## What is finite element method?

Finite element method (FEM) is a mathematical technique that primarily implements the numerical approach and finds approximate solutions to the engineering problems. The finite element method is widely used in many civil engineering applications, especially in the domain of structural mechanics to find solutions to partial differential equations (PDEs), which governs the motion of structures under external influences like loads.

Readers can refer to numerous books available which will help them to clear the fundaments of the FEM technique. Books such as 'Introduction to Finite Element Method' by J.N. Reddy, published by McGraw-Hill, and 'Finite Element Structural Analysis' by T.Y. Yang, published by Prentice-Hall, are a must-read. These books provide the fundamental concepts required to solve any problems related to FEM. These books will also provide an insight about National Agency for Finite Element Methods (NAFEMS), which introduces the readers to the standards associated with finite element analysis (FEA) and simulations. One can get expertise in solving a given engineering problem based on the posterior of such concepts in various engineering applications.

## Steps used in finite element analysis (FEA)

The finite element analysis or FEA is the set of procedures applied to solve an engineering problem and to arrive at a particular solution. The FEA uses the tools and techniques of the FEM. Both of them have the same meaning and can be used interchangeably.

For finding solutions to an engineering problem by FEM, a proper finite element modeling of the problem should be framed. There is a step-by-step procedure that should be applied. These steps are outlined below.

## Stiffness matrix formulation from PDEs

Every physical system in nature undergoes time-dependent behaviors, or, they can also be termed as unsteady. The motion of such systems is governed by partial differential equations or PDEs. A solution to these PDEs results in the determination of unknown parameters that govern the motion of the system. To formulate a stiffness matrix for a body, the very first step is to know the governing differential equation of the system. The different steps followed for the stiffness matrix formulation are:

1. Form the PDEs of the system: The methodology for formulation of a PDE of a system lies in the nature of the system. If the system is lumped, the corresponding differential equation can be formed by simple Newtonian mechanics for a continuous system a more specialized branch of mechanics like Hamiltonian mechanics and Lagrangian mechanics are applied.
2. Assume a trial solution: Having the differential equation of the system, a trial solution of the equation is chosen. Since the solution is unknown an approximate or a guess solution is applied. The trial solution mostly chosen is a piece-wise polynomial equation, also known as basis functions.
3. Estimate the error: As the solution chosen is a guess, it is not completely differentiable, and it will not satisfy the given differential equation completely. Hence, substituting the trial solution into the differential equation results in an error known as a residue, which needs to be minimized. There are many methods that aids in residual minimization, one such popular method is the Galerkin method. The Galerkin method minimizes the domain residual and reduces the order of the differential equation into a lower order known as weak formulation. Which can be solved easily.

## Computer-aided engineering (CAE)

Computer-aided engineering is the application of the computer to compute solutions using the finite element method. This is done with the use of the software. The software implements the finite element method using a set of algorithms, which are processed by computers to determine the solutions. Various FEA software is available which are meant to handle simple to complex problems. ABAQUS, ANSYS, LS-DYNA, NX-Siemens, and COMSOL Multiphysics are some of the FEA software used widely in industries and academics.

A finite element software automatically discretizes the given domain into multiple number of finite elements (the choice of the finite elements depends on the user), based on various algorithms, the procedure is known as meshing. After meshing, the user signifies the boundary conditions and other relevant inputs. The built solver solves the problem and provides the solution in the form of graphical representation.

The software provides the flexibility to animate as well as simulate the solutions under the given applications of loads and boundary conditions, which gives a through insight about the behavior of the system when it would be subjected to actual working conditions.

## Context and Applications

This topic is extensively studied in both undergraduate and post-graduate courses of

• Bachelors of Technology in Mechanical Engineering
• Bachelors of Technology in Civil Engineering
• Masters of Technology in Mechanical Engineering
• Masters of Technology in Civil Engineering

## Practice Problems

1. Which of the following is finite element software?

1. ANSYS
2. COMSOL Multiphysics
3. ABAQUS
4. All of these

Correct option- d

Explanation: ABAQUS, ANSYS, and COMSOL Multiphysics are all finite element software. This software contains algorithms of FEM which aids in performing the finite element analysis. The specialty of this software is it can handle millions of finite elements and account for better accuracy of solutions.

2. Which of the following is also known as basis functions?

1. Polynomial function
3. Cubic function
4. None of these

Correct option- a

Explanation: The polynomial functions are also known as basis functions. The basis functions are greatly used in numerical analysis like the FEM. They are also known as blending functions. These functions act as an extensive tool for mathematical interpolations.

3. Which of the following is the first step in the FEM?

1. Formation of PDEs
2. Discretization
3. Formation of the stiffness matrix
4. None of these

Correct option- b

Explanation: Discretization of the domain is the first and primary step in the FEA. In this step, the domain is subdivided into multiple subdomains by using specific finite elements. Each of the finite elements is characterized by nodes and degree of freedom.

4. Which of the following is true for the Galerkin method?

1. It minimizes domain residual.
2. It minimizes the complexity of the problem.
3. It solves the PDEs.
4. All of these

Correct option- a

Explanation: The Galerkin method is used to minimize the domain residual. The Galerkin method is an integral method, which does so by introducing specific weights. It also transforms the governing PDEs into weak forms.

5. Which of the following is true for a stiffness matrix?

1. It relates the force and deformation
2. It is derived by considering the material properties
3. It denotes the amount of deformation a member undergoes
4. Both a and b

Correct option- d

Explanation: The stiffness matrix relates the force and deformations of a member. They are derived using the material properties of the member. The stiffness matrix acts as an essential parameter for solving the equation and aids in the determination of the unknowns, especially deformations. For instance, the force and deformation matrix can be related to the stiffness matrix.

• Finite difference method (FDM)
• Finite volume method (FVM)

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