Electrical properties of an L-shaped conducting plate

This Python code implements a finite-element solution of the two-dimensionalLaplace equation for an L-shaped conducting plate. The computational domain isa square [0,l]×[0,l] with a square subdomain (a,l]×(a,l] removed, resulting in are-entrant corner geometry. The electric potential satisfies the Laplace equation inside the domain, withDirichlet boundary conditions applied on two adjacent outer edges: Φ = 0 onx = l for 0 ≤ y ≤ a, and Φ = 1 on y = l for 0 ≤ x ≤ a. All remaining boundariesare treated as insulating. The problem models steady-state current flow in aconducting plate driven by a fixed potential difference. An adaptive unstructured triangular mesh is generated using Delaunaytriangulation. The mesh is locally refined near the re-entrant corner and nearthe current-injecting boundaries in order to accurately resolve the strongfield gradients induced by the singular geometry. The finite-element discretization uses linear basis functions. The globalstiffness matrix is assembled explicitly and the resulting sparse linear systemis solved using SciPy’s sparse direct solver. The electric field is obtained asthe gradient of the computed potential. The total current is evaluated by integrating the normal component of theelectric field along the boundary x = l. From this flux, a dimensionlessresistance R* is computed. The code allows one to calculate R* as a function ofthe geometric parameter a/(2l), as well as to visualize the spatialdistribution of the electric potential. The script reproduces all numerical results presented in the accompanyingarticle, including potential maps and the resistance curve R*(a). No externaldependencies beyond NumPy, SciPy, and Matplotlib are required.