Wave equation

2D wave equation with Dirichlet boundary conditions

Consider the time dependent solution to 2D wave equation on annulus shaped domain

\( \begin{align*} \frac{ \partial^2 u}{\partial t^2} &= c^2 \nabla^2 u &&\text{in } \Omega, \\ u &= 0 &&\text{on } \partial \Omega_O,\\ u &= f(t) &&\text{on } \partial \Omega_I, \end{align*} \)

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

\( \begin{align*} f(t)= u_o \sin \omega_o t. \end{align*} \)

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.

 1 // // identifier to be added to nodes on the inner boundary
 2 int CENTRE = -10;
 3 
 4  // build circle domain
 5 BallShape<Vec2d> domain({0, 0}, outer_radius);
 6 auto discretization = domain.discretizeBoundaryWithStep(dx);
 7 
 8 // build source domain
 9 BallShape<Vec2d> empty({0, 0}, inner_radius);
10 auto discretization_empty = empty.discretizeBoundaryWithStep(dx, CENTRE);  
11     
12 // substract the source domain
13 discretization -= discretization_empty;

Next the domain is populated with nodes in acordance with the desired density function.

 1 // Lambda function for setting the density of nodes
 2 auto fill_density = [=](const Vec2d& p) {
 3     double r = p.norm();
 4     double default_value = dx;
 5     double dens = default_value;
 6      double r1 = 15*inner_radius;
 7      double r2 = 0.8*outer_radius;
 8      if (r < r1) dens = linear(inner_radius, 0.8*default_value, r1, default_value, r );
 9      if (r > r2) dens = linear(r2, default_value, outer_radius, 0.8* default_value, r);
10      return dens;
11 };
12 
13 GeneralFill<Vec2d> fill;
14 fill.seed(fill_seed);
15 discretization.fill(fill, fill_density);
16 
17  // find support
18 FindClosest find_support(n);
19 discretization.findSupport(find_support);