We consider the use of controllability techniques for the numerical solution of time-harmonic elastic wave equations. Instead of solving directly the time-harmonic equation, we return to the corresponding time-dependent equation and look for time-periodic solution. The basic approach is to time-integrate the wave equation from initial conditions until the time-periodic solution is reached. Unfortunately, the convergence of such an approach is usually slow. We accelerate the convergence with a control technique by representing the original time-harmonic equation as an exact controllability problem for the time-dependent wave equation. This involves finding such initial conditions that after one timeperiod the solution and its time derivative would coincide with the initial conditions.
Spatial discretization is done with spectral element method. It allows convenient treatment of complex geometries and varying material properties. The basis functions are higher order Lagrange interpolation polynomials, and the nodes of these functions are placed at Gauss-Lobatto collocation points. The integrals in the weak form of the equation are evaluated with the corresponding Gauss-Lobatto quadrature formulas. As a consequence of the choice, spectral element discretization leads to diagonal mass matrices which significantly improves the computational efficiency of the explicit time-integration used. Moreover, when using higher order elements, same accuracy is reached with less degrees of freedom than when using lower order finite elements.
After discretization, exact controllability problem is reformulated as a least-squares optimization problem, which is solved with a preconditioned conjugate gradient algorithm. Each conjugate gradient iteration requires computation of the gradient of the least-squares functional, which involves the solution of the state equation and the corresponding adjoint equation, solution of a linear system with the preconditioner, and some matrix-vector operations. Computation of the gradient of the functional is an essential point of the method, and we have done it with the adjoint state technique directly for the discretized problem. Algebraic multigrid method is used for preconditioning the conjugate gradient algorithm.