Automated dimensional analysis for PDEs

TL;DR

This paper introduces a systematic framework for integrating physical units into finite element methods, enabling automated nondimensionalization and improved numerical stability across various PDE systems.

Contribution

It presents a symbolic Quantity class and a graph-based traversal for automatic dimensional analysis within UFL, enhancing stability and error detection in PDE solvers.

Findings

Improved condition number for Navier--Stokes saddle-point systems.

Detection of floating-point cancellation errors in hyperelasticity.

Effective handling of coupled multiphysics problems with scaling.

Abstract

Physical units are fundamental to scientific computing. However, many finite element frameworks lack built-in support for dimensional analysis. In this work, we present a systematic framework for integrating physical units into the Unified Form Language (UFL). We implement a symbolic Quantity class to track units within variational forms. The implementation exploits the abelian group structure of physical dimensions. We represent them as vectors in $\mathbb{Q}^n$ to simplify operations and improve performance. A graph-based visitor pattern traverses the expression tree to automate consistency checks and factorization. We demonstrate that this automated nondimensionalization functions as the simplest form of Full Operator Preconditioning. It acts as a physics-aware diagonal preconditioner that equilibrates linear systems prior to assembly. Numerical experiments with the Navier--Stokes equations show that this improves the condition number of the saddle-point matrix. Analysis of Neo-Hooke hyperelasticity highlights the detection of floating-point cancellation errors in small deformation regimes. Finally, the Poisson--Nernst--Planck system example illustrates the handling of coupled multiphysics problems with derived scaling parameters. Although the implementation targets the FEniCSx framework, the concepts are general and easily adaptable to other finite element libraries using UFL, such as Firedrake or DUNE.