EJIIM for Calculation of effective Elastic Moduli in 3D Linear Elasticity

The Explicit Jump Immersed Interface Method solves boundary value problems by embedding arbitrary domains in rectangular parallelepipeds and extending the boundary value problem to such a domain. Now the boundary conditions become jump conditions, and separate equations for the discretized solution of the pde and for the jump conditions are written. In the first set of equations, the jumps correct standard difference formulas, in the second set of equations solution values from interior grid points are used to compute quantities on the interface. For constant coefficient pde, the first set can be solved very efficiently by direct Fast Fourier transform inversion, and this is used to efficiently evaluate the smaller Schur-complement matrix-vector multiplication in a conjugate gradient approach to find the jumps. Here this approch is applied to the three-dimensional equations of linear elasticity. The resulting method is second order convergent for the displacements in the maximum norm as the grid is refined. It is applied to calculate the effective elastic moduli of fibrous and porous microstructures in three dimensions using the strain energy equivalence principle. From these effective moduli, best estimates under various symmetry assumptions are calculated.