FEA Basics

Introduction to FEA

Finite element analysis (FEA) is one of the mostly widely used engineering analysis techniques in the world today.  Engineers employ FEA to simulate how a physical system (usually an engineered product or manufacturing process) will respond to expected loading conditions. Practical applications of  FEA include structural analysis of bridges and buildings, impact or crash analysis of automobiles, aerodynamic analysis of airplanes and airfoils, electromagnetic analysis of AC and DC motors, injection molding simulation of plastic parts, fluid flow analysis in channels and pipes and heat transfer through residential and commercial buildings. Finite element analysis is based on the fundamental physical principles that govern the behavior of these physical systems. Since biological systems must obey the same fundamental physical principles, the physical response of biological systems to known loading conditions can also be predicted using FEA. For simplicity this brief introduction to finite element modeling will be restricted to elasticity or structural analysis applications.

Like all analysis models, a finite element model is an abstraction of a more complicated physical system. The physical world is invariably too complex to model at every level of geometric and phenomenological detail. For example, to predict how a long slender object might deflect due to an applied transverse load, the object might be modeled as a “beam” which involves a number of simplifying assumptions according to the mechanics of beam theory as to how the object will deform under an applied load.  In the case of beam theory, these assumptions enable analytical equations to be derived that give the beam response to loads, such as shown in .

Figure 1. Bending of a simple beam

However, for more complicated problems closed-form analytical solutions cannot be obtained, or if obtained, lack sufficient accuracy to be of any value. Computer-based techniques must be employed to simulate how such systems will behave (i.e. physically respond) under known imposed loads.  Consider the ladder shown in Figure 2. A finite element model using “beam” finite elements is shown next to the ladder. The beam finite elements, shown in blue and purple, are represented as straight lines defined by points, called nodes, shown in green in the figure. Each beam element can have distinct material properties (e.g. Young’s modulus of elasticity), a distinct cross sectional area, and distinct moments of inertia about its bending axes. Forces and/or moments may be applied to a set of nodes and a different set of nodes may be constrained against deflections and/or rotations. This model is then solved by the “solver” of a finite element analysis program to yield the deflections, moments, and stresses through out the structure.   

Figure 3

Many different types of finite elements exist to enable modeling of physical systems at various levels of detail. The beam element model of the ladder above is not able to represent geometric details such as how the rungs of the ladder are attached to the ladder rails which may produce stress concentrations. Other types of common finite elements include shell or plate elements for modeling thin, three-dimensional structures, planar elements for modeling systems that can be assumed to respond in a two-dimensional plane due to in-plane loading conditions and little variation of geometry, material properties, and constraints in the out-of-plane direction, and three dimensional solid elements.  Examples of planar elements are shown in Figure 3. Corresponding three dimensional solid element versions of these elements are the tetrahedral element, the brick element, and a hexahedral element (a six-sided volume).

Figure 4 shows an object modeled with tetrahedral elements. The subdivision of the object into a contiguous set of finite elements connected together by nodes is called a finite element mesh.  Note that size of the finite elements may vary spatially; smaller elements or a high element density is often used to improve the accuracy of the solution in regions where the stress gradients are high. Most commercial programs have mesh generator tools to enable automatic meshing of geometry with non-uniform mesh density for certain types of elements. Specifically, robust automatic mesh generators exist for triangle element meshing of complex two dimensional geometry and tetrahedral element meshing of complex (but properly defined) three dimensional geometry. However, the price often paid for automatic meshing of three dimensional objects with simple tetrahedral elements is loss of accuracy and/or the computational resources required to solve the problem.

Automatic meshing of thin, geometrically complex, objects is a particular problem. Imagine that the object shown above were hollow with a wall thickness approximately equal to or even less than the distance between two connected nodes on the outside surface in the high element density region of the object. If the structure were this thin, then the tetrahedral elements shown in the object in the lower element density regions would have a very high aspect ratio. This can significantly reduce the accuracy of the solution. Further, thin structures exhibit bending behavior. To accurately capture bending behavior with four-noded solid tetrahedral elements, one would need to mesh the object such that the average size of each tetrahedral element is about 1/4th the wall thickness or less. The net result is that a lot more nodes and typically elements would be required to model such a thin structure. This would dramatically increase the analysis time and costs and might exceed available hardware and software resources.

One solution to this problem would be to mesh a thin structure with shell or plate elements. An example of a shell element mesh of an I-Beam is illustrated in Figure 5. Shell or plate elements are specifically designed to model quite accurately bending in thin three-dimensional structures. This requires meshing of the mid-plane surface (the imaginary surface lying half way between the outer and inner surfaces of the wall) with each element having a prescribed thickness to define the wall thickness through out the structure. Unfortunately, for highly irregular, thin three-dimensional geometric domains or entities found in many organic systems, automatic determination of mid-plane surfaces from a digitally reconstructed geometric model is highly problematic at best with current technology. 

As mentioned before, elements are defined by a set of nodes which dictate the shape of the element. A simple beam element is defined by two nodes. In a finite element model, each node has specific degrees of freedom. The degrees of freedom are the variables that are directly solved for by the finite element solver. 

The adjacent figure shows that each node of a three dimensional beam element has six degrees of freedom- three translations corresponding to the three principle coordinate directions and three rotations corresponding to rotations about the three principle coordinate axes. It is important to know that beam and shell or plate elements have both translational and rotational degrees of freedom, whereas planar and solid elements have only translational degrees of freedom.

Figure 6

Thus each node of a planar element has two degrees of freedom corresponding to translations in the two planar coordinate directions, whereas as each node in solid element mesh has three degrees of freedom corresponding to translations in the three coordinate directions.  Thus a three-dimensional solid tetrahedral finite element mesh consisting of 10,000 nodes has 30,000 degrees of freedom.   A three-dimensional shell element mesh consisting of 5,000 nodes would also have 30,000 degrees of freedom, albeit a completely different set of degrees of freedom which would include 15,000 rotational degrees of freedom.

There are five basic steps involved in developing a finite element model of a physical system:

• Geometry definition
• Discretization (i.e. meshing) of the geometry with a finite element mesh
• Specification and assignment of material properties to finite elements
• Specification of kinematic constraints
• Specification of loading conditions

After a model is properly constrained and loaded, it is then submitted to the finite element solver. At this point the definition of the geometry or physical domain, the subdivision of the domain into a valid finite element mesh with the specification and assignment of material properties, the application of boundary conditions or constraints and the specification of loads are reflected mathematically by a set of simultaneous algebraic equations that are solved for by the finite element solver using linear algebra techniques.  These algebraic equations have the form:

where [K] is the stiffness matrix of order N, {D} is the vector of N active and unknown nodal degrees of freedom, and {F} is a vector of known nodal loads. The stiffness matrix [K] represents the one-, two-, or three-dimensional structural stiffness of the system, depending on the dimensionality of the problem. N is the order of the matrix [K] and is equal to the number of active nodal degrees of freedom.

The finite element solver must solve this system of simultaneous equations to yield values for {D}, the unknown nodal degrees of freedom. Once the set of values for {D} is known, the finite element program computes unknown reaction loads, the deformations of the system through out the domain, and strains and stresses throughout the domain.  The number of simultaneous equations, or active degrees of freedom, can be quite large, often greater than ten thousand and even a million in some applications. Accordingly, the finite element solver might have to solve a set of a million simultaneous algebraic equations. Obviously, the utility and ubiquitous of the finite element method has grown in direct proportion to the ever increasing speed, power and ubiquity of digital computers and associated software.

 

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