Evaluation of the singular integrals of the three-dimensional thermoelasticity
The article deals with the solving of the problem of coupled thermoelastic vibrations of massive bodies. Numerical solution is sought by the boundary integral equation method. The main attention is paid to the definition of singular parts of integrals i.e. to integration along those boundary elements on which the pole is located. Two approaches are proposed for evaluation of singular integrals. The first approach is based on the expansion of integral equations kernels in a power series. For the realization of this approach, compact expressions approximating all components of the kernels are obtained. The obtained expressions has weak or strong singularities when the distance between the source point and integration point goes to zero similar to the behaviour of the corresponding elastostatic expressions. Therefore, such expressions can be successfully used for singular integrals evaluation. A number of numerical experiments have been performed, which confirms the robustness of the approach in a wide frequency range and allows us to trace the dependence between the number of retained members and the accuracy of the calculations. The second approach is related to the analytical calculation of integrals over a flat circle with a centre at the pole. In this paper the exact formulas were obtained that allow us to compute effectively the corresponding integrals over the boundary elements on which the pole is located. Among other things, the obtained exact expressions prove the existence of corresponding singular integrals in the sense of the Cauchy principal value. As a matter of a fact the above relations form the main part of a numerical algorithm aimed at solving the problem of coupled three-dimensional harmonic thermoelastic vibrations of massive elements of constructions using the of boundary element method.
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