Application of the finite element method to second order hyperbolic type partial differential equations

Abstract

In this dissertation various models with variational forms similar to that of the wave equation are considered, i.e. second order hyperbolic type partial differential equations. These models include several linear vibration problems and heat conduction models taking phase-lag into account. Clearly numerical methods need to be used to solve these problems and the Finite Element Method (FEM) is used in this study. Before applying such a method, existence of a solution needs to be established. Therefore, a review of the work by Van Rensburg and Van der Merwe (2002) on general second order hyperbolic type problems was done. The results were not only presented, but additional remarks and a discussion which assists in applying the theory were also included. To obtain convergence results and error estimates when FEM is applied to the various models, general convergence results were presented. For this the article by Basson and Van Rensburg (2013) was used. The first model considered consists of two serially connected Timoshenko beams. One of the beams was modelled as embedded in an elastic material, while the other beam is either free or subjected to a prescribed external load. This model can be adapted for a single beam with di fferent loads on separate parts. To apply the convergence theory though it was necessary to use the double beam model, while a single beam model can be used when FEM is applied. This was demonstrated when these models were used to model a plant with a tap root system. In this biological application various things were investigated, including different forms of FEM, a comparison of the results for the static double beam and static single beam, and the dynamics of the beam. These experiments indicated that the two models compare well and gave insight into how the parameter modelling the resistance of the soil in influences key aspects of how the plant reacts due to external forces. Models for rigid bodies attached to beams were also investigated. The equations used to describe the dynamics of a beam with a tip body were derived, with special attention given to the interface conditions. Consequently, a model problem for an intermediate rigid body between two Timoshenko beams was investigated. Hyperbolic heat conduction models were also considered and the application to bio-heat transfer in skin was discussed. Specifically, a model from the work by Dekka and Dutta (2019) was investigated. Their approach to existence of solutions was scrutinized and it was found that their application of existence results from the 2002 article by Van Rensburg and Van der Merwe is incomplete. Due to this the exposition of the theory is improved in the dissertation. For all the mentioned models, the existence and uniqueness of a solution were obtained by defining the relevant function spaces and proving the required properties. Convergence was also established from the general convergence results and the systems of ordinary differential equations were obtained which can be used to obtain numerical approximations.