Goal-oriented error estimation and adaptivity for fluid-structure interaction using exact linearized adjoints

Computer Methods in Applied Mechanics and Engineering

Volumn 200, Issue 37-40, 2011, Pages 2738-2757

  van der Zee, K G ^a   van Brummelen, E H ^b   Akkerman, I ^c   de Borst, R ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

A posteriori error estimation; Adaptive mesh refinement; Duality; Exact linearized adjoint; Fluid structure interaction; Shape derivative

Indexed keywords

ADAPTIVE MESH-REFINEMENT; ADJOINTS; DUALITY; POSTERIORI ERROR ESTIMATION; SHAPE DERIVATIVES;

ESTIMATION; FLUID STRUCTURE INTERACTION; LINEARIZATION;

FLUIDS;

EID: 79958712953     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2010.12.010     Document Type: Article

References (92)

1. Ainsworth M., Oden J.T. A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics 2000, Wiley.
2. Babuška I., Miller A. The post-processing approach in the finite element method - Part 1: Calculation of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Methods Engrg. 1984, 20:1085-1109.
3. Babuska I., Strouboulis T. The Finite Element Method and Its Reliability. Numerical Mathematics and Scientific Computation 2001, Oxford University Press.
4. Bayada G., Chambat M., Cid B., Vazquez C. On the existence of solution for a nonhomogeneous Stokes-rod coupled problem. Nonlinear Anal. 2004, 59(1-2):1-19.
5. Bazilevs Y., Calo V.M., Hughes T.J.R., Zhang Y. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 2008, 43(3):3-37.
6. Bazilevs Y., Calo V.M., Zhang Y., Hughes T.J.R. Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput. Mech. 2006, 38:310-322.
7. Becker R., Rannacher R. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 2001, 10:1-102.
8. Bello J.A., Fernández-Cara E., Lemoin J., Simon J. The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow. SIAM J. Control Optim. 1997, 35(2):626-640.
9. Boisgérault S., Zolésio J.-P. Boundary variations in the Navier-Stokes equations and Lagrangian functionals. Pure and Applied Mathematics 2001, vol. 216. Marcel Dekker, Inc. J. Cagnol, M.P. Polis, J.-P. Zolésio (Eds.).
10. Braack M., Ern A. A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 2003, 1(2):221-238.
11. Burman E., Fernández M.A. Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility. Comput. Methods Appl. Mech. Engrg. 2009, 198:766-784.
12. Carey V., Estep D., Tavener S. A posteriori analysis and adaptive error control for multiscale operator decomposition solution of elliptic systems I: Triangular systems. SIAM J. Numer. Anal. 2009, 47(1):740-761.
13. Causin P., Gerbeau J.-F., Nobile F. Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 2005, 194:4506-4527.
14. Cheng A., Coutand D., Shkoller S. Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 2007, 39(3):742-800.
15. Coutand D., Shkoller S. Motion of an elastic solid inside an incompressible viscous fluid. Arch. Ration. Mech. Anal. 2005, 176:25-102.
16. Dautray R., Lions J.-L. Mathematical analysis and numerical methods for science and technology. Functional and Variational Methods 1988, vol. 2. Springer-Verlag.
17. M.C. Delfour, J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, SIAM Series on Advances in Design and Control, vol. 4, Society for Industrial and Applied Mathematics, 2001.
18. Donea J., Huerta A., Ponthot J.-P., Rodríguez-Ferran A. Arbitrary Lagrangian-Eulerian methods. Fundamentals 2004, vol. 1:413-437. John Wiley & Sons, (Chapter 14). E. Stein, R. de Borst, T.J.R. Hughes (Eds.).
19. Dörfler W. A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 1996, 33(3):1106-1124.
20. Dunne T. An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaptation. Int. J. Numer. Methods Fluids 2006, 51:1017-1039.
21. Dziri R., Zolésio J.-P. Tube derivative of noncylindrical shape functionals and variational formulations. Lecture Notes in Pure and Applied Mathematics 2007, vol. 252:411-442. Chapman & Hall/CRC, (Chapter 23). R. Glowinski, J.-P. Zolésio (Eds.).
22. Eriksson K., Estep D., Hansbo P., Johnson C. Introduction to adaptive methods for differential equations. Acta Numer. 1995, 1-54.
23. D. Estep, V. Carey, V. Ginting, S. Tavener, T. Wildey, A posteriori error analysis of multiscale operator decomposition methods for multiphysics models, Journal of Physics: Conference Series 125 (Art. No. 012075).
24. Estep D., Ginting V., Ropp D., Shadid J.N., Tavener S. An a posteriori-a priori analysis of multiscale operator splitting. SIAM J. Numer. Anal. 2008, 46(3):1116-1146.
25. Estep D., Tavener S., Wildey T. A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem. SIAM J. Numer. Anal. 2008, 46(4):2068-2089.
26. T. Fanion, M.A. Fernández, P. Le Tallec, Deriving adequate formulations for fluid-structure interaction problems: from ALE to transpiration, in: A. Dervieux (Ed.), Fluid-Structure Interaction, Innovative Technology Series, Kogan Page Science, 2003, pp. 51-80 (Chapter 3, Derived from Revue européenne des éléments finis, Fluid-Structure Interaction, 9 (6-7) (2000).
27. Farhat C. CFD-based nonlinear computational aeroelasticity. Fluids 2004, vol. 3:459-480. John Wiley & Sons, (Chapter 13). E. Stein, R. de Borst, T.J.R. Hughes (Eds.).
28. Farhat C., Lesoinne M., Le Tallec P. Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput. Methods Appl. Mech. Engrg. 1998, 157:95-114.
29. Farhat C., van der Zee K.G., Geuzaine P. Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity. Comput. Methods Appl. Mech. Engrg. 2006, 195(17-18):1973-2001.
30. M.A. Fernández, J.-F. Gerbeau, A. Gloria, M. Vidrascu, A Partitioned Newton Method for the Interaction of a Fluid and a 3D Shell Structure, Tech. Rep. 6623, Institute National de Recherche en Informatique et en Automatique (INRIA), Paris, 2008.
31. Fernández M.A., Gerbeau J.-F., Grandmont C. A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Engrg. 2007, 69:794-821.
32. Fernández M.A., Le Tallec P. Linear stability analysis in fluid-structure interaction with transpiration. Part I: Formulation and mathematical analysis. Comput. Methods Appl. Mech. Engrg. 2003, 192:4805-4835.
33. Fernández M.A., Moubachir M. Sensitivity analysis for an incompressible aeroelastic system. Math. Models Methods Appl. Sci. 2002, 12(8):1109-1130.
34. Fernández M.A., Moubachir M. A Newton method using exact Jacobians for solving fluid-structure interaction. Comput. Struct. 2005, 83:127-142.
35. Fick P.W., van Brummelen E.H., van der Zee K.G. On the adjoint-consistent formulation of interface conditions in goal-oriented error estimation and adaptivity for fluid-structure interaction. Comput. Methods Appl. Mech. Engrg. 2010, 199:3369-3385.
36. Flucher M., Rumpf M. Bernoulli's free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 1997, 486:165-204.
37. Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System. Modeling, Simulation & Applications 2009, vol. 1. Springer, Milano, Italy. L. Formaggia, A. Quarteroni, A. Veneziani (Eds.).
38. Georgoulis E.H., Hall E., Houston P. Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM J. Sci. Comput. 2007, 30(1):246-271.
39. Gerbeau J.-F., Vidrascu M., Frey P. Fluid-structure interaction in blood flows on geometries based on medical imaging. Comput. Struct. 2005, 83:155-165.
40. Ghattas O., Li X. A variational finite element method for stationary fluid-solid interaction. J. Comput. Phys. 1995, 121:347-356.
41. Giles M.B., Süli E. Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 2002, 145-236.
42. Grandmont C. Existence et unicité de solutions d'un problème de couplage fluide-structure bidimensionnel stationnaire. C.R. Math. Acad. Sci. Paris 1998, 326:651-656.
43. Grandmont C. Existence for a three-dimensional steady state fluid-structure interaction problem. J. Math. Fluid Mech. 2002, 4:76-94.
44. Grandmont C. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 2008, 40(2):716-737.
45. Grätsch T., Bathe K.J. Goal-oriented error estimation in the analysis of fluid flows with structural interactions. Comput. Methods Appl. Mech. Engrg. 2006, 195:5673-5684.
46. Grisvard P. Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics 1985, vol. 24. Pitman Publishing.
47. Haslinger J., Neittaanmäki P. Finite Element Approximation for Optimal Shape, Material and Topology Design 1996, John Wiley and Sons. second ed.
48. Heil M. An efficient solver for the fully-coupled solution of large-displacement fluid-structure interaction problems. Comput. Methods Appl. Mech. Engrg. 2004, 193:1-23.
49. Hoffman J., Johnson C. Computability and adaptivity in CFD. Fluids 2004, vol. 3:183-206. John Wiley & Sons, (Chapter 7). E. Stein, R. de Borst, T.J.R. Hughes (Eds.).
50. Kärkkäinen K.T., Tiihonen T. Free surfaces: shape sensitivity analysis and numerical methods. Int. J. Numer. Methods Engrg. 1999, 44:1079-1098.
51. K.T. Kärkkäinen, T. Tiihonen, Shape calculus and free boundary problems, in: P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, D. Knörzer (Eds.), European Congres on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004, Jyväskylä, 2004.
52. M. Larson, R. Söderlund, Adaptive Finite Element Approximation of Two Way Coupled Problems, Research Report in Mathematics 10, Umeå University, Umeå, Sweden, 2007.
53. Larson M.G., Bengzon F. Adaptive finite element approximation of multiphysics problems. Commun. Numer. Methods Engrg. 2008, 24:505-521.
54. Larson M.G., Målqvist A. Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations. Comput. Methods Appl. Mech. Engrg. 2007, 196:3546-3561.
55. Larson M.G., Söderlund R., Bengzon F. Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer. Inter. J. Numer. Methods Fluids 2008, 57:1397-1420.
56. Le Tallec P., Mouro J. Fluid structure interaction with large structural displacements. Comput. Methods Appl. Mech. Engrg. 2001, 190:3039-3067.
57. Leicht T., Hartmann R. Anisotropic mesh refinement for discontinuous Galerkin methods in two-dimensional aerodynamic flow simulations. Int. J. Numer. Methods Fluids 2008, 56:2111-2138.
58. Lowe T.W., Pedley T.J. Computation of Stokes flow in a channel with a collapsible segment. J. Fluid Struct. 1995, 9:885-905.
59. Melbø H., Kvamsdal T. Goal oriented error estimators for Stokes equations based on variationally consistent postprocessing. Comput. Methods Appl. Mech. Engrg. 2003, 192:613-633.
60. Morin P., Siebert K.G., Veeser A. A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 2008, 18(5):707-737.
61. Moubachir M., Zolésio J.-P. Moving Shape Analysis and Control, Applications to Fluid Structure Interactions. Pure and Applied Mathematics 2006, Chapman & Hall/CRC.
62. Oden J.T., Babuška I., Nobile F., Feng Y., Tempone R. Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty. Comput. Methods Appl. Mech. Engrg. 2005, 194:195-204.
63. Oden J.T., Prudhomme S. Estimation of modeling error in computational mechanics. J. Comput. Phys. 2002, 182:496-515.
64. Oden J.T., Prudhomme S., Romkes A., Bauman P.T. Multiscale modeling of physical phenomena: adaptive control of models. SIAM J. Sci. Comput. 2006, 28(6):2359-2389.
65. Pironneau O. Optimal Shape Design for Elliptic Systems. Springer Series in Computational Physics 1984, Springer, Berlin.
66. O. Pironneau, Optimal shape design by local boundary variations, in: A. Cellina, A. Ornelas (Eds.), Optimal Shape Design: Lectures Given at the Joint CIM, CIME Summer School held in Tróia, Portugal, June 1-6, 1998, Lecture Notes in Mathematics, vol. 1740, Springer, Fondazione CIME, 2000, pp. 343-384.
67. S. Prudhomme, Adaptive Control of Error and Stability of h-p Approximations of the Transient Navier-Stokes Equations, Ph.D. Thesis, The University of Texas at Austin, Austin, TX, USA, August 1999.
68. Prudhomme S., Ben Dhia H., Bauman P.T., Elkhodja N., Oden J.T. Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method. Comput. Methods Appl. Mech. Engrg. 2008, 197:3399-3409.
69. Prudhomme S., Oden J.T. Computable error estimators and adaptive techniques for fluid flow problems. Lecture Notes in Computational Science and Engineering 2003, vol. 25:207-268. Springer-Verlag, Heidelberg. T.J. Barth, H. Deconinck (Eds.).
70. Saavedra P., Scott L.R. Variational formulation of a model free-boundary problem. Math. Comput. 1991, 57(196):451-475.
71. Sforza D.M., Putman C.M., Cebral J.R. Hemodynamics of cerebral aneurysms. Annu. Rev. Fluid Mech. 2009, 41:91-107.
72. Simon J. Differentiation with respect to the domain in boundary-value-problems. Numer. Funct. Anal. Optim. 1980, 2(7-8):649-687.
73. Sokołowski J., Zolésio J.-P. Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics 1992, vol. 16. Springer-Verlag.
74. Šolin P., Demkowicz L.F. Goal-oriented hp-adaptivity for elliptic problems. Comput. Methods Appl. Mech. Engrg. 2004, 193:449-468.
75. Stein E., Rüter M. Finite element methods for elasticity with error-controlled discretization and model adaptivity. Solids and Structures 2004, vol. 2:5-58. John Wiley & Sons, (Chapter 2). E. Stein, R. de Borst, T.J.R. Hughes (Eds.).
76. Stevenson R. The completion of locally refined simplicial partitions created by bisection. Math. Comput. 2008, 77(261):227-241.
77. Süli E., Houston P. Adaptive finite element approximation of hyperbolic problems. Lecture Notes in Computational Science and Engineering 2003, vol. 25:269-344. Springer-Verlag, Heidelberg. T.J. Barth, H. Deconinck (Eds.).
78. Taylor C.A. Blood flow. Fluids 2004, vol. 3:527-543. John Wiley & Sons, (Chapter16). E. Stein, R. de Borst, T.J.R. Hughes (Eds.).
79. Tezduyar T.E., Sathe S., Stein K., Aureli L. Modeling of fluid-structure interactions with the space-time techniques. Lecture Notes in Computational Science and Engineering 2006, vol. 53:50-81. Springer-Verlag, Heidelberg. H.-J. Bungartz, M. Schäfer (Eds.).
80. van Brummelen E.H. Added mass effects of compressible and incompressible flows in fluid-structure interaction. Trans. ASME J. Appl. Mech. 2009, 76(2):021206.
81. van Brummelen E.H., de Borst R. On the nonnormality of subiteration for a fluid-structure interaction problem. SIAM J. Sci. Comput. 2005, 27(2):599-621.
82. van Brummelen E.H., van der Zee K.G., de Borst R. Space/time multigrid for a fluid-structure-interaction problem. Appl. Numer. Math. 2008, 58:1951-1971.
83. K.G. van der Zee, Goal-adaptive Discretization of Fluid-structure Interaction, Ph.D. Thesis, Technische Universiteit Delft, Delft, The Netherlands, June 2009.
84. van der Zee K.G., van Brummelen E.H., de Borst R. Goal-oriented error estimation for Stokes flow interacting with a flexible channel. Int. J. Numer. Methods Fluids 2008, 56:1551-1557.
85. van der Zee K.G., van Brummelen E.H., de Borst R. Goal-oriented error estimation and adaptivity for free-boundary problems: the domain-map linearization approach. SIAM J. Sci. Comput. 2010, 32(2):1064-1092.
86. van der Zee K.G., van Brummelen E.H., de Borst R. Goal-oriented error estimation and adaptivity for free-boundary problems: the shape linearization approach. SIAM J. Sci. Comput. 2010, 32(2):1093-1118.
87. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics 1996, Wiley-Teubner, New York, Stuttgart.
88. W.A. Wall, Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen, Ph.D. Thesis, Universität Stuttgart, Institut für Baustatik, Stuttgart, Germany, November 1999.
89. Wall W.A., Gerstenberger A., Gamnitzer P., Förster C., Ramm E. Large deformation fluid-structure interaction - advances in ALE methods and new fixed grid approaches. Lecture Notes in Computational Science and Engineering 2006, vol. 53:195-232. Springer-Verlag, Heidelberg. H.-J. Bungartz, M. Schäfer (Eds.).
90. Yong D. Strings, chains, and ropes. SIAM Rev. 2006, 48(4):771-781.
91. Zhang Z., Babuška I. A numerical method for steady state free boundary problems. SIAM J. Numer. Anal. 1996, 33(6):2184-2214.
92. Zolésio J.-P. Introduction to shape optimization problems and free boundary problems. NATO ASI Series C: Mathematical and Physical Sciences 1992, vol. 380:397-457. Kluwer Academic Publishers. M.C. Delfour, G. Sabidussi (Eds.).