# Fine Element Analysis (FEA)

Partial differential equations (PDEs) are often utilized to express space- and time-dependent problems in physics, however, when applied in geometry and for the solving of other related problems, these equations are often insufficient. To address this, a number of discretization methods, such as the finite element method (FEM) - also referred to as fine element analysis (FEA) technique - are used to approximate the PDEs with numerical model equations. By utilizing FEA, engineering computations are better able to obtain important information regarding the response of physical systems to certain conditions.

Understanding FEA

The basic explanation for FEA is that these computations transform input information into outfit information to improve the overall engineering decision-making process. While these computations do not provide any new information or improve the quality of the information they receive, they provide useful information on the unknown elements of certain engineering processes. Although these unknowns can be infinite in a continuum, the FEA technique reduces such unknowns to a finite number.

By dividing the solution region into smaller parts, or elements, FEA techniques expresses unknown field variables into approximating functions within each element. Approximating functions are further defined in terms of field variables of nodes or nodal points. Therefore the most important preliminary aspects of the FEA process are selecting the elements and their respective properties, as well as the nodal unknowns.

Mathematically, this relationship is defined as:

[k]e {d}e = {F}e

[k]e represents stiffness matrix

{d}e represents nodal displacement vector of the element

{F}e represents nodal force vector

FEA Steps

1. Field variable(s) and element(s) selection
2. Discretize the continua
3. Select interpolation functions
4. Determine element properties
5. Assemble element properties to determine global properties
6. Impose boundary conditions
7. Solve system equations to acquire nodal unknowns
8. Make additional calculations that are needed to acquire values1

Industrial Applications of FEA

FEA is a useful analytical technique to determine the properties within solid mechanics, as well as for the structural analysis of equipment that incorporates fluid flow, heat transfer, as well as electric and magnetic fields. The diversity and flexibility of this analytical tool is used by civil engineers to analyze beams, space frames, plates, shells, folded plates, foundation, problems associated with rock mechanics and the analysis of seepage fluid through porous media. As this analytical technique has progressively advanced over the last several decades, its incorporation with powerful computer processors and continuous software development allows for its application to go beyond traditional engineering purposes into medical and geospatial analysis procedures.

FEA of Piezoelectric Materials

Piezoelectric devices are often composed of one or more functional materials, which can act by either mechanical, electrical, magnetic, thermal, chemical or optical mechanisms2. As the complexity of these devices often couple these properties together to accomplish a given electronic purpose, these materials therefore have the potential change their shape, respond to external stimuli and adjust their physical, geometrical and rheological properties as needed. FEA therefore plays an important role in demonstrating the numerical variation of any stresses, electric potential or displacement that may occur within piezoelectric materials and devices.

FEA in Failure Analysis

Some of the earliest uses of FEA was to analyze the amount of structural stress that human bones endured to further enhance biomechanic applications. For example, during the 1970s, researchers utilized FEA to study crack stability in aerial turbine blades in an effort to adapt this technique for pre-production manufacturing processes. By understanding the localization of defects and other potential failures of systems, FEA proved to be the most cost-effective aspect of the manufacturing process at this time and continues to save researchers thousands of dollars today3. FEA has grown to stimulate both operational and environmental conditions that may be exposed to a given structure or device without requiring a physical prototype.

References:

1. “Finite Element Analysis” S. S. Bhavikatti, 2005, New Age International
2. “Finite Element Analysis of Functionally Graded Piezoelectric Spheres” – InTechOpen
3. “Finite Element Analysis Applications in Failure Analysis: Case Studies” – InTechOpen

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Written by

### Benedette Cuffari

After completing her Bachelor of Science in Toxicology with two minors in Spanish and Chemistry in 2016, Benedette continued her studies to complete her Master of Science in Toxicology in May of 2018. During graduate school, Benedette investigated the dermatotoxicity of mechlorethamine and bendamustine; two nitrogen mustard alkylating agents that are used in anticancer therapy.

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