FEM

What is FEM at all ?

The Finite Element Method (FEM), also called FEA (for Finite Element Analysis), is actually an approximate mathematical method for solving problems which can be determined by differential equations. Most problems in statical analysis, fluid mecanics, and heat transfer are problems of this class. The main idea of FEM is to break a complicated problem with irregular edge conditions into small pieces (elements) of a finite size. Each piece is considered to be part of the main problem, thus connected to the other pieces via the global state information (i.e. deformation) of the element nodes, which are common nodes with the neighborhood elements. For the small element itself, the internal phisical laws (i.e. Hooks law for elastic deformation problems) can be sinplified, because the element shape is simple, mostly a triangel or a rectangle. The global problem can be transformed into a matrix of simple element equations which are connected by the condition, that common nodes undergo the same change of global state. Forces (perhaps loads in a structural problem, heat in a heat transfer problem etc.) which act on the edge of the global thing can be simplified as acting at discreet nodes. This altogether gives a big system of mostly linear equations which can be easily solved by computer. The result is the change of global state for each node (i.e. the new node coordinates after the deforming of a structure, or the new node temperatures in a heat transfer problem). Having this, further information for the small elements itself can be obtained , i.e. the element stresses in each direction.

Linear problems, e.g. structural problems ruled by Hookˇäs law, can be solved with one solver run, also Eigenvalue search for ideal-elastic buckling loads.

Non-linear problems, where you have to consider the change of geometry of material properties during the load application, have to be solved in more than one step, mostly iterative , using the results of the last run as start value for the next run, until certain exit conditions are fulfilled.Such problems include large deformation (cable and membrane structures) and local plastic material behavior, sometimes both.

The over critical buckling analysis normally is performed to find the ultimate maximum load of a structure before total failure, so plastic material behavior is to consider. Since the loads are already over the "Euler" buckling loads, large deflections are tackled by iterative solving.

Solving a problem with FEM requires the following steps:

1. Simplify the real physical problem down to an mathematically solvable problem, try to find wich effects rule the problem class, and choose the solution method.

2. Check with a small example wether your program can cope with the problem class

3. Define coordinates and material properties of the model, split the model into finite elements of sensible size, apply edge conditions (loads, supports, symmetrie etc) and check/define element properties (material thickness, bending stiffness, mass etc.)

4. Try to make an independent approximate analysis of the problem ; run the FEM solver for FEM results, check magnitude of results and main parameters

5. Watch FEM result plots and listings in detail, find out if elements are small enough to answer your specific problem; experiment with element size and watch how results depend from it, conclude the result and keep in mind what kind of accuracy may be expected

Most modern FEM programs integrate the main tasks (preprocessing, solving, postprocessing) very mean while very convinient.

(by Frank Heyder)