Goal-oriented error estimation and adaptivity for free-boundary problems: The shape-linearization approach

SIAM Journal on Scientific Computing

Volumn 32, Issue 2, 2010, Pages 1093-1118

  Van Der Zee, K G ^a   Van Brummelen, E H ^b   De Borst, R ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

A Posteriori Error Estimation; Adaptive Mesh Refinement; Bernoulli Free boundary Problem; Goal Oriented Error Estimation; Linearized Adjoint; Shape Derivative; Shape Differential Calculus

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; CALCULATIONS; DIFFERENTIATION (CALCULUS); ERROR ANALYSIS; MESH GENERATION;

A-POSTERIORI ERROR ESTIMATIONS; ADAPTIVE MESH REFINEMENT; ADJOINTS; BERNOULLI; BERNOULLI FREE-BOUNDARY PROBLEM; FREE-BOUNDARY PROBLEMS; GOAL-ORIENTED ERROR ESTIMATION; LINEARIZED ADJOINT; SHAPE DERIVATIVES; SHAPE DIFFERENTIAL CALCULUS;

LINEARIZATION;

EID: 77950838818     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/080741239     Document Type: Article

References (45)

1. R. BECKER AND R. RANNACHER, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1-102.
2. F. BOUCHON, S. CLAIN, AND R. TOUZANI, A perturbation method for the numerical solution of the Bernoulli problem, J. Comput. Math., 26 (2008), pp. 23-36.
3. C. CUVELIER AND R. M. S. M. SCHULKES, Some numerical methods for the computation of capillary free boundaries governed by the Navier-Stokes equations, SIAM Rev., 32 (1990), pp. 355-423.
4. R. DAUTRAY AND J.-L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2: Functional and Variational Methods, Springer, Berlin, 1988.
5. M. C. DELFOUR AND J.-P. ZOLÉSIO, Structure of shape derivatives for nonsmooth domains, J. Funct. Anal., 104 (1992), pp. 1-33.
6. M. C. DELFOUR AND J.-P. ZOLÉSIO, Tangential differential equations for dynamical thin/shallow shells, J. Differential Equations, 128 (1996), pp. 125-167.
7. M. C. DELFOUR AND J.-P. ZOLESIO, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, SIAM Ser. Adv. Design and Control 4, Society for Industrial and Applied Mathematics, Philadelphia, 2001.
8. M. C. DELFOUR AND J.-P. ZOLÉSIO, Tangential calculus and shape derivatives, in Shape Optimization and Optimal Design: Proceedings of the IFIP Conference, Lect. Notes Pure Appl. Math. 216, J. Cagnol, M. P. Polis, and J.-P. Zolésio, eds., Marcel Dekker, New York, 2001, pp. 37-60.
9. F. R. DESAINT AND J.-P. ZOLÉSIO, Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), pp. 234-269.
10. W. DETTMER AND D. PERIĆ, A computational framework for free surface fluid flows accounting for surface tension, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 3038-3071.
11. R. DJELLOULI, C. FARHAT, J. MANDEL, AND P. VANĚK, Continuous Fréchet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problem, IMA J. Appl. Math., 63 (1999), pp. 51-69.
12. K. EPPLER AND H. HARBRECHT, Efficient treatment of stationary free boundary problems, Appl. Numer. Math., 56 (2006), pp. 1326-1339.
13. K. EPPLER, H. HARBRECHT, AND R. SCHNEIDER, On convergence in elliptic shape optimization, SIAM J. Control Optim., 46 (2007), pp. 61-83.
14. L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Stud. Adv. Math. 5, Chapman & Hall/CRC, Boca Raton, FL, 1992.
15. J. FERCHICHI AND J.-P. ZOLÉSIO, Shape sensitivity for the Laplace-Beltrami operator with singularities, J. Differential Equations, 196 (2004), pp . 340-384.
16. M. FLUCHER AND M. RUMPF, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math., 486 ( 1997), pp. 165-204.
17. G. FREMIOT AND J. SOKOLOWSKI, The structure theorem for the Eulerian derivative of shape functionals defined in domains with cracks, Sib. Math . J., 41 (2000), pp. 974-993.
18. Translation of Sibirsk. Mat. Zh., 41 (2000), pp. 1183-1203
19. J.-F. GERBEAU AND T. LELIÈVRE, Generalized Navier boundary condition and geometric conservation law for surface tension, Comput. Method s Appl. Mech. Engrg., 198 (2009), pp. 644-656.
20. M. B. GILES AND E. SÜLI, Adjoint methods for PDEs: A posteriori error analysis and post-processing by duality, Acta Numer., 11 (2002), pp. 145-236.
21. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman Publishing, London, 1985.
22. J. S. HADAMARD, Mémoire sur le probltème d'analyse relatif à l'équilibre des plaques élastiques encastrées, in uvres de Jacques Hadamard, C.N.R.S . Editions, Paris, 1968, pp. 515-641.
23. Originally published in Mém. Sav. Étrang., 33, 1907.
24. I. HARARI AND T. J. R. HUGHES, What are C and h?: Inequalities for the analysis and design of finite element methods, Comput. Methods Appl. M ech. Engrg., 97 (1992), pp. 157-192.
25. J. HASLINGER AND P. NEITTAANMÄKI, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd ed., Wiley, New York, 1996.
26. K. ITO, K. KUNISCH, AND G. H. PEICHL, Variational approach to shape derivatives for a class of Bernoulli problems, J. Math. Anal. Appl., 314 (200 6), pp. 126-149.
27. K. ITO, K. KUNISCH, AND G. H. PEICHL, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), pp. 517-539.
28. K. T. KÄRKKÄINEN, Shape Sensitivity Analysis for Numerical Solution of Free Boundary Problems, Ph.D. thesis, University of Jyväskylä, Jyväskylä , Finland, Jyväskylä Studies in Computing 58, 2005.
29. K. T. KÄRKKÄINEN AND T. TIIHONEN, Free surfaces: Shape sensitivity analysis and numerical methods, Internat. J. Numer. Methods Engrg., 44 (1999), pp. 1079-1098.
30. K. T. KÄRKKÄINEN AND T. TIIHONEN, Shape calculus and free boundary problems, in Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004, P. Neittaanmäki, T. Rossi , S. Korotov, E. Oñate, J. Périaux, and D. Knörzer, eds., Jyväskylä, Finland, 2004.
31. A. LAURAIN, Structure of shape derivatives in non-smooth domains and applications, Adv. Math. Sci. Appl., 15 (2005), pp. 199-226.
32. G. MEJAK, Numerical solution of Bernoulli-type free boundary value problems by variable domain method, Internat. J. Numer. Method Engrg., 37 (1994), pp. 4219-4245.
33. A. NOVRUZI AND M. PIERRE, Structure of shape derivatives, J. Evol. Equ., 2 (2002), pp. 365-382.
34. O. PIRONNEAU, Optimal Shape Design for Elliptic Systems, Springer Ser. Comput. Phys., Springer, Berlin, 1984.
35. S. PRUDHOMME AND J. T. ODEN, Computable error estimators and adaptive techniques for fluid flow problems, in Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Lect. Notes Comput. Sci. Eng. 25, T. J. Barth and H. Deconinck, eds., Springer, Berlin, 2003, pp. 207-268.
36. P. H. SAKSONO AND D. PERIĆ, On finite element modelling of surface tension. Variational formulation and applications-Part I: Quasistatic problems, Comput. Mech., 38 (2006), pp. 265-281.
37. J. SIMON, Differentiation with respect to the domain in boundary-value-problems, Numer. Funct. Anal. Optim., 2 (1980), pp. 649-687.
38. J. SOKOLOWSKI AND J.-P. ZOLÉSIO, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992.
39. E. SÜLI AND P. HOUSTON, Adaptive finite element approximation of hyperbolic problems, in Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Lect. Notes Comput. Sci. Eng. 25, T. J. Barth and H. Deconinck, eds., Springer, Berlin, 2003, pp. 269-344.
40. J. I. ToiVANEN, J. HASLINGER, AND R. A. E. MÄKINEN, Shape optimization of systems governed by Bernoulli free boundary problems, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 3802-3815.
41. E. H. VAN BRUMMELEN, Mesh association by projection along smoothed-normal-vector fields: Association of closed manifolds, Internat. J. Numer. Methods Engrg., 73 (2008), pp. 493-520.
42. K. G. VAN DER ZEE, Goal-Adaptive Discretization of Fluid-Structure Interaction,Ph.D. thesis, Technische Universiteit Delft, Delft, The Netherlands, June 2009. Available online from http://repository.tudelft.nl.
43. K. G. VAN DER ZEE, E. H. VAN BRUMMELEN, I. AKKERMAN, AND R. DE BORST, Goal-oriented error estimation and adaptivity for fluid-structure interaction using exact linearized adjoints, Comput. Methods Appl. Mech. Engrg., submitted.
44. K. G. VAN DER ZEE, E. H. VAN BRUMMELEN, AND R. DE BORST, Goal-oriented error estimation and adaptivity for free-boundary problems: The domain-map linearization approach, SIAM J. Sci. Comput., 32 (2010), pp. 1064-1092.
45. J.-P. ZOLÉSIO, Introduction to shape optimization problems and free boundary problems,in Shape Optimization and Free Boundaries, NATO ASI Ser. C: Math. Phys. Sci. 380, M. C. Delfour and G. Sabidussi, eds., Kluwer Academic Publishers, Dordrect, 1992, pp. 397-457.