An iterative method for the interaction of large number of rigid plates on an elastic half-space

An iterative method for the interaction of large number of rigid plates of arbitrary shape on an elastic half-space is presented. The direct method for the solution of the interaction problem and calculation of the impedance matrix requires a large number of computational operations if the number of plates is large. Traditional methods require a sacrifice in modeling details. The concept of the iterative method is similar to Jacobi and Gauss-Seidel methods which are used for the regular matrix equations except that the operations are performed on matrix blocks instead of numbers. The initial solution is that of isolated plates and each subsequent iteration includes the effect of secondary reflection from neighboring plates. Unless the separation distance is unusually small, one or two iterations provide an excellent approximation. The Jacobi scheme, using matrix blocks, is physically equivalent to accounting for the diffraction from other plates, one iteration at a time. The Gauss-Seidel scheme, with updates made instantly, is far more efficient but does not explain the wave scattering phenomenon. Numerically, the proposed method results in significant reduction of the computational effort and increase in efficiency. It is presented for both static and dynamic interactions with excellent results.; Convergence of the method is investigated and illustrated through a case study in terms of the separation of the plates for the static case and in terms of the frequency and separation for the dynamic case. Over all, the method shows an efficient convergence for both Jacobi and Gauss-Seidel based algorithms. The Gauss-Seidel based algorithm demonstrates a faster convergence as is the case with the regular algebric systems of equations. In general, the method could be a remarkable tool for Soil-Structure Interaction applications where foundations of the structures are modeled by rigid plates.