Modular and flexible spectral-element waveform modelling in two and three dimensions

In this paper, we present a series of mathematical abstractions for seismologically relevant wave equations discretized using finite-element methods, and demonstrate how these abstractions can be implemented efficiently in computer code. Our motivation is to mitigate the combinatorial complexity present when considering geophysical waveform modelling and inversion, where a variety of spatial discretizations, material models, and boundary conditions must be considered simultaneously. We accomplish this goal by first considering three distinct classes of abstract mathematical models: (1) those representing the physics of an underlying wave equation, (2) those describing the discretization of the chosen equation onto a finite-dimensional basis and (3) those describing any spatial transforms. A full representation of the discrete wave equation can then be constructed using a hierarchical nesting of models from each class. Additionally, each class is functionally orthogonal to the others, and with certain restrictions models within one class can be interchanged independently from changes in another. We then show how this recasting of the relevant equations can be implemented concisely in computer software using an abstract object-oriented design, and discuss how recent developments in the numerical and computational sciences can be naturally incorporated. This builds to a set of results where we demonstrate how the developments presented can lead to an implementation capable of multiphysics waveform simulations in completely unstructured domains, on both hypercubical and simplical spectral-element meshes, in both two and three dimensions, while remaining concise, efficient and maintainable.