Finding Particles in a Distributed, Unstructured Mesh
It can be difficult to figure out where you are in an unstructured mesh of triangles or tetrahedra; worse when the mesh is distributed in parallel. We have to solve this problem for particle-in-cell type codes such as Underworld3. This is how we do it.
Particles in Underworld3
Underworld is built around the idea of active Lagrangian tracer particles that carry history and composition information as the material deforms. How do we combine this information with our symbolic mathematical framework ?
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.
Underworld3 published in Journal of Open Source Software
The aim of underworld3 is to provide strong support to users in developing sophisticated mathematical models, and provide the ability to interrogate those models during development and at run-time. Underworld3 encodes the mathematical structure of the equations it solves in symbolic form.
New features of the surface-coupling framework in Underworld 2
Coupling in Underworld 2 Underworld 2's UWGeodynamic module enables the coupling of tectonics and surface processes models, most notably, with Underworld 2 handling the tectonics and Badlands (Salles, Ding, & Brocard, 2018) handling the surface processes. This coupling framework has been employed to study landscape evolution and topography
