What is Finite Element Method or Finite Element Analysis? You might have seen engineers and scientists do simulations of stress in solids and fluid flow simulations in computer softwares like ANSYS or Abaqus, but what is actually happening in reality there?
Well, the answer to that is Finite Element Method or Analysis (FEM or FEA) is what is happening.
Simulations is the fourth step in an engineering design process and there are a lot of softwares that specialize in this step. And Finite Element Method is the backbone of all engineering simulations, whether it is stress analysis on a beam or fluid flow analysis in a channel.
Basically, Finite Element Method is the division of a large system or component on which direct application of analysis equations (for example: Stress Equations) is impossible because of geometry of components, into very small members of a certain size(called Finite Elements).
After this division, the analysis equations are applied on these individual members. These members are generated in such a way that they form a network spanning throughout the entire component. Finally, the solutions from all members are combined to get the overall results of that component in question.
The calculations of finite element method are very complicated and time consuming, and cannot be solved manually. That is why, this methodology is highly dependent on computers.
To explain it, we are going to consider an example of a bridge. And for the load, we are going to place a point load on the center of the beam, and we are going to perform stress analysis on it via Finite Element Methods of Finite Element Analysis Techniques. The figure is shown below:
EXAMPLE BRIDGE FIGURE
So, how it is all done? Let’s find out in 3 Steps. Now, these steps are too complicated and time consuming to be done by hand, and therefore, computers are used for the job.
Step 1: Defining Finite Elements
As mentioned above, the elements into which the component is divided into, is called a Finite Element.
Elements can be defined in various shapes. They can be two dimensional squares, triangles or any other geometrical shape.
Also, the size of elements translates to number of elements. As smaller the elements, the greater will be the number of elements and hence, more accurate the final result.
However, that also means more calculations to be done, and for that an optimum range is chosen on basis of how much accuracy is needed and how much processing power to perform those calculations is available.
Nature or shape of elements is defined by the points by which it is defined. For Example, as shown in figure below, we take elements composed of 8 points. These points are called as Nodes.
These nodes are able to move in a chosen co-ordinate system, which here as the element is 2 dimensional rectangle, means that nodes can move horizontal & vertical directions.
Image above shows a single finite element which in our case is a rectangle defined by 8 points (blue dots).
All points except the end points of the entire component which are fixed, have the ability to translate. These boundary conditions also need to be defined in finite element methods problem.
As they state the region over which analysis is going to be performed.
The figure above, shows the entire bridge divided into rectangular elements, called finite elements. Green color on sides represents the ground on either side of a river. Elements are of same shape and size.
The picture shown above does not look the stresses map that will be shown at the end of the article, but that is due to reason that the elements shown above is just a drawn approximation.
It is not an image obtained from an FEM software like ANSYS etc. However, the final image is.
Now, when the point load is going to act on bridge, it will make the nodes or these points move or displace because of elastic properties of the material. So, we need to define the stiffness of the material also.
[Recommended Reading Guide on Mechanical Properties of Materials]
Developing the Equation for Nodes
The key phase of defining the element is defining the displacement of a node as a function of its co-ordinates.
Now, as there are 8 nodes for each element, there will be 8 equations to describe one element. And these equations can be transformed into a matrix, which is called as element matrix.
After this, solid mechanics formulas are used. First, via displacements, stresses are calculated, then from stresses, strain energy is calculated.
From strain energy, potential energy is calculated, and then from minimum potential energy, two system equations are generated which define the entirety of an element.
The potential energies calculated are also in a matrix form of same order, and is denoted in this example as M.
The final two equations look like this:
Because the equations analogous similarity with Hooke’s Law (F=k.x), the matrix is sometimes termed as Stiffness Matrix. The only difference is that these equations operate on a matrix, instead of single value.
Step 2: Combining the Matrices for all Elements & Solving the System Matrix
Now, after developing the equations for one element (which in our case is stiffness matrix) by repeating the same steps again, until the matrices for all elements have been defined.
The important step now is to combine all these matrices into a single large matrix that represents the stiffness of whole system.
For this a simple merging technique is employed. As, it is clear that some nodes will be common in consecutive finite elements, and values for these nodes will be present in both matrices of neighboring elements.
These common nodes make solving the entire matrix of the system very easy.
As, the rows of all matrices represent simultaneous equations that describe the displacement of nodes of an element. So, we can simply solve the first equation and plug the solution into the next and then carry on this process until the end of matrix.
When first matrix has been solved, we can plug its result in matrix of its neighboring element, as they have some nodal values in common.
This process is repeated until all of the matrices have been solved,
At the end, we are left with a solution for a single node.
This final solution is of critical importance, and is used as sort of key for unlocking the analysis of the system.
Step 3: Using the Final Solution as a Key
Now, for the final step, we take that key solution and plug it into the equations of matrices of the finite elements, until the displacement for the all the elements because of that point load is obtained.
Then, from these displacement results, corresponding stresses are calculated.
These results for better understanding are then mapped according to colors that indicate the greater or lower stresses generated because of that point load.
There are thousands and sometimes millions of calculations that comprise the entirety of finite element method, and the explanation given here is without going too deep into mathematics of it all.
There are a lot complicated mathematical procedures like Weak Form Formulation etc. that are not described here.
However, I do intend to describe them in future articles.
As mentioned above, finite element method includes tons of calculations that are done in order to get the contours of stresses as shown in figure above.
Therefore, computers are used to perform the finite element method or analysis, and not just any computers, but computers with high processing power.
Students don’t need to perform finite element methods analysis on very complicated systems, and computers of moderate processing power can do the job for them.
However, professionals need high end computers to their job efficiently.
Check out our guides for buying budget laptops for CAD for students and professional engineers by clicking the links below:
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