**Finite Element Analysis** (**FEA**) or Finite Element Method is a widely used approach for solving integral and partial differential equations (PDE) when solving problems of applied physics in mathematical modelling and engineering. The solution process is based on either eliminating the differential equation completely for stationary problems, or decomposing the PDE into an approximate system of ordinary differential equations, which are then solved using some standard technique such as the Euler method, Runge–Kutta method, etc.

## A short history of FEA

Though Euler made the first steps in solving systems of differential equations in the 18-the century, Alexander Hrennikoff and Richard Courant in the 1940s were the first to investigate mechanics problems solving with the help of these methods. The current finite element method is considered to start with “Stiffness and Deflection Analysis of Complex Structures,” published in 1956 by Turner, Clough, Martin, and Topp. [Clough RW. The finite element method in plane stress analysis. Proceedings of the Second ASCE Conference on Electronic Computation, Pittsburgh, PA, 1960.]

The term “finite element” was coined by Ray Clough in 1960 in a paper given at the ASCE Conference on Electronic Computation in Pittsburgh. Clough emphasizes, “That first FEM paper attracted very little attention, but [Clough RW. Stress analysis of a gravity dam by the finite element method. Proceedings of the Symposium on the Use of Computers in Civil Engineering, Laboratorio Nacional de Engenharia Civil, Lisbon, Portugal,1962 (see also RILEM Bull. No. 19; June 1963)]… did attract some attention.” (Clough) Also, he emphasizes the pivotal role that Ed Wilson played in the computer program for FEM development.

## Partial Differential Equations

We can describe nearly everything around us with differential equations in partial differentials. Ordinary (with one argument) and partial (with several arguments) differential equations are used very often. An ordinary differential equation is a case of a partial differential equation, but the behavior of solutions is quite different in general.

Equation ** F (x, y(x), y′(x), . . . , y(n)) = 0** is an ordinary differential equation of n-th order for the unknown function y(x), where F is given.

## What kind of problems can FEA solve

With the help of FEA, engineers can immediately increase material accuracy when designing a product, seeing how all physical stresses may affect the design, and determining how stresses within one piece will affect the materials in another separate, but connected, part.

One of the most valued benefits of FEA is the ability to model an object, showing how critical factors affect the entire structure and why failures might occur.

Stress as one of such factors also can be modeled, and various stress scenarios simulated accurately, allowing engineers to identify potential weaknesses in the design and optimize it accordingly. Stress distributions can be analyzed and simulated through such common techniques as: Finite Element Analysis (FEA), Computational Fluid Dynamics (CFD), Multibody Dynamics (MBD), and some others. Using FEA, engineers can perform stress testing, accurately predicting stress distributions, deformations, and failure modes in complex components and assemblies, ensuring structural integrity and reliability. By conducting virtual stress tests using FEA, engineers can reduce the need for costly physical prototypes and accelerate the development of new products or systems. Overall, FEA serves as a valuable tool for stress testing, enabling engineers to make informed design decisions and ensure the structural robustness of engineering components and systems. Stress modeling is often used for lifting cranes, fatigue analysis for machines and machine parts, platform supports, brake or rotor lifetime certification, forensic analysis and validation, pressure vessel analysis, airport bridges, machine design, and many other purposes.

**Optimize Your Designs with FEA**

Reduce prototyping costs and time-to-market by simulating critical factors like stress and deformation. Ensure your products perform flawlessly with FEA.

FEA simulators help remove multiple repetitions of initial prototyping. By initially simulating the system in FEA software, the designer can model different designs and materials in hours versus the days or weeks of complex prototyping.

FEA is paramount in modern engineering. It empowers engineers to simulate and evaluate the performance of designs before physical prototypes are built, thereby reducing costs, time, and risks associated with product development. FEA is crucial in optimizing designs, identifying potential weaknesses, and ensuring that engineering solutions meet safety, performance, and regulatory standards.

## FEA terminology

A comprehensive understanding of FEA terminology is essential for practitioners. It forms the language through which engineers communicate and interpret analysis results. FEA terminology provides the framework for accurate analysis and effective collaboration among multidisciplinary teams, from boundary conditions and meshing techniques to material properties and convergence criteria. By mastering FEA terminology, engineers can effectively interpret simulation results, troubleshoot issues, and make informed decisions to improve the performance and reliability of engineering designs.

In FEA, the researched structure is divided into separate parts – ** finite elements** connected into nodes. These nodes are then connected by mathematical equations called shape functions. Together with the element type, the applicable material properties, and geometric characteristics, these nodes and shape functions describe what is generally called a finite element. There are many different types of elements used in FEA. These elements are developed independently and vary from one finite element (FE) software to another. In general, there are three groups of elements which are one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) elements. Each finite element characterizes by:

- a number of dimensions of the space used (1-D, 2-D, 3-D);
- geometric shape, usually being a simple geometric figure (right line section, triangle, square, tetrahedron);
- nodes set situated usually, but not always at lines or surfaces of elements chapter, being typical for neighboring elements;
- a set of degrees of freedom used, relating more often to nodes (but not obligatory to nodes) – displacements, bends, etc.
- rules, defining the relationship between finite element nodes’ displacements and system nodes. For instance, element nodes can be attached to system nodes rigidly or with joints;
- the system of approximation functions, defined inside the finite element area and letting roughly express the displacements components in any point of the element through its degrees of freedom;
- physical law defining the relationship between internal effort and displacements;
- the class of tasks to which a given type of finite element is applicable;
- the set of allowed loads and effects that can be applied directly to the finite element and the way of their setting. The collection of interconnected and basement-attached finite elements forms a design diagram called finite elements diagram or finite element model;
- and last but not least – the list of limitations and usage recommendations.

** Mesh** – is an array of finite element corner points (or nodes) and edges representation of a larger geometrical domain. Mesh forms the basis for defining element properties such as material properties, boundary conditions, and loads.

Mesh refinement plays a critical role in achieving accurate FEA results. As the mesh is refined, meaning the elements become smaller and more numerous, the model captures finer details of the geometry and behavior of the system being analyzed. This leads to more accurate predictions of stress distributions, deformations, and other physical phenomena.

** Node** – is a computational point that defines the shape of each finite element, the intersection of two or more mesh lines or edges. Each node possesses specific coordinates in space, defining its precise location within the domain. These coordinates determine the shape and size of the finite elements surrounding the node, influencing the overall accuracy and fidelity of the FEA simulation.

** The degree of freedom** relates to a translation or a rotation along three axes at each node of an element. There can be up to 6 degrees of freedom per node depending on the element type. Each degree of freedom represents a specific type of movement or deformation that a structure can undergo, typically categorized as the range of motions the node can experience along different axes. Degrees of freedom play a fundamental role in finite element analysis because by appropriately assigning and constraining them, engineers can simulate various loading conditions, boundary conditions, and structural responses with precision and reliability.

## Finite Elements Software

Different types of software are used to simulate other engineering and physical situations. FEA software is used to predict construction behavior. The most complicated task in performing a plate buckling check for a large structure, like a complete ship design, on a general Finite Element Analysis model is to define numerous plates and the dimensions of these plates to be verified. To guarantee the correct results, the model has to have a fine mesh with small enough finite elements. But at the same time, each plate field must be treated as one separate structural member for plate buckling checks. With the help of a specific tool, it is possible to break the boundaries of general FEA Analysis and enable the code checking directly in Simcenter 3D, Femap, Ansys, etc., by enabling the automatic recognition of structural items. The recognition of plates, stiffeners, and girders is based on mesh connectivity and can be performed on any structure which is built with 2D or, in some cases, even 3D elements. The structural members are defined automatically and mesh independently. This allows an engineer to have a model with fine mesh for precise results of the general finite element analysis and a list of structural members for code checking.

Since checks are done on structural items and not on finite elements, the best solution for both execution time and accuracy of the results would be to use the extension for general FEA programs capable of the automatic recognition of the structural members mesh independently.

SDC Verifier is software that follows this methodology. The best solution to avoid double work is to have the same environment for general FEA and code checks. In addition to the time-saving benefits, usage of code checking extensions for the General FEA programs allows to:

- Check the quality of modeling with the help of recognition tools;
- Understand the behavior of a studied structure by analyzing all possible loading;
- Analyze the critical parameters for the checks;
- Quickly improve the design by using the thickness factors and modifying the plate dimensions or with the help of powerful editors in the general FEA tools and instant update of the simulation data code-checking extension.
- Compare different design approaches to loading conditions in one user-friendly CAE environment condition and define the governing ones.

ANSYS is one of the most used finite element analysis software. It has lots of kinds of tools to simulate different engineering and physical situations, such as fluids in CFX and Fluent, static and dynamic motion analyses, steady-state and transient thermal and structural analyses, and modal analyses in which you can obtain free-vibrational frequencies, a bunch of vibrational analyses in ANSYS Mechanical. All of these different physical situations can be combined in ANSYS Workbench.

Most engineers related to engineering software have worked with Autodesk Inc. from their CAD software such as Autocad and Inventor.

Nastran is one of the most iconic finite element analysis software that was created by NASA. NASTRAN, is an acronym formed from **NA**SA **STR**ucture **AN**alysis. Nastran’s source codes are available in various kinds of other software such as NX Nastran, MSC Nastran, etc.

Simcenter Femap is an advanced simulation application for creating, editing and inspecting finite element models of complex products or systems. You can use advanced workflows in Simcenter Femap to model components, assemblies, or systems, to then determine a model’s behavioral response when subjected to real-world conditions. In addition, Simcenter Femap provides powerful data-driven and graphical results visualization and evaluation, which in combination with the industry-leading Simcenter Nastran (formely known as NX NASTRAN), delivers a comprehensive CAE solution that improves product performance.

There are also a lot of other types of FEA software for any type of task.

**SDC Verifier as a Finite Element Analysis Software**

SDC Verifier – a powerful modeling and standard checking software that works independently and within ANSYS, Femap, Simcenter 3D, helps to design and automatically verify a model according to multiple industry standards and generate reports. With the unique recognition tools, structural items such as beam members, plates, welds, stiffeners, panels, and joints are found on the model and directly verified according to any specific requirements. It helps to automatically verify the model according to multiple industry standards without the need to create any additional models or spreadsheets and generate all the required documentation in extensive calculation reports.

SDC Verifier speeds up the design phase by storing the full calculation process. A recalculation after a modification requires just a single click. All calculation settings are reapplied to the new model and FEM results. The Code of the checks is completely open, so you can modify the formulas of a standard or write your own custom rules.

SDC Verifier seamlessly addresses challenges in FEA by offering advanced design, verification, and validation tools, customizable checks and templates, specialized modules for fatigue analysis, seamless solving and integration with leading FEA software, and a user-friendly interface. By empowering engineers with these capabilities, SDC Verifier enhances the efficiency, accuracy, and reliability of finite element analysis processes, ultimately enabling better-informed design decisions and optimized product performance.

## Further reading

The finite elements analysis industry is developing rapidly, and engineers have to track new trends continuously in order to keep the working process efficient. Still, there are many resources that form the core of industry knowledge. *The Finite Element Method: Its Basis and Fundamentals* by O.C. Zienkiewicz, R.L. Taylor, and J.Z. Zhu and *Practical Finite Element Analysis* by Nitin S Gokhale, Anand N Thite, Sanjay S Deshpande, and Sanjeev V Bedekar are some examples of such sources. Professional communities, forums, and blogs can also be advantageous.

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