http://orcid.org/0000-0001-6366-3539
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ABSTRACT
The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.
Nomenclature
| A | = | coefficient matrix in (Eq. 9) |
| b1 | = | := ATe |
| aj, bj | = | coefficients in Fourier series |
| ck | = | coefficients in polynomial expansion |
| c | = | n-dimensional vector of coefficients |
| D | = | := ATA |
| e | = | the right-hand side in (Eq. 9) |
| f(x) | = | initial temperature |
| g(t) | = | right boundary temperature |
| = | boundary temperatures | |
| h(x) | = | final time temperature |
| ℋ | = | heat operator |
| ℋ* | = | adjoint heat operator |
| ℓ | = | length of rod |
| m0 | = | the number of adjoint Trefftz functions |
| = | the number of coefficients | |
| m3 | = | the number of measured data |
| n | = | the number of total coefficients |
| nq | = | the number of linear equations |
| s | = | level of noise |
| t | = | time |
| tf | = | final time |
| ti | = | |
| u(x, t) | = | temperature |
| vk(x, t) | = | adjoint Trefftz functions |
| x | = | space variable |
| xi | = | := iℓ∕(m3+1) |
| Greek symbols | = | |
| ε | = | convergence criterion |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
| m | = | index |
| T | = | transpose |
| Γ | = | boundary contour |
| Ω | = | domain |
Nomenclature
| A | = | coefficient matrix in (Eq. 9) |
| b1 | = | := ATe |
| aj, bj | = | coefficients in Fourier series |
| ck | = | coefficients in polynomial expansion |
| c | = | n-dimensional vector of coefficients |
| D | = | := ATA |
| e | = | the right-hand side in (Eq. 9) |
| f(x) | = | initial temperature |
| g(t) | = | right boundary temperature |
| = | boundary temperatures | |
| h(x) | = | final time temperature |
| ℋ | = | heat operator |
| ℋ* | = | adjoint heat operator |
| ℓ | = | length of rod |
| m0 | = | the number of adjoint Trefftz functions |
| = | the number of coefficients | |
| m3 | = | the number of measured data |
| n | = | the number of total coefficients |
| nq | = | the number of linear equations |
| s | = | level of noise |
| t | = | time |
| tf | = | final time |
| ti | = | |
| u(x, t) | = | temperature |
| vk(x, t) | = | adjoint Trefftz functions |
| x | = | space variable |
| xi | = | := iℓ∕(m3+1) |
| Greek symbols | = | |
| ε | = | convergence criterion |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
| m | = | index |
| T | = | transpose |
| Γ | = | boundary contour |
| Ω | = | domain |