Application of combined BEM-FEM algorithms in numerical modelling of diffusion problems

Abstract

A thermal diffusion process proceedings in heterogeneous domain Ω = Ω₁

$$ X \in \Omega :\quad {C_1}\partial {}_t{T_1}(X,t) = div[{\lambda_1}grad{T_1}(X,t)] + q{}_{{v1}}(X,t) $$

((1))

$$ X \in \Omega :\quad {C_2}{\partial_t}{T_2}(X,t) = div[{\lambda_2}gard\,{T_2}(X,t)] + {q_{{v2}}}(X,t) $$

((2))

and the following boundary-initial conditions

$$ X \in {\Gamma_{{12}}}:\quad - {\lambda_1}{\partial_n}{T_1}(X,t) = \frac{{{T_1}(X,t) - {T_2}(X,t)}}{{R(X,t)}} = {\lambda_2}\partial {}_n{T_2}(X,t) $$

((3))

$$ X \in {\Gamma_{{10}}}:\quad \Phi [{T_1},{\partial_n}{T_1}(X,t)] = 0,\quad X \in {\Gamma_{{20}}}:\quad \Phi [{T_2},{\partial_n}{T_2}(X,t)] = 0 $$

((4))

$$ t = 0:\quad T{}_1(X,0) = {T_{{10}}}(X)\quad \quad {T_2}(X,0) = {T_{{20}}}(X) $$

((5))

where C _e , λ _e , q _Ve , e=1, 2 are the thermophysical parameters and capacities of internal heat sources, ∂ _t T=∂T/∂t, ∂ _n T is a normal derivative at the point X∈Γ, R(X, t) is a thermal resistance between sub-domains considered, Γ₁₀, Γ₂₀ form the outer surface of the system. In the case R(X, t)=0 the heat flux continuity condition resolves itself into the condition additionally determining a continuity of temperature field. The domain considered is shown in Fig. 1.