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Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material). Using a finite elements discretization (FEM) of the PDE and functional we obtain an integer programming problem. Due to approximation error of FEM discretization, optimization problem becomes mesh-depended and possess false, physically inadequate optimums, while functional value heavily depends on fineness of discretization scheme used to compute it. To alleviate this problem, we propose regularization of given functional by error estimate of FEM discretization. This regularization provides robustness of solutions and improves obtained functional values as well. While the idea is broadly applicable, in this paper we apply our method to the heat conduction optimization. This type of problems are of practical importance in design of heat conduction channels, heat sinks and other types of heat guides.
This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori error estima
We derive a posteriori error estimates in the $L_infty((0,T];L_infty(Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estim
Many practical problems occur due to the boundary value problem. This paper evaluates the finite element solution of the boundary value problem of Poissons equation and proposes a novel a posteriori local error estimation based on the Hypercircle met
The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Youngs modulus is an affine function of a co
Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standa