Surface deformation due to loading of a layered elastic half‐space: a rapid numerical kernel based on a circular loading element

SUMMARY This study is motivated by a desire to develop a fast numerical algorithm for computing the surface deformation field induced by surface pressure loading on a layered, isotropic, elastic half‐space. The approach that we pursue here is based on a circular loading element. That is, an arbitrary surface pressure field applied within a finite surface domain will be represented by a large number of circular loading elements, all with the same radius, in which the applied downwards pressure (normal stress) is piecewise uniform: that is, the load within each individual circle is laterally uniform. The key practical requirement associated with this approach is that we need to be able to solve for the displacement field due to a single circular load, at very large numbers of points (or ‘stations’), at very low computational cost. This elemental problem is axisymmetric, and so the displacement vector field consists of radial and vertical components both of which are functions only of the radial coordinate r . We achieve high computational speeds using a novel two‐stage approach that we call the sparse evaluation and massive interpolation (SEMI) method. First, we use a high accuracy but computationally expensive method to compute the displacement vectors at a limited number of r values (called control points or knots), and then we use a variety of fast interpolation methods to determine the displacements at much larger numbers of intervening points. The accurate solutions achieved at the control points are framed in terms of cylindrical vector functions, Hankel transforms and propagator matrices. Adaptive Gauss quadrature is used to handle the oscillatory nature of the integrands in an optimal manner. To extend these exact solutions via interpolation we divide the r ‐axis into three zones, and employ a different interpolation algorithm in each zone. The magnitude of the errors associated with the interpolation is controlled by the number, M , of control points. For M = 54, the maximum RMS relative error associated with the SEMI method is less than 0.2 per cent, and it is possible to evaluate the displacement field at 100 000 stations about 1200 times faster than if the direct (exact) solution was evaluated at each station; for M = 99 which corresponds to a maximum RMS relative error less than 0.03 per cent, the SEMI method is about 700 times faster than the direct solution.