Grid-based volume integration for elasticity

A regular grid direct volume integral method has recently been proposed for domain integrals stemming from Poisson or nonlinear Laplace problems. The volume integral is exactly decomposed into a non-singular boundary integral, plus a remainder volume integral that can be accurately evaluated using a regular grid overlaying the problem domain. For elasticity, achieving the analogous decomposition requires the ‘Galerkin vector’ HH that satisfies E(H)=UE(H)=U, where EE is the elasticity equation and UU the Kelvin fundamental solution. Herein, Fourier transforms are employed to derive formulas for HH and the corresponding traction kernel THTH, for both two and three dimensional isotropic elasticity. The three dimensional formulas, and their numerical implementation, are validated by solving relatively simple body force elasticity problems with known solutions. Results for a fast (P-FFT) boundary integral formulation are also presented.