Boundary element algorithms for the solution of steady-state and transient heat conduction are presented. The algorithms are designed for efficient coupling with computational fluid dynamic discretizations and feature piecewise linear elements with offset nodal points. The steady-state algorithm employs the fundamental solution approach; the integration kernels are computed analytically based on linear shape functions, linear elements, and variably offset nodal points. The analytic expressions for both singular and nonsingular integrands are presented. The transient algorithm employs the transient fundamental solution; the temporal integration is performed analytically and the nonsingular spatial integration is performed numerically using Gaussian quadrature. A series solution to the integration is derived for the instance of a singular integrand. The boundary-only character of the algorithm is maintained by integrating the influence coefficients from initial time. Numerical results are compared to analytical solutions to verify the current boundary element algorithms. The steady-state and transient algorithms are numerically shown to be second-order accurate in space and time, respectively.
