Generating Finite Element Codes combining Adaptive Octrees with Complex Geometries

TL;DR

This paper introduces a domain-specific language (DSL) that simplifies the development and deployment of finite element methods on complex, adaptively refined meshes, enabling rapid prototyping and efficient large-scale computations.

Contribution

The paper presents a high-level DSL that abstracts FEM implementation complexity, simplifies handling complex geometries, and maintains good parallel performance for PDE solutions.

Findings

DSL enables rapid prototyping of FEM formulations.

Generated code scales efficiently to thousands of processes.

The approach simplifies complex boundary integral computations.

Abstract

We present a high-level domain-specific language (DSL) interface to drive an adaptive incomplete $k$-d tree-based framework for finite element (FEM) solutions to PDEs. This DSL provides three key advances: (a) it abstracts out the complexity of implementing non-trivial FEM formulations, (b) it simplifies deploying these formulations on arbitrarily complicated and adaptively refined meshes, and (c) it exhibits good parallel performance. Taken together, the DSL interface allows end-users to rapidly and efficiently prototype new mathematical approaches, and deploy them on large clusters for solving practical problems. We illustrate this DSL by implementing a workflow for solving PDEs using the recently developed shifted boundary method (SBM). The SBM requires approximating relatively complicated integrals over boundary surfaces. Using a high-level DSL greatly simplifies this process and allows rapid exploration of variations. We demonstrate these tools on a variety of 2-D and 3-D configurations. With fewer than 20 lines of input, we can produce a parallel code that scales well to thousands of processes. This generated code is made accessible and readable to be easily modified and hand-tuned, making this tool useful even to experts with the target software.