Mathematical Physics - Volume II - Numerical Methods

2. Finite element method

Finite element method is one of the basic methods for obtaining approximate solutions for various types of boundary problems. It involves the discretization of the solution domain into subdomains called finite elements. Approximate solutions in these finite element sets is then obtained using variation principles. Thanks to this general approach, finite element method became a widely used tool in solving numerous problems in various fields, such as mathematical physics, aerospace engineering, structural analysis, etc. The best way to understand the concepts of finite element method is to start with a one dimensional problem, hence such problems will be the first to be analysed here. 2.1 Finite element application to solving of one-dimensional prob lems In physics, most problems can be represented using two functions which need to be determined, the state variable u and the flux P . These two functions are related to each other via constitutive equations, which contain all necessary data about the material wherein the process takes place. These equations can be related to various laws of physics, but are always of the same mathematical shape, which can be represented in the following way, for the case of elastic stresses in a bar, where u is the displacement, and P is the stress: P ( z ) = − k ( z ) d u ( z ) d z . (2.1) The constitutive equation shown above is known as Hooke’s law modulus (material property related to its stiffness). This equation (2.1) indicates that the non-uniformity of a state variable (displacement) results in the occurrence of a flux (stress in this case), i.e. that this flux is proportional