Approximation of waves in composite media.

*(English)*Zbl 0682.73028
The mathematics of finite elements and applications, VI, MAFELAP 1987, Proc. 6th Conf., Uxbridge/UK 1987, 55-74 (1988).

[For the entire collection see Zbl 0652.00018.]

We are concerned with the numerical simulation of waves in a composite, isotropic medium \(\Omega \subset {\mathbb{R}}^ 2\) consisting of an elastic solid subdomain \(\Omega_ s\) surrounding a fluid-filled porous subdomain \(\Omega_ p\). The elastic wave equations and Biot’s low frequency dynamic equations are used to describe the propagation of waves in \(\Omega_ s\) and \(\Omega_ p\), respectively. Boundary conditions allowing conservation of energy flux are applied at the contact surface between \(\Omega_ s\) and \(\Omega_ p\), while absorbing boundary conditions are employed at the artificial boundaries of the model and a stress-free condition is used at the free surface of the domain.

In this paper we show the results obtained by the implementation of a finite element technique and compare the results with the corresponding ones for an inhomogeneous solid in which the porous medium is replaced by an elastic solid with physical properties partially equivalent to those of the porous medium in a sense to be explained later.

In §2 we describe the composite model and then derive a weak form of it. Then we discuss the finite element spaces used to discretize the problem and introduce the finite element procedure. Results concerning the existence and the uniqueness of the solution of the differential and finite element systems and the convergence of the approximate solution to that of the differential system are summarized and referenced. In §3 we describe the experiments performed and present the results of these experiments. Conclusions and some suggestions of continued research on the subject are also included.

We are concerned with the numerical simulation of waves in a composite, isotropic medium \(\Omega \subset {\mathbb{R}}^ 2\) consisting of an elastic solid subdomain \(\Omega_ s\) surrounding a fluid-filled porous subdomain \(\Omega_ p\). The elastic wave equations and Biot’s low frequency dynamic equations are used to describe the propagation of waves in \(\Omega_ s\) and \(\Omega_ p\), respectively. Boundary conditions allowing conservation of energy flux are applied at the contact surface between \(\Omega_ s\) and \(\Omega_ p\), while absorbing boundary conditions are employed at the artificial boundaries of the model and a stress-free condition is used at the free surface of the domain.

In this paper we show the results obtained by the implementation of a finite element technique and compare the results with the corresponding ones for an inhomogeneous solid in which the porous medium is replaced by an elastic solid with physical properties partially equivalent to those of the porous medium in a sense to be explained later.

In §2 we describe the composite model and then derive a weak form of it. Then we discuss the finite element spaces used to discretize the problem and introduce the finite element procedure. Results concerning the existence and the uniqueness of the solution of the differential and finite element systems and the convergence of the approximate solution to that of the differential system are summarized and referenced. In §3 we describe the experiments performed and present the results of these experiments. Conclusions and some suggestions of continued research on the subject are also included.

##### MSC:

74J99 | Waves in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

74E30 | Composite and mixture properties |

76S05 | Flows in porous media; filtration; seepage |