We present a convergence analysis of a cell-based finite volume (FV) discretization scheme applied to a problem of control in the coefficients of a generalized Laplace equation modelling, for example, a steady state heat conduction. Such problems arise in applications dealing with geometric optimal design, in particular shape and topology optimization, and are most often solved numerically utilizing a finite element approach. Within the FV framework for control in the coefficients problems the main difficulty we face is the need to analyze the convergence of fluxes defined on the faces of cells, whereas the convergence of the coefficients happens only with respect to the “volumetric” Lebesgue measure. Additionally, depending on whether the stationarity conditions are stated for the discretized or the original continuous problem, two distinct concepts of stationarity at a discrete level arise. We provide characterizations of limit points, with respect to FV mesh size, of globally optimal solutions and two types of stationary points to the discretized problems. We illustrate the practical behaviour of our cell-based FV discretization algorithm on a numerical example.
Mots-clés : topology optimization, finite volume methods
@article{M2AN_2011__45_6_1059_0, author = {Evgrafov, Anton and Gregersen, Misha Marie and S{\o}rensen, Mads Peter}, title = {Convergence of {Cell} {Based} {Finite} {Volume} {Discretizations} for {Problems} of {Control} in the {Conduction} {Coefficients}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1059--1080}, publisher = {EDP-Sciences}, volume = {45}, number = {6}, year = {2011}, doi = {10.1051/m2an/2011012}, mrnumber = {2833173}, zbl = {1269.65107}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2011012/} }
TY - JOUR AU - Evgrafov, Anton AU - Gregersen, Misha Marie AU - Sørensen, Mads Peter TI - Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 1059 EP - 1080 VL - 45 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011012/ DO - 10.1051/m2an/2011012 LA - en ID - M2AN_2011__45_6_1059_0 ER -
%0 Journal Article %A Evgrafov, Anton %A Gregersen, Misha Marie %A Sørensen, Mads Peter %T Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 1059-1080 %V 45 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011012/ %R 10.1051/m2an/2011012 %G en %F M2AN_2011__45_6_1059_0
Evgrafov, Anton; Gregersen, Misha Marie; Sørensen, Mads Peter. Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 6, pp. 1059-1080. doi : 10.1051/m2an/2011012. http://www.numdam.org/articles/10.1051/m2an/2011012/
[1] http://www.openfoam.com.
[2] Topology optimization of large scale Stokes flow problems. Struct. Multidisc. Optim. 35 (2008) 175-180. | MR | Zbl
, , and ,[3] Lectures on elliptic boundary value problems. Van Nostrand, Princeton, N.J. (1965). | MR | Zbl
,[4] Conception optimale de structures, Mathématiques et Applications 58. Springer (2007). | MR | Zbl
,[5] An optimal design problem with perimeter penalization. Calc. Var. Partial Differential Equations 1 (1993) 55-69. | MR | Zbl
and ,[6] Topology optimization of microfluidic mixers. Int. J. Numer. Methods Fluids 61 (2008) 498-513. | MR | Zbl
, and ,[7] Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. SIAM (2006) 648. ISBN 9780898716009. | MR | Zbl
, and ,[8] Nonlinear Programming. John Wiley & Sons, Inc, New York (1993). | MR | Zbl
, and ,[9] Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Engrg. 71 (1988) 197-224. CODEN CMMECC. ISSN 0045-7825. | MR | Zbl
and ,[10] Topology Optimization: Theory, Methods, and Applications. Springer-Verlag, Berlin (2003). 370. ISBN 3-540-42992-1. | MR | Zbl
and ,[11] Perturbation Analysis of Optimization Problems. Springer-Verlag, New York (2000), p. 601. ISBN 0-387-98705-3. | MR | Zbl
and ,[12] Topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Fluids 41 (2003) 77-107. CODEN IJNFDW. ISSN 0271-2091. | MR | Zbl
and ,[13] Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer-Verlag, Berlin (1989). x+308 ISBN 3-540-50491-5. | MR | Zbl
,[14] Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comput. 79 (2010) 1303-1330. | MR
and ,[15] Measure theory and fine properties of functions. CRC Press (1992). | MR | Zbl
and ,[16] On the limits of porous materials in the topology optimization of Stokes flows. Appl. Math. Optim. 52 (2005) 263-267. | MR | Zbl
,[17] Topology optimization of slightly compressible fluids. Z. Angew. Math. Mech. 86 (2005) 46-62. | MR | Zbl
,[18] Topology optimization of fluid problems by the lattice Boltzmann method, in IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials: Status and Perspectives, edited by M.P. Bendsøe, N. Olhoff and O. Sigmund. Springer, Netherlands (2006) 559-568.
, and ,[19] Topology optimization of fluid domains: Kinetic theory approach. Z. Angew. Math. Mech. 88 (2008) 129-141. | MR | Zbl
, and ,[20] Topology optimization for nano-scale heat transfer. Int. J. Numer. Methods Engrg. 77 (2009) 285. ISSN 00295981. | Zbl
, , and ,[21] Finite volume methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions 7. North Holland (2000) 713-1020. | MR | Zbl
, and ,[22] A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal 26 (2006) 326-353. http://imajna.oxfordjournals.org/cgi/content/abstract/26/2/326. | MR | Zbl
, and ,[23] Analysis tools for finite volume schemes. Acta Math. Univ. Comenianae LXXVI (2007) 111-136. | MR | Zbl
, , and ,[24] Topology optimization of three-dimensional linear elastic structures with a constraint on “perimeter”. Comput. Struct. 73 (1999) 583-594. CODEN CMSTCJ. ISSN 0045-7949. | MR | Zbl
, and ,[25] Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1935-1972. http://link.aip.org/link/?SNA/37/1935/1. | MR | Zbl
, and ,[26] Topology optimization of heat conduction problems using the finite volume method. Struct. Multidisc. Optim. 31 (2006) 251-259. ISSN 1615-147X. | MR | Zbl
, and ,[27] Topology optimization of channel flow problems. Struct. Multidisc. Optim. 30 (2005) 181-192. | MR | Zbl
, and ,[28] Topology and shape optimization of induced-charge electro-osmotic micropumps. New J. Phys. 11 (2009) 075019. http://stacks.iop.org/1367-2630/11/i=7/a=075019.
, , and ,[29] Perimeter constrained topology optimization of continuum structures, in IUTAM Symposium on Optimization of Mechanical Systems (Stuttgart, 1995). Solid Mech. Appl. 43. Kluwer Acad. Publ., Dordrecht (1996) 113-120. | MR | Zbl
, and ,[30] Autoduct: topology optimization for fluid flow, in Proceedings of Konferenz für angewandte Optimierung. Karlsruhe (2006).
, and ,[31] Topology optimization of flexible micro-fluidic devices. Struct. Multidisc. Optim. 42 (2010) 495-516. ISSN 1615-147X. http://dx.doi.org/10.1007/s00158-010-0526-6.
, , and ,[32] Applied shape optimization for fluids. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (2001) xvi+251. ISBN 0-19-850743-7. | MR | Zbl
and ,[33] Bionic optimization of air-guiding systems, in Proceedings of SAE 2004 World Congress & Exhibition. Detroit, MI, USA, Society of Automotive Engineering, Inc (2004) 95-100.
, and ,[34] Design of micro-fluidic bio-reactors using topology optimization. J. Comput. Theoret. Nano. 4 (2007) 814-816.
and ,[35] A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow. Int. J. Numer. Meth. Engrg. 65 (2006) 975-1001. | MR | Zbl
, and ,[36] A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Internat. J. Numer. Methods Fluids 58 (2008). | MR | Zbl
,[37] Computation of topological sensitivities in fluid dynamics: Cost function versatility, in ECCOMAS CFD 2006, Delft (2006).
, and ,[38] Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998) xxii+273. ISBN 0-7923-5170-3. | MR | Zbl
, and ,[39] Some convergence results in perimeter-controlled topology optimization. Comput. Methods Appl. Mech. Engrg. 171 (1999) 123-140. | MR | Zbl
,[40] A parallel Schur complement solver for the solution of the adjoint steady-state lattice Boltzmann equations: application to design optimization. Int. J. Comput. Fluid Dynamics 22 (2008) 464-475. | MR | Zbl
, and ,[41] Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Comput. Fluids 38 (2009) 910-923. | MR | Zbl
, and ,[42] A parametric level-set approach for topology optimization of flow domains. Struct. Multidisc. Optim. 41 (2010) 117-131. ISSN 1615-147X. http://dx.doi.org/10.1007/s00158-009-0405-1. | MR | Zbl
, , and ,[43] The method of moving asymptotes-a new method for structural optimization. Int. J. Numer. Methods Engrg. 24 (1987) 359-373. CODEN IJNMBH. ISSN 0029-5981. | MR | Zbl
,[44] A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12 (2002) 555-573. ISSN 1095-7189. | MR | Zbl
,[45] Convergence of an algorithm in optimal design. Struct. Optim. 13 (1997) 195-198.
,[46] Megapixel topology optimization on a graphics processing unit. SIAM Rev. 5 (2009) 707-721. | MR | Zbl
and ,[47] Applied Functional Analysis: Main Principles and Their Applications, 1st edition. Springer (1995). ISBN 0387944222. | MR | Zbl
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