CAE-Companion-2018-2019

Theory WISSEN CAE

Principles and Applications of FDM, FVMand FEM

1 Introduction The behaviour of mechanical systems in general is described by a set of partial differential equations (PDE, in case of a dis- tributed parameter system) or by a set of ordinary differential equations (ODE, in case of a discrete parameter system). They have to be fulfilled within the domain of the considered (structural) problem subjected to detailed geometrical and other boundary conditions. A closed solution in general is not possible. Hence approximated numerical solutions are applied during the analyzing phase within the engineering design process. Suitable tools of computer aided engineering (CAE) are the Finite DifferenceMethod (FDM), the Finite Vol- umeMethod (FVM), and the Finite Element Method (FEM). They should be discussed briefly in the following sections with respect to their principles and major focus of application. A final section provides a top level comparison of the three considered methods. 2 The Finite DifferenceMethod (FDM) The Finite DifferenceMethod (FDM) is probably the oldest of the three considered methods. It is based on an approxi- mation of the derivative expressions of the PDE or ODE by appropriate finite differences. As an example a set of two points in space (or time) may be considered, where some function values are provided. In this case the slope of the underlaying function may be approximated by the difference of the function values at these two points divided by the spatial (or time) distance between the locations. Higher order derivatives are equivalently obtained. In case of a PDE (or ODE) problem description the function values at those loca- tions are the unknown to be evaluated. Then an appropriate set of ”measurement points” is defined within the considered domain and the PDE (or ODE) is formulated based on the just described finite differences with respect to the provided boundary conditions. As a result a system of algebraic equa- tions for the unknown function values is obtained and finally solved. The more ”measurement points” are defined within the considered domain, the better in general the obtained function values approximate the solution of the PDE (or ODE). A typical application of this approach within engineering analysis is the investigation of a system behaviour at the time domain. Classical time integration methods are formulated on the base of FDM, i.e. Runge Kutta and the central differ- ence time integrator.

Figure 1: Derivatives Approximated by Finite Differences

3 The Finite VolumeMethod (FVM) The Finite VolumeMethod (FVM) is newer than the FDM. As indicated by its name, a subdivision of the entire domain of the considered problem is applied into a set of finite volumes of simple geometry like triangles, quadrilaterals, tetrahedrons, hexahedrons, etc. Within each of these volumes the consid- ered problem-specific PDE (or ODE) may be easily integrated by assuming average values for the unknown functions (i.e. at the center of those finite primitives). Possible derivative expres- sions of the PDE or ODEmay be approximated by appropriate finite differences obtained from themean functional values between the centers of neighbouring finite volumes. This approach is equivalent to the one of the FDMdescribed at the previous section. In a similar way the possible flux of physical quantities through the finite volume boundaries is treated. The mean function values at the finite volumes are the unknowns to be evaluated at the FVM. In this way for every finite volume a set of algebraic equations is obtained. All of them together with the problemdefining boundary conditions (asmentioned at the introductional section) describe a systemof algebraic equations to be subsequently solved for the unknownmean function values. Themore finite volumes are definedwithin the considered domain (! mesh refinement), the better in general the obtained function values approximate the solution of the PDE (or ODE). A typical application of this approachwithin engineering analysis is the investigation of flow fields in fluid dynamics. Hence the FVM is today themajor tool for computa- tional fluid dynamics (CFD).

Figure 2: Subdivision of the Spatial Domain into Finite Volumes

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