Finite Element Method : An Introduction

Finite element method(FEM) is sometimes referred to as finite element analysis, is a computational technique used to obtain approximate solutions of boundary value problems in engineering.

So what is boundary value problem? Boundary value problem is a mathematical problem in which one or more dependent variables satisfies a differential equation within a known domain. Let us take an example of problem of calculation of temperature distribution in a cuboid block. The differential equation responsible for the distribution of heat in that block is given by:

Here u is the temperature as a function of x,y and z as well as t. It is a function of coordinate in space as well as time. Likewise, alpha is the rate of diffusion of the medium.

In case you are wondering, the above differential equation takes into account of heat transfer in all three coordinate axes. Similarly, the value of q (internal heat generation ) is assumed to be zero.

For more info about heat equation, you should consult textbook related to heat transfer or read this wikipedia article.

The solution of above equation will give temperature distribution over coordinate space. i.e. the value of temperature at any given point in the body can be deduced by solving that differential equation.

To calculate solution of the above heat equation, we need to provide some boundary condition i.e. some known value of temperature at particular coordinate space. So, the equation along with boundary condition is known as boundary value problem.

Exact solution of the equation can be hard and complex to carry out. So finite element analysis comes into play. Finite element analysis or finite element method employs various mathematical and computational tools to approximate the solutions of boundary value problems. Note that, finite element method gives approximate solution.

Finite Element Method : Why approximate if you can exact ?

So, what is the need of approximating solutions?

Solutions are approximated simply because they are faster and easier. Finding exact solutions to differential equations is time intensive task. Complex mathematical tools are taken into account to calculate exact solutions. Exact solutions might be easier if the engineering design is simple. As engineering design is not always simple, finding exact solution is too difficult. Similarly, due to growing use of digital computers to automate and assist mechanical design as well as manufacturing process. Algorithms to derive solutions should be well suited to computers as well. Finite element methods are easier to program in computers and using the heavy computing power of computers, the approximate solutions come closer to exact solutions

Finite Element Method: What are the steps of finite element analysis.

Up until now, you know finite element method is simply technique or collection of technique used to approximate solutions to boundary value problems. I will now talk about process of carrying out finite element analysis. The steps of carrying out finite element method are common to all analysis, whether it is structural , heat transfer, fluid flow or other problem. The steps that are employed by any general finite element analysis softwares are briefly explained below.

Preprocessing

This process is described as defining the model and includes:

• Defining the geometric domain of the problem
• Defining the element types to be used
• Define the material properties of element
• Defining the geometric properties of elements
• Defining element connectivity i.e. meshing the model
• Defining the physical constraints (boundary condition)
• Defining the loadings.

In layman’s term, preprocessing step is the step in which simulation is prepared. In finite element method, the model ( entity that needs to be solved) is meshed into simpler elements i.e. a model of bar is meshed into pieces of simple cuboids or triangles for easier calculation. Thus dividing the problem into smaller parts is done in preprocessing steps. Different parameters of model are also defined in this step. If we were to find out the temparature distribution in a bar as shown below.

The bar itself is divided into numerous smaller cubes. In preprocessing, boundary value is defined. For example, in the above figure, boundary value might be the temperature at right end, or any known parameters.

Similarly, different parameters of material used in the model is also set in preprocessing. The parameters in the above figure might be conductivity of material used.

Simply put, preprocessing steps include defining of geometry of model, meshing, defining boundary conditions, defining material properties and defining output parameters.

Solution Step

In this step, calculations is made on the basis of all definitions. The finite element software assembles various governing algebraic equations in matrix form and computes the unknown values of the primary variables. The computed values are further substituted and computed to calculate other variables.

What actually happens is, the computer solves smaller standard elements included in the mesh. The solution of each mesh is interpolated and general solution is acquired. The larger the amount of elements constituting the model, the more precise would be the solution.

Post-processing step

In this step, the software employing finite element method analyses and evaluates the solution results. The software component responsible for post processing employs various sorting, printing and plotting methods to show the output in the most intuitive way possible.

Here in this picture, the magnitude of deformation of various points along the model is shown in colorful way. The red and black color indicates maximum deformation. This is done by post processor routines present in finite element method software.

In short, post processing step in finite element methods analyse,arrange and illustrate outputs variables in the best way possible so that it can be interpreted effectively.

This is how finite elements method is carried out. Finite element analysis is widely used in various engineering fields. It is extensively used in aerospace industries to design, analyse and test various aircraft parts such as wings and fuselage.

Also Check out this link for an example of finite element analysis using ANSYS http://geniuserc.com/finite-element-analysis-using-ansys/

Click here to know how aircraft wings are designed and manufactured.

Feel free to write queries in the comment section and I will get back to you as soon as possible.