Boundary reconstruction in two-dimensional steady state anisotropic heat conduction using a regularized meshless method (original) (raw)
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Boundary reconstruction in two-dimensional steady-state anisotropic heat conduction
Research Square (Research Square), 2023
We study the reconstruction of an unknown/inaccesible smooth inner boundary from the knowledge of the Dirichlet condition (temperature) on the entire boundary of a doubly connected domain occupied by a two-dimensional homogeneous anisotropic solid and an additional Neumann condition (normal heat flux) on the known, accessible, and smooth outer boundary in the framework of steady-state heat conduction with heat sources. This inverse geometric problem is approached through an operator that maps an admissible inner boundary belonging to the space of 2π−periodic and twice continuously differentiable functions into the Neumann data on the outer boundary which is assumed to be continuous. We prove that this operator is differentiable and hence a gradient-based method that employs the anisotropic single layer representation of the solution to an appropriate Dirichlet problem for the two-dimensional anisotropic heat conduction, is developed for approximating the unknown inner boundary. Numerical results are presented for both exact and perturbed Neumann data on the outer boundary and show the convergence, stability, and robustness of the proposed method.
Advances in Mathematical Physics, 2015
We propose a new meshless method to solve a backward inverse heat conduction problem. The numerical scheme, based on the fundamental solution of the heat equation and radial basis functions (RBFs), is used to obtain a numerical solution. Since the coefficients matrix is ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulted system of linear equations. Also, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. A test problem demonstrates the stability, accuracy, and efficiency of the proposed method.
Inverse Problems in Engineering Mechanics III, 2002
A 3-D finite element method (FEM) formulation for the detection of unknown steady boundary conditions in heat conduction is presented. The present FEM formulation is capable of determining temperatures and heat fluxes on the boundaries where such quantities are unknown or inaccessible, provided such quantities are sufficiently over-specified on other boundaries. A regularized form of the method is also presented. The regularization is necessary for solving problems where the over-specified boundary data contains errors. Details of the discretization and regularization and sample results for a 3-D problem are presented. KEYWORDS inverse problems, finite element method, inverse heat conduction
Singularities in anisotropic steady-state heat conduction using a boundary element method
International Journal for Numerical Methods in Engineering, 2002
In many heat conduction problems, boundaries with sharp corners or abrupt changes in the boundary conditions give rise to singularities of various types which tend to slow down the rate of convergence with decreasing mesh size of any standard numerical method used for obtaining the solution. In this paper, it is shown how this di culty may be overcome in the case of an anisotropic medium by a modiÿed boundary element method. The standard boundary element method is modiÿed to take account of the form of the singularity, without appreciably increasing the amount of computation involved. Two test examples, the ÿrst with a singularity caused by an abrupt change in a boundary condition and the second with a singularity caused by a sharp re-entrant corner, are investigated and numerical results are presented. Copyright ? 2002 John Wiley & Sons, Ltd.
Applied Mathematics and Computation, 2013
This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the quasireversibility regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.
Inverse Determination of Steady Boundary Conditions in Heat Transfer and Elasticity
In the case of steady heat conduction and steady elasticity there is often a problem of not knowing temperatures and heat fluxes or tractions and deformations boundary conditions on some parts of the boundary. However, there should be a sufficient amount of over-specified boundary conditions available on at least a portion of the remaining boundary. . In this case, the entire process of determining the boundary conditions is of a truly non-destructive nature since there is no need to have values of the field variables at points inside the object. These problems can be solved either non-iteratively using a boundary element formulation or iteratively using a finite element formulation. In the case of a non-iterative formulation the solution process is very simple and numerical results are practically guaranteed. In the case of an iterative formulation, a judicious application of appropriate regularization algorithms must be performed. Furthermore, the over-specified boundary conditions must be highly accurate. Both algorithms have been demonstrated to work on simply-connected and multiply-connected two-and-three-dimensional configurations.
A boundary integral method for solving inverse heat conduction problem
Journal of Inverse and Ill-Posed Problems, 2006
In this paper, a boundary integral method is used to solve an inverse heat conduction problem. An algorithm for the inverse problem of the one dimensional case is given by using the fundamental solution. Numerical results show that our algorithm is effective.
Iranian Journal of Science and Technology, Transactions A: Science, 2017
In this paper, we are concerned with the numerical solutions of the inverse heat conduction problems (IHCP) in one and two dimensions with free boundary conditions. For the one-dimensional problem, we first apply the Landau's transformation to replace the physical domain with a rectangular one. Reciprocally, some nonlinear terms appear thus an iterative scheme based on the application of the satisfier function is proposed for solving the problem. Second, we treat with the nonlinear two-dimensional problem by providing a collocation technique which takes advantage of the satisfier functions. Throughout this work, the presented schemes make the reader free of solving any nonlinear system of algebraic equations. Moreover, an admissible regularization strategy, namely, the Landweber's iterations method is used to overcome the numerical instability and achieve the acceptable approximations. Illustrative examples are included to show the efficiency of the presented algorithms.
Identification of the heterogeneous conductivity in an inverse heat conduction problem
International Journal for Numerical Methods in Engineering
This work deals with the problem of determining a non-homogeneous heat conductivity profile in a steady-state heat conduction boundary-value problem with mixed Dirichlet-Neumann boundary conditions over a bounded domain in R n , from the knowledge of the state over the whole domain. We develop a method based on a variational approach leading to an optimality equation which is then projected into a finite dimensional space. Discretization yields a linear although severely ill-posed equation which is then regularized via appropriate ad-hoc penalizers resulting a in a generalized Tikhonov-Phillips functional. No smoothness assumptions are imposed on the conductivity. Numerical examples for the case in which the conductivity can take only two prescribed values (a two-materials case) show that the approach is able to produce very good reconstructions of the exact solution.
Inverse Problems, 2010
The enclosure method was originally introduced for inverse problems of concerning non destructive evaluation governed by elliptic equations. It was developed as one of useful approach in inverse problems and applied for various equations. In this article, an application of the enclosure method to an inverse initial boundary value problem for a parabolic equation with a discontinuous coefficients is given. A simple method to extract the depth of unknown inclusions in a heat conductive body from a single set of the temperature and heat flux on the boundary observed over a finite time interval is introduced. Other related results with infinitely many data are also reported. One of them gives the minimum radius of the open ball centered at a given point that contains the inclusions. The formula for the minimum radius is newly discovered.