Singular boundary method for 2D and 3D heat source reconstruction

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Abstract

This paper presents the singular boundary method (SBM) in conjunction with the regularization method to recover the heat source in steady-state heat conduction problems from boundary temperature and heat flux measurements. To avoid the internal measurements, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis. As a result, the source inverse problems can be transformed to a direct fourth-order governing equation, which can be tackled by the SBM. The SBM utilizes the fundamental solutions to approximate the solutions and desingularizes the source singularities of fundamental solutions with origin intensity factors (OIFs), which are derived for both 2D and 3D cases. The regularization method is introduced to deal with noisy boundary. The feasibility and accuracy of the proposed method is validated by both 2D and 3D cases.

Introduction

Let Ω be a bounded domain in d(d=2,3), where d is the dimensionality of the space and Γ=Ω is its boundary. The steady-state heat conduction is governed by Poisson equation L0T=2T(x)=f(x),x=(x1,,xd)Ωwith boundary conditions T(x)=hD(x),xΓD,q(x)=T(x)n=hN(x),xΓN,where T is the temperature, x means the spatial coordinates, Ω a bounded domain, q the heat flux, and n the outward normal vector, hD(x) and hN(x) the measured data specified on the boundary Γ. In this case, the right hand side f(x) denoting a heat source is sometimes unknown.

The identification of the source terms has received considerable attention in science, engineering and bioengineering. Some studies [1] required internal measurements of the temperature as supplement information, while in most of time, these measurements are intrusive. Nevertheless, non-invasive measurement only on the boundary is mathematically insufficient to guarantee the uniqueness of the source terms. Thus, some a priori information on the source should be assumed. El Badia [2] considered an inverse source problem for an anisotropic elliptic equation for which the source is a combination of monopolar and dipolar sources. Collara et al. [3] assumed that there is only one harmonic source, piecewise constant sources in their work, and identified it in non-homogeneous media with boundary measurements. In this study, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis. Thus the source terms can be annihilated and the problem is transformed into a homogeneous fourth-order partial differential equation. Thus we reformulate the problem as

Formulation The source term satisfies a second order operator L1, namely, L1f(x)=0, then the problem of Eq. (1) can be reconstructed as L1L0T=0,xΩ,T(x)=hD(x),xΓD,q(x)=hN(x),xΓN. The operator L1 can be Laplace, Helmholtz or modified Helmholtz operators.

Then the higher-order direct problem (3), (4) can be solved with numerical methods, such as finite element method, boundary element method, method of fundamental solutions and other meshless methods [4], [5]. In this study, the singular boundary method (SBM) is applied to solve this problem with merely boundary discretization. The SBM is a boundary collocation method, which employs the fundamental solutions as the kernel functions. The key issue in the SBM is to desingularize the source singularities of the fundamental solutions with a finite value called origin intensity factor (OIF). Thus, the SBM avoids the sophisticated mathematics associated with the singular and nearly-singular integration in the BEM and perplexing choice of an optimal auxiliary boundary in the MFS [6], [7]. In the following, the formulation of the SBM will be established to identify the source terms.

Section snippets

Singular boundary method

In the SBM, the solution of partial differential equation (3) is approximated by a linear combination of the fundamental solutions with respect to source points: T(x)=j=1NαjG0(x,sj)+j=1NβjG1(x,sj), q(x)=j=1NαjG0(x,sj)nx+j=1NβjG1(x,sj)nx,where x and sj denote the field point and the jth source point, G0 and G1 are the fundamental solutions of the operators L0 and L1(shown in Table 1). αjj=1,,N and βjj=1,,N are the unknown coefficients, which are determined by imposing the boundary

Numerical examples

This section tests two cases to illustrate the applicability and accuracy of the SBM with proposed OIFs to inverse source problem. For the evaluation of numerical accuracy, the root mean square error (RMSE) is employed with definition as RMSE(f)=j=1Ntf̃jfj2j=1Ntfj2,where f̃j and fj are the exact and numerical solutions of source terms at the test point xj, respectively, and Nt denotes the total number of test points in the domain. A relative noise with the intensity ε is imposed on the

Conclusions

In this study, the SBM in conjunction with the regularization method is employed to identify the source term in the steady-state heat conduction. The inverse problem is transformed to a direct fourth-order problem based on the assumption that the heat source is assumed to satisfy a second-order partial differential equation. Then the SBM approximates the solutions with a linear combination of fundamental solutions with respect to the source points, and the source terms can be derived from

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant nos. 11662003, 11602114).

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