How Finite Element Method (FEM) Works?

 

Background

Over the period of time the Finite Element Method (FEM), also known as Finite Element Analysis (FEA) has become a default tool in the Computer-Aided Engineering (CAE) design process of products and systems. Engineers, all over the world, are deploying this powerful tool to analysis real life engineering problems. The applications of FEM are huge and diverse ranging from structural, thermal and fluid flow problems, to electrostatics and magnetostatics and so and so on.

The significance this essential tool in today’s modern world cannot be emphasize more. However, it is ironic that the engineering students (the potential future users of this tool), are often taught to perform design and synthesis for simple problems with ideal applications only. They are taught to learn engineering principles, build mathematical models and solve these simple problems. On the contrary, after graduation they are expected to solve the real world problem which are complex in nature and often expressed by scary-looking mathematical equations (so-called governing differential equations) which may not be solved easily or its exact solution (close-form solution) may does not exist. Here comes a much need FEM tool in the arsenal of fresh engineers to rescue them… 

In this article, I would explain how this amazing tool works to solve complicate engineering problems using a typical structural problem. The target audience of this article are undergrad students and/or and new engineers who have a little or experience with FEM. The article may also serve as a reminder of the basic concept behind the FEM for the experience users.

FEM is based on "Divide and Conquer" Philosophy

The underlying idea of FEM is similar to “divide and conquer”. We treat a complex problem as a set of simple problems which can be solved easily. Let me explain it further with an example shown below in Figure 1. A prismatic cantilever beam is subjected to a tip load P at its free end, and we want to compute its deflection. The term prismatic is a fancy terms dedicated for the structural components which have constant cross-section along its length. The general differential equation of beam that governs it is as following:

Even though above equations seems complicate, but, it can be solve it for the given square cross-sectional prismatic beam by applying the following boundary conditions.

We were lucky to have solution for the above case. Such way of approaching the problem by solving differential equations derived for a continuous domain is called analytical modeling. In reality, the cross-section of beam may be not rectangle type or not even prismatic or may be it is made of composite materials etc. There are unlimited possibilities for even a single structural beam which could not be coped with analytical models. That is why the exact solution of the real world engineering problem generally do not exist. This is alarming for us being engineers.

Figure 1. The defection of prismatic cantilever beam using analytical method (Differential Equation) and numerical method (i.e. FEM)

In case of FEM, we decompose the continuous domain of the problem, which is the cantilever beam in our case, into simple subdomains, called finite elements or shortly elements, hence is the name of this approach the Finite Element Methods (FEM). Every finite element has nodes through which it connects with other elements. In above example, each element consists of two nodes. Every finite element represent an algebraic equation for a subdomain in the finite or discretized domain. In other words, the whole finite domain is now expressed mathematically in terms a set of simultaneous algebraic equations, generally, denoted in vector and matrix notation as following:

Where vector {F} is called force vector, {U} is load vector and [K] is called stiffness matrix. The size vectors and matrix (i.e. a set of simultaneous equations) depends on the number of elements in the domain. 

The stiffness matrix represent the system properties of the domain, and displacement vector represents the behavior of the domain subjected to force vector. The displacement vector is also called nodal displacement vector, state vector or degree-of-freedom (dof) vector. The solution of unknown displacement vector is obtained by solving for given the known load vector and stiffness matrix as following.

It might be also interesting to you that the above notation used for representing a set of simultaneous algebraic equation is only well understood in mechanical and civil engineering disciplines, but also very popular in other disciplines where it represents other quantities as depicted in this following Table 1.

Table 1. Use of FEM in other disciplines.

Application Problem	Displacement vector represents	Force vector represents
Solid mechanics	Displacement	Mechanical force
Heat conduction	Temperature	Heat flux
Acoustics	Displacement potential	Particle velocity
Flow problems	Pressure	Particle velocity
Electrostatics	Electric potential	Charge density
Magnetostatics	Magnetic potential	Magnetic intensity

Coming back to our discussion on the solution of simultaneous algebraic equations, let me highlight that we have already learnt how to solve these equations manually in our elementary mathematics class during school education. We were able to solve a set of 2 or 3 simultaneous algebraic equations only, however, the solution of a big set is humanly impossible, at least to me.

Fortunately, the computers are very good at doing boring jobs. Computer do arithmetic operations that are only needed to solve simultaneous equations involving the inversion of stiffness matrix. Numerical algorithm are available to inverse a stiffness matrix considering time and memory constraints. Modern computers have the capability of solving rapidly huge set of simultaneous equations as all it require is the computational power and efficient algorithms. Consequently, with the development of computer, we are feel empower to analysis the huge and complicated engineering systems and their interactions with the power of FEM. 

Being users of FEM, it is very important for use to understand that such numerical solution is an approximate solution. The accuracy of FEM solution depends on the number and the type of elements. Here I will explain the role of the number of elements only as the type of elements is beyond the scope of this article. The number of elements in domain increases as we enhance the discretization of domain from course to fine. The accuracy of solution generally increase with increase in the number of elements, however, a stage will come that an increase in the number of elements will no longer increases the accuracy of the solution, such condition is said to be mesh convergence. A further increase in number of elements will increase the size of the set of simultaneous equations, which will become a computational intensive and time consuming task, but, with no proportionate improvement in the solution. We, being users, should have to strive for a reasonable approximate solution at optimal utilization of resources.

About the Author: Dr. Khazar Hayat is a professional engineer with almost 15+ year of experience in research, design, analysis and development of products made of fiber reinforced plastics composites (FRPCs). Currently, he is working as an Associate Professor at Mechanical Engineering Department, The University of Lahore, Pakistan, can be reaching by emailing at khazarhayat@gmail.com.