1. Introduction#
Dynamic soil-structure interaction problems (ISS), or even soil-fluid-structure interaction, are often solved using a substructuring technique where the complete system is broken down into two or more sub-domains. In particular, for cases of non-linear ISS, it is broken down into a linear domain (the ground in the far field) and a non-linear domain (the building and possibly a part of the surrounding ground).
The advantage of using this decomposition technique is essentially based on the possibility of using the numerical methods most suited to each field. Thus, the linear ground in the far field that corresponds to an infinite domain can be modelled with the border element method (BEM for the English acronym) and the non-linear part, which is rather confined to a bounded domain, often with a complex geometry for the structure part, can be modelled with the finite element method (FEM). This numerical approach BEM - FEM is in fact the one used in code_aster — MISS3D chaining.
In this context, two particularities are worth recalling. First, the fact that any non-linear problem must generally be solved with a transitory calculation. And then, the fact that MISS3D, based on a frequency formulation of the boundary element method, makes it possible to model the unbounded linear ground using an impedance matrix (ground stiffness that depends on the frequency) projected on a basis of modes representative of the kinematics of the interface between the linear and non-linear domain (also known by misnomer as interface ISS). In view of these two considerations, the coupled problem is entirely formulated in the time domain, but a convolution product coming from the frequency dependence of the impedance appears at the interface level.
To evaluate this convolution product, the literature proposes the time-frequency method or the method of hidden variables which, using the impedance function or its inverse, are based on a formulation in the frequency domain. Formulations of the impedance function are also found in the Laplace domain, which can be combined with convolutional quadrature methods. However, the method developed in CALC_MISS (the Laplace-Temps method) which is the one presented in this document makes it possible to evaluate convolution integrals by revealing the terms of inertia, damping and stiffness that characterize dynamic problems.
1.1. Problem: evaluation of convolutional integrals#
As mentioned earlier, the Laplace-Temps method involves the evaluation of a convolution product. The main difficulty of its calculation comes from the singular nature of the convolution kernel (the impedance matrix) which in fact makes it difficult to evaluate it with a simpler and classical approach linked to algorithms of the FFT type.
To quickly illustrate the origin of this product, we will assume that the system is linear in order to thus be able to start from the equations formulated in the Fourier domain:
where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrices of the system, \({\widehat{Z}}_{s}(\omega )\) the ground impedance matrix, \({\widehat{F}}_{s}(\omega )\) the equivalent seismic force, and \({\widehat{u}}_{\Gamma }(\omega )\), the vector of unknowns moving at the interface level. Then, the convolution product appears directly by applying the Fourier transform to the previous equation:
with \({R}_{\Gamma }(t)={\int }_{0}^{t}Z(t-\tau ){u}_{\Gamma }(\tau )\text{}d\tau ,\text{}0\le t\le T\) and \(Z(t)\) the analog ground impedance operator in time.
If the structure part (plus possibly the ground in the near field) were non-linear, the convolution product would also appear in the equations of the unknowns of interface ISS because the soil-structure interaction efforts only depend on the nature of the unbounded soil domain, which is itself assumed to be always linear.