raygeo.mesh.laplace

Functions

solve_laplace()

solve_laplace(
    mesh: types.TriangleMesh,
    max_iter: int = 1000,
    tolerance: float = 1e-08,
) -> Sequence[float]

Solve the Laplace equation Δu=0 on a triangle mesh.

Returns a scalar field with one value per vertex. Outer boundary vertices are fixed to u=1.0 and inner boundary vertices to u=0.0.

Parameter	Type	Description
mesh	types.TriangleMesh	TriangleMesh with boundary tags.
max_iter	int = 1000	Maximum conjugate gradient iterations.
tolerance	float = 1e-08	Convergence tolerance for CG residual.
Returns	Sequence[float]	List of scalar u values, one per vertex.
Complexity		O(i * n) where i = CG iterations, n = mesh vertices

Stiffness matrix edge weights on the mesh — line thickness ∝ |Kᵢⱼ|

Laplace solution on a multi-island domain — contour lines morph smoothly between four inner islands and the outer boundary

Laplace solution — contours morph smoothly from hole to boundary

Laplace solution on an L-shaped domain

solve_laplace_with_history()

solve_laplace_with_history(
    mesh: types.TriangleMesh,
    max_iter: int = 1000,
    tolerance: float = 1e-08,
) -> tuple[Sequence[float], Sequence[float]]

Solve the Laplace equation and return convergence history.

Identical to solve_laplace() but also returns the residual norm after each conjugate gradient iteration for convergence analysis.

Parameter	Type	Description
mesh	types.TriangleMesh	TriangleMesh with boundary tags.
max_iter	int = 1000	Maximum conjugate gradient iterations.
tolerance	float = 1e-08	Convergence tolerance for CG residual.
Returns	tuple[Sequence[float], Sequence[float]]	Tuple of (solution, residuals).
Complexity		O(i * n) where i = CG iterations, n = mesh vertices

Conjugate gradient convergence — residual norm per iteration