Controllability method for solving the frequency-domain elastodynamic equations (Ludovic Metivier, University of Grenoble)

01.04.2025 14:00

Computing numerical solution of frequency-domain wave
equations is a challenging problem. The corresponding linear system is
ill-conditioned, and preconditioners remaining efficient for large scale
3D problems, where one tries to model the wave propagation over several
hundreds of wavelengths in each direction, are difficult to design. For
practical applications, such as in seismic imaging, the common practice
has thus been to work with parallel direct solvers dedicated to the
solution of sparse linear systems. This approach is all the more
interesting when multiple right-hand-sides have to be considered, as it is
the case in seismic imaging. Recently, these approaches have been
coupled to iterative approaches through domain decomposition strategies,
yielding interesting perspectives in terms of 3D large scale
applications . In this work, we are interested yet in another approach,
named controllability approach, introduced by Bristeau et al in 1998 for
acouctic problems. The innovation relies on considering the problem of
finding a harmonic solution of the wave equations as finding a
time-domain solution satisfying a periodicitiy criterion: the solution
after one period needs to be as close as possible to the solution at
time 0. In this controllability problem, the control parameters are the
initial conditions on the solution, and one can express the
time-harmonic solution as a linear combination of the "optimal" initial
conditions. In this work, we implement this strategy for the 3D
elastodynamics equations, relying on an existing time-domain code for
these equations named SEM46, which implements a spectral element
discretization and is highly optimized to run large scale computational
machines. We show how the controllability approach can be adapted to the
3D elastodynamics equations and how we can benefit from SEM46 to
construct time-harmonic solutions for 3D large scale realistic problems.

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Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

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