Multivariate numerical derivative by solving an inverse heat source problem

A method for approximating multivariate numerical derivatives is presented from multidimensional noise data in this paper. Starting from solving a direct heat conduction problem using the multidimensional noise data as an initial condition, we conclude estimations of the partial derivatives by solving an inverse heat source problem with an over-specified condition, which is the difference of the solution to the direct problem and the given noise data. Then, solvability and conditional stability of the proposed method are discussed for multivariate numerical derivatives, and a regularized optimization is adopted for overcoming instability of the inverse heat source problem. For achieving partial derivatives successfully and saving amount of computation, we reduce the multidimensional problem to a one-dimensional case, and give a corresponding algorithm with a posterior strategy for choosing regularization parameters. Finally, numerical examples show that the proposed method is feasible and stable to noise data.     

Determination of a time-dependent heat source from nonlocal boundary conditions

In this paper, we investigate the inverse heat source problem of finding the time-dependent source function together with the temperature. Three general nonlocal conditions are considered for the boundary and overdetermination conditions resulting in six different cases. The boundary element method combined with Tikhonov regularization is employed in order to obtain an accurate and stable numerical solution.     

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

A meshless model for transient heat conduction in functionally graded materials

A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter. First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem. Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively. Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined. Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients. The accuracy of the proposed method is assessed through two numerical examples.     

The method of fundamental solutions for a biharmonic inverse boundary determination problem

In this paper, a nonlinear inverse boundary value problem associated to the biharmonic equation is investigated. This problem consists of determining an unknown boundary portion of a solution domain by using additional data on the remaining known part of the boundary. The method of fundamental solutions (MFS), in combination with the Tikhonov zeroth order regularization technique, are employed. It is shown that the MFS regularization numerical technique produces a stable and accurate numerical solution for an optimal choice of the regularization parameter. A. Zeb on study leave visiting the University of Leeds.     

An improved boundary distributed source method for two-dimensional Laplace equations

In this work, the boundary distributed source (BDS) method [EABE 34(11): 914-919] based on the method of fundamental solutions (MFS) is considered for the solution of two-dimensional Laplace equations. The BDS is a truly mesh-free method and quite easy to implement since the source points and field points are collocated on the domain boundary while the conventional MFS requires a fictitious boundary where the source points locate. The main idea of the BDS is that to avoid the singularities of the fundamental solutions the concentrated point sources in the conventional MFS are replaced by distributed sources over circles centered at the source points. In the original BDS, all elements of the system matrix can be derived analytically in a very simple form for the Dirichlet boundary conditions and off-diagonal elements for the Neumann boundary conditions, while the diagonal elements for the Neumann boundary conditions can be obtained indirectly from the constant potential field. This work suggests a simple way to determine the diagonal elements for the Neumann boundary conditions by invoking that the boundary integration of the normal gradient of the potential should vanish. Several numerical examples are addressed to show the feasibility and the accuracy of the proposed method.     

The method of fundamental solutions for inverse 2D Stokes problems

A numerical scheme based on the method of fundamental solutions is proposed for the solution of two-dimensional boundary inverse Stokes problems, which involve over-specified or under-specified boundary conditions. The coefficients of the fundamental solutions for the inverse problems are determined by properly selecting the number of collocation points using all the known boundary values of the field variables. The boundary points of the inverse problems are collocated using the Stokeslet as the source points. Validation results obtained for two test cases of inverse Stokes flow in a circular cavity, without involving any iterative procedure, indicate the proposed method is able to predict results close to the analytical solutions. The effects of the number and the radius of the source points on the accuracy of numerical predictions have also been investigated. The capability of the method is demonstrated by solving different types of inverse problems obtained by assuming mixed combinations of field variables on varying number of under- and over-specified boundary segments.     

Hermite spectral method for solving inverse heat source problems in multiple dimensions

In this paper, we in multiple dimensions. We present a stability estimate for determining the source term in the multiple dimensional heat equation in an unbounded domain, and the regularization parameter is chosen by a discrepancy principle. Error estimate between the exact solution and its regularization solution is given. Numerical experiments for the one-dimension and two-dimension cases show the effectiveness of the method.     

A method of fundamental solutions for transient heat conduction in layered materials

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

Iterative computational identification of a space-wise dependent source in parabolic equation

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time – the final overdetermination. More general problems are associated with some integral observation of the solution in time – the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spatial variables. Capabilities of proposed methods are illustrated by numerical examples for a two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation in space is used, whilst the time discretization is based on fully implicit two-level schemes.     

A Trefftz method in space and time using exponential basis functions: Application to direct and inverse heat conduction problems

In this paper we present a Trefftz method based on using exponential basis functions (EBFs) to solve one (1D) and two (2D) dimensional transient problems. We focus on direct and inverse heat conduction problems, the latter being the more challenging ones, to show the capabilities of the method. A summation of exponential basis functions (EBFs), satisfying the governing equation in time and space, with unknown coefficients is considered for the solution. The unknown coefficients are determined by the satisfaction of the prescribed time dependent boundary and initial conditions through a collocation method. Several 1D and 2D direct and inverse heat conduction problems are solved. Some numerical evidence is provided for the convergence and sensitivity of the method with respect to the noise levels of the measured data and time steps.     

An advanced meshless LBIE/RBF method for solving two-dimensional incompressible fluid flows

The present work presents a meshless local boundary integral equation (LBIE) method for the solution of two-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations. The method uses, for its meshless implementation, nodal points spread over the analyzed domain and employs in an efficient way the radial basis functions (RBF) for the interpolation of the interior and boundary variables. The inverse matrix of the RBF is computed only once for every nodal point and the interpolation functions are evaluated by the inner product of the inverse matrix with the weight vector associated to the integration point. This technique leads to a fast and efficient meshless approach, the locality of the method is maintained and the system matrices are banded with small bandwidth. The velocity–vorticity approach of the Navier–Stokes equations is adopted and the LBIEs are derived for the velocity and the vorticity field, resulting in a very stable and accurate implementation. The evaluation of the volume integrals is accomplished via a very efficient and accurate technique by triangularizing the local area of the nodal point to the minimum number of well formed triangles. Numerical examples illustrate the proposed methodology and demonstrate its accuracy.     

A novel computational method for solving Troesch's problem with high-sensitivity parameter

Troesch's problem is a nonlinear boundary value problem arising in the confinement of a plasma column by radiation pressure, and also in the theory of gas porous electrodes. It is well known that finding numerical solutions to this problem is challenging, especially when the sensitivity parameter is large. In this article, we present an efficient and accurate numerical method for solving Troesch's problem. The method presented in this work is capable of computing the solution, even for extremely high-sensitivity parameter. The method is based on the the Newton–Raphson–Kantorovich approximation method in function space combined with the standard finite difference method. Although, available numerical solvers fail to provide accurate numerical solutions when the sensitivity parameter λ becomes large (λ exceeds 100) [1–5], the method proposed here is able to provide accurate numerical solutions for extremely large values of this sensitivity parameter, up to λ = 500. Numerical experiments are provided to show the accuracy of the method compared to existing solvers, as well as its capability to compute the solution for high values of the sensitivity parameter λ.     

The boundary-element method for the determination of a heat source dependent on one variable

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions of the direct problem and a supplementary condition, called an overdetermination. In this problem, if the heat source is taken to be space-dependent only, then the overdetermination is the temperature measurement at a given single instant, whilst if the heat source is time-dependent only, then the overdetermination is the transient temperature measurement recorded by a single thermocouple installed in the interior of the heat conductor. These measurements ensure that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method with the discrepancy principle or the L-curve criterion for the choice of the regularization parameter. The boundary-element method (BEM) is developed for solving numerically the inverse problem and numerical results for some benchmark test examples are obtained and discussed     

A corrective smoothed particle method for boundary value problems in heat conduction

Combining the kernel estimate with the Taylor series expansion is proposed to develop a Corrective Smoothed Particle Method (CSPM). This algorithm resolves the general problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothed Particle Hydrodynamics (SSPH). In addition, the method’s ability to model derivatives of any order could make it applicable for any time‐dependent boundary value problems. An example of the applications studied in this paper is unsteady heat conduction, which is governed by second‐order derivatives. Numerical results demonstrate that besides the capability of directly imposing boundary conditions, the present method enhances the solution accuracy not only near or on the boundary but also inside the domain. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States.     

A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity

In this paper we derive a generalized fundamental solution for the BEM solution of problems of steady state heat conduction with arbitrarily spatially varying thermal conductivity. This is accomplished with the aid of a singular nonsymmetric generalized forcing function, D, with special sampling properties. Generalized fundamental solutions, E, are derived as locally radially symmetric responses to this nonsymmetric singular forcing function, D, at a source point ξ. Both E and D are defined in terms of the thermal conductivity of the medium. Although locally radially symmetric, E varies within the domain as the source point, ξ changes position. A boundary integral equation is formulated. Examples of generalized fundamental solutions are provided for various thermal conductivities along with the corresponding forcing function, D. Here, four numerical examples are provided. Excellent results are obtained with our formulation for variations of thermal conductivity ranging from quadratic and cubic in one dimension to exponential in two dimensions. Problems are solved in regular and irregular regions. Current work is under way investigating extensions of this general approach to further applications where nonhomogeneous property variations are an important consideration.