Source of Errors in a FEM based Solution

When solving an engineering problem using FEM, we begin with developing its mathematical model. The mathematical model cannot cover all details of the physical problems, therefore, we need to simplify it and include the essential aspects of physical problem only. The mathematical model involves the solution of the governing differential equation for the system to get its exact solution which is often not possible.

The next step is to solve this mathematical model using FEM. For this purpose, we discretized the model domain into finite elements. This is equivalent to transforming the governing differential equation into a set of simultaneous algebraic equations which are solved for unknown variables to get the required numerical solution with the help of computer.

In doing above procedure, various kinds of errors are involved as shown in Figure 1.

1. When we represent a physical system with a mathematical model, we make several assumption. That’s why a mathematical model cannot fully describe the all details of a physical system. This is called the modelling error. The better the assumption we make in building a mathematical model of a physical system, the lower will be the modelling error.
2. Solution of the governing differential equation of a mathematical model is exact if we are able to solve it. Often, we resume to FEM to get a numerical solution which is an approximation. Here is the involving of discretization error. It can be reduced by increasing the mesh density at the cost of computational resources. 
3. Lastly, whenever we compute a numerical solution, a third type of error is involved, called the numerical error. It is due to the limitation of computer memory to store numerical values. It depends on the specification of the computer system and can never be zero.

Figure 1. Sources of errors (i.e. modelling error, discretization error & numerical error) involved in getting FEM based solution of a physical problem.

Being engineer, we need to be familiar with these error sources. Let me further explain these errors with a following example.

Let’s we want to analysis the structure of an overhead storage tank as shown in Figure 2(a). The actual physical system is complex in nature and it would be rather difficult to develop a mathematical model that cover all details of this physical system. Remember we are engineers, not mathematicians, so we begin to simplify this problem to develop its mathematical model. Here are few assumptions that can make our lives at peace:

• Assume that the ground behaves as a rigid body.
• The inlet and outlet water piping and its connections does not contribute to the structure performance
• The effect of staircase can be neglected
• The weight water body can be distributed among the columns. 

By applying the above assumptions, we are now able to simplify the problem as shown in Figure 2b. Whenever, we build a mathematical model of the physical problem, the modelling error is involved that depends on nature of assumptions we establish to simplify the problem. The mathematical model involves the solution of governing differential equation which do not have close-form solution in the most of practical applications. But do remember that we get an exact solution if we are able to solve the differential equation.

Figure 2. An overhead water storage tank, (a) physical problem, (b) mathematical model, and (c) discretized model.

Let’s resume to FEM to solve this mathematical model to get an approximate solution. We discretize the problem domain into finite elements as shown in Figure 2(c). During discretization, there is involvement of discretization error as we are getting an approximate numerical solution. If we discretize the model very fine, then, the discretization error will be small but the computational cost will be high. Typically, we begin with a coarse mesh, then, we keep on refining the mesh density while keeping an eye on the solution. At the onset, the solution changes drastically with increase in the mesh density until it begins to converge. At this point, a further increase in the mesh size does not affect the solution, and it is said the mesh convergence is reached. So, this discretization error is the second source of error that we should keep in mind while doing FEA of a physical problem.

The last source of error is the numerical error. This error always exist whenever we make use of computer as it has limited memory to store the numerical values being computed. This error cannot be avoided in the numerical computations due finite precision involving floating point or integer values.

In nutshell, whenever we apply FEM to solve an engineering problem, we must understand that the numerical solution we are getting is infect with various types of errors i.e. modelling error, discretization error and numerical error. Consequently, we should always doubt the solution we are getting from FEM and always try to verify or validate the results whenever possible before making a critical decision.

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.