4A700 - Finite element method

Contents

Balance laws as used in mechanical modelling often take the form of partial differential equations. Together with the associated boundary conditions, these equations form a boundary value problem. For example, one can think of static equilibrium (balance of momentum), flow problems (mass balance and momentum balance), heat problems (energy balance) or combinations of these. Analytical solutions of these problems usually can only be obtained for relatively simple equations and geometries. For practical analyses we are therefore mostly limited to approximate, numerical solutions. A very popular method to obtain such approximations is the Finite Element Method. It divides the considered domain into small subdomains (elements), on which the solution is approximated by relatively low-order polynomials. The boundary value problem can then be rewritten as a system of equations in terms of the coefficients of these polynomials, which can be solved by (linear) algebra techniques.
In the first part of the course, the approach described above is elaborated for a one-dimensional, linear problem. Via a weighted-residuals formulation, the differential equation is rewritten in its weak form. Subsequent discretisation using shape functions results in a set of linear algebraic equations. Special attention is given to the selection of shape functions, the treatment of boundary conditions, convergence of the solution and to numerical quadrature.
The techniques developed in one dimension are subsequently extended to two and three dimensions - first for scalar equations (e.g. stationary heat conduction), and then for vectorial problems (e.g. static equilibrium in elasticity). Several element classes are discussed, but emphasis is on quadrilateral elements.

Finally, some more advanced topics are touched upon, such as aspects of time discretisation (for instationary problems), coupled problems and mixed formulations and physical nonlinearity.
During the exercises which form part of the course, the theory is applied by implementing (parts of) a finite element program in MATLAB, performing analyses and interpreting results. For this purpose MATLAB code is provided which can be installed on the notebook. Computer problems also form part of the examination. for which a notebook is thus required.

Learning objectives

Generating approximate solutions of boundary value problems using Galerkin's method and thus providing insight in the underlying principles of finite element codes.