FEA Program

The Basic Principle of Finite Element Analysis

Finite element analysis is a method that is designed to obtain approximate solutions to complex engineering problems. It is a numerical technique that is based on the premise that complex engineering problems can be solved by subdividing them into smaller problems or into more manageable elements and then solving them separately. According to Rao (2011, p. 11), the behaviour of a model can be described using complex partial differential equation. By considering the behaviours of the finite elements that comprise a model, these equations can be reduced into less complex linear equations that can be solved with ease using the standard methods of matrix algebra. Thus, finite element method applies these standard methods of matrix algebra to solve equations derived from a set of finite elements that make a model.

According to Rao (2011, p. 11), any particular variable in a complex problem has an infinite number of values since it is a function of each point or node within a structure. Therefore, a problem describing a model comprise of an infinite number of unknown values. The finite element method is used to solve such a problem in three major phases: Pre-processing phase, analysis phase and post-processing phase (Rao, 2011, p. 11). The first step in the Pre-processing phase involves subdividing a model or solution region into finite elements. This is done by taking into account any symmetry, loadings, material changes or boundary conditions. The unknown variables within each element are then expressed in terms of less complex set of functions describing each element. These functions are then defined in terms of the unknown variables at specific points known as nodes (Rao, 2011, p. 11). Nodes or nodal points reflect the changes in geometry, material properties, applied loads and constraint conditions. These points usually lie on the boundaries between elements. The less complex functions representing finite elements are then selected to represent the differences in variables within an element. This process is known as meshing. As Rao (2011, p. 12) points out, polynomials are used as the functions for the variables since they are easier to differentiate and integrate. The degree of polynomials selected is dependent on the number of unknown variables at each nodal point, the number of nodal points that are assigned to each finite element and the continuity requirements imposed at interpolation boundaries and at the nodes.

After the finite element mesh has been successfully established, matrix functions expressing the properties of each finite element are formed. This can be done in three different methods namely; direct method, weighed residual method and the variation method (Huebner et al, 2001, p. 301). Direct method is used for simple problems only. The weighed residual method is done by evaluating the equations assigned to the finite elements directly from its governing directional equation. The variation method involves reducing the potential energy of a system and as Huebner et al (2001, p. 301) point out, it is a more straight forward approach compared to the weighed residual method. According to Huebner et al (2001, p. 301), the weighed residual method is more accurate and powerful since any complex problem can be described by one or more differential functions. The last step in the pre-processing stage involves incorporation of boundary conditions. The boundary conditions are applied at the nodal points.

The second phase is the analysis or solution phase. This phase involves calculation of matrix equations for each finite element. As Huebner et al (2001, p. 301) explain, when nodal points are only at the corners of elements, these elements are known as simplex elements and the solution process for the equations assigned on them involves exact evaluation. However, when there are nodal points in between the corner nodes, the elements are known as high order elements and equations assigned on them are evaluated using numerical integration (Huebner et al, 2001, p. 302). The system equations are solved either by elimination method or wavefront method to give the unknown values at the nodal points. The post-processing phase involves calculation of displacements and strains or stresses. Simplex elements are evaluated at nodes while high-order elements are evaluated at integration points. Finally, the nodal values are averaged and the results are presented in printed or plotted format.

Application of FEM software in industry

According to Desai (2012, p. 31), FEM was originally developed for the purpose of analyzing aircraft structures. However, it has increasingly become applicable to a wide variety of problems in engineering. Currently, FEM is used in industries to solve a wide variety of boundary value problems which can be grouped into three categories namely, (1) time-independent or steady state or equilibrium problems (2) eigenvalue problems and (3) transient or propagation problems. According to Desai (2012, p. 31), FEM is mostly used in time-independent problems. In a steady-state problem, FEM is used to find the stress distribution or the steady-state displacement if it is a solid mechanical problem, heat flux or temperature distribution if it is a heat transfer problem and velocity or pressure distribution if it is a fluid mechanics problem.

In eigenvalue problems also, time will not explicitly appear (Desai, 2012, p. 31). These kinds of problems are extensions of steady-state problems but with additional need to determine critical values of certain parameters, on top of the corresponding time-independent configurations. In these kinds of problems, FEM helps to determine the buckling loads and mode shapes of the natural frequencies if it is a structure or a solid mechanics problem, resonance characteristic if it is an electrical circuit problem and stability of laminar flows if it is a fluid mechanics problem (Desai, 2012, p. 31). The transient or propagation problems are time-dependent. These problems arise when, for instance, one tries to determine the response of a body under sudden heating or cooling in a heat transfer field or while under time-varying force in solid mechanics area.

A brief description of the application of FEM in various engineering fields is essential in order to understand how these problems are solved. In an aerospace engineering and structural mechanics, FEM application include steady state conditions in shell structures, plates, beams, torsion and stress analysis of various structures (Desai, 2012, p. 32). An Eigenvalue analysis includes analysis of natural frequency of structures, vibrations, visco-elastic damping and stability of structures. The transient analysis includes stress wave propagation, dynamic response of models to periodic loads and thermo-elastic and visco-elastic problems. FEM applications in mechanical engineering include transient and steady thermo analysis in fluids and solids, automotive design and analysis, stress analysis in solids and manufacturing process simulation (Desai, 2012, p. 32). Applications in geotechnical engineering include slope stability analysis, stress analysis, seepage of fluids in solids and soils, soil structure interactions, analysis of tunnels, dams and boreholes and propagation of stress waves.

In fluid mechanics, hydraulic and water resources engineering, applications of FEM include analysis of steady and transient seepage and porous media and aquifers, pollution and salinity studies of surface and sub-surface water problems, analysis of fluids movements in containers, internal and external flow analysis, analysis of water distribution networks and sediment transport analysis (Desai, 2012, p. 32). Applications in nuclear engineering include steady and dynamic analysis of thermo-elastic and viso-elastic reactor components, reactor containment structures and steady and transient temperature distribution analysis of reactors. In electronics and electrical engineering, FEM applications include analysis of electromagnetic, electrical network, thermo-sonic wire bond, insulation design in high voltage equipments, heat analysis in electronic and electrical equipments, molding process analysis in encapsulation of integrated circuits and dynamic analysis in motors. In metallurgical engineering, FEM is applied in metallurgical process simulation, casting and molding (Desai, 2012, p. 32). It is also applied in simulation of chemical processes, chemical reaction simulations and in transport processes such as diffusion and advection. It is widely applied in environmental engineering in areas of air pollution modeling, pollutant transport modeling, environmental process simulation and land-fill analysis. FEM is applied in meteorology in wind predictions, climate predictions and monsoon prediction. Finally it is applied in bioengineering in prediction of blood circulation, human organs simulation and total synthesis of human body (Desai, 2012, p. 32). Generally, the mode of application of FEM extends to all engineering fields in industries.

References

Desai, Y. M. 2012. Finite Element Method with applications in Engineering, Pearson Education, New Delhi

Huebner, K. H., Dewhirst, D. L., Smith, D. E. & Byrom, T. G. (2001), The Finite Element Method for Engineers, John Wiley & Sons, London

RAO, S. S. 2011. The Finite Element Method in Engineering. Butterworth-Heinemann, New York