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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

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 Geodynamic Modelling Workshop. Perth, 7 May 2025

You are invited to a full-day workshop on Underworld — a computational Geodynamics / Tectonics modelling framework. This is an excellent opportunity for researchers, students, and professionals interested in computational geodynamics and Earth science modelling to learn about Underworld.

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

Underworld3

Introducing Underworld3: Mathematically Self-Describing Modelling in Python for Desktop, HPC and Cloud.

30 Years of Citcom, Ellipsis and Underworld

CITCOM is a geodynamics modelling code based on the finite element method that is designed for planetary evolution modelling where large spatial variations and strong non-linearities occur in the material properties.