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EN
The aim of this paper is to create an optimal shape of the 2D domain that is described by the Non-Uniform Rational B-Splines (NURBS) curves. This work presents a method based on the topological derivative for the Laplace equation that determines the sensitivity of a given cost function to the change of its topology. As a numerical approach, the boundary element method is considered. To check the effectiveness of the proposed approach, the example of computations was carried out.
EN
In order to achieve the desired topology we often have to remove material of the area considered. This work presents the author's algorithm which can be used in the reconstruction of the boundary of domain after elimination of a certain amount of material. The paper introduces some details about the procedure that allows one to achieve the expected shape of a domain. The topological-shape sensitivity method for the Laplace equation is used to obtain an optimal topology, whereas numerical methodology utilizes the boundary element method. In the conclusion of the paper the example of computation is shown.
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.
EN
A non-homogeneous system being the composition of burn wound and healthy tissue is considered. The heat exchange between sub-domains and environment is described by the system of partial differential equations (the Pennes equations) supplemented by the assumed boundary conditions. Additional problems associated with sensitivity analysis with respect to thermal parameters occurring in the mathematical model are formulated. Both the basic problem and additional ones concerning the sensitivity with respect to selected parameters are solved using the boundary element method. In the final part of the paper the results of computations are shown.
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.
EN
In the paper, the position of the boundary between burned and healthy tissue is described by the NURBS curve. The temperature field in the domain is calculated by means of the boundary element method. The influence of discretization on the temperature distribution in the burned and healthy skin tissue is analysed. Different numbers of boundary elements and internal cells are taken into account. In the final part of the paper the examples of computations are shown.
EN
In the paper the burned and healthy layers of skin tissue are considered. The temperature distribution in these layers is described by the system of two Pennes equations. The governing equations are supplemented by the boundary conditions. On the external surface the Robin condition is known. On the surface between burned and healthy skin the ideal contact is considered, while on the internal surface limiting the system the body temperature is taken into account. The problem is solved by means of the boundary element method.
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Content available remote Shape sensitivity analysis of temperature distribution in a non-homogeneous domain
EN
The heated non-homogeneous domain from the two sub-domains compound is considered. The temperature distribution is described by the system of two Laplace equations. At the surface Γ c between sub-domains the ideal contact is assumed, at the remaining surfaces the Dirichlet, Neumann and Robin conditions are taken into account. The problem is solved by means of the boundary element method. To estimate the changes of temperature due to the change of local geometry of internal boundary Γ c the implicit variant of shape sensitivity analysis is applied. In the final part, the results of computations are shown and the conclusions are formulated.
EN
In the paper two sub-domains which are in thermal contact are considered. The temperature field in these domains is described by the system of two Laplace equations supplemented by the boundary conditions. The position of surface between sub-domains is unknown. Additional information necessary to solve the identification problem results from a knowledge of external surface temperature distribution. The direct problem is solved using the boundary element method. To solve the inverse problem formulated the gradient method is applied. In the final part of the paper the results of computations are shown. The algorithm proposed here can be used, among others, in the medical practice (e.g. in burns therapy).
10
Content available remote Shape sensitivity analysis : implicit approach using boundary element method
EN
The Laplace equation (2D problem) supplemented by boundary conditions is analyzed. To estimate the changes of temperature in the 2D domain due to the change of local geometry of the boundary, the implicit method of sensitivity analysis is used. In the final part of the paper, the example of numerical computations is shown.
EN
In the paper, a 2D domain in which the temperature field is described by the Laplace equation and the assumed boundary conditions is considered. To estimate the changes of temperature due to the change of the boundary local geometry, the implicit approach of shape sensitivity analysis is used. In the final part of paper, examples of numerical computations are shown and conclusions are formulated.
12
Content available remote Sensors location for estimation of temperature dependent thermal conductivity
EN
Nonlinear Poisson equation is considered in which the thermal conductivity is a function of temperature ë(T )=p1T+p2, where p1, p2 are the unknown parameters. To solve the inverse problem the additional information connected with the knowledge of temperature T at the set of points (sensors) selected from the domain considered is necessary. The fundamental problem is the selection of sensors location and here the algorithm assuring the optimal sensors location is presented.
13
Content available remote Determination of temperature field using the bem and b-splines (nurbs)
EN
Temperature field determination in the domain of complex shape is presented. The boundary of the domain considered is described by the NURBS curves. The temperature field in this domain is calculated by means of the boundary element method.
14
Content available remote Experiment design for estimation of temperature dependent thermal conductivity
EN
The nonlinear Poisson equation is considered, in which the thermal conductivity is a function of temperature λ (T ) = p1T+p2, where p1, p2 are the unknown parameters. To solve the inverse problem consisting in the identification of p1 and p2 the additional information connected with the knowledge of temperature T at the set of points (sensors) selected from the domain considered is necessary. The fundamental problem is the selection of sensors location and here the algorithm assuring the optimal sensors location is proposed. In the final part of the paper the results of computations are shown.
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Content available remote Experiment design for parameters estimation of nonlinear Poisson equation. Part 1
EN
In this part of the paper the nonlinear Poisson equation is considered, this means the conductivity is a function of the form D(x) = p1x1x2+p2, where p1, p2 are the parameters and x = {x1, x2}, − 1 ≤ x1, x2 ≤ 1 are the spatial co-ordinates. Sensitivity analysis with respect to parameters p1, p2 using the direct differentiation approach is discussed. The basic problem and additional ones are solved using the finite difference method. In the final part of the paper the results of computations are shown.
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Content available remote Experiment design for parameters estimation of nonlinear Poisson equation. Part 2
EN
In this part of the paper the inverse problem consisting in the identification of unknown parameters p1, p2 appearing in conductivity D(x) = p1x1x2+p2 where x = {x1, x2}, − 1 ≤ x1, x2 ≤ 1 is analyzed. To solve this problem the additional information connected with the knowledge of function U(x) at the set of points (sensors) selected from the domain considered is necessary. The fundamental problem is the selection of sensors location and here the algorithm assuring the optimal sensors location is presented. In the final part of the paper the results of computations are shown.
EN
The temperature dependent thermal conductivity of metal is identified. To solve the problem the gradient method is used. On the stage of numerical computations boundary element method is applied, at the same time in order to simplify the algorithm the Kirchhoff transformation is introduced. The example of computations is also shown.
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