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1

Applying the topological derivative and the boundary element method to design of shape of the heat exchanger

Freus K., Freus S.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 2

5--12

EN

In this work, the topological derivative for the Laplace equation is used to solve a design problem. This derivative describes the sensitivity of the problem when a small hole is formed at an arbitrary point of the domain. The goal of this work is to design topology of the domain when the Robin condition is imposed on the holes. Physically, the holes can be construed as cooling channels. For finding the solution of the governing equation the boundary element method is applied. The final part of the paper presents the design of the heat exchanger and results of computations.

2

Determination of an optimal shape of domain using the topological derivative and the Boundary Element Method

Freus K., Freus S.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 4

41--48

EN

In the paper, the topological derivative for the Laplace equation is taken into account. The governing equation is solved by means of the Boundary Element Method. The topological-shape sensitivity method is used to determine the points showing the lowest sensitivities. On the selected points, material is eliminated by opening a hole, using the appropriate iterative process. This one is halted when a given amount of material is removed. The objective of this work is to obtain an optimal topology of the domain considered. In the final part of the paper, the example of computations is shown.

3

Adaptive finite elements based on sensitivities fortopological mesh changes

Friederich J., Leugering G., Steinmann P.

Control and Cybernetics

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2014

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Vol. 43, no. 2

279--306

EN

We propose a novel approach to adaptive refinement in FEM based on local sensitivities for node insertion. To this end, we consider refinement as a continuous graph operation, for instance by splitting nodes along edges. Thereby, we introduce the concept of the topological mesh derivative for a given objective function. For its calculation, we rely on the first-order asymptotic expansion of the Galerkin solution of a symmetric linear second-order elliptic PDE. In this work, we apply this concept to the total potential energy, which is related to the approximation error in the energy norm. In fact, our approach yields local sensitivities for minimization of the energy error by refinement. Moreover, we prove that our indicator is equivalent to the classical explicit a posteriori error estimator in a certain sense. Numerical results suggest that our method leads to efficient and competitive adaptive refinement.

4

Crack detection by the topological gradient method

Amstutz S., Horchani I., Masmoudi M.

Control and Cybernetics

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2005

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Vol. 34, no 1

81-101

EN

The topological sensitivity analysis consists in studying the behavior of a shape functional when modifying the topology of the domain. In general, the perturbation under consideration is the creation of a small hole. In this paper, the topological asymptotic expansion is obtained for the Laplace equation with respect to the insertion of a short crack inside a plane domain. This result is illustrated by some numerical experiments in the context of crack detection.