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.

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.

In the chapter, problems connected with the numerical modeling of bioheat transfer processes are presented. In particular the non-homogeneous system of a burn wound and healthy tissue is considered. The heat exchange between sub-domains and environment is described by a system of partial differential equations (the Pennes equations) supplemented by adequate boundary conditions. The first goal of the research is the estimation of the changes of temperature fields due to perturbations in thermal parameters using the direct method of sensitivity analysis. Both the basic problem and additional ones concerning the sensitivity with respect to selected parameters are solved using the boundary element method. The second goal is the problem of burn wound shape identification. The additional information necessary to solve such a task results from the knowledge of temperature distribution on the external surface of skin tissue. At the stage of solving the inverse problem, a gradient method has been used. In the final part of the chapter the results of computations are shown.

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.

The nonlinear Poisson equation is considered, in w hich 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 iden tification of p1 and p2 the additional information connected with the knowledge of tempera ture T at the set of points (sensors) selected from the domain considered is necessary. T he 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 compu tations are shown. 1. Formulation of the problem The following 2D problem is considered [ ] : λ ( ) ( ) ( ) 0 : ( ) ( ) b x T T x + Q x = x T x = T x ∈Ω ∇ ∇ ∈Γ (1) where T is the temperature, x = (x1, x2) are the spatial coordinates, λ(T ) is the thermal conductivity, Q (x) is the source function, Tb (x) is known boundary temperature. We assume that 1 2 λ ( ) T = T + p p (2) where p1, p2 are the coefficients. When the direct problem is considered then all geom etrical and thermophysical parameters appearing in the mathematical model (1) are known. In the paper the inverse parametric problem is disc us ed in which it is assumed that the coefficients p1, p2 are unknown. To solve the inverse problem the addi tional information is necessary. So, we assume that the emperatures at the selected points x ∈Ω are given 1 2 ( ) 1 , 2 , . . . , i i i d d T = T , , i = N x x (3) where N is the number of sensors. Please cite this article as: Ewa Majchrzak, Katarzyna Freus, Sebastian Freus, Experiment design for estimation of temperature dependent thermal conductivity, Scientific Research of the Institute of Mathematics and Computer Science, 2010, Volume 9, Issue 1, pages 83-88. The website: http://www.amcm.pcz.pl/ E. Majchrzak, K. Freus, S. Freus 84 The accuracy of identification depends significantl y on the choice of sensors location and this problem is here discussed. 2. Algorithm of optimal sensors location Let X = {x, x,...,x } denotes the set of spatial points at which measur ements may be taken. The practical design problem consists in selection of corresponding weights w1, w2,...,wM which define the best experimental conditions [1]. To solve this problem the following iterative algorithm unde r the assumption that number of unknown parameters equals 2 and number of sensors e quals N can be applied [2]. At first, the sensitivity matrix is constructed

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.

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.

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.

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

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. 1. Formulation of the problem The following 2D problem is considered [ ] : λ ( ) ( ) ( ) 0 : ( ) ( ) b x T T x + Q x = x T x = T x ∈Ω ∇ ∇ ∈Γ (1) where T is the temperature, x = (x1, x2) are the spatial coordinates, λ(T ) is the thermal conductivity, Q (x) is the source function, Tb (x) is known boundary temperature. We assume that 1 2 λ ( ) T = T + p p (2) where p1, p2 are the coefficients. When the direct problem is considered then all geometrical and thermophysical parameters appearing in the mathematical model (1) are known. In the paper the inverse parametric problem is discussed in which it is assumed that the coefficients p1, p2 are unknown. To solve the inverse problem the additional information is necessary. So, we assume that the temperatures at the selected points x ∈Ω are given 1 2 ( ) 1 , 2 , . . . , i i i d d T = T , , i = N x x (3) where N is the number of sensors. Please cite this article as: Ewa Majchrzak, Katarzyna Freus, Sebastian Freus, Experiment design for estimation of temperature dependent thermal conductivity, Scientific Research of the Institute of Mathematics and Computer Science, 2010, Volume 9, Issue 1, pages 83-88. The website: http://www.amcm.pcz.pl/ E. Majchrzak, K. Freus, S. Freus 84 The accuracy of identification depends significantly on the choice of sensors location and this problem is here discussed. 2. Algorithm of optimal sensors location Let X = {x, x,...,x } denotes the set of spatial points at which measurements may be taken. The practical design problem consists in selection of corresponding weights w1, w2,...,wM which define the best experimental conditions [1]. To solve this problem the following iterative algorithm under the assumption that number of unknown parameters equals 2 and number of sensors equals N can be applied [2]. At first, the sensitivity matrix is constructed

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. 1. Formulation of the problem The two-dimensional elliptic equation is considered [ ] 0 ) ( ) ( ) ( : = + ∇ ∇ Ω ∈ x Q x U x D x (1) where D (x) is the coefficient of conductivity, Q (x) is the source function, x = = {x1, x2}, Ω = {x1, x2: −1≤x1≤1, −1≤x2≤1}. The function Q (x) is defined as follows [ ] 2 2 2 1 ) 75 . 0 ( ) 75 . 0 ( exp 10 ) ( − + − = x x x Q (2) while the conductivity coefficient is expressed as 1 2 1 2 ( ) D x = + p p x x (3) where p1, p2 are the parameters (p1 = 1.05, p2 = 4.09). The equation (1) is supplemented by Dirichlet boundary condition ( ) 0 x : U x = ∈Γ (4) It should be pointed out that the mathematical model presented above taken from [1] is connected with computer-assisted tomography. The aim of investigations is to solve the problem formulated and to determine the sensitivity functions ∂U/∂p1, ∂U/∂p2 using the direct differentiation method. Please cite this article as: Ewa Majchrzak, Katarzyna Freus, Sebastian Freus, Experiment design for parameters estimation of nonlinear Poisson equation Part I, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 1, pages 113-118. The website: http://www.amcm.pcz.pl/ E. Majchrzak, K. Freus, S. Freus 114 2. Sensitivity models The equation (1) in Cartesian co-ordinate takes the form 0 ) , ( ) , ( ) ( ) , ( ) ( 2 1 2 2 1 2 2 1 1 2 1 2 1 2 2 1 1 1 = +       ∂ ∂ + ∂ ∂ +       ∂ ∂ + ∂ ∂ x x Q x x x U p x x p x x x x U p x x p x (5)

In the paper the temperature field determination i n the domain of complex shape is presented. The boundary of the domain considered is described by the NURBS curves and the temperature field in this domain is calcula ted by means of the boundary element me- thod. Such approach allows to determine the changes of temperature due to the local change of boundary configuration. In the final part of the paper the examples of computations are shown. 1. Boundary element method for Laplace equation