Dirk Vandepitte

The theory of Biot is most commonly used to model the behaviour of poroelastic materials. Nowadays, the Finite Element Method is the most well-established method to predict the dynamic response of those materials. This method subdivides the problem domain into a large number of small elemental domains in which the dynamic behaviour is approximated using simple polynomial shape functions. Due to the high number of unknowns per node and the fine meshes required to accurately capture the wave fields, the use of FEM is practically limited to low-frequency applications. Recently, an efficient Wave Based Method has been proposed to solve the Biot equations. This method is based on a Trefftz approach and takes the physics of the three wave types existing in poroelastic media into account, resulting in a faster convergence. This paper discusses applications of this method for multilayered structures and axisymmetric problems.

Bone fracture healing is a complex process in which angiogenesis or the development of a blood vessel network plays a crucial role. In this paper, a mathematical model is presented that simulates the biological aspects of fracture healing including the formation of individual blood vessels. The model consists of partial differential equations, several of which describe the evolution in density of the most important cell types, growth factors, tissues and nutrients. The other equations determine the growth of blood vessels as a result of the movement of leading endothelial (tip) cells. Branching and anastomoses are accounted for in the model. The model is applied to a normal fracture healing case and subjected to a sensitivity analysis. The spatiotemporal evolution of soft tissues and bone, as well as the development of a blood vessel network are corroborated by comparison with experimental data. Moreover, this study shows that the proposed mathematical framework can be a useful tool in the research of impaired healing and the design of treatment strategies.

In the past decades element‐based numerical modelling techniques have become a commonly used and invaluable tool for the analysis of acoustic radiation and scattering problems. However, the pollution errors associated with the element discretisation inherent to these methods increase with the increasing size of the problem domain. As a result, the applicability of these methods for radiation and scattering problems in which the source and receiver positions are located far from each other is often prohibited. The Wave Based Method (WBM) is an alternative deterministic prediction method for the analysis of steady‐state acoustic problems. It is based on an indirect Trefftz approach in that wave functions, which are exact solutions of the underlying differential equation, are used to describe the dynamic response. The enhanced computational efficiency of the WBM as compared to the element based methods has been shown already for the analysis of both finite and (semi‐)infinite acoustic problems. This paper introduces a novel WBM‐based methodology to model the acoustic source‐receiver transfer path functions in a multiple scatterer environment. A sound propagation validation case illustrates the potential of the proposed approach.