Difference between revisions of "Wave equation"
From Medusa: Coordinate Free Mehless Method implementation
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| − | ==2D wave equation with | + | ==2D wave equation with Dirichlet boundary conditions == |
Consider the time dependent solution to 2D wave equation on annulus shaped domain | Consider the time dependent solution to 2D wave equation on annulus shaped domain | ||
Revision as of 20:26, 7 May 2019
2D wave equation with Dirichlet boundary conditions
Consider the time dependent solution to 2D wave equation on annulus shaped domain
<math> \begin{align*}
\frac{ \partial^2 u}{\partial t^2} &= c^2 \nabla^2 u &&\text{in } \Omega, \\
u &= 0 &&\text{on } \partial \Omega_O,\\
u &= f(t) &&\text{on } \partial \Omega_I,
\end{align*} </math>
where $\partial\Omega_I$ denotes the inner and $\partial\Omega_O$ the outer boundary of the domain. Through the boundary condition on the inner boundary the source is introduced to the problem as a function of time
<math> \begin{align*} f(t)= u_o \sin \omega_o t. \end{align*} </math>
First the domain is constructed by subtracting a smaller circle domain from a larger one. Boundaries of the domain are populated in the same step.
// // identifier to be added to nodes on the inner boundary
int CENTRE = -10;
// build circle domain
BallShape<Vec2d> domain({0, 0}, outer_radius);
auto discretization = domain.discretizeBoundaryWithStep(dx);
// build source domain
BallShape<Vec2d> empty({0, 0}, inner_radius);
auto discretization_empty = empty.discretizeBoundaryWithStep(dx, CENTRE);
// substract the source domain
discretization -= discretization_empty;
Next the domain is populated with nodes in acordance with the desired density function.
// Lambda function for setting the density of nodes
auto fill_density = [=](const Vec2d& p) {
double r = p.norm();
double default_value = dx;
double dens = default_value;
double r1 = 15*inner_radius;
double r2 = 0.8*outer_radius;
if (r < r1) dens = linear(inner_radius, 0.8*default_value, r1, default_value, r );
if (r > r2) dens = linear(r2, default_value, outer_radius, 0.8* default_value, r);
return dens;
};
GeneralFill<Vec2d> fill;
fill.seed(fill_seed);
discretization.fill(fill, fill_density);
// find support
FindClosest find_support(n);
discretization.findSupport(find_support);