NIST-02 (Reentrant Corner)¶

This is a reentrant corner problem causing a singularity in the solution.

Model problem¶

Equation solved: Laplace equation

(1) $-\Delta u = 0.$

Domain of interest: $(-1, 1)^2$ with a section removed from the clockwise side of the positive $x$ axis.

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

$u(x, y) = r^{\alpha}\sin(\alpha \theta)$

where $\alpha = \pi / \omega$ , $r = \sqrt{x^2+y^2}$ , and $\theta = tan^{-1}(y/x)$ . Here $\omega$ determines the angle of the re-entrant corner.

Material parameters¶

This benchmark has four different versions, we use the global variable PARAM (below) to switch among them.

int PARAM = 1;     // PARAM determines which parameter values you wish to use for the strength of the singularity in
                   // the current (nist-2) Reentrant Corner problem.
                   // PARAM      strength         OMEGA            ALPHA
                   // 0:            1             5*Pi/4           4/5
                   // 1:            2             3*Pi/2           2/3
                   // 2:            3             7*Pi/4           4/7
                   // 3:            4             2*Pi             1/2