Symbolic Time Derivatives in Underworld3
In Underworld3, the time derivative is a symbolic object. It appears in the solver's strong form as a SymPy expression, alongside the constitutive stress and the body force.
Constitutive Models in Symbolic Form
A constitutive model is a Python class where the relationship between fluxes and gradients is encoded in SymPy. At every stage the mathematics is visible, inspectable, and differentiable. The framework handles Jacobians, C code generation, and PETSc integration. You handle the physics.
Physical Units in Computational Geodynamics
Non-dimensionalisation (rewriting problems to make them re-scalable from lab to the real world) is a specialised task that all modellers confront at some point. It is important for accurate and efficient numerical solutions. Can we take away the pain that comes with the task ?
Mesh Variables and PETSc Vectors: Keeping Arrays in Sync
One of the less glamorous but important problems in a finite element framework is this: how does the user assign values to a field variable, and how does the framework ensure that PETSc sees those values correctly, in parallel, without the user needing to keep track themselves?
How Underworld3 Turns SymPy into C
What actually happens between the moment you write a mathematical expression in underworld3's python layer and the moment PETSc receives a finite element term in the form of compiled C code ?
Our Journey from Underworld2 to Underworld3
Underworld is a code for geodynamics — mantle convection, lithospheric deformation, subduction, ice flow. We solve coupled, nonlinear PDEs with complex rheologies using Lagrangian particles to track material history. The project has been running for 20 years: why did we start again ?
AI and Scientific Software: What We Learned Rebuilding Underworld3
Underworld3 has about 50,000 lines of Python/ Cython wrapping PETSc, SymPy, and a just-in-time compiler. I began a trial of AI coding tools in 2025 and they have gradually become central to the way our team works. This is a story of co-evolution as much as it is about adoption of a new set of tools.
