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Boundary integral formulation for 2D and 3D thermal problems exhibiting a linearly varying stochastic conductivity. (English) Zbl 0858.73078
This work examines steady-state heat conduction in a stochastic heterogeneous medium, where the thermal conductivity varies linearly along one direction, and its slope is a sum of a constant part and a zero-mean random part. As the first step, the governing Laplace equation is solved using a coordinate transformation of independent spatial variables, and the exact Green functions in both two and three dimensions are obtained for a linearly varying conductivity profile. In addition, a boundary integral equation statement in which the Green functions appear as kernels is concurrently obtained.
Next, material stochasticity is introduced, and the perturbation approach is employed for deriving the mean value and covariance of the Green functions up to the second order. Perturbations are also used to solve the discretized boundary integral equation, so that mean vectors and covariance matrices can be obtained for the response, temperature and heat flux. An example involving steady-state temperature distribution in a block where conductivity varies due to a buried heat source illustrates the method. Finally, comparisons are made with Monte Carlo simulations.

74S15 Boundary element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
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