Application of the Boundary Integral Equation (Boundary Element) Method to Time Domain Transient Heat Conduction Problems

Application of the boundary integral equation (BIE) or boundary element method to two-dimensional transient heat flow problems using higher-order spatial shape functions is presented. Many different formulations have been proposed for the treatment of heat conduction (diffusion) problems by the BIE method, the most efficient of which is the one which employs a time dependent fundamental solution. The formulation adopted for this analysis employs space and time dependent fundamental solutions to derive the boundary integral equation in the time domain. It is an implicit time-domain formulation and is valid for both regular and unbounded domains. A time stepping scheme (time integral method) is then used to solve the boundary initial value problem by marching forward in time. Constant and linear temporal interpolation and quadratic shape functions are used to approximate field quantities in the time and space domains, respectively. Temporal and spatial integrations are carried out to form a system of linear equations. At the end of each time step, these equations are solved to obtain unknown values at that time. Validity of the BIE formulation is demonstrated by solving some test cases whose analytical solutions are known. Two-dimensional heat flow in a cooled turbine rotor blade is carried out as a practical application.