Application of finite element methods for scattering and vibration problems

The scattering of waves by an inclusion of arbitrary shape is a problem of interest in several fields of engineering. Several of the analytical/numerical techniques that have been developed in recent years, although easy to implement and of wide applicability, are not satisfactory for truly arbitrary geometries, materials that are inhomogeneous and/or anisotropic and when nearby boundaries are involved. For such problems, we have found it convenient to develop a hybrid method using the finite element approximation in a bounded region enclosing the inclusion and the fields in the infinite exterior domain are expanded in wavefunctions that satisfy radiation conditions. Fields and appropriate derivatives are matched at physical and mathematical boundaries leading to a solution of the fields in the entire region. A slightly different approach has been developed for thin elastic shells of arbitrary shape immersed in water. For this problem the modal impedance of the shell is found using shell theory and a finite element technique. The impedance is then used in the integral representations for the exterior problem and solved by the usual T‐matrix approach. Some of the examples that will be discussed are (1) scattering of elastic waves by voids and solid inclusions embedded in a solid, (2) scattering of elastic waves by a transversely isotropic piezoelectric inclusion, (3) scattering of acoustic waves by thin elastic shells, laminated composite shells, and stiffened shells in water, and (4) natural vibration frequencies and mode shapes of objects of arbitrary shape.