CAE-Companion-2018-2019

Theory WISSEN CAE

4 The Finite Element Method (FEM) The Finite Element Method (FEM) to some extend is similar to the FVMdiscussed in the previous section. Hence it subdivides the entire domain of the considered problem into a set of primitive domains like triangles, quadrilaterals, tetrahedrons, hexahedrons, etc. These are called Finite Elements. Within each of these elements the unknown field is interpolated by a combination of (initially unknown) function values and spatial shape functions. These shape functions are specific to each finite element. Based on this approach the considered PDE or ODE may be integrated over the elemental domain with the still unknown function values as parameter variables. For each element a set of algebraic equations is obtained. All of them together with the problem defining boundary conditions (as mentioned at the introduc- tional section) form a system of algebraic equations to be subsequently solved for the unknown parameter values. The more finite elements are defined within the considered do- main (! mesh refinement), the better in general the obtained function values approximate the solution of the PDE (or ODE). A typical application of this approach within engineering anal- ysis is the investigation of mechanical structures with respect to their static or dynamic behaviour like deformation and mechanical stress distribution. But the wide field of applica- tion of the FEM includes as well safety aspects (i.e. computer crash analysis at the automotive industry) and the simulation of production processes like sheet metal forming.

5 Properties of theMethods The three methods FDM, FVM and FEM as described in the previous sections contain some similarities. I.e. the obtained solution of the considered problem is always an approximation depending on the applied number of objects, either function evaluation points, finite volumes or finite elements. Hence the quality of the result always depends on the applied effort. But in details each of the method behave in a different way depending on its individual methodical approach. The FDM is purely based on the differential description of the PDE or ODE defining the considered problem. The FVM and FEM are based on a (numerical) integration of the underlaying PDE or ODE. Practical tests reveal a high precision for the FDM compared to (physical) measurements. The real strength of the FEM lies on the flexibility with respect to its applica- tion. The FVM is found to be somewhere located between FDM and FEM, both with respect to flexibility and preci- sion. Over the past decades these properties have surely defined the major field of engineering application for the three methods FDM, FVM and FEM.

Figure 4: Flexibility and rrecision of the discussed methods (see [4])

References [1] Bathe, K.J., Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996 [2] Ferziger, J.H., Peric, M., Computational Methods for Fluid Dynamics, Springer Verlag, Berlin, 1996 [3] Hughes, T.J.R. The Finite Element Method, Prent. Hal. Int. Ed., Englewood Cliffs, New Jersey, 1987 [4] Lecheler, S. Numerische Stömungssimulation, Vieweg-Teubner Verlag, Wiesbaden, 2009 [5] Wissmann, J., Sarnes, K.-D., Finite Elemente in der Struk- turmechanik, Springer Verlag, Berlin, 2005

Figure 3: Deformed Finite Element model and border stresses

CAEWissen by courtesy of Prof. Dr.-Ing. DetlevMaurer, University of Applied Sciences Landshut

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