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1
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, 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.
3
Identification of internal hole parameters on the basis of boundary temperature
EN
The Laplace equation describing temperature field in 2D domain with an internal hole of circle shape supplemented by adequate boundary conditions is considered. On the basis of known temperature at the fragment of boundary the position of circle center or its radius is identified. To solve the inverse problem discussed the least square criterion is formulated, and next the gradient method coupled with the boundary element method is applied. To determine sensitivity coefficients the shape sensitivity analysis is used. In the final part of the paper the examples of computations are shown.
4
Shape sensitivity analysis with respect to the parameters of internal hole
EN
The Laplace equation describing temperature field in 2D domain supplemented by adequate boundary conditions is considered. The aim of investigations is to estimate the changes of temperature due to changes of shape parameter (e.g. radius or position of internal hole). To solve the problem, the implicit differentiation method of shape sensitivity analysis coupled with the boundary element method is applied.
5
Shape sensitivity analysis of eigenvalues revisited
EN
The paper can be considered as a complement to previous papers of the authors. An insight into applied asymptotic analysis of boundary value problems in singularly perturbed domains is presented. As a result, the asymptotic expansions of eigenvalues are obtained and discussed in terms of integral attributes of the geometrical perturbations including the virtual mass tensor, polarization tensor etc. The results are presented in such a way that can be easily employed in numerical methods for shape optimization and inverse problems.
6
Influence of casting dimensions on the solidification process
EN
The non-homogeneous casting-mould domain is considered and the perturbation of the solidification process due to the changes of geometrical dimensions is analyzed. The methods of shape sensitivity analysis are used. On the stage of numerical realization the finite difference method is applied. In the final part of the paper the example of computations is shown.
8
Shape sensitivity analysis of temperature distribution in heating tissue
EN
In the paper the numerical analysis of heat transfer process proceeding in domain of biological tissue is presented (2D problem). The temperature distribution results from the action of external heat source emanating the boundary heat flux. The sensitivity analysis of this process with respect to the geometrical parameters of biological tissue is considered. The explicit differentiation method basing on the material derivative approach is used. On the stage of numerical computations the boundary element method is applied. In the final part of the paper the results obtained are shown.
9
Application of boundary element method to shape sensitivity analysis
EN
In the paper the boundary element method to shape sensitivity analysis is applied. The Laplace equation is analyzed and the aim of investigations is to estimate the changes of temperature in the 2D domain due to the change of local geometry of the boundary. Here the implicit differentiation method of shape sensitivity analysis is used. In the final part of the paper the example of numerical computations is shown.
10
Topological derivative for linear elastic plate bending problems
EN
This study concerns the application of the Topological-Shape Sensitivity Method as a systematic procedure to determine the Topological Derivative for linear elastic plate bending problems within the framework of Kirchhoff's kinematic approach. This method, based on classical Shape Sensitivity Analysis, leads to a constructive procedure to obtain the Topological Derivative. Utilising the well known terminology of structural optimization, we adopt, the total potential strain energy as the cost function and the equilibrium equation as the constraint. Variational formulation as well as the direct differentiation method are used to perform the shape derivative of the cost function. Finally, in order to obtain a uniform distribution of bending moments in several plate problems, the Topological Derivative was approximated, by the Finite Element Method, and used to find the best place to insert holes. A simple hard-kill like topology algorithm, which furnishes satisfactory qualitative results in agreement with those reported in the literature, is also shown.
EN
In the paper the influence of the casting geometrical parameters on the course of solidification process is discussed. In order to solve the problem, the methods of sensitivity analysis are applied. Here the direct approach of shape sensitivity analysis is used and then the special methods of the governing equations and boundary initial conditions differentiation should be introduced. The basic problem and additional one connected with the shape sensitivity analysis are solved using the boundary element method. Th e BEM algorithm for parabolic equations is adapted to the solidification process modelling by the introduction of numerical procedure called the artificial heat source method. In the final part of the paper the results of computations are presented.
PL
W artykule analizowano wpływ parametrów geometrycznych odlewu na przebieg procesu krzepnięcia. W celu rozwiązania tak sformułowanego zadania zastosowano metodę analizy wrażliwości. Wykorzystano tzw. podejście bezpośrednie bazujące na definicji pochodnej materialnej, które polega na różniczkowaniu równań tworzących opis matematyczny procesu względem parametru kształtu. Zadanie podstawowe i dodatkowe związane z analizą wrażliwości rozwiązano stosując metodę elementów brzegowych uzupełnioną procedurą sztucznego źródła ciepła. W końcowej części artykułu przedstawiono wyniki obliczeń numerycznych.
EN
In the paper the numerical analysis of heat transfer process proceeding in domain of biological tissue is presented. In particular, the 1D problem is considered. The temperature distribution results from the action of the disk emanating the constant heat flux. The problem corresponds, among others, to chronic heating caused by implants being a heat source [1]. The sensitivity analysis of this process with respect to the thickness of biological tissue is presented. On the stage of numerical computations the boundary element method is used. In the final part of the paper the results obtained are shown.
EN
In the paper the sensitivity analysis of soliditication process with respect to the casting and mould thicknesses (problem 1D) is presented. The aim of investigations is to estimate the change of temperature in the system considered due to a change of position of contact surface between casting and mould sub-domains.
PL
W pracy przedstawiono analizę wrażliwości procesu krzepnięcia ze względu na grubość odlewu i formy odlewniczej (zadanie 1D). Głównym celem badań było oszacowanie zmian temperatury w rozważanym układzie spowodowanych zmianą położenia granicy kontaktu między odlewem i formą.
14
Shape sensitivity analysis in nonlinear transient heat transfer
EN
In the paper the 1D model of nonlinear transient heat transfer process is assumed. Thermophysical parameters of the material considered are strongly temperature dependent. The sensitivity analysis of this process with respect to the thickness of plate is presented. On the stage of numerical computations the finite difference method is used. In the final part of the paper the results obtained are shown.
15
EN
This study concerns the application of shape sensitivity analysis as a systematic methodology to determine the energy release rate of cracked shells, within the framework of a linear elastic approach that takes into account the effect of transverse shear deformation. This methodology and the direct method of shape sensitivity analysis is applied to shells with an arbitrary middle surface and leads to an explicit general expression for the shape sensitivity of the total potential strain energy. In elastic shells with cracks, crack initiation is simulated by a change of shape characterized by a suitable tangential velocity distribution over the middle surface of the shell. In this case, a useful expression of energy release rate is expressed in terms of the strain-stress state and the adopted shape change velocity field. Finally, shape sensitivity analysis is applied to the circular cylindrical shell and thus the condition of null divergence of the corresponding Eshelby tensor is verified.
PL
Badania opisane w pracy dotyczą zastosowania analizy wrażliwości kształtu jako systematycznej metodologii wyznaczania tempa uwalnianej energii powłok z pęknięciami w ramach liniowego podejścia uwzględniającego efekt deformacji od ścinania poprzecznego. Ta metodologia i bezpośrednia analiza wrażliwości kształtu została zastosowana do powłok o dowolnej powierzchni środkowej, pozwalając na znalezienie jawnego i ogólnego wyrażenia na pochodną całkowitej energii odkształcenia. W podatnych powłokach z pęknięciami symulację inicjacji pęknięcia dokonano na podstawie zmiany kształtu określonej odpowiednim rozkładem prędkości powierzchni środkowej powłoki. W takim przypadku użyteczną formułę określającą tempo uwalnianej energii wyznaczono w funkcji stanu naprężenia i odkształcenia oraz zmian rozkładu pola prędkości powierzchni środkowej. Na koniec, analizę wrażliwości kształtu zastosowano do szczególnego przypadku powłoki cylindrycznej, gdzie warunek zerowej dywergencji odpowiadającego tensora Eshelby'ego został potwierdzony.
16
Sensitivity analysis of solidification with respect to the mould thickness
EN
In the paper the sensitivity analysis of solidification process with respect to the mould thickness is presented. On the stage of numerical computations the boundary element method is used. In the final part of the paper the results obtained are shown.
PL
W pracy przedstawiono analizę wrażliwości procesu krzepnięcia ze względu na grubość formy odlewniczej. Na etapie obliczeń wykorzystano metodę elementów brzegowych. W końcowej części artykułu pokazano wyniki obliczeń.
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