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Automated dimensional analysis for PDEs

Michal Habera, Andreas Zilian

TL;DR

This work embeds dimensional analysis into the Unified Form Language to automatically nondimensionalize PDE variational forms and verify dimensional consistency. By introducing a Quantity class and a graph-based unit transformation with homogeneous factorization and normalization, the framework behaves as a physics-aware diagonal preconditioner (Full Operator Preconditioning) applied at the form level, not just after assembly. Demonstrations on Poisson, incompressible Navier–Stokes, Neo–Hooke elasticity, and Poisson–Nernst–Planck illustrate how dimensionless groups emerge and guide conditioning and coupling insights. The approach is implemented in the Dolfiny package for FEniCSx but is designed to be transferable to other UFL-based libraries, with broad implications for robustness and efficiency in multiphysics simulations.

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.

Automated dimensional analysis for PDEs

TL;DR

This work embeds dimensional analysis into the Unified Form Language to automatically nondimensionalize PDE variational forms and verify dimensional consistency. By introducing a Quantity class and a graph-based unit transformation with homogeneous factorization and normalization, the framework behaves as a physics-aware diagonal preconditioner (Full Operator Preconditioning) applied at the form level, not just after assembly. Demonstrations on Poisson, incompressible Navier–Stokes, Neo–Hooke elasticity, and Poisson–Nernst–Planck illustrate how dimensionless groups emerge and guide conditioning and coupling insights. The approach is implemented in the Dolfiny package for FEniCSx but is designed to be transferable to other UFL-based libraries, with broad implications for robustness and efficiency in multiphysics simulations.

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 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.
Paper Structure (33 sections, 1 theorem, 89 equations, 18 figures, 1 table)

This paper contains 33 sections, 1 theorem, 89 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Existing libraries for physical units
  3. Brief mathematical background
  4. Definitions
  5. Dimensional analysis principles
  6. Buckingham Pi Theorem
  7. Example (Flow around a sphere).
  8. Implementation in a domain specific language for PDEs
  9. The Unified Form Language
  10. Example (Poisson equation in UFL).
  11. Quantity class
  12. Example (Quantities with units for the Poisson equation).
  13. Transformation
  14. Example (Transforming the Poisson equation).
  15. Factorization
  16. ...and 18 more sections

Key Result

Theorem 1

Let a physical system be described by a relation $F(q_1, q_2, \ldots, q_n) = 0$ involving $n$ physical quantities that depend on $k$ base dimensions. Then the system can be equivalently described by some $G(\Pi_1, \Pi_2, \ldots, \Pi_p) = 0$, where $\ker(\mathbf M) = \text{span}\{\mathbf d(\Pi_1), \m

Figures (18)

  • Figure 1: Implementation of the Poisson equation weak form in the Unified Form Language (UFL).
  • Figure 2: Example of defining reference quantities with physical units using the Quantity class.
  • Figure 3: Comparison of the UFL expression DAG before and after inserting quantity scaling factors. Blue nodes show the added Quantity objects and their effect on the computation graph.
  • Figure 4: Quantity factorized DAG with dimensional scales annotated on each node. This representation exposes the dimensional homogeneity of the expression $\nabla u \cdot \nabla \delta u$ with final units $u_\mathrm{ref}^2 l_\mathrm{ref}^{-2}$.
  • Figure 5: Factorized DAG for the nonhomogeneous sum $\nabla u \cdot \nabla \delta u + \kappa^2 u \delta u$. The sum node at the root shows inconsistency between its operands $u_\mathrm{ref}^2 l_\mathrm{ref}^{-2}$ and $\kappa^2 u_\mathrm{ref}^2$.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Theorem 1: Buckingham Pi