NIST-05 (Battery):¶

Model problem¶

This problem models heat conduction in a battery with nonhomogeneous materials. The solution has multiple point singularities in the interior of the domain. The domain is the rectangle shown in the following figure; the numbered regions indicate the areas of different material constants.

Equation solved:

$-\frac{\partial }{\partial x}\left(p(x, y)\frac{\partial u}{\partial x}\right) -\frac{\partial }{\partial y}\left(q(x, y)\frac{\partial u}{\partial y}\right) - f(x, y) = 0.$

Boundary conditions: Zero Neumann on left edge, and Newton on the rest of the boundary:

$p(x, y)\frac{\partial u}{\partial x}\nu_1 + q(x, y)\frac{\partial u}{\partial y}\nu_2 = g_{left}(x, y) \ \mbox{on} \ \Gamma_{left},$

$p(x, y)\frac{\partial u}{\partial x}\nu_1 + q(x, y)\frac{\partial u}{\partial y}\nu_2 + c(x, y)u = g_{right}(x, y) \ \mbox{on} \ \Gamma_{right},$

$p(x, y)\frac{\partial u}{\partial x}\nu_1 + q(x, y)\frac{\partial u}{\partial y}\nu_2 + c(x, y)u = g_{top}(x, y) \ \mbox{on} \ \Gamma_{top},$

$p(x, y)\frac{\partial u}{\partial x}\nu_1 + q(x, y)\frac{\partial u}{\partial y}\nu_2 + c(x, y)u = g_{bottom}(x, y) \ \mbox{on} \ \Gamma_{bottom}.$

Here $p(x, y)$ , $q(x, y)$ , and the right hand side $f(x, y)$ are constant coefficient functions in different materials.

Domain of interest: $(0, 8.4) \times (0, 24)$ .

Exact solution: Unknown.

Material parameters¶

// Problem parameters.
const int OMEGA_1 = 1;
const int OMEGA_2 = 2;
const int OMEGA_3 = 3;
const int OMEGA_4 = 4;
const int OMEGA_5 = 5;

const double P_1 = 25.0;
const double P_2 = 7.0;
const double P_3 = 5.0;
const double P_4 = 0.2;
const double P_5 = 0.05;

const double Q_1 = 25.0;
const double Q_2 = 0.8;
const double Q_3 = 0.0001;
const double Q_4 = 0.2;
const double Q_5 = 0.05;

const double F_1 = 0.0;
const double F_2 = 1.0;
const double F_3 = 1.0;
const double F_4 = 0.0;
const double F_5 = 0.0;

Boundary condition parameters¶

// Boundary condition coefficients for the four sides.
const double C_LEFT = 0.0;
const double C_TOP = 1.0;
const double C_RIGHT = 2.0;
const double C_BOTTOM = 3.0;

const double G_N_LEFT = 0.0;
const double G_N_TOP = 3.0;
const double G_N_RIGHT = 2.0;
const double G_N_BOTTOM = 1.0;