Computational Science – ICCS 2020: 20th International Conference, Amsterdam, The Netherlands, June 3–5, 2020, Proceedings, Part VII [1st ed.] 9783030504359, 9783030504366

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LNCS 12143

Valeria V. Krzhizhanovskaya · Gábor Závodszky · Michael H. Lees · Jack J. Dongarra · Peter M. A. Sloot · Sérgio Brissos · João Teixeira (Eds.)

Computational Science – ICCS 2020 20th International Conference Amsterdam, The Netherlands, June 3–5, 2020 Proceedings, Part VII

Lecture Notes in Computer Science Founding Editors Gerhard Goos Karlsruhe Institute of Technology, Karlsruhe, Germany Juris Hartmanis Cornell University, Ithaca, NY, USA

Editorial Board Members Elisa Bertino Purdue University, West Lafayette, IN, USA Wen Gao Peking University, Beijing, China Bernhard Steffen TU Dortmund University, Dortmund, Germany Gerhard Woeginger RWTH Aachen, Aachen, Germany Moti Yung Columbia University, New York, NY, USA

12143

More information about this series at http://www.springer.com/series/7407

Valeria V. Krzhizhanovskaya Gábor Závodszky Michael H. Lees Jack J. Dongarra Peter M. A. Sloot Sérgio Brissos João Teixeira (Eds.) •

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Computational Science – ICCS 2020 20th International Conference Amsterdam, The Netherlands, June 3–5, 2020 Proceedings, Part VII

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Editors Valeria V. Krzhizhanovskaya University of Amsterdam Amsterdam, The Netherlands

Gábor Závodszky University of Amsterdam Amsterdam, The Netherlands

Michael H. Lees University of Amsterdam Amsterdam, The Netherlands

Jack J. Dongarra University of Tennessee Knoxville, TN, USA

Peter M. A. Sloot University of Amsterdam Amsterdam, The Netherlands

Sérgio Brissos Intellegibilis Setúbal, Portugal

ITMO University Saint Petersburg, Russia Nanyang Technological University Singapore, Singapore João Teixeira Intellegibilis Setúbal, Portugal

ISSN 0302-9743 ISSN 1611-3349 (electronic) Lecture Notes in Computer Science ISBN 978-3-030-50435-9 ISBN 978-3-030-50436-6 (eBook) https://doi.org/10.1007/978-3-030-50436-6 LNCS Sublibrary: SL1 – Theoretical Computer Science and General Issues © Springer Nature Switzerland AG 2020 The chapter “APE: A Command-Line Tool and API for Automated Workflow Composition” is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/ licenses/by/4.0/). For further details see license information in the chapter. This work is subject to copyright. All rights are reserved 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

Preface

Twenty Years of Computational Science Welcome to the 20th Annual International Conference on Computational Science (ICCS – https://www.iccs-meeting.org/iccs2020/). During the preparation for this 20th edition of ICCS we were considering all kinds of nice ways to celebrate two decennia of computational science. Afterall when we started this international conference series, we never expected it to be so successful and running for so long at so many different locations across the globe! So we worked on a mind-blowing line up of renowned keynotes, music by scientists, awards, a play written by and performed by computational scientists, press attendance, a lovely venue… you name it, we had it all in place. Then corona hit us. After many long debates and considerations, we decided to cancel the physical event but still support our scientists and allow for publication of their accepted peer-reviewed work. We are proud to present the proceedings you are reading as a result of that. ICCS 2020 is jointly organized by the University of Amsterdam, NTU Singapore, and the University of Tennessee. The International Conference on Computational Science is an annual conference that brings together researchers and scientists from mathematics and computer science as basic computing disciplines, as well as researchers from various application areas who are pioneering computational methods in sciences such as physics, chemistry, life sciences, engineering, arts and humanitarian fields, to discuss problems and solutions in the area, to identify new issues, and to shape future directions for research. Since its inception in 2001, ICCS has attracted increasingly higher quality and numbers of attendees and papers, and 2020 was no exception, with over 350 papers accepted for publication. The proceedings series have become a major intellectual resource for computational science researchers, defining and advancing the state of the art in this field. The theme for ICCS 2020, “Twenty Years of Computational Science”, highlights the role of Computational Science over the last 20 years, its numerous achievements, and its future challenges. This conference was a unique event focusing on recent developments in: scalable scientific algorithms, advanced software tools, computational grids, advanced numerical methods, and novel application areas. These innovative novel models, algorithms, and tools drive new science through efficient application in areas such as physical systems, computational and systems biology, environmental systems, finance, and others. This year we had 719 submissions (230 submissions to the main track and 489 to the thematic tracks). In the main track, 101 full papers were accepted (44%). In the thematic tracks, 249 full papers were accepted (51%). A high acceptance rate in the thematic tracks is explained by the nature of these, where many experts in a particular field are personally invited by track organizers to participate in their sessions.

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Preface

ICCS relies strongly on the vital contributions of our thematic track organizers to attract high-quality papers in many subject areas. We would like to thank all committee members from the main and thematic tracks for their contribution to ensure a high standard for the accepted papers. We would also like to thank Springer, Elsevier, the Informatics Institute of the University of Amsterdam, the Institute for Advanced Study of the University of Amsterdam, the SURFsara Supercomputing Centre, the Netherlands eScience Center, the VECMA Project, and Intellegibilis for their support. Finally, we very much appreciate all the Local Organizing Committee members for their hard work to prepare this conference. We are proud to note that ICCS is an A-rank conference in the CORE classification. We wish you good health in these troubled times and hope to see you next year for ICCS 2021. June 2020

Valeria V. Krzhizhanovskaya Gábor Závodszky Michael Lees Jack Dongarra Peter M. A. Sloot Sérgio Brissos João Teixeira

Organization

Thematic Tracks and Organizers Advances in High-Performance Computational Earth Sciences: Applications and Frameworks – IHPCES Takashi Shimokawabe Kohei Fujita Dominik Bartuschat Agent-Based Simulations, Adaptive Algorithms and Solvers – ABS-AAS Maciej Paszynski David Pardo Victor Calo Robert Schaefer Quanling Deng Applications of Computational Methods in Artificial Intelligence and Machine Learning – ACMAIML Kourosh Modarresi Raja Velu Paul Hofmann Biomedical and Bioinformatics Challenges for Computer Science – BBC Mario Cannataro Giuseppe Agapito Mauro Castelli Riccardo Dondi Rodrigo Weber dos Santos Italo Zoppis Classifier Learning from Difficult Data – CLD2 Michał Woźniak Bartosz Krawczyk Paweł Ksieniewicz Complex Social Systems through the Lens of Computational Science – CSOC Debraj Roy Michael Lees Tatiana Filatova

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Organization

Computational Health – CompHealth Sergey Kovalchuk Stefan Thurner Georgiy Bobashev Computational Methods for Emerging Problems in (dis-)Information Analysis – DisA Michal Choras Konstantinos Demestichas Computational Optimization, Modelling and Simulation – COMS Xin-She Yang Slawomir Koziel Leifur Leifsson Computational Science in IoT and Smart Systems – IoTSS Vaidy Sunderam Dariusz Mrozek Computer Graphics, Image Processing and Artificial Intelligence – CGIPAI Andres Iglesias Lihua You Alexander Malyshev Hassan Ugail Data-Driven Computational Sciences – DDCS Craig C. Douglas Ana Cortes Hiroshi Fujiwara Robert Lodder Abani Patra Han Yu Machine Learning and Data Assimilation for Dynamical Systems – MLDADS Rossella Arcucci Yi-Ke Guo Meshfree Methods in Computational Sciences – MESHFREE Vaclav Skala Samsul Ariffin Abdul Karim Marco Evangelos Biancolini Robert Schaback

Organization

Rongjiang Pan Edward J. Kansa Multiscale Modelling and Simulation – MMS Derek Groen Stefano Casarin Alfons Hoekstra Bartosz Bosak Diana Suleimenova Quantum Computing Workshop – QCW Katarzyna Rycerz Marian Bubak Simulations of Flow and Transport: Modeling, Algorithms and Computation – SOFTMAC Shuyu Sun Jingfa Li James Liu Smart Systems: Bringing Together Computer Vision, Sensor Networks and Machine Learning – SmartSys Pedro J. S. Cardoso João M. F. Rodrigues Roberto Lam Janio Monteiro Software Engineering for Computational Science – SE4Science Jeffrey Carver Neil Chue Hong Carlos Martinez-Ortiz Solving Problems with Uncertainties – SPU Vassil Alexandrov Aneta Karaivanova Teaching Computational Science – WTCS Angela Shiflet Alfredo Tirado-Ramos Evguenia Alexandrova

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Organization

Uncertainty Quantification for Computational Models – UNEQUIvOCAL Wouter Edeling Anna Nikishova Peter Coveney

Program Committee and Reviewers Ahmad Abdelfattah Samsul Ariffin Abdul Karim Evgenia Adamopoulou Jaime Afonso Martins Giuseppe Agapito Ram Akella Elisabete Alberdi Celaya Luis Alexandre Vassil Alexandrov Evguenia Alexandrova Hesham H. Ali Julen Alvarez-Aramberri Domingos Alves Julio Amador Diaz Lopez Stanislaw Ambroszkiewicz Tomasz Andrysiak Michael Antolovich Hartwig Anzt Hideo Aochi Hamid Arabnejad Rossella Arcucci Khurshid Asghar Marina Balakhontceva Bartosz Balis Krzysztof Banas João Barroso Dominik Bartuschat Nuno Basurto Pouria Behnoudfar Joern Behrens Adrian Bekasiewicz Gebrai Bekdas Stefano Beretta Benjamin Berkels Martino Bernard

Daniel Berrar Sanjukta Bhowmick Marco Evangelos Biancolini Georgiy Bobashev Bartosz Bosak Marian Bubak Jérémy Buisson Robert Burduk Michael Burkhart Allah Bux Aleksander Byrski Cristiano Cabrita Xing Cai Barbara Calabrese Jose Camata Mario Cannataro Alberto Cano Pedro Jorge Sequeira Cardoso Jeffrey Carver Stefano Casarin Manuel Castañón-Puga Mauro Castelli Eduardo Cesar Nicholas Chancellor Patrikakis Charalampos Ehtzaz Chaudhry Chuanfa Chen Siew Ann Cheong Andrey Chernykh Lock-Yue Chew Su Fong Chien Marta Chinnici Sung-Bae Cho Michal Choras Loo Chu Kiong

Neil Chue Hong Svetlana Chuprina Paola Cinnella Noélia Correia Adriano Cortes Ana Cortes Enrique Costa-Montenegro David Coster Helene Coullon Peter Coveney Attila Csikasz-Nagy Loïc Cudennec Javier Cuenca Yifeng Cui António Cunha Ben Czaja Pawel Czarnul Flávio Martins Bhaskar Dasgupta Konstantinos Demestichas Quanling Deng Nilanjan Dey Khaldoon Dhou Jamie Diner Jacek Dlugopolski Simona Domesová Riccardo Dondi Craig C. Douglas Linda Douw Rafal Drezewski Hans du Buf Vitor Duarte Richard Dwight Wouter Edeling Waleed Ejaz Dina El-Reedy

Organization

Amgad Elsayed Nahid Emad Chriatian Engelmann Gökhan Ertaylan Alex Fedoseyev Luis Manuel Fernández Antonino Fiannaca Christos Filelis-Papadopoulos Rupert Ford Piotr Frackiewicz Martin Frank Ruy Freitas Reis Karl Frinkle Haibin Fu Kohei Fujita Hiroshi Fujiwara Takeshi Fukaya Wlodzimierz Funika Takashi Furumura Ernst Fusch Mohamed Gaber David Gal Marco Gallieri Teresa Galvao Akemi Galvez Salvador García Bartlomiej Gardas Delia Garijo Frédéric Gava Piotr Gawron Bernhard Geiger Alex Gerbessiotis Ivo Goncalves Antonio Gonzalez Pardo Jorge González-Domínguez Yuriy Gorbachev Pawel Gorecki Michael Gowanlock Manuel Grana George Gravvanis Derek Groen Lutz Gross Sophia Grundner-Culemann

Pedro Guerreiro Tobias Guggemos Xiaohu Guo Piotr Gurgul Filip Guzy Pietro Hiram Guzzi Zulfiqar Habib Panagiotis Hadjidoukas Masatoshi Hanai John Hanley Erik Hanson Habibollah Haron Carina Haupt Claire Heaney Alexander Heinecke Jurjen Rienk Helmus Álvaro Herrero Bogumila Hnatkowska Maximilian Höb Erlend Hodneland Olivier Hoenen Paul Hofmann Che-Lun Hung Andres Iglesias Takeshi Iwashita Alireza Jahani Momin Jamil Vytautas Jancauskas João Janeiro Peter Janku Fredrik Jansson Jirí Jaroš Caroline Jay Shalu Jhanwar Zhigang Jia Chao Jin Zhong Jin David Johnson Guido Juckeland Maria Juliano Edward J. Kansa Aneta Karaivanova Takahiro Katagiri Timo Kehrer Wayne Kelly Christoph Kessler

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Jakub Klikowski Harald Koestler Ivana Kolingerova Georgy Kopanitsa Gregor Kosec Sotiris Kotsiantis Ilias Kotsireas Sergey Kovalchuk Michal Koziarski Slawomir Koziel Rafal Kozik Bartosz Krawczyk Elisabeth Krueger Valeria Krzhizhanovskaya Pawel Ksieniewicz Marek Kubalcík Sebastian Kuckuk Eileen Kuehn Michael Kuhn Michal Kulczewski Krzysztof Kurowski Massimo La Rosa Yu-Kun Lai Jalal Lakhlili Roberto Lam Anna-Lena Lamprecht Rubin Landau Johannes Langguth Elisabeth Larsson Michael Lees Leifur Leifsson Kenneth Leiter Roy Lettieri Andrew Lewis Jingfa Li Khang-Jie Liew Hong Liu Hui Liu Yen-Chen Liu Zhao Liu Pengcheng Liu James Liu Marcelo Lobosco Robert Lodder Marcin Los Stephane Louise

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Organization

Frederic Loulergue Paul Lu Stefan Luding Onnie Luk Scott MacLachlan Luca Magri Imran Mahmood Zuzana Majdisova Alexander Malyshev Muazzam Maqsood Livia Marcellino Tomas Margalef Tiziana Margaria Svetozar Margenov Urszula Markowska-Kaczmar Osni Marques Carmen Marquez Carlos Martinez-Ortiz Paula Martins Flávio Martins Luke Mason Pawel Matuszyk Valerie Maxville Wagner Meira Jr. Roderick Melnik Valentin Melnikov Ivan Merelli Choras Michal Leandro Minku Jaroslaw Miszczak Janio Monteiro Kourosh Modarresi Fernando Monteiro James Montgomery Andrew Moore Dariusz Mrozek Peter Mueller Khan Muhammad Judit Muñoz Philip Nadler Hiromichi Nagao Jethro Nagawkar Kengo Nakajima Ionel Michael Navon Philipp Neumann

Mai Nguyen Hoang Nguyen Nancy Nichols Anna Nikishova Hitoshi Nishizawa Brayton Noll Algirdas Noreika Enrique Onieva Kenji Ono Eneko Osaba Aziz Ouaarab Serban Ovidiu Raymond Padmos Wojciech Palacz Ivan Palomares Rongjiang Pan Joao Papa Nikela Papadopoulou Marcin Paprzycki David Pardo Anna Paszynska Maciej Paszynski Abani Patra Dana Petcu Serge Petiton Bernhard Pfahringer Frank Phillipson Juan C. Pichel Anna Pietrenko-Dabrowska Laércio L. Pilla Armando Pinho Tomasz Piontek Yuri Pirola Igor Podolak Cristina Portales Simon Portegies Zwart Roland Potthast Ela Pustulka-Hunt Vladimir Puzyrev Alexander Pyayt Rick Quax Cesar Quilodran Casas Barbara Quintela Ajaykumar Rajasekharan Celia Ramos

Lukasz Rauch Vishal Raul Robin Richardson Heike Riel Sophie Robert Luis M. Rocha Joao Rodrigues Daniel Rodriguez Albert Romkes Debraj Roy Katarzyna Rycerz Alberto Sanchez Gabriele Santin Alex Savio Robert Schaback Robert Schaefer Rafal Scherer Ulf D. Schiller Bertil Schmidt Martin Schreiber Alexander Schug Gabriela Schütz Marinella Sciortino Diego Sevilla Angela Shiflet Takashi Shimokawabe Marcin Sieniek Nazareen Sikkandar Basha Anna Sikora Janaína De Andrade Silva Diana Sima Robert Sinkovits Haozhen Situ Leszek Siwik Vaclav Skala Peter Sloot Renata Slota Grazyna Slusarczyk Sucha Smanchat Marek Smieja Maciej Smolka Bartlomiej Sniezynski Isabel Sofia Brito Katarzyna Stapor Bogdan Staszewski

Organization

Jerzy Stefanowski Dennis Stevenson Tomasz Stopa Achim Streit Barbara Strug Pawel Strumillo Dante Suarez Vishwas H. V. Subba Rao Bongwon Suh Diana Suleimenova Ray Sun Shuyu Sun Vaidy Sunderam Martin Swain Alessandro Taberna Ryszard Tadeusiewicz Daisuke Takahashi Zaid Tashman Osamu Tatebe Carlos Tavares Calafate Kasim Tersic Yonatan Afework Tesfahunegn Jannis Teunissen Stefan Thurner

Nestor Tiglao Alfredo Tirado-Ramos Arkadiusz Tomczyk Mariusz Topolski Paolo Trunfio Ka-Wai Tsang Hassan Ugail Eirik Valseth Pavel Varacha Pierangelo Veltri Raja Velu Colin Venters Gytis Vilutis Peng Wang Jianwu Wang Shuangbu Wang Rodrigo Weber dos Santos Katarzyna Wegrzyn-Wolska Mei Wen Lars Wienbrandt Mark Wijzenbroek Peter Woehrmann Szymon Wojciechowski

Maciej Woloszyn Michal Wozniak Maciej Wozniak Yu Xia Dunhui Xiao Huilin Xing Miguel Xochicale Feng Xu Wei Xue Yoshifumi Yamamoto Dongjia Yan Xin-She Yang Dongwei Ye Wee Ping Yeo Lihua You Han Yu Gábor Závodszky Yao Zhang H. Zhang Jinghui Zhong Sotirios Ziavras Italo Zoppis Chiara Zucco Pawel Zyblewski Karol Zyczkowski

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Contents – Part VII

Simulations of Flow and Transport: Modeling, Algorithms and Computation Decoupled and Energy Stable Time-Marching Scheme for the Interfacial Flow with Soluble Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guangpu Zhu and Aifen Li A Numerical Algorithm to Solve the Two-Phase Flow in Porous Media Including Foam Displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filipe Fernandes de Paula, Thiago Quinelato, Iury Igreja, and Grigori Chapiro A Three-Dimensional, One-Field, Fictitious Domain Method for Fluid-Structure Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yongxing Wang, Peter K. Jimack, and Mark A. Walkley Multi Axes Sliding Mesh Approach for Compressible Viscous Flows . . . . . . Masashi Yamakawa, Satoshi Chikaguchi, Shinichi Asao, and Shotaro Hamato Monolithic Arbitrary Lagrangian–Eulerian Finite Element Method for a Multi-domain Blood Flow–Aortic Wall Interaction Problem . . . . . . . . . Pengtao Sun, Chen-Song Zhang, Rihui Lan, and Lin Li Morphing Numerical Simulation of Incompressible Flows Using Seamless Immersed Boundary Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kyohei Tajiri, Mitsuru Tanaka, Masashi Yamakawa, and Hidetoshi Nishida

3

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32 46

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deal.II Implementation of a Two-Field Finite Element Solver for Poroelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhuoran Wang and Jiangguo Liu

88

Numerical Investigation of Solute Transport in Fractured Porous Media Using the Discrete Fracture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohamed F. El-Amin, Jisheng Kou, and Shuyu Sun

102

Adaptive Multiscale Model Reduction for Nonlinear Parabolic Equations Using GMsFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yiran Wang, Eric Chung, and Shubin Fu

116

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Contents – Part VII

Parallel Shared-Memory Isogeometric Residual Minimization (iGRM) for Three-Dimensional Advection-Diffusion Problems . . . . . . . . . . . . . . . . . Marcin Łoś, Judit Munoz-Matute, Krzysztof Podsiadło, Maciej Paszyński, and Keshav Pingali

133

Numerical Simulation of Heat Transfer in an Enclosure with Time-Periodic Heat Generation Using Finite-Difference Method . . . . . . . . . . . . . . . . . . . . Igor Miroshnichenko and Mikhail Sheremet

149

Development of an Object-Oriented Programming Tool Based on FEM for Numerical Simulation of Mineral-Slurry Transport . . . . . . . . . . . . . . . . . Sergio Peralta, Jhon Cordova, Cesar Celis, and Danmer Maza

163

Descending Flight Simulation of Tiltrotor Aircraft at Different Descent Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ayato Takii, Masashi Yamakawa, and Shinichi Asao

178

The Quantization Algorithm Impact in Hydrological Applications: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alessio De Rango, Luca Furnari, Donato D’Ambrosio, Alfonso Senatore, Salvatore Straface, and Giuseppe Mendicino

191

An Expanded Mixed Finite Element Method for Space Fractional Darcy Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huangxin Chen and Shuyu Sun

205

Prediction of the Free Jet Noise Using Quasi-gas Dynamic Equations and Acoustic Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrey Epikhin and Matvey Kraposhin

217

Simulation Based Exploration of Bacterial Cross Talk Between Spatially Separated Colonies in a Multispecies Biofilm Community . . . . . . . . . . . . . . Pavel Zarva and Hermann J. Eberl

228

Massively Parallel Stencil Strategies for Radiation Transport Moment Model Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marco Berghoff, Martin Frank, and Benjamin Seibold

242

Hybrid Mixed Methods Applied to Miscible Displacements with Adverse Mobility Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iury Igreja and Gabriel de Miranda

257

Smart Systems: Bringing Together Computer Vision, Sensor Networks and Machine Learning Learn More from Context: Joint Modeling of Local and Global Attention for Aspect Sentiment Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Siyuan Wang, Peng Liu, Jinqiao Shi, Xuebin Wang, Can Zhao, and Zelin Yin

271

Contents – Part VII

ArtPDGAN: Creating Artistic Pencil Drawing with Key Map Using Generative Adversarial Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SuChang Li, Kan Li, Ilyes Kacher, Yuichiro Taira, Bungo Yanatori, and Imari Sato Interactive Travel Aid for the Visually Impaired: from Depth Maps to Sonic Patterns and Verbal Messages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piotr Skulimowski and Pawel Strumillo Ontology-Driven Edge Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Konstantin Ryabinin and Svetlana Chuprina Combined Metrics for Quality Assessment of 3D Printed Surfaces for Aesthetic Purposes: Towards Higher Accordance with Subjective Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jarosław Fastowicz, Piotr Lech, and Krzysztof Okarma

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285

299 312

326

Path Markup Language for Indoor Navigation . . . . . . . . . . . . . . . . . . . . . . Yang Cai, Florian Alber, and Sean Hackett

340

Smart Fire Alarm System with Person Detection and Thermal Camera. . . . . . Yibing Ma, Xuetao Feng, Jile Jiao, Zhongdong Peng, Shenger Qian, Hui Xue, and Hua Li

353

Data Mining for Big Dataset-Related Thermal Analysis of High Performance Computing (HPC) Data Center . . . . . . . . . . . . . . . . . . . . . . . . Davide De Chiara, Marta Chinnici, and Ah-Lian Kor A Comparison of Multiple Objective Algorithms in the Context of a Dial a Ride Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pedro M. M. Guerreiro, Pedro J. S. Cardoso, and Hortênsio C. L. Fernandes

367

382

Software Engineering for Computational Science Lessons Learned in a Decade of Research Software Engineering GPU Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ben van Werkhoven, Willem Jan Palenstijn, and Alessio Sclocco Unit Tests of Scientific Software: A Study on SWMM . . . . . . . . . . . . . . . . Zedong Peng, Xuanyi Lin, and Nan Niu NUMA-Awareness as a Plug-In for an Eventify-Based Fast Multipole Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laura Morgenstern, David Haensel, Andreas Beckmann, and Ivo Kabadshow

399 413

428

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Contents – Part VII

Boosting Group-Level Synergies by Using a Shared Modeling Framework. . . Yunus Sevinchan, Benjamin Herdeanu, Harald Mack, Lukas Riedel, and Kurt Roth

442

Testing Research Software: A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . Nasir U. Eisty, Danny Perez, Jeffrey C. Carver, J. David Moulton, and Hai Ah Nam

457

APE: A Command-Line Tool and API for Automated Workflow Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vedran Kasalica and Anna-Lena Lamprecht

464

Solving Problems with Uncertainties An Ontological Approach to Knowledge Building by Data Integration. . . . . . Salvatore Flavio Pileggi, Hayden Crain, and Sadok Ben Yahia A Simple Stochastic Process Model for River Environmental Assessment Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hidekazu Yoshioka, Motoh Tsujimura, Kunihiko Hamagami, and Yumi Yoshioka A Posteriori Error Estimation via Differences of Numerical Solutions . . . . . . Aleksey K. Alekseev, Alexander E. Bondarev, and Artem E. Kuvshinnikov

479

494

508

Global Sensitivity Analysis of Various Numerical Schemes for the Heston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emanouil Atanassov, Sergei Kucherenko, and Aneta Karaivanova

520

Robust Single Machine Scheduling with Random Blocks in an Uncertain Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wojciech Bożejko, Paweł Rajba, and Mieczysław Wodecki

529

Empirical Analysis of Stochastic Methods of Linear Algebra . . . . . . . . . . . . Mustafa Emre Şahin, Anton Lebedev, and Vassil Alexandrov Wind Field Parallelization Based on Python Multiprocessing to Reduce Forest Fire Propagation Prediction Uncertainty . . . . . . . . . . . . . . . . . . . . . . Gemma Sanjuan, Tomas Margalef, and Ana Cortés Risk Profiles of Financial Service Portfolio for Women Segment Using Machine Learning Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jessica Ivonne Lozano-Medina, Laura Hervert-Escobar, and Neil Hernandez-Gress Multidimensional BSDEs with Mixed Reflections and Balance Sheet Optimal Switching Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rachid Belfadli, M’hamed Eddahbi, Imade Fakhouri, and Youssef Ouknine

539

550

561

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Contents – Part VII

xix

Teaching Computational Science Modeling and Automatic Code Generation Tool for Teaching Concurrent and Parallel Programming by Finite State Processes . . . . . . . . . . . . . . . . . . Edwin Monteiro, Kelvinn Pereira, and Raimundo Barreto

593

Automatic Feedback Provision in Teaching Computational Science . . . . . . . . Hans Fangohr, Neil O’Brien, Ondrej Hovorka, Thomas Kluyver, Nick Hale, Anil Prabhakar, and Arti Kashyap

608

Computational Science vs. Zombies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valerie Maxville and Brodie Sandford

622

Supporting Education in Algorithms of Computational Mathematics by Dynamic Visualizations Using Computer Algebra System . . . . . . . . . . . . Włodzimierz Wojas and Jan Krupa

634

Teaching About the Social Construction of Reality Using a Model of Information Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loren Demerath, James Reid, and E. Dante Suarez

648

Bringing Harmony to Computational Science Pedagogy . . . . . . . . . . . . . . . . Richard Roth and William Pierce

661

UNcErtainty QUantIficatiOn for ComputationAl MdeLs Intrusive Polynomial Chaos for CFD Using OpenFOAM . . . . . . . . . . . . . . . Jigar Parekh and Roel Verstappen Distributions of a General Reduced-Order Dependence Measure and Conditional Independence Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mariusz Kubkowski, Małgorzata Łazȩcka, and Jan Mielniczuk MCMC for Bayesian Uncertainty Quantification from Time-Series Data . . . . Philip Maybank, Patrick Peltzer, Uwe Naumann, and Ingo Bojak Uncertainty Quantification for Multiscale Fusion Plasma Simulations with VECMA Toolkit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jalal Lakhlili, Olivier Hoenen, Onnie O. Luk, and David P. Coster Sensitivity Analysis of Soil Parameters in Crop Model Supported with High-Throughput Computing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Gasanov, Anna Petrovskaia, Artyom Nikitin, Sergey Matveev, Polina Tregubova, Maria Pukalchik, and Ivan Oseledets A Bluff-and-Fix Algorithm for Polynomial Chaos Methods . . . . . . . . . . . . . Laura Lyman and Gianluca Iaccarino

677

692 707

719

731

742

xx

Contents – Part VII

Markov Chain Monte Carlo Methods for Fluid Flow Forecasting in the Subsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, and Arunasalam Rahunanthan Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

757

773

Simulations of Flow and Transport: Modeling, Algorithms and Computation

Decoupled and Energy Stable Time-Marching Scheme for the Interfacial Flow with Soluble Surfactants Guangpu Zhu

and Aifen Li(&)

School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China [email protected]

Abstract. In this work, we develop an efficient energy stable scheme for the hydrodynamics coupled phase-field surfactant model with variable densities. The thermodynamically consistent model consists of two Cahn–Hilliard–type equations and incompressible Navier–Stokes equation. We use two scalar auxiliary variables to transform nonlinear parts in the free energy functional into quadratic forms, and then they can be treated efficiently and semi-implicitly. A splitting method based on pressure stabilization is used to solve the Navier– Stokes equation. By some subtle explicit-implicit treatments to nonlinear convection and stress terms, we construct a first-order energy stable scheme for the two-phase system with soluble surfactants. The developed scheme is efficient and easy-to-implement. At each time step, computations of phase-field variables, the velocity and pressure are decoupled. We rigorously prove that the proposed scheme is unconditionally energy stable. Numerical results confirm that our scheme is accurate and energy stable. Keywords: Surfactant Navier–stokes


Interfacial flow
Phase-field modeling


1 Introduction Surfactants, interface active agents, are known to lower the interfacial tension and allow for the formation of emulsion [1, 2]. Commonly-used surfactants are amphiphilic compounds, meaning they contain both hydrophilic heads and hydrophobic tails [1, 3]. This special molecular composition enables surfactants to selectively absorb on fluid interfaces. Surfactants play a crucial role in everyday life and many industrial processes, such as the cleanser essence, the crude oil recovery and pharmaceutical materials, thus having an understanding of their behavior is a necessity. Numerical simulation is taking an increasingly significant position in investigating the interfacial phenomena, as it can provide easier access to some quantities such as surfactant concentration, pressure and velocity, which are difficult to measure experimentally. However, the computational modeling of interfacial dynamics with surfactants remains a challenging task.

© Springer Nature Switzerland AG 2020 V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 3–17, 2020. https://doi.org/10.1007/978-3-030-50436-6_1

4

G. Zhu and A. Li

The phase-field model is an effective modeling and simulation tool in investigating interfacial phenomena and it has been extensively used with much successes [4]. This method introduces a phase-field variable to distinguish two pure phases. The interface is treated as a thin layer, inside which the phase-field variable varies continuously [5, 6]. Unlike shape interface models, the phase-field model does not need to track the interface explicitly, and the interface can be implicitly and automatically captured by the evolution of phase-field variable. Therefore, the computations and analysis of the phase-field model are easier than other models [7, 8]. The phase-field model was first used to study the dynamics of phase separation with surfactants in [9]. Two phase-field variables were introduced in their work. Since then, a variety of phase-field surfactant models have been proposed and reviews of these models can refer to [10–12]. Here we only highlight two representative works. The authors in [13] introduced the logarithmic Floy-Huggins potential to restrict the range of surfactant concentration. A nonlinear coupling surface energy potential was used to account for the high surfactant concentration along the fluid interface. An enthalpic term was also adopted to stabilize the phase-field model and control the surfactant solubility in the bulk phases. Their model can describe realistic adsorption isotherms, e.g., Langmuir isotherm, in thermodynamic equilibrium. In [14], the authors analyzed the well-posedness of the phase-field surfactant model proposed in [13], and provided strong evidence that the model was mathematically ill-posed for a large set of physically relevant parameters. They made critical modifications to the model and substantially increased the domain of validity. In this study, we will use this modified model to describe a binary fluid-surfactant system. Numerically, it is a challenging issue to discretize the strong couplings between two phase-field variables. The introduction of hydrodynamics will further increase the complexity for the development of numerical schemes. Several attempts have been made to solve the interfacial flows with surfactants [15–18], but none of them can provide the energy stability for numerical schemes in theory. Most recently, we constructed a first-order and a second-order schemes, which are linear and totally decoupled, for a phase-field surfactant model with fluid flow [19]. However, this study only considered the case of matched density and viscosity, which greatly reduces difficulties in algorithm developments. Thus, the main purpose of this study is to construct an efficient, easy-to-implement and energy stable scheme for the hydrodynamics coupled phase-field surfactant model with variable densities. The rest of this paper is organized as follows. In Sect. 2, we describe a hydrodynamics coupled phase-field surfactant model with variable densities. In Sect. 3, we develop an efficient energy stable scheme carry out the energy stability for the proposed scheme. Several numerical experiments are investigated in Sect. 4 and the paper is finally concluded in Sect. 5.

2 Governing Equation In this section, we consider a typical phase-field surfactant model in [14, 19] for a twophase system with surfactants

Decoupled and Energy Stable Time-Marching Scheme

Z Ef ðu; /; wÞ ¼

5

!  2 2 2 w /  1 Cn2 w/ þ dX; ð2:1Þ jr/j2 þ F ð/Þ þ PiGðwÞ  4 4 4Ex

where F ð/Þ is the double well potential and GðwÞ the logarithmic Flory–Huggins potential, 

/2  1 F ð/ Þ ¼ 4

2 ;

GðwÞ ¼ w ln w þ ð1  wÞ lnð1  wÞ:

Two phase-field variables are used in the free energy functional. The first phase-field variable / uses two constants (–1 and 1) to distinguish two phases, and it varies continuously across the interface between –1 and 1. The other phase-field variable w is used to represent the surfactant concentration. The parameter Cn determines the interfacial thickness and Pi is a temperature-dependent parameter. More details of the free energy functional can refer to [6] and [19]. Although both the double well potential and the Flory–Huggins potential are bounded from below, the latter is not always positive in the whole domain. Thus, we add a zero term PiB  PiB to the free energy functional, and rewrite (2.1) into  2 2 ! Z  2 2 w /  1 Cn w/ Ef ð/; wÞ ¼  dX jr/j2 þ F ð/Þ þ PiðGðwÞ þ BÞ þ 4 4Ex 4  PiBjXj; ð2:2Þ where the positive constant B ensures GðwÞ þ B [ 0; and B = 1 is adopted in this study. Note that the free energy is not changed due to the introduction of the zero term PiB  PiB: We now use the scalar auxiliary variable (SAV) approach [12, 20] to transform the free functional into a new form. Through the simple substitution of scalar variables, the nonlinear parts of the free energy are transformed into quadratic forms of new scalar variables. More precisely, we define two scalar variables U¼

pffiffiffiffiffiffiffiffiffiffiffiffi Eu ð/Þ;

V¼

pffiffiffiffiffiffiffiffiffiffiffiffi Ev ðwÞ;

ð2:3Þ

Where Z Eu ð/Þ ¼

Z F ð/ÞdX;

Ev ðwÞ ¼

ðGðwÞ þ BÞdX:

Then the free energy can be transformed into Z Ef ð/; w; U; V Þ ¼

!  2 2 We Cn2 w/2 2 2 w / 1 qjuj þ jr/j  þ dX þ U 2 + PiV 2  PiBjXj; 2 4 4 4Ex

ð2:4Þ

6

G. Zhu and A. Li

Through the functional derivatives of Ef with respect to phase-field variables / and w, we can obtain chemical potentials w/ and ww Cn2 U w/ 1  w/W; Ut ¼ pffiffiffiffiffiffiffiffiffiffiffiffi D/ þ pffiffiffiffiffiffiffiffiffiffiffiffi F 0 ð/Þ þ w/ ¼  2Ex 2 Eu ð/Þ 2 Eu ð/Þ

Z

F 0 ð/Þ/t dX; ð2:5Þ

PiV /2 W2  ; ww ¼ pffiffiffiffiffiffiffiffiffiffiffiffi G0 ðwÞ þ 4Ex 4 Ev ðwÞ

1 Vt ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 Ev ðwÞ

Z

G0 ðwÞwt dX:

ð2:6Þ

Note that /2  1 are denoted as Win (2.5) and (2.6). Evolutions of phase-field variables / and w can be described by the conserved Cahn–Hilliard–type equations [14, 21], 1 Dw/ ; Pe/

ð2:7Þ

1 r
Mw rww ; Pew

ð2:8Þ

/t þ r
ðu/Þ ¼ wt þ r
ðuwÞ ¼

where Pe/ and Pew are Péclet numbers. A degenerate mobility Mw ¼ wð1  wÞ; which vanishes at the extreme points w ¼ 0 and w ¼ 1; is adopted to combine with the logarithmic chemical potential ww : Eqs. (2.6)–(2.9) are coupled to the Navier–Stokes equation in the form [4, 14] qut þ qu
ru þ J
ru 

  1 1 r
gDðuÞ þ rp þ /rw/ þ wrww ¼ 0; Re ReCaCn

r
u ¼ 0;

ð2:9Þ ð2:10Þ

   where DðuÞ ¼ ru þ rT u; and J ¼ kq  1 rw/ 2Pe/ : u is the velocity field, p is the pressure, Re is the Reynolds number and Ca is the Capillary number. We usually assume the density q and viscosity g has the following linear relations, q¼

1  kq 1 þ kq /þ ; 2 2

g¼

1  kg 1 þ kg /þ : 2 2

where kq and kg are density and viscosity ratios, respectively. In particular, if we consider the body force, e.g., the gravitational force, the dimensionless momentum equation read qut þ qu
ru þ J
ru 

 1 1  r
gDðuÞ þ rp þ /rw/ þ wrww  qg ¼ 0; Re BoCn ð2:11Þ

Decoupled and Energy Stable Time-Marching Scheme

7

where Bo ¼ ReCa is the Bond number, and g is the unit vector denoting the direction of body force. Periodic boundary conditions or the following boundary conditions @n /n þ 1 ¼ rwn/þ 1
n ¼ rwnwþ 1
n ¼ u ¼ @n pn þ 1 ¼ 0; on C; can be used to close the above governing system. Here C denotes boundaries of the domain. The total energy Etot of the hydrodynamic system (2.5)–(2.10) is the sum of kinetic energy Ek and free energy Ef  2 ! We Cn2 w/2 w /2  1 2 2 qjuj þ  dX jr/j þ 2 4 4Ex 4

Z Etot ðu; /; w; U; V Þ ¼

þ U 2 + PiV 2  PiBjXj; where W e ¼ ReCaCn, and we can easily derive the following energy dissipation law. Z d 1 Etot ¼  dt Pe/ Z 1  Pew

  rw/ 2 dX pffiffiffiffiffiffiffi   Mw rww 2 dX  CaCn 2

Z

pffiffiffi   gDðuÞ2 dX  0:

Next, we will develop an efficient time-marching scheme for the above governing system and carry out the energy estimate. To simplify the presentation, in the next section, we will take (2.9) as an example to construct the desired scheme.

3 Numerical Scheme 3.1

Energy Stable First-Order Scheme

We now present a first-order time-marching scheme to solve the governing system in Sect. 2. To deal with the case of nonmatching density, a cut-off function [4] is defined as ( ~n þ 1

/

¼

/n þ 1   sign /n þ 1

j/n þ 1 j  1; j/n þ 1 j [ 1:

Given wn ;/n ; un and pn, the scheme (3.1) calculates wn þ 1 ;/n þ 1 ; un+1 and pn+1 for n  0 in three steps. In step 1, we update wn þ 1 and /n þ 1 by solving   wn þ 1  wn 1 þ r
un wn  r
Mwn rwnwþ 1 ¼ 0; Pew dt

ð3:1aÞ

8

G. Zhu and A. Li

PiV n þ 1 0 n ð/n Þ2 ðW n Þ2 ffi  ; wnwþ 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð w Þ þ G 4Ex 4 Ev ðwn Þ Vn þ1  Vn 1 ffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi dt 2 Ev ðwn Þ

Z

G0 ðwn Þ

wn þ 1  wn dX; dt

  /n þ 1  /n 1 þ r
un /n  Dwn/þ 1 ¼ 0; Pe/ dt wn/þ 1 ¼ 

ð3:1bÞ

ð3:1cÞ

ð3:1dÞ

 Cn2 Un þ 1 wn þ 1 /n þ 1 1 n þ 1 n  n þ 1 n 0 D/n þ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  w ð / Þ þ W / þ /n ; F n 2 2 2Ex Eu ð/ Þ ð3:1eÞ Un þ 1  Un 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt 2 Eu ð/n Þ

Z

F 0 ð/n Þ

/n þ 1  /n dX; dt

ð3:1fÞ

with periodic boundary conditions or the following boundary conditions @n wnwþ 1 ¼ @n /n þ 1 ¼ @n wn/þ 1 ¼ 0; on C; where un is the intermediate velocity un ¼ un 

dtwn dt/n nþ1 rw  rwn/þ 1 : w Weqn Weqn

ð3:1gÞ

In step 2, we update un+1 by solving [22] 8 nþ1      un  n n  1 nu > > þ q u
run þ 1 þ Jn þ 1
run þ 1  r
gn D un þ 1 þ r 2pn  pn1 >q > Re dt > < 1þk   1  n q n nþ1 nþ1 r
un un þ 1 ¼ 0; þ þ / rw/ þ w rww > > > 4 We > > : nþ1 u ¼ 0; on C;

ð3:1hÞ where

Decoupled and Energy Stable Time-Marching Scheme

Jn þ 1 ¼ g

nþ1

kq  1 1  kq ~ n þ 1 1 þ kq / ; rwn/þ 1 ; qn þ 1 ¼ þ 2Pe/ 2 2 1  kg ~ n þ 1 1 þ kg / : ¼ þ 2 2

9

ð3:1iÞ

In step 3, we update pn+1 by solving the pressure Poisson equation with a constant coefficient [4, 23] 8   <  D p n þ 1  p n ¼  v r
un þ 1 ; dt : rpn þ 1
n ¼ 0; on C;

ð3:1jÞ

  where v ¼ 12 min 11 ; kq :

  Remark 3.1. (1) Computations of wn þ 1 ; /n þ 1 ; un+1 and pn+1 are decoupled, which indicate that the scheme (3.1) is efficient and easy-to-implement. At each time step, un+1 and pn+1 can be obtained by solving only two elliptic equations; Moreover, Vn+1 and Un+1 do not involve any extra computational cost, since they can be calculated explicitly once we obtain wn þ 1 and /n þ 1 : (2) In the explicit convective velocity un ; we introduce a firstorder stabilization term [24], which plays a dominant role in decoupling the computation   of wn þ 1 ; /n þ 1 from un+1 and constructing the unconditionally energy stable scheme. Theorem 3.1. The scheme (3.1) is unconditionally energy stable, and satisfies the following discrete energy dissipation law: nþ1 n Etot  Etot 



 2 ffi  dt qffiffiffiffiffiffinffi n þ 1

2 dt n þ 1 2 dtCaCn

pffiffiffiffi  gn D un þ 1  0; ð3:2Þ

Mw rww 

rw/  Pew Pe/ 2

where n Etot ¼

dt2 W e We n Cn2 q ; jun j2 þ krpn k2 þ kr/n k2 þ ðU n Þ2 þ PiðV n Þ2 2 2v 4 1 1  n w ; j/n j2  wn ; jW n j2  PiBjXj; þ 4Ex 4

here k
k denotes the L2-norm in X. Now we will rigorously prove the discrete energy dissipation law in (3.2). We first introduce an intermediate kinetic energy [25] as n Ek; ¼

W e  n n n q u ; u : 2

n The difference between Ekn þ 1 and Ek; is estimated as

ð3:3Þ

10

G. Zhu and A. Li

W e  n þ 1  n þ 1 2 W e  n  n 2 n q q ; u Ekn þ 1  Ek; ¼ ; u  2 2    2 W e n  n þ 1 2  n 2 W e nþ1 q ; u q ¼  u  q n ; u n þ 1  þ 2 2 2 We  n þ 1  2  n n þ 1  n þ 1 W e  n  n þ 1 n q ; u q ¼ We q u   u ; u  un  þ  qn ; un þ 1  : 2 2

ð3:4Þ Substituting (3.1i) into (3.1a), we obtain the following identity nþ1

q

    dt 1  kq r
/n un  dtr
Jn þ 1 : q ¼ 2 n

ð3:5Þ

We can also easily derive from (3.1g) that      W eqn un þ 1  un þ dt wn rwnwþ 1 þ /rwn/þ 1 ¼ W eqn un þ 1  un :

ð3:6Þ

Using the identity (3.6), we have         W eqn un þ 1  un ¼ dtCaCnr
gn D un þ 1  dtW er 2pn  pn1  dtW e qn un
run þ 1    dtW e 1 þ kq  dtW eJn þ 1
run þ 1  r
un un þ 1 : 4

ð3:7Þ By taking the L2 inner product of (3.7) with un+1, and using (3.4) and the following identities 2    dtW e     r
qn un ; un þ 1   dtW e qn un
run þ 1 ; un þ 1 ¼ 2      2  n n   n þ 1 2 dtW e 1 þ kq  dtW e 1  kq    ¼ r
/ u ; u r
un ; un þ 1  ; þ 4 4    nþ1   1
r un þ 1 þ r
Jn þ 1 un þ 1 ; un þ 1 ¼ 0: J 2

Decoupled and Energy Stable Time-Marching Scheme

11

we can derive that  2    ffi  dtCaCn

pffiffiffiffi gn D un þ 1 dtW e qn un
run þ 1 ; un þ 1 2    2  nþ1  dtW e 1 þ kq  nþ1 nþ1   dtW e J r
un ; un þ 1 
ru ; u   4    dtW e pn þ 1  2pn þ pn1 ; r
un þ 1 þ dtWe pn þ 1 ; r
un þ 1   2 dtW e 1  kq  2   W e  n  n þ 1   un   q; u r
/n un þ r
Jn þ 1 ; un þ 1  2 2 2  n þ 1  2  nþ1  ffiffiffiffi ffi p dtCaCn

gn D u

dtW e p ¼  2pn þ pn1 ; r
un þ 1 2 2  nþ1  W e  n  nþ1 q ; u þ dtW e p ; r
un þ 1   un  : 2 ð3:8Þ

n Ekn þ 1  Ek; ¼

Using the Eq. (3.1g), we obtain  2    We n  n W e  n  n 2 n q ; u jun j2 ¼ W eqn un  un ; un  q ; u  un  Ek;  Ekn ¼ 2 2  We   2 n n qn ; un  un  ¼ dt w r
wnwþ 1 þ / rwn/þ 1 ; un  2   2  n n þ 1  n n þ 1 W e  n  n q ; u  un  : ¼ dt r
wu ; ww þ dt r
/u ; w/  2 ð3:9Þ Summing up Eqs. (3.8) and (3.9), we get  2   ffi  dtCaCn

pffiffiffiffi gn D un þ 1 dtW e pn þ 1  2pn þ pn1 ; r
un þ 1 Ekn þ 1  Ekn ¼  2    nþ1      ; r
un þ 1 þ dt r
wun ; wnwþ 1 þ dt r
/un ; wn/þ 1 þ dtW e p 2 W e  n  n 2 W e  n  n þ 1 q ; u q ; u  un  :   un   2 2 ð3:10Þ By taking the L2 inner product of (3.1j) with dt2We(pn+1 − 2pn+ pn − 1)/v and withdt Wepn+1/v separately, we obtain 2



    

 dt2 W e 

r pn þ 1  pn 2  r pn  pn1 2 þ r pn þ 1  2pn þ pn1 2 2v   ¼ dtW e pn þ 1  2pn þ pn1 ; r
un þ 1 ; ð3:11Þ

12

G. Zhu and A. Li

and



   dt2 W e 

rpn þ 1 2 krpn k2 þ r pn þ 1  pn 2 ¼ dtW e pn þ 1 ; r
un þ 1 : 2v

ð3:12Þ

Combining (3.11) and (3.12), yields      dtW e pn þ 1  2pn þ pn1 ; r
un þ 1 þ dtW e pn þ 1 ; r
un þ 1 2

 

dt2 W e 

rpn þ 1 2 krpn k2  dt W e r pn  pn1 2 ¼ 2v 2v  

dt2 W e

r pn þ 1  2pn þ pn1 2 : þ 2v

ð3:13Þ

We take the difference of (3.1j) at step tn+1 and tn, pair the resulting equation with dt We(pn+1 − 2pn+ pn − 1)/(2v) then take integration by parts for both sides to derive 2



    

dt2 W e

r pn þ 1  2pn þ pn1 2  v W e un þ 1  un 2  W e qn ; un þ 1  un  2 : ð3:14Þ 2v 2 4

Summing up Eqs. (3.10), (3.13) and (3.14), and using the triangle inequality 2 W e  n  n 2 W e  n  n þ 1 2 W e  n  n þ 1 q; u q ; u  un   q ; u  un  þ  un  : 2 2 4 ð3:15Þ we can derive that  2 dt2 W e  n þ 1 2 ffi  dtCaCn

rp

krpn k2

pffiffiffiffi Ekn þ 1  Ekn   gn D un þ 1  2 2v ð3:16Þ    n n þ 1  n n þ 1 þ dt r
wu ; ww þ dt r
/u ; w/ : By taking the inner product of (3.1a) with dtwnwþ 1 ; we can easily derive that 

   dt

qffiffiffiffiffiffinffi n þ 1 2 wn þ 1  wn ; wnwþ 1 þ dt r
un wn ; wnwþ 1 ¼ 

Mw rww : ð3:17Þ Pew

  By taking the inner product of (3.1b) with  wn þ 1  wn ; we can derive that    1  n 2 nþ1  wn þ 1  wn ; wnwþ 1 ¼  Pi V n þ 1 an ; wn þ 1  wn   wn j/ j ; w 4Ex 1  n 2 nþ1 n w : þ jW j ; w 4

ð3:18Þ

Decoupled and Energy Stable Time-Marching Scheme

where an ¼ G0 ðwn Þ obtain

13

ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Ev ðwn Þ: Taking the inner product of (3.1c) with 2dtPiVn+1 to

h 2  2 i   Pi V n þ 1 ðV n Þ2 þ V n þ 1  V n ¼ Pi V n þ 1 an ; wn þ 1  wn :

ð3:19Þ

Summing up Eqs. (3.17)–(3.19), we get

h   2  2 i  dt

qffiffiffiffiffiffinffi n þ 1 2 Pi V n þ 1 ðV n Þ2 þ V n þ 1  V n ¼ dt r un wn ; wnwþ 1 

Mw rww

Pew   1 1  j/n j2 ; wn þ 1  wn þ jW n j2 ; wn þ 1  wn : 4Ex 4

ð3:20Þ By taking the inner product of (3.1d) with dtwn/þ 1 ; we have 

   dt

n þ 1 2 /n þ 1  /n ; wn/þ 1 þ dt r
un /n ; wn/þ 1 ¼ 

rw/ : Pe/

ð3:21Þ

  By taking the inner product of (3.1e) with  /n þ 1  /n ; we can derive that     Cn2  n þ 1 r/ ; r/n þ 1  r/n  U n þ 1 bn ; /n þ 1  /n  /n þ 1  /n ; wn/þ 1 ¼  2  1    1  nþ1 nþ1 nþ1 w  / ; /  /n þ wn þ 1 W n /n þ 1 þ /n ; /n þ 1  /n 2Ex 2



  Cn2 

r/n þ 1 2 kr/n k2 þ r/n þ 1  r/n 2  U n þ 1 bn ; /n þ 1  /n ¼ 4   2 i 1 h n þ 1  n þ 1 2  n þ 1 w  ; / ; j/n j2 þ wn þ 1 ; /n þ 1  /n   w 4Ex   2 i 1 h n þ 1  n þ 1 2  n þ 1 w þ ; W ; jW n j2  wn þ 1 ; W n þ 1  W n  :  w 4 ð3:22Þ where bn ¼ F 0 ð/n Þ obtain h

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eu ð/n Þ: Taking the inner product of (3.1f) with 2dtUn+1 to

2  2 i  n þ 1 n n þ 1  b ; /  /n : U n þ 1 ðU n Þ2 þ U n þ 1  U n ¼ U

ð3:23Þ

Summing up Eqs. (3.20)–(3.23), and dropping off some positive terms, we have

14

G. Zhu and A. Li

h i h i

  Cn2 

r/n þ 1 2 kr/n k2 þ U n þ 1 2 ðU n Þ2 þ Pi V n þ 1 2 ðV n Þ2 4 i 1 h i  2  1 h n þ 1  n þ 1 2  n w w n þ 1 ; W n þ 1   w n ; j W n j 2 ; /  w ; j/n j2  þ 4Ex 4



      dt

qffiffiffiffiffiffinffi n þ 1 2 dt n þ 1 2 

Mw rww 

rww dt r un wn ; wnwþ 1  dt r un /n ; wn/þ 1 : Pew Pew

ð3:24Þ Finally, combining (3.16) and (3.24), we arrive at the desired result.

4 Numerical Results To implement the scheme (3.1), we use a finite difference method on staggered grids to discretize space. We pay special attention to the discretization of the convection terms in the Cahn-Hilliard and Navier-Stokes equations. A composite high resolution scheme, known as the MINMOD scheme, is used to reduce the undershoot and overshoot around the interface. The computations of wn þ 1 ;/n þ 1 ; un+1 and pn+1 can be totally decoupled if we replace wn þ 1 in (3.1e) with wn : The simplified scheme is extremely efficient and easy-to-implement. However, this simplification will definitely destroy the unconditional energy stability of our scheme. The implementation of such a simplified scheme requires small time step-sizes to obtain the desired accuracy and energy stability. The above scheme is adopted in [26] and numerical results demonstrate the energy stability of the proposed scheme. Here we will not present these results due to the limit of article length. We simulate the droplet deformation under the horizontal body force and a shear flow in a computational domain X = [0, 3]  [0, 1]. Periodic boundary conditions are applied on the left and right sides. A circular droplet with the radius of r = 0.3 is initially placed at (1, 0.5). Other simulation parameters are listed as follows: Pe/ ¼ 10; Pew ¼ 100; Re ¼ 10; Bo ¼ 1; Cn ¼ 0:01; Ex ¼ 1; Pi ¼ 0:1227; kq ¼ kv ¼ 10: Figure 1 shows the time evolution plots of droplet deformation and surfactant concentration. The droplet continuously deforms and moves forward under the action of the shear flow and the body force. We can divide the whole process into two stages based on the droplet deformation and surfactant migration. At the first stage, the body force has limited effect on the droplet deformation compared with the shear flow. Surfactants gradually migrate toward droplet tips, as shown in Fig. 1(b), resulting in the non-uniformity of interfacial tension along the interface. As we mentioned before, the surfactant concentration gradient induces the Marangoni stress, which will resist the further migration of surfactants. However, the Marangoni stress is not large enough to balance the effect of shear flow, and surfactants continue to move toward tips. In Fig. 1 (c), surfactants are swept into the bulk phases when concentration reaches the maximum at the droplet tips. At the second stage, the body force plays an important role in

Decoupled and Energy Stable Time-Marching Scheme

15

the droplet deformation and surfactant migration. In Fig. 1(d), surfactants on the tip A are slowly swept towards the ABC segment under the effect of the body force. Surfactants along the ADC segment continuously move to the tips under the combined action of the shear flow and the body force.

Fig. 1. Evolutions of pressure field (background color), quiver plot of velocity (u, v), phase-field variables / and w. For each subfigure, the right is the profile of w. (wb = 1.5  10−2).

1

0.8

0.6

0.4

0.2

0 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Fig. 2. Profiles of phase-field variable / at t = 1 (left) and t = 2 (right). (black dash line: wb= 110−6; blue solid line: wb= 1.5  10−2; red solid line: wb= 510−2) (Color figure online)

16

G. Zhu and A. Li

Figure 2 demonstrates the profiles of phase-field variable / at three different wb values. A more prolate profile of / is observed for a higher surfactant bulk concentration, which confirms the effect of surfactants in reducing the interfacial tension.

5 Conclusion The numerical approximation of incompressible and immiscible two-phase flows with soluble surfactants is the main topic in this paper. An efficient, accurate and energy stable time-marching scheme is constructed using the SAV approach for the hydrodynamics coupled phase-field surfactant model with variable densities. We rigorously prove the unconditional energy stability of the semi-implicit scheme. Numerical results demonstrate the energy stability of the proposed scheme.

References 1. Khatri, S., Tornberg, A.-K.: An embedded boundary method for soluble surfactants with interface tracking for two-phase flows. J. Comput. Phys. 256, 768–790 (2014) 2. Yang, X.: Numerical approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system. J. Sci. Comput. 74, 1533–1553 (2018) 3. Liu, H., Zhang, Y.: Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 229(24), 9166–9187 (2010) 4. Shen, J., Yang, X.: Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53(1), 279–296 (2015) 5. Yue, P., Feng, J.J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2014) 6. Zhu, G., Kou, J., Yao, J., Li, A., Sun, S.: A phase-field moving contact line model with soluble surfactants. J. Comput. Phys. 405, 109170 (2020) 7. Xu, X., Di, Y., Yu, H.: Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines. J. Fluid Mech. 849, 805–833 (2018) 8. Zhu, G., Chen, H., Yao, J., Sun, S.: Efficient energy-stable schemes for the hydrodynamics coupled phase-field model. Appl. Math. Model. 70, 82–108 (2019) 9. Laradji, M., Guo, H., Grant, M., Zuckermann, M.J.: The effect of surfactants on the dynamics of phase separation. J. Phys.: Condens. Matt. 4(32), 6715 (1992) 10. Li, Y., Kim, J.: A comparison study of phase-field models for an immiscible binary mixture with surfactant. Eur. Phys. J. B 85(10), 340 (2012) 11. Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid– surfactant phase field model. Comput. Method Appl. M 318, 1005–1029 (2017) 12. Zhu, G., Kou, J., Sun, S., Yao, J., Li, A.: Decoupled, energy stable schemes for a phase-field surfactant model. Comput. Phys. Commun. 233, 67–77 (2018) 13. Van der Sman, R., Van der Graaf, S.: Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol. Acta 46(1), 3–11 (2016) 14. Engblom, S., Do-Quang, M., Amberg, G., Tornberg, A.-K.: On diffuse interface modeling and simulation of surfactants in two-phase fluid flow. Commun. Comput. Phys. 14(4), 879– 915 (2013)

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15. Booty, M., Siegel, M.: A hybrid numerical method for interfacial fluid flow with soluble surfactant. J. Comput. Phys. 229(10), 3864–3883 (2010) 16. Pätzold, G., Dawson, K.: Numerical simulation of phase separation in the presence of surfactants and hydrodynamics. Phys. Rev. E 52(6), 6908 (1995) 17. Teigen, K.E., Song, P., Lowengrub, J., Voigt, A.: A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230(2), 375–393 (2011) 18. Yun, A., Li, Y., Kim, J.: A new phase-field model for a water–oil-surfactant system. Appl. Math. Comput. 229, 422–432 (2014) 19. Zhu, G., Kou, J., Sun, S., Yao, J., Li, A.: Numerical approximation of a phase-field surfactant model with fluid flow. J. Sci. Comput. 80(1), 223–247 (2019) 20. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018) 21. Zhu, G., Kou, J., Yao, B., Wu, Y.-S., Yao, J., Sun, S.: Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants. J. Fluid Mech. 879, 327–359 (2019) 22. Feng, X., Kou, J., Sun, S.: A novel energy stable numerical scheme for Navier-Stokes-CahnHilliard two-phase flow model with variable densities and viscosities. In: Shi, Y., et al. (eds.) ICCS 2018. LNCS, vol. 10862, pp. 113–128. Springer, Cham (2018). https://doi.org/10. 1007/978-3-319-93713-7_9 23. Gao, M., Wang, X.-P.: An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity. J. Comput. Phys. 272, 704–718 (2014) 24. Shen, J., Yang, X.: Decoupled energy stable schemes for phase-field models of two-phase complex fluids. SIAM J. Sci. Comput. 36(1), B122–B145 (2014) 25. Kou, J., Sun, S.: Thermodynamically consistent modeling and simulation of multicomponent two-phase flow with partial miscibility. Comput. Method Appl. M 331, 623–649 (2018) 26. Zhu, G., Li, A.: Interfacial dynamics with soluble surfactants: a phase-field two-phase flow model with variable densities. Adv. Geo-Energy Res. 4(1), 86–98 (2020)

A Numerical Algorithm to Solve the Two-Phase Flow in Porous Media Including Foam Displacement Filipe Fernandes de Paula1,2(B) , Thiago Quinelato1 , Iury Igreja1,2 , and Grigori Chapiro1,2 2

1 Laborat´ orio de Matem´ atica Aplicada – LAMAP, Juiz de Fora, MG, Brazil Graduate Program on Computational Modeling, Federal University of Juiz de Fora, Juiz de Fora, MG, Brazil [email protected], {thiago.quinelato,iuryigreja,grigori}@ice.ufjf.br

Abstract. This work is dedicated to simulating the Enhanced Oil Recovery (EOR) process of foam injection in a fully saturated reservoir. The presence of foam in the gas-water mixture acts in controlling the mobility of the gas phase, contributing to reduce the effects of fingering and gravity override. A fractional flow formulation based on global pressure is used, resulting in a system of Partial Differential Equations (PDEs) that describe two coupled problems of distinct kinds: elliptic and hyperbolic. The numerical methodology is based on splitting the system of equations into two sub-systems that group equations of the same kind and on applying a hybrid finite element method to solve the elliptic problem and a high-order finite volume method to solve the hyperbolic equations. Numerical results show good efficiency of the algorithm, as well as the remarkable ability of the foam to increase reservoir sweep efficiency by reducing gravity override and fingering effects. Keywords: EOR · Hybrid mixed methods Foam injection · Mobility reduction

1

· Finite volume methods ·

Introduction

The enhanced oil recovery by injection of gas is a technique that is used since the 1930’s [14]. The sweep efficiency of gas, however, can be affected by gravity (by This research was carried out in association with the R&D project registered as ANP 20715-9, “Modelagem matem´ atica e computacional de inje¸ca ˜o de espuma usada em recupera¸ca ˜o avan¸cada de petr´ oleo” (Universidade Federal de Juiz de Fora (UFJF)/Shell Brasil/ANP) - Mathematical and computational modeling of foam injection as an enhanced oil recovery technique applied to Brazil pre-salt reservoirs, sponsored by Shell Brasil under the ANP R&D levy as “Compromisso de Investimentos com Pesquisa e Desenvolvimento”. This project is carried out in partnership with Petrobras. G. Chapiro was supported in part by CNPq grant 303245/2019-0. c Springer Nature Switzerland AG 2020  V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 18–31, 2020. https://doi.org/10.1007/978-3-030-50436-6_2

Numerical Algorithm for Two-Phase Flow in Porous Media with Foam

19

the gravity override phenomenon, that occurs when the injected gas accumulates in the upper layers of the reservoir) and by the development of preferential paths (viscous fingering), due to gas lower density and viscosity. These obstacles can be surpassed by the creation of foam, that can be defined as the agglomeration of gas bubbles separated by thin liquid films (lamellae), since foam apparent viscosity is much higher than the viscosity of gas [11,12,23]. The usage of foam in oil recovery is mainly motivated by the reduction of the gas phase mobility [15]. Population balance models can be used to simulate foam creation, destruction and flow through porous media. In this approach, it is common to define the foam texture (nf ), a quantity that represents the number of bubbles (or lamellae) per unit volume. Mechanisms of foam creation and coalescence play an important role in the model [15,17]. The major hypothesis adopted for the lamella-coalescence mechanism is that bubbles collapse near the limiting water ∗ ) or, equivalently, the limiting capillary pressure. In this context, saturation (Sw a foam model based on the well-known steady-state behavior of foam in porous media was proposed by Ashoori et al. in [2]. It considers a large, nearly constant, reduction in gas mobility at high water saturation and an abrupt weakening or collapse of foam at a limiting water saturation. Foam texture in local equilibrium (nLE D ), where bubble generation and destruction reach a local equilibrium state, depends only on the water saturation (Sw ):  ∗ 0, Sw ≤ S w LE   nD (Sw ) = , (1) ∗ ∗ tanh A(Sw − Sw ) , Sw > Sw with constant A. The dynamic foam net generation is given by a first-order approach, introduced in [24] and later related to the local-equilibrium bubble texture in [2], with a time constant 1/Kc , as follows:   (2) rg − rc = Kc nmax nLE D (Sw ) − nD , where rg and rc are the foam generation and coalescence rates, respectively, nmax is the maximum foam texture and nD = nf /nmax is the dimensionless foam texture. This model represents a simplification of foam behavior in porous media without significantly sacrificing the physical phenomena [24]. Other models that associate bubble generation and destruction to the limiting capillary pressure or the gradient of gas pressure can be found in [15,17]. In this model, the reduction in the mobility of gas by foam is viewed as a reduction of the gas relative permeability krg : krg (Sw , nD ) =

0 krg (Sw ) , 18500nD + 1

(3)

0 where krg is the gas relative permeability when no foam is formed. Another view on the mobility reduction is based on the apparent viscosity of foam [13,15,17,24]. Comprehensive reviews on the mechanisms of bubble creation, destruction, and also on the reduction of gas mobility can be found in [9,21].

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The following system of equations describes an incompressible two-phase flow in a porous medium: ∂ φSg (rg − rc ) (φSg nD ) + ∇ · (ug nD ) = , ∂t nmax ∂ (φSα ) + ∇ · uα = 0, α ∈ {g,w}, ∂t uα = −Kλα (∇pα − ρα g) , α ∈ {g,w},

(4) (5) (6)

where (4) is a population balance equation for foam texture [6] and (5)–(6) account for fractional flow and hydrodynamics. We use φ to denote the porosity of the medium, and ρα , Sα , uα , and pα to denote density, saturation, superficial velocity, and pressure, respectively, of phase α. Also, K = K(x) is the intrinsic permeability tensor, and g is the gravity vector. From the fractional flow theory, λw = krw /μw and λg = krg /μg denote the mobility of water and gas phases, respectively, where the viscosity of water and gas are given by μw and μg . It is assumed that the porous medium is fully saturated, i.e., Sw + Sg = 1. The numerical approach for solving this system of PDEs should be capable of handling several complexities due to discontinuity, non-linearity, stiffness, natural instabilities, among others. The numerical methods should also preserve important properties, such as local conservation of mass, shock capture, nonoscillatory solutions, accurate approximations, and reduced numerical diffusion effects. To the extent of our knowledge, this problem is usually solved using explicitin-time finite difference schemes [1,16,21,25]. Also, the use of commercial software is prevalent in the literature [7,20, and references within]. The most common approach in commercial software is to represent the effect of foam by a factor that reduces the mobility of the gas phase; therefore, bubble creation and destruction are not represented explicitly [9,21]. An effective numerical scheme to solve this kind of model and to address its inherent complexities is based on rewriting the problem in terms of global pressure, as in [4]. In this scheme, one has two distinct coupled problems: an elliptic problem and a degenerate hyperbolic problem. The next step is to decouple the system of PDEs into two subsystems of equations, each one of a different nature. In doing so, each subsystem can be solved by specialized methods, such as finite element and finite volume methods, according to their mathematical properties and the relation between precision and computational efficiency required in the resolution of each step. In this sense, for the spatial discretization of the hyperbolic problems we can employ the finite volume method, for instance; for time discretization, a common choice is a finite difference method, while a hybrid finite element method can be applied to the elliptic equations. In this context, we develop a staggered algorithm to decouple the hydrodynamics from the hyperbolic system, resulting in a scheme that uses a locally conservative hybrid mixed finite element method to approximate the velocity and pressure fields and a high-order finite volume scheme to solve the hyperbolic equations. The two problems are solved in different time scales. Thus, the

Numerical Algorithm for Two-Phase Flow in Porous Media with Foam

21

proposed staggered algorithm is employed to simulate two-phase (water and gas) flow in a heterogeneous porous medium. We compare pure gas-water injection with the gas-water-foam flow. The results show a reduction of gravity override and viscous fingering effects when the foam is present. This work is organized as follows: in Sect. 2, we define a fractional flow formulation for Eqs. (4)–(6) using the concept of global pressure; in Sect. 3, we present an algorithm to solve the problem using a hybrid mixed finite element method for the elliptic problem and a high-order finite volume method methodology to solve the hyperbolic equations; numerical results are shown in Sect. 4. Finally, in Sect. 5, we present some concluding remarks.

2

Model Problem

To build a fractional flow model for the water-gas-foam flow in porous media we follow the global pressure approach from [4]. The global pressure is defined as  Sw dPc λw λw dη and fw = = fw , (7) p = pg − dη λ λ w + λg 1−Sgr where Pc (Sw ) = pg − pw is the capillary pressure. From (6) and (7) the total velocity u is written as [4]:  λg ρg + λw ρw g, (8) u = ug + uw = −Kλ ∇p − G (Sw ) , with G (Sw ) = λ where g = −9.81ˆ j m/s2 and jˆ is the unitary vector in vertical direction. It follows directly from the previous definitions that uw = fw u + Kλg fw ∇Pc − b, ug = fg u − Kλg fw ∇Pc + b,  where b = Kλg fw ρg − ρw g. Let Ω ⊂ Rd , d = 2 or 3, have Lipschitz boundary Γ = ∂Ω. Using the hypothesis of rigid porous medium, the total fluid velocity u, the global pressure p, the water saturation Sw , and the foam texture nD satisfy, in Ω × (0, T ], the following system of equations:  u = −Kλ ∇p − G (Sw ) , (9) ∇ · u = 0, (10) ∂S ∂f i + − ∇ · (C∇S) = Φ, ∂t ∂xi i=1 d

φ

where i denotes a spatial direction, and   Sw fw ui − bi S= , fi = , Sg nD nD fg ui + nD bi  −1 0 Cijkl = Bik Djl , B= , nD 0

(11)

⎤ 0 Φ = ⎣ φSg (rg − rc ) ⎦ , nmax dPc , Djl = Kjl λg fw dSw ⎡

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and boundary and initial conditions u·n=u ¯ ¯ S=S

on ΓN × (0, T ], on

ΓN−

× (0, T ],

on ΓD × (0, T ],

p = p¯ S (x, 0) = S

0

(12)

in Ω,

where Γ = ΓN ∪ ΓD , ΓN ∩ ΓD = ∅, with ΓN denoting the boundary region with Neumann condition (specified injection velocity), ΓD defining the boundary  region with Dirichlet condition on the potential, ΓN− = x ∈ ΓN ; u ¯(x) < 0 and n is the unit outer normal vector to Γ . For simplicity, we assume ΓD = ∅ and ΓN− = ∅.

3

Numerical Method

In this section, we introduce the sequential algorithm that combines two kinds of numerical methods to solve (9)–(11). The hydrodynamics (9)–(10) is approximated using a naturally stable mixed finite element method introduced in [22]. This method is locally conservative, relying on the strong imposition of the continuity of normal fluxes and on a discontinuous pressure field. The combination of a hybrid formulation with a static condensation technique reduces the number of degrees of freedom in the global problem. The transport system (11) is solved using the KNP method, a conservative, high-order, central-upwind finite volume scheme introduced in [18] that shows reduced numerical diffusion effects. The KNP scheme is an extension of the KT method [19] that generalizes the numerical flux using more precise information about the local propagation velocities. At the same time, the KNP scheme has an upwind nature, since it respects the directions of wave propagation by measuring the one-sided local velocities. KNP is a semi-discrete method based on the REA (Reconstruct Evolve Average) algorithm of Godunov [8]. Furthermore, the KNP scheme allows for using small steps in time without requiring an excessive refinement of the spatial mesh, since the numerical diffusion does not depend on the time step. After discretization in space, the resulting system of ODEs is integrated in time using a BDF (Backward Differentiation Formula), an implicit, multi-step method that is especially indicated to solve stiff equations [5]. 3.1

The Sequential Algorithm

The system of Eqs. (9)–(11) is strongly coupled. It is possible, however, to apply a staggered algorithm to solve an approximate problem composed of two subsystems: an elliptic one, with time step Δtu , and a hyperbolic one, with time step Δts . Each of them is solved separately using adaptive time steps, as described in Algorithm 1, i.e., one can use smaller time steps to bound the error in the approximations under a certain tolerance. In addition, it is often recommended that Δtu > Δts , as the time scale of the hydrodynamics is usually much slower

Numerical Algorithm for Two-Phase Flow in Porous Media with Foam

23

than of the transport. In each iteration, approximations for velocity un+1 and pressure pn+1 fields at t = tn+1 are computed from   n un+1 = −Kλ ∇pn+1 − G (Sw ) in Ω, (13a) ∇ · un+1 = 0

in Ω,

(13b)

supplemented by the boundary conditions (12). Then an iterative algorithm is used to find approximations for water saturan+1 tion (Sw ) and foam texture (nn+1 D ) by solving the following system of PDEs n n+1 ] (for simplicity, we omit the superscript n + 1): in Ω × (t , t   ∂S n+1 ∂f n+1 i + − ∇ · C∇S n+1 = Φn+1 , ∂t ∂xi i=1 d

φ

(14)

with boundary and initial conditions ¯ on Γ − × (tn , tn+1 ], S=S N

S (x, tn ) = S n in Ω.

(15)

Algorithm 1: Sequential algorithm to solve (13)–(15). 0 Set initial conditions Sw and n0D ; n ← 0; t ← 0; ts ← 0; do Compute velocity (un+1 ) and pressure (pn+1 ) fields using (13); t = t + Δtu ; k ← 0; S n+1,0 ← S n ; do Compute (S n+1,k+1 ) using (14) and (15) with un+1 ; ts = ts + Δts ; k = k + 1; while ts < t; S n+1 ← S n+1,k ; n = n + 1; while t < T ;

In the following sections we comment on the methods employed to solve each problem. 3.2

Hybrid Mixed Finite Element Method for Darcy Flow

When a mixed finite element formulation is used to approximate the Darcy system (13), it is necessary to simultaneously fulfill the compatibility condition between spaces and to impose the continuity of the normal vector across

24

F. F. de Paula et al.

interelement edges. In addition, the resulting linear system is indefinite, which can restrict the numerical solvers that could be applied. By using a hybrid formulation, the continuity requirement is imposed via Lagrange multipliers, defined on the interelement edges. Furthermore, if the local problems are solvable, it is possible to eliminate all degrees of freedom related to local problems using a static condensation technique, resulting in a considerable reduction of the computational cost, since the global system involves only the degrees of freedom of the Lagrange multiplier. Also, in this case, the global problem is positivedefinite. Once the approximation for the Lagrange multipliers is found, the original degrees of freedom (associated with velocity and pressure) can be computed in local, independent problems. We first introduce some notations and definitions, for simplicity restricting ourselves to Ω ⊂ R2 . The three-dimensional case follows directly. Let L2 (Ω) denote the Hilbert space of square-integrable functions in Ω, with the usual inner product (·, ·)Ω , and let H(div, Ω) be the space of vector functions having each component and divergence in L2 (Ω). Assuming Ω is a polygon, we define a partition Th of Ω composed of quadrilaterals and use K to denote an arbitrary element of the partition. The set of edges of K is denoted by ∂K, the set of edges in Th is denoted by Eh , and Eh∂ denotes the set of all boundary edges, i.e., those with all points in Γ . Finally, the set of interior edges is denoted by Eh0 = Eh \ Eh∂ . For every element K ∈ Th , there exists c > 0 such that h ≤ che , where he is the diameter of the edge e ∈ ∂K and h, the mesh parameter, is the element diameter. For each edge of an element K we associate a unit outward normal vector nK . The (discontinuous) RT spaces of index k [22] are here denoted by Uhk × Phk . We define the following sets of functions on the mesh skeleton:   Mkh = μh ∈ L2 (Eh ); μh |e ∈ pk (e), ∀e ∈ Eh , μh |e = p¯, ∀e ∈ Eh∂ ∩ ΓD , (16)   ¯ kh = μh ∈ L2 (Eh ); μh |e ∈ pk (e), ∀e ∈ Eh , μh |e = 0, ∀e ∈ Eh∂ ∩ ΓD , (17) M where pk (e) denotes the set of polynomial functions of degree up to k on e. From these definitions, we can write the following hybrid mixed formulation for the hydrodynamics problem (13): n and nnD , find the pair [uh , ph ] ∈ Uhk × Phk and the Lagrange multiplier Given Sw k ¯ k, λh ∈ Mh such that, for all [v h , qh , μh ] ∈ Uhk × Phk × M h  

n λh v h · nK ds = (G (Sw ) , v h )Ω (18) (Auh , v h )K − (ph , ∇ · v h )K + K∈Th

∂K



−(qh , ∇ · uh )K = −(f, qh )Ω

K∈Th

 K∈Th

∂K

n n where A = A(Sw , nnD ) = (Kλ(Sw , nnD ))−1 .

(19)

 μh uh · nK ds =

u ¯ μh ds, ΓN

(20)

Numerical Algorithm for Two-Phase Flow in Porous Media with Foam

25

To solve the hybrid formulation (18)–(19) we apply the static condensation technique that consists in a set of algebraic operations, done at the element level, to eliminate all degrees of freedom corresponding to the variables uh and ph , leading to a global system with the degrees of freedom associated with the multipliers only. We can observe that static condensation causes a major reduction in the size of the global problem, which is now rewritten in terms of the multiplier only. Also, the new system of equations is positive-definite, allowing for using simpler and more robust solvers. In the end, a hybrid formulation associated with static condensation leads to a great reduction of the computational cost required to solve the global problem. In this work, the deal.II library [3] is used to solve this hydrodynamics problem. 3.3

High Order Central-Upwind Scheme for the Transport Problem

The numerical methodology used to approximate the water saturation and bubble texture Eqs. (14) and (15) is a high-order non-oscillatory central-upwind finite volume method proposed in [18] and here referred to as KNP. Like many other finite volume methods, the KNP scheme is based on a grid of control volumes (or cells). The upwind nature of KNP is because it respects the directions of wave propagation by measuring the one-sided local speeds, given by   max/min max/min max/min , ΛS + ,0 , al±1/2,i = max / min ΛS − l±1/2,i

l±1/2,i

on direction i and a cell of index l, where l+1/2 is the right (resp. top) face and j− 1/2 is the left (resp. bottom) face of a cell, S − l±1/2,i is the local reconstruction of S at the left (resp. bottom) side of a face, and S + l±1/2,i is the local reconstruction max are the maximum of S at the right (resp. top) side of a face; ΛX and Λmin X and minimum eigenvalues, respectively, of the Jacobian ∂f i /∂S at S = X. The result of spatial discretization using KNP is the system of ODEs in conservative form:  d 

H l−1/2,i − H l+1/2,i dS l = φ + P l,i + Φl , (21) dt hi i=1 where Φl = Φ (S l ), hi is the cell size in the i-th direction, with the convective numerical fluxes given by     − + min amax l±1/2,i f i S l±1/2,i − al±1/2,i f i S l±1/2,i H l±1/2,i = min amax l±1/2,i − al±1/2,i +

min  amax l±1/2,i al±1/2,i

amax l±1/2,i

−

amin l±1/2,i

 − S+ l±1/2,i − S l±1/2,i ,

(22)

26

F. F. de Paula et al.

and diffusive numerical fluxes given by   ˜ l+1/2,i S l+1,i − D ˜ l−1/2,i S l + D ˜ l−1/2,i S l−1,i ˜ l+1/2,i + D D Pl,i (t) = , h2i

(23)

˜ l±1/2,i is defined as the harmonic mean of (DB)l and (DB)l±1,i . The where D scheme (21)–(23), combined with minmod reconstruction of the type ˜l (x) = S nl + S

d

dnl,i (xi − xl,i ),

i=1



dnl,i

S nl,i − S nl−1,i S nl+1,i − S nl−1,i S nl+1,i − S nl,i = minmod θ , ,θ hi 2hi hi



˜l is a piecewise linear approximation is a TVD scheme if 1 ≤ θ ≤ 2 [18], where S n ˜ to the solution at time t , i.e., S l (x) ≈ S nl (x). Then we can use the fact that xl±1/2,i = xl,i ± hi /2 to find S ± l±1/2,i . Various numerical methods can be used to solve the system of ODEs (21). In this work, a variable order, adaptive step Backward Differentiation Formula (BDF) was chosen. This stable, implicit scheme allows for taking larger time steps than an explicit method would require, which reduces computational cost. In our numerical simulations we used the implementation of the BDF scheme from the CVode package, available in the SUNDIALS library [10].

4

Numerical Results

Applying the numerical methods described in Sect. 3, we now present results of numerical experiments that aim to assess the influence of foam and gravity effects in two-phase flow. Two scenarios are simulated: flow without and with foam. In the first scenario, we consider that only a mixture of water and gas is flowing through the porous medium, setting nD = 0 in the hyperbolic problem and solving only for Sw in (11). The hydrodynamics and the mobility of the gas phase 0 ). In the second scenario, we assume that surfactant remain unchanged (krg = krg is readily available in the water phase, allowing for foam creation and changes in the mobility of gas phase. This scenario is simulated using the full problem (9)–(11). In both scenarios, the capillary pressure and relative permeabilities are 

1 − Sw − Sgr Pc = 330 Sw − Swc



0.01 , krw =

 4 2 − S Sw − Swc 1 − S w gr 0 , krg = . 1 − Swc − Sgr 1 − Swc − Sgr

The permeability is assumed isotropic K = κ(x)I, where κ(x) is the permeability field of layers 1 (case A, Fig. 1(a)) and 36 (case B, Fig. 1(b)) of the SPE10 project1 , rotated to the xy plane. The right boundary is chosen to be the Dirichlet 1

https://www.spe.org/web/csp/datasets/set02.htm.

Numerical Algorithm for Two-Phase Flow in Porous Media with Foam

27

type (ΓD ) with p¯ = 0, while left, top and bottom boundaries are set to Neumann ¯ < 0 for the left boundary and u ¯ = 0 for the top and bottom condition (ΓN ) with u boundaries. Coefficients and numerical parameters used in the simulations are shown in Table 1. Table 1. Simulation parameters. Parameter

Value −3

Parameter

Value

Porosity

0.25

Water viscosity (μw ) [Pa s]

1.0 × 10

Gas viscosity (μg ) [Pa s]

2.0 × 10−5 A

Water residual saturation (Swc ) 0.2

400

Kc [1/s]

1.0 × 10−6

Gas residual saturation (Sgr )

0.0

Dimensions [m]

3.67 × 1.0

∗ Critical water saturation (Sw )

0.37

Final time [s]

1.0 × 104

Max foam texture (nmax ) [m−3 ] 8.0 × 1013 Injection velocity (¯ u) [m s−1 ] 0 Initial water saturation (Sw )   Injected water saturation S¯w

Δtu [s]

20.0

3.0 × 10−5 Number of cells

220 × 60

1.0

Minmod parameter (θ) 1.0

0.372

Absolute tolerance

1.0 × 10−6

Initial foam texture (n0D )

0.0

Relative tolerance

1.0 × 10−4

Injected foam texture (¯ nD )

0.0

RT index (k)

0

The water saturation profiles for case A at t = 2 000 s and t = 10 000 s are shown in Figs. 2 and 3, respectively. The gravity effects are much more pronounced in the no-foam simulation. Also, as expected, the water phase displacement occurs more slowly in the foam presence due to the gas mobility reduction caused by foam. Note that, without foam, the gas breakthrough has already taken place at t = 10 000 s (Fig. 3), which does not occur when foam is present. Moreover, for the foam model adopted [2], viscous fingering and gravity override are reduced with foam as time advances. As a result, a better sweep efficiency of the medium is observed when foam is present, as can be seen in Fig. 6(a). In our experiments, foam injection increased total water recovery by approximately 100%.

(a) Case A.

(b) Case B.

Fig. 1. Permeability map of layer 1 (a) and layer 36 (b) of the SPE10.

28

F. F. de Paula et al.

Fig. 2. Case A: water saturation at t = 2 000 s. Left column: without gravity effects; right column: with gravity effects; top row: without foam; bottom row: with foam.

Fig. 3. Case A: water saturation at t = 10 000 s. Left column: without gravity effects; right column: with gravity effects; top row: without foam; bottom row: with foam.

Fig. 4. Case B: water saturation at t = 2 000 s. Left column: without gravity effects; right column: with gravity effects; top row: without foam; bottom row: with foam.

Numerical Algorithm for Two-Phase Flow in Porous Media with Foam

29

Fig. 5. Case B: water saturation at t = 10 000 s. Left column: without gravity effects; right column: with gravity effects; top row: without foam; bottom row: with foam.

In case B, the permeability field has a more evident preferential channel in the lower region (see Fig. 1(b)). The results for this channelized porous formation (Figs. 4 and 5) reinforce the foam’s ability to reduce the effects of gravity override and viscous fingering, according to the model used, even though this case presents a more pronounced preferential path. The water cumulative production curves for case B (Fig. 6(b)) show that gravity effects on the production are much more pronounced when no foam is present; this is due to the influence of gravity on diverting the flow from the high permeability channel, resulting in higher sweeping efficiency. Moreover, foam injection in this case increases total water production by approximately 50%, when gravity is considered, and by about 100% when gravity is neglected.

(a) Case A.

(b) Case B.

Fig. 6. Water cumulative production for cases A and B.

30

5

F. F. de Paula et al.

Conclusions and Remarks

In this work, we presented a locally conservative numerical algorithm to solve the gas-water flow including foam injection. The system of PDEs that models this phenomenon was derived considering a fractional flow formulation based on the global pressure. The numerical staggered approach proposed combines a high-order central upwind finite volume method for the hyperbolic equations adopting BDF time integrations with a hybrid finite element method to solve the Darcy’s problem employing Raviart-Thomas spaces. The proposed methodology was applied to simulate regimes with pure gaswater injection and gas-water-foam flow. In this context, we have established a comparison between these two regimes considering two layers of SPE10 project with different heterogeneous permeability fields. The results, based on the model proposed in [2], point to the foam’s ability to reduce the gravity override and viscous fingering even in cases of porous media with rather pronounced preferential channels. Acknowledgements. The authors are thankful to Professor Pacelli P. L. Zitha for fruitful preliminary discussions.

References 1. Afsharpoor, A.: Mechanistic foam modeling and simulations: gas injection during surfactant-alternating-gas processes using foam-catastrophe theory. Ph.D. thesis, Louisiana State University (2009) 2. Ashoori, E., Marchesin, D., Rossen, W.R.: Roles of transient and local equilibrium foam behavior in porous media: traveling wave. Colloids Surf. A Phys. Chemical Eng. Asp. 377, 228–242 (2011). https://doi.org/10.1016/j.colsurfa.2010.12.042 3. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II - a general-purpose objectoriented finite element library. ACM Trans. Math. Softw. 33(4) (2007). https:// doi.org/10.1145/1268776.1268779 4. Chavent, G., Jaffr´e, J.: Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows Through Porous Media, vol. 17. North-Holland, Amsterdam (1986) 5. Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Natl. Acad. Sci. U. S. A. 38(3), 235 (1952) 6. Falls, A.H., Hirasaki, G.J., Patzek, T.W., Gauglitz, D.A., Miller, D.D., Ratoulowski, T.: Development of a mechanistic foam simulator: the population balance and generation by snap-off. SPE Reserv. Eng. 3, 884–892 (1988). https:// doi.org/10.2118/14961-PA 7. Farajzadeh, R., Lotfollahi, M., Eftekhari, A.A., Rossen, W.R., Hirasaki, G.J.: Effect of permeability on implicit-texture foam model parameters and the limiting capillary pressure. Energy Fuels 29(5), 3011–3018 (2015). https://doi.org/ 10.1021/acs.energyfuels.5b00248 8. Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 89(3), 271–306 (1959)

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9. Hematpur, H., Mahmood, S.M., Nasr, N.H., Elraies, K.A.: Foam flow in porous media: concepts, models and challenges. J. Nat. Gas Sci. Eng. 53, 163–180 (2018). https://doi.org/10.1016/j.jngse.2018.02.017 10. Hindmarsh, A.C., et al.: SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31(3), 363–396 (2005) 11. Hirasaki, G.J.: A review of the steam foam process mechanisms (1989). SPE 19518 12. Hirasaki, G.J.: The steam-foam process. J. Pet. Technol. 41(5), 449–456 (1989). https://doi.org/10.2118/19505-PA 13. Hirasaki, G.J., Lawson, J.B.: Mechanisms of foam flow in porous media: apparent viscosity in smooth capillaries. SPE J. 25(02), 176–190 (1985) 14. Jones, S.A., Getrouw, N., Vincent-Bonnieu, S.: Foam flow in a model porous medium: I. The effect of foam coarsening. Soft Matter 14, 3490–3496 (2018). https://doi.org/10.1039/C7SM01903C 15. Kam, S.I.: Improved mechanistic foam simulation with foam catastrophe theory. Colloids Surf. A Physicochem. Eng. Asp. 318(1), 62–77 (2008). https://doi.org/ 10.1016/j.colsurfa.2007.12.017 16. Kam, S.I., Nguyen, Q.P., Li, Q., Rossen, W.R.: Dynamic simulations with an improved model for foam generation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2004) 17. Kovscek, A.R., Patzek, T.W., Radke, C.J.: A mechanistic population balance model for transient and steady-state foam flow in boise sandstone. Chem. Eng. Sci. 50(23), 3783–3799 (1995). https://doi.org/10.1016/0009-2509(95)00199-F 18. Kurganov, A., Noelle, S., Petrova, G.: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23(3), 707–740 (2001). https://doi.org/10.1137/S1064827500373413 19. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160(1), 241– 282 (2000). https://doi.org/10.1006/jcph.2000.6459 20. Ma, K., Farajzadeh, R., Lopez-Salinas, J.L., Miller, C.A., Biswal, S.L., Hirasaki, G.J.: Non-uniqueness, numerical artifacts, and parameter sensitivity in simulating steady-state and transient foam flow through porous media. Transp. Porous Media 102(3), 325–348 (2014). https://doi.org/10.1007/s11242-014-0276-9 21. Ma, K., Ren, G., Mateen, K., Morel, D., Cordelier, P.: Modeling techniques for foam flow in porous media. SPE J. 20(3), 453–470 (2015) 22. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods. LNM, vol. 606, pp. 292–315. Springer, Heidelberg (1977). https:// doi.org/10.1007/BFb0064470 23. Smith, D.H. (ed.): Surfactant-Based Mobility Control. No. 373 in ACS Symp. Ser., Am. Chem. Soc., Washington, D.C. (1988) 24. Zitha, P.L.J.: A new stochastic bubble population model for foam in porous media. In: SPE/DOE Symposium on Improved Oil Recovery. Society of Petroleum Engineers (2006) 25. Zitha, P.L.J., Du, D.X., Uijttenhout, M., Nguyen, Q.P.: Numerical analysis of a new stochastic bubble population foam model. In: SPE/DOE Symposium on Improved Oil Recovery. Society of Petroleum Engineers (2006)

A Three-Dimensional, One-Field, Fictitious Domain Method for Fluid-Structure Interactions Yongxing Wang(B) , Peter K. Jimack , and Mark A. Walkley School of Computing, University of Leeds, Leeds LS2 9JT, UK [email protected]

Abstract. In this article we consider the three-dimensional numerical simulation of Fluid-Structure Interaction (FSI) problems involving large solid deformations. The one-field Fictitious Domain Method (FDM) is introduced in the framework of an operator splitting scheme. Threedimensional numerical examples are presented in order to validate the proposed approach: demonstrating energy stability and mesh convergence; and extending two dimensional benchmarks from the FSI literature. New three dimensional benchmarks are also proposed. Keywords: Fluid-Structure Interaction · Finite element · Fictitious domain · Immersed Finite Element · One-field · Monolithic scheme · Eulerian formulation

1

Introduction

Numerical simulation of Fluid-Structure Interaction (FSI) problems is a computational challenge due to its strong nonlinearity, especially in the case of large solid deformations. This challenge is exacerbated in three dimensions due to the need for efficient numerical algorithms to handle the large number of degrees of freedom that are inevitably required. In this paper we generalize our recent onefield Fictitious Domain Method (FDM) [22,23] from two to three dimensions, enhance the efficiency and robustness of the proposed time-stepping scheme, and demonstrate the resulting algorithm’s capabilities on a number of challenging test problems. We also provide potential benchmark problems to allow results to be compared against those from other schemes in the future. Lagrangian and Arbitrary Lagrangian-Eulerian (ALE) methods are widely adopted when considering a relatively small solid deformation [6,11,15]. Discrete remeshing can be used for large deformations [10,18], however this can be very costly in the case of three dimensions and can present challenges for mass conservation. The cut finite element method (cutFEM) [4,5] may also be applied to solve FSI problems [14,19], although it is not trivial to deal with the discontinuous integral across the elements cut by the moving fluid-solid interface, especially in three dimensional cases. The Fictitious Domain Method (FDM) [1,3,8,12,13] uses two meshes to represent the fluid and solid separately, which c Springer Nature Switzerland AG 2020  V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 32–45, 2020. https://doi.org/10.1007/978-3-030-50436-6_3

One-Field FDM for FSI

33

can easily handle large deformation of the solid. However the FDM approach solves a very large equation system: both the velocity in the whole domain (fluid and solid) and the displacement in the solid domain, coupled via a distributed Lagrange multiplier (DLM) which is also an unknown variable. The Immersed Finite Element Method (IFEM) [2,17,24,27] also uses two meshes but only solves for velocity in the whole domain, while the solid information is assembled on the right-hand side of the fluid equation as a prescribed force term. This IFEM approach achieves FSI behaviour through this forcing term, and is therefore relatively efficient in three dimensional simulations compared to the DLM approach. Performance of the IFEM method depends strongly on the fluid and solid properties and usually works well when the solid behaves similarly to the fluid (such as a relatively soft solid) [20]. It has been successfully used, for example, in the area of biomechanics [16,27]. The one-field FDM approach [22,23] similarly only solves for one velocity field in the whole domain. However, this proposed method assembles the solid equations and implicitly includes them in the equation system. The one-field FDM approach has the same generality and robustness as the FDM/DLM: both of them solve the fluid equations and solid equations as one system. However the former needs to solve only for one velocity field while the latter solves for fluid velocity, solid displacement and Lagrange multiplier. The proposed onefield FDM may also be regarded as a special linearisation of the implicit IFEM, which however is more robust compared with explicit IFEM and more efficient compared with the implicit IFEM [24], allowing a wide range of solid parameters to be considered and naturally dealing with the case of different densities between fluid and solid [22,23]. In short, the one-field FDM combines the FDM/DLM advantage of robustness and the classical/explicit IFEM advantage of efficiency. The scheme has been validated through comparison with idealised two-dimensional test cases and against experimental data and simulation results drawn from the literature [21]. In this article, the one-field FDM is extended, implemented and validated in three dimensions for the first time through the use of a newly applied operator splitting scheme. The paper is organized as follows. The control equations and a general finite element weak formulation are introduced in Sect. 2.1 and 2.2 respectively, followed by time discretization in Sect. 2.3. The operator splitting scheme is introduced in Sect. 2.4, followed by the linearisation (implementation detail) in Sect. 2.5 and the numerical algorithm for the final linear equation system in Sect. 2.6. Several three-dimensional numerical tests are given in Sect. 3, and conclusions are presented in Sect. 4.

2

One-Field Fictitious Domain Method

In this section, we review the one-field fictitious domain method [22] and develop it further based upon a three-step operator splitting scheme and the case novel block-matrix preconditioners. The system is described in a manner that is independent of the spatial dimensions, thus ensuring its capability in three dimensions, which is the primary purpose of this paper.

34

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f

s

Fig. 1. Schematic diagram of FSI, Ω = Ω t ∪ Ω t .

2.1

Control Partial Differentiation Equations

In the following context, Ωtf ⊂ Rd and Ωts ⊂ Rd (d = 3 in this article) denote the moving fluid and solid domain respectively, with the moving interface Γt = f s f s Ω t ∩ Ω t as shown schematically in Fig. 1. Ω = Ω t ∪ Ω t is a fixed domain with outer boundary Γ = ΓD ∪ ΓN , with ΓD and ΓN being Dirichlet and Neumann boundaries respectively. We denote by X the reference coordinates of the solid, by x = x(·, t) the current coordinates of the solid, and by x0 the initial coordinates of the solid. Notice that we choose X to be the stress-free configuration, which may be different to the initial configuration x0 . Let ρ, μ, u, σ and g denote the density, viscosity, velocity, stress tensor and acceleration due to gravity respectively. We assume both an incompressible fluid in Ωtf and incompressible solid in Ωts . The conservation of momentum and conservation of mass therefore take the same form as follows. Momentum equation: ρ

du = ∇ · σ + ρg, dt

(1)

and continuity equation: ∇ · u = 0. An incompressible Newtonian constitutive equation in σ = σ f = τ f − pf I,

(2) Ωtf

can be expressed as: (3)

with τ f = μf Duf being the deviatoric part of stress σ f , and Du = ∇u + ∇T u. An incompressible neo-Hookean solid with viscosity μs is assumed in Ωts [3], and the constitutive equation may be expressed as: σ = σ s = τ s − ps I, 



(4)

with τ s = c1 FFT − I + μs Dus being the deviatoric part of stress σ s , and ∂x ∂x ∂x0 F = ∂X = ∂x = ∇0 x∇X x0 being the deformation tensor of the solid, and 0 ∂X c1 is a solid material parameter. Finally the system is completed with continuity of velocity uf = us and normal stress σ f ns = σ s ns on interface Γt , and standard Dirichlet/Neumann boundary (on ΓD /ΓN ) and initial conditions.

One-Field FDM for FSI

2.2

35

Finite Element Weak Form

In the following context, let L2 (ω) be the square integrable   functions in domain ∂u 1 2 ω, and H (ω) = u : u, ∂xi ∈ L (ω) for i = 1, · · · , d . We also denote by H01 (ω) the subspace of H 1 (ω) whose functions have zero values on the Dirichlet 2 2 boundary of ω, and denote by  L0 (ω) the fsubspace of L (ω) whose functions have f p in Ωt . Given v ∈ H01 (Ω)d , we perform the zero mean value. Let p = ps in Ωts following symbolic operations:     f Eq. (1) (σ) · v ≡ Eq. (1) σ · v + Eq. (1) (σ s ) · v Ωtf

Ω

 ≡

  Eq. (1) σ f · v +

Ω



Ωts

Ωts



  Eq. (1) (σ s ) − Eq. (1) σ f · v.

Integrating the stress terms by parts, the above operations, using constitutive equations (3) and (4), give:       du du ·v+ ·v τ f : ∇v − p∇ · v + ρs − ρf ρf Ω dt Ω Ω Ωts dt     (5)  s   s  ¯ · v, + τ − τ f : ∇v = ρ − ρf g · v + ρf g · v + h Ωts

Ωts

Ω

ΓN

¯ denotes the prescribed normal stress on ΓN . Note that the integrals on where h the interface Γt are cancelled out due to the continuity of normal stress: σ f ns = σ s ns , because they are internal forces for the whole FSI system. Combining with the following symbolic operations for q ∈ L2 (Ω),    − Eq. (2)q − Eq. (2)q ≡ − Eq. (2)q, Ωtf

Ωts

Ω

leads to the weak form of the FSI system as follows. Given u0 and Ω0s , find u(t) ∈ H 1 (Ω)d , p(t) ∈ L2 (Ω) and Ωts , such that for ∀v ∈ H01 (Ω)d , ∀q ∈ L2 (Ω), the following equation holds:    ∂u μf f ρ ·v+ρ (u · ∇) u · v + Du : Dv − p∇ · v 2 Ω Ω ∂t Ω Ω    μδ du ·v+ q∇ · u + ρδ Du : Dv − 2 Ωts Ω Ωts dt      T  ¯ · v, + c1 FF − I : ∇v = ρf g · v + ρδ g·v+ h 

f

Ωts

Ω

Ωts

(6)

ΓN

where ρδ = ρs − ρf and μδ = μs − μf , and the integral over Ωts , d(·) dt is the time derivative with respect to a frame moving with the solid velocity us = u|Ω s . t

36

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Y. Wang et al.

Discretisation in Time

Using the backward Euler method to discretise in time, Eq. (6) may be approximated as follows. s Given un , pn and Ωns , find un+1 ∈ H 1 (Ω)d , pn+1 ∈ L2 (Ω) and Ωn+1 , such d 1 2 that for ∀v ∈ H0 (Ω) , ∀q ∈ L (Ω), the following equation holds:   un+1 − un f f ·v+ρ (un+1 · ∇) un+1 · v ρ Δt Ω Ω    μf + Dun+1 : Dv − pn+1 ∇ · v − q∇ · un+1 2 Ω Ω Ω   (7) μδ un+1 − un ·v+ + ρδ Dun+1 : Dv s Δt 2 Ωn+1 Ωs   n+1    T  ¯ · v, + c1 FF − I : ∇v = ρf g · v + ρδ g·v+ h s Ωn+1

s Ωn+1

Ω

ΓN

s and Ωn+1 is updated from Ωns by the following formula: s = {x : x = xn + Δtun+1 , xn ∈ Ωns } . Ωn+1

2.4

(8)

An Operator Splitting Scheme

The formulation of (7) is implicit. However we shall solve it semi-implicitly via the following operator spitting scheme which is based upon [9]. (1) Convection step:     un+1/3 − un ·v+ un+1/3 · ∇ un+1/3 · v = 0. Δt Ω Ω

(9)

(2) Diffusion step: 

 un+2/3 − un+1/3 μf ·v+ Dun+2/3 : Dv Δt 2 Ω Ω   un+2/3 − un μδ ·v+ Dn un+2/3 : Dn v + ρδ Δt 2 Ωns s Ωn   Fn+2/3 FTn+2/3 − I : ∇n v + c1 ρf

s Ωn



g · v + ρδ

= ρf Ω

 s Ωn

(10)

 ¯ · v. h

g·v+ ΓN

(3) Pressure step:    un+1 − un+2/3 ·v− pn+1 ∇ · v − q∇ · un+1 = 0. ρf Δt Ω Ω Ω

(11)

One-Field FDM for FSI

37

In the above, ∇n (·) represents the divergence in the current coordinates at t = tn and Dn = ∇n + ∇Tn . Note that the variables un+1/3 and un+2/3 are just intermediate values, not specifically the velocity at time t = tn + Δt 3 or t = tn + 2Δt . The notation F or F is interpreted as follows: n+1/3 n+2/3 3 Ft =

∂xt = ∇X (xn + ut Δt) , ∂X

(12)

with t = n + 1/3 or n + 2/3. Using this splitting scheme, standard approaches can be taken to solve the pure convection equation (9) (see [9]), and iterative methods with an efficient preconditioner can be applied to solve the “degenerate” Stokes Equations (11) (see Sect. 2.6 of [7,21]). The main challenge is in how to approximate the term Fn+2/3 FTn+2/3 − I in Eq. (10), which is nonlinearly related to the solid displacement and hence to the solid velocity. In the following subsection we focus on expressing and linearising Fn+2/3 FTn+2/3 − I in terms of velocity un . 2.5

Linearisation of the Diffusion Step

The specific choice of linearisation is the core of this proposed one-field FDM approach, and is what makes it distinctive from all other schemes. Let Fn+2/3 FTn+2/3 − I be denoted by Ft FTt − I = st with t = n + 2/3, then st can be computed as follows:   st = Ft FTt − I = ∇X xt ∇TX xt − I . (13) Using the chain rule, this last equation can also be expressed as:

or

st = ∇n xt ∇X xn ∇TX xn ∇Tn xt − I + ∇n xt ∇Tn xt − ∇n xt ∇Tn xt

(14)

  st = ∇n xt ∇Tn xt − I + ∇n xt ∇X xn ∇TX xn − I ∇Tn xt .

(15)

Then st can be expressed based on the previous coordinate xn as follows: st = ∇n xt ∇Tn xt − I + ∇n xt sn ∇Tn xt . Using xt = xn + Δtut (see (12)), this can finally be expressed as:   st = Δt ∇n ut + ∇Tn ut + Δt∇n ut ∇Tn ut + sn + Δt2 ∇n ut sn ∇Tn ut + Δt∇n ut sn + Δtsn ∇Tn ut .

(16)

(17)

There are two nonlinear terms in this equation, which may be linearised as ∇n ut ∇Tn ut = ∇n ut ∇Tn un + ∇n un ∇Tn ut − ∇n un ∇Tn un ,

(18)

and ∇n ut sn ∇Tn ut = ∇n ut sn ∇Tn un + ∇n un sn ∇Tn ut − ∇n un sn ∇Tn un .

(19)

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Substituting sn+2/3 = Fn+2/3 FTn+2/3 − I, using expressions (17), (18) and   (19), into the diffusion step (10), and neglecting terms of O Δt2 , after some algebra we produce the one-field FDM formulation:   un+2/3 − un+1/3 μf ·v+ Dun+2/3 : Dv ρf Δt 2 Ω Ω   un+2/3 − un μδ + Δtc1 ·v+ + ρδ Dn un+2/3 : Dn v Δt 2 s s Ωn Ωn (20)    + Δtc1 ∇n un+2/3 sn + sn ∇Tn un+2/3 : ∇n v 

s Ωn

Ω

2.6





g · v + ρδ

= ρf

s Ωn

 ¯ · v − c1 h

g·v+ ΓN

s Ωn

sn : ∇n v.

Iterative Linear Algebra Solver

In this section, we shall discuss the numerical algorithms in order to solve the final linear equations from the diffusion step (20) and pressure step (11). For convection step, we use the Taylor-Galerkin method in this paper [9]. Let us write Eq. (20) in an operator matrix form as follows: An un+2/3 = bn ,

(21)

where An = M/Δt + K + PTn (Msn /Δt + Ksn ) Pn , and bn = f + PTn fns + Mun+1/3 /Δt + PTn Msn Pn un /Δt. The above matrix operators are defined as:     μf f Mu · v = ρ u · v, Ku · v = Du : Dv, 2 Ω Ω Ω Ω      μδ + Δtc1 ∇n usn + sn ∇Tn u : ∇n v, Ksn u · v = Du : Dv + Δtc1 2 s s s Ωn Ωn Ωn       ¯ · v, f · v = ρf g·v+ fns · v = ρδ g · v − c1 sn : ∇n v, h Ω

Ω

ΓN

s Ωn

s Ωn

s Ωn

where u, v ∈ H 1 (ω)d with ω being Ω or Ωns . Finally Pn is a restriction from H 1 (Ω)d to H 1 (Ωns )d (PTn is the corresponding injection from H 1 (Ωns )d to H 1 (Ω)d ): Pn u = us = u|Ω s . We use the finite element interpolation to n approximate Pn after discretisation in space. A preconditioned Conjugate Gradient method can efficiently solve Eq. (21). We use the incomplete Cholesky decomposition of matrix M/Δt + K as a preconditioner in order to solve Eq. (21). Very good convergence performance can be observed from our numerical tests (although the precise performance of the linear algebraic solver is not the topic of this article).

One-Field FDM for FSI

39

Similarly, the “degenerate Stokes” problem (pressure step (11)) can also be expressed in an operator matrix form: 



Mun+2/3 /Δt M/Δt B un+1 , (22) = pn+1 0 BT 0   where, ∀v in H 1 (Ω)d and q ∈ L2 (Ω), Ω (Bv) q = − Ω q∇ · v. We use the MinRes algorithm [7] to solve the system with the following preconditioner: 

  M (Δp p) q = ∇p · ∇q, ∀p, q ∈ H 1 (Ω). (23) , where Δp Ω Ω We justify this since we can derive a Schur complement in the form of S = BT M−1 B. The operators that are discretised in this form imply that S will be spectrally equivalent to a discrete Laplacian. Hence we expect this preconditioner will be effective for this system, similarly to analysis for Stokes equation [7].

3

Numerical Experiments

In the following numerical tests, the convection and diffusion steps are discretised with quadratic finite elements (tri-quadratic hexahedra and quadratic tetrahedra), and the pressure step is discretised with the Taylor-Hood element. For stability it is sufficient that μδ ≥ 0 [21,22], however for simplicity, and to be consistent with [2,17,27] for example, we assume μδ = 0 in these tests. 3.1

Oscillating Ball

In this section, we consider a 3D oscillating ball which is an extension of the 2D disc in [23,28]. We use this example to test stability of the proposed approach by investigating the evolution of total energy:   ρf ρδ 2 2 |un | + |un | Etotal (tn ) = 2 Ω 2 Ωns (24)  n    Δtμf  c1 trFn FTn − d . + Duk : Duk + s 2 2 ΩX Ω k=1

The four different energy contributions/terms in the above equation have the following respective meanings: Kinetic energy of fluid plus fictitious fluid, kinetic energy of solid minus fictitious fluid, viscous dissipation (over n time steps of size Δt) and the potential energy of the solid. The ball is initially located at the centre of Ω = [0, 1] × [0, 1] × [0, 0.6] with a radius of 0.2. Using the property of symmetry this computation is carried out on 1/8 of domain Ω: [0, 0.5] × [0, 0.5] × [0, 0.3]. The initial velocities of x and y components are the same as that used in [22,28], which are prescribed by the stream function (25) Φ = Φ0 sin(ax)sin(by),

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(a) Velocity.

(b) Pressure.

Fig. 2. Distribution of velocity norm and pressure on the fluid mesh at t = 0.21 (the ball is maximally stretched), Δt = 5.0 × 10−3 .

(a) Displacement on the solid mesh.

(b) Evolution of the energy ratio Etotal (tn )/Etotal (t0 ).

Fig. 3. Solid deformation corresponding to Fig. 2 and Evolution of the energy ratio.

with Φ0 = 5.0×10−2 and a = b = 2π. The z component of velocity is initially set to be 0. In this test, ρf = 1, μf = 0.01, ρs = 1.5 and c1 = 1. In order to visualise the mesh and deformation of the solid, a snapshot of fluid velocity and pressure are presented in Fig. 2, and the corresponding deformed solid is displayed in Fig. 3(a). It can be seen from Fig. 3(b) that the total energy is nonincreasing, which is an indication of stability. In addition, the total energy converges to the initial system energy as we reduce the size of the time step, which shows the desired energy conservation property of the proposed scheme. 3.2

Oscillating Cylinder

In this test we consider a cylindrical pillar oscillating in a cuboid channel as shown in Fig. 4, which is a 3D extension of the 2D leaflet in [13,22,25]. We use this example to test the mesh convergence of the proposed scheme. The size of the cuboid is: length L = 3, height H = 1 and width W = 0.5. The cylinder is located at the center of the cuboid’s base, with radius of r = 0.05 and height h = 0.8. We use a symmetry boundary condition on the top, front and back

One-Field FDM for FSI

41

Table 1. Material properties for the oscillating cylinder and oscillating tri-leaflets. Fluid f

ρ = 100 kg/m f

Solid 3

μ = 10 N · s/m

ρs = 100 kg/m3 2

c1 = 107 N/m2

surfaces of the cuboid. All the velocity components are fixed to be zero at the bottom of the cuboid, and the inlet and outlet flow are defined by: ux = 15y (2 − y/H) sin (2πt) ,

uy = uz = 0.

(26)

We use the same material properties as used in [13,22,25] for the 2D leaflet (see Table 1), which is a natural extension of the corresponding 2D problem with similar boundary conditions. We use a tri-quadratic hexahedras fluid mesh of size 10×20×60 (width×height×length) for a coarse mesh, 16×32×96 for a medium mesh and 20 × 40 × 120 for a fine mesh. We use a linear tetrahedral solid mesh of 10304 elements with 2675 vertices for a coarse mesh, 19040 elements with 4786 vertices for a medium mesh and 38080 elements with 8883 vertices for a fine mesh. A stable small time step Δt = 1.0 × 10−4 is adopted for all the cases. In order to visualise the results of this simulation, snapshots of the velocity norm and stream lines in the background domain and the solid deformations are presented in Fig. 5 and 6 respectively. The displacement of initial point (1.55, 0.8, 0.5) (the top of the cylinder) for three different meshes is plotted in Fig. 7 as a function of time, from which mesh convergence with regard to the displacement is observed (the medium and fine mesh results are almost indistinguishable in these plots).

Fig. 4. Sketch of the oscillating cylinder in a cuboid.

3.3

Oscillating Tri-Leaflets

In this section we consider a 3D circular tube with flexible, opening tri-leaflets. A similar case has been studied in [26]. The computational domain is shown in Fig. 8 with L = 2 and R = 0.5 in this test. Note that there is a small gap (with the angle α = 0.4◦ as shown in Fig. 8(b)) between the three parts of the tri-leaflets in order to avoid contact, which is not currently included in our

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Fig. 5. Velocity norm and stream lines using a medium mesh at t = 2.3. The shadow shows the deformation of the solid corresponding to the case of t = 2.3 in Fig. 6.

(a) Horizontal displacement against time.

Fig. 6. Solid deformation at three different stages.

(b) Vertical displacement against time.

Fig. 7. Displacement at point (1.55, 0.8, 0.25) versus time.

model. The tube walls are no-slip boundaries, and the inlet and outlet flow are prescribed by:  ux = 15r (1 − r/R) (1 + r/R) sin (2πt) , r = y 2 + z 2 , uy = uz = 0,

(a) Sketch of the oscillating tri-leaflets in a tube.

(b) Geometry of the tri-leaflets with α = 0.4◦ .

Fig. 8. Computational domain for the test problem of tri-leaflets.

One-Field FDM for FSI

43

which is an extension of formula (26). The material properties for the fluid and solid are presented in Table 1. Two different meshes are used to test this problem: a coarse mesh of 12750 tri-quadratic hexahedra with 106151 nodes and a fine mesh of 90000 tri-quadratic hexahedra with 735861 nodes in order to discretise the cube; a coarse mesh of 7390 linear tetrahedra with 2657 vertices and a fine mesh of 27460 linear tetrahedra with 8917 vertices in order to discretise the trileaflets. The density of the background mesh can be observed in Fig. 9, which also presents a snapshot of the velocity norm and stream lines. The solid mesh can be observed in Fig. 11, which also shows the deformation of the tri-leaflets with horizontal velocity (x component) at different stages in order to visualise the pattern of the oscillation. The displacement at one of the tri-leaflet tips is plotted as a function of time in Fig. 10, from which it can be seen that the coarse mesh leads to small oscillation, however it is not present in the fine mesh simulation. The maximal fluid velocity is ux = 18.2 at the centre of the channel when the tri-leaflets are completely open, and the maximal solid velocity at the leaflet tip is ux = 18.2 when the tri-leaflets are completely close.

Fig. 9. Snapshot of velocity norm and stream lines in background domain and x-component velocity on the solid mesh at t = 0.2.

(a) t = 0.4,

(b) t = 0.55,

Fig. 10. The x-component displacement at the tip of each of the trileaflets.

(c) t = 0.8,

Fig. 11. The velocity of x component at different times for the tri-leaflets using the coarse mesh.

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4

Conclusions

In this article the one-field Fictitious Domain Method (FDM) [22] is extended in three ways: through an efficient operator splitting scheme, the implementation of block-matrix preconditioning and into three space dimensions. One numerical example is presented in order to validate the energy conservation, a second is used to test mesh convergence, and the last numerical example is extended from a two-dimensional benchmark for comparison and, we believe, can act as a 3D benchmark for future comparison. It can be seen from these tests that the onefield FDM approach may be adopted to simulate a variety of FSI problems with large solid deformation in three space dimensions. We know from our 2D tests that, for soft solids, execution times are comparable with an IFEM implementation: and considerably faster as the solid becomes more stiff (capable of using a larger time step). Consequently we propose the one-field FDM as a general approach that combines that robustness of FDM/DLM and the computational efficiency of IFEM.

References 1. Baaijens, F.P.: A fictitious domain/mortar element method for fluid-structure interaction. Int. J. Numer. Methods Fluids 35(7), 743–761 (2001) 2. Boffi, D., Cavallini, N., Gastaldi, L.: The finite element immersed boundary method with distributed Lagrange multiplier. SIAM J. Numer. Anal. 53(6), 2584–2604 (2015) 3. Boffi, D., Gastaldi, L.: A fictitious domain approach with Lagrange multiplier for fluid-structure interactions. Numerische Mathematik 135(3), 711–732 (2016). https://doi.org/10.1007/s00211-016-0814-1 4. Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes problem. ESAIM Math. Model. Numer. Anal. 48(3), 859–874 (2014) 5. B¨ urman, E., Hansbo, P., Larson, M.G., Massing, A.: Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM Math. Model. Numer. Anal. 52(6), 2247–2282 (2018) 6. Degroote, J., Bathe, K.J., Vierendeels, J.: Performance of a new partitioned procedure versus a monolithic procedure in fluid–structure interaction. Comput. Struct. 87(11–12), 793–801 (2009) 7. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers. Oxford University Press (OUP), Oxford (2014) 8. Glowinski, R., Pan, T., Hesla, T., Joseph, D., P´eriaux, J.: A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169(2), 363–426 (2001) 9. Glowinski, R.: Finite Element Methods for Incompressible Viscous Flow. Elsevier, Amsterdam (2003) 10. Hecht, F., Pironneau, O.: An energy stable monolithic Eulerian fluid-structure finite element method. Int. J. Numer. Methods Fluids 85, 430–446 (2017) 11. Heil, M., Hazel, A.L., Boyle, J.: Solvers for large-displacement fluid-structure interaction problems: segregated versus monolithic approaches. Comput. Mech. 43(1), 91–101 (2008)

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12. Hesch, C., Gil, A., Carre˜ no, A.A., Bonet, J., Betsch, P.: A mortar approach for fluid–structure interaction problems: immersed strategies for deformable and rigid bodies. Comput. Methods Appl. Mech. Eng. 278, 853–882 (2014) 13. Kadapa, C., Dettmer, W., Peri´c, D.: A fictitious domain/distributed Lagrange multiplier based fluid–structure interaction scheme with hierarchical B-Spline grids. Comput. Methods Appl. Mech. Eng. 301, 1–27 (2016) 14. Kadapa, C., Dettmer, W., Peri´c, D.: A stabilised immersed framework on hierarchical b-spline grids for fluid-flexible structure interaction with solid-solid contact. Comput. Methods Appl. Mech. Eng. 335, 472–489 (2018) 15. K¨ uttler, U., Wall, W.A.: Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput. Mech. 43(1), 61–72 (2008) 16. Liu, W.K., et al.: Immersed finite element method and its applications to biological systems. Comput. Methods Appl. Mech. Eng. 195(13–16), 1722–1749 (2006) 17. Peskin, C.S.: The immersed boundary method. Acta Numerica 11, 479–517 (2002) 18. Pironneau, O.: Numerical study of a monolithic fluid–structure formulation. In: Frediani, A., Mohammadi, B., Pironneau, O., Cipolla, V. (eds.) Variational Analysis and Aerospace Engineering. SOIA, vol. 116, pp. 401–420. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45680-5 15 19. Schott, B., Ager, C., Wall, W.A.: Monolithic cut finite element based approaches for fluid-structure interaction. Int. J. Numer. Methods Eng. 119(8), 757–796 (2019) 20. Wang, X., Zhang, L.T.: Interpolation functions in the immersed boundary and finite element methods. Comput. Mech. 45(4), 321–334 (2009) 21. Wang, Y.: A one-field fictitious domain method for fluid-structure interactions. Ph.D. thesis, University of Leeds (2018) 22. Wang, Y., Jimack, P.K., Walkley, M.A.: A one-field monolithic fictitious domain method for fluid–structure interactions. Comput. Methods Appl. Mech. Eng. 317, 1146–1168 (2017) 23. Wang, Y., Jimack, P.K., Walkley, M.A.: Energy analysis for the one-field fictitious domain method for fluid-structure interactions. Appl. Numer. Math. 140, 165–182 (2019) 24. Wang, Y., Jimack, P.K., Walkley, M.A.: A theoretical and numerical investigation of a family of immersed finite element methods. J. Fluids Struct. 91, 102754 (2019) 25. Yu, Z.: A DLM/FD method for fluid/flexible-body interactions. J. Comput. Phys. 207(1), 1–27 (2005) 26. Yu, Z., Shao, X.: A three-dimensional fictitious domain method for the simulation of fluid-structure interactions. J. Hydrodyn. Ser. B 22(5), 178–183 (2010). https:// doi.org/10.1016/S1001-6058(09)60190-6 27. Zhang, L., Gerstenberger, A., Wang, X., Liu, W.K.: Immersed finite element method. Comput. Methods Appl. Mech. Eng. 193(21), 2051–2067 (2004) 28. Zhao, H., Freund, J.B., Moser, R.D.: A fixed-mesh method for incompressible flow– structure systems with finite solid deformations. J. Comput. Phys. 227(6), 3114– 3140 (2008)

Multi Axes Sliding Mesh Approach for Compressible Viscous Flows Masashi Yamakawa1(&), Satoshi Chikaguchi1, Shinichi Asao2, and Shotaro Hamato1 1

Kyoto Institute of Technology, Matsugasaki Sakyo-ku, Kyoto 606-8585, Japan [email protected] 2 College of Industrial Technology, Nishikoya, Amagasaki, Hyogo 661-0047, Japan

Abstract. To compute flows around a body with a rotating or movable part like a tiltrotor aircraft, the multi axes sliding mesh approach has been proposed. This approach is based on the unstructured moving grid finite volume method, which has adopted the space-time unified domain for control volume. Thus, it can accurately express such a moving mesh. However, due to the difficulty of mesh control in viscous flows and the need to maintain the stability of computation, it is restricted to only inviscid flows. In this paper, the multi axes sliding mesh approach was extended to viscous flows to understand detailed flow phenomena around a complicated moving body. The strategies to solve several issues not present in inviscid flow computations are described. To show the validity of the approach in viscous flows, it was applied to the flow field of a sphere in uniform flow. Multiple domains that slide individually were placed around the sphere, and it was confirmed that the sliding mesh did not affect the flow field. The usability of the approach is expected to be applied to practical viscous flow computations. Keywords: Computational fluid dynamics Sliding mesh approach
Viscous flows


Unstructured moving mesh


1 Introduction Numerical simulations of flows around a body with movable parts like a rotorcraft or sports athlete has a high utility value for various fields. However, handling a moving mesh is challenging in a body-fitted coordinate system. When the movable scope of its parts is small, the moving mesh method using a tension spring [1] can be used. On the other hand, for large motions, the mesh method is restricted. It is almost impossible to express a rotary motion such as the rotor part of a helicopter by using the moving mesh method with spring. To resolve this issue, the sliding mesh approach [2] was proposed. In this approach, the motion of a body is expressed by sliding the boundary of adjacent divided computational domains. This is different from the overset grid method in which one domain is put on another domain. An information exchange of physical values between domains is then conducted by interpolation, which might not satisfy physical conservation laws. On the other hand, by using the sliding mesh approach for the information exchange, the physical value can be conserved. One of the simplest © Springer Nature Switzerland AG 2020 V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 46–59, 2020. https://doi.org/10.1007/978-3-030-50436-6_4

Multi Axes Sliding Mesh Approach for Compressible Viscous Flows

47

applications of the sliding mesh approach is the divided cylindrical computational domain for axial direction. Its rotating cylinder has been applied to, for example, the simulation of a flow around a multistage turbine cascade. Also, one domain can be also embedded in another domain. In this case, the embedded sub domain should be cylindrical or spherical. Furthermore, there should not be a gap between two domains during the rotation of the embedded domain. Although the sliding mesh approach is very useful, it is difficult to express complicated motion. For example, the rotor part of a helicopter is expressed with comparative ease, but to express the rotor blade of a tiltrotor like the Osprey V-22 is impossible. This is because the rotor blade rotates, and moreover, an engine nacelle having a rotor blade also rotates on different axis to change the flight mode. In this case, the flows around a tiltrotor during rotor-blade mode and fixed-wing mode are computed [3] individually. In a simulation focused on changing flight modes, its computations [4] were conducted for fixed degrees of the engine nacelle at 0, 30, 60, and 90° as calculating a moving engine nacelle was quite difficult. For this issue, we proposed the multi axes sliding mesh approach [5], in which the moving engine nacelle is expressed in the middle size computational domain. The small size domain including the rotating blade is then embedded in the middle size domain with both domains embedded in the large size main domain. Furthermore, we succeeded in rotating the small and middle domains individually. However, the approach is conducted under inviscid flows to prioritize reproducibility of complicated motion. Therefore, the turbulent flow transition phenomenon in the wake of rotor could not be calculated. The objective of this paper is to apply the multi axes sliding mesh approach to viscous flows. The formulation of the approach and its validity when applying a flow around a sphere will be shown.

2 Numerical Approach 2.1

Governing Equation

For the governing equation, the following three-dimensional (3D) Navier–Stokes equation for compressible flows written in conservation law form is adopted.
 @q @E @F @G 1 @Ev @Fv @Gv þ þ þ ¼ þ þ @t @x @y @z Re @x @y @z

ð1Þ

Where 3 3 2 3 2 3 2 3 3 3 2 2 2 qu qw qv q 0 0 0 2 7 6 sxx 7 6 syx 7 6 szx 7 7 6 quw 6 qu 7 6 quv 6 qu þ p 7 7 7 6 7 6 7 6 7 7 7 6 6 6 6 7; Ev ¼ 6 sxy 7; Fv ¼ 6 syy 7; Gv ¼ 6 szy 7 7 7; F ¼ 6 qv2 þ p 7; G ¼ 6 qvw 6 q¼6 7 7 6 7 6 7 6 7 7 6 2 6 qv 7; E ¼ 6 quv 6 4 sxz 5 4 syz 5 4 szz 5 5 5 4 qw þ p 5 4 qw 5 4 qvw 4 quw fE5 fF5 fG5 wðe þ pÞ e vðe þ pÞ uðe þ pÞ 2

ð2Þ

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The unknown variables q, u, v, w, and e show the gas density, velocity components in the x, y, and z directions, and total energy per unit volume, respectively. The working fluid is assumed to be a perfect gas, and the pressure p is defined by    1  p ¼ ðc  1Þ e  q u2 þ v2 þ w2 2

ð3Þ

fE5, fF5, and fG5 are shown in Eq. (4). Here, l and lt are the coefficients of molecular viscosity and eddy viscosity, respectively. Pr, Prt, and Re are the Prandtl number, turbulent Prandtl number, and Reynolds number, respectively. The ratio of specific heats c is typically taken as being 1.4. In this study, Pr = 0.72 and Prt= 0.9 are obtained.

2.2

fE5 ¼ usxx þ vsxy þ wsxz þ


 1 l l @T 2 þ t c  1 Pr Prt @x

fF5 ¼ usyx þ vsyy þ wsyz þ


 1 l l @T 2 þ t c  1 Pr Prt @y

fG5 ¼ uszx þ vszy þ wszz þ


 1 l l @T 2 þ t c  1 Pr Prt @z

ð4Þ

Numerical Schemes

The sliding mesh approach is a type of moving mesh approach. In this study, the unstructured moving grid finite volume method [6] is adopted. The method assures a geometric conservation law [7] as well as a physical conservation law. A control volume in the space-time unified domain (x, y, z, t), which is four-dimensional (4D) for 3D flows, is then used. This approach has been mainly applied to Euler equations for inviscid compressible flows. In this paper, the approach is discretized for compressible viscous Fv flows. For the discretization, Eq. (1), which is written in divergence form, is integrated as Z

e v dX ¼ 0; reF

ð5Þ

X

where ev ¼ F


 1 1 1 E  E v ; F  Fv ; G  Gv ; q : Re Re Re

ð6Þ

Since the approach is based on a cell-centered finite volume method, the flow variables are defined at the center of the cell in the (x, y, z) space. Thus, the control volume becomes a 4D polyhedron in the (x, y, z, t)-domain. For the control volume, Eq. (4) is rewritten using the Gauss theorem as:

Multi Axes Sliding Mesh Approach for Compressible Viscous Flows

Z X

e v dX ¼ reF

Z

ev
e F n dV ¼

XNs þ 2 l¼1

ðqnt þ UÞl ¼ 0;

49

ð7Þ

V

where U ¼ H  Hv , H ¼ Enx þ Fny þ Gnz ; Hv ¼

ð8Þ

 1  Ev nx þ Fv ny þ Gv nz ; Re

Here, Ns indicates the number of boundary surfaces of the element. l is the volume of trajectory generated by the moving boundary surface of the element from t = n to t = n + 1. Then, Eq. (7) is rewritten as Eq. (9), and by solving Eq. (9), new q is obtained. qn þ 1 ðnt ÞNs þ 2 þ qn ðnt ÞNs þ 1 þ

Ns h i X qn þ 1=2 nt þ Un þ 1=2 ¼ 0 l

l¼1

qn þ 1=2 ¼

 1  nþ1 q þ qn 2

Un þ 1=2 ¼

 1  nþ1 U þ Un 2

ð9Þ

The inviscid flux vectors are evaluated using the Roe flux difference splitting scheme [8] with the MUSCL scheme as well as the Venkatakrishnan limiter [9]. The vectors are discretized by central difference. To solve the implicit algorithm, the LU-SGS implicit scheme is adopted. 2.3

Evaluation on a Boundary

On a boundary, the first derivative of a physical value cannot be evaluated using central difference. For example, discretization of the first derivative for primitive variable u is described. Figure 1 shows a discretization outline of the first derivative. The first derivative for primitive variable u is obtained by solving the follow equations. Au_ ¼ b;

ð10Þ

where 0

xc  x w A ¼ @ 0:5ðxv2  xv3 Þ  xv1 0:5ðxv3  xv1 Þ  xv2

yc  yw 0:5ðyv2  yv3 Þ  yv1 0:5ðyv3  yv1 Þ  yv2

1 zc  zw 0:5ðzv2  zv3 Þ  zv1 A; 0:5ðzv3  zv1 Þ  zv2

ð11Þ

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Fig. 1. Discretization outline of the first derivative

0

1 0 1 uc  uw ux u_ ¼ @ uy A; b ¼ @ 0:5ðuv2  uv3 Þ  uv1 A: uz 0:5ðuv3  uv1 Þ  uv2 Here, v indicates vertex, c indicates the center of an element, and w indicates the center of a boundary surface for an element. Also, uw is evaluated as following equation, uw ¼

 1 uc þ ughost : 2

ð12Þ

The vertex of the primitive variable uvi is calculated using the following weighted average method, where uwj is the physical value at the cell center of the triangle constructed by vertex wviw and rij is the distance between the vertex and center point of each cell around it. N P

uvi ¼

j2 i

uw j r1ij

N P j2 i

rij ¼

;

ð13Þ

1 rij

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2  2 xwj  xvi þ ywj  yvi þ zwj  zvi :

ð14Þ

3 Sliding Mesh Approach 3.1

Multi Axes Sliding Mesh Approach

In the sliding mesh approach, a sliding boundary surface exists. Here, the embedded sub computational domain is rotated in the main domain. In a 3D system, the embedded sub domain should have an almost spherical or cylindrical configuration. Although the

Multi Axes Sliding Mesh Approach for Compressible Viscous Flows

51

computational cost using the approach is not expensive, the movable range of vertices is limited. In other words, the motions of an object are restricted. Thus, to improve flexibility, the axes of the rotating sub domain are added in the approach. However, to avoid an interaction between sub domains that have individual axes, one sub domain is embedded in the other sub domain, as shown in Fig. 2. In this figure, computational domain 3 is embedded in computational domain 2, which is embedded in computational domain 1. The whole domain can be moved using the moving computational domain (MCD) method [10]. The advantage of this method is that it does not require a spring method to move the object, so it is less likely to create extremely skewed elements. Basically, the multi axes sliding mesh approach has the potential to express any object motion combined with the MCD method without destroying computational mesh.

Fig. 2. Multi axes sliding mesh approach

The physical values on the sliding plane interpolate with each other through the plane. Interpolation values are determined depending on the area where domain elements overlap. Specifically, the value is calculated in accordance with the area of the overlapping part Sij between the elements of the sliding plane, as shown in Fig. 3. The value of the part is defined with Eq. (15). P

qj Sij j2i qbi ¼ P Sij

ð15Þ

j2i

Where cell j.

P j2i

shows the sum of cell j adjacent to cell i. Then, qj is the physical value of

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Fig. 3. Overlapping part in slide element

3.2

Multi Axes Sliding Mesh for Viscous Flow Computation

3.2.1 Evaluation of the First Derivative on a Sliding Surface As the sliding surface is also a boundary, a specific evaluation of the first derivative of the primitive variable on the sliding surface is required along with the evaluation of the boundary. However, unlike boundaries, there is an element on the opposite side of a surface. Thus, the physical value of that element should be used to calculate the first derivative of the primitive variable. First, a ghost cell j adjacent to element i through the boundary surface of element i is generated. Then, qbj calculated as Eq. (16) is interpolated in element j. Here, element k is adjacent to element i across the sliding surface. Figure 4 shows a schematic diagram around the sliding surface. P

qk Sjk qbj ¼ P : Sjk k2j

ð16Þ

k2j

For example, the first derivative of primitive variable u is calculated using the following central difference Eq. (17) and the partially deformed Gauss-Green’s theorem (18).

Fig. 4. Evaluation of the first derivative on a sliding surface

Multi Axes Sliding Mesh Approach for Compressible Viscous Flows

53

(

 )
 @u 1 @u @u ¼ þ ; @ x ij 2 @x i @x j

ð17Þ


  Ns
@u 1 X hfi ui þ hfj uj kðjÞ þ hbj ubj ð1  kðjÞÞ ¼ nij : @ x i VX i j 2 i hf i þ hf j kðjÞ þ hbj ð1  kðjÞÞ

ð18Þ

Where hbj is calculated using hk, which is the distance between the center point of element k and the center of the adjacent surface of elements i and j, as shown in Eq. (19), P

hk Sjk hbj ¼ P : Sjk k2j

ð19Þ

k2j

In Eq. (18), if adjacent element j is a ghost cell, k(j) = 0, else, k(j) = 1. 3.2.2

Evaluation of the First Derivative on an Element Having both a Sliding Surface and Boundary In this subsection, an evaluation of the first derivative on an element that has both a sliding surface and boundary is described. First, the primitive variable for a vertex located on both the sliding surface and boundary is calculated. The first derivative of the primitive variable is then calculated using Eqs. (10) to (14). For example, the calculation procedure of the primitive variable uvi is shown in Eqs. (20) to (22). Its schematic figure of this case is shown in Fig. 5. N P

uvi ¼

j2 i

uw j r1ij kv ð jÞ þ ub j rb1ij ð1  kv ð jÞÞ

N P 1 j2 i

rb i j

rij

kv ð j Þ þ

1 rb ij

ð 1  kv ð j Þ Þ





;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 ¼ xb j  xvi þ yb j  yvi þ zb j  zvi ;

P P xc k Sjk yc k Sjk zc k Sjk k2j k2j k2j xb j ¼ P ; yb j ¼ P ; zb j ¼ P : Sjk Sjk Sjk

ð20Þ

ð21Þ

P

k2j

k2j

ð22Þ

k2j

Where ubj is the primitive variable in the center of the ghost cell of element j that has vertex i. The variable is then calculated from Eq. (15). (xck, yck, zck) is the coordinates in the center of element k located adjacent to element j across the sliding surface. Finally, element j has vertex i. If element j is a ghost cell, kv (j) = 0, else, kv (j) = 1.

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Fig. 5. Evaluation of the first derivative on a sliding surface and boundary

3.2.3 Prism Element on Sliding Surface When viscous flows are computed using an unstructured mesh, it is necessary to use quite thin prism elements in the boundary layer. However, if the shape of the body boundary is curved, part of an element might overlap the sliding element and static element as shown in Fig. 6. If there is no overlap between the elements, the physical value cannot be interpolated. Such a problem occurs when the difference between both volumes is not small. Thus, the volume difference should be as small as possible.

Fig. 6. Sliding mesh near a body surface

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4 Verification of the Multi Axes Sliding Mesh Approach 4.1

Application to a Flow Around a Sphere

The multi axes sliding mesh approach is applied to a viscous flow around a sphere. Figure 7 shows a schematic figure of the flow. The sphere is placed in a uniform flow with two sliding cylinders, which have rotation axes in different directions. Each sliding cylindrical mesh rotates around the static sphere, so the sliding mesh must not affect the flow. To confirm the validity of the approach, it is compared with the flow around a sphere in a single mesh. In Fig. 8, case 1 shows a schema of multi axes sliding cylinders around a sphere and case 2 shows its comparison.

Fig. 7. Multi axes sliding mesh around a sphere

Fig. 8. Schema of comparative computation

4.2

Initial Mesh and Computational Conditions

Figure 9 shows the initial mesh for case 1. The total number of meshes is 4,219,268. Figure 10 shows a single mesh for comparison (case 2). The number of meshes is 4,578,854. Their elements were created by using MEGG3D [11]. The diameter of the computational domain (domain 3 in case 1, whole domain in case 2) is 40 times that of the sphere.

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Fig. 9. Case 1: Multi axes sliding mesh (Left: atmosphere, Right: sliding cylinders)

Fig. 10. Case 2: Single mesh (Left: atmosphere, Right: whole mesh)

Table 1 Conditions to verify the interpolation on the sliding mesh surface in consideration of viscosity Name Initial conditions Density Velocity (x-direction) Velocity (y-direction) Velocity (z-direction) Pressure Other conditions Time step size Reynolds number Rotational speed of domains Radius (domain 1, domain 2) Height (domain 1, domain 2)

Symbol Value q u v w p

1.0 0.1 0.0 0.0 1.0/c

Dt Re x1 , x2 r1, r2 h1, h2

0.001 10,000 0.05, 0.03 0.7, 0.75 0.25, 1.5

Multi Axes Sliding Mesh Approach for Compressible Viscous Flows

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Computational conditions are shown in Table 1. The rotations of domains 1 and 2 in case 1 are dominated by Eqs. (23) and (24), respectively. Therefore, while both domain 1 and its axis rotates, only domain 2 rotates and its axis remains fixed. 0

1 0 cos x t x0 @ y0 A ¼ @ sin x t 0 z0

 sin x t cos x t 0




4.3

x0 y0




¼

10 0 1 0 0 A @ 0 cos x t 1 0 sin x t

cos x t sin x t

 sin x t cos x t

10 1 0 x  sin x t A @ y A; cos x t z


 x : y

ð23Þ

ð24Þ

Computational Result

Figure 11 shows the conditions of the sliding mesh as the result. The sliding motion was confirmed to have no skewed and crushed elements. Under the sliding mesh environment, a flow around a sphere was computed. Figure 12 shows the velocity contours. Around the sphere, the flow in case 1 corresponded reasonably well with that in case 2. Thus, highly accurate interpolation was seen on the sliding surface, confirming that the sliding mesh did not affect the flow. The pressure drag coefficient of the sphere surface in case 1 was compared with that of case 2, other calculation results [12], and experimental results [13] as shown in Table 2. The discrepancy between case 1 and other calculation results is around 1.0%. Furthermore, the deviation from the experimental results is less than 3.0%, which also shows the validity of the sliding approach. The discrepancy between case 2 and the other calculation and experimental results is larger than case 1 despite no moving and sliding mesh around the sphere. This is possibly due to the cylindrical sliding domain potentially generating a regular mesh. Figure 13 shows the averaged pressure drag coefficient of case 1 and case 2 on a sphere surface. As the flow is unsteady, the time-averaged drag coefficient is used. Both match in front of the sphere, but there is a slight difference in wake. In general, a complicated flow containing vortices occurs behind a sphere. Thus, the mesh behind the sphere should be generated delicately. However, interpolation between the first layer of the static mesh and sliding mesh might affect such a sensitive flow.

Fig. 11. Conditions of the sliding mesh

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Case 1

Case 2 Fig. 12. Velocity contours around sphere

Table 2. Drag coefficient of sphere Case 1 Case 2 Calculated value Experimental value Drag coefficient 0.389 0.379 0.393 0.40

Fig. 13. Averaged pressure drag coefficient on sphere surface

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5 Conclusion In this paper, the multi axes sliding mesh approach for compressible viscous flows was formulated. In particular, the interpolation process between prism elements on sliding surfaces was described. As a result of the computation of the flow around a sphere, the sliding motion of multiple cylinders without skewed and crushed elements were confirmed. The results also showed that there the sliding mesh had no affect on flow. A comparison of other experimental and computational results showed the validity of the multi axes sliding mesh approach. This approach could potentially be applied to complicated motions like a bicycle rider is computing. Acknowledgments. This publication was subsidized by JKA through its promotion funds from KEIRIN RACE.

References 1. Murayama, M., Nakahashi, K., Matsushima, K.: Unstructured dynamic mesh for large movement and deformation. AIAA Paper 2002-0122 (2002, to appear) 2. Bakker, A., et al.: Sliding mesh simulation of laminar flow in stirred reactors. Chem. Eng. Res. Des. 75(1), 42–44 (1997) 3. Chaderjian, N.M., Field, M.: Advances in rotor performance and turbulent wake simulation using DES and adaptive mesh refinement. In: 7th International Conference on Computational Fluid Dynamics (2012) 4. Ying, Z., Liang, Y., Shuo, Y.: Numerical study on flow fields and aerodynamics of tilt rotor aircraft in conversion mode based on embedded grid and actuator model. Chin. J. Aeronaut. 28(1), 93–102 (2015) 5. Takii, A., Yamakawa, M., Asao, S., Tajiri, K.: Six degrees of freedom numerical simulation of tilt-rotor plane. In: Rodrigues, J.M.F., Cardoso, P.J.S., Monteiro, J., Lam, R., Krzhizhanovskaya, V.V., Lees, M.H., Dongarra, J.J., Sloot, P.M.A. (eds.) ICCS 2019. LNCS, vol. 11536, pp. 506–519. Springer, Cham (2019). https://doi.org/10.1007/978-3-03022734-0_37 6. Yamakawa, M., et al.: Unstructured moving-grid finite-volume method for unsteady shocked flows. J. Comput. Fluids Eng. 10(1), 24–30 (2005) 7. Obayashi, S.: Freestream capturing for moving coordinates in three dimensions. AIAA J. 30, 1125–1128 (1992) 8. Roe, P.L.: Approximate Riemann solvers parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981) 9. Venkatakrishnan, V.: On the accuracy of limiters and convergence to steady state solutions. AIAA Paper, 93-0880 (1993) 10. Asao, S., et al.: Simulations of a falling sphere with concentration in an infinite long pipe using a new moving mesh system. Appl. Thermal Eng. 72, 29–33 (2014) 11. Ito, Y.: Challenges in unstructured mesh generation for practical and efficient computational fluid dynamics simulations. Comput. Fluids 85, 47–52 (2013) 12. Constantinescu, G.S., Squires, K.D.: LES and DES investigations of turbulent flow over a sphere at Re = 10,000. Flow Turbul. Combust. 70, 267–298 (2003) 13. Achenbach, E.: Experiments on the flow past spheres at very high Reynolds numbers. J. Fluid Mech. 54, 565–575 (1972)

Monolithic Arbitrary Lagrangian–Eulerian Finite Element Method for a Multi-domain Blood Flow–Aortic Wall Interaction Problem Pengtao Sun1(B) , Chen-Song Zhang2 , Rihui Lan1 , and Lin Li3 1

2

Department of Mathematical Sciences, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154, USA [email protected] LSEC and NCMIS, Academy of Mathematics and System Science, Beijing, China 3 Department of Mathematics, Peking University, Beijing, China https://sun.faculty.unlv.edu/ Abstract. In this paper, an arbitrary Lagrangian–Eulerian (ALE) finite element method in the monolithic approach is developed for a multidomain blood flow–aortic wall interaction problem with multiple moving interfaces. An advanced fully discrete ALE-mixed finite element approximation is defined to solve the present fluid–structure interaction (FSI) problem in the cardiovascular environment, in which two fields of structures are involved with two fields of fluid flow, inducing three moving interfaces in between for the interactions. Numerical experiments are carried out for a realistic cardiovascular problem with the implantation of vascular stent graft to demonstrate the strength of our developed ALE-mixed finite element method. Keywords: Cardiovascular fluid–structure interaction (FSI) Multi-domain · Vascular aneurysm · Stent graft · Arbitrary Lagrangian–Eulerian (ALE) · Mixed finite element

1

·

Introduction

In this paper, we consider the problem of interactions between the incompressible free viscous fluid flow and the deformable elastic structure of multiple fields. This problem is of great importance in a wide range of applications, such as in the cardiovascular area of physiology, where a significant example of this type of problem can be described as the blood fluid–vessel interaction problem. During a cardiovascular cycle, any abnormal mechanical change on the elastic property of the aortic vessel could induce some severe cardiovascular diseases (CVDs), such P. Sun and R. Lan were partially supported by NSF Grant DMS-1418806. C.-S. Zhang was supported by the National Key Research and Development Program of China (Grant No. 2016YFB0201304), NSFC 11971472, and the Key Research Program of Frontier Sciences of CAS. c Springer Nature Switzerland AG 2020  V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 60–74, 2020. https://doi.org/10.1007/978-3-030-50436-6_5

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as the vascular stenosis, the vascular rupture, the aneurysm, the aortic dissection, and etc. Hence, a comprehensive modeling study and an accurate numerical simulation on the interactions between the blood fluid flow and the elastic aortic wall become extremely important and urgent to help medical professionals understand how the CVDs occur, grow, and fatally affect the patients, as well as to help cardiovascular physicians to find out the best medical treatment. In the past several decades, the vascular stent graft has become one of the most popular fixation devices to carry out surgery for endovascular aortic repair (EVAR) in order to rescue CVD patients. For instance, without the need for an open surgery, the implanted vascular stent graft may cure the vascular stenosis by expanding the constrictive aortic lumen back to normal and then supporting therein as the backbone of the aorta; can treat the aortic dissection (see the left of Fig. 1) by substituting for the dissected aortic wall that is induced by an intima tear; and can remedy the abdominal aortic aneurysm (AAA) (see the middle of Fig. 1) by replacing the bulgy part of the aortic wall that is caused by a lack of elasticity in the aorta. In all cases, because of the elastic expansion of the vascular stent and the residual elasticity energy remained in the diseased aortic wall, most of the blood fluid that is trapped in the void space between the stent graft and the pathologically changed aortic wall are supposed to be squeezed back to the blood vessel lumen through the stent graft. But, there might be still a small amount of blood fluid remaining in the corner of aorta and will be eventually solidified as the thrombi. Thus, all extra blood fluids that are trapped in the shrinking void spaces are supposed to be expelled. Then the diseased blood vessel would be gradually cured to a great extent, demonstrated as either bouncing back for the aortic aneurysm case, or no longer deteriorated and then eventually normalized in an improving cardiovascular environment for the aortic dissection case (see Fig. 1, right).

Fig. 1. The CT-scanned data of CVD patients. Left: The aortic dissection, where the green part illustrates the false aortic lumen. Middle: The abdominal aortic aneurysm. Right: Different aortic dissection cases treated by the implanted vascular stent graft. (Color figure online)

However, the EVAR surgery has been long lacking a quantitative study about how and under what circumstance the CVDs are given rise to and then grow, as well as how they are changed after the implantation of a vascular stent graft,

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i.e., an accurate and efficient mathematical modeling and numerical simulation of the hemodynamic FSI are still largely missing in the investigations before and after the EVAR surgery in the modern medical field of CVDs. Moreover, it has been increasingly recognized that a personalized vascular stent graft is crucially needed for CVD patients, which cannot be done either without a quantitative numerical study of the hemodynamic FSI on the feasibility of the vascular stent graft in the cardiovascular environment. In this paper, we study the effect of using an elastic material model to describe the deformable aortic wall and stent graft, and their interactions with the blood fluid flow, where, due to the loss of elasticity of the aortic wall, which is however still within the range of linear elasticity, the blood vessel lumen is inflated to form an abdominal aortic aneurysm, and the vascular stent graft is implanted into the aneurysm to separate the blood fluid into two parts. Thus four domains, i.e., two fluid domains and two structure domains, are separated by three interfaces. To model the blood fluid, we consider the Navier–Stokes equations, under the assumptions of incompressibility and Newtonian rheology, which is conventionally defined in Eulerian description. The elastic structure equation, which is conventionally defined in Lagrangian description, can be generally defined by various elastic materials such as the classical Hookean-type material model. In addition, in such a set of governing equations of FSI problem, the time and space dependence of the primary unknowns and of the moving interfaces between the fluid and the elastic structure play a significant role for the dynamic interactions in between, where, the well defined interface conditions among four domains are crucial as well. Regarding the numerical methodology to be studied in this paper, we develop an advanced numerical method based upon the arbitrary Lagrangian–Eulerian (ALE) mapping within a monolithic approach to solve the present FSI problem. In the first place, we prefer the monolithic approach [10], in view of its unconditional stability and the immunity of any systematic error in the implementation of interface conditions for any kind of FSI problem. Moreover, a high-performance preconditioning linear algebraic solver can also be developed and parallelized for the monolithic system without doing an alternating iteration by subdomains [1]. In contrast, the partitioned approach [2], which decouples the FSI system and iteratively solves the fluid and the structure equations via an iteration-by-subdomain, is conditionally stable and conditionally convergent under a particular range of the physical parameters of FSI model, e.g., when fluid and structural densities are of the same order. This is called the added-mass effect [4] specifically induced by the partitioned approach. Unfortunately, the hemodynamic FSI problems are within the particular range of the added-mass effect, making the partitioned approach very difficult to use. Hence again, the monolithic approach is the primarily reliable method to be studied in this paper. On the top of the monolithic approach, we develop an ALE method for the present FSI problem. As a type of body-fitted mesh method, ALE technique [10] has become the most accurate and also the most popular approach for solving FSI problems and other general interface problems [6,8], where the mesh on the interface is accommodated to be shared by both the fluid and the structure, and thus to automatically satisfy the interface conditions across the interface.

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The structure of the paper is the following: in Sect. 2 we define a type of dynamic FSI model. The ALE mapping and the weak form in ALE description are defined in Sect. 3. We further develop the monolithic ALE finite element method for the present FSI model in semi- and fully discrete schemes in Sect. 4. Numerical experiments for a realistic cardiovascular problem with the implantation of vascular stent graft are carried out in Sect. 5.

2

A Model of Fluid-Structure Interaction (FSI) Problem

Let Ω be an open bounded domain in Rd (d = 2, 3) with a convex polygonal boundary ∂Ω, I = (0, T ] (T > 0). Four subdomains, Ωtk := Ωi (t) ⊂ Ω (k = s

f1 , f2 , s1 , s2 ) (0 ≤ t ≤ T ), satisfying Ωtf1 ∪ Ωtf2 ∪ Ωts1 ∪ Ωts2 = Ω, Ωtfi ∩ Ωt j = ∅ (i, j = 1, 2), respectively represent two fluid domains: Ωtf1 is embraced by the implanted stent graft while Ωtf2 occupies the interior of the aortic aneurysm, both of which are filled with an incompressible and viscous blood fluid, and, two elastic structure domains: Ωts1 is the aortic wall while Ωts2 is formed by the implanted vascular stent graft. As illustrated in Fig. 2, four subdomains are separated by three interfaces: Γt12 := ∂Ωtf1 ∩ ∂Ωts2 , Γt21 := ∂Ωtf2 ∩ ∂Ωts1 , Γt22 := ∂Ωtf2 ∩ ∂Ωts2 , which may move/deform along with t ∈ I, resulting that Ωtk (k = f1 , f2 , s1 , s2 ) also change with t ∈ I and are termed as the current (Eulerian) domains with ˆ k := respect to x k , in contrast to their initial (reference/Lagrangian) domains, Ω k Ω0 (k = f1 , f2 , s1 , s2 ) with respect to xˆ k , where, a flow map is defined from ˆ k to Ωtk (k = f1 , f2 , s1 , s2 ), as: xˆ k → x k (ˆ Ω x k , t) such that x k (ˆ x k , t) = xˆ k + ˆ k is the displacement field in the Lagrangian frame. ˆ k (ˆ x k , t), ∀t ∈ I, where u u ˆ x k , t) which equals ψ(x k (ˆ ˆ = x k , t), t), and ∇ In what follows, we set ψˆ = ψ(ˆ f f1 f2 ˆ s s1 s2 12 21 ˆ ˆ ∇xˆ k (k = s1 , s2 ). Let Ωt = Ωt ∪ Ωt , Ω = Ω ∪ Ω , Γt = Γt ∪ Γt ∪ Γt22 .

Fig. 2. Two schematic domains of multi-domain FSI for the CVD case of aneurysm, where, Ωtf1 : the blood vessel lumen, Ωtf2 : the blood fluid squeezed inside the aneurysm, Ωts1 : the aortic wall, Ωts2 : the vascular graft, and three interfaces Γt12 := ∂Ωtf1 ∩ ∂Ωts2 , Γt21 := ∂Ωtf2 ∩ ∂Ωts1 , Γt22 := ∂Ωtf2 ∩ ∂Ωts2 .

In this paper, we are interested in studying a pressure-driven flow through the deformable channel with a two-way coupling between the incompressible fluid and the elastic material, where the fluid motion is defined in Eulerian description and the structure motion is defined in Lagrangian description as follows

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Fluid motion: ⎧ Dv f ρf Dt − ∇ · σ f = ρf f f , ⎪ ⎪ ⎪ ⎪ ∇ · v f = 0, ⎨ v f = v bf , ⎪ ⎪ ⎪ σ f n f = 0, ⎪ ⎩ v f = v 0f , Elastic structure motion: ⎧ ∂ 2 uˆ s ˆ ˆ s = ρˆs fˆ s , ⎪ ⎪ ρˆs ∂t2 − ∇ · σ ⎨ ˆ bs , ˆs = u u ⎪ ˆ sn ˆ s = 0, σ ⎪ ⎩ ˆs u ˆ 0s , ∂∂t ˆs = u = vˆ 0s , u Interface conditions:  v f = v s, σf n f = σs n f ,

in Ωtf × I, in Ωtf × I, on [∂Ωtf \Γt ]D × I, on [∂Ωtf \Γt ]N × I, ˆf , in Ω

(a) (b) (c) (d) (e)

ˆ s × I, in Ω ˆ s \Γˆ ]D × I, on [∂ Ω ˆ s \Γˆ ]N × I, on [∂ Ω s ˆ , in Ω

(f ) (g) (h) (i)

on Γt × I, on Γt × I,

(j) (k)

(1)

ˆ s is the displacement where, v f is the velocity of the free fluid defined in Ωtf , u ˆs u = vˆ s (ˆ x s , t) = v s (x s (ˆ x s , t), t), ρf and of elastic structure that leads to ∂∂t ρˆs are the constant densities of the incompressible free fluid and the elastic D· denotes the classical concept of material derivative as structure, respectively. Dt Dψ ∂ψ = + (ψ · ∇)ψ. The stress tensor of each phase is defined as Dt ∂t T

σ f := σ f (v f ) = 2μf D(v f ) − pf I , D(v f ) := (∇v f + (∇v f ) )/2, ˆ ·u ˆu ˆu ˆ s := σ ˆ s I , ε(u ˆ s (u ˆ s ) = 2μs ε(u ˆ s ) + λs ∇ ˆ s ) := (∇ ˆ s + (∇ ˆ s )T )/2, σ

(2)

where, pf denotes the fluid pressure, μf , μs and λs are constant physical parameters representing the fluid viscosity, the shear modulus, and the Lam´e constant of elastic structures, respectively. Here different elastic structures may have differ ent elastic parameters, i.e., μs Ωˆ sk = μsk , λs Ωˆ sk = λsk , ρˆs Ωˆ sk = ρˆsk (k = 1, 2), ˆ s Ωˆ sk = u ˆ sk (k = 1, 2). inducing different structure displacements u Remark 1. The equation of elastic structure can be also reformulated in terms ˆs u ˆs = of the structure velocity, v s , as follows by substituting ∂∂t = vˆ s and u  t 0 ˆ s + 0 vˆ s dτ into (1(f)) [10] u vs ˆ s × I, ˆ ˆ v s ) = ρˆs fˆ s + ∇ ˆ ·σ ˆ s (u ˆ 0s ), in Ω ρˆs ∂ˆ ∂t − ∇ · σ s (ˆ b ˆ s \Γˆ ]D × I, on [∂ Ω x , t) = vˆ s , vˆ s (ˆ ˆ s \Γˆ ]N × I, ˆ sn ˆ s = 0, σ on [∂ Ω 0 s ˆ , ˆ s, vˆ s (ˆ x , 0) = u in Ω  



t ˆ · t vˆ s dτ I . ˆ s (ˆ where σ v s ) = 2μs ε 0 vˆ s dτ + λs ∇ 0

3

(a) (b) (c) (d)

(3)

ALE Mapping and the Weak Form in ALE Description

The core of ALE method is the introduction of ALE mapping, which is essentially ˆ f ))d , defined a type of time-dependent bijective affine mapping, X t ∈ (W 1,∞ (Ω

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ˆ f ⊂ Rd to the current (Eulerian) domain from the initial (Lagrangian) domain Ω f d Ωt ⊂ R , as follows [8,9] ˆ f → Ωtf , Xt : Ω xˆ f → x f (ˆ x f , t),

(4)

∈ (W 1,∞ (Ωtf ))d . The key reason why this mapping works well for and X −1 t FSI problems and other standard interface problems is because the following x f , t), t) and its ALE time derivative, Proposition 1 holds for any v = v (x f (ˆ  dv  ∂v = (x , t) + w (x , t) · ∇v (x , t), where w f is the domain velocity of f f f f dt x ∂t ˆ f

Ωt and w f =

∂X t ∂t

◦ (X t )−1 (x f ).

  ˆ = Proposition 1 [9]. v ∈ H 1 (Ωt ) and dv ∈ H 1 (Ωt ) if and only if v dt ˆ xf 1 ˆ ˆ(ˆ v xf , t) = v ◦ Xt ∈ H (Ω), and vice versa. Only the current (Eulerian) domain that involves a moving boundary/interface needs the ALE mapping to produce a moving mesh therein which can accommodate the motion of the boundary/interface without breaking the mesh connectivity along the time. In practice, one way to define the affineˆ f → Ωtf , is the harmonic extension technique. For type ALE mapping, X t : Ω instance, to compute the ALE mapping for the fluid mesh, X t , in the present FSI problem, we solve the following Laplace equation ⎧ ˆf , (a) ⎨ Δxˆ f X t = 0, in Ω (5) ˆ s , on Γˆ , (b) Xt = u ⎩ ˆ f \Γˆ . (c) X t = 0, on ∂ Ω Once the ALE mapping X t is computed, then the fluid mesh and its moving t ˆ f = ∂X velocity, w f , can be updated, accordingly, as: x f = xˆ f + X t , w ∂t and −1 ˆ f ◦ (X t ) . Due to Proposition 1, such defined ALE mapping, X t , wf = w dv  can guarantee that v f ∈ (H 1 (Ωtf ))d and dtf xˆ ∈ (H 1 (Ωtf ))d if and only if f ˆ f ))d , and vice versa, which is a sufficient and necessary vˆ f = v f ◦ X t ∈ (H 1 (Ω condition for definitions of the following functional spaces ˆ ◦ (X t )−1 , ψ ˆ ∈ (H 1 (Ω ˆ f ))d }, : Ωtf → Rd , ψ f = ψ f f f f ∈ V t : ψ f = 0 on [∂Ωt \Γt ]D }, ∈ V ft : ψ f = v bf on [∂Ωtf \Γt ]D }, ˆ = ψ ◦ X t on Γˆ , ψ ∈ V f ∩ L2 (Γt )}, ˆ s ))d : ψ ∈ (H 1 (Ω s f f t s s ˆ ˆ = 0 on [∂ Ω ˆ :ψ ˆ ∈V \ Γ ] }, D s ˆs ˆ ˆ s \Γˆ ]D }, ˆ bs on [∂ Ω s ∈ V : ψs = v b ˆ f )}, Qft := {qf : Ωtf → R, qf = qˆf ◦ (X t )−1 , qˆf ∈ L20 (Ω f ˆ on Γˆ , ψ ˆ ∈V ˆ s ∩ L2 (Γˆ )}, ˆ := {ξˆ ∈ (H 1 (Ω ˆ f ))d : ξˆ = 0 on ∂ Ω ˆ f \Γˆ ; ξˆ = ψ W s s b ˆ f := {ξˆ ∈ (H 1 (Ω ˆ f ))d : ξˆ = 0 on ∂ Ω ˆ f }, W 0 (6)

V ft V ft,0 V ft,b ˆs V ˆs V 0 ˆs V

:= {ψ f := {ψ f := {ψ f ˆ := {ψ s ˆ := {ψ s ˆ := {ψ

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ˆf ) : which make v f ∈ V ft,b ⊂ (H 1 (Ωtf ))d , pf ∈ Qft ⊂ L20 (Ωtf ) = {ˆ qf ∈ L2 (Ω  s 1 s d ˆ ⊂ (H (Ω ˆ )) . ˆf dˆ xf = 0}, and vˆ s ∈ V ˆf q b Ω dv  In view of the definition of ALE time derivative dtf xˆ , the interface condition (1(k)) and the time differentiation of the harmonic ALE Eq. (5), the weak form of FSI problem (1)–(3) can be defined as follows. ˆ s × Qf × W ˆ f) ˆ f ) ∈ (H 2 ∩ L∞ )(0, T ; V ft,b × V Weak Form. Find (v f , vˆ s , pf , w t b b such that  dv f  ˆ f ◦ X −1 , ψ f )Ω f + ρf (v f − w ρf ( t ) · ∇v f , ψ f Ω f ˆf x t t dt  + 2μf D(v f ), D(ψ f ) Ω f − (pf , ∇ · ψ f )Ω f + (∇ · v f , qf )Ω f t t t  

 t ∂ vˆ s ˆ ˆ ) , ψ s )Ωˆs + 2μs ε + ρˆs ( vˆ s dτ , ε(ψ s ∂t ˆs 0 Ω (7)

 t 

 ˆ ˆ ˆ ˆ ˆ ˆ ˆ f , ∇ξ ) ˆ = ρf (f , ψ ) f + λs ∇ · + (∇w vˆ s dτ , ∇ · ψ s

0

ˆ ) ˆ s − 2μs + ρˆs (fˆ s , ψ s Ω



f Ωf

ˆs Ω

ˆ ) ˆ 0s ), ε(ψ ε(u s

f



ˆs Ω

ˆ ˆ ·u ˆ ·ψ ˆ 0s , ∇ − λs ∇ s

ˆ , qf , ξˆ ) ∈ V f × V ˆ s × Qf × ∀(ψ f , ψ s f t 0 t,0

f Ωt



,

ˆs Ω ˆ f. W 0

Introduce the following trilinear function

1 (8) (u · ∇v , w )Ω f − (u · ∇w , v )Ω f . β(u, v , w ) = t t 2 Applying the Green’s theorem, the incompressibility (1(b)), the interface conditions (1(j)) and (5(b)), we obtain 

(v f − w f ) · ∇v f , ψ f



f

Ωt

= β(v f , v f , ψ f )Ω f − β(w f , v f , ψ f )Ω f + t

t

 1 v f ∇ · w f , ψ f Ωf . t 2

Then (7) can be reformulated as   dv f  ˆ f ◦ X −1 ρf , ψ + β(v f , v f , ψ f )Ω f − β(w t , v f , ψ f )Ωtf t dt xˆ f f Ω f t   1 −1 ˆ f ◦ X t ), ψ f Ω f + 2μf D(v f ), D(ψ f ) Ω f v f ∇ · (w + t t 2 ˆw ˆ ξˆf ) ˆ ˆ f,∇ − (pf , ∇ · ψ f )Ω f + (∇ · v f , qf )Ω f + (∇ Ωf t t  

 t ∂ vˆ s ˆ ˆ ) , ψ s )Ωˆs + 2μs ε + ρˆs ( vˆ s dτ , ε(ψ s ∂t ˆs 0 Ω

 t 

 ˆ ˆ ) ˆs ˆ· ˆ ·ψ + λs ∇ = ρf (f f , ψ f )Ω f + ρˆs (fˆ s , ψ vˆ s dτ , ∇ s s Ω 0

− 2μs



ˆs Ω

ˆ ) ˆ 0s ), ε(ψ ε(u s



ˆs Ω

t

ˆ ˆ ·u ˆ ·ψ ˆ 0s , ∇ − λs ∇ s

ˆ , qf , ξˆ ) ∈ V f × V ˆ s × Qf × ∀(ψ f , ψ s f t 0 t,0

,

ˆs Ω ˆ f. W 0

(9)

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4

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The Monolithic ALE Finite Element Approximation

We first introduce finite element spaces to discretize the functional spaces defined in (6). Conventionally, the finite element approximation to the linear elasticity equation is defined in a Lagrange-type piecewise polynomial (finite element) ˆ s , to accommodate its numerical solution vˆ s,h . In order to satisfy ˆs ⊂V space, V h b the interface condition (1(j)), we need not only the meshes in Ωtf and Ωts are conforming through Γt , but also the finite element space of fluid (Navier-Stokes) equations, V ft,h × Qft,h ⊂ V ft × Qft , matches the finite element space of the ˆ s , through Γˆ , i.e., a Lagrange-type finite element space structure equation, V shall be adopted to accommodate (v f,h , pf,h ) as well. So in this paper, we use the stable Stokes-pair, i.e., (H 1 )d -type mixed element, to discretize the NavierStokes equations, e.g., the Taylor-Hood (P 2 P 1 ) element, where the quadratic piecewise polynomials, P 2 , form the finite element V ft,h ⊂ V ft to hold v f,h , and the linear piecewise polynomials, P 1 , form Qft,h ⊂ Qft to hold pf,h . To

match V ft,h , we choose the same quadratic piecewise polynomial set P 2 to form ˆ s ⊂ V s that holds vˆ s,h as the finite element solution to the linear elasticity V h b ˆs ⊂V ˆ s, W ˆ f ⊂W ˆ f, W ˆ f ⊂W ˆ f. equation. In addition, V ft,h,0 ⊂ V ft,0 , V h,0 0 h b h,0 0 For any t ∈ [0, T ], we consider a discretization of the mapping X t by means ˆ f → Ωtf . We of piecewise linear Lagrangian finite elements, denoted by X h,t : Ω 1,∞ ˆ f d (Ω )) and X −1 assume that X h,t is smooth and invertible, X h,t ∈ (W h,t ∈ f d −1 1,∞ ˆ f,h ◦ X h,t = (Ωt )) . Then, we have the discrete mesh velocity w f,h = w (W ∂X h,t dv f h −1 ◦ (X h,t ) (x f ), and the discrete ALE time derivative (x f , t) = ∂t ∂v f ∂t (x f , t)

dt

ˆf x

ˆ f,h = w f,h ◦ X h,t = vˆ s,h on Γˆ . + w f,h (x f , t) · ∇v f (x f , t). Clearly, w So, differentiating (5) with time, we can similarly define a discrete ALE mapping ˆ f,h as follows for w ⎧ ˆf , ˆ f,h = 0, in Ω (a) ⎨ Δxˆ f w ˆ f,h = vˆ s , on Γˆ , (b) w ⎩ f ˆ ˆ ˆ f,h = 0, on ∂ Ω \Γ . (c) w 4.1

(10)

The Semi-discrete Scheme of ALE Finite Element Method

Choosing P 2 P 1 mixed element, P 2 element and P 1 element to construct the ˆ s and W ˆ f , respectively, we define the finite element spaces V ft,h × Qft,h , V h h following semi-discrete ALE finite element approximation to FSI problem (1)– (3) based on the weak form (9).

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ˆ ×Qf × ˆs,h , pf,h , w ˆ f,h ) ∈ (H 2 ∩L∞ )(0, T ; Vft,h × V ALE-FEM 1. Find (vf,h , v h t,h f ˆ W ) such that h



ρf

 h ∂vf,h  ,ψ ∂t ˆxf f,h

Ωtf

+

ˆ f,h ◦ X−1 + β(vf,h , vf,h , ψ f,h )Ω f − β(w h,t , vf,h , ψ f,h )Ω f t

1 ˆ f,h ◦ X−1 vf,h ∇ · (w ), ψ f,h h,t 2 Ωtf

t



 + 2μf D(vf,h ), D(ψ f,h ) Ω f t

ˆw ˆ ξˆf,h ) ˆ ˆ f,h , ∇ − (pf,h , ∇ · ψ f,h )Ω f + (∇ · vf,h , qf,h )Ω f + (∇ Ωf t t  

 t ∂ˆ vs,h ˆ ˆ ) ˆs,h dτ , ε(ψ v , ψ s,h )Ωˆs + 2μs ε + ρˆs ( s,h ∂t ˆs 0 Ω

 t 

 ˆ ˆ ) ˆs ˆ· ˆ ·ψ ˆs,h dτ , ∇ v + λs ∇ = ρf (ff , ψ f,h )Ω f + ρˆs (ˆfs , ψ s,h s,h Ω 0

− 2μs



ˆs Ω

ˆ ) ε(ˆ u0s ), ε(ψ s,h



ˆs Ω

t

ˆ ˆ ·u ˆ ·ψ ˆ 0s , ∇ − λs ∇ s,h

f ˆ , qf,h , ξˆ ) ∈ Vf ˆs ∀(ψ f,h , ψ s,h f,h t,h,0 × Vh,0 × Qt,h ×

,

ˆs Ω f ˆ . W h,0

(11) 4.2

The Fully Discrete Scheme of ALE Finite Element Method

Now we develop the fully discrete monolithic ALE finite element approximation for the FSI model (1). Introduce a uniform partition 0 = t0 < t1 < · · · < tN = T with the time-step size Δt = T /N . Set tn = nΔt, ϕn = ϕ(x n , tn ) for n = 0, 1, · · · , N . Define the following backward Euler time differences based on the discrete ALE mapping X h,t :

 n+1 du  dt x ˆf n+1  ∂u ∂t

t n+1 ≈ dX = t u

≈ dt un+1

=

1 Δt 1 Δt



un+1 − un ◦ X n+1,n , (a)  n+1 u (b) − un ,

(12)

where, X m,n = X h,tn ◦ (X h,tm )−1 . By the Taylor’s expansion in a subtle way, we can obtain the first-order convergence with respect to Δt for (12(a)) [5]. Based on the semi-discrete scheme (11), now we can define a fully discrete ALE finite element method (FEM) for (1) as follows.

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f f n+1 ˆs ˆf ˆn+1 ˆ n+1 ALE-FEM 2. Find (vn+1 s,h , pf,h , w f,h ) ∈ Vn+1,h × Vh × Qn+1,h × Wh f,h , v such that for n = 0, 1, · · · , N − 1,

ρf −1 ˆ n+1 − β(w f,h ◦ Xh,t

n+1

   t n+1 dX t vf,h , ψ f,h

, vn+1 f,h , ψ f,h )Ω f

  + 2μf D(vn+1 f,h ), D(ψ f,h )

f

Ωn+1

n+1

f

Ωn+1

n+1 + β(vn+1 f,h , vf,h , ψ f,h )Ω f

n+1

 1  n+1 −1 ˆ n+1 vf,h ∇ · (w + f,h ◦ Xh,tn+1 ), ψ f,h Ω f 2 n+1

− (pn+1 f,h , ∇ · ψ f,h )Ω f

ˆ ˆn+1 + ρˆs (dt v ˆs s,h , ψ s,h )Ω

n+1



+ (∇ · vn+1 f,h , qf,h )Ω f

n+1

    ˆ ) + μs Δt ε ˆ vn+1 , ε(ψ s,h s,h

ˆs Ω

(13)

λs Δt  ˆ n+1 ˆ ˆ  n+1 ˆw ˆˆ ˆ n+1 ∇·ˆ vs,h , ∇ · ψ s,h + + (∇ , ψ f,h )Ω f ˆ f = ρf (ff f,h , ∇ξ f,h )Ω ˆs Ω t 2     Δt λ n+1 s n n ˆ ) ˆ s − μs Δt ε(ˆ ˆ ) ˆ ˆ ·ˆ ˆ ·ψ ∇ vs,h , ∇ + ρˆs (ˆfs , ψ vs,h ), ε(ψ − s,h Ω s,h s,h ˆ s ˆs Ω Ω 2     n n ˆ ˆ ˆ ·u ˆ ·ψ ˆs , ∇ − 2μs ε(ˆ , us ), ε(ψ s,h ) s − λs ∇ s,h s ˆ Ω

ˆ Ω

f ˆ , qf,h , ξˆ ) ∈ Vf ˆs ˆf ∀(ψ f,h , ψ s,h f,h n+1,h,0 × Vh,0 × Qn+1,h × Wh,0 .

4.3

Nonlinear Iteration Algorithm

ˆ s,h , Assume that the mesh at the time step tn is denoted by Tnh = Tnf,h ∪ T ˆ ˆ where Ts,h is the Lagrangian structural mesh in Ωs which is always fixed, Tf,h is the ALE fluid mesh in Ωtf which needs to be updated all the time through the ˆ f,h + X h,t , where, X h,t is only subject to discrete ALE mapping, i.e., Tnf,h = T n n n n−1 Δt ˆ ˆ s,h = 2 (ˆ v ns,h + vˆ n−1 ) + u on Γˆ for the sake of the structure displacement u s,h s,h n n−1 Δt ˆ f,h + w ˆ f,h ) + X h,tn−1 . Due to a conforming mesh across Γt , or, X h,tn = 2 (w a small deformation displacement of the structure, such specific ALE mapping always guarantees a shape-regular fluid mesh in Ωtf . Suppose all the necessary solution variables at the time step tn are given as:

ˆ nf,h , X h,tn . χn := v nf,h , vˆ ns,h , pnf,h , w In the following, we define an implicit iterative scheme for the FSI simulation at the current (n + 1)-th time step. Algorithm 1. Nonlinear iteration at the (n+1)-th time step for FSI. n+1,0 f 1. Let j = 1, χn+1,0 = χn , and Tf,h = Tnf,h which partitions Ωn+1,0 = Ωnf . n+1,j 2. Solve the following linearized mixed finite element equation for χ with χn+1,j−1 given at the j − 1 iteration: n+1,j n+1,j n+1,j n+1,j ˆ s × Qf ˆf ˆs,h ˆ f,h Find (vf,h ,v , pf,h , w ) ∈ Vfn+1,j,h × V h n+1,j,h × Wh such that

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t n+1,j , ψ f,h dX t vf,h

 f

Ωn+1,j−1

ˆ n+1,j−1 − β(w f,h

+ β(vn+1,j−1 , vn+1,j , ψ f,h )Ω f f,h f,h

n+1,j−1

◦

n+1,j X−1 , ψ f,h )Ω f h,tn+1 ,j−1 , vf,h

n+1,j−1

 1 −1 n+1,j ˆ n+1,j−1 ∇ · (w + ◦ X )v , ψ h,t ,j−1 f,h f,h f,h f n+1 2 Ωn+1,j−1   ), D(ψ f,h ) f − (pn+1,j , ∇ · ψ f,h )Ω f + 2μf D(vn+1,j f,h f,h Ωn+1,j−1

vn+1,j , qf,h )Ω f f,h

+ (∇ ·     ˆ ) ˆn+1,j , ε(ψ + μs Δt ε v s,h s,h



n+1,j−1

ˆ )ˆ ˆn+1,j ρˆs (dt v ,ψ s,h Ωs s,h

+  λs Δt ˆ n+1,j ˆ ˆ  ˆs,h , ∇ · ψ s,h ∇·v + ˆs ˆs 2 Ω Ω n+1,j n+1,j ˆ ˆ = ρf (f ,ψ ) f + ρˆs (f , ψ ) ˆs n+1,j−1

(14)

ˆw ˆ ξˆ ) ˆ ˆ n+1,j + (∇ ,∇ s f,h Ωf f,h Ω s,h Ω f,h f n+1,j−1     Δt λ s ˆ ˆ ·v ˆ ˆ ˆn ∇ − μs Δt ε(ˆ vn − s,h ), ε(ψ s,h ) s,h , ∇ · ψ s,h ˆs ˆs 2 Ω    Ω  n n ˆ ˆ ˆ ˆ ˆ s , ∇ · ψ s,h us ), ε(ψ s,h ) − 2μs ε(ˆ − λs ∇ · u , ˆs Ω

ˆs Ω

f ˆ , qf,h , ξˆ ) ∈ Vf ˆs ˆf ∀(ψ f,h , ψ s,h f,h n+1,j,h,0 × Vh,0 × Qn+1,j,h × Wh,0 .

ˆ n+1,j + w ˆ nf,h ) + Xh,tn , and update the ALE fluid mesh 3. Let Xh,tn+1 ,j = Δt 2 (wf,h n+1,j ˆ f,h + Xh,t ,j which partitions Ω f Tf,h =T n+1 n+1,j . 4. For a given tolerance ε, determine whether or not the following stopping criteria hold for the relatively iterative errors: vn+1,j − vn+1,j−1 (H 1 (Ω f ))d f,h f,h vn+1,j−1 (H 1 (Ω f ))d f,h

t

t

+

− pn+1,j−1 L2 (Ω f ) pn+1,j f,h f,h pn+1,j−1 L2 (Ω f ) f,h

t

+

t

+

ˆn+1,j−1 ˆ vn+1,j −v (H 1 (Ωˆ s ))d s,h s,h ˆ vn+1,j−1 (H 1 (Ωˆ s ))d s,h

ˆ n+1,j ˆ n+1,j−1 w −w (H 1 (Ωˆ f ))d f,h f,h ˆ n+1,j−1 w (H 1 (Ωˆ f ))d f,h

(15) ≤ ε.

If yes, go to Step 5; otherwise, let j ← j + 1 and go back to Step 2 to continue the nonlinear iteration. n+1,j which 5. Let χn+1 = χn+1,j , and update the ALE fluid mesh Tn+1 f,h = Tf,h f partitions Ωn+1 .

5

Numerical Experiments

A hypothetical CVD patient with a growing aneurysm is used as an illustration example. The patient’s CT imaging data (See Fig. 3) are collected to construct a geometrical model (See the right of Fig. 2). The cardiovascular parameters of this patient are estimated based on the patient’s routine test results such as blood work, blood pressure, diabetes. Usually, if an aneurysm becomes about 2 in. in

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Fig. 3. The computational domain (Right) is generated from a patient’s CT data (Left).

Fig. 4. Left: the overall computational mesh. Right: the cross section in flow direction from the left to the right, where a stent graft separates the blood fluid to two fields.

diameter and continues to grow, or begins to cause symptoms, the patient may need surgery to repair the artery before the aneurysm bursts. An endovascular aortic repair (EVAR) surgery involves replacing the weakened section of the vessel with an artificial tube, called a vascular stent graft. A computational mesh for an aneurysm CVD patient implanted with a stent graft is shown in Fig. 4, where the vascular stent graft, that is represented by an artificial blood vessel, expands inside the artery and eventually sticks to the inlet end of the aortic wall. Hence, a large blood fluid cavity is formed between the aortic aneurysm and the stent graft. Note that the blood fluid is thus separated by the stent graft into two parts: one inside the stent graft lumen driven by the incoming blood flow, while the other part is squeezed into the aneurismal cavity, waiting for being expelled due to the residual elasticity energy that remains in the diseased aorta. To numerically model this multi-domain cardiovascular FSI problem in an accurate and effective fashion, an important technical issue needs to be clarified as follows. As shown on the right of Fig. 4, we extend the outlet end of the aortic wall further in order to connect two blood fluid fields together through the gap between the aortic wall and the stent graft. Thus, two blood fluid fields which respectively flow through the stent graft lumen and the aneurysm cavity, are eventually contained in one single connected fluid body and share the same inlet and outlet. This is crucial for the success of our FSI simulation, since the blood fluid that is contained inside the aneurysm cavity no longer owns an inlet thus no more incoming flow after the stent graft expands and squeezes onto the

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Time=0.001s

Time=0.05s

Fig. 5. Velocity magnitude, vector and streamline fields for the 4-field FSI between blood fluids, aortic wall and stent graft in the early stage.

inlet end of the aortic wall, therefore the incompressible fluid equations that is adopted to model the blood fluid flow will be violated in this fluid part if it is not connected with the main blood flow stream through the stent graft lumen. Another important technical issue is that, to make ALE method successfully work for this multi-domain FSI simulation all the time, we need to avoid two structure parts, the aortic wall and the stent graft, to come into contact with each other. Otherwise, there is no space left for the blood fluid to flow through the contacting places of these two structure parts, which will induce a failure of ALE method since the ALE fluid mesh in these contacting area will lose many fluid mesh cells due to the physical contact of two structure parts, recalling that the preservation of mesh connectivity all the time is a unique feature and a warrant for success of ALE method. To deal with such a possible situation during the multi-domain FSI simulation, we set up a threshold for the distance in the normal direction between the aortic wall and the stent graft, namely, we do not let them physically get in touch with each other but just treat they “contact” at a pseudo zero distance in the normal direction once the threshold is reached. In addition, they also do not make any relative shift in the tangential direction since they are merged together on the inlet end of the blood vessel, as shown in Fig. 4. By that way, we do not have to handle the complicated contacting problem occurring between two structures at this time. Simultaneously, the ALE fluid mesh is still

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73

feasible within the threshold gap, thus our developed ALE mixed finite element method still works well for the multi-domain cardiovascular FSI problem. Such idea to avoid the contact between multiple structures is also similarly adopted in [3] where a thin layer of fluid is introduced around structures, such that there is no real “contact” between both phases. In the realistic numerical simulation, the forces are still transferred via the remaining small layer of fluid. Our approach is stable under refinement of the temporal and spatial discretization. We will carry out a more comprehensive study in the future for this approach by comparing with experiments and numerical benchmarks, in depth. In this work, for the first time we successfully apply our monolithic ALE FEM [7,10] and FSI codes to this multi-domain (artery, stent graft, blood fluid in the vessel lumen and the aneurysmal cavity) FSI problem with three moving interfaces in a much subtle fashion. Given a time-dependent incoming velocity profile on the inlet to simulate the pulsatile blood flow, some numerical results are gained for the velocity field of this multi-domain FSI problem in its early stage in forms of magnitude contour, vector and streamline, as shown in Fig. 5. We can clearly see that the larger velocity magnitudes occur in the stent graft lumen rather than in the aneurysm cavity, which is because the incoming blood fluid continues to flow through the inlet to the blood vessel lumen while the contained blood fluid inside the aneurysm cavity is less active due to the lack of residual elasticity inside the diseased aortic wall, but after a realistically long term run, the aortic wall of the aneurysm, driven by the residual elasticity, shall be able to be shrunk back to some extent, and then squeeze the remaining blood out of the aneurysm cavity through the small gap, as illustrated in Fig. 5 after 0.05 s in the computation.

References 1. Barker, A., Cai, X.: Scalable parallel methods for monolithic coupling in fluidstructure interaction with application to blood flow modeling. J. Comput. Phys. 229(3), 642–659 (2010) 2. Causin, P., Gerbeau, J., Nobile, F.: Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Eng. 194(42), 4506–4527 (2005) 3. Frei, S., Richter, T., Wick, T.: Eulerian techniques for fluid-structure interactions: Part II – applications. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds.) Numerical Mathematics and Advanced Applications - ENUMATH 2013. LNCSE, vol. 103, pp. 755–762. Springer, Cham (2015). https://doi.org/10.1007/ 978-3-319-10705-9 75 4. Idelsohn, S., Del, P., Rossi, R., O˜ nate, E.: Fluid-structure interaction problems with strong added-mass effect. Int. J. Numer. Methods Eng. 80, 1261–1294 (2009) 5. Lan, R., Sun, P.: A monolithic arbitrary Lagrangian-Eulerian finite element analysis for a Stokes/parabolic moving interface problem. J. Sci. Comput. (2020, in press) 6. Lan, R., Sun, P., Mu, M.: Mixed finite element analysis for an elliptic/mixedelliptic coupling interface problem with jump coefficients. Procedia Comput. Sci. 108, 1913–1922 (2017)

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7. Leng, W., Zhang, C., Sun, P., Gao, B., Xu, J.: Numerical simulation of an immersed rotating structure in fluid for hemodynamic applications. J. Comput. Sci. 30, 79–89 (2019) 8. Mart´ın, J.S., Smaranda, L., Takahashi, T.: Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time. J. Comput. Appl. Math. 230, 521–545 (2009) 9. Nobile, F., Formaggia, L.: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East West J. Numer. Math. 7, 105–132 (2010) 10. Yang, K., Sun, P., Wang, L., Xu, J., Zhang, L.: Modeling and simulation for fluidrotating structure interaction. Comput. Methods Appl. Mech. Eng. 311, 788–814 (2016)

Morphing Numerical Simulation of Incompressible Flows Using Seamless Immersed Boundary Method Kyohei Tajiri(B) , Mitsuru Tanaka , Masashi Yamakawa , and Hidetoshi Nishida Department of Mechanophysics, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, Japan [email protected]

Abstract. In this paper, we proposed the morphing simulation method on the Cartesian grid in order to realize flow simulations for shape optimization with lower cost and versatility. In conventional morphing simulations, a simulation is performed while deforming a model shape and the computational grid using the boundary fitting grid. However, it is necessary to deform the computational grid each time, and it is difficult to apply to a model with complicated shape. The present method does not require grid regeneration or deformation. In order to apply the present method to models with various shapes on the Cartesian grid, the seamless immersed boundary method (SIBM) is used. Normally, when the SIBM is applied to a deformed object, the velocity condition on the boundary is imposed by the moving velocity of the boundary. In the present method, the velocity condition is imposed by zero velocity even if the object is deformed because the purpose of the present morphing simulation is to obtain simulation results for a stationary object. In order to verify the present method, two-dimensional simulations for the flow around an object were performed. In order to obtain drag coefficients of multiple models, the object was deformed in turn from the initial model to each model in the present morphing simulation. By using the present method, the drag coefficients for some models could be obtained by one simulation. It is concluded that the flow simulation for shape optimization can be performed very easily by using the present morphing simulation method. Keywords: Computational fluid dynamics · Morphing simulation method · Immersed boundary method · Incompressible flow · Shape optimization

1

Introduction

There are many products around us that are closely related to the flow phenomenon. Improvements in the performance of these products are always Supported by organization x. c Springer Nature Switzerland AG 2020  V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 75–87, 2020. https://doi.org/10.1007/978-3-030-50436-6_6

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expected. On the other hand, reducing the time and cost required to develop these products is also an important issue. Shape optimization through flow simulations at the stage of design is one of these efforts. By determining the optimum shape from many candidate product shapes (candidate models) at the early stage of product development, the effort of the redesign is reduced. As a result, development costs are reduced. Conventionally, flow simulations have been performed for each of these many candidate models. However, in recent years, the cost required for flow simulations has increased because the number of candidate models has increased in order to develop higher performance products. In order to reduce the number of these simulations, simulations are performed while deforming the model shape and the computational grid in shape optimization using flow simulations [1]. In this method, the number of flow simulations for shape optimization can be reduced, and the optimum shape can be determined in the flow simulation because results for many models can be obtained in one simulation. However, it is necessary to deform the computational grid each time, and it is difficult to apply to a model with complicated shape. In addition, the simulation on the boundary fitted grid can be expected to have high computational accuracy, however, the computational efficiency is inferior to the simulation on the Cartesian grid. In this paper, in order to realize flow simulations for shape optimization with lower cost and versatility, a method is proposed to perform simulation while deforming a model on the Cartesian grid that does not require grid regeneration or deformation. We call this method the morphing simulation method. In order to apply the present method to models with various shapes on the Cartesian grid, the seamless immersed boundary method (SIBM) [2], which is an improved method of the immersed boundary method (IBM) [3] is used. In the IBM, additional force terms are added to the momentum equations to satisfy the velocity conditions on the virtual boundary points where the computational grid and the boundary of the object intersect. In order to apply the IBM to an object with arbitrary shape, it is only necessary to know the position of the virtual boundary on the grid. Therefore, the IBM can be easily applied to an object with a complicated shape. As for the estimation of the additional forcing term, there are mainly two methods, that is, the feedback [4,5] and direct [6] forcing term estimations. Generally, the direct forcing term estimation is adopted because of the simplicity of the algorithm. However, the conventional IBM with the direct forcing term estimation generates the unphysical pressure oscillations near the virtual boundary because of the pressure jump between inside and outside of the virtual boundary. The SIBM was proposed in order to remove these unphysical pressure oscillations. In the past study, the SIBM was applied not only to stationary objects but also to moving or scaling objects [7,8]. Therefore, it is possible to use the SIBM in the morphing simulation method proposed in this paper. Normally, when the SIBM is applied to a moving or scaling object, the velocity condition in the estimation of the additional forcing term is determined by the moving velocity of the object. In the present method, the additional forcing term is estimated under the condition that the velocity is

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zero even if the object is deformed because the purpose of the present morphing simulation is to obtain simulation results for a stationary object. In this paper, the morphing simulation by the present method is performed for some models and compared with the conventional static SIBM where simulation is performed for each model and the effectiveness of the present method is discussed.

2 2.1

Morphing Numerical Simulation Using Seamless Immersed Boundary Method Governing Equations

The governing equations are the continuity equation and the incompressible Navier-Stokes equations. Moreover, the forcing term is added to the NavierStokes equation for the SIBM. The non-dimensional continuity equation and incompressible Navier-Stokes equations are written as, ∂ui = 0, ∂xi ∂ui ∂p = Fi − + Gi , ∂t ∂xi ∂ui 1 ∂ 2 ui + , Fi = −uj ∂xj Re ∂xj ∂xj

(1) (2) (3)

where, Re denotes the Reynolds number defined by Re = L0 U0 /ν0 . U0 , L0 and ν0 are the reference velocity, the reference length and the kinematic viscosity, respectively. ui = (u, v) and p are the velocity components and the pressure. Gi in Eq. 2 denotes the additional forcing term for the SIBM. Fi denotes the convective and diffusion terms. 2.2

Numerical Method

The incompressible Navier-Stokes equations (Eq. 2) are solved by the second order finite difference method on the collocated grid arrangement. The convective terms are discretized by the fully conservative finite difference method [9] and is written, for example, as,      ∂u  ∂u  1 ∂u  v  v  = +v  ∂y I,J 2 ∂y I,J+ 1 ∂y I,J− 1 2 2   uI,J+1 − uI,J uI,J − uI,J−1 1 + vI,J− 12 = vI,J+ 12 , (4) 2 Δ Δ where, v is the y component of velocity I, J are the grid index and Δ is grid spacing. The velocity at the midpoint (for example, J + 12 ) of the grid is calculated by linear interpolation. The diffusive and pressure terms are discretized

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by the conventional second order centered finite difference method. For the time integration, the fractional step approach [12] based on the forward Euler method is applied. For the incompressible Navier-Stokes equations in the SIBM, the fractional step approach can be written by u∗i = uni + ΔtFin ,   ∂pn ∗ n un+1 = u + Δt − + G i i , i ∂xi

(5) (6)

where u∗i denotes the fractional step velocity and Δt is the time increment. The resulting pressure equation is solved by the successive over-relaxation (SOR) method. 2.3

Seamless Immersed Boundary Method

In order to adopt the SIBM, the additional forcing term in the momentum equations, Gi , should be estimated. In the SIBM, the additional forcing term is estimated by the direct forcing term estimation [2]. The direct forcing term estimation is shown in Fig. 1. We explain in two-dimensions but the extension to three-dimensions is straightforward. For the forward Euler time integration, the forcing term can be determined by Gni = −Fin +

¯ n+1 − un U ∂pn i , + i ∂xi Δt

(7)

¯ n+1 denotes the velocity linearly interpolated from the velocity on the where U i near grid point and the velocity (uvb ) determined by the velocity condition on the virtual boundary. Namely, the forcing term is specified as the velocity com¯ n+1 . In the IBM, the grid =U ponents at next time step satisfy the relation, un+1 i i points added forcing term are restricted near the virtual boundary only (show Fig. 1(a)). In this approach, the non-negligible velocity appears inside the virtual boundary. Also, the pressure distributions near the virtual boundary show the unphysical oscillations because of the pressure jump. In the SIBM, the forcing term is added not only on the grid points near the virtual boundary but also in the region inside the virtual boundary shown in Fig. 1(b) in order to remove the unphysical oscillations near the virtual boundary. In the region inside the virtual ¯b , ¯ n+1 = U boundary, the forcing term is determined by satisfying the relation, U i ¯b is the velocity which satisfies the velocity condition at the grid point. where U When applying the SIBM to a stationary object, the velocity condition on and inside the virtual boundary is zero velocity. As mentioned above, an algorithm of the SIBM is very simple and can easily be extended to three dimensions. Therefore, it is applied to flow around moving or scaling objects [7,8]. When applying the SIBM to a moving or scaling object, the velocity condition on and inside the virtual boundary are obtained by the moving velocity of the object at that point. Moreover, there are also examples of application to turbulence flow [10,11].

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(a) Conventional IBM.

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(b) Seamless IBM.

Fig. 1. Grid points added forcing terms.

2.4

Morphing Numerical Simulation

The morphing of a model on the Cartesian grid is shown in Fig. 2. In the present method, the object is deformed in turn from the first model to the model that requires simulation results. In the SIBM, only the position of the virtual boundary of the object on the fixed grid is updated even if the object deforms. In the SIBM, the virtual boundary of the object with arbitrary shape is represented by boundary nodes in the two-dimensional simulation as shown in Fig. 2. The boundary between these boundary nodes is approximated by straight lines. By determining the intersection between the boundary and the grid that is the virtual boundary point, SIBM can be applied to an object having an arbitrary shape. In the three-dimensional simulation, the virtual boundary of the object with arbitrary shape is represented by triangular polygons and boundary nodes [7]. In the present morphing simulation, the object is deformed from one model to another model by moving these boundary nodes every time step. Once the position of the boundary nodes at each time step is determined, it is easy to apply the SIBM to the model. In the present method, the boundary nodes for the model before deformation is linearly moved to the position of the boundary nodes for the next model. Therefore, the algorithm in the present method is extremely easy. Normally, in the SIBM for the moving or deforming object, the additional forcing term is determined by the moving velocity of the object or boundary. In the present method, the additional forcing term is estimated under the condition that the velocity is zero even if the object is deformed. It is because the purpose of the present morphing simulation is to obtain simulation results for a stationary each model.

3

Application to Two-Dimensional Model

In this paper, in order to verify the present method, two-dimensional simulations for the flow around an object are performed. In order to obtain drag coefficients

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Fig. 2. Morphing of model on the Cartesian grid.

of multiple models, the object is deformed in turn from the initial model to each model in the present morphing simulation. In another case, the morphing has downtime in each model. In this paper, two-dimensional flows around square, circular and elliptic cylinder whose drag coefficients can be compared with the reference results are considered. The computational domain is shown in Fig. 3. In this simulation, a model is set as shown in Fig. 4 and the model is deformed in order from 1 to 5. Each process indicates morphing processes. For example, the model is deformed from 1 to 2 in the process 1, and the model is deformed from 3 to 2 in the process 3. In the present morphing simulation, firstly, the static SIBM simulation is performed for the model 1 and then the processes 1 to 4 are performed in the morphing simulation. This deformation may be larger than the deformation in general shape optimization. In each model, the length of the side of the square cylinder, the diameter of the circular cylinder, and the length of the major axis of the elliptic cylinder are the reference length L = 1. The length of the minor axis of the elliptic cylinder is 0.5. As a result, the processes 1 and 2 are scaling down the model and the processes 3 and 4 are scaling up the model. In addition, the processes 1 and 4 are two-dimensional deformations and the processes 2 and 3 are one-dimensional deformations. As for the computational conditions, the impulsive start determined by the uniform flow (u = 1, v = 0) is adopted. On the inflow boundary, the velocity is fixed by the uniform flow and the pressure is imposed by the Neumann condition obtained by the normal momentum equation. On the outflow and side boundaries (right, top and bottom boundaries), the velocity is extrapolated from the inner points and the pressure is obtained by the Sommerfeld radiation condition [13]. On the virtual boundary and inside the boundary, the velocity condition is the velocity is zero. The Reynolds number is set as Re = 40. The flow around each model is steady flow under this Reynolds number. In order to reduce the number of grid points, the hierarchical Cartesian grid with level 4 is introduced. The grid resolution near the model is Δ = 1/80. The number of boundary nodes in each model is 400 and the distance between the nodes in the case of the square cylinder

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(models 1, 5) which is the largest model is smaller than the grid spacing. In addition, the boundary node also exists at each vertex of the square cylinder as shown in Fig. 2.

Fig. 3. Computational domain.

Fig. 4. Configuration of model morphing.

Firstly, in order to obtain the reference results for each model, the conventional static simulations by SIBM without morphing are performed under the above conditions. In Table 1, the drag coefficient is shown with the reference results [14–16]. In this paper, the drag coefficient is estimated by  ∂ui i −2 O (Gx − uj ∂x − ∂u ∂t )ds j , (8) CD = ρ0 U02 L where O denotes the region to which the forcing term is added in the SIBM. ρ0 and U0 denote the reference density and velocity of the flow. The drag coefficient by the conventional static SIBM is in good agreement with the reference result in each model. Therefore, these drag coefficients are used as reference results for verifying the present morphing simulation method.

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K. Tajiri et al. Table 1. Drag coefficient of each model in static SIBM. Square cylinder Circular cylinder Elliptic cylinder Static SIBM (Presents) 1.728

1.568

1.631

Sen et al. [14]

1.787

–

–

Dennis et al. [15]

–

1.522

–

Sen et al. [16]

–

–

1.567

In this paper, the morphing simulation is performed under some deformation speeds. The deformation speed is set by the non-dimensional time for each process in Fig. 4. In the present simulation, there is no difference in deformation time between processes. Then, simulations are performed in the case of nondimensional time is 1, 2, 4, 8 and 16 for the processes (Case 1 to 5). In other words, the deformation speed is slower in Case 5 than in Case 1. In addition, in order to investigate the possibility that the deformation time can be set shorter, a simulation is performed in which downtime of deformation is set after the deformation to each model. In this simulation, the deformation time is the same 1 as Case 1, and the downtime of deformation is 1 (Case 6). That is, the total time for each process in Case 6 is shorter than Case 3 to 5. The above conditions are summarized in Table 2. Table 2. Non-dimensional time for a process in each condition. Total time for a process Deformation time Downtime Case 1

1

–

Case 2

2

–

Case 3

4

–

Case 4

8

–

Case 5 16

–

Case 6

1

1

Figures 5, 6, 7 and 8 show the pressure contours of each model. Note that the pressure contours of the models 2 and 4 by the static SIBM is same. In all cases, the pressure contours obtained by the present method are similar to those obtained by the static SIBM. In particular, those in Cases 5 and 6 are in good agreement with those in the static SIBM. Figure 9 shows time histories of the comparison of the drag coefficients at each deformation speed. The horizontal axis shows the non-dimensional time converted into the model number. For example, the model number is 2 when the horizontal axis is 2 and the model is being deformed from 2 to 3 (process 2)

Morphing Numerical Simulation Using SIBM

(Static SIBM)

(Case 1)

(Case 5)

(Case 6)

Fig. 5. Pressure contours of model 2.

(Static SIBM)

(Case 1)

(Case 5)

(Case 6)

Fig. 6. Pressure contours of model 3.

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(Static SIBM)

(Case 1)

(Case 5)

(Case 6)

Fig. 7. Pressure contours of model 4.

when the horizontal axis is between 2 and 3. Table 3 shows the comparison of the drag coefficients of each model at each deformation speed. In Case 1–5, the drag coefficients of each model are closer to the reference values as the deformation speed is slower. When the model is deformed, oscillations of the drag coefficient due to the virtual boundary moving across the grid are observed. These oscillations are remarkable in the case of two-dimensional deformation. In particular, the results for each model in Case 4 and 5 are close to the reference results. Therefore, it was shown that the results equivalent to the results by the conventional static SIBM can be obtained by the present morphing method. In Case 6, the drag coefficients are shown different from the reference value just like Case 1 immediately after deformation, however, the values become to the same level as in Cases 4 and 5 in the deformation downtime. Therefore, it was shown that the present morphing simulation with downtime can set the deformation speed faster than the present morphing simulation without downtime. Table 4 shows the comparison of the rate of computational time of each morphing process at each deformation speed. Each computational time is based on the computational time of the process 1 in Case 1. It can be observed that the longer the non-dimensional time for deformation is, the longer the computational time is. The computational time of Case 6 is almost the same as Case 2 where the non-dimensional time of each process is the same. According to the above results, it was shown that the present method can be accelerated by adding in the morphing process the downtime.

Morphing Numerical Simulation Using SIBM

(Static SIBM)

(Case 1)

(Case 5)

(Case 6)

Fig. 8. Pressure contours of model 5.

(Case 1 to 5)

(Case 4 to 6)

Fig. 9. Comparison of drag coefficients at each deformation speed.

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K. Tajiri et al. Table 3. Drag coefficient of each model. Model 2 Model 3 Model 4 Model 5 Case 1

1.175

1.288

1.802

2.327

Case 2

1.342

1.461

1.677

2.048

Case 3

1.428

1.567

1.615

1.908

Case 4

1.487

1.617

1.587

1.840

Case 5

1.525

1.638

1.577

1.801

Case 6

1.500

1.607

1.527

1.760

Static SIBM 1.568

1.631

1.568

1.728

Table 4. Rate of computational time of each morphing process. Process 1 Process 2 Process 3 Process 4

4

Case 1 1.00

0.67

1.04

1.88

Case 2 1.46

1.19

1.54

2.19

Case 3 2.02

2.03

2.22

2.55

Case 4 2.77

3.05

3.13

3.20

Case 5 3.68

4.50

4.49

4.03

Case 6 1.25

1.01

1.46

2.10

Conclusions

In this paper, we proposed the morphing simulation method on the Cartesian grid in order to realize flow simulations for shape optimization with lower cost and versatility. By using SIBM that is the Cartesian grid approach, the present method could be applied very easily to an object with arbitrary shape. In order to verify the present method, the two-dimensional simulations for the flow around an object were performed. In order to obtain drag coefficients of multiple models, the object was deformed in turn from the initial model to each model in the present morphing simulation. By using the present method, the drag coefficients for some models could be obtained by one simulation. These drag coefficients became closer to the reference values by decreasing the deformation speed of the model. Furthermore, by setting the downtime after the deformation, drag coefficients close to the reference values were obtained even when the deformation speed was high. Therefore, it can be concluded that the flow simulation for shape optimization can be performed very easily and the number of times of flow simulation for many models can be significantly reduced by using the present morphing simulation method.

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References 1. Hino, T.: Shape optimization of practical ship hull forms using Navier-Stokes analysis. In: Proceedings of the 7th International Conference on Numerical Ship Hydro (1999) 2. Nishida, H., Sasao, K.: Incompressible flow simulations using virtual boundary method with new direct forcing terms estimation. In: Deconinck, H., Dick, E. (eds.) Computational Fluid Dynamics 2006, pp. 371–376. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-92779-2 57 3. Peskin, C.S., McQueen, D.M.: A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. 81(2), 372–405 (1989) 4. Goldstein, D., Handler, R., Sirovich, L.: Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 105(2), 354–366 (1993) 5. Saiki, E.M., Biringen, S.: Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method. J. Comput. Phys. 123(2), 450–465 (1996) 6. Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yosof, J.: Combined immersedboundary finite-difference methods for three-dimensional complex simulations. J. Comput. Phys. 161(1), 35–60 (2000) 7. Nishida, H., Tajiri, K.: Numerical simulation of incompressible flows around a fish model at low reynolds number using seamless virtual boundary method. J. Fluid Sci. Technol. 4(3), 500–511 (2009) 8. Nishida, H., Tajiri, K., Tanaka, M.: Seamless immersed boundary method for flow around a scaled object. In: Proceedings 12th Asian Computational Fluid Dynamics Conference, pp. 1–9 (2018) 9. Morinishi, Y., Lund, T.S., Vasilyev, O.V., Moin, P.: Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143(1), 90–124 (1998) 10. Tajiri, K., Nishida, H., Tanaka, M.: Large eddy simulation of turbulent flow using seamless immersed boundary method. In: Proceedings of the 8th International Conference on Computational Fluid Dynamics, ICCFD8-197, pp. 1–13 (2014) 11. Tajiri, K., Nishida, H., Tanaka, M.: Property of seamless immersed boundary method for large eddy simulation of incompressible turbulent flows. J. Fluid Sci. Technol. 9(2), 1–8 (2014) 12. Rhie, C.M., Chow, W.L.: Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21(11), 1525–1532 (1983) 13. Kawakami, K., Nishida, H., Satofuka, N.: An open boundary condition for the numerical analysis of unsteady incompressible flow using the vorticitystreamfunction formulation. Trans. Jpn. Soc. Mech. Eng. Ser. B 60(574), 1891– 1896 (1994). (Japanese) 14. Sen, S., Mittal, S., Biswas, G.: Flow past a square cylinder at low Reynolds numbers. Int. J. Numer. Methods Fluids 67(9), 1160–1174 (2011) 15. Dennis, S.C.R., Chang, G.Z.: Numerical solutions for steady flow past a circular cylinder at Reynolds num-bers up to 100. J. Fluid Mech. 42(3), 471–489 (1970) 16. Sen, S., Mittal, S., Biswas, G.: Steady separated flow past elliptic cylinders using a stabilized finite-element method. Comput. Model. Eng. Sci. 86(1), 1–26 (2012)

deal.II Implementation of a Two-Field Finite Element Solver for Poroelasticity Zhuoran Wang and Jiangguo Liu(B) Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA {wangz,liu}@math.colostate.edu

Abstract. This paper presents a finite element solver for poroelasticity in the 2-field approach and its implementation on the deal.II platform. Numerical experiments on benchmarks are presented to demonstrate the accuracy and efficiency of this new solver. Keywords: Darcy flow · deal.II · Finite element methods · Hexahedral meshes · Poroelasticity · Quadrilateral meshes · Weak Galerkin

1

Introduction

Poroelasticity is an important problem in science and engineering. The Biot’s model for linear poroelasticity has been well accepted and commonly used. It couples solid displacement u and fluid pressure p through the following partial differential equations (PDEs)  −∇ · (2με(u) + λ(∇ · u)I) + α∇p = f , (1) ∂t (c0 p + α∇ · u) + ∇ · (−K∇p) = s,   where ε(u) = 12 ∇u + (∇u)T is the strain tensor with λ > 0, μ > 0 being the Lam´e constants, f a given body force, K a conductivity tensor, s a known fluid source, α (usually close to 1) the Biot-Williams constant, and c0 ≥ 0 the constrained storage capacity. Appropriate boundary and initial conditions are posed to close the system. Finite element methods (FEMs) are common tools for solving the Biot’s model. Depending on the unknown quantities to be solved, poroelasticity solvers are usually grouped into 3 types: – 2-field : Solid displacement and fluid pressure are to be solved; – 3-field : Solid displacement, fluid pressure and velocity are to be solved; – 4-field : Solid stress & displacement, fluid pressure & velocity are to be solved. Liu and Wang were partially supported by US National Science Foundation grant DMS-1819252. We thank Dr. Wolfgang Bangerth for the computing resources. c Springer Nature Switzerland AG 2020  V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 88–101, 2020. https://doi.org/10.1007/978-3-030-50436-6_7

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A major issue in numerical solvers for poroelasticity is the poroelasticity locking, which usually appears as nonphysical pressure oscillations. This happens when the porous media are low-permeable or low-compressible [12,28,36]. Early on, the continuous Galerkin (CG) FEMs were applied respectively to solve for displacement and pressure. But it was soon recognized that such solvers were subject to poroelasticity locking and the 2-field approach was nearly abandoned. The mixed finite element methods can be used to solve for pressure and velocity simultaneously and meanwhile coupled with a FEM for linear elasticity that is free of Poisson-locking. Therefore, the 3-field approach has been the main stream [5,25–27,33,34]. The 4-field approach is certainly worth of investigation, but it just involves too many unknowns (degrees of freedom) [35]. The weak Galerkin (WG) finite element methods [31] have emerged as a new class of numerical methods with nice features that can be applied to a wide variety of problems including Darcy flow and linear elasticity [14,18,24,30]. Certainly, WG solvers can be developed for linear poroelasticity [17], they are free of poroelasticity locking but may involve a lot of degrees of freedom. Recently, our efforts have been devoted to reviving the 2-field approach for development of efficient and robust finite element solvers for poroelasticity [13]. This may involve incorporation of WG FEMs with WG FEMs or classical FEMs. In this paper, we continue such efforts to develop a poroelasticity solver that couples the WG finite elements for Darcy flow and the classical Lagrangian elements with reduced integration for linear elasticity. Moreover, we provide an accessible efficient implementation of this new solver on deal.II, a popular finite element package [3].

2

Discretization of Linear Elasticity by Lagrangian Elements with Reduced Integration

This section discusses discretization of linear elasticity by Lagrangian Qd1 finite elements (d = 2, 3) with reduced integration that is needed for our new FE solver for poroelasticity. For convenience of presentation, we consider the linear elasticity in its usual form  −∇ · σ = f (x), x ∈ Ω, (2) u|Γ D = uD , (σn)|Γ N = tN , where Ω is a 2d- or 3d-bounded domain occupied by a homogeneous and isotropic elastic body, f a body force, uD , tN respectively Dirichlet and Neumann data, n the outward unit normal vector on the domain boundary that has a nonD N overlapping decomposition ∂Ω  = Γ ∪Γ  . As mentioned in Sect. 1, u is the 1 T solid displacement, ε(u) = 2 ∇u + (∇u) the strain tensor, and σ = 2μ ε(u)+ λ(∇·u)I the Cauchy stress tensor with I being the identity matrix. The Lam´e constants λ, μ are given by λ=

Eν , (1 + ν)(1 − 2ν)

μ=

E , 2(1 + ν)

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where E is the elasticity modulus and ν ∈ (0, 12 ) is Poisson’s ratio. One major issue in finite element solvers for linear elasticity is that as the elastic material becomes nearly incompressible or ν → 12 , mathematically as λ → ∞, a FE solver may fail to produce correct results. This often appears as loss of convergence rates in displacement errors or spurious behaviors in numerical stress and dilation (divergence of displacement). This is the so-called “Poisson locking” [6]. It is well known that the classical linear (bilinear, trilinear) Lagrangian finite elements are subject to Poisson locking. Many remedies for Poisson locking have been developed. Reduced integration is probably the easiest technique aiming at a quick fix for the classical Lagrangian elements, although the theory was less elegant [7,9,22]. In this paper, we adopt the remedy in [9] and extend it to 3-dim. In other words, we consider vector-valued Lagrangian bilinear and trilinear finite elements with reduced integration CG.Qd1 (R.I.) (here d = 2, 3) for solving linear elasticity and provide deal.II implementation of these solvers. Specifically, the 1-point Gaussian quadrature is employed for handling the dilation term. Let E be a convex quadrilateral with vertices Pi (xi , yi )(i = 1, 2, 3, 4) that are oriented counterclockwise. A bilinear mapping F from (ˆ x, yˆ) in the reference ˆ = [0, 1]2 to (x, y) ∈ E is established. Its Jacobian determinant is element E ˆ we have 4 scalar-valued bilinear basis functions denoted as J(ˆ x, yˆ). On E, φˆ4 (ˆ x, yˆ) = (1 − x ˆ)ˆ y, φˆ3 (ˆ x, yˆ) = x ˆyˆ, ˆ ˆ φ1 (ˆ x, yˆ) = (1 − x ˆ)(1 − yˆ), φ2 (ˆ x, yˆ) = x ˆ(1 − yˆ).

(3)

They are mapped to the quadrilateral E as rational functions of x, y: x, yˆ), i = 1, 2, 3, 4. φi (x, y) = φˆi (ˆ On E, we have 8 node-based vector-valued local basis functions:                 0 0 0 0 φ1 φ φ φ , , 2 , , 3 , , 4 , . 0 φ1 0 φ2 0 φ3 0 φ4

(4)

(5)

They span CG.Q21 (E). The notation is a bit confusing, since the shape functions are now rationals instead of polynomials. For any v ∈ CG.Q21 (E), we consider   1 1 ∇·v = v(x, y)dxdy = v(x, y)J(ˆ x, yˆ)dˆ xdˆ y, (6) |E| E |E| Eˆ where |E| is the volume of E. Let Vh be the space of vector-valued shape functions constructed from the CG.Q21 elements on a quasi-uniform quadrilateral mesh Eh . Let Vh0 be the subspace of Vh consisting of shape functions that vanish on Γ D . A finite element scheme for linear elasticity in the strain-div formulation seeks uh ∈ Vh so that ASD h (uh , v) = Fh (v), where ASD h (uh , v) =

 E∈Eh

∀v ∈ Vh0 ,

2μ (ε(uh ), ε(v))E + λ(∇ · uh , ∇ · v)E ,

(7) (8)

deal.II Implementation of a 2-Field FE Solver for Poroelasticity

Fh (v) =

 E∈Eh

3



(f , v)E +

tN , v .

91

(9)

γ∈ΓhN

WG Finite Element Discretization for Darcy Flow

This section briefly discusses the weak Galerkin finite element discretization for Darcy flow that is needed for our new 2-field solver for linear poroelasticity. Among the existing finite element solvers for Darcy flow [4,8,10,11,15,18,19], [20,21,23,29,31,32], the newly developed weak Galerkin solvers have some nice features that are attractive for large-scale computing tasks. In particular, the WG(Qk , Qk ; RT[k] ) methods (with integer k ≥ 0) approximate the primal unknown pressure by using polynomial shape function of degree at most k separately defined in element interiors and on edges/faces. Their discrete weak gradients are reconstructed in the unmapped Raviart-Thomas spaces RT[k] and used to approximate the classical gradient in the variational form. The WG Darcy solvers based on these novel notions (i) are locally mass-conservative; (ii) provide continuous normal fluxes; (iii) result in SPD linear systems that are easy to be solved. In [32], we discussed deal.II implementation of such WG Darcy solvers for 0 ≤ k ≤ 5. The numerical tests on SPE10 Model 2 have demonstrated the aforementioned nice features and practical usefulness of the novel WG methodology. In this section, we briefly review the basic concepts of weak Galerkin by recapping WG(Q0 , Q0 ; RT[0] ) for Darcy flow on quadrilateral meshes. For ease of presentation, we consider the Darcy flow problem in its usual form ⎧ ⎨ ∇ · (−K∇p) ≡ ∇ · u = s, p| D = pD , on Γ D (10) ⎩ Γ u · n = uN , on Γ N , where Ω is a polygonal domain, p the primal unknown pressure, u the Darcy velocity, K conductivity tensor (medium permeability divided fluid dynamic viscosity) that is uniformly SPD over the domain, s a known source, pD a Dirichlet boundary condition, uN a Neumann boundary condition, n the outward unit normal vector on ∂Ω, which has a nonoverlapping decomposition Γ D ∪ Γ N . First we define the lowest-order unmapped Raviart-Thomas space as RT[0] (E) = Span(w1 , w2 , w3 , w4 ), where w1 =

  1 , 0

w2 =

  0 , 1

 w3 =

 X , 0

(11) 

w4 =

 0 , Y

(12)

and X = x − xc , Y = y − yc are the normalized coordinates using the element center (xc , yc ). For a given quadrilateral element E, we consider 5 discrete weak functions φi (0 ≤ i ≤ 4) as follows:

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– φ0 for element interior: It takes value 1 in the interior E ◦ but 0 on the boundary E ∂ ; – φi (1 ≤ i ≤ 4) for the four sides respectively: Each takes value 1 on the i-th edge but 0 on all other three edges and in the interior. The discrete weak gradient ∇w φ is established in RT[0] (E) via integration by parts [31]:    (∇w φ) · w = φ∂ (w · n) − φ◦ (∇ · w), ∀w ∈ RT[0] (E). (13) E

E∂

E◦

For implementation, this involves solving a size-4 SPD linear system. However, when E becomes a rectangle [x1 , x2 ] × [y1 , y2 ] with Δx = x2 − x1 , Δy = y2 − y1 , one can obtain these discrete weak gradients explicitly: ⎧ −12 −12 ∇w φ0 = 0w1 + 0w2 + (Δx) 2 w3 + (Δy)2 w4 , ⎪ ⎪ ⎪ −1 6 ⎪ 0w4 , ⎪ ⎨ ∇w φ1 = Δx w1 + 0w2 + (Δx)2 w3 + 1 6 ∇w φ2 = Δx w1 + 0w2 + (Δx) 0w4 , (14) 2 w3 + ⎪ −1 6 ⎪ ⎪ ∇ φ = 0w + w + 0w + w , w 3 1 3 ⎪ Δy 2 (Δy)2 4 ⎪ ⎩ ∇ φ = 0w + 1 w + 6 0w + w 4 1 2 3 Δy (Δy)2 w4 . These discrete weak gradients are used to approximate the classical gradient in the variational form for the Darcy flow problem. Let Eh be a quasi-uniform convex quadrilateral mesh for the given polygonal domain Ω. Let ΓhD be the set of all edges on the Dirichlet boundary Γ D and ΓhN be the set of all edges on the Neumann boundary Γ N . Let Sh be the space of discrete shape functions on Eh that are degree 0 polynomials in element interiors and also degree 0 polynomials on edges. Let Sh0 be the subspace of functions in Sh that vanish on ΓhD . For (10), we seek ph = {p◦h , p∂h } ∈ Sh such that p∂h |ΓhD = Q∂h (pD ) (the L2 -projection of Dirichlet boundary data into the space of piecewise constants on ΓhD ) and Ah (ph , q) = F(q), where

∀q = {q ◦ , q ∂ } ∈ Sh0 ,

  K ∇w ph · ∇w q, Ah (ph , q) = μ E∈Eh E     F(q) = sq ◦ − uN q ∂ . E∈Eh

E◦

γ∈ΓhN

(15)

(16)

(17)

γ

Clearly, (15) is a large-size sparse SPD system. After the numerical pressure ph is solved from (15), an elementwise numerical velocity is obtained by a local L2 -projection back into the subspace RT[0] : uh = Qh (−K∇w ph ).

(18)

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The projection can be skipped if K is an elementwise constant scalar matrix. Furthermore, the bulk normal flux on any edge is defined as  uh · ne . (19) e∈E ∂

It has been proved [21] that such a WG solver is locally conservative and guarantees normal flux continuity.

4

Coupling WG(Q0 , Q0 ; RT[0] ) and CG.Q21 (R.I.) for Poroelasticity

In this section, the continuous Galerkin Qd1 (d = 2, 3) elements with reduced integration and the weak Galerkin WG(Q0 , Q0 ; RT[0] ) elements are combined with the implicit Euler temporal discretization to solve linear poroelasticity problems. Assume a given domain Ω is already partitioned into a quasi-uniform quadrilateral mesh Eh . For a given time period [0, T ], let 0 = t(0) < t(1) < . . . < t(n−1) < t(n) < . . . < t(N ) = T be a temporal partition. We denote Δtn = t(n) − t(n−1) for n = 1, 2, . . . , N . Let Vh and Vh0 be the spaces of vector-valued shape functions based on the (n) (n−1) first-order CG elements. Let uh , uh ∈ Vh be the approximations to solid (n) and t(n−1) , respectively. displacement at time moments t Let Sh and Sh0 be the spaces of scalar-valued discrete weak functions constructed in Sect. 3 based on the WG(Q0 , Q0 ; RT[0] ) elements. Similarly, let (n) (n−1) ph , ph ∈ Sh be the approximations to fluid pressure at time moments t(n) (n−1) and t , respectively. Note that the discrete weak trial function has two parts: (n)

(n),◦

ph = {ph (n),◦

(n),∂

, ph

},

(20)

(n),∂

where ph lives in element interiors and ph lives on the mesh skeleton. Applying the implicit Euler discretization, we establish the following timemarching scheme, for any v ∈ Vh0 and any q ∈ Sh0 , ⎧   (n) (n) (n),◦ ⎪ 2μ ε(u ), ε(v) + λ(∇ · uh , ∇ · v) − α(ph , ∇ · v) = (f (n) , v), ⎪ h ⎪ ⎨     (n),◦ (n) (n) c0 ph , q ◦ + Δtn K∇ph , ∇q + α(∇ · uh , q ◦ ) ⎪     ⎪ ⎪ (n−1),◦ (n−1) ⎩ , q ◦ + Δtn s(n) , q ◦ + α(∇ · uh , q ◦ ), = c0 ph

(21)

for n = 1, 2, . . . , N , where ∇ · v is the elementwise average that represents the reduced integration technique. The above two equations are further augmented with appropriate boundary and initial conditions. This results in a large monolithic system at each time step. Theses errors are calculated to assess the accuracy of our poroelasticity solver:

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– L2 ([0, T ]; L2 (Ω))-norm for interior pressure errors

p − p◦h 2L2 (L2 ) =

N 

((n),◦) 2

L2 (Ω) ,

Δtn p(n) − ph

(22)

n=1

– L2 ([0, T ]; L2 (Ω))-norm for displacement errors

u − uh 2L2 (L2 ) =

N 

(n)

Δtn u(n) − uh 2L2 (Ω) ,

(23)

n=1

– L2 ([0, T ]; H 1 (Ω))-norm for displacement errors

u −

uh 2L2 (H 1 )

=

N 

(n)

Δtn ∇u(n) − ∇uh 2L2 (Ω) ,

(24)

n=1

– L2 ([0, T ]; L2 (Ω))-norm for stress errors

σ − σh 2L2 (L2 ) =

N 

(n)

Δtn σ (n) − σh 2L2 (Ω) .

(25)

n=1

5

Code Excerpts with Comments

This section provides some code excerpts with comments. More details can be found in our code modules for deal.II (subject to minor changes). We want to point that the elasticity discretization can also be replaced by the so-called EQ1 or BR1 elements [3,16], which are now available in deal.II Version 9.1. 5.1

Code Excerpts for WG(Q0 , Q0 ; RT[0] )

There was a discussion on this in [32]. Here we recap the most important concepts very briefly. Note that FE RaviartThomas is a Raviart-Thomas space for vectorvalued functions, FESystem defines WG finite element spaces in the interiors and on edges/faces. Shown below is the code for the lowest-order WG finite elements. 88 89 90 91

227 228 229 230

FE_RaviartThomas fe_rt; DoFHandler dof_handler_rt; FESystem fe; DoFHandler dof_handler;

fe_rt (0); dof_handler_rt (triangulation); fe (FE_DGQ(0), 1, FE_FaceQ(0), 1); dof_handler (triangulation);

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5.2

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Code Excerpts for CG.Q21 with Reduced Integration

This part shows how we use CG.Q21 with reduced integration to discretize linear elasticity. FE Q defines the finite element space for displacement vectors. Each component of the vector is in the FE Q space. 88

FE_Q(1),dim;

Here, the reduced integration technique with one-point Gaussian quadrature is used to calculate the dilation (divergence of displacement). 88

5.3

QGauss

reduced_integration_quadrature_formula(1);

Code Excerpts for Coupled Discretizations for Poroelasticity

We couple CG.Q21 (R.I.) and WG(Q0 , Q0 ; RT[0] ) to solve linear poroelasticity. FESystem defines the finite element spaces for displacement, interior pressure, and face pressure. Shown below is the coupled finite elements. 88 89 90 91

88 89

FE_RaviartThomas fe_rt; DoFHandler dof_handler_rt; FESystem fe; DoFHandler dof_handler;

fe_rt (0), dof_handler_rt (triangulation),

90 91 92 93 94

fe (FE_Q(1),dim, FE_DGQ(0), 1, FE_FaceQ(0), 1), dof_handler (triangulation),

We use block structures to store matrices and variables. The following piece defines the degrees of freedom associated with displacement, interior pressure, and face pressure. 88 89 90 91 92 93 94

std::vector dofs_per_block (3); DoFTools::count_dofs_per_block (dof_handler, dofs_per_block, block_component); const unsigned int n_u = dofs_per_block[0], n_p_interior = dofs_per_block[1], n_p_face = dofs_per_block[2], n_p = dofs_per_block[1]+ dofs_per_block[2];

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The implementation for the WG Darcy solver discussed in [32] is naturally re-used and incorporated. The following piece calculates the coupling terms with reduced integration in the local matrix. However, we only use the reduced integration for divergence of vector-valued shape functions. 88 89 90 91 92 93 94 95 96 97

for (unsigned int q_index = 0; q_index < n_q_points_reduced_integration; ++q_index){ for (unsigned int i = 0; i < dofs_per_cell; ++i){ const double div_i_reduced_integration = fe_values_reduced_integration [displacements_reduced_integration].divergence(i, q_index); for (unsigned int j = 0; j < dofs_per_cell; ++j){ const double div_j_reduced_integration = fe_values_reduced_integration [displacements_reduced_integration].divergence(j, q_index);

98

local_matrix(i, j) += - alpha * fe_values_reduced_integration [pressure_interior_reduced_integration].value(j, q_index) * div_i_reduced_integration + alpha* (div_j_reduced_integration * fe_values_reduced_integration [pressure_interior_reduced_integration].value (i,q_index))) * fe_values_reduced_integration.JxW(q_index);

99 100 101 102 103 104 105 106 107

}}}

Finally, this piece hands the coupling term in the local right-hand side. 88 89 90 91 92 93 94 95 96 97

6

for (unsigned int q=0; q 0 is the final time. We denote the solution and the source term at t = tn by u(·, tn ) and f (u(·, tn )) respectively. The variational formulation for the problem (1) is: find u(·, t) ∈ H01 (Ω) such that

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∂u ,v ∂t

 + A(u, v) = f, v

in Ω × [0, T ],

∀v ∈ H01 (Ω), (2)

u(x, 0) = g(x) in Ω, u(x, t) = 0 on ∂Ω × [0, T ].

 where A(u, v) = Ω κ∇u · ∇v dx. In order to discretize (2) in time, we need to apply some time differencing methods. For simplicity, we first apply the implicit Euler scheme with time step Δt > 0 and in Sect. 3, we will consider the exponential time differencing method (ETD). We obtain the following discretization for each time tn = nΔt, n = 1, 2, · · · , N (T = N Δt), u(·, tn ) − u(·, tn−1 ) = div(κ∇u(·, tn )) + f (u(·, tn )). Δt Let T h be a partition of the domain Ω into fine finite elements. Here h > 0 is the fine grid mesh size. The coarse partition, T H of the domain Ω, is formed such that each element  in T H is a connected union of fine-grid blocks. More precisely, H ∀Kj ∈ T , Kj = F ∈Ij F for some Ij ⊂ T h . The quantity H > 0 is the coarse mesh size. We will consider the rectangular coarse elements and the methodology can be used with general coarse elements. An illustration of the mesh notations is shown in the Fig. 1. We denote the interior nodes of T H by xi , i = 1, · · · , Nin , where Nin is the number of interior nodes. The coarse elements of T H are denoted by Kj , j = 1, 2, · · · , Ne , where Ne is the number of coarse elements. We define the coarse neighborhood of the nodes xi by Di := ∪{Kj ∈ TH : xi ∈ Kj }. 2.2

The GMsFEM and the Multiscale Basis Functions

In this paper, we will apply the GMsFEM to solve nonlinear parabolic equations. The method is motivated by the finite element framework. First, a variational formulation is defined. Then we construct some multiscale basis functions. Once the fine grid is given, we can compute the fine-grid solution. Let γ1 , · · · , γn be the standard finite element basis, and define Vf = span{γ1 , · · · , γn } to be the fine space. We obtained the fine solution denoted by unf at t = tn by solving  1  n  uf , v + A unf , v = Δt



 1 n−1 n u + f (uf ), v , Δt f

∀v ∈ Vf ,

(3)

u0f = gh , where gh is the Vf based approximation of g. The construction of multiscale basis functions follows two general steps. First, we construct snapshot basis functions in order to build a set of possible modes of the solutions. In the second step, we construct multiscale basis functions with a suitable spectral problem defined in the snapshot space. We take the first few dominated eigenfunctions as basis functions. Using the multiscale basis functions, we obtain a reduced model.

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119

Fig. 1. Left: an illustration of fine and coarse grids. Right: an illustration of a coarse neighborhood, coarse element, and oversampled domain

More specifically, once the coarse and fine grids are given, one may construct the multiscale basis functions to approximate the solution of (2). To obtain the multiscale basis functions, we first define the snapshot space. For each coarse neighborhood Di , define Jh (Di ) as the set of the fine nodes of T h lying on ∂Di and denote the its cardinality by Li ∈ N+ . For each fine-grid node xj ∈ Jh (Di ), we define a fine-grid function δjh on Jh (Di ) as δjh (xk ) = δj,k . Here δj,k = 1 if j = k and δj,k = 0 if j = k. For each j = 1, · · · , Li , we define the snapshot basis (i) functions ψj (j = 1, · · · , Li ) as the solution of the following system 

(i) = 0 in Di −div κ∇ψj (4) (i) ψj = δjh on ∂Di . (i)

The local snapshot space V snap corresponding to the coarse neighborhood Di (i) (i) is defined as follows Vsnap := span{ψj : j = 1, · · · , Li } and the snapshot space Nin (i) reads Vsnap := i=1 Vsnap . In the second step, a dimension reduction is performed on Vsnap . For each i = 1, · · · , Nin , we solve the following spectral problem:

(i) (i) (i) (i) κ∇φj · ∇v = λj κ ˆ φj v ∀v ∈ Vsnap , j = 1, . . . , Li (5) Di

Di

Nin

2

in where κ ˆ := κ i=1 H 2 |∇χi | and {χi }N i=1 is a set of partition of unity that solves the following system:

−∇ · (κ∇χi ) = 0 in K ⊂ Di χi = pi on each ∂K with K ⊂ Di χi = 0 on ∂Di where pi is some polynomial functions and we can choose linear functions for simplicity. Assume that the eigenvalues obtained from (5) are arranged in ascending order and we may use the first 1 < li ≤ Li (with li ∈ N+ ) eigenfunctions (i) (related to the smallest li eigenvalues) to form the local multiscale space Voff :=

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(i)

snap{χi φj : j = 1, · · · , Li }. The mulitiscale space Voff is the direct sum of the Nin (i) local mulitiscale spaces, namely Voff := i=1 Voff . Once the multiscale space Voff is constructed, we can find the GMsFEM solution unoff at t = tn by solving the following equation   1 n−1 1 n n n u , v + A (uoff , v) = u + f (uoff ), v , Δt off Δt off (6) u0off , v = g, v, ∀v ∈ Voff . 2.3

Online Enrichment

We will present the constructions of online basis functions [1] in this section. Online Adaptive Algorithm. In this subsection, we will introduce the method of online enrichment. After obtaining the multiscale space Voff , one may add some online basis functions based on local residuals. Let unoff ∈ Voff be the solution Di , we define obtained in (6) at time t = tn . Given a coarse neighborhood  Vi := H01 (Di ) ∩ Vsnap equipped with the norm v 2Vi := Di κ|∇v|2 . We also define the local residual operator Rin : Vi → R by n Rn i (v; uoff ) :=



Di



    1 n−1 1 n uoff + f (un uoff v , κ∇un off ) v − off · ∇v + Δt Δt Di

∀v ∈ Vi . (7)

The operator norm Rin , denoted by Rin Vi∗ , gives a measure of the quantity of residual. The online basis functions are computed during the time-marching process for a given fixed time t = tn , contrary to the offline basis functions that are pre-computed. Suppose one needs to add one new online basis φ into the space Vi . The analysis in [1] suggests that the required online basis φ ∈ Vi is the solution to the following equation A(φ, v) = Rni (v; un,τ off )

∀v ∈ Vi .

(8)

We refer to τ ∈ N as the level of the enrichment and denote the solution of (6) n,0 by un,τ off . Remark that Voff := Voff for time level n ∈ N. Let I ⊂ {1, 2, . . . , Nin } be the index set over some non-lapping coarse neighborhoods. For each i ∈ I, n,τ +1 n,τ = Voff ⊕ we obtain a online basis φi ∈ Vi by solving (8) and define Voff n,τ +1 . span {φi : i ∈ I}. After that, solve (6) in Voff Two Online Adaptive Methods. In this section, we compare two ways to obtain online basis functions which are denoted by online adaptive method 1 and online adaptive method 2 respectively. Online adaptive method 1 is adding online basis using online adaptive method from offline space in each time step, which means basis functions obtained in last time step are not used in current time step. Online adaptive method 2 is keeping online basis functions in each time step. Using this accumulation strategy, we can skip online enrichment after a certain time period when the residual defined in (7) is under given tolerance. We also presents the results of these two methods in Fig. 3 and Fig. 4 respectively.

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121

Numerical Results. In this section, we present some numerical examples to demonstrate the efficiency of our proposed method. The computational domain is Ω = (0, 1)2 ⊂ R2 and T = 1. The medium κ1 and κ2 are shown in Fig. 2, where the contrasts are 104 and 105 for κ1 and κ2 respectively. Without special descriptions, we use κ1 . For each function to be approximated, we define the following quantities ena and en2 at t = tn to measure energy error and L2 error respectively. ena =

unf − unoff V (Ω) unf V (Ω)

en2 =

unf − unoff L2 (Ω) unf L2 (D)

where unf is the fine-grid solution (reference solution) and unoff is the approximation obtained by the GMsFEM method. We define the energy norm and L2 norm of u by

u 2V (Ω) =

∇u 2 Ω

u 2L2 =

u 2 . Ω

Example 2.1. In this example, we compare the error using adaptive online method 1 and uniform enrichment under different numbers of initial basis functions. We set the mesh size to be H = 1/16 and h = 1/256. The time step is Δt = 10−3 and the final time is T = 1. The initial condition is u(x, y, t)|t=0 = 4(0.5 − x)(0.5 − y). We set the permeability to be κ1 . We set the source term f = 12 (u3 − u), where = 0.01. We present the numerical results for the GMsFEM at time t = 0.1 in Table 1, 2, and 3. For comparison, we present the results where online enrichment is not applied in Table 4. We observe that the adaptive online enrichment converges faster. Furthermore, as we compare Table 4 and Table 1, we note that the online enrichment does not improve the error if we only have one offline basis function per neighborhood. Because the first eigenvalue is small, the error decreases in the online iteration is small. In particular, for each iteration, the error decrease slightly. As we increase the number of initial offline basis, the convergence is very fast and one online iteration is sufficient to reduce the error significantly. Example 2.2. We compare online Method 1 and 2 under different tolerance. We keep H, h and the initial condition the same as in Example 2.1. We choose intial number of basis to be 450, which means we choose two initial basis per neighborhood. We keep the source term as f = 12 (u3 − u). When = 0.01, we choose the time step Δt to be 10−4 . We plot the error and DOF from online Method 1 in Fig. 3 and compare with results from online Method 2 in Fig. 4. From Fig. 3 and 4, we can see the error and DOF reached stability at t = 0.01. In Fig. 4, we can see the DOF keeps increasing before turning steady. The error remains at a relatively low level without adding online basis after some time. As a cost, online method 2 suffers bigger errors than method 1 with same tolerance. We also apply our online adaptive method 2 under permeability κ2 in Fig. 5. The errors are relatively low for two kinds of permeability.

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(a) κ1

(b) κ2

Fig. 2. Permeability field Table 1. The errors for online enrichment when number of initial basis = 1. Left: Adaptive enrichment. Right: Uniform enrichment. ea

DOF 225 460 550

e2

14.47% 19.55% 2.23% 1.14% 1.20% 0.6%

DOF 225 450 675

ea

e2

14.48% 19.56% 8.39% 6.54% 2.45% 1.1%

Table 2. The errors for online enrichment when number of initial basis = 2. Left: Adaptive enrichment. Right: Uniform enrichment DOF 450 681

ea

e2

4.66% 2.64% 1.65% 0.52%

DOF 450 675

ea

e2

4.65% 2.64% 1.10% 0.669%

Table 3. The errors for online enrichment when number of initial basis = 3. Left: Adaptive enrichment. Right: Uniform enrichment DOF 675 903

ea

e2

2.89% 1.07% 0.944% 0.511%

DOF 675 900

ea

e2

2.89% 1.07% 1.13% 0.894%

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123

Table 4. The errors for different  in source term without online enrichment. Up: Energy error. Down: L2 error Source function t = 0.1 t = 0.2  = 0.1  = 0.01

5.97% 5.94% 15.1% 15.3%

Source function t = 0.1 t = 0.2  = 0.1  = 0.01

4.57% 4.57% 11.9% 12.0%

(a) error with tolerance 10−4

(b) DOF with tolerance 10−4

(c) error with tolerance 10−3

(d) DOF with tolerance 10−3

Fig. 3. Error and DOF obtained by online method 1 in Example 2.2

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(a) error with tolerance 10−4

(b) DOF with tolerance 10−4

(c) error with tolerance 10−3

(d) DOF with tolerance 10−3

(e) error with tolerance 10−2

(f) DOF with tolerance 10−2

Fig. 4. Error and DOF obtained by online method 2 in Example 2.2

Adaptive Multiscale Model Reduction for Nonlinear Parabolic Equations

(a) error with κ1

(b) DOF with with κ1

(c) error with κ2

(d) DOF with with κ2

125

Fig. 5. Error and DOF obtained by online method 2 in Example 2.2

3

Application to the Allen-Cahn Equation

In this section, we apply our proposed method to the Allen Cahn equation. We use the Exponential Time Differencing (ETD) for time dsicretization. To deal with the nonlinear term, DEIM is applied. We will present the two methods in the following subsections. 3.1

Derivation of Exponential Time Differencing

Let τ be the time step. Using ETD, unoff is the solution to (9) unoff , v + τ A(unoff , v) = exp(− u0off , v

= g, v

τ f (un−1 off ) )un−1 off , v

2 un−1 off ∀v ∈ Voff

Next, we will derive this equation. We have ut − div(κ∇u) +

1 f (u) = 0

2

(9)

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Multiplying the equation by integrating factor ep(u) , we have ep(u) ut + ep(u)

1 f (u) = ep(u) div(κ∇u)

2

We require the above to become d(ep(u) u) = ep(u) div(κ∇u) dt

(10)

By solving d(ep(u) u) d = ep(u) ut + ep(u) ( p(u))u, dt dt we have

p(u(tn , ·)) − u(0, ·)) =

tn

0

1 f (u) .

2 u

Using Backward Euler method in (10), we have un − τ div(κ∇un ) = e−p(u)n un−1

(11)

where p(u)n = p(u(tn )−u(tn−1 )). To solve (11), we approximate (11) as follows: − τ2

e−p(u)n un−1 ≈ e

f (u(tn−1 )) u(tn−1 )

u(tn−1 ).

(12)

τ f (un−1 off ) )un−1 off .

2 un−1 off

(13)

Using above approximation, we have unoff − τ div(κ∇unoff ) = exp(−

3.2

DEIM Method

When we evaluate the nonlinear term, the complexity is O(α(n) + c · n), where α is some function and c is a constant. To reduce the complexity, we approximate local and global nonlinear functions with the Discrete Empirical Interpolation Method (DEIM) [2]. DEIM is based on approximating a nonlinear function by means of an interpolatory projection of a few selected snapshots of the function. The idea is to represent a function over the domain while using empirical snapshots and information at some locations (or components). The key to complexity reduction is to replace the orthogonal projection of POD with the interpolation projection of DEIM in the same POD basis. We briefly review the DEIM. Let f (τ ) be the nonlinear function. We are desired to find an approximation of f (τ ) at a reduced cost. To obtain a reduced order approximation of f (τ ), we first define a reduced dimentional space for it. We would like to find m basis vectors (where m is much smaller than n), φ1 , · · · , φm , such that we can write f (τ ) = Φd(τ ),

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where Φ = (φ1 , · · · , φm ). We employ POD to obtain Φ and use DEIM (refer Table 5) to compute d(τ ) as follows. In particular, we solve d(τ ) by using m rows of Φ. This can be formalized using the matrix P P = [e℘1 , . . . , e℘m ] ∈ Rn×m , where e℘i = [0, · · · , 1, 0, · · · , 0] ∈ Rn is the ℘th i column of the identity matrix In ∈ Rn×n for i = 1, · · · , m. Using P T f (τ ) = P T Φd(τ ), we can get the approximation for f (τ ) as follows:  −1 T f (τ ) ≈ f˜(τ ) = Φd(τ ) = Φ PT Φ P f (τ ) Table 5. DEIM algorithm DEIM

Algorithm

Φ = (φ1 , · · · , φm ) obtained by applying POD on a sequence of ns functions evaluations → − Output The interpolation indices λ = (λ1 , · · · , λm )T 1. Set [ρ, λ1 ] = max{|φ1 |} → − 2. Set Φ = [φ1 ], P = [eλ1 ], and λ = (λ1 ) 3. for i = 2, · · · , m, do Solve (P T Φ)w = P T φi for some i Compute r = φi − Φw Compute [ρ, λi ] = max{|r|} − → → − λ Set Φ = [Φ, φi ], P = [P, eλi ], and λ = λi end for Input

3.3

Numerical Results

Example 3.3. In this example, we apply the DEIM under the same setting as in Example 2.2 and we did not use the online enrichment procedure. We compare the results in Fig. 8. To test the DEIM, we first consider the solution using DEIM where the snapshot are obtained by the same equation. First, we set = 0.01. We first solve the same equation and obtain the snapshot Φ. Secondly, we use DEIM to solve the equation again. The two results are presents in Fig. 6. The first picture are the errors we get when DEIM are not used while used in second one. The errors of these two cases differs a little since the snapshot obtained in the same equation. Then we consider the cases where the snapshots are obtained: 1. 2. 3. 4.

Different Different Different Different

right hand side functions. initial conditions. permeability field. time steps.

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Different Right Hand Side. Since the solution for different can have some similarities, we can use the solution from one to solve the other. In particular, since it will be more time-consuming to solve the case when is smaller. We can use the f (u) for = 0.09 to compute the solution for = 0.1 since solutions for these two cases can only vary a little. I show the results in Fig. 7. Different Initial Conditions. In this section, we consider using the snapshot from different initial conditions, we record the results in Fig. 9. We first choose the initial condition to be compared Fig. 9 and Fig. 6, we can see that different initial conditions can have less impact on the final solution since the solution is close to the one where the snapshot is obtained in the same equation. Different Permeability Field. In this section, we consider using the snapshot from different permeability, we record the results in Fig. 10. For reference, the first two figures plots the fine solution and multiscale solution without using DEIM. And we construct snapshot from another permeability κ1 and we apply it to compute the solution in κ2 . The last figure shows the of using DEIM is relatively small. Different Time Steps. In this section, we construct the snapshot by using nonlinear function obtained in previous time step for example when t < 0.05. Then we apply it to DEIM to solve the equation in 0.05 < t < 0.06. We use these way to solve the equation with permeability κ1 and κ2 respectively. We plot the results in Fig. 11 and 12. From these figures, we can see that DEIM have different effects applied to different permeability. With κ1 , the error increases significantly when DEIM are applied. But with κ2 , the error decreased to a lower level when we use DEIM.

(a) not using DEIM

(b) using DEIM

Fig. 6. Error for same 

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(a) not using DEIM

(b) using DEIM

Fig. 7. Error for different 

(a) fine solution when  = 0.01

(b) multiscale solution when  = 0.01

Fig. 8. Comparing fine and multiscale solutions.

(a) solution

(b) error

Fig. 9. Using DEIM for different initial conditions

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(a) fine solution under permeability κ2

(b) multiscale solution under permeability a2

(c) solution under permeability κ2 using DEIM

(d) error for solution using DEIM

Fig. 10. Using DEIM for different permeability field

(a) solution

(b) error

Fig. 11. Using DEIM for under different time step for κ1

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(b) error

Fig. 12. Using DEIM for under different time step for κ2

Acknowledgement. The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project numbers 14302018 and 14304719) and CUHK Faculty of Science Direct Grant 2018-19.

References 1. Chung, E.T., Efendiev, Y., Leung, W.T.: Residual-driven online generalized multiscale finite element methods. J. Comput. Phys. 302, 176–190 (2015) 2. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010) 3. Chung, E., Efendiev, Y., Hou, T.Y.: Adaptive multiscale model reduction with generalized multiscale finite element methods. J. Comput. Phys. 320, 69–95 (2016) 4. Efendiev, Y., Gildin, E., Yang, Y.: Online adaptive local-global model reduction for flows in heterogeneous porous media. Computation 4, 22 (2016) 5. Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Benner, P., Sorensen, D.C., Mehrmann, V. (eds.) Dimension Reduction of Large-Scale Systems. LNCS, vol. 45, pp. 261–306. Springer, Berlin (2005). https://doi.org/10. 1007/3-540-27909-1 10 6. Kunisch, K., Volkwein, S.: Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102, 345–371 (1999). https://doi.org/10.1023/A:1021732508059 7. Graham, W.R., Peraire, J., Tang, K.Y.: Optimal control of vortex shedding using loworder models. Part I-Open-loop model development. Int. J. Numer. Methods Eng. 44, 945–972 (1999) 8. Ravindran, S.S.: A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Methods Fluids 34, 425–448 (2000) 9. Prud’homme, C., et al.: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluids Eng. 124, 70–80 (2002)

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10. Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T., Rovas, D.V.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Ser. Math. 331, 153–158 (2000) 11. Maday, Y., Patera, A.T., Turinici, G.: A priori convergence theory for reducedbasis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17, 437–446 (2002). https://doi.org/10.1023/A:1015145924517 12. Veroy, K., Rovas, D.V., Patera, A.T.: A posterior error estimation for reducedbasis approximation of parametrized elliptic coercive partial differential equations: “convex inverse” bound conditioners. ESAIM Control Optim. Calc. Var. 8, 1007– 1028 (2002) 13. Nguyen, N.C., Rozza, G., Patera, A.T.: Reduced basis approximation dand a posteriori error estimationd for the time-dependent viscous Burgers’ equation. Calcolo 46, 157–185 (2009) 14. Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized Multiscale Finite Element Methods. arXiv:1301.2866. http://arxiv.org/submit/631572

Parallel Shared-Memory Isogeometric Residual Minimization (iGRM) for Three-Dimensional Advection-Diffusion Problems Marcin L  o´s1 , Judit Munoz-Matute2 , Krzysztof Podsiadlo1 , Maciej Paszy´ nski1(B) , and Keshav Pingali3 1

AGH University of Science and Technology, Krak´ ow, Poland [email protected] 2 The University of the Basque Country, Bilbao, Spain 3 The University of Texas at Austin, Austin, USA

Abstract. In this paper, we present a residual minimization method for three-dimensional isogeometric analysis simulations of advectiondiffusion equations. First, we apply the implicit time integration scheme for the three-dimensional advection-diffusion equation. Namely, we utilize the Douglas-Gunn time integration scheme. Second, in every time step, we apply the residual minimization method for stabilization of the numerical solution. Third, we use isogeometric analysis with B-spline basis functions for the numerical discretization. We perform alternating directions splitting of the resulting system of linear equations, so the computational cost of the sequential LU factorization is linear O(N ). We test our method on the three-dimensional simulation of the advectiondiffusion problem. We parallelize the solver for shared-memory machine using the GALOIS framework. Keywords: Isogeometric analysis · Implicit dynamics · Advection-diffusion problems · Linear computational cost solvers · GALOIS framework

1

· Direct

Introduction

The alternating direction implicit method (ADI) is a popular method for performing finite difference simulations on regular grids. The first papers concerning the ADI method were published in 1960 [1,3,5,19]. This method is still popular for fast solutions of different classes of problems with finite difference method [8,9]. In its basic version, the method introduces intermediate time steps, and the differential operator splits into the x, y (and z in 3D) components. As a result of this operation, on the left-hand side, we only deal with derivatives in one direction, while the rest of the operator is on the right-hand side. The resulting system of linear equations has a multi-diagonal form, so the factorization of this system is possible with a linear O(N ) computational cost. It is a common misunderstanding that the direction splitting solvers are limited to simple geometries. c Springer Nature Switzerland AG 2020  V. V. Krzhizhanovskaya et al. (Eds.): ICCS 2020, LNCS 12143, pp. 133–148, 2020. https://doi.org/10.1007/978-3-030-50436-6_10

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They can be also applied to discretizations in extremely complicated geometries, as described in [10]. In this paper, we generalize this method for three-dimensional simulations of the time-dependent advection-diffusion problem with the residual minimization method. We use the basic version of the direction splitting algorithm, working on a regular computational cube, since this approach is straightforward and it is enough to proof our claims that the residual minimization stabilizes the advection-diffusion simulations. In particular, we apply the residual minimization method with isogeometric finite element method simulations over a threedimensional cube shape computational grids with tensor product B-spline basis functions. The resulting system of linear equations can be factorized in a linear O(N ) computational cost when executed in sequential mode. We use the finite element method discretizations with B-spline basis functions. This setup, as opposed to the traditional finite difference discretization, allows us to apply the residual minimization method to stabilize our simulations. The isogeometric analysis (IGA) [4] is a modern method for performing finite element method (FEM) simulations with B-splines and NURBS. In enables higher order and continuity B-spline based approximations of the modeled phenomena. The direction splitting method has been rediscovered to solve the isogeometric L2 projection problem over regular grids with tensor product B-spline basis functions [6,7]. The direction splitting, in this case, is performed with respect to space, and the splitting is possible by exploiting the Kronecker product structure of the Gram matrix with tensor product structure of the B-spline basis functions. The L2 projections with IGA-FEM were applied for performing fast and smooth simulations of explicit dynamics [11–16,20]. This is because the explicit dynamics with isogeometric discretization is equivalent to the solution of a sequence of isogeometric L2 projections. In this paper, we focus on the advection-diffusion equation used for simulation of the propagation of a pollutant from a chimney. We introduce implicit time integration scheme, that allows for the alternating direction splitting of the advectiondiffusion equation. We discover that the numerical simulations are unstable, and deliver some unexpected oscillations and reflections. Next, we utilize the residual minimization method in a way that it preserves the Kronecker product structure of the matrix and enables stabilized linear computational cost solutions. The actual mathematical theory concerning the stability of the numerical method for weak formulations is based on the famous “Babu´ska-Brezzi condition” (BBC) developed in years 1971–1974 at the same time by Ivo Babu´ska, and Franco Brezzi [25–27]. The condition states that a weak problem is stable when |b(u, v)| ≥ γuU , ∀u ∈ U. (1) sup v∈V vV However, the inf-sup condition in the above form concerns the abstract formulation where we consider all the test functions from v ∈ V and look for solution at u ∈ U (e.g. U = V ). The above condition is satisfied also if we restrict to the space of trial functions uh ∈ Uh ⊂ U

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v∈V

|b(uh , v)| ≥ γuh Uh . vV

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However, if we use test functions from the finite dimensional test space Vh = span{vh } ⊂ V |b(uh , vh )| sup ≥ γh uh Uh , (3) vh ∈Vh vh Vh we do not have a guarantee that the supremum (3) will be equal to the original supremum (1), since we have restricted V to Vh . The optimality of the method depends on the quality of the polynomial test functions defining the space Vh = span{vh } and how far are they from the supremum defined in (1). There are many method for stabilization of different PDEs [28–31]. In 2010, the Discontinuous Petrov Galerkin (DPG) method was proposed, with the modern summary of the method described in [32]. The DPG method utilizes the residual minimization with broken test spaces. In other words, it first generates a system of linear equations      l G −B r = . (4) u 0 BT 0 This system of linear equations has the inner product block G over the test space, the two blocks with the actual weak form B and B T , and the zero block 0. The test space is larger than the trial space, and the inner product and the weak form blocks are rather sparse matrices. Therefore, the dimension of the system of linear equations is at least two times larger than the original system of equations arising from standard Galerkin method. In the DPG method, the test space is broken in order to obtain a block-diagonal matrix G and the Schur complements can be locally computed over each finite element. The price to pay is the presence of the additional fluxes on the element interfaces, resulting from breaking the test spaces, so the system over each finite element looks like ⎤⎡ ⎤ ⎡ ⎤ ⎡ G −B1 −B2 r l ⎣B1T 0 0 ⎦ ⎣u⎦ = ⎣0⎦ . (5) 0 t 0 B2T 0 We do not know any other reason of breaking the test spaces in the DPG method other then reduction of the computational cost of the solver. In this paper, we want to avoid dealing with fluxes and broken spaces since it is technically very complicated. Thus, we stay with the unbroken global system (4) and then we have to face one of the two possible methods. The first one would be to apply adaptive finite element method, but then the cost of factorization in 3D would be up to four times slower than in the standard finite element method and broken DPG (without the static condensation). This is because depending on the structure of the refined mesh, we will have a computational cost of the multi-frontal solver varying between O(N ) to O(N 2 ) [33], and our N is two times bigger than in the original weak problem, and 22 = 4. This could be an option that we will discuss in a future paper.

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Another method that we exploit in this paper is to keep a tensor product structure of the computational patch of elements with tensor product B-spline basis functions, decompose the system matrix into a Kronecker product structure, and utilize a linear computational cost alternating directions solver. Even for the system (4) resulting from the residual minimization we successfully perform direction splitting to obtain a Kronecker product structure of the matrix to maintain the linear computational cost of the alternating directions method. In order the stabilize the time-dependent advection-diffusion simulations, we perform the following steps. First, we apply the time integration scheme. We use the Douglas-Gunn second order time integration scheme [2]. Second, we stabilize a system from every time step by employing the residual minimization method [34–36]. Finally, we perform numerical discretization with isogeometric analysis [4], using tensor product B-spline basis functions over a three-dimensional cube shape patch of elements. The novelties of this paper with regard to our previous work are the following. In [11], we described parallel object-oriented JAVA based implementation of the explicit dynamics version of the alternating directions solver, without any residual minimization stabilization, and for two-dimensional problems only. In [12], we described sequential Fortran based implementation of the explicit dynamics solver, with applications of the elastic wave propagation, without implicit time integration schemes and any residual minimization stabilization. In [16], we described the parallel distributed memory implementation of the explicit dynamics solver, again without implicit time integration scheme and residual minimization method. In [14], we described the parallel shared-memory implementation of the explicit dynamics solver, with the same restrictions as before. In [13,17] we applied the explicit dynamics solver for two and three-dimensional tumor growth simulations. In all of these papers, we did not used implicit time integration schemes, and we did not perform operator splitting on top of the residual minimization method. In [20], we investigate different time integration schemes for two-dimensional residual minimization method for advection-diffusion problems. We do not go for three-dimensional computations, and we do not apply parallel computations there. In this paper, we apply the residual minimization with direction splitting for the first time in three-dimensions. We also investigate the parallel scalability of our solver, using the GALOIS framework for parallelization. For more details on the GALOIS framework itself, we refer to [21–24]. The structure of this paper is the following. We start in Sect. 2 with the derivation of the isogeometric alternating direction implicit method for the advection-diffusion problem. The following Sect. 3 derives the residual minimization method formulation of the advection-diffusion problem in three-dimensions. Next, in Sect. 4, we present the linear computational cost numerical results. We summarize the paper with conclusions in Sect. 5.

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Model Problem of Three-Dimensional Advection-Diffusion

Let Ω = Ωx × Ωy × Ωz ⊂ R3 an open bounded domain and I = (0, T ] ⊂ R, we consider the three-dimensional linear advection-diffusion equation ⎧ in Ω × I, ⎪ ⎨ ut − ∇ · (α∇u) + β · ∇u = f u=0 on Γ × I, (6) ⎪ ⎩ u(0) = u0 in Ω, where Ωx , Ωy and Ωz are intervals in R. Here, ut := ∂u/∂t, Γ = ∂Ω denotes the boundary of the spatial domain Ω, f : Ω × I −→ R is a given source and u0 : Ω −→ R is a given initial condition. We consider constant diffusivity α and constant velocity field β = [βx βy βz ]. We split the advection-diffusion operator Lu = −∇ · (α∇u) + β · ∇u as Lu = L1 u + L2 u + L3 u where L1 u := −α

∂u ∂u ∂u ∂u ∂u ∂u + βx , L2 u := −α 2 + βy , L3 u := −α 2 + βz . 2 ∂x ∂x ∂y ∂y ∂z ∂z

Based on this operator splitting, we consider different Alternating Direction Implicit (ADI) schemes to discretize problem (6). First, we perform a uniform partition of the time interval I¯ = [0, T ] as 0 = t0 < t1 < . . . < tN −1 < tN = T, and denote τ := tn+1 − tn , ∀n = 0, . . . , N − 1. In the Douglas-Gunn scheme, we integrate the solution from time step tn to tn+1 in three substeps as follows: ⎧ τ τ ⎪ (1 + L1 )un+1/3 = τ f n+1/2 + (1 − L1 − τ L2 − τ L3 )un , ⎪ ⎪ 2 2 ⎪ ⎨ τ τ (1 + L2 )un+2/3 = un+1/3 + L2 un , (7) 2 2 ⎪ ⎪ ⎪ ⎪ ⎩ (1 + τ L3 )un+1 = un+2/3 + τ L3 un . 2 2 The variational formulation of scheme (7) is     ⎧  ∂un+1/3 ∂un+1/3 ∂v τ ∂un ∂v τ τ ⎪ n+1/3 ⎪ ⎪ α , + β , v = (un , v) − α , , v) + (u x ⎪ ⎪ 2 ∂x ∂x 2 ∂x 2 ∂x ∂x ⎪ ⎪ ⎪ ⎪     ⎪ n n ∂v n n ∂v ⎪ ∂u ∂u ∂u ∂u τ ⎪ ⎪ βx ,v − τ α , − τ βy ,v − τ α , − ⎪ ⎪ ⎪ 2 ∂x ∂y ∂y ∂y ∂z ∂z ⎪ ⎪     ⎪  ⎨ ∂un ∂un+2/3 ∂un+2/3 ∂v τ τ n+1/2 n+2/3 , v + τ (f α , + β , v , v), (u , v) + − τ β z y ⎪ ⎪ ∂z 2 ∂y ∂y 2 ∂y ⎪ ⎪ ⎪    ⎪ n ∂v n n+1 ∂v ⎪ ⎪ ∂u τ τ ∂u τ ∂u ⎪ = (un+1/3 , v) + ⎪ , , v , (un+1 , v) + , α + βy α ⎪ ⎪ 2 ∂y ∂y 2 ∂y 2 ∂z ∂z ⎪ ⎪ ⎪    ⎪ n+1 n ∂v n ⎪ ⎪ ∂u ∂u ∂u τ τ τ ⎪ + ⎩ βz , v = (un+2/3 , v) + α , + βz ,v , 2 ∂z 2 ∂z ∂z 2 ∂z

(8)

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where (·, ·) denotes the inner product of L2 (Ω). Finally, expressing problem (8) in matrix form we have

⎧ τ x x x ⎪ M (K + + G ) ⊗ M y ⊗ M z un+1/3 ⎪ ⎪ 2 ⎪ ⎪

⎪ τ ⎪ x x x ⎪ (K = M − + G ) ⊗ M y ⊗ M z un ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ − τ M x ⊗ (K y + Gy ) ⊗ M z un − τ M x ⊗ M y ⊗ (K z + Gz )un + τ F n+1/2 ⎪ ⎪ ⎪

⎨ x y τ M ⊗ M + (K y + Gy ) ⊗ M z un+2/3 (9) 2 ⎪ ⎪ ⎪ τ x y z n+1/3 x y y z n ⎪ ⎪ =M ⊗M ⊗M u + M ⊗ (K + G ) ⊗ M u , ⎪ ⎪ ⎪ ⎪

2 ⎪ τ ⎪ x y z z z ⎪ M ⊗ M ⊗ M + (K + G ) un+1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ = M x ⊗ M y ⊗ M z un+2/3 + M x ⊗ M y ⊗ τ (K z + Gz )un , 2 where M x,y,z , K x,y,z and Gx,y,z are the 1D mass, stiffness and advection matrices, respectively.

3

Isogeometric Residual Minimization Method

In our method, in every time step we solve the problem with identical left-handside: Find u ∈ U such that b (u, v) = l (v) ∀v ∈ V,     ∂u ∂v ∂u b (u, v) = (u, v) + τ /2 βi , v + αi , , ∂xi ∂xi ∂xi

(10) (11)

Here i ∈ {1, 2, 3}, so we have denoted here (x1 , x2 , x3 ) = (x, y, z), and i means that we are not using the Einstein summation here. The right-hand-side depends on the sub-step and the time integration scheme used. In the Douglas-Gunn time integration scheme, in the first, second and third sub-step the right-hand side is defined as:         ⎧ ∂w ∂w τ ∂w ∂v τ ∂w ∂v ⎪ ⎪ , , v , , v l (w, v) = (w, v) − α − β − τ α − τ β x y ⎪ ⎪ 2 ∂x ∂x 2 ∂x ∂y ∂y ∂y ⎪ ⎪ ⎪     ⎪ ⎪ ∂w ∂v ∂w ⎪ ⎪ , , v + τ (f n+1/2 , v), −τ α − τ βz ⎪ ⎨ ∂z ∂z ∂z     ⎪ ∂w τ ∂w ∂v τ ⎪ ⎪ , ,v , l (w, v) = (w, v) + α + βy ⎪ ⎪ 2 ∂y ∂y 2 ∂y ⎪ ⎪ ⎪     ⎪ ⎪ ∂w ∂v τ ∂w τ ⎪ ⎪ ⎩ l (w, v) = (w, v) + , ,v . α + βz 2

∂z ∂z

2

∂z

(12) In our advection-diffusion problem we seek the solution in space      2  ∂v 2 v +