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Everything you want to know about FE, but are afraid to ask

By Dr Auday Al-Rawe
When Stephen Hawking was writing A brief history of time his publisher warned him that every equation included would half the readership.  If Hawking can describe the working of the universe without equations, it must be possible to do the same for the method of Finite Element Analysis (FEA).  This description is by no means complete but the aim is to set out the basic ideas and give an impression of what can be achieved.

Finite element analysis is a simulation process that allows engineers to calculate, with some accuracy, how a component or structure will respond to a load.  The ability to predict a performance in advance of detailed design has obvious benefits in the modern design or construction process.

Mathematically, the structure to be analysed is subdivided into a mesh of finite sized elements of simple shape that are defined by corner nodes.  Within each element, the variation of displacement, under a given load, is assumed to be determined by simple polynomial shape functions and displacement at the element nodes.  From this, the equations of equilibrium are assembled in a matrix form which can be easily programmed and solved on a computer.  After applying the appropriate boundary conditions (model restraints), displacement at the nodes are found by solving the matrix stiffness equation.  Once the nodal displacements are known, element stresses and strains can be calculated to assess the designated component.

The simplest type of finite element is a spring.  A spring has the property of stiffness – how much it will stretch for a given load.  This stiffness is assumed linear for most engineering analysis, that is, Hooke’s law applies.  The spring has two end points – nodes at which the displacements are evaluated and the forces applied, and these nodes can move along the axis of the spring.  The nodes have one degree of freedom, axial displacement.

Using simple matrix methods it is possible to calculate the force/displacement behaviour of complex arrangements of springs – interesting but not very useful.
The power of the finite element method lies in the ability to represent continuous sections of material as arrays of springs.  A simple triangular section of material is represented as three lumped stiffness’s, or springs.  The clever bit is working out what the actual stiffness of the springs should be, and it is this process that is the primary function of commercial programs.

There is a wide range of element types, each formulated with a specific job in mind.  Solid elements are used for concrete castings and similar components, shells for plate and sheet structures and beams for components such as portal frames. Each element has a number of nodes, and these nodes can have up to six degrees of freedom (three translations and three rotations).  The mathematics and computation involved in evaluating the behaviour of these elements is far from trivial, but is well within the capabilities of modern computing methods.

A typical model may consist of anything up to 100,000 elements so solving the problem to establish the structural response can be very involved process.

FEA Process
Building a finite element model comprises a number of distinct stages, each being performed in a particular order.

Project specification
As with any engineering project the most important stage of an analysis study is the project specification.  Why is the analysis being undertaken? What confidence is there in the input data? How accurate does the answer have to be? How much time is there to solve the problem? How to simulate loads and boundary conditions? All these questions must be answered before the computer is switched on.

Model definition
The geometry of the component must be defined, either in an FE package or brought in from a CAD program.  This definition must include load and fixing areas, and is often a simplified version of a component or a structure.

The model must be meshed
This may take from seconds, in the case of simple CAD geometries, to several hours, where complex structures are concerned.  Meshing is the process of dividing a component into discrete elements such as tetrahedral elements for solids, or quads and triangular elements for shell structures.  In most FEA tools, the generation of elements could be either manual or automatic. The meshing stage requires decisions to choose an appropriate element type suitable for the frame of the designed structure (i.e., solid, shell or beam elements).

Loads and fixings (boundary conditions) must be defined on the model, and finally the appropriate material data must be attached to different parts of the model.

The solution
Thanks to modern software, this part is usually fully automatic.  The FE solver can be logically divided into three main parts, the pre-solver, the mathematical calculations and the post-solver.  The first one reads the model, formulates the mathematical representation of the model, and check all parameters defined before proceeding to the calculations.  If the model is correct, an element stiffness matrix is formed for the problem and the solver proceeds to the mathematical calculations, which calculate the result (displacement, temperature, pressure, etc).  The post-solver then receives the result to calculate strains, stresses, heat fluxes, etc, for each node within the component ready for presentation.

The solution may take seconds or hours depending on model size and complexity.  This is an area where great strides are being made and supercomputer performances are being achieved on PC’s.  Such is PC performance power today that large and complex problems can be practically solved on a desktop machine.

Range of solutions
Although only solutions to structural problems have been discussed, the finite element method can be applied to a whole range of other engineering phenomena.

Thermal analysis allows engineers to investigate the thermal behaviour of bodies when subject to heat inputs and environmental effect, and this can be linked to structural analysis to investigate thermal stability or thermal stressing. Dynamic analysis can evaluate mode shapes and natural frequencies, useful for the investigation of vibration problems.  More exacting studies can take account of forced vibrations and damping effects.

In the initial explanation the fact that elements stiffness obeys Hooke’s law was mentioned; however, most engineering materials are non-linear in their behaviour.  This can be simulated using specialist software and much work has been done on problems as varied as multiple bodies in contact and the time-dependent response of rubber materials.

Solutions exist for many, many effects and the range of problems FE has been used to investigate is vast.

Results presentation
The results from finite elements analysis were in the past only available as listings from a line printer – the colour-filled plots common today could only be achieved through labour-intensive hand methods. Results presentations now include deformed shape plots, colour-filled pictures, mode shapes, and natural frequencies, as well as animations. Slices can be made through 3D models to facilitate the viewing of internal stress patterns. The principal stresses or strains and yield stresses or strains according to failure theories (such as Von Mises and Tresca) can be calculated for further assessment.

The quality of results presentation allows lessons to be learned about the analysed component, which would be impossible or too expensive to investigate with experimental techniques.

Links to CAD & beyond...
As we move through the 21st Century, design by virtual simulation will become more and more realistic.  Nearly all CAD systems offer either integrated FE codes or interface to FE codes, allowing design to go beyond the geometry stage.  It is now part of the design process to investigate thje function of a component, and this investigation will lead to designs that work effectively, with little actual development being done using physical prototypes.

Other mature simulation technologies exist, mainly dealing with mechanism and fluid flow analysis, and it is the integration of these with finite element analysis that is the next great challenge of simulation software.