A moving mesh fictitious domain approach for shape optimization problems
ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 1, pp. 31-45.
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     title = {A moving mesh fictitious domain approach for shape optimization problems},
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     url = {http://www.numdam.org/item/M2AN_2000__34_1_31_0/}
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Mäkinen, Raino A. E.; Rossi, Tuomo; Toivanen, Jari. A moving mesh fictitious domain approach for shape optimization problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 1, pp. 31-45. http://www.numdam.org/item/M2AN_2000__34_1_31_0/

[1] G.P. Astrakhantsev, Method of fictitious domains for a second-order elliptic equation with natural boundary conditions. USSR Comput. Math. Math. Phys. 18 (1978) 114-121. | MR | Zbl

[2] C. Atamian, G.V. Dinh, R. Glowinski, J. He and J. Périaux, On some imbedding methods applied to fluid dynamics and electro-magnetics. Comput. Methods Appl. Mech. Engrg. 91 (1991) 1271-1299. | MR

[3] I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. | MR | Zbl

[4] A. Bespalov, Yu.A. Kuznetsov, O. Pironneau and M.-G. Vallet, Fictitious domain with separable preconditioners versus unstructured adapted meshes. Impact Comput. Sci. Eng. 4 (1992) 217-249. | MR | Zbl

[5] C. Börgers, A triangulation algorithm for fast elliptic solvers based on domain imbedding. SIAM J. Numer. Anal. 27 (1990). 1187-1196. | MR | Zbl

[6] C. Börgers and O.B. Widlund, On finite element domain imbedding methods. SIAM J. Numer. Anal. 27 1990) 963-978. | MR | Zbl

[7] V. Braibant and C. Fleury, Shape optimal design using B-splines. Comput. Methods Appl. Mech. Engrg. 44 (1984) 247-267. | Zbl

[8] J.H. Bramble, The Lagrangian multiplier method for Dirichlet's problem. Math. Comp. 37 (1981) 1-11. | MR | Zbl

[9] J.H. Bramble, J.E. Pasciak and A.H. Schatz, The construction of preconditioners for elliptic problems by substructuring, I. Math. Comp. 47 (1986) 103-134. | MR | Zbl

[10] R.A. Brockman, Geometric sensitivity analysis with isoparametric finite elements. Comm. Appl. Numer. Math. 3 (1987) 495-499. | Zbl

[11] T.F. Chan, Analysis of preconditioners for domain decomposition. SIAM J. Numer. Anal. 24 (1987) 382-390. | MR | Zbl

[12] J. Daňková and J. Haslinger, Fictitious domain approach used in shape optimization: Neumann boudary condition, in Control of Partial Differential Equations and Applications (Laredo, 1994), Lecture Notes in Pure and Appl. Math., Dekker, New York 174 (1996) 43-49. | MR | Zbl

[13] J. Daňková and J. Haslinger, Numerical realization of a fictitious domain approach used in shape optimization. I. Distributed controls. Appl. Math. 41 (1996) 123-147. | MR | Zbl

[14] P. Duysinx, W.H. Zhang and C. Fleury, Sensitivity analysis with unstructured free mesh generators in 2-D and 3-D shape optimization, in Structural Optimization 93, Vol. 2, Rio de Janeiro (1993) 205-212.

[15] P.E. Gill, W. Murray and M.H. Wright, Practical Optimization. Academic Press, New York (1981). | MR | Zbl

[16] R. Glowinski, T. Hesla, D.D. Joseph, T.-W. Pan and J. Périaux, Distributed Lagrange multiplier methods for particulate flows, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M. F.Wheeler Eds., Wiley, Chichester (1997) 270-279. | Zbl

[17] R. Glowinski and Yu.A. Kuznetsov, On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrande multiplier method. CR. Acad. Sci. Paris Sér. I Math. 327 (1998) 693-698. | MR | Zbl

[18] R. Glowinski, T.-W. Pan, A.J. Kearsley and J. Périaux, Numerical simulation and optimal shape for viscous flow by a fictitious domain method. Internat J. Numer. Methods Fluids 20 (1995) 695-711. | MR | Zbl

[19] R. Glowinski, T.-W. Pan and J. Périaux, A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg. 111 (1994) 283-303. | MR | Zbl

[20] A. Greenbaum, Iterative Methods for Solving Linear Systems, Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, USA 17 (1997). | MR | Zbl

[21] J. Haslinger, Imbedding/control approach for solving optimal shape design problems. East-West J. Numer. Math. 1 (1993) 111-119. | MR | Zbl

[22] J. Haslinger, Comparison of different fictitious domain approaches used in shape optimization. Tech. Rep. 15, Laboratory of Scientific Computing, University of Jyväskylä (1996).

[23] J. Haslinger, K.H. Hoffmann and M. Kočvara, Control/fictitious domain method for solving optimal shape design problems. RAIRO Modél. Math. Anal. Numér. 27 (1993) 157-182. | Numdam | MR | Zbl

[24] J. Haslinger and D. Jedelský, Genetic algorithms and fictitious domain based approaches in shape optimization. Structural Optimization 12 (1996) 257-264.

[25] J. Haslinger and A. Klarbring, Fictitious domain/mixed finite element approach for a class of optimal shape design problems. RAIRO Modél. Math. Anal. Numér. 29 (1995) 435-450. | Numdam | MR | Zbl

[26] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd ed., Wiley, Chichester (1996). | MR | Zbl

[27] J. He, Méthodes de domaines fictifs en méchanique des fluides applications aux écoulements potentiels instationnaires autour d'obstacles mobiles. Ph.D. thesis, Université Paris VI (1994).

[28] E. Heikkola, Y. Kuznetsov, T. Rossi and P. Tarvainen, Efficient preconditioners based on fictitious domains for elliptic FE-problems with Lagrange multipliers, in ENUMATH 97 - Proceedings of the 2nd European Conference on Numerical Mathematics and Advanced Applications, H.G. Bock, G- Kanschat, R. Rannacher, F. Brezzi, R. Glowinski, Yu.A. Kuznetsov and J.Périaux Eds., World Scientific Publishing Co., Inc., River Edge, NJ (1998) 646-661. | MR | Zbl

[29] K. Kunisch and G. Peichl, Shape optimization for mixed boundary value problems based on an embedding method. Dynam. Contin. Discrete Impuls. Systems 4 (1998) 439-478. | MR | Zbl

[30] Yu.A. Kuznetsov, Efficient iterative solvers for elliptic finite element problems on nonmatching grids. Russian J. Numer. Anal. Math. Modelling 10 (1995) 187-211. | MR | Zbl

[31] Yu.A. Kuznetsov, Iterative analysis of finite element problems with Lagrange multipliers, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler Eds., Wiley, Chichester (1997) 170-178. | Zbl

[32] Yu.A. Kuznetsov and M.F. Wheeler, Optimal order substructuring preconditioners for mixed finite element methods on nonmaching grids, East-West J. Numer. Math. 3 (1995) 127-143. | MR | Zbl

[33] R. Mäkinen, Finite-element design sensitivity analysis for non-linear potential problems. Comm. Appl. Numer. Math. 6 (1990) 343-350. | MR | Zbl

[34] G.I. Marchuk, Yu.A. Kuznetsov and A.M. Matsokin, Fictitious domain and domain decomposition methods. Soviet J. Numer. Anal. Math. Modelling 1 (1986) 3-35. | MR | Zbl

[35] NAG, The NAG Fortran Library Manual: Mark 18. NAG Ltd, Oxford (1997).

[36] C.C. Paige and M.A. Saunders, Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12 (1975) 617-629. | MR | Zbl

[37] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York (1984). | MR | Zbl

[38] W. Proskurowski and P. S. Vassilevski, Preconditioning capacitance matrix problems in domain imbedding. SIAM J. Sci. Comput. 15 (1994) 77-88. | MR | Zbl

[39] T. Rossi, Fictitious Domain Methods with Separable Preconditioners. Ph.D. thesis, Department of Mathematics, University of Jyväskylä (1995). | Zbl

[40] T. Rossi and J. Toivanen, A parallel fast direct solver for block tridiagonal Systems with separable matrices of arbitrary dimension. SIAM J. Sci. Comput. 20 (1999) 1778-1793. | MR | Zbl

[41] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer-Verlag, Berlin (1992). | MR | Zbl

[42] P.N. Swarztrauber, The methods of cyclic reduction and Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Rev. 19 (1977) 490-501. | MR | Zbl

[43] J. Toivanen, Fictitious Domain Method Applied to Shape Optimization. Ph.D. thesis, Department of Mathematics, University of Jyväskylä (1997). | MR | Zbl

[44] L. Tomas, Optimisation de Forme et Domaines Fictifs: Analyse de Nouvelles Formulations et Aspects Algorithmiques. Ph.D. thesis, École Centrale de Lyon (1997).