Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients

Anton Evgrafov; Misha Marie Gregersen; Mads Peter Sørensen

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 6, page 1059-1080
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Evgrafov, Anton, Gregersen, Misha Marie, and Sørensen, Mads Peter. "Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1059-1080. <http://eudml.org/doc/276352>.

@article{Evgrafov2011,
abstract = { 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. },
author = {Evgrafov, Anton, Gregersen, Misha Marie, Sørensen, Mads Peter},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Topology optimization; finite volume methods; topology optimization; nonlinear elliptic equation; convergence; Laplace equation; steady state heat conduction; numerical example},
language = {eng},
month = {6},
number = {6},
pages = {1059-1080},
publisher = {EDP Sciences},
title = {Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients},
url = {http://eudml.org/doc/276352},
volume = {45},
year = {2011},
}

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
DA - 2011/6//
PB - EDP Sciences
VL - 45
IS - 6
SP - 1059
EP - 1080
AB - 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.
LA - eng
KW - Topology optimization; finite volume methods; topology optimization; nonlinear elliptic equation; convergence; Laplace equation; steady state heat conduction; numerical example
UR - http://eudml.org/doc/276352
ER -

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