Finite Element Method (FEM) and Finite Different Method (FDM)
Comparison of two methods for the simulation of casting processes
Both methods FEM and FDM are numeric approximations to make computable certain physical / technical procedures. Often such procedures can only be described by (partial) differential equations, that are not analytically solvable for the general case. Both methods have in common, that the geometry to be calculated has to be fractionated in small basic elements.
In FDM one can only use cubes as the basic elements, whereas in FEM one can use arbitrary basic elements, the sides haven't even to be straight. If one is dependent on using cubes as the only element, this leads automatically to a bad approximation of the geometry, because technical objects aren't usually designed pure rectangular. First of all this seems only an optical problem, but as is easy to see, it is far more than that. For example you are not able to enmesh a hollow cylinder with thin walls with a constant wallthickness when you use only cubes (see figure). This of course leads to the result, that doing a solidification calculation, this wall cannot have a homogeneous temperature distribution across the diameter, if the calculation is not than that unprecise, that you can't even see the difference in the wallthickness.
- 48 nodes - 108 elements
- 176 nodes - 296 elements
- 632 nodes - 876 elements
(the number of elements will be doubled, if triangles are used)
- 8 nodes - 4 elements
- 32 nodes - 16 elements
- 192 nodes - 128 elements
Using cubes to model the geometry bears still further problems. FDM and FEM both are approximations, because one has to make an assumption on how the distribution of the value to be calculated is in the basic elements. Usually it is assumed to be linear. The more the actual distribution differs from this assumption, the bigger the error becomes naturally. This difference is big right there, where the local gradient shows bigger changes; this is normally only in certain areas of the model. The FEM can meet this problem in making a local refinement right in those areas, whereas the mesh may be a bit more gross in the areas, where the gradient doesn't change that much. In FDM this possibility is not feasible to this degree, as a refinement isn't possible only local, but will have to be made throughout the entire geometry. These are the reasons that the FDM has to handle greater amounts of elements and that means that the computing capacity has to be increased by far in order to reach the same accuracy in the results as would do a FEM calculation.
Another eminent advantage of FEM over FDM is the possibility to calculate with one model of the geometry different types of simulations, including:
The exact contour describing in FEM of the casting, core or mould can furthermore used for rapid prototyping in different proceedings.
- residual stress analysis
- distortion calculation
- strength analysis
- modal analysis