Implementation

Cauchy is a header-only C++20 codebase with 7 core headers, 6 validation entry points, and 19 Google Test cases. The architecture mirrors the LBM-2D solver for consistency across the portfolio.

Module Reference

File	Lines	Purpose	Key Types
fea_types.hpp	216	Core types, enums, globals, JSON parser	Material, Node, Mesh, CaseType, PlaneType
elements.hpp	380	Element stiffness matrices	BarElement, Q4Element, Q8Element, T3Element
sparse.hpp	170	Sparse matrix formats	COOMatrix, CSRMatrix, DenseMatrix
solver.hpp	173	Linear solvers	CholeskySolver, CGSolver
mesh.hpp	400	Mesh generation	generate_structured_quad, generate_structured_quad8, generate_cook_quad8, load_json
postprocess.hpp	288	Stress recovery, JSON output	ElementStress, compute_all_stresses
adaptivity.hpp	350	ZZ error estimator, adaptive refinement	SPRResult, ElementError, refine_mesh, adaptive_loop
convergence.hpp	140	GCI computation, h-refinement wrapper	GCIResult, ConvergenceSample, run_study
fea.hpp	240	Assembly + solve pipeline	assemble, solve, compute_strain_energy

Include Hierarchy

fea.hpp
  fea_types.hpp    (types, enums, globals)
  elements.hpp     (bar + Q4 + Q8 + T3 stiffness matrices)
  sparse.hpp       (COO/CSR sparse matrix)
  solver.hpp       (Cholesky + CG)
  mesh.hpp         (structured quad mesher + Q8 generators + JSON input)
  postprocess.hpp  (stress recovery, Von Mises)
  adaptivity.hpp   (ZZ SPR, error indicators, red-green refinement)
  convergence.hpp  (GCI, h-refinement)

Entry Point Pattern

Each main_*.cpp follows a consistent 7-step pipeline:

1. Set globals (g_nx, g_ny, g_case, g_analysis)
2. Generate or load mesh (Q4 or Q8 via --q8 flag)
3. Assemble global stiffness matrix (COO, OpenMP parallel)
4. Apply boundary conditions (penalty method, including Q8 midside nodes)
5. Solve (Cholesky or Conjugate Gradient)
6. Post-process (stress recovery, energy balance)
7. Write JSON output, report statistics

Build Instructions

Adaptive Refinement Pipeline

The adaptivity module (adaptivity.hpp) implements:

1. SPR Recovery: Weighted average of Gauss-point stresses at nodes from surrounding elements
2. Error Indicators: Element-wise = integral of |recovered - FE stress|^2
3. Marking: "Largest first" strategy with threshold theta (0.5 default)
4. Red-Green Refinement: Split marked Q4 elements into 4 (red) or 2 (green), add midside nodes
5. Adaptive Loop: Solve -> recover -> estimate -> mark -> refine -> repeat

22 Google Test cases across 10 test suites:

Suite	Tests	What It Validates
BarElementTest	2	Stiffness matrix correctness, symmetry
Q4ElementTest	4	Shape functions, Jacobian, stiffness symmetry, negative Jacobian detection
Q8ElementTest	3	Shape functions, 3x3 Gauss, 16x16 stiffness symmetry
SparseMatrixTest	3	COO-to-CSR conversion, SpMV, duplicate merging
CholeskyTest	2	Identity and diagonal system solves
CGTest	2	Diagonal system convergence, real FE system convergence
MeshTest	3	Node counts, coordinates, element connectivity
PatchTest	1	Constant stress recovery (element verification)
EnergyBalanceTest	1	Strain energy == work done (U == W)
SolverComparisonTest	1	Cholesky and CG produce identical results

CI Pipeline

GitHub Actions runs on every push: builds on Ubuntu and macOS, runs the full Google Test suite. Matrix configuration ensures cross-platform compatibility.