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at IFEA | Finite Element Analysis?

The Finite Element Analysis (FEA) is the simulation of any given physical phenomenon using the numerical technique called Finite Element Method (FEM). Engineers use FEA software to reduce the number of physical prototypes and experiments and optimize components in their design phase to develop better products, faster while saving on expenses.

It is necessary to use mathematics to comprehensively understand and quantify any physical phenomena such as structural or fluid behavior, thermal transport, wave propagation, the growth of biological cells, etc. Most of these processes are described using Partial Differential Equations (PDEs). However, for a computer to solve these PDEs, numerical techniques have been developed over the last few decades and one of the prominent ones, today, is the Finite Element Analysis.

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Differential equations not only describe natural phenomena but also physical phenomena encountered in engineering mechanics. These partial differential equations (PDEs) are complicated equations that need to be solved in order to compute relevant quantities of a structure (like stresses (ϵ), strains (ϵ), etc.) in order to estimate the structural behavior under a given load. It is important to know that FEA only gives an approximate solution to the problem and is a numerical approach to get the real result of these partial differential equations. Simplified, FEA is a numerical method used for the prediction of how a part or assembly behaves under given conditions. It is used as the basis for modern simulation software and helps engineers to find weak spots, areas of tension, etc. in their designs. The results of a simulation-based on the FEA method are usually depicted via a color scale that shows, for example, the pressure distribution over the object.

Depending on one’s perspective, FEA can be said to have its origin in the work of Euler, as early as the 16th century. However, the earliest mathematical papers on Finite Element Analysis can be found in the works of Schellbach [1851] and Courant [1943].

FEA was independently developed by engineers in different industries to address structural mechanics problems related to aerospace and civil engineering. The development for real-life applications started around the mid-1950s as papers by Turner, Clough, Martin & Topp [1956], Argyris [1957], and Babuska & Aziz [1972] show. The books by Zienkiewicz [1971] and Strang & Fix [1973] also laid the foundations for future developments in FEA software.

Figure 1: FEA Simulation of a piston rod. The different colors are indicators of variable values that help predict mechanical behavior.

Divide and Conquer

To be able to make simulations, a mesh, consisting of up to millions of small elements that together form the shape of the structure, needs to be created. Calculations are made for every single element. Combining the individual results gives us the final result of the structure. The approximations we just mentioned are usually polynomial and in fact, interpolations over the element(s). This means we know values at certain points within the element but not at every point. These ‘certain points’ are called nodal points and are often located at the boundary of the element. The accuracy with which the variable changes is expressed by some approximation for eg. linear, quadratic, cubic, etc. In order to get a better understanding of approximation techniques, we will look at a one-dimensional bar. Consider the true temperature distribution T(x) along the bar in the picture below:

Figure 2: Temperature distribution along a bar length with linear approximation between the nodal values.

Let’s assume we know the temperature of this bar at 5 specific positions (Numbers 1-5 in the illustration). Now the question is: How can we predict the temperature in between these points? A linear approximation is quite good but there are better possibilities to represent the real temperature distribution. If we choose a square approximation, the temperature distribution along the bar is much more smooth. Nevertheless, we see that irrespective of the polynomial degree, the distribution over the rod is known once we know the values at the nodal points. If we would have an infinite bar, we would have an infinite amount of unknowns (DEGREES OF FREEDOM (DOF)). But in this case, we have a strong hold.