The finite element method (FEM) and finite element analysis (FEA) work together to give engineers insight into the structural behavior of particular designs, so they can locate weak points and improve them.

**FEM:** Developed by engineers in the mid-1950s, FEM provides a numerical solution for a complex problem, which allows for some level of error. Usually, it’s used when a math equation is too complex to be solved in a typical fashion. A simple way to understand FEM is to look at it as separating a large problem into a series of smaller ones (“finite elements”). This makes the overall problem easier to investigate. Engineers use FEM when they need to develop an adoptable design that’s practical but not necessarily perfect for a particular application.

**FEA:** The mathematical equations behind FEM are applied to create a simulation, or what’s known as a finite element analysis (FEA). This simulation is used to provide a structural analysis of how a particular product or design would react under stress in the real world. The simulation breaks down the entire model into smaller elements within a mesh, which engineers use to test how the different elements of a design interact and perform under simulated stressors.

In other words, FEA is a virtual model that helps engineers experiment with a particular structural design, usually with software. Together, FEA and the underlying math of FEM are used to predict the structural behavior and integrity of a design.

“Complex mathematics is required in order to understand the physical phenomena that occur all around us. These include things like fluid dynamics, wave propagation, and thermal analysis,” writes Trevor English in an article for Interesting Engineering. “Analyzing most of these phenomena can be done using partial differential equations, but in complex situations where multiple highly variable equations are needed, Finite Element Analysis is the leading mathematical technique.”

#### Benefits of FEM and FEA

**Improved accuracy and enhanced design:** FEM and FEA allow for enhanced accuracy of structural analysis because they give insight into how the various elements of a design are interacting in minute detail. They also allow engineers to investigate both the interior and exterior of a design.

**Faster and inexpensive testing:** Because FEM and FEA allow engineers to create virtual simulations, they reduce the need for physical prototypes and testing, which saves time and reduces costs.

#### Applications of FEM and FEA

Traditionally, FEM was used to test designs within aerospace and civil engineering, but it is now expanding to other disciplines, including biomechanics, thermomechanical, fluid-structure interaction, biomedical engineering, ferroelectric, thermo-chemo-mechanical problems, piezoelectric, and electromagnetics.

The mathematical principles behind FEM can also be applied to other areas, like computational fluid dynamics (CFD) as well as the thermal dynamics of a structure.

“For example, if you know the temperature at one point in an object, how would you determine the exact temperature at other points of the object, dependent upon time?” writes Trevor English in Interesting Engineering. “Utilizing FEA, an approximation can be made for these points using different modes of accuracy. There’s a square approximation, a polynomial approximation, and a discrete approximation. Each of these techniques increases in accuracy and complexity.”

#### Learn the Finite Element Method (FEM)

The new course program from IEEE Educational Activities, Finite Element Method for Photonics, provides a comprehensive and up-to-date account of FEM in photonics devices, with an emphasis on practical, problem-solving applications and real-world examples. Engineers will come away from this program with an understanding of how mathematical concepts translate to computer code finite element-based methods.

Connect with an IEEE Content Specialist today to learn how to get access to this program for your organization.

Interested in the course for yourself? Visit the IEEE Learning Network (ILN).

#### Resources

*Gigantic, Michael. (10 September 2020). What Is Finite Element Analysis? Learning Hub.*

*English, Trevor. (7 November 2019). What Is Finite Element Analysis and How Does It Work? Interesting Engineering.*

*Harish, Ajay. (21 Mar 2019). Finite Element Method – FEM and FEA Explained. SimScale.*

*The Benefits of Finite Element Analysis in Manufacturing. Manor Tool.*

[…] easier to decipher. This equation is then used to create a simulation, or what’s known as the finite element analysis. This simulation is then used by engineers to test how different elements of a design react to […]

[…] As discussed in a previous post, FEM is a numerical solution for a complex problem, which breaks down a much larger problem into a series of smaller ones (“finite elements”), making the overall problem easier to pick apart. This equation is then used to create a simulation (known as the finite element analysis), which gives engineers a more detailed analysis into the design and how its various elements work together. […]

[…] Some of these innovations are being made with the help of the finite element method (FEM). As discussed in a previous post, FEM is a numerical solution for a complex problem, which is used to create simulations. These […]

[…] to test the structural integrity of their prototypes. These computer simulations are based on the finite element method (FEM) and finite element analysis (FEA), which work in unison to give engineers insight into the structural behavior of particular designs, […]

[…] the researchers performed an analysis using the finite element method (FEM). As we discussed in a previous post, FEM is a numerical solution that breaks down a much larger, complex problem into a series of […]