Desceliers C., Grimal Q., Haiat G., Naili S., Soize C.

This communication presents a hybrid method to simulate the transient elastic waves over a short time in multilayer semi-infinite media subjected to given transient loads. The medium is constituted of a finite number of unbounded multilayers with bounded thickness. This boundary value problem can be solved in the frequency domain and in the 3D-spectral domain using the Green functions. Another approach consists in solving the problem in the frequency domain and in a 2D-spectral domain (corresponding to the two space infinite dimensions) and using the finite element method for discretizing the bounded domain corresponding to the third space dimension. Such a method can induce numerical difficulties which can be avoided by using an adapted algebraic formulation which can be tricky to be implemented. For the transient response of the system over a short time, the numerical cost is smaller in solving directly the problem in the time domain than in the frequency domain for which the calculation would require a broad frequency band increasing the numerical cost. Therefore we present a method avoiding such difficulties. It is based on a time domain formulation conjugated with a 2D-space Fourier transform for the two space infinite dimensions and using a finite element discretisation for the third space finite dimension. First, the boundary value problem is written in 1D-space, 2D-spectral space and time domain. The weak formulation of the boundary problem is introduced. Then the finite element approximation for the 1D-space is constructed. An implicit time integration scheme is used for solving the differential equation. The 3D-space solution is then obtained by an inverse 2D-space Fourier transform. An example is presented for a 3-layers system constituted of an elastic solid layer sandwiched between two acoustic fluid layers and excited by an acoustic line source located in one of the two acoustic fluid layers.