Nonlinear Finite Elements in Dynamics

This report focuses on the implementation of the finite element method for nonlinear dynamical problems arising in solid mechanics. The theory of continuum mechanics is extensively covered in the literature and it is assumed the reader is already familiar with it.The report is clearly divided between a theoretical and a practical part. The theoretical part is not an extensive summary of what can be found in the literature. It details procedures and derives equations I have used for the implementation and which are not common in the literature. Although in the end, the equations look familiar, the way the implementation was thought out is fundamentally different.Usually, the transition from the general conservation equations of continuum mechanics to numerical models is achieved trough the introduction of Voigt notation. This in itself is fully justified as the representation of many tensor quantities are symmetric matrices and thus we want to avoid computing twice the same component. Moreover, this leads to simple equations at a first glance. However, the resulting matrices must be artificially constructed to be consistent with Voigt notation. They do not result from any mathematical rule. This is a first drawback. Moreover, these matrices are generally sparse (notably the matrix N,often found in finite element literature). Naturally, they could be stored as sparse matrices but given their very short lifetime, this operation may not be profitable. Not doing so would mean asking the computer to perform more operations than what would strictly be necessary as a number of entries are zero. The aim of this first part is to develop an alternative procedure that will stay as close as possible to the mathematical representation of the balance of linear momentum. It will therefore be our stating point. Furthermore,we will prove that it is possible to obtain the same numerical results by using standardized and compact equations.