# A Breakdown of the Finite Element Analysis

Engineers are increasingly embracing computer simulations 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, so they can locate weak points and improve them.

While FEM and FEA can help test your engineering designs, there are a number of equations, principles, issues, and types of simulations you can consider before getting started.

#### What is the Finite Element Method?

FEM offers a numerical solution for a complex problem, which allows for some level of error. It is usually used when a math equation is too complex to be solved in a typical fashion. FEM breaks a large problem into a series of smaller ones known as “finite elements,” which make an overall problem much easier to analyze.

#### What is the Finite Element Analysis?

These complex math problems are used to create a computer simulation, or FEA, which provides a visual analysis of how a particular product or design would react under stress in the real world. It does this by breaking down the entire model into a mesh where engineers test how the different elements of a design would interact and perform under simulated stressors.

#### What are Differential Equations?

Differential equations describe both natural phenomena and physical phenomena found in engineering mechanics. Known as “partial differential equations,” (PDEs), these complex equations must be resolved in order to get an accurate prediction of how the prototype will behave and react under certain conditions, such as stresses and strains.

#### What are the Different Types of Partial Differential Equations?

There are different types of PDEs, which are important to understand before moving ahead with the FEA. These include:

Elliptic
Solving elliptic PDEs involves two primary methods, including Finite Difference Analysis (FDA), an approximate technique for solving partial differential equations, and Variational, in which small energy differences can lead to drastically different outcomes. For example, if a ceramic vase falls over while on the floor, it is unlikely to break. However, if it falls from a table, it’s very likely to break since it delivers more energy to the floor.

Hyperbolic
These PDEs uphold solutions with discontinuities commonly associated with “jumps.” Wave equations are an example. These explain the movement of strings, wires, and fluid surfaces, such as waves in a body of water.

Parabolic
These PDEs depict time-dependent diffusion problems. Examples include particle diffusion and heat conduction.

Mesh Convergence
An important but often ignored issue in computational mechanics, mesh convergence can impact the accuracy of a FEA simulation. Mesh convergence is associated with how small elements have to be to guarantee the solutions of an analysis are not impacted by altering the size of the mesh. There are three main ways to measure mesh convergence: displacement error, strain error, and stress error. However, a few different errors can be defined to each of these, which can be compared and must reduce mesh refinement.

#### What are the Different Types of FEA?

Some of the major types of FEA include:

• Extended Finite Element Method (XFEM), which requires continuous displacements throughout elements.
• Generalized Finite Element Method (GFEM), which merges the features of classical FEM software with non-traditional meshless approaches.
• Mixed Finite Element Method, which is a merging of the use of automatic mesh refinement (h-refinement) with an increase in the arrangement of polynomials (p-refinement).
• Discontinuous Galerkin Finite Element Method (DG-FEM), which compared to traditional methods, many experts consider it to be a good alternative for solving hyperbolic equations.

#### Finite Element Method (FEM) for Photonics

Learn how FEM can be used to model and simulate photonic components/devices and analyze how they will behave in response to various outside influences. The Finite Element Method for Photonics course program 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 gain an understanding of how mathematical concepts translate to computer code finite element-based methods after completing this program.

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

(26 February 2021). Parabolic partial differential equation. Wikipedia

Rapp, E. Bastian (2017). Differential Equations. ScienceDirect.