COMPUTATIONAL FLUID AND SOLID MECHANICS

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COMPUTATIONAL FLUID AND SOLID MECHANICSProceedings First MIT Conference on Computational Fluid and Solid Mechanics June 12-15,2001

Editor: K.J. Bathe Massachusetts Institute of Technology, Cambridge, MA, USA

VOLUME 1

2001 ELSEVIERAmsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

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Preface

Mathematical modeling and numerical solution is today firmly established in science and engineering. Research conducted in almost all branches of scientific investigations and the design of systems in practically all disciplines of engineering can not be pursued effectively without, frequently, intensive analysis based on numerical computations. The world we live in has been classified by the human mind, for descriptive and analysis purposes, to consist of fluids and solids, continua and molecules; and the analyses of fluids and solids at the continuum and molecular scales have traditionally been pursued separately. Fundamentally, however, there are only molecules and particles for any material that interact on the microscopic and macroscopic scales. Therefore, to unify the analysis of physical systems and to reach a deeper understanding of the behavior of nature in scientific investigations, and of the behavior of designs in engineering endeavors, a new level of analysis is necessary. This new level of mathematical modeling and numerical solution does not merely involve the analysis of a single medium but must encompass the solution of multi-physics problems involving fluids, solids, and their interactions, involving multi-scale phenomena from the molecular to the macroscopic scales, and must include uncertainties in the given data and the solution results. Nature does not distinguish between fluids and solids and does not ever exactly repeat itself. This new level of analysis must also include, in engineering, the effective optimization of systems, and the modeling and analysis of complete life spans of engineering products, from design to fabrication, to possibly multiple repairs, to end of service. The objective of the M.I.T. Conferences ^ on Computational Fluid and Solid Mechanics is to bring together researchers and practitioners of mathematical modeling and numerical solution in order to focus on the current state of analysis of fluids, soUds, and multi-physics phenomena and

to lead towards the new level of mathematical modeling and numerical solution that we envisage. However, there is also a most valuable related objective indeed a "mission" - for the M.I.T. Conferences. When contemplating the future and carving a vision thereof, two needs stand clearly out. The first is the need to foster young researchers in computational mechanics, because they will revitaUze the field with new ideas and increased energy. The second need is to bring Industry and Academia together for a greater synthesis of efforts in research and developments. This mission expressed in 'To bring together Industry and Academia and To nurture the next generation in computational mechanics'' is of great importance in order to reach, already in the near future, the new level of mathematical modeling and numerical solution, and in order to provide an exciting research environment for the next generation in computational mechanics. We are very grateful for the support of the sponsors of the Conference, for providing the financial and intellectual support to attract speakers and bring together Industry and Academia. In the spirit of helping young researchers, fellowships have been awarded to about one hundred young researchers for travel, lodging and Conference expenses, and in addition. Conference fees have been waived for all students. The papers presented at the Conference and published in this book represent, in various areas, the state-of-the-art in the field. The papers have been largely attracted by the session organizers. We are very grateful for their efforts. Finally, we would like to thank Jean-Frangois Hiller, a student at M.I.T, for his help with the Conference, and also Elsevier Science, in particular James Milne, for the efforts and help provided to publish this book in excellent format and in due time for the Conference.K.J. BATHE, M.I.T.

^ A series of Conferences is planned.

Session Organizers

We would like to thank the Session Organizers for their help with the Conference. G. Astfalk, Hewlett-Packard Company, U.S.A. N. Bellomo, Politecnico di Torino, Italy Z. Bittnar, Prague Technical University, Czech Republic D. Boffi, University of Pavia, Italy S. Borgersen, SciMed, U.S.A. M. Borri, Politecnico di Milano, Italy M.A. Bradford, University of New South Wales, Australia M.L. Bucalem, University of Sao Paulo, Brazil J. Bull, The University of Newcastle upon Tyne, U.K. S.W. Chae, Korea University, South Korea D. Chapelle, INRIA, France C.N. Chen, National Cheng Kung University, Taiwan G. Cheng, Dalian University of Technology, PR. China H.Y. Choi, Hong-Ik University, South Korea K. Christensen, Hewlett-Packard Company, U.S.A. M.A. Christon, Sandia National Laboratories, U.S.A. R. Cosner, The Boeing Company, U.S.A. S. De, Massachusetts Institute of Technology., U.S.A. Y.C. Deng, General Motors, U.S.A. R.A. Dietrich, GKSS Forschungszentrum, Germany J. Dolbow, Duke University, U.S.A. E.H. Dowell, Duke University, U.S.A. R. Dreisbach, The Boeing Company, U.S.A. E.N. Dvorkin, SIDERCA, Argentina N. El-Abbasi, Massachusetts Institute of Technology, U.S.A. C. Felippa, University of Colorado, Boulder, U.S.A. D. Ferguson, The Boeing Company, U.S.A. D. M. Frangopol, University of Colorado, Boulder, U.S.A. L. Gastaldi, University of Pavia, Italy P. Gaudenzi, University of Rome, Italy A. Ghoniem, Massachusetts Institute of Technology, U.S.A. R. Glowinski, University of Houston, U.S.A. P. Gresho, Lawrence Livermore National Laboratory, U.S.A. N. Hadjiconstantinou, Massachusetts Institute of Technology, U.S.A. M. Hafez, University of California, Davis, U.S.A. K. Hall, Duke University, U.S.A. 0. Hassan, University of Wales, U.K. A. Ibrahimbegovic, ENS-Cachan, France S. Idelsohn, INTEC, Argentina A. Jameson, Stanford University, U.S.A. 1. Janajreh, Michelin, U.S.A. R.D. Kamm, Massachusetts Institute of Technology, U.S.A. S. Key, Sandia National Laboratories, U.S.A. W. Kirchhoff, Department of Energy, U.S.A. W.B. Kratzig, Ruhr-Universitat Bochum, Germany A. Krimotat, SC Solutions, Inc., U.S.A. C.S. Krishnamoorthy, Indian Institute of Technology, Madras, India (deceased) Y. Kuznetsov, University of Houston, U.S.A. L. Martinelli, Princeton University, U.S.A. H. Matthies, Technical University of Braunschweig, Germany S.A. Meguid, University of Toronto, Canada K. Meintjes, General Motors, U.S.A. C. Meyer, Columbia University, U.S.A. R. Ohayon, CNAM, France M. Papadrakakis, National Technical University of Athens, Greece K.C. Park, University of Colorado, Boulder, U.S.A. J. Periaux, Dassault Aviation, France O. Pironneau, Universite Pierre et Marie Curie, France E. Rank, Technical University of Munich, Germany A. Rezgui, Michelin, France C.Y Sa, General Motors, U.S.A. G. Schueller, University of Innsbruck, Austria T. Siegmund, Purdue University, U.S.A. J. Sladek, Slovak Academy of Sciences, Slovak Republic S. Sloan, University of Newcastle, Australia G. Steven, University of Sydney, Australia R. Sun, DaimlerChrysler, U.S.A. S. Sutton, Lawrence Livermore National Laboratory, U.S.A. B. Szabo, Washington University, St. Louis, U.S.A. J. Tedesco, University of Florida, U.S.A. T. Tezduyar, Rice University, U.S.A. B.H.V. Topping, Heriot-Watt University, U.K. F.J. Ulm, Massachusetts Institute of Technology, U.S.A. J.M. Vacherand, Michelin, France L. Wang, University of Hong Kong, Hong Kong X. Wang, Polytechnic University of New York, U.S.A. N. Weatherill, University of Wales, U.K. J. White, Massachusetts Institute of Technology, U.S.A. P. Wriggers, University of Hannover, Germany S. Xu, General Motors, U.S.A. T. Zohdi, University of Hannover, Germany

Fellowship Awardees

M. Al-Dojayli, University of Toronto, Canada B.N. Alemdar, Georgia Institute of Technology, U.S.A. M.A. Alves, Universidade do Porto, Portugal R. Angst, Technical University of Berlin, Germany D. Antoniak, Wroclaw University of Technology, Poland S. J. Antony, University of Surrey, U.K. A. Badeau, West Virginia University, U.S.A. W. Bao, The National University of Singapore, Singapore M. Bathe, Massachusetts Institute of Technology, U.S.A. A.C. Bauer, University of New York, Buffalo, U.S.A. C. Bisagni, Politecnico di Milano, Italy S. Butkewitsch, Federal University of Uberlandia, Brazil S. Cen, Tsinghua University, China G. Chaidron, CNAM, France M. Council, Chalmers University of Technology, Sweden A. Czekanski, University of Toronto, Canada C. E. Dalhuysen, Council for Scientific and Industrial Research, South Africa D. Dall'Acqua, Noetic Engineering Inc., Canada S. De, Massachusetts Institute of Technology, U.S.A. D. Demarco, SIDERCA, Argentina J. Dolbow, Duke University, U.S.A. J.E. Drews, Technische Universitat Braunschweig, Germany J.L. Drury, University of Michigan, U.S.A. C.A. Duarte, Altair Engineering, U.S.A. F. Dufour, CSIRO Exploration and Mining, Australia A. Ferent, INRIA, France M.A. Fernandez, INRIA, France Y. Fragakis, National Technical University of Athens, Greece A. Frangi, PoUtecnico di Milano, Italy T. Fujisawa, University of Tokyo, Japan J.R. Fernandez Garcia, Universidade de Santiago de Compostela, Spain J.F. Gerbeau, INRIA, France M. Gliick, Friedrich-Alexander University, Erlangen, Germany C. Gonzalez, Politecnica de Madrid, Spain K. Goto, University of Tokyo, Japan S. Govender, University of Natal, South Africa T. Gratsch, University of Kassel, Germany B. Gu, Massachusetts Institute of Technology, U.S.A. Y. T. Gu, National University of Singapore, Singapore S. Gupta, Indian Institute of Science, Bangalore, India M. Handrik, University of Zilina, Slovakia

L. Haubelt, Rice University, U.S.A. V. Havu, Helsinki University of Technology, Finland N. Impollonia, University of Messina, Italy R. lozzi. University of Rome, "La Sapienza", Italy H. Karaouni, Ecole Polytechnique, France R. Keck, University of Kaiserslautern, Germany C.W. Keierleber, University of Nebraska, Lincoln, U.S.A. K. Kolanek, Polish Academy of Sciences, Poland L. Ktibler, University of Erlangen-Niimberg, Erlangen, Germany D. Kuzmin, University of Dortmund, Germany N.D. Lagaros, National Technical University of Athens, Greece R. Garcia Lage, Instituto de Engenharia Mecanica, Portugal P.D. Ledger, University of Swansea, Wales, U.K. J. Li, Courant Institute, New York, U.S.A. J. Li, Massachusetts Institute of Technology, U.S.A. G. Limbert, University of Southampton, U.K. K. Liu, Polytechnic University of New York, U.S.A. M.B. Liu, National University of Singapore, Singapore J. Long, University of New York, Buffalo, U.S.A. I. Lubowiecka, Technical University of Gdansk, Poland A.A. Mailybaev, Moscow State Lomonosov University, Russia M. Malinen, Helsinki University of Technology, Finland E.A. Malsch, Columbia University, U.S.A. Y. Marzouk, Massachusetts Institute of Technology, U.S.A. M. Meyer, Technische Universitat Braunschweig, Germany B. Miller, Rzeszow University of Technology, Poland D.P. Mok, University of Stuttgart, Germany G. Morgenthal, University of Cambridge, U.K. M. Moubachir, Laboratoire Central des Fonts et Chaussees, France S.K. Nadarajah, Stanford University, U.S.A. J. Nemecek, Czech Technical University, Prague, Czech Republic T.S. Ng, Imperial College, U.K. N. Nuno, Universita di Parma, Italy M. Palacz, Polish Academy of Sciences, Poland H. Pan, Nanyang Technological University, Singapore G. Pedro, University of Victoria, Canada X. Peng, Northwestern University, U.S.A. R.C. Penmetsa, Wright State University, U.S.A. R. Premkumar, Indian Institute of Technology, Madras, India

Fellowship Awardees C. Prud'homme, Massachusetts Institute of Technology, U.S.A. K. Roe, Purdue University, U.S.A. S. Rugonyi, Massachusetts Institute of Technology, U.S.A. M.L. Munoz Ruiz, Universidad de Malaga, Spain N. Ruse, University of Stuttgart, Germany S. Sarkar, Indian Institute of Science, Bangalore, India C.A. Schenk, University of Innsbruck, Austria S. Shankaran, Stanford University, U.S.A. D. Slinchenko, University of Natal, South Africa D.O. Snyder, Utah State University, U.S.A. K.A. S0rensen, University of Swansea, Wales, U.K. A. Takahashi, University of Tokyo, Japan S. Ubal, Universidad Nacional del Litoral, Argentina U.V. Unnithan, Indian Institute of Technology, Chennai, India F. Valentin, National Laboratory of Brazil for Scientific Computing, Brazil R. Vodicka, Technical University of Kosice, Slovakia V.M. Wasekar, University of Cincinnati, U.S.A. S. Wijesinghe, Massachusetts Institute of Technology, U.S.A. M.W. Wilson, Georgia Institute of Technology, U.S.A. W. Witkowski, Technical University of Gdansk, Poland A.M. Yommi, Universidad Nacional del Litoral, Santa Fe, Argentina Y. Zhang, Dalian University of Technology, China K. Zhao, General Motors Corp., U.S.A.

Sponsors

The following organizations are gratefully acknowledged for their generous sponsorship of the Conference:

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The mission of the Conference: To bring together Industry and Academia and To nurture the next generation in computational mechanics

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Contents Volume 1

Preface Session Organizers Fellowship Awardees Sponsors

v vi vii ix

Plenary PapersAlum, N., Ye, W., Ramaswamy, D., Wang, X., White, J., Efficient simulation techniques for complicated micromachined devices Brezzi, R, Subgrid scales, augmented problems, and stabilizations Dreisbach, R.L., Cosner, R.R., Trends in the design analysis of aerospace vehicles Ingham, T.J., Issues in the seismic analysis of bridges Lions, J.L., Virtual control algorithms Makinouchi, A., Teodosiu, C, Numerical methods for prediction and evaluation of geometrical defects in sheet metal forming McQueen, DM., Peskin, C.S., Zhu, L., The Immersed Boundary Method for incompressible fluid-structure interaction Ottolini, R.M., Rohde, S.M., GMs journey to math: the virtual vehicle 2 8 11 16 20 21 26 31

Solids & StructuresAntony, SJ., Ghadiri, M., Shear resistance of granular media containing large inclusions: DEM simulations Araya, R., Le Tallec, R, Hierarchical a posteriori error estimates for heterogeneous incompressible elasticity Augusti, G., Mariano, P.M., Stazi, F.L., Localization phenomena in randomly microcracked bodies Austrell, P.-E., Olsson, A.K., Jonsson, M., A method to analyse the nonlinear dynamic behaviour of rubber components using standard FE codes Baar, Y., Hanskotter, U., Kintzel, O., Schwab, C, Simulation of large deformations in shell structures by the p-version of the finite element method Bardenhagen, S.G., Byutner, O., Bedrov, D., Smith, G.D., Simulation of frictional contact in three-dimensions using the Material Point Method 36 39 43 47 50 54

xii Bauchau, O.A., Bottasso, C.L., On the modeling of shells in multibody dynamics

Contents Volume 1

58 61 65 68 72 74 78 82 85 88 91 95 99 104 107 HI 114

Bay lot, J.T., Papados, P.P., Fragment impact pattern effect on momentum transferred to concrete targets Becache, E., Joly, P., Scarella, G., A fictitious domain method for unilateral contact problems in non-destructive testing Belforte, G., Franco, W., Sorli, M., Time-frequency pneumatic transmission line analysis Bohm, R, Duda, A., Wille, R., On some relevant technical aspects of tire modelling in general Borri, M., Bottasso, C.L., Trainelli, L, An index reduction method in non-holonomic system dynamics Boucard, PA., Application of the LATIN method to the calculation of response surfaces Brunet, M., Morestin, R, Walter, H., A unified failure approach for sheet-metals formability analysis Bull, J. W., Underground explosions: their effect on runway fatigue life and how to mitigate their effects Cacciola, P., Impollonia, N., Muscolino, G., Stochastic seismic analysis of R-FBI isolation system Carter, J.P, Wang, C.X., Geometric softening in geotechnical problems Cen, S., Long, Y., Yao, Z., A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates Chakraborty, S., Brown, D.A., Simulating static and dynamic lateral load testing of bridge foundations using nonlinear finite element models . . Chapelle, D., Rerent, A., Asymptotic analysis of the coupled model shells-3D solids Chapelle, D., Oliveira, D.L., Bucalem, M.L., Some experiments with the MITC9 element for Naghdis shell model Chen, X., Hisada, T, Frictional contact analysis of articular surfaces Choi, H.Y., Lee, S.H., Lee, LH., Haug, E., Finite element modeling of human head-neck complex for crashworthiness simulation Chun, B.K., Jinn, J.T., Lee, J.K., A constitutive model associated with permanent softening under multiple bend-unbending cycles in sheet metal forming and springback analysis Crouch, R.S., Remandez-Vega, J., Non-linear wave propagation in softening media through use of the scaled boundary finite element method . . . . Czekanski, A., Meguid, S.A., Time integration for dynamic contact problems: generalized-of scheme Dai, L., Semi-analytical solution to a mechanical system with friction Davi, G., Milazzo, A., A novel displacement variational boundary formulation David, S.A., Rosdrio, J.M., Investigation about nonlinearities in a robot with elastic members

120 125 128 132 134 137

Contents Volume 1 De, S., Kim, /., Srinivasan, M.A., Virtual surgery simulation using a collocation-based method of finite spheres Deeks,AJ.,WollJ.R, Efficient analysis of stress singularities using the scaled boundary finite-element method Djoudi, M.S., Bahai, K, Relocation of natural frequencies using physical parameter modifications Duddeck, F.M.E., Fourier transformed boundary integral equations for transient problems of elasticity and thermo-elasticity Dufour, E, Moresi, L., Muhlhaus, H., A fluid-like formulation for viscoelastic geological modeling stabilized for the elastic limit Dvorkin, E.N., Demarco, D., An Eulerian formulation for modehng stationary finite strain elasto-plastic metal forming processes Dvorkin, E.N., Toscano, R.G., Effects of internal/external pressure on the global buckling of pipelines El-AbbasU N., Bathe, K.J., On a new segment-to-segment contact algorithm El-Abbasi, N., Meguid, S.A., Modehng 2D contact surfaces using cubic splines Eelippa, C.A., Optimal triangular membrane elements with drilling freedoms FemdndeZ'Garcia, J.R., Sofonea, M., Viaho, J.M., Numerical analysis of a sliding viscoelastic contact problem with wear Frangi, A., Novati, G., Springhetti, R., Rovizzi, M., Numerical fracture mechanics in 3D by the symmetric boundary element method Galbraith, P.C., Thomas, D.N., Finn, M.J., Spring back of automotive assembhes Gambarotta, L., Massabd, R., Morbiducci, R., Constitutive and finite element modehng of human scalp skin for the simulation of cutaneous surgical procedures Gebbeken, N., Greulich, S., Pietzsch, A., Landmann, F, Material modelling in the dynamic regime: a discussion Gendron, G., Fortin, M., Goulet, R, Error estimation and edge-based mesh adaptation for solid mechanics problems Gharaibeh, E.S., McCartney, J.S., Erangopol, D.M., Reliability-based importance assessment of structural members Ghiocel, D.M., Mao, H., ProbabiUstic life prediction for mechanical components including HCF/LCF/creep interactions Giner, E., Fuenmayor, J., Besa, A., Tur, M., A discretization error estimator associated with the energy domain integral method in linear elastic fracture mechanics Gonzalez, C, Llorca, J., Micromechanical analysis of two-phase materials including plasticity and damage Goto, K., Yagawa, G, Miyamura, T, Accurate analysis of shell structures by a virtually meshless method Guilkey, J.E., Weiss, J.A., An implicit time integration strategy for use with the material point method Gupta, S., Manohar, C.S., Computation of reliabihty of stochastic structural dynamic systems using stochastic FEM and adaptive importance sampling with non-Gaussian sampling functions

xiii

140 142 146 150 153 156 159 165 168 171 173 177 180 184 186 192 198 201

206 211 214 216

220

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Contents Volume 1

Guz, LA., Soutis, C., Accuracy of analytical approaches to compressive fracture of layered solids under large deformations Hadjesfandiari, A.R., Dargush, G.F., Computational elasticity based on boundary eigensolutions Haldar, A., Lee, 5.K, Huh, / , Stochastic response of nonlinear structures Han, S., Xiao, M., A continuum mechanics based model for simulation of radiation wave from a crack Handrik, M., Kompis, V., Novak, P., Large strain, large rotation boundary integral multi-domain formulation using the Trefftz polynomial functions . . Hamau, M., Schweizerhof, K., About linear and quadratic 'Solid-Shell elements at large deformations Hartmann, U., Kruggel, R, Hierl, T., Lonsdale, G., Kloppel, R., Skull mechanic simulations with the prototype SimBio environment Havu,V,Hakula,H, An analysis of a bilinear reduced strain element in the case of an elliptic shell in a membrane dominated state of deformation Ibrahimbegovic, A., Recent developments in nonlinear analysis of shell problem and its finite element solution Ingham, T.J., Modeling of friction pendulum bearings for the seismic analysis of bridges lozzi, R., Gaudenzi, P., MITC finite elements for adaptive laminated composite shells Janajreh, L, Rezgui, A., Estenne, V., Tire tread pattern analysis for ultimate performance of hydroplaning Kanapady, R., Tamma, K.K., Design and framework of reduced instruction set codes for scalable computations for nonlinear structural dynamics Kang,M.-S.,Youn,S,-K., Dof splitting p-adaptive meshless method Kapinski, S., Modelling of friction in metal-forming processes Kashtalyan, M., Soutis, C., Modelling of intra- and interlaminar fracture in composite laminates loaded in tension Kawka, M., Bathe, K.J., Implicit integration for the solution of metal forming processes Kim, H.S., Tim, HJ., Kim, C.B., Computation of stress time history using FEM and flexible multibody dynamics Kong, J.S., Akgul, K, Frangopol, DM., Xi, Y., Probabilistic models for predicting the failure time of deteriorating structural systems Koteras, J.R., Gullemd, A.S., Porter, V.L., Scherzinger, W.M., Brown, K.H., PRESTO: impact dynamics with scalable contact using the SIERRA framework Kratzig,W.B.,Jun,D., Layered higher order concepts for D-adaptivity in shell theory Krishnamoorthy, C.S.,Annamalai, V, Vmu Unnithan, U., Superelement based adaptive finite element analysis for linear and nonlinear continua under distributed computing environment KUbler, L, Eberhard, P., Multibody system/finite element contact simulation with an energy-based switching criterion

224 227 232 235 238 240 243

247 251 255 259 264

268 272 276 279 283 287 290 294 297

302 306

Contents Volume 1 Laukkanen, A., Consistency of damage mechanics modeling of ductile material failure in reference to attribute transferability . . . LeBeau, K.H., Wadia-Fascetti, SJ., A model of deteriorating bridge structures Leitdo, VM.A., Analysis of 2-D elastostatic problems using radial basis functions Limbert, G., Taylor, M , An explicit three-dimensional finite element model of an incompressible transversely isotropic hyperelastic material: application to the study of the human anterior cruciate ligament Liu, G.R., Liu, M.B., Lam, K.Y., Zong, Z., Simulation of the explosive detonation process using SPH methodology Liu, G.R., Tu, Z.H., MFree2D: an adaptive stress analysis package based on mesh-free technology Lovadina, C, Energy estimates for linear elastic shells Lubowiecka, L, Chroscielewski, J., On the finite element analysis of flexible shell structures undergoing large overall motion Luo, A.C.J., A numerical investigation of chaotic motions in the stochastic layer of a parametrically excited, buckled beam . . Lyamin, A.V., Sloan, S.W., Limit analysis using finite elements and nonlinear programming Malinen, M., Pitkdranta, J., On degenerated shell finite elements and classical shell models Martikainen, J., Mdkinen, R.A.E., Rossi, T, Toivanen, J., A fictitious domain method for linear elasticity problems Massin, R, Al Mikdad, M., Thick shell elements with large displacements and rotations Mathisen, K.M., Tiller, L, Okstad, K.M., Adaptive ultimate load analysis of shell structures Matsumoto, T, Tanaka, M., Okayama, S., Boundary stress calculation for two-dimensional thermoelastic problems using displacement gradient boundary integral identity Mitchell, J.A., Gullerud, A.S., Scherzinger, W.M., Koteras, R., Porter, V.L., Adagio: non-hnear quasi-static structural response using the SIERRA framework Toukourou, M.M., Gakwaya, A., Yazdani, A., An object-oriented finite element implementation of large deformation frictional contact problems and applications Nemecek, J., Patzdk, B., Bittnar, Z., Parallel simulation of reinforced concrete column on a PC cluster Noguchi, H., Kawashima, T, Application of ALE-EFGM to analysis of membrane with sliding cable Nuno, N., Avanzolini, G., Modeling residual stresses at the stem-cement interface of an idealized cemented hip stem Obrecht, H., Briinig, M., Berger, S., Ricci, S., Nonlocal numerical modelling of the deformation and failure behavior of hydrostatic-stress-dependent ductile metals Olson, L, Throne, R., Estimation of tool/chip interface temperatures for on-line tool monitoring: an inverse problem approach .

xv

310 314 317

319 323 327 330 332 336 338 342 346 351 355

359 361

365 369 372 374

378 381

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Contents Volume 1

Pacoste, C, Eriksson, A., Instability problems in shell structures: some computational aspects Palacz, M, Krawczuk, M , Genetic algorithm for crack detection in beams Papadrakakis, M., Fragakis, K, A geometric-algebraic method for semi-definite problems in structural mechanics PatzdK B., RypU D., Bittnar, Z , Parallel algorithm for explicit dynamics with support for nonlocal constitutive models Pawlikowski, M., Skalski, K., Bossak, M , Piszczatowski, S,, Rheological effects and bone remodelling phenomenon in the hip joint implantation PeiLu,X., Computational synthesis on vehicle rollover protection Peng,X., Cao,J., Sensitivity study on material characterization of textile composites Penmetsa, R.C., Grandhi, R.V, Uncertainty analysis of large-scale structures using high fidelity models Perez-Gavildn, J.J., Aliabadi, M.H., A note on symmetric Galerkin BEM for multi-connected bodies Pradhan, S.C., Lam, K.Y., Ng,TY., Reddy, J.N., Vibration suppression of laminated composite plates using magnetostrictive inserts Pradlwarter, H.J., Schueller, G.I., PDFs of the stochastic non-linear response of MDOF-systems by local statistical linearization Proppe, C, Schueller, G.L, Effects of uncertainties on lifetime prediction of aircraft components Randolph, M.F., Computational and physical modelling of penetration resistance Rank, E., Duster, A., h- versus p-version finite element analysis for J2 flow theory Roe, K., Siegmund, T, Simulation of interface fatigue crack growth via a fracture process zone model Rosson, B.T, Keierleber, CM, Improved direct time integration method for impact analysis Rucker, M., Rank, E., The /7-version PEA: high performance with and without parallelization Ruiz, G., Pandolfi, A., Ortiz, M., Finite-element simulation of complex dynamic fracture processes in concrete Sdez, A., Dominguez, J., General traction BE formulation and implementation for 2-D anisotropic media Sanchez-Hubert, J., Boundary and internal layers in thin elastic shells Sanchez Palencia, E., General properties of thin shell solutions, propagation of singularities and their numerical incidence Savoia, M., Reliability analysis of structures against buckling according to fuzzy number theory Scheider, I., Simulation of cup-cone fracture in round bars using the cohesive zone model Schenk, C.A., Bergman, L.A., Response of a continuous system with stochastically varying surface roughness to a moving load

385 389 393 396 399 403 406 410 413 416 420 425 429 431 435 438 441 445 449 452 454 456 460 463

Contents Volume 1 Schroder, J., Miehe, C, Elastic stability problems in micro-macro transitions Semedo Gargdo, J.E., Mota Soares, CM., Mota Soares, C.A., Reddy, J.N., Modeling of adaptive composite structures using a layerwise theory Sladek, /., Sladek, V, Van Keer, R., The local boundary integral equation and its meshless implementation for elastodynamic problems Slinchenko, D., Verijenko, VE., Structural analysis of composite lattice structures on the basis of smearing stiffness Soric, J., Tonkovic, Z., Computer techniques for simulation of nonisothermal elastoplastic shell responses Stander, N., The successive response surface method applied to sheet-metal forming Szabo, BA.,Actis, R.L, Hierarchic modeling strategies for the control of the errors of idealization in FEA Tahar, B., Crouch, R.S., Techniques to ensure convergence of the closest point projection method in pressure dependent elasto-plasticity models Takahashi, A., Yagawa, G., Molecular dynamics calculation of 2 billion atoms on massively parallel processors Tedesco, J.W., Bloomquist, D., Latta, T.E., Impact stresses in A-Jacks concrete armor units Thompson, L.L., Thangavelu, S.R., A stabilized MITC finite element for accurate wave response in Reissner-Mindlin plates Tijssens, M.G.A., van der Giessen, E., Sluys, L.J., Modeling quasi-static fracture of heterogeneous materials with the cohesive surface methodology Tsukrov, I., Novak, J., Application of numerical conformal mapping to micromechanical modeling of elastic solids with holes of irregular shapes Tyler-Street, M., Francis, N., Davis, R., Kapp, J., Impact simulation of structural adhesive joints Vermeer, P.A., Ruse, N., On the stability of the tunnel excavation front Verruijt, A., Numerical aspects of analytical solutions of elastodynamic problems Vidrascu, M., Delingette, H., Ayache, N., Finite element modeling for surgery simulation Vlachoutsis, S., Clinckemaillie, J., Distributed memory parallel computing for crash and stamp simulations Vodicka, R., The first-kind and the second-kind boundary integral equation systems for some kinds of contact problems with friction Wagner, W., Klinkel, S., Gruttmann, E, On the computation of finite strain plasticity problems with a 3D-shell element Wang, J.G., Liu, G.R., Radial point interpolation method for no-yielding surface models Wang, X., Bathe, K.J., Walczak, J., A stress integration algorithm for /s-dependent elasto-plasticity models Whittle, AJ., Hsieh, Y.M., Pinto, E, Chatzigiannelis, ., Numerical and analytical modeling of ground deformations due to shallow tunneling in soft soils

xvii

468 471 473 475 478 481 486

490 496 499 502 509

513 517 521 524 527 530

533 536 538 542 546

xviii

Contents Volume 1

Witkowski, W, Lubowiecka, /., Identification of chaotic responses in a stable Duffing system by artificial neural network Yang, C., Soh, A. -K., Special membrane elements with internal defects Zarka, 7., Kamouni, //., Fatigue analysis during one-parametered loadings Zdunek, A., Non-linear stability analysis of stiffened shells using solid elements and the p-version FE-method Zhang, K, Lin, J., Random vibration of structures under multi-support seismic excitations Zhao, K., On simulation of a forming process to minimize springback Zhou, X., Tamma, K.K., Sha, D., Linear multi-step and optimal dissipative single-step algorithms for structural dynamics Zhu, P., Abe, M, Fujino, K, A 3D contact-friction model for pounding at bridges during earthquakes Zohdi, T.L, Wriggers, P., Computational testing of microheterogeneous materials

550 554 559 562 566 568 571 575 579

Optimization & DesignAl-Dojayli, M., Meguid, S.A., Shape optimization of frictional contact problems using genetic algorithm Bartoli, G., Borri, C, Facchini, L, Paiar, F, Simulation of non-gaussian wind pressures and estimation of design loads Bisagni, C, Optimization of helicopter subfloor components under crashworthiness requirements Bull,J.W., Some results from the Self-Designing Structures research programme Butkewitsch, S., On the use of 'meta-models to account for multidisciplinarity and uncertainty in design analysis and optimization Cardona, A., Design of cams using a general purpose mechanism analysis program Cheng, G., Guo, X., On singular topologies and related optimization algorithm Connell, M., Tullberg, O., Kettil, P, Wiberg, N.-E., Interactive design and investigation of physical bridges using virtual models Consolazio, G.R., Chung, J.H., Gurley, K.R., Design of an inertial safety barrier using explicit finite element simulation DalVAcqua, D., Lipsett, A.W., Faulkner, M.G, Kaiser, T.M.Y, An efficient thermomechanical modeling strategy for progressing cavity pumps and positive displacement motors Doxsee Jr, L.E., Using Pro/MECHANICA for non-linear problems in engineering design Dreisbach, R.L, Peak, R.S., Enhancing engineering design and analysis interoperability. Part 3: Steps toward multi-functional optimization . . Ghiocel, DM., Stochastic process/field models for turbomachinery applications 584 588 591 595 599 603 606 608 612 616 620 624 628

Contents Volume 1 Gu, Z, Zhao, G., Chen, Z, Optimum design and sensitivity analysis of piezoelectric trusses Hagiwara, L, Shi, Q.Z., Vehicle crashworthiness design using a most probable optimal design method Harte, R., Montag, U., Computer simulations and crack-damage evaluation for the durability design of the world-largest cooling tower shell at Niederaussem power station Hartmann, D., Baitsch, M., Weber, H., Structural optimization in consideration of stochastic phenomena - a new wave in engineering Hollowell, W.T., Summers, S.M., NHTSAs supporting role in the partnership for a new generation of vehicles Ivdnyi, P., Topping, B.H.V., Muylle, J., Towards a CAD design of cable-membrane structures on parallel platforms James, R.J., Zhang, L, Schaaf, DM., Wemcke, G.A., The effect of hydrodynamic loading on the structural reliability of culvert valves in lock systems Kolanek, K., Stocki, R., Jendo, S., Kleiber, M., An efficiency of numerical algorithms for discrete reliability-based structural optimization Krishnamoorthy, C.S., Genetic algorithms and high performance computing for engineering design optimization Launis, S.S., Keskinen, E.K., Cotsaftis, M., Dynamics of wearing contact in groundwood manufacturing system Liu, S., Lian, Z , Zheng, X, Design optimization of materials with microstructure Liu, C, Wang, T.-L., Shahawy, M., Load lateral distribution for multigirder bridges Maleki, S., Effects of diaphragms on seismic response of skewed bridges Matsuho, A.S., Frangopol, D.M., Applications of artificial-life techniques to reliability engineering Maute, K., Nikbay, M., Farhat, C, HPC for the optimization of aeroelastic systems Miller, B., Ziemiahski, L., Updating of a plane frame using neural networks Ogawa, Y., Ochiai, T, Kawahara, M., Shape optimization problem based on optimal control theory by using speed method Papadrakakis, M., Lagaros, N.D., Reliability based optimization using neural networks Papadrakakis, M., Lagaros, N.D., Fragakis, Y., Parallel computational strategies for structural optimization Peak, R.S., Wilson, MM, Enhancing engineering design and analysis interoperability. Part 2: A high diversity example Peri, D., Campana, E.F, Di Mascio, A., Development of CFD-based design optimization architecture Peterson, DM., The functional virtual prototype: an innovation framework for a zero prototype design process Prasad Varma Thampan, C.K., Krishnamoorthy, C.S., An HPC model for GA methodologies applied to reliability-based structural optimization

xix

633 637

641 645 649 652 655 660 663 668 672 676 681 685 688 692 696 698 701 704 708 711 714

XX

Contents Volume 1

Rovas, D.V, Leurent, T, Prud'homme, C , Patera, A.T., Reduced-basis output bound methods for heat transfer problems Schramm, U., Multi-discipUnary optimization for NVH and crashworthiness Sedaghati, R., Tabarrok, B., Suleman, A., Optimum design of frame structures undergoing large deflections against system instability Senecal, PK., Reitz, R.D., CFD modeling applied to internal combustion engine optimization and design Shan, C, Difficulties and characteristics of structural topology optimization Shankaran, 5., Jameson, A., Analysis and design of two-dimensional sails Sheikh, S.R., Sun, M., Hamdani, H., Existence of a lift plateau for airfoils pitching at rapid pitching rates Stander, N., Burger, M., Shape optimization for crashworthiness featuring adaptive mesh topology Steven, G.P, Proos, K., Xie, Y.M., Multi-criteria evolutionary structural optimization involving inertia Wilson, MM, Peak, R.S., Fulton, R.E., Enhancing engineering design and analysis interoperability. Part 1: Constrained objects Wolfe, R.W,Heninger,R., Retrofit design and strategy of the San Francisco-Oakland Bay Bridge continuous truss spans support towers based on ADINA Wu, J., Zhang, R.R., Radons, S., Vibration transmissibility of printed circuit boards by calibrated PEA modeling

718 721 725 729 733 737 739 743 747 750

755 758

Plenary Papers

Efficient simulation techniques for complicated micromachined devicesN. Alu^u^ W. Ye^ D. Ramaswamy^ X. Wang^ J. White'='*^ Department of General Engineering, University of Illinois, Urbana, IL 61801-2996, USA ^Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA ^ Department of Electrical Engineering and Computer Science, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract In this short paper, we briefly describe techniques currently used for simulating micromachined devices. We first survey the fast 3-D solvers that make possible fluid and field analysis of entire micromachined devices and then describe efficient techniques for coupled-domain simulation. We describe the matrix-implicit multilevel-Newton method for coupling solvers which use different techniques, and we describe a mixed-regime approach to improve the individual solver's efficiencies. Several micromachined device examples are used to demonstrate these recently developed methods. Keywords: M E M S ; Fast Stokes; CAD; Pre-corrected FFT; Simulation; Mixed regime

1. Introduction In this short paper, we briefly describe techniques currently used for simulating micromachined devices. We first survey the recently developed fast 3-D solvers that make possible the fluid and field analysis of entire micromachined devices. Then, we discuss the recently developed techniques for efficient coupled domain and mixed regime analysis, as they have made it possible to efficiently simulate devices whose operation involves several physical domains. In each section, we present computational results on real micromachined devices both to make clear the problem scale and to demonstrate the efficiency of these new techniques.

2. Fast 3-D solvers The exterior fluid and electrostatic force on a surfacemicromachined device can, in principle, be computed using finite-difference or finite-element methods. Such methods are becoming less popular, primarily due to the development of fast 3-D solvers which are much more efficient in this setting. In particular, for surface-micromachined * Corresponding author. E-mail: white@rle-vlsi.mit.edu 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

devices: (1) exterior forces need only be evaluated on poly silicon surfaces, (2) the geometries are innately 3-D and extremely complicated, (3) the exterior fields usually satisfy linear space-invariant partial differential equations. Since forces are not needed in the volume of the exterior, only on the surface, the exterior volume-filling grid for finite-element and finite difference methods seems inefficient. In addition, the geometrically complicated nature of micro-machined devices makes generating such an exterior volume grid difficult. The electrostatic problem is linear and space invariant, and so the Laplace's equation that describes the exterior electrostatics can be replaced with an integral equation which relates the surface potentials to the surface normal electric fields. In many cases, the fluid forces are reasonably well described by the linear Stoke's equation, and so an integral formulation involving only surface quantities can be used to determine fluid traction forces. The electrostatic potential and the fluid velocity, assuming Stoke's flow, both satisfy an integral equation over the poly silicon surface given by Green's theorem: u(x) / G{x,x)^ \ ^ 9n M(x)dfl, (1)

9n

N. Alum et al. /First MIT Conference on Computational Fluid and Solid Mechanics where u is either the electrostatic potential or the fluid velocity, ;c is a point on the surface, and d/dn is the derivative in the direction normal to the polysilicon surface. Discretization of the above integral equation leads to a dense system of equations which becomes prohibitively expensive to form and solve for complicated problems. To see this, consider the electrostatics problem of determining the surface charge given the potential on conductors. A simple discretization for the electrostatics problem is to divide the polysilicon surfaces into n flat panels over which the charge density is assumed constant. A system of equations for the panel charges is then derived by insisting that the correct potential be generated at a set of n test, or collocation, points. The discretized system is then Pq = ^ (2) Short-range stiiiimed direct!J

Fig. 1. A cluster of collocation points separated from a cluster of panels. products [4,5]. Perhaps the first practical use of such methods combined the fast multipole algorithms for charged particle computations with the above simple discretization scheme to compute 3-D capacitance and electrostatic forces [6]. Higher-order elements and improved efficiency for higher accuracy have been the recent developments [8,10]. The many different physical domains involved in micromachined devices has focussed attention on fast techniques which are Green's function independent, such as the precorrect-FFT schemes [3,9]. 2.1. Example fluid simulation As an example of using a fast solver, consider determining the quality factor of a comb-drive resonator packaged in air. To compute the quality factor, it is necessary to determine the drag force on the comb. The small spatial scale of micromachined combs implies that flow in these devices typically have very low Reynolds numbers, and therefore convection can often be ignored. In addition, fluid compression can be ignored for devices which use lateral actuation, like many of the comb-drive based structures fabricated using micromachining. The result of these two simpUfications is that fluid damping forces on laterally actuated microdevices can be accurately analyzed by solving the incompressible Stokes equation, rather than by solving the compressible Navier-Stokes equation. That the fluid can be treated as Stokes flow, and that the quantity of interest is the surface traction force, makes it possible to use a surface integral formulation to compute comb drag [11]. Then, the methods described above can be used to rapidly solve a discretization of the integral equation [12,13]. In Fig. 2, the discretization of a comb is shown. Notice that only the surface is discretized, yet still the number of unknowns in the system exceeds 50,000. An accelerated Stoke's flow solver completed the simulation in under 20 min, direct methods would have taken weeks and required over 16 gigabytes of memory. The simulated traction force in the motion direction is shown in Fig. 3. Note the surprisingly high contribution to the force from the structure sides. It should be noted that the quality factor computed from the numerical drag force analysis matched measure quahty factor for this structure to better than 10% [14].

where q is the n-length vector of panel charges, ^ is the w-length vector of known collocation point potentials. Since the Green's function for electrostatics is the reciprocal of the separation distance between x and x\

'' = f

panel.

4n.!.

X^

' ^'

(3>

where xt is the iih collocation point. Since the integral in (3) is nonzero for every panel-collocation-point pair, every entry in P is nonzero. If direct factorization is used to solve (2), then the memory required to store the dense matrix will grow like n^ and the matrix solve time will increase like n^. If instead, a preconditioned Krylov-subspace method like GMRES [1] is used to solve (2), then it is possible to reduce the solve time to order n^, but the memory requirement will not decrease. In order to develop algorithms that use memory and time that grows more slowly with problem size, it is essential not to form the matrix explicitly. Instead, one can exploit the fact that Krylov-subspace methods for solving systems of equations only require matrix-vector products and not an explicit representation of the matrix. For example, note that for P in (2), computing Pq is equivalent to computing n potentials due to n charged panels and this can be accomplished approximately in nearly order n operations [2,3]. To see how to perform such a reduction in cost, consider Fig. 1. The short-range interaction between close-by panels must be computed directly, but the interaction between the cluster of panels and distant panels can be approximated. In particular, as Fig. 1 shows, the distant interaction can be computed by summing the clustered panel charges into a single multipole expansion (denoted by M in the figure), and then the multipole expansion can be used to evaluate distant potentials. Several researchers simultaneously observed the powerful combination of integral equation approaches, Krylovsubspace matrix solution algorithms, and fast matrix-vector

N. Alum et al. /First MIT Conference on Computational Fluid and Solid Mechanics

2.5

2.5

Fig. 2. A discretized comb drive resonator over a substrate.

R

Fx

1

-2351.96 -4937.22 -7522.49 -10107.7 -12693 ^ -15278.3 17863.5 20448.8 23034.1 II -25619.3 28204.6 -30789.8 -33375.1 -35960.4 -38545.6

' ~ ^

E-05

0.00015

0.0001

5E-05

Fig. 3. Drag force distribution on the resonator, bottom (substrate-side) view. 3. Coupled-domain mixed-regime simulation Self-consistent electromechanical analysis of micromachined polysilicon devices typically involves determining mechanical displacements which balance elastic forces in the polysilicon with electrostatic pressure forces on polysilicon surface. The technique of choice for determining elastic forces in the polysilicon is to use finite-element methods to generate a nonlinear system equations of the form Fiu)P{u,q)=0 (4)

where w is a vector of finite-element node displacements, F relates node displacements to stresses, and P is the force produced by the vector representing the discretized surface charge q. Note that as the structure deforms, the pressure changes direction, so P is also a function of u. One can

N. Aluru et al. /First MIT Conference on Computational Fluid and Solid Mechanics view this mechanical analysis as a 'black box' which takes an input, q, and produces an output u as in HMiq) (5)150

200 h

In order to determine the charge density on the polysilicon surface due to a set of appHed voltages, one can use a fast solver, as described above. One can view the electrostatic analysis as a 'black box' which takes, as input, geometric displacements, w, and produces, as output, a vector of discretized surface charges, ^, as in q=HE{U)

100

(6)

Self-consistent analysis is then to find a u and q which satisfies both (5) and (6). 3.1. Multilevel-Newton -50 h A simple relaxation approach to determining a self-consistent solution to (5) and (6) is to successively use (5) to update displacements and then using (6) to update charge. Applying (5) implies solving the nonlinear equation, (4), typically using Newton's method [15]. Although the relaxation method is simple, it often does not converge. Instead, one can apply Newton's method to the system of equations

-100-50 0 50

Fig. 4. Comb drive accelerometer. tion. Computing Huiq + oid\) means using an inner loop Newton method to solve (4), which is expensive, though improvements can be made [19]. An important advantage of matrix-free multilevel-Newton methods is that it is not necessary to modify either the mechanical or electrostatic analysis programs. 3.2. Mixed regime simulation

q u

HE(U)

HM{q)_

=

0 0

(7)

in which case the updates to charge and displacement are given by solving

/L

dHE\ _du I Aq Au\-HEU HAA

(8)

^q

The above method is referred to as a multi-level Newton method [16,17], because forming the right-hand side in (8) involves using an inner Newton's method to apply HM. In order to solve (8), one can apply a Krylov-subspace iterative method such as GMRES. The important aspect of GMRES is that an explicit representation of the matrix is not required, only the ability to perform matrix-vector products. As is clear from examining (8), to compute these products one need only compute (dHM/dq)Aq and (dHE/du)Au. These products can be approximated by finite differences as in ^HM ^ dq ^ Huiq+aAq) a Huiq) (9)

In many micromachined devices, such as the mechanical structure in Fig. 4, much of the structure acts as a rigid body. Therefore, many finite-element degrees of freedom can be eliminated and replaced with a rigid body with only 6 degrees of freedom i/rigid = {^, 0. V^, ^R^ jR, zR). The u in (4) is then ^elastic U Mrigid. The rigid/elastic mechanical solver greatly reduces the size of the stiffness matrix with the bulk shrinking to a dense 6 x 6 block (see Fig. 5). The surface of the rigid body still has to be discretized finely to properly resolve the electrostatic forces. The rigid/elastic interface should be intruded into the rigid block for a small area around the tether-block mass interface in order to avoid sharp singularities in stress across the tether-block interface. 3.3. Tilting mirror example A coupled domain mixed regime solver was tested against the experimental data of a scanning mirror (see Figs. 6 and 7) [20] with 12 x 50 x 1.1 |xm SiN hinges (Young's Modulus = 243.2 MPa, Poisson's Ratio = 0.28)

where is a very small number. Therefore, this matrix-free multilevel-Newton method [18] can treat the individual solvers as black boxes. The black box solvers are called once in the outer Newton loop to compute the right hand side in (8) and then called once per each GMRES itera-

N. Alum et al. /First MIT Conference on Computational Fluid and Solid MechanicsRigid/elastic ; fully elastic (8x10x2 block 2x2x3 hinges)

Ov

12 500

251

.2237.5 +v 37.5 -V All dim in microns Fig. 7. Cross-section of scanning mirror.o experiment ; - simulation (30x30x3 block 3x4x3 hinges)

5

10 15 differential voltage in volts

20

Fig. 5. Elastic/rigid matrix reduction. and 500 x 600 x 25 [xm SiN on Si central plate kept at 0 v. The ground electrodes are kept at 37.5 v volts. The plot (Fig. 8) shows a close match of the simulation in the linear regime and convergence failure corresponding to pullin is obtained at 12.13 v as opposed to 13.4 v of the experimental data. On an average each load step took 80 min (Digital Alpha 433 MHz). For a coarse mesh the elastic/rigid simulation is compared with the fully elastic simulation (Fig. 5) to show a very close match. The CPU time for 10 load steps for the fully elastic case was 16.8 h as opposed to 58 min for the rigid/elastic case.

2

4 6 8 10 12 Differential voltage in v for scanning mirror

Fig. 8. Mirror tilt with differential voltage v. for coupled-domain analysis, and mixed-regime techniques. It is now possible to simulate the coupled-domain behavior of an entire micromachined design in under an hour on a workstation rather than days or weeks on a supercomputer. The next step is to use these tools to automatically generate macromodels of micromachined devices, and make possible accurate simulation of systems which use micromachined devices.

4. Conclusions Simulation of entire microdevices is becoming more routine in engineering design thanks to a combination of fast integral equation solvers, multilevel-Newton methods

0

"^

-200

Fig. 6. Scanning mirror (coarse mesh).

A^. Aluru et al. /First MIT Conference on Computational Fluid and Solid Mechanics Acknowledgements The authors would like to thank the many students who have developed codes described above including Keith Nabors, Joel Phillips, and Joe Kanapka. This work was supported by the DARPA composite CAD, microfluidics and muri programs, as well as grants from the Semiconductor Research Corporation and the National Science Foundation. [11] pole method for the Laplace equation in three dimensions. Acta Numer 1997, pp. 229-269. Pozrikidis C. Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, Cambridge, 1992. Aluru NR, White J. A fast integral equation technique for analysis of micro flow sensors based on drag force calculations. International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, April 1998, pp. 283-286. Ye W, Kanapka J, Wang X, White J. Efficiency and accuracy improvements for FastStokes, a precorrected-FFT accelerated 3-D Stokes Solver. International Conference on ModeHng and Simulation of Microsystems, Semiconductors, Sensors and Actuators, San Juan, April 1999. Ye W, Wang X, Hemmert W, Freeman DM, White J. Viscous drag on a lateral micro-resonator: fast 3-D fluid simulation and measured data. IEEE Solid-State Sensor and Actuator Workshop, Hilton-Head Island, SC, June 1999. Bathe KJ. Finite Element Procedures, Prentice-Hall, Englewood Chffs, NJ, 1996. Rabbat NB, Sangiovanni-VincenteUi A, Hsieh HY. A Multilevel-Newton algorithm with macromodeling and latency for the analysis of large scale nonlinear circuits in the time domain. IEEE Trans, on Circuits and Systems, CAS-26(9):733-741, Sept. 1979. Brown PN, Saad Y Hybrid Krylov Methods for Nonlinear Systems of Equations, SIAM J Sci Statist Comput 1990;11: 450-481. Aluru NR, White J. A coupled numerical technique for selfconsistent analysis of micro-electro-mechanical systems, microelectromechanical systems (MEMS). ASME Dynamic Systems and Control (DSC) Series, New York 1996;59: 275-280. Ramaswamy D, Aluru N, White J. Fast coupled-domain, mixed-regime electromechanical simulation. Proc. International Conference on Solid-State Sensors and Actuators (Transducers '99), Sendai Japan, June, 1999, pp. 314-317. Dickensheets DL, Kino GS. Silicon - Micromachined Scanning Confocal Optical Microscope. J Microelectromech Syst Vol. 7, No. 1, March 1998.

[12]

[13] References [1] Youcef Saad, Schultz MH. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Statist Comput 1986;7(3): 105-126. [2] Barnes J, Hut P. A hierarchical 0{N\ogN) force-calculation algorithm. Nature 1986;324:446-449. [3] Hockney RW, Eastwood JW. Computer simulation using particles. New York: Adam Hilger, 1988. [4] Rokhlin V. Rapid solution of integral equation of classical potential theory J Comput Phys 1985;60:187-207. [5] Hackbusch W, Nowak ZP. On the fast matrix multiplication in the boundary element method by panel clustering, Numer Math 1989;54:463-491. [6] Nabors K, White J. Fastcap: a multipole accelerated 3-D capacitance extraction program. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, November 1991;10:1447-1459. [7] Nabors K, Korsmeyer FT, Leighton FT, White J. Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional first-kind integral equations of potential theory. SIAM J Sci Statist Comput 1994;15(3):713-735. [8] Bachtold M, Korvink JO, Bakes H. The Adaptive, Multipole-Accelerated BEM for the Computation of Electrostatic Forces, Proc. CAD for MEMS, Zurich, 1997, pp. 14. [9] Phillips JR, White JK. A precorrected-FFT method for electro-static analysis of complicated 3-D structures. IEEE Trans, on Computer-Aided Design, October 1997; 16(10): 1059-1072. [10] Greengard L, RokhUn V. A new version of the fast multi-

[14]

[15] [16]

[17]

[18]

[19]

[20]

Subgrid scales, augmented problems, and stabilizationsFranco Brezzi *Dipartimento di Matematica and I.A.N.-C.N.R., Via Ferrata 1 27100 Pavia, Italy

Abstract We present an overview of some recent approaches to deal with instabiUties of numerical schemes and/or subgrid phenomena. The basic idea is that of enlarging (as much as one can) the finite element space, then to do an element-by-element preprocessing, and finally solve a problem with the same number of unknowns as the one we started with, but having better numerical properties. Keywords: Residual free bubble; Stabilization

1. Introduction In a number of applications, subgrid scales cannot be neglected. Sometimes, they are just a spurious by-product of a discretized scheme that lacks the necessary stability properties. In other cases, they are related to physical phenomena that actually take place on a very small scale, but still have an important effect on the solution. In recent times, it was discovered that some mathematical tricks to deal with these problems can help in both situations. One of these tricks is based on the so-called Residual Free Bubbles (RFB). In what follows, we are going to discuss its application, by considering two typical examples, one for each category: the case of advection diffusion problems and the case of composite materials. For dealing with these problems, in a typical mathematical fashion, we shall choose very simple toy problems that will, however, still retain some of the basic difficulties of their bigger industrial counterparts. In particular, we consider: 1: Advection-dominated scalar equations: find umV:= H^(Q) such that Lu:= -sAu-{-c-S/u = f in ^ , w = 0 on dQ. (1.1) Here Q is, say, a convex polygon, c a given vector-valued smooth function (convective term), / a given smooth forcing term, and s a positive scalar (diffusion coefficient). Clearly, x = (xi,X2). The numerical approximation of the problem becomes nontrivial when the product of s times a characteristic length of the problem (for instance, the * E-mail: brezzi@dragon.ian.pv.cnr.it 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

diameter of Q) is much smaller than |c| in a non-negligible part of the domain. The variational formulation of (1.1) is find u e V such thatC(u, v) := I eVu -Vvdx -\- /

-i"

/

C-VUV&K

(1.2)

doc Vi; V.

2: Linear elliptic problems with composite materials: find M in V := H^(^) such that: Lu := - V . (a{x)Vu) = / in ^ , M = 0 on dQ. (1.3)

As before, Q is, say, a convex polygon, and / a given smooth forcing term. The (given) scalar function a{x) is assumed to be greater than a given positive constant ao in the whole domain Q, and represents, somehow, the characteristics of a composite material. The numerical approximation of (1.3) becomes nontrivial when a has a fine structure, exhibiting sharp changes on a scale that is much smaller than the diameter of ^ . The variational formulation of (1.3) is find M e V such that (M, V) := / a(jc)Vw Vvdx

-I

fvdx

VUG V

(1.4)

The first example corresponds to problems where an unsuited numerical scheme can generate spurious oscillations in the numerical solution, which are not present in the exact solution (that in general, will just exhibit a boundary layer

F. Brezzi/First MIT Conference on Computational Fluid and Solid Mechanics near the part of the boundary where c n > 0, where n is the outward unit vector normal to 9 ^ . On the contrary, the second example corresponds to problems where a fine structure is already present, all over the domain, and needs to be captured by the numerical scheme, at an affordable cost. In the sequel, we are going to give the basic idea of a general strategy that can prove useful, possibly in different ways, for both types of problems. V e Bh(K) and obtain, from (2.4) that the restriction wf of UB to K is the unique solution of the following local bubble equation: find UB ^ Bh(K) such that C(u^s, V) = -C(UH, V) + (/, V) Wv e Bh(K). (2.5) Equation (2.5), if solvable, would allow to express each wf in terms of Uh. At the formal level, we can introduce the solution operator SK, that associates to every function g (for instance in L^(K)) the solution SK(g) e H^{K) of C{SK{g),v) = {g,v) yveH^(K) (2.6) and write the solution i/f of (2.5) as wf = SK^/ - Luh). We are now ready to go back to (2.4), take v = Vh, and substitute in UA = Uh + UB its expression as given by (2.5) and (2.6) to obtain C{uh, Vh) - Y^C{SK{Luh), Vh) = (/,^/.)-X!>^(

, Po'^trcle ~^^ File

1

Fig. 5. East Bay Bridge analysis, process.

TJ. Ingham /First MIT Conference on Computational Fluid and Solid Mechanics

19

_ W W^ Im

Vi^^ ffietmn. i^eot^

kin ^ B

Deflection Damping Factor Re Lajfout Sdtrfamr ^rMod^io yge Cap" to tem aph cap mpedegxem^k^

o.i v The ctef)K:tai and dampBig facfty ^-e onKf used "]> fix piie ird^ce matriK aid hytrid models aid 0.015915 X for pfe cap Irrf)edarc8 matrix mocteis

u u

Fig. 6. Access database for model generation, pile modeling and layout form. 5. Conclusions The seismic analysis of large bridges presents many choices regarding the level of detail to include in a global model and the analysis of critical components. The use of automated methods for data storage, model generation, and the manipulation of results is an important factor in the complexity of the models that can be practicably handled.

[4]

[5]

[6] References [1] ADINA Theory and Modeling Guide. ADINA R&D, Cambridge, MA, 1999. [2] Baker G, Ingham T, Heathcote D. Seismic retrofit of Vincent Thomas suspension bridge. Transportation Research Record No. 1624. Transportation Research Board, 1998. [3] Ingham TJ. ModeUng of friction pendulum bearings for

[7]

[8]

the seismic analysis of bridges. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. Ingham TJ, Rodriguez S, Donikian R, Chan J. Seismic analysis of bridges with pile foundations. Comput Struct 1999;72:49-62. Ingham TJ, Rodriguez S, Nader M. Seismic modeling and analysis of the Golden Gate Bridge. Proceedings of the Structural Engineers World Congress, San Francisco, CA, 1998. Nader M, Manzanarez R, Ingham T, Baker G. Seismic Design Strategy for the New San Francisco Oakland Bay Bridge Suspension Span. Proceedings of the 16th International Bridge Conference, Pittsburgh, PA, 1999. Rodriguez S, Ingham TJ. Seismic Protective Systems for the Stiffening Trusses of the Golden Gate Bridge. Proceedings of the National Seismic Conference on Bridges and Highways, San Diego, CA, 1995. SC-Porthole7 Program. SC Solutions, Santa Clara, CA.

20

Virtual control algorithmsJ.L. Lions *Institut de France, 23 quai de Conti, 75006 Paris, France

Abstract Some recent advances in the development of virtual control algorithms for the approximate solution of boundary value problems are presented. Keywords: Virtual control algorithms; Controllability; Domain decomposition; Heterogeneous decomposition

Let us consider an equation

A(u) = f

(1)

in a domain ^ c R'^, where A is an elliptic operator (linear or not, scalar or vectorial), and where u is subject to boundary conditions, not specified here. We embed the problem in a family of relaxed problems By = g + k (2)

(4) Heterogeneous decompositions: follows a paper by Gervasio et al. [5], to appear in Numerische Mathematik. (5) High precision with low order finite elements: [6], to appear. (6) Time decomposition: [7], [8]. Cf. also a paper in preparation with Y. Maday. (7) Towards meshless methods: paper in preparation.

in a domain Q (which can coincide with Q, or not), where B is an elliptic operator, related to A but 'simpler' than A, where y is subject to adequate boundary conditions on 9 ^ . In (2) the RHS contains two terms. The function g is constructed depending on / and the function X (scalar or vectorial) is a virtual control. It is to be chosen in such a way that y allows to recover the solution u of (1), exacdy (resp. approximately). In control theory terminology, it corresponds to exact (resp. approximate) controllability. This type of idea, of course made precise, allows a lot of flexibility in the construction of algorithms for the approximation of the solution of (1), the so-called virtual control algorithms. The idea was introduced in a note by JL Lions and O Pironneau [1] and since then it has been applied to a number of situations. The lecture will try to present the main ideas of the following ones. (1) Domain decomposition methods: see [1] above and [2]. (2) Decomposition of operators: [3]. (3) Decomposition of energy spaces: [4]. *Tel.: +33 (1) 4427-1708; Fax: +33 (1) 4427-1704; E-mail: jacques-louis.lions@college-de-france.fr 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

References [1] Lions JL, Pironneau O. Algorithmes paralleles pour la solution de problemes aux limites. C.R.A.S. Paris 1998;327(I):947-952. [2] Lions JL, Pironneau O. Domain decomposition methods for CAD. C.R.A.S. Paris 1999;328(I):73-80. [3] Lions JL, Pironneau O. Virtual control, replicas and decomposition of operators. C.R.A.S. Paris 2000;330(I):47-54. [4] Glowinski R, Lions JL, Pironneau O. Decomposition of energy spaces and applications. C.R.A.S. Paris 1999;329(I):445-452. [5] Gervasio P, Lions JL, Quarteroni A. Heterogeneous coupling by virtual control methods. Numer Math, to appear. [6] Lions JL, Pironneau O. to appear. [7] Lions JL. Virtual and effective control for distributed systems and the decomposition of everything. J Anal Math, Hebrew Univ. of Jerusalem 2000;80:257-297. [8] Lions JL. Remarks on the control of everything. Eccomass, Barcelona, September 2000.

21

Numerical methods for prediction and evaluation of geometrical defects in sheet metal formingA. Makinouchi^'*, C. Teodosiu^

^ The Institute of Physical and Chemical Research RIKEN, Materials Fabrication Laboratory, 2-1 Hirosawa, Wako 351-0199, Japa ^ LPMTM CNRS, University Paris Nord, Villetaneuse , France

Abstract This paper presents a short overview of the state-of-the-art prediction and evaluation of geometrical defects in sheet metal forming, focusing on recent advances in the finite element (FE) simulation, on the benchmark tests organized to obtain reference experimental data for appraising ability of simulation codes, and on the attempt to define numerical measures for quantitatively evaluating various geometrical defects. Keywords: Sheet metal forming; Geometrical defects; Springback; Benchmark test

1. Introduction Sheet forming simulation is becoming a key technology for automotive manufacturers, sheet metal parts producers and stamping tool makers, aiming at predicting forming defects by using finite element software, in order to replace the actual tryout of stamping dies by a computer tryout. The main types of defects occurring in sheet metal forming are tearing, surface deflection, wrinkling, and springback (see Fig. 1). The last three types are also called geometrical defects. Among the three geometrical defects springback is a very sensitive forming defect, as the cumulative geometrical inaccuracy of the stamped parts may lead to serious trouble during assembling of various parts. Moreover, this difficulty tends to increase with the recent use of aluminum alloys and high-strength steels by the car manufacturers. Fig. 2 illustrates the main types of geometrical defects produced by springback (edited by Yoshida [1]).

2. Requirement from industries In 1998, the authors visited automotive industries and sheet steel suppliers in Europe, Japan and the United States, * Corresponding author. Tel.: +81 (48) 467-9314; Fax: +81 (48) 462-4657; E-mail: akitake@postman.riken.go.jp 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to discuss with engineers and researchers working at die shops and in sheet stamping sections. The reason of this visit was to prepare our keynote paper on the advance in FEM simulation and its related technologies in sheet metal forming for the CIRP Annual Meeting [2]. The visited companies were Daimler Benz, Renault Automobiles, Volvo Car Corporation and SOLLAC in Europe, Mazda, Nissan Motor, Toyota Motor and Nippon Steel in Japan, and Ford Motor, Chrysler Corporation, US Steel and National Steel in the United States. A large number of international conferences have been devoted to the sheet metal forming simulation, and an extensive literature has been published on this topic throughout the last two decades. However, the information obtained from these sources was not considered sufficient to address the above issues, because the very trend of sheet forming simulation had undergone significant changes during the last ten years. Indeed, most engineers working in automakers and sheet suppliers are software users, and their opinion does rarely appear in publications. Therefore, the authors considered that a direct contact with the technical staff involved in sheet metal forming simulations was a highly necessary prerequisite for learning the actual evaluation of the software used for industrial applications. A quite interesting bulk of information has been obtained in this way. Although a wide variety of FE codes are employed in the industries, these codes may be divided into five categories based on the formulation and solution

22

A. Makinouchi, C. Teodosiu /First MIT Conference on Computational Fluid and Solid Mechanics

Tearing

Surface deflection

Wrinkling

Springback

Fig. 1. Main types of defects encountered in sheet metal forming. Table 1 Assessment of FE codes by industrial researchers and engineers for each category classified by formulation and solution strategy Solution strategy Formulation FE codes Dynamic explicit Incremental method LS-DYNA3D PAM-STAMP OPTRIS All the companies Static explicit ITAS3D Static implicit Large step method MTLFRM AUTO FORM One step method SIMEX ISOPUNCH A F ONE STEP FAST FORM3D Renault Benz Volvo Sollac National Steel

Company employing codes

Nissan Nippon Steel

Ford

Benz Volvo Ford Chrysler Nissan Sollac

Defects predicted: wrinkling thickness/tearing surface defects geometrical defects after springback

A, X o, A X A, X A, X X X

: satisfactorily predicted; A = possible to simulate but poor results; x = impossible to simulate. strategy used. The assessment of the codes by industrial researchers and engineers is summarized for each category in Table 1. Inspection of this table reveals that the tearing and wrinkling are rather satisfactorily predicted, while prediction of the springback is very poor, while the surface deflection is not simulated. Most of the engineers strongly emphasized the importance of an accurate springback prediction.

A. Makinouchi, C. Teodosiu /First MIT Conference on Computational Fluid and Solid Mechanics Rail Panel

23

Springback angle

Side wall curl

Twisting

Warping

Shape fixing defect at punch bottom

Fig. 2. Geometrical defects produced by springback.

3. FE approach to simulate geometrical defects We shall recall here briefly some of the merits and drawbacks of three main types of FE approaches employed in the simulation of sheet metal forming, namely the dynamic explicit, the static implicit, and the static explicit codes. The dynamic-explicit codes are very robust and efficient for large-scale problems. The central difference expUcit scheme is used to integrate the equations of motion, whereas the non-equilibrated forces are transformed into inertial forces at each step. Lumped mass matrices are used, and hence no system of equations has to be solved. In spite of its success for industrial applications, dynamic explicit codes have also some intrinsic drawbacks. Thus, in order to reduce the number of steps necessary to simulate the almost quasi-static deformation processes, several numerical artifacts have to be employed, e.g. the increase

of the mass density and of the punch velocity by at least one order of magnitude and the introduction of artificial damping in order to limit the inertial effects. Moreover, the results obtained when simulating the springback depend on the type and dimensions of the finite elements and even of the number of integration points [3]. Thus, the simulation of forming defects requires a considerable experience on the user side for adequately designing the finite element mesh and choosing the scaling parameters for mass, velocity and damping (see, e.g. [4]). The static-implicit approach may seem ideally suited for metal forming problems, since the equilibrium equations are solved iteratively, thus ensuring that the equilibrium conditions are fulfilled at every step. However, in practice, complex nonlinear problems involving many contacts, may result in slow, or even lack of convergence. In the static-explicit approach, the rate forms of the

24

A. Makinouchi, C. Teodosiu /First MIT Conference on Computational Fluid and Solid Mechanics 5. Numerical representation of geometrical defects Assuming that a powerful FE code could accurately predict all geometrical defects illustrated in Fig. 2, this will be still not enough for the present requirements of the stamping industry. Indeed, the final goal of simulations is to quantitatively evaluate the geometry of stamped parts and, on this basis, to find the optimized die shapes that are able to produce parts of the exactly designed shape. To meet such requirements, it is essential to have clear definitions of forming defects and of the intrinsic values used to evaluate each geometrical defect. This problem is also a major concern in the 3DS Project. The surface of each defect model possesses some global features, which describe the overall distortions, such as the surface being 'bent' or 'twisted', and local features, which describe local distortions and their locations. There are many ways of defining such measures. One of the most promising way is to describe the local intrinsic character of the surface by the Gaussian curvature, and to represent the global features by the aggregate normal vectors to the surface [11].

kinematic, constitutive and equilibrium equations are integrated by a simple forward Euler scheme, involving no iterations (see, e.g. [5]). This implies that equilibrium equations are satisfied only in rate form, and thus the obtained solution can gradually drift away from the true one. In order to reduce the errors involved by linearizing the incremental analysis, a relatively large number of small incremental steps have to be used. The main advantage of this approach is its robustness, since it requires no iterative processes. Furthermore, by the very existence of intrinsic deviations from perfect equilibrium, the static-explicit algorithm is able to simulate defects arising from local instabilities, like wrinkling (see, e.g. [6]), while the static implicit codes are hardly able to treat such situations, unless such instabilities are allowed for by special numerical techniques, which require a considerable computational effort.

4. Benchmark tests to evaluate ability of FE codes for prediction of geometrical defects At several international conferences, like the VDI International Conference held at Zurich, Switzerland in 1991 [7], NUMISHEET'93 at Isehara, Japan in 1993 [8], NUMISHEET'96 at Dearborn, USA in 1996 [9], and NUMISHEET'99 at Besangon, France in 1999 [10], benchmark tests were organized in order to appraise the capability of FE codes to predict forming defects. The experimental benchmark tests have been concurrently performed by several teams over the world, in order to obtain reference data. However, most of the benchmark experimental results obtained by different participants disagreed greatly with each other and thus provided rather poor reference data for evaluating the codes. It is eventually possible to find out a posteriori the reasons for this scattering of experimental data. However, because the benchmark results are evaluated by the conference organizing committee, which dissolves after the event, it has been practically impossible to further analyze the discrepancies noticed during the conference. For the purpose of solving this problem, a three-year international research project named Digital Die Design System (3DS) started its activity in 2(XX), under the framework of the international collaborative program. Intelligent Manufacturing System (IMS). Fourteen industrial partners and seven academic and research institutes participate to the project from Canada, European Union and Japan, the present authors being deeply involved with the technical management of this project. The obtaining of reliable experimental data, with a controlled and minimized scatter, is one of main targets of the project. Such carefully performed and comprehensively documented experimental tests are expected to become a worldwide recognized database for the validation of numerical methods and codes dealing with the simulation of sheet metal forming processes.

6. Conclusions A short overview of recent activity in numerical methods to predict and evaluate geometrical defects in sheet metal forming is presented. Although FE codes were introduced into many industries, further intensive research effort is necessary to approach to the final goal: designing the optimum tool geometry directly by simulation.

References [1] Yoshida K (Ed). Handbook of Ease or Difficulty in Press Forming, Tokyo, 1987. (English translation, Ann Arbor, MI: National Center for Manufacturing Science, Inc., 1993.) [2] Makinouchi A, Teodosiu C, Nakagawa T. Advances in FEM simulation and its related technologies in sheet metal forming. Ann CIRP 1998;47(2):641-649. [3] Mattiasson K, Thilderkvist P, Strange A, Samuelsson A. Simulation of springback in sheet metal forming. In: Shen S, Dawson PR (Eds), Proc. NUMIFORM'95. Rotterdam: Balkema, 1995, pp. 115-124. [4] Lee SW, Yang DY. An assessment of numerical parameters influencing springback in explicit finite element analysis of sheet metal forming processes. J. Mater Process Technol 1998:80-81:60-67. [5] Kawka M, Makinouchi A. Shell-element formulation in the static explicit FEM code for the simulation of sheet stamping. J Mater Process Technol 1995;50: 105-115. [6] Kawka M, Olejnik L, Rosochowski A, Sunaga H, Makinouchi A. Modeling wrinkling phenomena in sheet metal forming. Proceedings of AEPA'98, 1998. [7] Proceedings of VDI International Conference. FE Simula-

A. Makinouchi, C. Teodosiu/First MIT Conference on Computational Fluid and Solid Mechanics tion of 3-D Sheet Metal Forming Processes in Automotive Industry, Zurich, Switzerland, 1991. [8] Proceedings of NUMISHEET'93, Isehara, Japan, 1993. [9] Proceedings of NUMISHEET'96, Dearborn, USA, 1996.

25

[10] Proceedings of NUMISHEET'99, Besan9on, France, 1999. [11] Kase K, Makinouchi A, Nakagawa T, Suzuki H, Kimura F. Shape error evaluation method of free-form surfaces. Comput-Aided Design 1999;31(8):495-505.

26

The Immersed Boundary Method for incompressible fluid-structure interactionDavid M. McQueen, Charles S. Peskin *, Luoding ZhuCoumnt Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA

Abstract In this paper the Immersed Boundary Method is presented, with some recent developments. The method is used to analyze fluid-structure interaction problems. Different aspects of the method are illustrated by applying it to blood flow in the heart and a flapping filament (flag-in-wind) problem. Keywords: Immersed Boundary Method; Fluid-structure interaction; Cardiac fluid dynamics; Flapping filament; Flag in wind; Computational fluid dynamics; Incompressible elasticity; Heart valves

1. Introduction In the study of fluid-structure interaction, it is useful to think of the structure as a part of the fluid where additional forces are applied, and where additional mass may be localized. In this paper, we consider the case of a viscous incompressible fluid that interacts with an immersed structure that is made of an incompressible viscoelastic material. To keep things as simple as possible, we assume that the viscosity is Newtonian and uniform throughout the system. This restriction can certainly be removed, but we shall not address that complication here. The mass density of the ambient fluid is also assumed to be uniform, but the structure is allowed to have a nonuniform mass density which may be greater or lower than that of the fluid. Instead of separating the system into its two components coupled by boundary conditions, as is conventionally done, we use the incompressible Navier-Stokes equations, with a nonuniform mass density and an applied elastic force density, to describe the coupled motion of the hydroelastic system in a unified way. In order to do this, however, we need to supplement the Navier-Stokes equations by a Lagrangian description of the elastic material, from which the elastic force density and the nonuniform mass density that appear in the Navier-Stokes equations may be calculated. Moreover, we need a mathematical apparatus to translate in either direction between Lagrangian quantities * Corresponding author. Tel.: +1 (212) 998-3126; Fax: -Hi (212) 995-4121; E-mail: peskin@cims.nyu.edu 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

and the corresponding Eulerian quantities. This apparatus is conveniently provided by the Dirac delta function. The equations of motion that result from this point of view directly motivate a numerical method known as the "Immersed Boundary Method" [1-5]. This name emphasizes an important feature of the method: that it can handle not only immersed elastic structures that displace a finite volume, but also immersed elastic boundaries like heart valve leaflets (for which the method was originally designed), insect wings, sails, and parachutes, all of which may be idealized as surfaces which, despite having zero volume, nevertheless apply finite forces to the fluid in which they are immersed. Clearly, the Dirac delta function is particularly well suited to this situation.

2. Equations of motion As described in Section 1, we use an Eulerian description of the system as a whole (fluid -h structure) supplemented by a Lagrangian description of the structure. The independent variables of the Eulerian description are the Cartesian coordinates x and the time t, and the independent variables of the Lagrangian description are curvilinear material coordinates q,r,s and again the time t. The Eulerian description of the system as a whole involves the velocity field w(jc, r), the hydrostatic pressure field p(x,t), th^ mass density p{x, t) and the Eulerian elastic force density/(jc, 0The Lagrangian description of the immersed elastic material involves its configuration X{q,r,s,t), its Lagrangian

D.M. McQueen et al. /First MIT Conference on Computational Fluid and Solid Mechanics elastic force density F{q, r,s,t), and its Lagrangian additional mass density M(q,r,s), the integral of which over any chunk of the material gives the mass of that chunk minus the mass of the fluid displaced. Since both the mass and volume of any such chunk of the immersed elastic material are conserved, M is independent of time. Note that M = 0 in the case of a neutrally buoyant structure, and that M will be negative at any material point for which the mass density of the immersed elastic material is less than that of the ambient fluid. To complete the Lagrangian description of the elastic material, we need to specify the elastic potential energy functional, E[X], which is used in the calculation of the elastic forces from the configuration X(, , ,t) at any given time. The mass density po of the ambient fluid and the viscosity /x of the system as a whole are constant parameters. With this notation, our equations of motion read as follows: p{x,t) (-^JrU'Vu\+Vp W u=0 fix, t)= F{q, r, s, t) 8 (x - X(q, r, s, t)) dq dr ds = ixV^u +f{x, t) (1)(2)

27

(3)

p{x, t) = po-\- / M(q, r, s) 8 (x - X(q, r, s, t)) dq dr ds(4)

Note that Eq. (1) also involves the non-uniform mass density p{x, t). Since the fluid and the structure are both incompressible, it must be the case that p{x, t) at any given material point is independent of time, i.e., that Dp/Dr = 0, where D/Dr is the material derivative: 9/9f -I- a V. This constraint is implicit in Eqs. (4) and (5); it does not have to be imposed separately. Eqs. (3) and (4) provide conversions from the Lagrangian force and mass densities F{q, r,s,t) and M(q, r, s) to the corresponding Eulerian force and mass densities,/(x, t) and p(x, t), respectively. The relationship between corresponding den