Simulations of seismic wave propagation using a spectral-element method in a Lagrangian framework with logarithmic strain

We show how the linearized equations that govern the motion of a body that undergoes deformation can be generalized to capture geometrical non-linearities in a spectral-element formulation. Generalizing the equations adds little complexity, the main addition being that we have to track the deformation gradient. Geometrical changes due to deformation are captured using the logarithmic strain. We test the geometrically non-linear formulation by considering numerical experiments in seismic wave propagation and cantilever beam bending and compare the results with the linearized formulation. In cases where finite deformation occurs, the effect of solving the geometrically non-linear equations can be significant while in cases where deformation is smaller the result is similar to solving the linearized equations. We find that the time it takes to run the geometrically non-linear simulations is on the same order of magnitude as running the linearized simulations. The limited amount of added cost and complexity suggests that we might as well solve the geometrically non-linear equations since it does not assume anything about the size of deformations.