Dimensional analysis in UFL

We integrate physical units into the Unified Form Language (UFL) and demonstrate practical applications. Our approach enables semi-automated, physics-based scaling of terms in partial differential equations (PDEs). This scaling serves as an effective preconditioner for multi-physics problems, improving conditioning that can be further enhanced with standard algebraic methods. Additionally, dimensional analysis based on the Buckingham Pi theorem is presented. This allows for automated discovery of dimensionless numbers and non-dimensionalisation of UFL forms. This functionality is implemented in the dolfiny framework.