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ABSTRACT
In this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.
Nomenclature
| A | = | matrix defined in Eq. (8) |
| b | = | vector defined in Eq. (8) |
| bi | = | exact boundary data |
| = | boundary data with noise | |
| c(ξ) | = | density function in Eq. (5) |
| D | = | matrix defined in Eq. (9) |
| d | = | distance between the virtual and real boundary |
| M | = | the number of collocation points |
| N | = | the number of elements |
| n | = | outward normal vector in Eq. (3) |
| p | = | vector defined in Eq. (11) |
| = | Neumann boundary condition | |
| r | = | vector defined in Eq. (11) |
| R | = | random variable in Eq. (12) |
| = | Dirichlet boundary condition | |
| u(x) | = | temperature |
| u* | = | fundemental solution |
| x | = | space variable |
| y | = | space variable |
| Greek symbols | = | |
| α | = | coefficients in Eq. (11) |
| β | = | coefficients in Eq. (11) |
| δ | = | noise level in Eq. (12) |
| ξ | = | source point |
| ε | = | stopping criterion |
| θ | = | angle |
| Ω | = | bounded domain |
| γ | = | the boundary of ω |
| Γ1 | = | known boundary |
| Γ2 | = | unknown boundary |
| = | virtual boundary | |
| Subscripts and superscripts | = | |
| i | = | index |
| k | = | index |
Nomenclature
| A | = | matrix defined in Eq. (8) |
| b | = | vector defined in Eq. (8) |
| bi | = | exact boundary data |
| = | boundary data with noise | |
| c(ξ) | = | density function in Eq. (5) |
| D | = | matrix defined in Eq. (9) |
| d | = | distance between the virtual and real boundary |
| M | = | the number of collocation points |
| N | = | the number of elements |
| n | = | outward normal vector in Eq. (3) |
| p | = | vector defined in Eq. (11) |
| = | Neumann boundary condition | |
| r | = | vector defined in Eq. (11) |
| R | = | random variable in Eq. (12) |
| = | Dirichlet boundary condition | |
| u(x) | = | temperature |
| u* | = | fundemental solution |
| x | = | space variable |
| y | = | space variable |
| Greek symbols | = | |
| α | = | coefficients in Eq. (11) |
| β | = | coefficients in Eq. (11) |
| δ | = | noise level in Eq. (12) |
| ξ | = | source point |
| ε | = | stopping criterion |
| θ | = | angle |
| Ω | = | bounded domain |
| γ | = | the boundary of ω |
| Γ1 | = | known boundary |
| Γ2 | = | unknown boundary |
| = | virtual boundary | |
| Subscripts and superscripts | = | |
| i | = | index |
| k | = | index |