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# Numerical simulations

## Non conforming interface projections

When one applies a domain decomposition technique such as the mortar method, integrals of discrete functions defined over (the trace of) the first mesh multiplied by discrete functions defined over (the trace of) the second mesh over the coupling interface (that will be called Gamma) must be computed. In order to properly perform this integration, the intersection between both meshes needs to be done. In the following simulation we use the fast algorithm introduced by M. J. Gander and C. Japhet in the paper "An algorithm with optimal complexity for non-matching grid projections".

In this first example we consider two different meshes of the interface Gamma = [-1,1] x [-1,1]x{0} (see figures (a) and (b)). Both meshes exactly represent the interface Gamma. In figure (c) we show the intersection between both meshes that we use to compute the projection matrices (see the associated animation to have more details on how the algorithm works).

(a)  First mesh of the interface Gamma=[-1,1]x[-1,1]x{0}.
(b)  Second mesh of the interface Gamma=[-1,1]x[-1,1]x{0}.
(c)  Intersections used to compute the projection matrix when the meshes displayed above are considered. Click to animate [72 Kb] .

When both meshes do not represent the same geometry, following the ideas introduced by B. Flemish, J. M. Melenk and B. I. Wohlmuth, we locally project one mesh over the other and we apply a (slight) modification of the previous algorithm. In figures (d) and (e) we present two meshes of Gamma=[-1,1]x[-1,1]x{0} that do not exactly represent the geometry. In figure (f) we represent the intersections performed by the algorithm to compute the projection matrix when the second mesh is locally projected into the first one. In figure (g) we show the results in the other case. The matrices obtained by both procedures are not identical (see the associated animations to have more details on how the algorithm works).

(d)  First mesh of the interface Gamma=[-1,1]x[-1,1]x{0}.
(e)  Second mesh of the interface Gamma=[-1,1]x[-1,1]x{0}.

(f)  Intersections used to compute the projection matrix when the second mesh is projected over the first. Click to animate [72 Kb] .
(g)  Intersections used to compute the projection matrix when the first mesh is projected over the second. Click to animate [64 Kb] .

## DG and retarded potentials for acoustics on unbounded domains

In this study we are coupling an interior discretization based on a DG approximation in space combined with explicit finite differences in time with a boundary method (the retarded potential method) to solve the transient wave equation on unbounded domains. This yields to a global discretization that is stable under the usual CFL condition on the interior domain.

On this numerical example we simulate the propagation of a spherical wave generated by an initial condition on the 3D space. The interior domain is the L-shaped domain on the figures which is not convex (note that the generated waves are not only out-going with respect the the boundary of the interior domain, but also in-going). The behavior of the wave outside the interior domain is taken into accout by the retarded potential method. Since we have not considered any scatterer, no spurious reflexions due to the artifitial boundaries should be created. The numerical results are in perfect agreement with the exact solucion.

From left to right, from top to bottom: (a) the pressure field in the interior domain (DG unknown discretized with P2 polynomials eventually discontinuous), (b) the trace of the pressure field on the artificial boundary (RP unknown discretized with P1 continuous polynomials), (c) the first component of the velocity field in the interior domain (DG unknown discretized with P2 polynomials eventually discontinuous), (d) the normal trace of the velocity field on the artificial boundary (RP unknown discretized with P0 polynomials).

## Fictitious domains method in elastodynamics

When we discretize elastodynamic equations with the Q1divx Q0 element the fictitious domain method with a diagonal crack fails. The amplitud of the transmited wave does not converge to zero.
Enriching the finite element space for the velocity field, that is, using the Q1divx P1disc element, the fictitious domain method provides a good approximation of the solution for any crack configuration.

## Space-time mesh refinement for elastodynamics

The conservative method conserves a discrete energy that is equivalent to the L2-norm of the numerical solution when the usual CFL condition is satisfied. It is thus very robust (the numerical scheme is stable under the usual CFL condition) and allows to use quasi-optimal settings on every region: the ratio between the time discretization step and the space discretization step can be kept (almost) constant. How ever, it introduces high frequency spurious waves (whose amplitud goes to zero, but rather slow) that can pollute the numerical solution on the regions where the ratio of refinement is larger than two. See the associated papers for the details.
The post-processed scheme is obtained applying a post-processing by averaging in time to the solution obtained by the conservative scheme. In consequence, this method has the same stability properties as the conservative scheme. The amplitud of the high frequency spurious waves is drastically reduced obtaining a good approximation for space-time mesh refinements with arbitrary rate.

## Fictitious domain method and space-time mesh refinement for elastodynamics

On this numerical example we combine both, the fictitious domain method and the local time stepping technique. The ratio of refinement on each box is 4. In consequence, we are using a mesh that is 16 times finer  on the tips of the cracks, if we compare it to the mesh we are using is most of the computational domain. The singularities on the solution are well resolved. We use the Q1div x P1disc elements on the regions interacting with cracks (where we are using the fictitious domain method). On the other regions we use the Q1div x Q0 element that is cheaper.
This is a detail of the previous numerical experiment. One cas see the size of the elements on the coarse mesh. On the refined regions the resolution is much higher since the elements are much smaller.
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