The Finite Element Method (FEM) is an approximate numerical technique for the solution of physical systems governed by partial and ordinary differential equations – or more simply a tool to ascertain stress/heat transfer/fluid motion…. in a system.
A physical system can normally be described mathematically as a set of i) governing equations (ordinary or partial differential equations), ii) loading and boundary conditions and iii) physical properties/geometry.
Using the FEM the differential governing equations are converted into algebraic form and an approximate solution for the unknown (e.g. displacement or temperature), formed from the sum of a number of easily manipulated, linearly independent algebraic functions multiplied by unknown scalar weighting values, is then substituted into the mathematical model.
A system of equations for each descretised element of the component is then assembled into global stiffness and mass matrices to which the boundary and loading conditions may be applied. These weighting values are then manipulated to minimise the residual/energy of the system and obtain the solution.
Why Finite Elements?
Origen primarily offers finite element analysis (FEA) services to complement both its failure analysis and stress measurement services – but FEA is also offered as a design service, one of FEM’s primary uses.
Failure often occurs in complex structures and components where stress analysis is complicated by non standard geometry or the interaction of a number of loading cases. In these cases closed form analytic solutions are seldom available and numerical stress analysis techniques, such as finite elements, are required to ascertain the stress profile in the component. As such FEA is an extremely useful tool in both fatigue and fracture mechanics studies, as well as in failure investigations.
FEA both complements, and is complemented by, experimental measurement of the stresses/vibration (see stress measurement) in a component FEA allows the results from a finite number of measurement positions to be extended to the entire component. In turn experimental stress measurement allows the effect of actual loading to be quantified accurately and allows the finite element model to be ‘calibrated’.
The dynamic response of the structure often dominates – in these cases FEA is almost indispensable.
Finite element techniques can be used to study solid mechanics, fluid dynamics, heat transfer, electromagnetics, quantum mechanics and acoustics in both linear and non linear as well as multidimensional problems.
How Does One Approach A Finite Element Analysis?
Initially the geometry of the system is modelled in either two or three dimensions. This numerical representation of the component is then broken into discrete blocks (or elements) – in a ‘meshing’ process. Material properties are then added to the model and boundary and loading conditions are then applied to the model in various steps. For simple components/geometries this can be carried out manually using a text editor, but in most cases is done using the pre-processor of the finite element package employed or by using commercial drawing/meshing computer packages.
Once this numerical model has been created the model is ‘run‘ to obtain solutions. Depending on the complexity of the structure/system and loading these runs may take seconds to weeks to complete. Advances in the speed of processors and the ever increasing size of hard drives has reduced the time of these runs significantly (or simply extended complexity of FEA models that can be analysed). Post processing of these results then allows the ‘pretty pictures’ and ‘animated GIFs’ for which FEA is renowned, to be created.