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.
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.
How many processors should we use to solve Problem X?
Parallel computation puts many CPUs to work on solving a problem much more quickly than one CPU alone. But this only works if the tasks are carefully scheduled and the additional CPUs are not waiting around for something to do. How do we choose the right number of processors for a given problem ?
Underworld and Singularity
TL; DR: Underworld is now Singularity-enabled, making it easier and somewhat quicker to use compared to traditional HPC installs. We demonstrate results using over 10,000 CPUs and more than one billion unknowns for solving the Stokes equation with Underworld 2.16
