Chapter 3. The efficacy of finite element analysis (FEA) as a design tool for
3.3 Basic concepts and essential elements of FEA
FEA is the application of the finite element method in which the object or system is represented by a geometrically similar model consisting of multiple-linked, simplified representations of discrete regions. This numerical analysis uses a complex system of points called nodes, which form a grid called a mesh. This mesh is programmed to contain the material and structural properties, which defines how the structure will react to certain loading conditions. Nodes are assigned at a certain density throughout the material depending on the anticipated diffusion or stress levels of a particular area (Pathare & Opara, 2014; Moaveni, 2008; Roduit et al., 2005; Cook et al., 2002; Gupta & Meek, 1996). The basic idea of FEA is to find
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solutions to a complicated problem by replacing it with a simpler one. The domain (usually a physical structure) in FEA is discretised into several subdomains or finite elements and the process is known as discretisation. FEA techniques are used to obtain approximate solutions to boundary value problems (otherwise known as field problems) in engineering (Reddy, 2014; Moaveni, 2008). In a boundary value problem, the field is the domain of interest and it usually represents a physical structure (Reddy, 2014; Hughes, 2012; Moaveni, 2008).
Generally, the concept of FEA involves a piecewise polynomial interpolation. Over the entire structure, the field quantity becomes interpolated in a piecewise form and a set of simultaneous algebraic equation is generated at the nodes, which is associated with the elements (Eq. (3.1)), as in the case of a structural analysis. The functions of all the elements are then assembled to form the global matrix equation i.e., the governing algebraic equations that define and represent the entire structure under study (Eq. (3.2)).
K e
u e f e (3.1)where,
K e is the elementary stiffness matrix, dependent and determined by the geometry, element and material properties,
e
u is the elementary displacement vector which defines the nodes motion under loading and
f e is the elementary force vector which defines the applied force on the element.
[ ]K u f (3.2)
where,
K is the global stiffness matrix,
u is the vector of the unknown nodal displacements (or temperature in thermal analysis) and
f is the vector of the applied nodal forces (or heat flux in thermal analysis).The governing algebraic equations can be solved for the dependent variable at each node after applying the boundary conditions after which the strain and stress can be calculated based on the displacement of nodes associated with the element. Many engineering phenomena can be expressed using the governing equations and boundary conditions. However, it is important to mention that the governing equations are associated and depend on the physics of the problem being analysed and not with the method of the solution (Nikishkov, 2004). Hence, the solved governing equation depends on the physical problem to be analysed.
FEA is performed with a great accuracy since the actual shape, load and boundary conditions, as well as the material properties can be specified and applied. In order to formulate the FEA of a physical problem, certain steps are common to all analyses whether structural, heat transfer, fluid flow, or any other problem. In
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practice, three principal steps are involved in FEA: pre-processing, analysis (solver), and post-processing (Srirekha & Bashetty, 2010; Cook et al., 2002).
3.3.1 Pre-processing
All the tasks prior to the numerical simulation process are referred to as pre- processing. These include the conceptualisation of the problem, meshing, and building the computational model. The pre-processing stage involves the creation of the model to be used for the analysis. The structure is created using the computer aided design (CAD) program that can come with the FEA software or be provided by another software developer. The success of the entire FEA process crucially depends at this stage on the skill of the analyst to determine the level of the simplification to be introduced or included into the model when compared with the physical situation. For a successful analysis, the software user must be able to determine the mesh and element types to be used in the model. The structure is divided into a number of discrete sub-regions known as “elements”, connected at discrete points known as “nodes”. The structure represented by nodes and elements is called the “mesh”. The elements not only represent subdivisions of the structure, but also the mechanical properties and behaviour of the structure. Most FEA software packages have the ability to perform meshing and define the shape and properties of the structure to be analysed simultaneously (Xia & Sun, 2002; Cook et al., 2002). An example of a meshed corrugated paperboard is shown in Figure 3.1.
Complex regions of the structure such as curves require a higher number of elements to accurately represent the geometry, whereas regions with simple geometry can be represented by fewer elements. Choosing an appropriate element for a structure requires some factors such as; prior knowledge of FEA, knowledge of the behaviour and properties of the structure, the elements available in the FEA software and the characteristics of the elements. In the pre-processing stage, once the meshing of the structure is completed, the constraints, loads, boundary condition and the material properties of the structure are defined. In addition, the entire structure is fully defined by the geometric model at this stage.
3.3.2 Analysis
The analysis step is the processing stage whereby the computer is used to solve the set of mathematical equations. After the meshing, the dataset such as the geometry, constraints, load, and mechanical properties of the structure generated as inputs are put into the finite element code to generate matrix equations for each element. These are then reassembled together to form a global matrix equation for the structure. That is, these datasets are used as input to the FEA code, which constructs and solves a system of linear and nonlinear algebraic equations until convergence is achieved. Nodal results such as displacement values at different nodes, temperature values at different nodes in a heat transfer problem or velocity values in a fluid dynamics analysis are also obtained at this stage (Cook et al., 2002). One of the
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problems that might be encountered in the analysis stage is solving large linear systems. In modern FEA packages, algebraic or geometric multigrid methods are employed to accelerate the iterative solution process for large linear systems. Another problem that may be experienced with the FEA solver is the nonlinearity of the model to be analysed. However, certain nonlinearities may be neglected to obtain linear equations that can be solved easily. For these purposes, in most FEA software packages, segregated and continuation solvers are developed. A third problem that may be faced at the analysis stage is the model instability, which results in a poor approximation of the mathematical model being produced. One way to solve this problem is through better adaptive meshing which is an important factor in improving model behaviour.
3.3.3 Post-processing
This is the final stage involved in FEA. Raw data generated in the analysis step makes it difficult to interpret. In the post-processing stage, these data are evaluated and used to create either 2D or 3D representations such as; the deflected shape of the structure, stress plots, and other animations, which are useful in better understanding the behaviour of the problem being analysed. For example, depending on the FEA software package available, colour may be used to indicate the value of some component of stress or displacement on the analysed structure. An example of a typical model of ventilated corrugated paperboard package under buckling is shown in Figure 3.2. An outward buckling occurred on the length side of the package and was observed to originate from the middle of the length side face. The deep red colour observed on the face of the length side of the package indicates region where the greatest buckling occurred. Usually, the pre- and post- processing stages are part of the same FEA software package. An overview of the steps involved in the process of finite element analysis is shown in Figure 3.3. Estimating the error in the post-processing stage of a finite element simulation is an important task, which can be achieved by solving the equations of the mathematical model using different sizes of mesh to obtain a convergence of the numerical solution. Furthermore, sensitivity analysis of the model is very important particularly to different stages and input parameters such as the initial conditions, boundary conditions, applied loads, material properties and varying constraints. Recently, optimisation software packages are being combined into FEA software packages and are being used in iterative calculations in order to optimise critical shape or dimensions of an analysed structure.