Finite-Element Discretization

How Kronos turns the continuous equations of equilibrium, MHD, and thermomechanics into large sparse systems a computer can solve.

From continuous to discrete

The PDEs of equilibrium, MHD, and thermomechanics are continuous; a computer solves discrete systems. Finite-element discretization is the bridge: it represents the solution on a mesh using basis functions, converting a PDE into a large sparse system of algebraic equations. This is the numerical backbone of Kronos's non-stochastic physics workloads.

The method

The domain is divided into elements; the unknown field is approximated as a weighted sum of basis functions over those elements. The PDE is cast in weak form and required to hold against test functions, producing a system whose matrix is sparse because each basis function overlaps only its neighbors. Solving that system yields the discrete solution.

# Weak form -> discrete system (schematic):
# find u_h = sum_j u_j phi_j  such that
# integral( grad v . grad u_h ) dV = integral( v f ) dV  for all test v
# assembles to  K u = b  (K sparse stiffness matrix)

Why mesh quality matters

Accuracy depends on the mesh. The breeder's negative-triangularity boundary and the burner's high-field end plugs demand fine, well-shaped elements where fields vary sharply, and coarser elements elsewhere for efficiency. A poor mesh either misses the physics or wastes compute, so mesh generation and adaptivity are part of the workload, not a preliminary.

• Mesh resolution concentrated where fields vary
• Sparse stiffness matrices from local basis overlap
• Iterative solvers with preconditioning for large systems
• Adaptive refinement guided by error estimates

The resulting sparse systems are large and are solved iteratively, with preconditioning to accelerate convergence. These solves are the communication-heavy, strong-scaling-limited workloads that drive the demand for a low-latency interconnect, because the iterative solver exchanges data across partitions every step.

Discretization applies across both machines and several physics. Breeder equilibrium, MHD stability, and thermomechanical response of the CrMoNbV vessel, and the burner's field and structural solves, all rest on the same finite-element foundation, differing in geometry, mesh, and the equations assembled onto it.