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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.
