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/
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