realized using a raytracing technique provided by Grasshopper. The procedure is implemented in a specifically coded module (cf. Listing 6.2). As input, the support points, the normal vectors of the part surface at the support points, the part geometry, and the ray vectors are required. After geometry type conversion from Brep, which is a 3D solid format native to Rhino, to a mesh in the style of STL, the ray vectors are iteratively positioned at every support point. Rays are created and intersection of the ray with the part geometry is solved. If one of the rays does not intersect with the part, this path is considered free and therefore the point

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accessible by standard removal tools such as pliers or chisels. The custom module outputs two arrays, one holding the accessible points, one the inaccessible. Only the accessible ones are passed to the next step. The ray direction vectors originate in the same point, and are orientated to fill a sphere evenly. The complete set of ray vectors can be considered a ball of vectors. The Grasshopper routine to create equally spaced points on sphere is adopted from [513]. The vectors are then derived from the points on the surface and the center point of the sphere. At the end of the interface point creation, an array holding 3D points is given, who need support due to the overhang angle restriction, are within the design domain and hence reachable by the support structure, and are accessible by tools for support removal. This step can be extended to incorporate the simulation results and adding extra points. Those extra points may indicate stress concentrations requiring additional support even though the overhang angle restriction is not violated. As this is not the case with the demonstration use cases of this thesis, an explicit implementation of those routines is not performed, though.

6.2.3

Tree sampling

After the determination of the interface support points, they are sampled to represent the leaves of individual tree supports. To be able to distinguish individual tree supports, the data structure options of Grasshopper are utilized. Grasshopper features three distinct data structures [514]: single item, list of items, and tree of items. Based on the data structure, modules execute differently. The most important difference, which is also utilized in the tree sampling, is between lists and tree data structures: While a module provided with a list uses all list items in a single operation, the module executes every branch when given a data tree. For example, when the Mass Addition module has a list of numbers as input, it will add up all these numbers, whereas when operating on a tree, the module will output a list of numbers representing the sum of the numbers stored in each branch [514]. While this kind of data structure has a broad range of individual functionalities, within this thesis it is used to store multiple sets of points for the individual tree supports. That way, following operations such as the actual tree topology generation will execute for every defined tree support dataset, similar to a for-loop iterating through a nested array. Other characteristics of tree data structures are not relevant in this thesis, however, the interested reader is referred to e.g. [515]. The support point sampling is realized in a custom module, which is presented in Listing 6.3. The input includes the support points, part geometry, and the simulation results. The first step of the sampling consists of sorting the points according to their grid row into a nested list. In the implementation, grid rows are identified in Y-direction, i.e. points with the same X-coordinate belong to the same row. This can be changed, however, as the

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choice is arbitrary. After sorting the points is finished, the sampling of the respective rows is performed. Sampling can be done with different goals: On the one hand, sampling can aim to create the minimum amount of tree supports. Therefore, the maximum tree size is applied until not enough interface points are left. On the other hand, an even distribution of tree supports can be achieved. Because the design rules determined in Section 5 facilitate an even load distribution, the second approach is chosen. The appropriate number of points per tree support is determined based on the total number of points in the respective row as well as the maximum crown width. A symmetric topology is prioritized. Then, the list containing the current row’s points is divided and the point sets are stored in a data tree structure. The support points represent the leaves of the tree structure. They are therefore referred to as tree leaves. When there are fewer points than the defined number left in the row, the remaining points are assigned to a last tree support. Listing 6.3: Sampling of support points to individual tree input data

POINT SAMPLING TO TREES Accessible support points at part interface Part geometry Simulation results Output: Tree leaves Tree roots Stresses at tree leaves Input:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

initialize [tree leaves, tree roots, stress] global arrays for every point do assign point to list according to grid row for every row do calculate max. number of tree leaves per tree while unprocessed points in row do divide row points into trees with defined number of leaves assign leaves to [tree leaves] global array if less row points than max. number of tree leaves left then assign remaining row points to [tree leaves] global array for every tree do identify stress values at tree leaves derive load-based root assign root to [tree roots] global array assign stresses to [stress] global array return [tree leaves, tree roots, stress] global arrays

Using a rectangular grid and the row-wise sampling approach, the tree supports can be considered 2D trees, as all defining points (leaves, root) share a plane. The choice of avoiding complete 3D tree supports is due to two reasons: First, the support design rules have been developed using a 2D topology optimization setup. To be able to adopt those rules,

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the cross section of the tree in the third dimension thus cannot be changed. Second, a 2D tree support design facilitates manual removal. As consecutive step to the point sorting into the data tree, the corresponding tree roots are calculated. The calculation is based on the simulation results, since the root position has been found dependent on the load distribution (cf. Section 5.2). For every tree support, the stress values at the tree leaves are determined by interpolating the simulation results. Then, the center of load along the Y-direction (horizontal axis of the tree due to row definition along Y-direction) for the tree is derived. The root is then defined by the load center and the bottom of the design space and stored in a second data tree structure, matching the leaves data tree. Finally, the tree leaves, tree roots, as well as the stresses at the leaves are returned by the module.

6.2.4

Tree design

With the tree leaves, i.e. the sampled interface points, and roots required for the space colonization algorithm available, the actual tree design procedure takes place. Again, a custom Grasshopper module is developed.

Figure 6.8: Tree design procedure

An overview of the procedure is given in Figure 6.8. First, the trunk is created by directly applying the design rules of Section 5.2. Then, the complete tree topology with branches

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is generated by the adapted space colonization algorithm. The last step consists of the 3D modeling of the tree topology to provide a manufacturable solid support. The trunk consists of one single line segment defined by the root point and a tree node. The line is significantly longer than the distance between two tree nodes applied in space colonization. Based on the tree leaves (cf. Figure 6.9a), the tree crown width 𝑤𝑐 is calculated. Using 𝑤𝑐 in combination with the crown geometry parameter 𝑟ℎ𝑤 , the corresponding crown height ℎ𝑐 is derived. To finally determine the trunk length, ℎ𝑐 is subtracted from the minimum Z-coordinate of the tree leaves, creating the end point of the trunk. Using this end point and the root as start point, the line is generated (cf. Figure 6.9b).

Figure 6.9: Data preparation steps for space colonization

The space colonization algorithm is applied to derive the tree topology. However, the algorithm described in Section 6.1.3 needs to be adapted to the support design rules. Listing 6.4 provides a summary of the complete, modified algorithm. As input data, the tree leaves, the trunk data, and the segment length, i.e. the distance between two adjacent tree nodes, is required. To ensure that the trunk is not prolonged by the algorithm, a split is enforced prior to the actual space colonization procedure (cf. Figure 6.9c). The split is realized by organizing the tree leaves into two sets, depending on whether they are located left or right of the root. Since the tree leaves have an equal X-coordinate, the left or right positioning is determined via the Y-coordinate. Then, the actual tree topology generation is performed for both sets of tree leaves, taking the upper trunk node as root for the space colonization algorithm. In the end, the results are combined into one (line 26) prior to returning the tree topology to the main function.

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Listing 6.4: Tree topology design by space colonization

TREE TOPOLOGY BY SPACE COLONIZATION Tree leaves Trunk data Segment length Output: Tree topology (straight branches) Tree branch segments Input:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

initialize [tree straight branches, segmented branches, segment directions] global arrays split tree leaves into left and right section for every section of leaves do while not all leaves are reached do for every leaf do find the nearest existing branch segment identify branch tip segments for every branch segment do find all leaves considering the current branch segment as nearest branch if there are leaves then derive new direction vector create new segment if new segment extends existing branch then check design restrictions if valid solution exits then adjust segment (optional) update straight branch else enforce earlier branch initiation else initialize new branch divide base branch into parent and child branch update global arrays identify reached leaves adjust branch tips to leaves update global arrays combine section arrays return [tree straight branches, segmented branches] global arrays

Following the base space colonization algorithm, the nearest branch to every tree leaf is identified. Furthermore, the segments representing the current branches’ tips are noted. After this preparation, the algorithm iterates through every branch segment already created. First, it is checked if there are any leaves that consider the current branch segment end node the nearest one. Given this case, the new direction vector is calculated and a new brand segment is created. The further processing of the new segment depends on whether

Tree support modeling procedure

167

it extends an existing branch, i.e. is connected to a branch tip segment, or initiates a branch split. When a branch is extended by the new segment, a straight branch consisting of one single line is created, and the overhang angle 𝛼𝑡 and angle to parent branch 𝛽𝑡 of this straight branch are checked for design restriction violations (cf. Figure 6.10a). Using this simplification is required as the segments forming a branch change the direction iteratively when influenced by more than one leaf, leading to arched branches at the micro-level. However, for engineering application, a straight branch is desired for clear load transmission. Therefore, the macro-level straight branch is derived from the first node of the first segment and the last node of the last segment. Furthermore, it may be beneficial at this step to ensure that the new branch is still within the design domain. For most cases, this is implicitly realized through the sampling and functionality of the leaves as attraction points. However, if the direct way from trunk to leaves is not free (e.g., tree support is required to grow around a protrusion) or the design domain itself is quite narrow, there is a possibility of design domain violation. Since neither of the described challenges is present in the use cases investigated in this thesis, no explicit design domain restriction is implemented, though. The algorithm adjusts the branch’s direction up to five times by shifting the direction of the last segment 5° towards the global Z-axis, should either of the angle design restrictions be violated. If a valid solution is found, the segment is adjusted accordingly and the current straight branch is updated. Else, the branch origin is moved down by enforcing the split of the parent branch at an earlier segment.

Figure 6.10: Situations encountered during the tree generation by space colonization

If the new segment denotes a branch split, a new branch is initialized (cf. Figure 6.10b). The branch the new segment develops from is divided into a parent branch (segments below split point) and the second child branch (any remaining segments above split point). The global arrays are updated and the procedure continues. No check of the angle restrictions takes place.

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After processing all branch segments existent at the start of the iteration, the leaves are investigated to identify those leaves considered reached (cf. Figure 6.10c). They are then excluded from the following iterations. The procedure is repeated until every leave has been reached. In a last post-processing step, the final branch tips are adjusted to meet the tree leaves exactly (not within the reaching distance) and all global arrays are finalized. When both the left and right side is processed, the individual tree structure sides are combined to a single tree structure (cf. Figure 6.11). The duplicate trunk is removed. This step finalizes the tree topology development.

Figure 6.11: Joining of individual solutions for the left and right tree structure sides

The last major step in the tree support design is the 3D modeling. Based on the tree topology created via space colonization, rectangular cross sections with varying width are applied. The choice of cross section geometry is due to the fact that a 2D topology optimization setup was used to determine the design rules. Transferring a 2D result to a 3D case requires the missing dimension to be constant, which is achieved by fixing the height of the rectangular cross section. Most of the 3D modeling is done within the custom module for tree design. Listing 6.5 displays the respective procedure. Prior to the actual modeling, the load to be compensated by the supports is determined at every interface point by subtracting the yield strength of the material from the stress computed by the process simulation. Support structures need to counter-act plastic deformation, and therefore the major part of the stresses is taken by the material itself. Only stresses exceeding the yield strength are critical. Then, starting at the highest branch level, i.e. at the part-support interface, the branch widths are derived. At the highest level, the branch width is based on the load at the support point. Here, Equation (5.5) is utilized to determine 𝑟𝑙𝑤 . Given 𝑟𝑙𝑤 and the branch length 𝑙𝑡 , the branch width 𝑤𝑡 is determined. If the tree support consists of the trunk only, the application of 𝑟𝑙𝑤 leads to unrealistic widths because of the great length. Therefore, the result of the calculation is reduced with an empirical factor. Furthermore, a minimum branch width restriction is applied to account for manufacturability. After the branch

Tree support modeling procedure

169

widths at the highest level are determined, the remaining branches’ widths are derived by iterating down the tree levels. Here, instead of using the method applied to the highestlevel branches, the width of the parent branch is defined as the sum of the child branches’ widths. The idea behind this is to keep the area of the cross section constant in relation to the load. The parent branch needs to be able to cope with both loads acting on the child branches. In accordance with the observations stated in Section 5.2.2, a factor of 𝑟𝑡 = 0.9 is applied to the sum. Listing 6.5: 3D modeling of the branches’ cuboid surfaces

3D MODELING OF TREE Tree topology Stresses at leaves Yield strength of material Output: Cuboid surfaces of branches Input:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

initialize [3D tree] global array for every leaf do determine load to be compensated by support for every branch starting from the highest branch level do if branch at highest tree level then derive load-based branch width else derive child-based branch width for every branch starting from trunk do create cross section curve at branch end from branch width if branch at highest tree level then translate cross section curve into part geometry along branch direction create cross section curve at branch start from branch width if branch not at trunk level then adjust outer corners of cross section curve to cross section curve of parent branch create ruled surface between both cross section curves add ruled surface to [3D tree] global array return [3D tree] global array

When the branch width assignment is finished, the actual geometrical modeling takes place. As main technique the algorithm uses the CreateRuledSurface function. Here, a surface between two curves is created. The curves may be closed or open, with the first one being the case in this thesis. To be able to apply the function, the cross sections need to be described by curves. Iterating from the lowest (trunk) branch level to the highest, rectangular curves with a fixed height and individual width defined in the prior step are created at the respective branch’s start and end node. The cross sections are oriented parallel to the build platform. Except for the trunk, the outer corners of the rectangles at the branch start node are moved to the corners of the parent’s cross section curve to ensure a

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smooth transition between the branches. At the highest level end node, the cross section curve is further translated into the part geometry along the branch’s direction to ensure complete fusion with the part. All transformations are visible in the validation result (cf. Table 6.3). Then the rules surface is created, which describes the lateral surface of the respective branch. At the branch top or bottom, the geometry is open, making the branches hollow shells at this state. This untrimmed cuboid surfaces are returned to Grasshopper. To create valid 3D geometries, two post-processing steps are added to the tree design procedure. First, the open lateral surfaces are closed to create volumes. Second, a boolean union of the individual branch geometries is performed. The result is a closed 3D model of the tree support.

6.2.5

Data export

To be able to save the tree supports as a CAD geometry importable to data preparation software, the geometries need to be passed from Grasshopper to the Rhino application. Otherwise, only a preview is provided. This is done manually and is called “Baking”. The module holding the final tree support geometries has to be selected; the “Bake” option is located in the context menu. Geometries can be baked to the default or any other custom layer in the Rhino document. Furthermore, if a list of geometries is baked, they can be grouped. Only when baking is done the geometries are actually in the Rhino document and can be selected for further processing, e.g. saving to an STL file. To finalize the tree support generation, both part (which may have been translated or rotated in the early stages) and tree supports need to be baked to the Rhino document. That way, they are correctly positioned towards each other. After baking, the parts are manually selected in Rhino and exported to an STL file.

6.3

Validation of design rule implementation

To validate the correct and appropriate integration of the support design rules into the space colonization algorithm, several topology optimization results are reproduced via the procedure described above. Table 6.3 shows the comparison of two examples: the rectangular design domain of 50 mm 𝑥 100 mm with 0° and 10° overhang angle, respectively. Both times the constant load of 200 MPa is applied. As this load is directly applied to the supports, no subtraction of the yield strength is required. The exact positions of the leaves and root are measured from the topology optimization result. The part (grey) is modeled as simple plate to provide the geometry required for point projection. The design domain is not shown in the tree generation figures for clarity of the visualization. The validation deals with the tree design step only; the required geometries and data are manually modeled and directly connected to the tree design module. Table 6.4 lists the remaining parameters of the tree design algorithm.

Validation of design rule implementation

171

Table 6.3: Comparison of topology optimization and tree generation results for the use case of 𝑎Ω = 100 mm and 𝑏Ω = 50 mm at constant 𝐹𝑡 = 200 MPa, with 𝛼𝑡 = 0° (top) and 𝛼𝑡 = 10° (bottom) Topology optimization result

Tree generation result

172 Blue lines: Tree topology and width measurements Grey lines: Contour of topology optimization result

Support Structure Design Grey: Part geometry Red: Tree support geometry Green lines: Tree topology Dark blue points: Leaves Light blue point: Root

Table 6.4: Tree design parameter in validation Parameter

Value

Segment length

𝑑𝑏

0.1 mm

Reach (kill) distance

𝑑𝑘

0.1 mm

𝑤𝑡,𝑚𝑖𝑛

0.25 mm

Min. branch width

Comparing the tree generation result and corresponding topology optimization, it is concluded that the current implementation is able to reproduce the topology optimization results well. This is true for both tree topology and 3D tree geometry, especially when keeping in mind that the tree topology on the topology optimization result is drawn by hand. Furthermore, the adjustments of the rectangular cross sections described above can be seen. While the inner edge of the branch is parallel to the tree topology line, the outer edge deviates as it is meeting the edge of the parent’s branch. In addition, the branch tips at the highest level are translated into the part for complete fusion. The only significant difference lies in the design of the branch splits. In the topology optimization results, there is more material used to smoothen the transitions. The design seen here can be described mathematically by the method of tensile triangles developed by Mattheck et al. [516, 517] for optimal notch design. Based on observations made in nature and especially from trees, the method of tensile triangles provides notch geometries that reduce the stresses and therefore prevent cracking due to stress concentration at the notch. While the method is widely known and used today, the automatic application is still difficult and hence not realized in this thesis.

7 Support Structure Performance Benchmark The evaluation of current support optimization approaches identified a lack of standardized benchmark procedures for the assessment and comparison of the support’s technical and economic performance (cf. Section 2.5.6). To close this gap and enable the comparison of different supports, a benchmark procedure is developed in this section. The focus of the benchmark procedure lies on the comparison of supports rather than the characterization of specific attributes, e.g. tensile strength. This is because not only the support geometry, but also the support strategy may be optimized, and the overall effect on the part production is of interest. The methodology for the benchmark development is sketched out in Figure 7.1, and is explained in detail in the following sections. The validity of both technical and economic benchmark methods has been proven in preliminary studies [518, 519].

Figure 7.1: Procedure of the benchmark development

7.1

Technical Performance

To give a qualitative assessment of the support performance, benchmark parts are designed in a two-step approach: First, visible and measurable consequences of insufficient support performance are identified to act as benchmark criteria. Second, common geometrical features in need of support are defined. Both aspects are combined into a set of benchmark parts, which further consider additional requirements arising from the benchmark concept. Insufficient support performance regarding the specific tasks may lead to various effects. If overhanging features are not sufficiently supported, the molten material sinks into the powder bed, following gravity. The dimensional error related to this phenomenon is called dross formation. Extensive dross formation increases the thickness of the next layer because of the shift towards the build platform of the present layer. This induces the risk of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_7

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delamination of the adjacent layers because the laser is not able to melt all powder particles as the layer thickness has increased too much. In the worst case, the overhanging feature is not built at all. Insufficient heat dissipation capabilities lead to discoloration, dimensional errors due to sintered powder, and result in distortion due to residual stresses. Missing fixation of the part in space enables the residual stresses to act freely within the material. With increasing stresses, support detachment from the part or build platform, distortion, cracks, and finally, part detachment from the build platform takes place. Analyzing the respective consequences of insufficient support performance all at once (cf. Table 7.1), it is evident that one visible defect may be caused by more than one support task failure. Therefore, a direct evaluation of a specific task is not possible, promoting the approach of qualitatively assessing the overall support performance instead of quantifying single properties. Table 7.1: Defects due to insufficient support performance with regard to the support tasks

Defect Delamination

Overhang support

Heat dissipation

Fixation in space

x

x

x

Distortion due to stresses Support detachment

x

Part detachment

x

Cracks

x

Dimensional inaccurcay

x

Discoloration Build failure

x x

x

Commonly, supports are applied to overhanging features. Those may consist of rectangular or square bar elements, thin walls, or boreholes [28]. In conclusion, surfaces are either flat or curved, and if the size of the element is approaching small dimensions, the surface transforms into an edge. This distinction is important due to the various support types as presented in Section 2.4.1. Additional to the evaluation criteria derived from the consequences of insufficient support performance as well as the base geometrical features, further requirements to the benchmark procedure are defined in Table 7.2. Regarding the supports, a minimal support length is defined to avoid any influence of the build platform on the benchmark parts. This also targets the stability of the optimized supports. Concerning the benchmark parts, it is necessary to take the build envelope of commercial PBF-LB/M systems into account. Small systems, which typically consist of a circular build platform, are e.g. the SLM 125

Technical Performance

175

(SLM Solutions Group AG, Lübeck, Germany), EOS M 100 (EOS GmbH, München, Germany), TruPrint 1000 (TRUMPF GmbH + Co. KG, Ditzingen, Germany), or the AconityMINI (Aconity GmbH, Herzogenrath, Germany). The smallest build volume of the considered systems is with the EOS M 100 system: 100 mm in diameter, and a height of 95 mm. These values define the maximum dimensions for the benchmark parts. Furthermore, the consolidation of all elements into one part imposes the risk of cross-effects. It is therefore beneficial to separate the functions into several benchmark parts, which also goes hand in hand with the dimensional restriction. Table 7.2: Requirement specification of the benchmark procedure

No. Item

Priority

Specification

Note

High

Min. 30 mm

Avoid influence of build platform

Support 1

Support length

Benchmark part 2

Part dimensions

High

Fitting ∅ 100 mm, 95 mm Enable manufacturing in height build volume small systems

3

Geometrical features

High

Contain Curved edges Curved surfaces Straight edges Straight surfaces

Common features

4

Support task inclusion

High

Contain Material accumulation to provoke overheating Free bar elements to provoke distortion

Include extreme conditions regarding support tasks

5

Functional consolidation

Medium Minimal

Avoid cross-effects

Avoid rare or costly techniques

Evaluation 6

Type of measuring techniques

High

Broadly available

7

Number of qualitative methods

Medium Minimal

Reduce subjective evaluation

176 8

Support Structure Performance Benchmark Number of measuring techniques

Low

Minimal

Reduce complexity of procedure

To make the benchmark procedure accessible to as many research groups as possible, it should involve broadly available measuring techniques. With all required equipment at hand, the benchmark can be performed in less time and with less cost. This also induces the goal of minimizing the number of involved measuring techniques. Furthermore, to ensure a high quality and reliability of the benchmark results, the use of quantitative evaluation methods should be reduced to the bare necessity, which results from the compromise of the availability of measuring techniques and the investigated criteria. With the requirements given in Table 7.2, the benchmark parts are designed and the applied evaluation techniques are defined. Even though both processes are performed parallel to each other as the measurement methods do have requirements towards the geometrical design and vice versa, they are presented in two separate sections for structural clarity.

7.1.1

Design of benchmark parts

For the benchmark procedure, five different parts are designed, forming a set of benchmark parts. Each part corresponds to one type of common geometrical feature. The size of the parts is approximately 50 mm in height and 20 mm in thickness, with varying width up to 90 mm. The choice of dimensioning ensures compliance with requirement no. 2. Furthermore, the separation with respect to geometrical feature enables the specific investigation of a single feature if desired. An overview of the parts and the respective characteristics is given in Table 7.3. The order of the parts does not suggest any order in manufacturing. The part S consists of inclined straight overhanging surfaces. The critical overhang angle in PBF-LB/M is below 30° (relative to build platform, [28]), therefore the interval of 0° to 30° is covered in steps of 10°. This approach results in four different surfaces, whose area projected on the build platform is held constant. Consequently, the support height is varying, with a minimum of ℎ𝑠 = 30 mm. To allow for representative size of the supported surfaces, part S is slightly thicker than the standard dimensions stated in the previous paragraph. All surfaces are connected by a middle block. Part E addresses edge overhangs in various ways: There are straight edges, an edge bend in Z-direction, as well as an edge with a curvature in the X-Y plane. By this arrangement, the support’s ability to follow curvatures is tested. To allow broader support arrangements to be examined as well, the edges are designed with triangular extension such that the supports can be attached to the side surfaces of the edges. In addition, the growing volume

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177

in Z-direction challenges the heat dissipation capability of the supports, provoking overheating. Furthermore, to investigate the stability of the supports, straight edges are arranged at different heights (15 mm, 30 mm, 40 mm). Table 7.3: Overview of benchmark parts

h x w x d [mm3 ]

Features

Part S / Straight surfaces 52.5 x 52 x 30

  

Straight surfaces Inclination angle of 0° − 30° Min. support height: 30 mm

Part E / Edges 50 x 66 x 20

   

Straight edges Support heights: 15 mm, 30 mm, 40 mm Edge with curvature in horizontal direction Edge with curvature in vertical direction

Part C / Curved surfaces 57 x 90 x 20

 

Holes (curved surface) Diameters: 56 mm, 36 mm

Part ST / Straight structural transition 50 x 25 x 20

 

Structural transition (edge) Min. support height: 33 mm

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Support Structure Performance Benchmark Features

h x w x d [mm3 ] Part CT / Curved structural transition 50 x 25 x 20

 

Structural transition (curved) Min. support height: 33 mm

The third part, part C, includes curved overhanging surfaces, i.e. holes. To be able to perform the first evaluation step of visual inspection (cf. Section 7.1.2), the features are aligned such that the symmetry axes are parallel. This arrangement leads to a restriction regarding the number of different diameters to not violate the dimensional requirements in Table 7.2. Therefore, two different diameters of 36 mm and 56 mm are applied. The smaller diameter is chosen such that self-support of the geometry due to the gradient of the surface is avoided, and the requirement of minimum support height is obeyed. The second diameter is defined with a significantly larger diameter to provide a different curvature to the benchmark. Part ST and CT represent structural transitions, where thin walls merge together with a significant volume on top. The geometries are adopted from [373]. A continuous as well as a discontinuous transition is included. The volume on top on the one hand challenges heat dissipation, and on the other hand applies an eccentric load to the thin walls, testing the support’s performance under compression. The exact dimensions of all benchmark parts are given in Appendix A.2.

7.1.2

Definition of measurement methods

Except for the dimensional accuracy, every defect listed in Table 7.1 can be observed visually. To keep the number of experimental measurements minimal, a two-step evaluation procedure of the technical support performance is designed. The first stage consists of a qualitative visual inspection of the parts, where the existence of the mentioned defects is noted by counting the occurrence in the five benchmark parts. While there is no further distinction for cracks, notches, and part or support detachment, the discoloration is rated as ‘light’ or ‘strong’. This differentiation is important as the discoloration is a visual hint at a microstructure, which is different from the standard due to the overheating, but not accommodated with material failure.

Economic Performance

179

The second evaluation step consists of the quantitative assessment of the criteria. To incorporate requirements no. 4 and 5, this stage is restricted to the evaluation of the dimensional accuracy, because it can be measured via imaging techniques or coordinate measuring machines with no need for complex experimental setups. In part S, the deviation of the 0° overhanging surfaces’ angle is measured. Furthermore, the edge deviation at the end of the beam is investigated to look for possibly accumulated particles indicating heat dissipation issues. The diameter accuracy of part C is included to evaluate dross formation at the top of the holes. Last, the deviation of the thin sidewalls in part ST and CT is measured at the lower and upper end, and the mean value is noted as an indicator of the structural transition’s quality. Part E is not included in this evaluation step as the edges do not provide measurable features leading to further insights regarding the support performance.

7.2

Economic Performance

When assessing the feasibility of an optimized product, cost evaluation is a key factor [520]. This is also true for optimized supports since the trade-off of increased performance and costs has to be considered before application. Therefore, cost models of the PBFLB/M process are analyzed concerning their support consideration, and a specific support cost model is developed.

7.2.1

Cost model for support structures in PBF-LB/M

Of the cost models presented in Table 2.7, various consider supports. However, this is done mainly by including the support volume [364, 367, 373, 377–379, 384] or time for removal [367, 369, 380, 381, 384]. Schröder et al. [382] as well as Rudolph & Emmelmann [365] simply apply a factor, which is not further specified, to the overall costs. In [367, 373, 381, 384], the time for the support design is also taken into account. These simplifications in the support-induced cost consideration are not able to accurately determine the real cost as the related cost may vary significantly throughout production steps, hence raising the need to consider every affected step: A part with large support volume may require a short removal time because the ease of removal is high, while a part with only few, but hard to reach supports will need an extensive amount of time for support removal. This variance from one part to another also prohibits the use of constant factors for precise support cost determination. Only Alexander et al. [384] as well as Piili et al. [376] include the supports in more detail than just the reference parameters mentioned above. In conclusion, support integration is not sufficient in existing cost models. Furthermore, the predictive models lack detail and validation versus calculative cost models. As no specific support cost model is available in the literature, an own model is developed. Even though the support cost prediction would be a useful tool in optimization, a calculative model is derived to avoid the issues of cost prediction approaches outlined in Section 2.6.

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Support Structure Performance Benchmark

In combination with the benchmark procedure presented in Section 7.1 this is deemed sufficient because the required production of the benchmark parts enables the direct assessment of time-based activities. As the most recent cost model with all production phases relevant to supports included (data preparation, manufacturing, post-processing), the model of Kranz [373] is chosen as base model. The different cost items provided are analyzed regarding possible influence of supports. In Figure 7.2, the general structure of the support cost model is pointed out, including an indication of the type of the respective cost factors. In the following section, the corresponding functions are determined to give a complete support cost model.

Figure 7.2: Support cost model structure and corresponding cost types

The additional costs caused by supports for a single part are categorized with respect to the corresponding production phase, i.e. support design phase costs (𝐶𝐷 ) occurring during the general PBF-LB/M preparation phase, manufacturing phase costs (𝐶𝑀 ), and post-processing phase costs (𝐶𝑃 ). The sum of each phase’s costs yields the overall support cost 𝐶𝑆 . 𝐶𝑆 [€] = 𝐶𝐷 [€] + 𝐶𝑀 [€] + 𝐶𝑃 [€]

(7.1)

The specific costs of the three phases are detailed in the following. Three general types of costs occur repeatedly. Those cost types are briefly described here to avoid repetitive explanations. Adopting the activity-based approach, each type contains a time component as well as an hourly rate specific to the type of cost. The first type is the operator cost (𝐶𝑜 ), denoting the involvement of personnel (engineer, technician, etc.). The cost rate consists of the gross wage of the employee 𝑐𝑤,𝑔 and the incidental wage 𝑐𝑤,𝑖 paid by the employer, and is related to the labor time 𝑡𝑙 : € € 𝐶𝑜 [€] = 𝑡𝑙 [h] ∗ (𝑐𝑤,𝑔 [ ] + 𝑐𝑤,𝑖 [ ]) h h

(7.2)

Economic Performance

181

Usually, the gross wage is given as an annual value rather than an hourly rate, together with the required work hours per week (𝑡𝑊𝑒𝑒𝑘 ). In that case, it can be translated to an hourly notation by mapping the annual gross wage starting from the workdays per year (𝑑𝑌𝑒𝑎𝑟 ) to the workdays per week (𝑑𝑊𝑒𝑒𝑘 ), as stated in Equation (7.3).

𝑐𝑤,𝑔

d 𝑑𝑊𝑒𝑒𝑘 [ ] € € week [ ] = 𝑐𝑤,𝑔 [ ] ∗ d h h a 𝑑 𝑌𝑒𝑎𝑟 [ a ] ∗ 𝑡𝑊𝑒𝑒𝑘 [week]

(7.3)

The second general type of cost describes the cost of the used technical equipment (𝐶𝑒 ). It is composed of the capital investment 𝐶𝑖 as well as the amortization period 𝑡𝑎 . It is then related to the time in use (𝑡𝑢,𝑒 ) and the overall utilization rate 𝑢𝑒 of the respective equipment to exactly determine the share in equipment costs, as equipment usually is applied to various tasks throughout its lifetime. 𝐶𝑒 [€] =

𝑡𝑢,𝑒 [h] 𝐶𝑖 [€] ∗ h 𝑢𝑒 [ ] 𝑡𝑎 [a] a

(7.4)

The last general cost type is the room cost (𝐶𝑟 ) associated with renting office or workshop space. It is determined by the area of space (𝐴) required and the respective costs, which are composed of the actual rent (𝑐𝑟,𝑟 ) and the operational costs (𝑐𝑟,𝑜 ) including additional expenses for electricity, heat, furniture, and maintenance. If the property is owned, not rented, it may be substituted by an equipment cost. Similar to 𝐶𝑒 , 𝐶𝑟 is related to the time in use (𝑡𝑢,𝑟 ) and the overall utilization of the room (𝑢𝑟 ). € € 𝑡𝑢,𝑟 [h] 2 2 m m 𝐶𝑟 [€] = ∗ 𝐴 [m2 ] ∗ (𝑐𝑟,𝑟 [ ] + 𝑐𝑟,𝑜 [ ]) h a a 𝑢𝑟 [ ] a

(7.5)

The annual expenses in terms of rent and operating costs are derived by summing up the monthly payments.

𝑐𝑟,𝑟

𝑐𝑟,𝑜

€ € months 2 2 m m [ ] = 𝑐𝑟,𝑟 [ ] ∗ 12 [ ] a month a € € months 2 2 m m [ ] = 𝑐𝑟,𝑜 [ ] ∗ 12 [ ] a month a

(7.6)

(7.7)

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Support Structure Performance Benchmark

Design phase costs To design supports during the data preparation step before manufacturing, an engineer or technician uses specialized software for a specific time. As demonstrated in Section 2.5, support optimization is done mainly computationally, too. Consequently, the costs of the design phase are composed of the operator cost (𝐶𝐷,𝑜 ), required equipment cost (𝐶𝐷,𝑒 ), and the necessary office space cost (𝐶𝐷,𝑟 ). 𝐶𝐷 [€] = 𝐶𝐷,𝑜 [€] + 𝐶𝐷,𝑒 [€] + 𝐶𝐷,𝑟 [€]

(7.8)

Applying Equations (7.2), (7.4), and (7.5), the exact costs are determined. Note that the time to position a part on the build platform is not included in 𝐶𝐷 , because this step is assumed independent of the support design. However, the method of support design may lead to different times applied to the operator cost compared to the equipment and room costs: Computational methods such as topology optimization require extensive computational times, but do not need to be constantly monitored by the operator. Therefore, the operator cost is derived by applying the time of operator occupation 𝑡𝐷,𝑙 to Equation (7.2), while 𝐶𝐷,𝑒 and 𝐶𝐶,𝑟 are determined using the time of equipment occupation 𝑡𝐷,𝑒 . When standard supports already implemented in the software are used, 𝑡𝐷,𝑙 = 𝑡𝐷,𝑒 applies. The equipment consists of the software (𝑆𝑊) and the computer hardware (𝐻𝑊) required to run the software. The hardware cost is a standard equipment type, but the software cost may appear in two different ways: If bought, the software is treated as equipment. However, software is often distributed by applying a licensing scheme for a restricted period. In that case, the license cost substitutes the investment cost related to the amortization period. Since the licensing of software is the common way of software distribution today, it is prioritized here. Indicating units only when deviating from the general definitions of Equation (7.2), (7.4), and (7.5), the different cost items of the support design phase are defined as follows: 𝐶𝐷,𝑜 [€] = 𝑡𝐷,𝑙 ∗ (𝑐𝐷,𝑤,𝑔 + 𝑐𝐷,𝑤,𝑖 )

(7.9)

𝐶𝐷,𝑒 [€] =

𝑡𝐷,𝑒 € 𝑡𝐷,𝑒 𝐶𝑖,𝐻𝑊 ∗ 𝑐𝑆𝑊 [ ] + ∗ 𝑢𝐷,𝑒 𝑎 𝑢𝐻𝑊 𝑡𝑎,𝐻𝑊

(7.10)

𝐶𝐷,𝑟 [€] =

𝑡𝐷,𝑒 ∗ 𝐴𝑜 ∗ (𝑐𝑟,𝑟,𝑜 + 𝑐𝑟,𝑜,𝑜 ) 𝑢𝐷,𝑜

(7.11)

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Manufacturing phase costs Following the approach of [373], the manufacturing phase costs are split into the material costs (𝐶𝑀,𝑚 ) and the production costs (𝐶𝑀,𝑝 ). The material costs take into consideration the raw material consumption of the supports as well as the possible sale of the detached supports as scrap metal. The difference in purchase (𝑐𝑚𝑎𝑡 ) and sale price (𝑐𝑠𝑐𝑟𝑎𝑝 ) is related to the weight of the supports, which is derived from the material’s density 𝜌𝑀,𝑠 and the respective support volume 𝑉𝑀,𝑠 . Note that this procedure underestimates the costs slightly, because it may not be possible to collect the whole support volume for reselling. 𝐶𝑀,𝑚 [€] = 𝜌𝑀,𝑠 [

kg € € ] ∗ 𝑉𝑀,𝑠 [m3 ] ∗ (𝑐𝑚𝑎𝑡 [ ] − 𝑐𝑠𝑐𝑟𝑎𝑝 [ ]) 3 m kg kg

(7.12)

The production costs combine all costs related to the manufacturing system, i.e. machine operating costs and consumables (filter, gas, energy) required for the PBF-LB/M process. The equipment, room, and service cost (𝑐𝑀,𝑠𝑒𝑟 ) of maintenance are summed up to yield the machine operating costs: 𝐶𝑀,𝑝,𝑚 [€] =

𝑡𝑀,𝑏 𝐶𝑖,𝑚 𝑡𝑀,𝑏 𝑡𝑀,𝑏 € ∗ + ∗ 𝐴𝑚 ∗ (𝑐𝑟,𝑟,𝑚 + 𝑐𝑟,𝑜,𝑚 ) + ∗ 𝑐𝑀,𝑠𝑒𝑟 [ ] 𝑢𝑀,𝑚 𝑡𝑎,𝑚 𝑢𝑀,𝑚 𝑢𝑀,𝑚 a

(7.13)

The build time 𝑡𝑀,𝑏 is a crucial factor to Equation (7.13) because of two possible influences of supports: First, the support volume has to be built in addition to the actual part. Second, supports can be used to set off the part from the build platform to facilitate easy detachment. In that case, the support application leads to additional layers to be manufactured, which have to be included in the build time as additional recoating occurs. The exact amount of extra layers due to supports is calculated from the difference in build height (Δℎ) due to the offset and the layer thickness (𝑡ℎ). For each layer, the time needed for depositing a new powder layer (𝑡𝑀,𝑝 ) and a possible in-built system delay 𝑡𝑀,𝑑𝑒𝑙 between layers, which some manufacturing systems apply to ensure adequate cooling of the previous layer, are considered. 𝑡𝑀,𝑏 [h] =

𝑉𝑀,𝑠 [m3 ] Δℎ [m] + ∗ (𝑡𝑀,𝑝 [h] + 𝑡𝑀,𝑑𝑒𝑙 [h]) m3 𝑡ℎ [m] 𝑀̇ [ ] h

(7.14)

If the melt rate 𝑀̇ is not known, it is determined from the scan velocity (𝑣𝑠 ), hatch distance (𝑑ℎ ), layer thickness (𝑡ℎ), and the number of lasers (𝑛𝑙 ) equipped in the manufacturing system or utilized in the build job, as the build platform may not be fully packed. An

184

Support Structure Performance Benchmark

important assumption regarding Equation (7.15) is the equal distribution of volume in the build chamber, i.e. all active lasers operate simultaneously. 𝑀̇ [

m3 m ] = 𝑣𝑠 [ ] ∗ 𝑑ℎ [m] ∗ 𝑡ℎ [m] ∗ 𝑛𝑙 h h

(7.15)

The costs of consumables incorporate the cost of the processing gas (𝐶𝑀,𝑝,𝑔 ) and the respective filter ( 𝐶𝑀,𝑝,𝑓 ), as well as the energy supplying the manufacturing system (𝐶𝑀,𝑝,𝑒 ). The filter is treated as standard equipment. The gas and energy costs are directly derived from the market price (𝑐𝑔 and 𝑐𝑒 ) as well as the gas flow of the manufacturing ̇ system 𝑓𝑀,𝑔 and the energy consumption 𝐸𝑀,𝑚 , respectively. Equation (7.16)-(7.18) assume that only the part investigated is manufactured in the build job. If this is not the case, the consumables costs are evenly distributed across all 𝑛 produced parts according to their respective volume. ̇ 𝐶𝑀,𝑝,𝑔 [€] = 𝑡𝑀,𝑏 [h] ∗ 𝑓𝑀,𝑔 [

𝑚3 € ] ∗ 𝑐𝑔 [ 3 ] ℎ 𝑚

€ 𝐶𝑀,𝑝,𝑒 [€] = 𝑡𝑀,𝑏 [h] ∗ 𝐸𝑀,𝑚 [kW] ∗ 𝑐𝑒 [ ] kWh 𝐶𝑀,𝑝,𝑓 [€] =

𝑡𝑀,𝑏 𝐶𝑖,𝑓 ∗ 𝑢𝑀,𝑚 𝑡𝑎,𝑓

(7.16) (7.17) (7.18)

The overall manufacturing phase costs are determined by the sum of the different cost items. 𝐶𝑀 [€] = 𝐶𝑀,𝑚 [€] + 𝐶𝑀,𝑝,𝑚 [€] + 𝐶𝑀,𝑝,𝑔 [€] + 𝐶𝑀,𝑝,𝑒 [€] + 𝐶𝑀,𝑝,𝑓 [€]

(7.19)

Post-processing phase costs During post-processing, the part and supports are detached from the build platform, followed by the support structure removal. Because both tasks may be performed in various ways, individual cost structures are required. In the context of this work, the following assumptions are made: 1.

Detachment is realized by wire eroding.

2.

Supports are removed manually.

In wire eroding, additional effort caused by supports is present when the part is oriented on the build platform such that supports increase the eroding distance 𝑑𝑃,𝑤 . Taking the extra time for wire eroding (𝑡𝑃,𝑤 ) defined by the wire eroding velocity 𝑣𝑃,𝑤 into account,

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185

the respective costs are derived in a manner similar to the production costs. Differentiating machine and consumable costs, the wire eroding costs consist of equipment, room, service, and energy cost. 𝑡𝑃,𝑤 [ℎ] =

𝑑𝑃,𝑤 [𝑚] 𝑚 𝑣𝑃,𝑤 [ ] ℎ

𝐶𝑃,𝑝,𝑤 [€] =

(7.20)

𝑡𝑃,𝑤 𝐶𝑖,𝑤 𝑡𝑃,𝑤 𝑡𝑃,𝑤 € ∗ + ∗ 𝐴𝑤 ∗ (𝑐𝑟,𝑟,𝑤 + 𝑐𝑟,𝑜,𝑤 ) + ∗ 𝑐𝑃,𝑠𝑒𝑟 [ ] 𝑢𝑃,𝑤 𝑡𝑎,𝑤 𝑢𝑃,𝑤 𝑢𝑃,𝑤 a

(7.21)

The energy cost of the wire eroding is defined by the following Equation (7.17): 𝐶𝑃,𝑤,𝑒 [€] = 𝑡𝑃,𝑤 ∗ 𝐸𝑃,𝑤 [kW] ∗ 𝑐𝑒 [

€ ] kWh

(7.22)

The manual support removal requires a technician to use a workshop and different tools for a specific time 𝑡𝑃,𝑟 . Therefore, the removal costs include the labor costs of the technician, the equipment cost of the tools, as well as the room costs of the workshop. The workshop and tools may be used for other activities, too, so the assignment of the respective utilization rates 𝑢𝑃,𝑤𝑠 and 𝑢𝑃,𝑡 is necessary. Following the general definitions of cost types, the cost items for the support removal costs are defined as 𝐶𝑃,𝑟,𝑜 [€] = 𝑡𝑃,𝑙 ∗ (𝑐𝑃,𝑤,𝑔 + 𝑐𝑃,𝑤,𝑖 )

(7.23)

𝐶𝑃,𝑟,𝑒 [€] =

𝑡𝑃,𝑙 𝐶𝑖,𝑡 ∗ 𝑢𝑃,𝑡 𝑡𝑎,𝑡

(7.24)

𝐶𝑃,𝑟,𝑟 [€] =

𝑡𝑃,𝑙 ∗ 𝐴𝑤𝑠 ∗ (𝑐𝑟,𝑟,𝑤𝑠 + 𝑐𝑟,𝑜,𝑤𝑠 ) 𝑢𝑃,𝑤𝑠

(7.25)

The overall post-processing costs equal the sum of the different cost items. 𝐶𝑃 [€] = 𝐶𝑃,𝑝,𝑤 [€] + 𝐶𝑃,𝑤,𝑒 [€] + 𝐶𝑃,𝑟,𝑜 [€] + 𝐶𝑃,𝑟,𝑒 [€] + 𝐶𝑃,𝑟,𝑟 [€]

(7.26)

A detailed cost modeling requires a great variety of parameters. An overview of all values necessary for the complete cost calculation is given in the table of Appendix A.3. Furthermore, the table provides values and their references applied in this work.

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Support Structure Performance Benchmark

7.2.2

Procedure for quick cost evaluation during the benchmark procedure

For the use of the full cost model developed in the previous section, a considerable number of parameters has to be known and evaluated during the benchmark procedure. If the economical point of view is not of high importance or some parameters are not known, the cost may be approximated by the evaluation of the significant cost drivers. This includes the time for activities involving personnel, namely the time for support design as well as support removal, material consumption, the ease of support removal, and the resulting surface quality. Personnel involvement is costly due to the high cost rates evolving from the wages. The material consumption including possibly trapped powder directly relates to the material costs, and indicates the additional build time. The ease of support removal is an important criterion regarding the post-processing effort at a given support volume. Last, the surface quality after support removal indicates the required surface finishing effort. While the evaluation of these parameters is by no means a detailed cost assessment, as can be easily identified when comparing to the complete cost model, it provides a feasible indication of cost development tendencies in the respective production phases. The times for support design 𝑡𝐷,𝑙 and removal 𝑡𝑃,𝑙 are tracked by the respective person using timers. 𝑡𝑃,𝑙 is further differentiated in the time for the sole support removal 𝑡𝑃,𝑙,𝑟𝑒𝑚 and the time needed to remove any residuals of the supports 𝑡𝑃,𝑙,𝑟𝑒𝑠 . The goal of the postprocessing efforts is a surface state where the support influence on the part surface does not affect further surface finishing by sandblasting, which is assumed a post-processing step independent of the supports. For material consumption evaluation, the benchmark parts are weighted prior and after support removal. By considering the difference of both values as overall material consumption, powder trapped within the support structures is included. The ease of removal as well as the surface quality afterwards is assessed via qualitative methods. The ease of removal is rated by the operating person according to the scale applied by Gralow et al. [274] and the respective values is assigned: (1) Close to no resistance (very easy to remove) (2) Low resistance (easy to remove) (3) Medium resistance (moderate effort to remove) (4) High resistance (hard or impossible to remove) The evaluation of the surface quality follows a similar approach. Here, the surface is visually inspected after support removal, but before the second post-processing step of residual removal. A rating according to the following scale is performed: (1) Residuals that can be removed by sandblasting (no further effect of supports) (2) Residuals that need to be removed by files (additional post-processing effort required)

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(3) Existence of craters, also known as ‘pitting’ (no surface quality enhancement possible) The use of qualitative scales enables easy, quick comparison. However, while the assessment can be aided by the provision of references in the evaluation sheet, it has to be kept in mind that these evaluations heavily rely on the person performing them when comparing different benchmarks. It is therefore advised to always include a reference support well known, such as the block or pin support. This advice is adhered in the demonstration of this thesis’ work (cf. Section 8). Here, the use of the quick assessment is also shown with the real-life application. Hence, examples for the different scales defined in this section are given in Section 8.

8 Demonstration of algorithmic support structures To exploit the capabilities of the tree supports developed in this thesis, the benchmark procedure presented in Section 7 is utilized. The full digital tree support design process sketched out in Section 6 (process simulation, support generation) is described. The parts are manufactured and evaluated following the benchmark procedure. The material applied is Ti-6Al-4V as already used throughout the thesis. The tree supports are compared to the standard block and cone supports most commonly applied in industrial applications. Note that in the following benchmark all measured times will be given in minutes. Due to the uncertainties related to manual work and time keeping, detailing results at the level of seconds is considered pseudo-accurate with no valid information.

8.1

Process simulation

To determine the process-induced stresses in the benchmark parts, part-scale process simulation is utilized. The simulation procedure is presented in Figure 8.1. The benchmark parts’ manufacturing processes are computed using the commercial software Simufact Additive 2020 FP1 (Simufact Engineering GmbH, Hamburg, Germany). The results are exported to a .csv spreadsheet, which is then further processed to separate mesh coordinates and the corresponding stresses using a MATLAB R2021a (MathWorks Inc., Natick, USA) script. For each benchmark part, an individual study is conducted. The material model of Section 4 is employed.

Figure 8.1: Procedure of PBF-LB/M process simulation and result post-processing

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_8

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Demonstration of algorithmic support structures

8.1.1 Simulation setup Simufact Additive is based on the inherent strain method, and follows the experimental calibration method of Setien et al. [212]. A pre-defined cantilever is manufactured with two different orientations on the build platform (along X- and Y-direction). Then, the supports of the cantilever are cut while the cantilever remains on the build platform, and the maximum displacement in Z-direction is measured. Both values of Z-direction deformation are the input of the software’s calibration routine, as well as initial guesses of the inherent strain vector. The routine then adjusts the vector to meet the experimental Zdisplacement. The exact routine is not disclosed, though. The software has been calibrated for the material and process parameter combination used for the benchmark (also applied in material modeling experiments, cf. Table 4.2) in prior works, and is available ready-touse. The part STL file is imported to Simufact Additive and positioned at the center of the virtual build platform. From all the process steps that may be included in the simulation (manufacturing, heat treatment, part detachment, support removal), only the manufacturing step is considered. No supports are assigned to the parts. Due to this, the computing of part E was not possible as the overhanging edges are hanging in free space to the simulation, which the software considers not realistic and refuses to solve. Hence, no PBF-LB/M simulation results are available for part E. After this general setup, the voxel mesh is defined. A uniform voxel mesh with an individual voxel size is set, see Table 8.1 for details. Following the guidelines of Simufact Additive, the voxel sizes of part ST and CT are chosen to match their wall thicknesses. To maintain comparability, the same voxel size is also applied to the other parts. However, this voxel size resulted in computation termination for part S because of difficulties in adding a new layer to the underlying distorted voxel mesh. Therefore, the voxel size automatically chosen by Simufact Additive is applied here, which is larger, resulting in significantly less computational layers. Furthermore, Simufact Additive uses adaptive meshing to reduce computational efforts. In adaptive meshing, only regions with small features or near the load application area (top layer) are discretized by the small voxels defined by the user. Larger regions not in direct contact with the applied loads are re-meshed and discretized by elements of a larger size. This procedure reduces the number of elements and therefore the number of nodes to be considered in the calculations. The coarsening of the mesh, i.e. the adjustment of element size, can be continuous, resulting in a deformed mesh. In Simufact Additive, the adaptive meshing is realized with a h-type re-meshing algorithm [521] that preserves the shape of the original elements, though. Here, larger elements are subdivided into smaller elements and vice versa. The number of different element sizes is defined by the level of coarsening, which may be set by the user.

Process simulation

191

Table 8.1: Mesh parameters of process simulation applied to the benchmark parts (cf. Table 7.3)

Voxel size [mm]

No. of voxels

Level of coarsening

No. of layers

S

0.797512

25,084

2

66

E

-

-

-

-

C

0.5

197,080

2

116

ST

0.5

68,880

2

100

CT

0.5

58,080

2

100

Part

Table 8.2: Computation times for PBF-LB/M process simulation and post-processing of the benchmark parts (cf. Table 7.3)

Setup time [min] Part

Computation time [min]

Simulation

Post-processing

Simulation

Post-processing

S

2

1

5

18

E

-

-

-

-

C

2

1

66

199

ST

2

1

20

44

CT

2

1

15

38

Σ

8

4

106

299

The simulations are run on the computer used for topology optimization (specifications stated in Table 5.4). The time required for the setup and computation of the respective studies is given in Table 8.2. The results are loaded into the in-built Arctool of Simufact Additive, which allows exporting the numerical results to spreadsheets. For each iteration of adding a computational layer, an individual result file is created. The spreadsheets are then processed with the script summarized in Listing 8.1. First, the data of the point coordinates and the effective (von Mises) stresses are extracted from the current spreadsheet. Because Simufact Additive stores the distorted mesh’s coordinates (given that any deformation is present), the current point coordinates are mapped to the initial map. That way, it is possible to identify and update individual nodes. The maximum stress of each node is saved. If a node is not existent in the current iteration due to the mesh coarsening, its value is interpolated from the nearest nodes in Z-direction. Finally, the maximum stresses and point coordinates are written to separate text files. The point coordinates are corrected by the translation due to the positioning at the build platform center to match the original coordinate system. The time required for the post-processing of the simulation results is given in Table 8.2. It is evident that the processing of the results required nearly triple the

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Demonstration of algorithmic support structures

amount of time of the actual simulations. This is assumed to be caused by the script not being optimized for computational efficiency. Furthermore, reading files requires many resources. Integrating the tracking of the maximum stress throughout the manufacturing process directly into the PBF-LB/M process simulation, as may be possible with Simufact Additive’s option to program custom scripts, could increase the computational efficiency. Listing 8.1: Processing of simulation results

SIMULATION RESULT PROCESSING Number of iterations in simulation Translation of part geometry in Simufact Additive in x,y-direction File path to simulation results File path of export files Output: Point coordinates Max. stress occurring at point during simulation Input:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

initialize [point coordinates, max. stresses] global arrays for every simulation iteration result do read .csv file extract [point connectivity, point coordinates, x,y,z displacement, effective stresses at points] to respective arrays map current to initial point coordinates by displacement values for points not in global, but in current array do add [point coordinates, effective stresses] to global arrays for points in current and global array do if effective stress > max. stress then update global array for points in global, but not in current array do interpolate value in current iteration if interpolated stress > max. stress then update global array remove point coordinate translation of Simufact Additive write [point coordinates, max. stresses] global arrays to separate text files

8.1.2 Results & discussion The results of the PBF-LB/M process simulations and the corresponding post-processing are presented in Table 8.3, which shows the local maximum stress throughout manufacturing. For ease of comparison, the color scale is the same for all figures. In part S and C, the staircase effect is clearly visible as stripes of high stresses denoting the overhang ends’ upper edge of the respective layers. Following the TGM theory, the stresses are at a maximum at the overhang ends’ upper edge because the heat dissipation is not as efficient as in regions closer to the main body, which leads to heat accumulation and corresponding thermal gradients. Interestingly, the stresses in the 0° overhang of part S are comparatively low, although the heat dissipation rate is the lowest of all overhangs. This is attributed to

Process simulation

193

the small thickness of the overhang and the corresponding short time of exposure to thermo-mechanical loads, as stress development is a time-dependent process. The overhang beam consists of only two layers of voxels. The maximum stress in part C is slightly higher than in the others, which are approximately in the same range. However, the total difference is small compared to the magnitude of the stress values, which are given in GPa. Furthermore, especially in part S, the interpolation scheme to update all nodes despite the adaptive meshing did not catch all nodes. This is visible by individual points not matching the surrounding values. Table 8.3: Overview of maximum stresses calculated by Simufact Additive

Maximum stress Part S / Straight surfaces 𝜎𝑚𝑎𝑥,𝑆 = 1.235 ∗ 109 Pa

194

Demonstration of algorithmic support structures Maximum stress Part C / Curved surfaces 𝜎𝑚𝑎𝑥,𝐶 = 1.307 ∗ 109 Pa

Part ST / Straight structural transition 𝜎𝑚𝑎𝑥,𝑆𝑇 = 1.266 ∗ 109 Pa

Support generation

195 Maximum stress Part CT / Curved structural transition 𝜎𝑚𝑎𝑥,𝐶𝑇 = 1.205 ∗ 109 Pa

8.2

Support generation

Because the benchmark procedure is set up as comparison rather than a quantitative assessment of technical properties, four different support configurations are investigated. On the one hand, the standard block and cone supports created without any manual adjustments are included. They provide a reference since those types of support are well known due to their broad application, see Section 2.4. Furthermore, they set up the scenario of an inexperienced operator, who is not proficient in the numerous input parameter to those support types. On the other hand, two different configurations of the algorithmic tree supports are added to the study: One tree support designed as similar as possible to the standard supports in terms of geometrical parameters. The second tree support aims at highlighting the optimization potential.

8.2.1 Block & cone supports The standard block and cone supports are generated via the data preparation software Materialise Magics. The parts are positioned directly on the build platform, i.e. supports connecting the part and build platform to ease part removal are prevented. The automatic support generation tool is used. Here, the surfaces in need of support are detected according to the overhang angle. By default, block supports are applied, the type of support can

196

Demonstration of algorithmic support structures

be changed after initial generation, though. This is done for the cone supports. The block support is created with a X,Y-hatching of ℎ𝑥𝑦,𝑏 = 1.5 mm (distance between walls in Xand Y-direction). Furthermore, the block support is fragmented every 𝑑𝑓 = 4.5 mm with a gap width of 𝑤𝑔 = 0.3 mm. The walls are manufactured by single laser passes, resulting in a wall thickness of about 0.15 mm. The cone supports are positioned with a distance of ℎ𝑥𝑦,𝑐 = 2 − 4 mm to each other. The diameter at the part-support interface is 𝑑𝑐,𝑖 = 0.1 − 0.5 mm, whereas the diameter at the bottom (build platform or part) may be slightly higher with 𝑑𝑐,𝑏 = 0.1 − 0.8 mm. Table 8.4: Overview of generated block and cone supports

Block support

Cone support Part S / Straight surfaces

𝑉𝑆,𝐵 = 15,376 mm³

𝑡𝐷,𝑆,𝐵 = 2 min

𝑉𝑆,𝐶 = 6,230 mm³

𝑡𝐷,𝑆,𝐶 = 2 min

Support generation

197

Block support

Cone support Part E / Edges

𝑉𝐸,𝐵 = 1,554 mm³

𝑡𝐷,𝐸,𝐵 = 3 min

𝑉𝐸,𝐶 = 2,335 mm³

𝑡𝐷,𝐸,𝐶 = 15 min

Part C / Curved surfaces 𝑉𝐶,𝐵 = 11,366 mm³

𝑡𝐷,𝐶,𝐵 = 2 min

𝑉𝐶,𝐶 = 4,426 mm³

𝑡𝐷,𝐶,𝐶 = 2 min

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Demonstration of algorithmic support structures Block support

Cone support

Part ST / Straight structural transition 𝑉𝑆𝑇,𝐵 = 3,867 mm³

𝑡𝐷,𝑆𝑇,𝐵 = 5 min

𝑉𝑆𝑇,𝐶 = 1,663 mm³

𝑡𝐷,𝑆𝑇,𝐶 = 5 min

Part CT / Curved structural transition 𝑉𝐶𝑇,𝐵 = 3,530 mm³

𝑡𝐷,𝐶𝑇,𝐵 = 5 min

𝑉𝐶𝑇,𝐶 = 1,382 mm³

∑ 𝑉𝐵 = 35,693 mm³

∑ 𝑉𝐶 = 16,036 mm³

∑ 𝑡𝐷,𝐵 = 17 min

∑ 𝑡𝐷,𝐶 = 29 min

𝑡𝐷,𝐶𝑇,𝐶 = 5 min

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199

The procedure for part E and the corresponding results differ from the other parts. First, as described in Section 2.4 and confirmed by a preliminary benchmark study [518], block supports are designed to support surfaces, not edges. However, the line support is an adaption of the block support for edge handling: The main body consists of a single wall in the same style as a block support’s wall. Short traverse walls reinforce the single wall for stability. The line support hence represents a segment of a block support that is applicable to edges. Since the block support performance for part E was not satisfactory in [518], it is substituted by line supports in this study. The second difference to the processing of the other parts is that Materialise Magics is not able to position cone supports along an edge automatically. However, the cone supports can be added manually. This is done for part E, resulting in a slightly higher support design time compared to the other configurations. The results of the support generation are shown in Table 8.4. A side view as well as a top view indicating the support pattern is displayed. Please note that the scaling of the graphics is not uniform. It stands out that the automatically generated cone supports are not regular in size and/or positioning. With Materialise GmbH not disclosing the underlying algorithm, no exact reason for this irregular behavior could be determined. Table 8.4 summarizes the design time as well as the volume of the block and cone supports. As already described, the design time for part E’s cone supports is significantly higher due to the manual application. Additionally, part ST and CT required a little more time because the surfaces to be supported had to be assigned manually as they do not violate the overhang restriction implemented in Materialise Magics. Regarding the support volume, the cone supports require significantly less volume for each part except part E. Overall, the cone supports require only 45 % of the block support volume.

8.2.2 Algorithmic tree supports To create the algorithmic tree supports, the design domains need to be defined. Because all surfaces to be supported have free space underneath them, the design domains are created by projecting the surface to the next surface (build platform or part) in negative Zdirection. For part E, where no surfaces but edges are of interest, the surfaces adjacent to the edges are used. The width of the small design domains is half the base width of the overhanging edges. The design domains are presented in Table 8.5. The time required for the design domain design 𝑡𝐷,𝐷 is also given, which is directly proportional to the number of different design domain geometries. As pointed out in Section 5, several geometrical parameters of the tree supports need to be defined. First, the hatching distance in X- and Y-direction 𝑑𝑔 = 𝑑𝑔𝑥 = 𝑑𝑔𝑦 is chosen. In the tree support literature related to PBF-LB/M, values of 1 mm [319], 2 mm [336], or

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Demonstration of algorithmic support structures

5 mm [302] are given. Additionally, the minimum branch width at the part-support interface is required. Here, studies have defined values of 𝑤𝑡,𝑚𝑖𝑛 = 0.4 − 1.5 mm [302, 313, 331, 485]. The validity of those ranges for the hatching distance as well as branch thickness is verified in a small experimental study [522]. Utilizing cone supports, different combinations of cone diameter 𝑑𝑐 = 𝑑𝑐,𝑖 = 𝑑𝑐,𝑏 (0.5 − 1 mm) and 𝑑𝑔 are applied to a thin plate, which is oriented with varying angles towards the build platform (0° − 45°). The values of 𝑑𝑔 are chosen in a way that the free overhang between two cone supports, i.e. bridge length, is in the range of 1 − 5 mm, and are therefore dependent on the cone diameter. The maximum cone diameter is limited to 1 mm, rather than 1.5 mm as in [302], following an expert interview conducted at the Fraunhofer Institution of Additive Production Technologies (Hamburg, Germany). The expert stated that 𝑑𝑐 > 1 mm are usually avoided in practice because of the excessive removal effort. Furthermore, two different sets of process parameters are investigated. Parameters analyzed are obvious defects of support and/or plate, as well as the geometrical accuracy consisting of support diameter, bridge length, and plate thickness. The main findings of the study (for more details, see [522]) are the following: 1.

No build failure due to the choice of support parameters occurred.

2.

The larger the support diameter, the higher the geometrical accuracy.

3.

A smaller overhang angle decreases the dimensional accuracy of the plate thickness. Those findings are in accordance with the literature, and prove the tree support parameters used in [302, 313, 319, 331, 336, 485] to be valid.

Finally, two configurations of the tree support geometrical parameters are chosen for the comparison. The first configuration is designed following the parameters of the block and cone support of Section 8.2.1. Here, the 𝑑𝑔 = 2 mm is defined. The second configuration exploits the optimization potential with 𝑑𝑔 = 3 mm . For both configurations, the remaining support parameters are fixed, see Table 8.6 as well as Table 6.4, as the tree design validation parameters are adopted.

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Table 8.5: Indication of design domain (red) and part geometry (blue)

Part and design domain geometry Part S / Straight surfaces

Part E / Edges

𝑡𝐷,𝐷,𝑆 = 5 min

𝑡𝐷,𝐷,𝐸 = 7 min

Part C / Curved surfaces 𝑡𝐷,𝐷,𝐶 = 2 min

Part ST / Straight structural transition

Part CT / Curved structural transition

𝑡𝐷,𝐷,𝑆𝑇 = 1 min

𝑡𝐷,𝐷,𝐶𝑇 = 1 min

∑ 𝑡𝐷,𝐷 = 16 min

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Table 8.6: Tree design parameter in demonstration

Parameter

Value

Hatch distance

𝑑𝑔

2 mm 3 mm

Offset

𝑑𝑜

2 mm

𝑤𝑡,𝑚𝑖𝑛

0.5 mm

𝑡𝑡

1 mm

Min. branch width Tree thickness

While part S, C, ST, and CT are subjected to the tree support design process described in Section 6.2, part E needed a manual setup again as the grid projection scheme does not fulfill the requirement of the support interface points to follow the overhanging edges. Hence, a separate procedure for the interface point creation is implemented. The Grasshopper module m+SelEdge located in the MeshPlus extension is used to pick the respective edges once, which has to be done manually. The edges are stored in an object container. Then, equidistant points along the edges are created, with 𝑑𝑔 defining the point distance. The interface points are passed to the sampling routine. After this step, the procedure of Section 6 is followed. Additionally, as no PBF-LB/M process simulation results exist for part E, leave stresses of 200 MPa are directly passed to the space colonization module. Also, the point coordinates are input from the interface points. Hence, the sampling procedure does only sample the interface points regarding the individual tree supports, but does not do any stress processing.

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203

Table 8.7: Overview of generated tree supports

Tree support 𝑑𝑔 = 2 mm

𝑑𝑔 = 3 mm Part S / Straight surfaces

𝑉𝑆,𝑇2 = 5018 mm³

𝑡𝐷,𝑆,𝑇2 = 5 min

𝑉𝑆,𝑇3 = 3820 mm³

𝑡𝐷,𝑆,𝑇3 = 5 min

Part E / Edges 𝑉𝐸,𝑇2 = 753 mm³

𝑡𝐷,𝐸,𝑇2 = 10 min

𝑉𝐸,𝑇3 = 553mm³

𝑡𝐷,𝐸,𝑇3 = 10 min

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Demonstration of algorithmic support structures Tree support 𝑑𝑔 = 2 mm

𝑑𝑔 = 3 mm Part C / Curved surfaces

𝑉𝐶,𝑇2 = 4364 mm³

𝑡𝐷,𝐶,𝑇2 = 2 min

𝑉𝐶,𝑇3 = 2990 mm³

𝑡𝐷,𝐶,𝑇3 = 2 min

Part ST / Straight structural transition 𝑉𝑆𝑇,𝑇2 = 1404 mm³

𝑡𝐷,𝑆𝑇,𝑇2 = 1 min

𝑉𝑆𝑇,𝑇3 = 715 mm³

𝑡𝐷,𝑆𝑇,𝑇3 = 1 min

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205

Tree support 𝑑𝑔 = 2 mm

𝑑𝑔 = 2 mm

Part CT / Curved structural transition 𝑉𝐶𝑇,𝑇2 = 1244 mm³

𝑡𝐷,𝐶𝑇,𝑇2 = 1 min

𝑉𝐶𝑇,𝑇3 = 604 mm³

∑ 𝑉𝑇2 = 12,783 mm³

∑ 𝑉𝑇3 = 8,682 mm³

∑ 𝑡𝐷,𝑇2 = 19 min

∑ 𝑡𝐷,𝑇3 = 19 min

𝑡𝐷,𝐶𝑇,𝑇3 = 1 min

The final results of the tree support generation are presented in Table 8.7. Compared to the standard supports, the tree supports achieved enormous reductions in support volume: The conservative tree support (𝑑𝑔 = 2 mm) decreases the support volume by 64 % compared to the block support, and by 21 % in relation to the cone support. The tree support with 𝑑𝑔 = 3 mm even further minimizes support volume with a reduction of 76 % and 46 % with regard to the block and cone support, respectively. In addition, the support design time for tree and cone supports is similar, and less than the time required to design the block supports. This comparison highlights the extraordinary potential of the tree supports to reduce the support-induced manufacturing cost. However, it has to be noted that the time for the process simulation adds to the total design costs of the tree supports. Further economic evaluation is performed in Section 8.4.

8.3

Specimen manufacturing and technical evaluation

The benchmark part sets are manufactured in two consecutive build jobs using a SLM500HL manufacturing system (SLM Solutions Group AG, Lübeck, Germany) with four lasers simultaneously processing a build envelope of 500 𝑥 280 𝑥 365 mm³. Each laser has its own field of work, which are overlapping to the adjacent laser’s field. The SLM500HL system has a powder handling system with in-built sieving. The powder material used is Ti-6Al-4V grade 5 (Tekna, Sherbrooke, Canada) with a nominal particle size

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Demonstration of algorithmic support structures

distribution of 20 − 53 µm. The powder is in a reused state. The inert process gas is Argon. Detachment from the build platform is done by wire eroding using a FANUC ROBOCUT 𝛼-C600iC system (FANUC Corp., Oshino-mura, Japan). For the manual support removal, chisel and hammer are utilized. The dimensional accuracy is measured using a Keyence VHX-5000 (Keyence Corp., Osaka, Japan) digital microscope.

8.3.1 Data preparation During data preparation, a 2 mm offset is applied to parts ST and CT for EDM to avoid the removal of the respective part’s base plate. The parts are positioned on the build platform such that laser overlap regions are avoided to reduce errors due to two lasers working on one part. In Figure 8.2, the three red blocks indicate the shared laser working fields. Red circles denote the holes for build platform fixation. Furthermore, one part geometry is assigned the same laser. To minimize the forces applied to the parts in manufacturing by the recoater blade, the parts are oriented at a 35° angle with respect to the recoater direction, which is along the Y-axis. Table 8.8: Process parameters for manufacturing

Parameter

Solid

Support

Laser power

𝑃𝐿

240 W

80 W

Scan velocity

𝑣𝑠

1,200 mm/s

400 mm/s

Hatch distance

𝑑ℎ

105 µm

Layer thickness

𝑡ℎ

60 µm

The process parameters applied are given in Table 8.8. The hatch distance 𝑑ℎ is set in accordance with the laser’s focus diameter. For the standard supports, individual support process parameters are used. The lower scan velocity enhances the accuracy of the laser movement, which is critical to the thin features of the block support. However, the energy input is equal with a volume energy density of 𝐸𝑣 = 31.75 J/mm³. The definition of 𝐸𝑣 is given in Equation (8.1). The tree supports are manufactured with the solid material parameter set as they are imported as solid part. 𝐸𝑣 =

𝑃𝐿 𝑡ℎ ∗ 𝑑ℎ ∗ 𝑣𝑠

(8.1)

Specimen manufacturing and technical evaluation

207

Figure 8.2: Part orientation on build platform (recoater movement along Y-axis)

8.3.2 Manufacturing results During the manufacturing of the benchmark parts, one major incident occurred: The part E supported by the line support failed. The support detached from the part and the recoater dragged the overhang of the flexible edge in its direction; see the section pointing towards the reader in Figure 8.3. In the following layers, the AM system proceeded to process the overhang. Because it was no longer connected to the support, the overhang bend upwards and severely damaged the recoater blade. In consequence, all parts in line with the recoater blade defect were affected as well. The incident was noticed and the damaged parts were removed from the on-going build process (cf. Figure 8.4, the damaged part on the right below the line was removed from the build job by mistake), enabling the successful production of the unaffected parts. Since this happened in the first of the two build jobs, the damaged parts (except for the line support part E) were added to the second build job to attain complete benchmark sets.

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Demonstration of algorithmic support structures

The overall manufacturing results are documented as figures in Appendix A.4. As first step, the visual inspection is performed. The results are summarized in Table 8.9 for the individual benchmark parts and support configurations. Apart from the block support part E, all benchmark parts were successfully built. The cone support benchmark set contains some defects: First, part S and E do have a crack in the benchmark geometry. Furthermore, in part S several cone supports broke, and in part E the two cone supports at the free outer section of the flexible edge detached from the benchmark part. Part E also shows light discoloration, which is true for the cone and tree support configurations. The tree support with 𝑑𝑔 = 3 mm shows slightly more prominent discoloration, but is still rated as light. The difference indicates a decreasing capability of heat dissipation, though.

Figure 8.3: Failure of line support / part E, Copyright © K. L. Bartsch

Figure 8.4: Line of defect recoater blade and damaged parts, Copyright © K. L. Bartsch

In addition to the observations documented in Table 8.9, slight bending of single tree supports is noticed for part E, ST, and CT. In part ST and CT, only the outer row of tree supports is affected, whereas in part E again the flexible edge is (see Figure 8.5). While the bending did not influence the manufacturing result, it is a sign to be considered in future tree support dimensioning.

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209

Table 8.9: 1st step of benchmark – visual inspection

Support Criterion

Block

Cone

Tree Tree 𝑑𝑔 = 2 mm 𝑑𝑔 = 3 mm

Part S / Straight surface Successfully built









Cracks



 (1)





Notches









Detached support



 (8)





Discoloration









Part E / Edges Successfully built









Cracks

-

 (1)





Notches

-







Detached support

-

 (2)





Discoloration

-

 light

 light

 light

Part C / Curved surfaces Successfully built









Cracks









Notches









Detached support









Discoloration









Part ST / Straight structural transition Successfully built









Cracks









Notches









Detached support









Discoloration









Part CT / Curved structural transition

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Demonstration of algorithmic support structures Support Tree Tree 𝑑𝑔 = 2 mm 𝑑𝑔 = 3 mm

Block

Cone

Successfully built









Cracks









Notches









Detached support









Discoloration









Criterion

Figure 8.5: Bending of tree support (2 mm) in part E, Copyright © K. L. Bartsch

8.3.3 Support removal After the visual inspection step, the supports are removed with the help of a hammer and chisel. Figures of the benchmark parts without supports are also included in Appendix A.4. The time needed is measured, and though not required for the economic assessment, the support removal is rated as proposed for the quick evaluation. The results are given in Table 8.10. Because the economic evaluation is based on complete benchmark part sets, the block support part E of the study in [518] is applied. Both studies were conducted with the same manufacturing equipment, although the support removal was done by pliers rather than chisel and hammer. The support removal is done by a beginner with little experience.

Table 8.10: 2nd step of benchmark – support removal

Support Criterion

Block

Cone

Tree Tree 𝑑𝑔 = 2 mm 𝑑𝑔 = 3 mm

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211

Part S / Straight surface Removal time

𝑡𝑃,𝑟,𝑆,𝑖

5 min

4 min

9 min

4 min

Difficulty of removal

𝑆𝐷,𝑆,𝑖

2

2

2

1

Residuals

𝑆𝑅,𝑆,𝑖

1

1

1

2

Part E / Edges Removal time

𝑡𝑃,𝑟,𝐸,𝑖

(5 min)

2 min

4 min

2 min

Difficulty of removal

𝑆𝐷,𝐸,𝑖

(4)

1

1

1

Residuals

𝑆𝑅,𝐸,𝑖

(2)

2

2

1

Part C / Curved surfaces Removal time

𝑡𝑃,𝑟,𝐶,𝑖

9 min

6 min

11 min

6 min

Difficulty of removal

𝑆𝐷,𝐶,𝑖

3

2

3

2

Residuals

𝑆𝑅,𝐶,𝑖

1

2

2

1

Part ST / Straight structural transition Removal time

𝑡𝑃,𝑟,𝑆𝑇,𝑖

4 min

2 min

4 min

2 min

Difficulty of removal

𝑆𝐷,𝑆𝑇,𝑖

3

2

3

2

Residuals

𝑆𝑅,𝑆𝑇,𝑖

1

2

1

1

Part CT / Curved structural transition Removal time

𝑡𝑃,𝑟,𝐶𝑇,𝑖

2 min

2 min

4 min

1 min

Difficulty of removal

𝑆𝐷,𝐶𝑇,𝑖

3

2

3

2

Residuals

𝑆𝑅,𝐶𝑇,𝑖

1

1

1

1

25 min

16 min

32 min

15 min

̅̅̅ 𝑆𝐷

3

1.8

2.4

1.6

̅̅̅ 𝑆𝑅

1.2

1.6

1.4

1.2

∑ 𝑡𝑃,𝑟

Regarding the removal time it is noted that the minimal time is required by both cone and liberal tree (𝑑𝑔 = 3 mm) support. The difference is considered insignificant due to the uncertainties related to the manual removal as well as manual time keeping. The most time is allocated by the conservative tree support. This does not exactly match up with the score of the difficulty of removal, where the block support scores higher but requires less time. One reason may be the order of processing the support configurations, which is: (1) Conservative tree support, (2) cone support, (3) liberal tree support, and (4) block support. Since the operator had little experience, it is assumed that a learning effect is included in

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Demonstration of algorithmic support structures

𝑡𝑃,𝑟 , and that 𝑡𝑃,𝑟 and 𝑆𝐷 will correlate for the conservative tree support in a second and third benchmark process. Comparing the individual parts of a benchmark set, a correlation with the support volume is noticed. This is deemed reasonable, as the benchmark parts do not include supported surfaces which are hard to reach with manual tools. Hence, only the support volume has a direct influence on the removal time, apart from the general support design. In terms of removal difficulty, the liberal tree support is the easiest to remove, with the cone support being very close in the score. This result corresponds to the number of partsupport interfaces as well as the interconnection of support features. The block support with the connected grid walls scores the highest in difficulty, and the tree support in general is less easy to remove than a cone support due to the connection of the branches. This is apparent in the conservative tree support having a higher difficulty score than the cone support, although the support volume is less. The liberal tree support does profit from the smaller number of interface points, leading to a similar score as the cone support despite the branch connections. None of the support configurations leads to damage due to support removal, as 𝑆𝑅 indicates. All configurations score between 1 and 2. The block and liberal tree support score the lowest, whereas the cone support has the highest score. This is caused by the cones with the maximum diameter at the bottom interface to the part, e.g. in part C. A cone with a diameter of 4 mm has a significantly larger interface area compared to the other configurations, which hinders the clean splitting of part and support by the chisel. A reduction of the maximum diameter will most likely decrease 𝑆𝑅 , but will also lead to increased support volume with all its dependent characteristics. Last, it is noted that the block support configuration without any perforation settings trapped a considerable amount of powder within its grid. During the unpacking of the build job from the AM machine, actions against trapped powder involved vacuuming and tapping the build platform’s back with a small hammer while it was set upright on its edge. During wire eroding, the water in the tank further washes out trapped powder, especially when cutting through supports. Still, during support removal a noticeable amount of powder was released. Aside from the safety issues arising from free Ti-6Al-4V powder, this indicates a material consumption close to the whole grid’s bounding box rather than the grid’s wall volume.

8.3.4 Dimensional accuracy The results of the dimensional accuracy measurements are presented in Table 8.11. As the 50 mm diameter section of part C barely fit the field of view of the microscope despite using the function for stitching several images together, the 30 mm diameter is also analyzed.

Economic evaluation

213

Table 8.11: 3rd step of benchmark – dimensional accuracy

Support Block

Criterion

Cone

Tree Tree 𝑑𝑔 = 2 mm 𝑑𝑔 = 3 mm

𝑃𝑎𝑟𝑡 𝑆 / 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 Edge deviation

Δ𝑒0°

1.05 mm

0.86 mm

1.94 mm

1.03 mm

Angle deviation

Δ𝛼0°

1°

1°

1°

1°

Part C / Curved surface Diameter deviation

Δ𝑑30

0.42 mm

1.12 mm

0.60 mm

2.22 mm

Δ𝑑50

0.43 mm

1.77 mm

2.65 mm

2.17 mm

Part ST / Straight structural transition Wall deviation

Δ𝑤𝑆𝑇

0.14 mm

0.25 mm

0.33 mm

0.23 mm

Part CT / Curved structural transition Wall deviation

Δ𝑤𝐶𝑇

0.14 mm

0.38 mm

0.22 mm

0.42 mm

The dimensional accuracy of the block support is higher than the other support configurations, apart from the edge deviation of part S. Among the others, no clear tendency is apparent, leading to the conclusion that the performance of the cone and tree supports are similar. The part C measurements of the conservative tree support (2 mm) show a significantly larger difference than the other support configurations. It is suspected that the value of Δ𝑑50 may be too high due to measuring errors.

8.4

Economic evaluation

In addition to the technical assessment, the economic evaluation proposed in Section 7.2 is performed, too. Each set of benchmark parts for a respective support configuration is treated as an individual print job for ease of calculation, though the manufacturing of the parts demonstrates that three or more benchmark part sets fit the build platform of the SLM500HL system. However, combining different support configurations in one build job introduces a dependence on the combination of sets, which would distort the cost comparison.

8.4.1 Input parameters The dynamic input parameters of the cost model given in the prior sections are summarized in Table 8.12. It is defined that the support configurations neither elevate the part from the build platform nor result in additional wire eroding distance. This is not exactly

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Demonstration of algorithmic support structures

true for the presented case as part ST and CT have been raised 2 mm, however, solid material was used to do so rather than the respective support structure. Also, the overhanging surfaces of part S do increase the wire eroding distance at the bottom of the build platform. Changing the positioning of the parts such that a benchmark part without outer supports is present at each end of the build platform would avoid this, though. Thus, the actual positioning is considered insignificant and neglected. Table 8.12: Dynamic input parameter of cost model

Support Criterion Design time

𝑡𝐷,𝑙 𝑡𝐷,𝑒

Tree 𝑑𝑔 = 2 mm

Block

Cone

17 min

29 min 16 min

Removal time

𝑡𝑃,𝑟

25 min

Support volume

𝑉𝑀,𝑠

35,693 mm³

Tree 𝑑𝑔 = 3 mm

47 min 452 min 32 min

15 min

16,036 mm³ 12,783 mm³ 8,682 mm³

While the block and cone support are created in one process with no computation time, leading to 𝑡𝐷,𝑙 = 𝑡𝐷,𝑒 , the tree support generation consists of several steps. Hence, it is not only necessary to distinguish the operator and equipment time, but all time components involving different softwares. For simplicity, it is assumed that the same CAD software is used for design domain and support generation. Additionally, the time with regard to the PBF-LB/M simulation software Simufact Additive as well as the Matlab software used for simulation result processing needs to be taken into account. Table 8.13 indicates the respective support design time components. Note that both operations in Rhino 6 / Grasshopper do not distinguish between operator and equipment time; the actual computation times of the Grasshopper tree support is approximately 10 s on the computer used in this thesis. Only the software involved in any kind of large computation requires differentiation. The static input parameters of the cost model are presented in Appendix A.3. Where parameters of the SLM500HL system are not known, e.g. the gas consumption, they are extrapolated from the parameters of the AM system used in [518] according to the difference in build volume. As the benchmark aims at comparing support structures rather than deriving the exact quantitative costs, this is deemed sufficient. Table 8.13: Support design time components

Software Simufact Additive Rhino 6 / Grasshopper

Operator time

Equipment time

8 min

114 min 35 min

Economic evaluation Matlab

215 4 min

303 min

8.4.2 Results In the economic evaluation, first the single-lot production scenario, which was experimentally performed, is investigated. In a second step, the findings are extended to other production scenarios with increasing lot size to demonstrate the impact of the achieved optimization to industrial PBF-LB/M manufacturing. Single-lot production The overall support cost of the different support configurations for single-lot production are displayed in Figure 8.6. The conservative tree support (2 mm) is the most expensive support configuration at 122.70 €. This is due to the highest removal cost, and the design cost of the tree supports being significantly higher because of the process simulation step. The least expensive support configuration is the cone support. The liberal tree support (3 mm) is less expensive than the block support, with a cost reduction of 4 %.

Figure 8.6: Overall support cost of respective configurations

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Demonstration of algorithmic support structures

Figure 8.7 presents a breakdown of the individual production phases’ contribution to the overall support cost. It is immediately noticed that while the cone and tree supports are dominated by the design cost, the block support’s main cost contributor is the manufacturing phase. The block support has by far the highest manufacturing cost with 60.98 € compared to the cone support (27.40 €) or tree supports (21.84 € and 14.84 €), which is related to the significantly higher support volume (cf. Table 8.1). Even for the standard cone support, which was not subjected to an extended design process, the design cost are slightly higher than the manufacturing cost. The breakdown of the tree support costs shows that the manufacturing cost is actually the smallest factor. This on the one hand demonstrates the effect of volume reduction on cost, on the other hand highlights the potential of further optimization especially in the design phase. The removal phase’s contribution is relatively stable throughout the support configurations. To exploit additional optimization potential, sensitivity analysis of several parameters is performed. First, the design phase is investigated, as the previous paragraph already mentioned the potential. As already stated in Section 2.1.2, there is an increasing integration of CAD design, process simulation, and data preparation into a single software, see e.g. the Siemens NX or Altair Inspire software. This development allows for the assumption that process simulation will soon be a standard process step in AM, and can therefore be removed from the calculation of additional costs due to supports. Removing all simulation-related cost factors from the design phase cost results in a 27 % cost reduction from 122.70 € to 89.45 € for the conservative tree support, and a 34 % cost reduction from 99.19 € to 65.95 € for the liberal tree support. Consequently, the conservative tree support would also be less expensive than the standard cone support, and cost optimization achieved by the liberal tree support would be increased to an 11 % cost reduction in comparison to the cone support for a single-lot production scenario.

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Figure 8.7: Contribution of the production phases to the overall support cost

Another influencing factor characterized by a high variance are the operators in the design and removal phase. Based on their skill as well as mental and physical condition, the time required to perform the individual tasks may vary significantly. When the design time is de- or increased, the overall support cost changes proportionally (cf. Figure 8.8). A 50 % decrease in the design time related to operator actions (𝑡𝐷,𝑙 ) results in a cost reduction of 9 % (9.31 €) for the block support as well as 26 % (25.15 €) for the liberal tree support, and vice versa in the case of an increase. Consequently, while a decrease in design operator time increases the cost gap between the block and liberal tree support, an increase quickly results in the tree support cost exceeding the block support cost. Furthermore, the production phases’ contributions to the overall support cost change accordingly. Similar observations are made when evaluating changes in the removal operator time. Because the contribution of the removal phase is larger in the block support compared to the liberal tree support, as indicated in Figure 8.7, the change in cost is larger for the block support than the liberal tree support: While a 50 % decrease of 𝑡𝑃,𝑟 reduces the liberal tree support cost to 91.92 € (−8 %), the block support cost is down to 91.73 € (−12 %).

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Figure 8.8: Design operator time variation

Serial production scenario Serial production is a way to optimize costs per part. On the one hand, the build capacity of the AM system is fully utilized, and on the other hand, the contribution of the design phase cost is minimized as the design phase only takes place once no matter the number of parts produced later on. The removal cost, however, scales linearly as every part has to be post-processed. Figure 8.9 displays the support cost development over the number of benchmark part sets within a single build job. As stated in the introduction of Section 0, the calculations are performed assuming only one set of benchmark parts is manufactured at a time. However, during the manufacturing of the benchmark parts, three sets have been produced simultaneously. The maximum number of benchmark part sets fitting the build envelope of the SLM500HL system is considered six sets. The overall support cost decreases exponentially with increasing number of sets. Especially the difference between one and two sets indicates the challenge single-lot production poses economically: For the block and cone support, a decrease of 9 % and 22 % is achieved. The conservative and liberal tree supports even reach a 29 % and 36 % cost reduction. This is due to the design phase contribution being the most prominent in the tree support configurations; hence, the distribution of the design phase costs benefits the tree supports the most. At two sets, the

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conservative tree support is already less expensive than the block support. For four or more sets, the liberal tree support takes over the position of the configuration with minimal overall cost. At six sets per build job, the liberal tree support cost is 13 % less than the cone support cost per set (41.03 € and 47.25 €).

Figure 8.9: Number of benchmark part sets in single build job

The cost reductions achieved for the tree supports are mostly due to the distribution of the design phase cost throughout the benchmark part sets, see Figure 8.10. Here, the contribution of the different production phases to the liberal tree support cost is displayed in relation to the number of benchmark sets in one build job. From five sets on, the design cost is not dominant anymore. Furthermore, the design cost contribution is reduced from 70 % for a single benchmark part set to 28 % for a fully utilized build envelope. If the use case is extended to serial production, the decrease of manufacturing cost is the dominant effect, since the design phase cost becomes negligible. Today, typical lot sizes in PBF-LB/M range from single-lot production up to 100 parts per year [14, 523]. Only few use cases reach the threshold of more than 10,000 parts per year, e.g. the fuel nozzle and implants of GE Additive [14]. Superimposing these production numbers to the benchmark use case at six benchmark sets per build job, perceiving one benchmark set as a single part, results in approximately 17 build jobs per year for the common lot size as well

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Demonstration of algorithmic support structures

as approximately 1667 build jobs per year for the mass production scenario. In the following, the effect of the lot size is analyzed by assuming linear scaling of the manufacturing and removal phase costs based on the build job configuration of six benchmark sets. Thus, additional scaling effects in material and consumables prices are neglected for the sake of simplicity.

Figure 8.10: Tree support (3 mm) production phase contribution for different number of benchmark part sets in single build job

Figure 8.11 visualizes the development of the cost per benchmark set for the different support types at the defined lot sizes. The difference between one build job (six sets) and 100 sets of benchmark parts is significantly higher than between 100 and 10,000 sets: For the cone support, the cost per benchmark set is reduced by 10 % from 47.25 € to 42.27 € when scaling up to 100 sets, whereas the additional step towards mass production comes at a further cost reduction of 1 % (41.96 €). A similar development is noticed for the remaining support types, e.g. the liberal tree support. Here, the first scaling to 100 sets comes at a cost reduction of 26 % (41.03 € to 30.08 € per benchmark set), the second scaling to 10,000 sets results in an additional cost reduction of 2 % to 29.40 €. The liberal tree support is characterized by the largest scaling effects, which is attributed to the fact that the manufacturing as well as removal phase costs of the liberal tree support are the the lowest of all support types. Compared to the standard cone support (block support costs are significantly higher), the liberal tree support generates only 72 % of the support-induced

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costs at 100 benchmark part sets. This leads to a total saving of 1,219 €. At 10,000 sets, the liberal tree support makes up only 70 % of the cone support costs per benchmark set, opening up the potential of reducing the total support costs by 125,600 €. Apart from the parameters investigated in this section, there is a considerable number of other influencing factors to the overall support cost: the performance of the hardware and/or software used in all production phases, their respective costs, and local constraints such as wages, rent, or energy prices. However, those factors do not only affect all considered support configurations, but the actual part’s cost, too. Therefore, no significant information is added to the benchmark when investigating additional parameters, as the benchmark procedure is not meant to derive the exact support cost, but to provide a comparison between different configurations. This is also why no part cost is given here to point out the contribution of the support cost to the overall part cost. The actual relationship between support and part cost is highly dependent on the part’s geometry and therefore not suited to assess the general performance of a novel support structure or support layout strategy.

Figure 8.11: Support cost per benchmark set for serial production

222 8.5

Demonstration of algorithmic support structures Benchmark summary & conclusion

Summarizing the technical and economic assessment, the block support is characterized by a mixed performance and comparatively high cost. The block support failed in the part testing for edge support, and demonstrates a low heat dissipation capability when using the part manufactured in [518] as a reference. However, the grid layout of the block support is highly resistant against shear stresses, which is apparent in the higher dimensional accuracy compared to the remaining support configurations. Furthermore, the block support is not that easy to remove, but barely damages the part-support interface. Thus, in accordance with the literature presented in Section 2, the block support is deemed suitable for large surfaces with multidimensional loads. On the downside, the support-related cost is significantly higher than the other support configurations, with an even higher material consumption due to excessive powder trapping. The parts supported by cone supports were built successfully, but cracks were found in both benchmark parts and supports. The cone support dissipated the most heat, which is associated with the volumetric nature of the cone support. It is easy to remove, but controlling the breaking point is challenging, leading to residuals requiring further processing. The cone support is able to support edges and surfaces, although the edge support needs manual positioning of the supports in the software utilized in this benchmark. This is a serious issue in the context of support design automation to avoid the effect of the operator’s experience level. Further challenges arise due to the reduced shear resistance of the cone geometry compared to the block support, which leads to a lower dimensional accuracy. One of the greatest benefits of the cone support, though, is the low cost. Again, the findings of the benchmark fit the literature. The tree support configurations were both successfully manufactured with no cracking occurring in part and support. The conservative tree support (𝑑𝑔 = 2 mm) displays a heat dissipation ability very close to the cone support despite a 21 % lower support volume. The liberal tree support (𝑑𝑔 = 3 mm) is slightly less efficient regarding heat dissipation, which is attributed to the even lower support volume. The tree supports are capable of supporting both edges and surfaces, but share the cone support’s challenge of shear resistance. The removal of the tree supports is more difficult than the cone support, but the amount of residuals is less. The respective support volumes also lead to comparatively low manufacturing costs. However, the additional steps in support design increase the design cost significantly in relation to the standard supports. The algorithms developed in this thesis are not optimized for efficiency, though, artificially increasing the computation times. Consequently, the main benefit of the algorithmic support design in single-lot production is the eradiction of the need for user experience, which is hard to quantify as it affects various cost factors such as wages, but also failed build jobs. However, in serial production scenarios the reduction of manufacturing and removal costs overpowers the

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increase in design costs. Here, just a full utilization of the build envelope results in the liberal tree support being the least expensive option already. For increasing lot sizes, the liberal tree support design achieves significant cost reductions in comparism to the standard cone support, not to mention the block support. The research hypothesis formulated in Section 3 states that load-based, algorithmic tree supports are able to perform at the same level as standard support types while simultaneously optimizing support material consumption as well as support application. Getting back to this research hypothesis, the benchmark shows that the algorithmic, load-based tree supports partially fulfill these requirements in the case of single-lot production, and completely reach the optimization goals for medium serial production.

In conclusion, the research hypothesis is proven valid.

Nevertheless, challenges in the application of the algorithmic tree supports remain to be investigated in future research. Already mentioned were the reduced resistance against shear stresses as well as the computation times increasing the support cost. Furthermore, the testing of design domains with higher complexity in its geometry is required to ensure general applicability.

9 Conclusion Additive manufacturing is a key technology in digital production and is considered essential for addressing several megatrends in global manufacturing, namely product customization, resource and energy efficiency in products, digital transformation, as well as decentralized and resilient supply chains. However, there are still barriers regarding the broad industrial application of AM, especially PBF-LB/M. These include the remaining lack of digitalization, associated investment and on-going cost, as well as the general complexity of the PBF-LB/M process, leading to considerable requirements regarding user experience to exploit the full potential of value creation by PBF-LB/M products. An analysis of the PBF-LB/M process’ state of digitalization revealed that most of the steps lacking digitalization are related to support structures. Therefore, advancing the support structure design and application by the digitalization of the design process presents an opportunity to address the identified barriers. Supports contribute to the production cost without adding value to the part, since they are removed after the manufacturing process. Therefore, decreasing the use of supports to the necessary minimum is an important task in support design to decrease cost. However, supports are crucial to the successful manufacturing of complex parts. Insufficient support performance leads to build failure, which is to be avoided at all cost. At the same time, extensive experience is required for appropriate support layout and dimensioning, as current software neither automates the support design process completely, nor considers the process-induced loads to supports. Hence, there is potential for significant optimization with the goal of advancing PBF-LB/M and increasing its reliability. A thorough investigation of the state of the art as well as current support optimization approaches revealed that the method of topology optimization is the most promising approach to create individual, efficient supports. However, several issues were identified: First, the studies in the literature depend on artificial loads based on assumptions. Furthermore, topology optimization was found to be very sensitive to its parameters and requiring experience for result interpretation, which would add to the already needed experience for support design. Last, current studies do not consider every support tasks but usually only one. Which support task is focused does vary, though. Furthermore, studies only use cost indicators for assessing the optimization success. In summary, fully digitalized and automated support design is not available in research. This thesis addresses the research gap by developing a digital and automated support design procedure, which is based on actual process-induced loads determined by PBF-LB/M process simulation. During preliminary studies combining process simulation and topology optimization for support design, it was noticed that the results resembled tree-like structures no matter the investigated physics. This finding was validated by literature and

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5_9

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own systematic analysis. This novel insight allowed developing a support design procedure where a part is first subjected to PBF-LB/M process simulation to derive the processinduced loads to the supports, followed by a support design algorithm based on the space colonization theory. To realize the support design approach, four different fields of expertise are dealt with in this thesis: material modeling, support design rule determination, support generation algorithm development, and the development of benchmark methods to enable the optimization success evaluation. Since commercial software on PBF-LB/M process simulation is already available in the software market, this topic is not subject to own innovations. First, because all numerical methods involved require material data, a model of a reference material is established. Due to its known proneness to residual stresses, the titanium alloy Ti-6Al-4V is chosen as reference material. A complete material model of the thermophysical, optical, and mechanical material properties relevant to the PBF-LB/M process is derived from literature as well as own experiments. A possible influence of the martensitic microstructure characteristic for PBF-LB/Ti-6Al-4V specimens on the thermo-physical properties is evaluated experimentally and found to be not existent, allowing for the use of literature data on non-AM Ti-6Al-4V. For the mechanical properties, where sufficient literature data on PBF-LB/Ti-6Al-4V could be compiled, the possible influence of the specimen’s orientation on the build platform, the surface condition, as well as the year of publication is checked. No direct correlation was found. In a second step, the design rules for the supports are determined using topology optimization, more specific the density-based SIMP approach. In a systematic study, the design domain size, aspect ratio, overhang angle and geometry, as well as load height and form was varied. The parameters defining a tree structure are identified and the topology optimization results are evaluated accordingly. Qualitative as well as quantitative observations are made and compiled in a set of design rules. The design rules are then implemented into a base algorithm for tree generation. Here, after indepth assessment of the current state of the art related to virtual tree design, the space colonization algorithm is identified as most suitable algorithm. The complete procedure for tree support generation consists of three individual steps (excluding data import and export): First, part surfaces in need of support are identified and part-support interface points are created. Then, the interface points are sampled to define subsets corresponding to individual tree structures. The third step consists of the tree topology generation by space colonization and the 3D modeling of the tree supports. The fourth field in need of innovation is the development of a benchmark procedure to enable the assessment of the optimization success regarding the technical and economic performance of the novel supports, as no standard procedure exists in literature. For the technical evaluation, a set of five benchmark parts is designed based on the support tasks

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of supporting overhanging features (surfaces, edges), dissipating heat, and fixating the part in space to counter-act deformation. The different aspects are addressed by provoking support failure. The technical benchmark is suitable to compare different support approaches, but does not quantify specific support properties. For the economic assessment, a support cost model is derived, taking into account all additional costs solely due to supports. This includes personnel, equipment, and room costs. Furthermore, the support design, manufacturing, and removal phase are considered. To validate the support design procedure, the benchmark is carried out experimentally. Two different tree support configurations are created, one more conservative and one liberal. For comparison, the standard block and cone support are chosen, as their performance is well known within the PBF-LB/M industry and research. Both tree support configurations require significantly less material than the standard supports. Benchmark sets with all four support types are manufactured once and evaluated according to the benchmark procedure. It is found that the tree supports are slightly better in support performance overall, as the block support experienced support failure in one part and various defects occurred in the cone support. However, the exact ranking of the support types differs from evaluation criterion to another. In terms of support cost, the tree supports were not able to outperfom the cone support in the present single-lot manufacturing scenario. The liberal tree support is able to beat the block support, though. The high support cost is due to the PBF-LB/M process simulation and the post-processing of the simulation results. Sensitivity analysis of the operator costs due to manual work as well as omission of the process simulation as support-related cost factor reveals significant potential of cost reduction in comparison to the standard supports. This potential is demonstrated in the analysis of serial production scenarios consisting of the full utilization of the build envelope compared to single-lot production, as well as lot sizes of 100 and 10,000 benchmark part sets. Here, the liberal tree support is able to achieve a six-figure total cost saving at minimum. In summary, the developed method for algorithm-based tree supports has successfully digitalized the support design process and eradicated the requirement of user experience while maintaining sufficient technical performance. A cost reduction is achieved for lot sizes larger than four benchmark part sets. The novel support design needs to be subject to further work to reach the state of a mature methodology. Future work needs to address the rework of the self-written codes to optimize for computational efficiency. Furthermore, the methodology requires validation in industrial applications and should be extended to other materials. Last, the investigation of the part-support interface cross-section geometry provides the opportunity to ease the support removal as well as achieve a higher surface quality, leading to less finishing efforts. In terms of the digitalization of the complete PBFLB/M process, the post-processing steps of part detachment and support removal need to be focused. When the PBF-LB/M process is fully digitalized, an increase in technology

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adoption especially by SME’s is expected, enabling the digital transformation of the manufacturing industry.

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Appendix Technical documentation of slicer test specimen

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Bartsch, Digitalization of design for support structures in laser powder bed fusion of metals, Light Engineering für die Praxis, https://doi.org/10.1007/978-3-031-22956-5

284

Appendix

Appendix Technical documentation of the benchmark parts

285

286

Appendix

Appendix

287

288

Appendix

Appendix

289

290

Appendix Static input parameter of cost model

Parameter Value

Unit

Description

Source

Operators 𝑐𝑤,𝑔,𝐷

46,366.03 €/a

Annual gross wage of design engineer

[524], E13 St. 2

𝑐𝑤,𝑔,𝑅

39,986.54 €/a

Annual gross wage of technician

[524], E11 St. 2

34.60

€/h

Average incidental wage 2018 (Germany)

[525]

𝑑𝑌𝑒𝑎𝑟

219

d/a

𝑑𝑊𝑒𝑒𝑘

5

d/week Average workdays per week in 2018 (Germany)

[526]

𝑡𝑊𝑒𝑒𝑘

38

h/week Average work hours per week in 2018 (Germany)

[526]

𝑢𝑂

5,256

h/a

Utilization office room, all workdays per year

Estimation

𝐴𝑂

3

m²

Standard office desk including room for chair (2 m 𝑥 1.5 m)

Estimation

𝐴𝑀

21

m²

Space for PBF-LB/M machine including room for technician (3.5 m 𝑥 6 m)

[527]

𝐴𝑊

3

m²

Space for wire eroding machine including room for technician (1.5 m 𝑥 2 m)

Estimation

𝑢𝑊𝑆

2,628

h/a

Utilization of workshop

Estimation

𝐴𝑊𝑆

2

m²

Standard workbench including space for technician (2 m 𝑥 1 m)

Estimation

14.9

€/m²

Average rent of industrial space in [528] Hamburg, Germany

3.11

€/m²

[529]

𝑐𝑤,𝑖,𝐷 𝑐𝑤,𝑖,𝑅

Workdays per year in 2018 (Germany)

[526]

Rooms

𝑐𝑅𝑅,𝑂 𝑐𝑅𝑅,𝑀 𝑐𝑅𝑅,𝑊 𝑐𝑅𝑅,𝑊𝑆 𝑐𝑅𝑂,𝑂

Appendix

291

Parameter Value

Unit

Description

Source

Average operational cost of space in Hamburg, Germany

𝑐𝑅𝑂,𝑀 𝑐𝑅𝑂,𝑊 𝑐𝑅𝑂,𝑊𝑆 Software – Materialise Magics 𝑢𝑆𝑊

2,628

h/a

Utilization of data preparation software Materialise Magics

Estimation

𝑐𝑆𝑊

9,000

€/a

License cost of data preparation software Materialise Magics

Quotation

Software – MATLAB R2021b 𝑢𝑆𝑊

2,628

h/a

Utilization of data processing software MATLAB R2021b

Estimation

𝑐𝑆𝑊

800

€/a

License cost of data processing software MATLAB 2021b

Online

Software – Rhino 6 𝑢𝑆𝑊

2,628

h/a

Utilization of CAD software Rhino Estimation 6

𝑐𝑆𝑊

995

€

Software cost of CAD software Rhino 6

Online

𝑡𝑎,𝑆𝑊

1

𝑎

Amortization period of CAD software

[530]

Software – Simufact Additive 𝑢𝑆𝑊

2,628

h/a

Utilization of data preparation software Materialise Magics

Estimation

𝑐𝑆𝑊

25,000

€

License cost of data preparation software Materialise Magics

Internal communication

𝑡𝑎,𝑆𝑊

1

𝑎

Amortization period of CAD software

[530]

Estimation

Computer Hardware 𝑢𝐻𝑊

6,570

h/a

Utilization of computer hardware, can run overnight

𝐶𝐼,𝐻𝑊

1,000

€

Investment cost of computer Quotation hardware (Lenovo Thinkpad L470)

𝑡𝑎,𝐻𝑊

3

a

Amortization period of computer hardware

[530]

292

Appendix

Parameter Value

Unit

Description

Source

Material 𝜌

4,430

kg/m²

Density of Ti-6Al-4V

Section 4

𝑐𝑀𝑎𝑡

250

€/kg

Price of Ti-6Al-4V powder

[373]

𝑐𝑆𝑐𝑟𝑎𝑝

5

€/kg

Scrap price of solid titanium

[531]

PBF-LB/M machine SLM500HL h/a

Utilization of PBF-LB/M machine, Estimation single shift production

𝑢𝑀

4,380

𝐶𝐼,𝑀

1,200,000 €

Investment cost of PBF-LB/M machine

𝑡𝑎,𝑀

5

𝑎

Amortization period of PBF-LB/M [532] machine

𝑐𝑀,𝑆𝑒𝑟

30,000

€/a

Service cost of PBF-LB/M machine

[373]

𝑡𝑃

6

s

Time for recoating of PBF-LB/M machine

Estimation

𝑡𝐷𝑒𝑙

2

s

Delay time of PBF-LB/M machine [373]

𝐶𝐼,𝐹

335

€

Investment cost of PBF-LB/M machine filter

[373]

𝑡𝑎,𝐹

0.02

a

Amortization period of filter

[373]

𝑐𝐺

7.5

€/h

Gas cost

Estimation, based on [373]

𝐸

8

kW

Maximum energy consumption of [527] PBF-LB/M machine

𝑝𝐸

0.28

€/kWh Average energy price in Hamburg, Online Germany

𝑢𝑇

1,314

h/a

Utilization of tools

𝐶𝐼,𝑇

500

€

Investment cost of one full set of Online tools

𝑡𝑎,𝑇

7

a

Amortization period of tools

Estimation, internal communication

Energy

Tools Estimation

[533]

Appendix

293

Experimental results of demonstration – block support After detachment

After removal Part S / Straight surfaces

Part C / Curved surfaces

Part ST / Straight structural transition

294

Appendix Part CT / Curved structural transition

Appendix

295

Experimental results of demonstration – cone support After detachment

After removal Part S / Straight surfaces

Part E / Edges

Part C / Curved surfaces

296

Appendix Part ST / Straight structural transition

Part CT / Curved structural transition

Appendix

297

Experimental results of demonstration – tree support (𝑑𝑔 = 2 mm) After detachment

After removal Part S / Straight surfaces

Part E / Edges

Part C / Curved surfaces

298

Appendix Part ST / Straight structural transition

Part CT / Curved structural transition

Appendix

299

Experimental results of demonstration – tree support (𝑑𝑔 = 3 mm) After detachment

After removal Part S / Straight surfaces

Part E / Edges

Part C / Curved surfaces

300

Appendix Part ST / Straight structural transition

Part CT / Curved structural transition