## What is finite element analysis?

Finite element analysis (FEA) is an integral part of computer-aided engineering (CAE). The FEA is a posteriori tool that requires extensive understanding and expertise to master. It has been used in many complex engineering and real-life applications. It began as an extension of the matrix method in structural analysis. It was initially used as a tool for structural engineering applications alone. Finite element analysis has numerous applications in certain domains such as in structural engineering and bio-medical engineering.

## Usage of FEA

Starting from simple linear static problems to complex non-linear dynamics problems, they are solved by the approach of FEA with ease. The approach is also sometimes referred to as the finite element method (FEM).

FEM is a powerful tool for finding approximate solutions of the partial differential equations (PDE) or transforming the partial differential equations to simple algebraic equations, which are then easily solved using the basic rules of mathematics.

The most significant application of finite element analysis is in the area of structural mechanics. Various CAE software implements the finite element method such as ANSYS, ABAQUS, NASTRAN, and NX-Siemens. This software uses the methodology of finite element method and some of them use another approximation methodology known as the finite difference method (FDM).

## Working of finite element analysis

The simulations performed in FEA are done by using a mesh of millions of small elements, combined to create the shape of the domain of interest. Calculations are performed on each element, and their results are combined as a whole to get the required approximate solution.

The calculations are approximated to be usually of polynomial, with interpolations across the small elements. The points where values are exactly obtained are known as nodal points. The position of the nodal points is always at the boundaries of the elements.

## Different types of PDEs

The finite element analysis is targeted at solving the PDEs and determining their approximate solutions. There are three main different types of PDEs, they are, elliptic, hyperbolic, and parabolic. The elliptic PDEs are smooth, the hyperbolic PDEs support solutions with discontinuities and the parabolic PDEs describe the time-dependent diffusion problems. The knowledge of the boundary conditions and initial conditions is necessary while approaching to solve the PDEs.

The elliptic equations can be solved using two approaches, one is the variational methods and the other is the FDM. Hyperbolic PDEs are associated with jumps or discontinuities in the solution. The FEM method is usually unsuitable for solving hyperbolic kind of equations.

## FEA procedures

Below are the summarizations of basic key ideas applied in the FEM for finding approximate solutions,

- The solutions of the field variables satisfying the boundary conditions and differential equations are unknown. The initial start of the FEA analysis begins with assuming a trial solution for the differential equations. The trial solutions are usually polynomials.
- As the solutions are assumed or guessed, it generally does not satisfy the boundary conditions completely. So, it forms a domain residual or error.
- The domain residual varies from one point to another within the domain and cannot be reduced to zero completely, but can be made zero along with some selected points.
- For real-life complex problems, choosing a continuous trial function over the entire domain is not relevant. Therefore, it is mandatory to discretize the domain into a finite segment called finite element using a piece-wise trial function.
- The trial function used in each segment or finite element is known as the element level shape function.
- Using the shape functions, the weighted sum of domain residual is estimated and combined with the other elements in the entire domain.

## Steps in finite element modeling

The basic steps to solve any problem by using the FEA and FEM are outlined below,

- Domain discretization
- Selection of displacement models
- Derivation of element stiffness matrices
- Assembling
- Solution for the unknowns
- Post-processing

**Domain discretization**

In domain discretization, the entire domain of the problem is divided into small elemental finite elements. The elements may be linear, quadratic, tetrahedron, or shell elements. Depending on the complexity of the problem, the elements may be one-dimensional, two-dimensional, or three-dimensional.

**Selection of displacement models**

In this stage, the approximation of the FEM or FEA begins. It is not possible to select a displacement model that can accurately model the actual displacement variation of the domain. So, ideally, a polynomial function is chosen to signify the displacement problem.

**Derivation element stiffness matrices**

The elemental stiffness matrices are derived from the materials and the properties of the elements, by the application of the minimum potential energy principle. The stiffness is related to the displacements to forces at the nodal points.

**Assembling**

This process includes the assemblage of all the individual stiffness matrices into a global stiffness matrix for the entire domain.

**Solution for the unknowns**

The finite element method reduces the order of the partial differential equation, the reduced order is thus called the weak form. The Galerkin method is the most popular method used in the finite element method to turn the strong form (higher-order) of the PDEs into the lower order (weak formulation). From the weak form, the algebraic equations are derived and by applying the boundary conditions, the equations are solved using simple mathematics.

**Post-processing**

The solutions are plotted graphically to conclude the required purpose of the problem. The process of using the data from the solution in creating a graphical representation is called post-processing.

## Finite element formulation from governing equation

Every physical phenomenon in nature can be represented by PDEs. Application of elementary mathematical procedures turns out to be tedious or almost impossible in determining the solution of these equations. Hence, the weighted residual method forms a fundamental part of the FEA to determine the approximate solution to these PDEs.

The use of the weighted residual method for the estimation of the approximate solutions can be done in the following ways,

- A trial function is assumed, generally, for the one-dimensional problem, a polynomial function is chosen, the polynomial function is given as, $f\left(x\right)={c}_{\circ}+{c}_{1}x+{c}_{2}{x}^{2}+....$
- This assumed function will neither satisfy the boundary conditions nor the differential equation. Also, the assumed function is not differentiable in the domain. This function is substituted into the differential equation to determine the domain residual.
- Determine the unknown parameters, ${c}_{\circ},{c}_{1},{c}_{2},....$ by using the boundary conditions and the initial conditions.

## Context and Applications

The applications of finite element analysis or FEA find a wide number of applications in different engineering problems. It has been used extensively in the structural engineering domain and different medical research applications.

The topic is a top priority area of research for the post-doctoral level for the development of a more accurate solution. Besides this, the topic is extensively used in the bachelor of science and master of science curriculum. The topic can be found in popular books published by McGraw-Hill and Prentice-hall.

## Practice Problems

- Which of the following is the preliminary step in finite element modeling?

- Formation of basis functions
- Formation of elementary stiffness matrices
- Discretization of domain
- Assuming a trial solution

Correct option- c

Explanation: The first step in the finite element method (FEM), is to divide the domain into multiple sub-domains called finite elements. The step is also called discretization.

2. The order of PDEs is reduced to a lower order. Which of the following options holds for the statement?

- The PDEs are represented as a weak form.
- The PDEs are known as first-order differential equations.
- The PDEs are known as second-order differential equations.
- None of these

Correct option- a

Explanation: The partial differential equations that govern the behavior of the domain is reduced to a lower order, known as the weak form. The popular Galerkin method is applied to turn the higher-order PDEs into a lower order.

3. Which of the following is CAE software?

- ANSYS
- NX-Siemens
- ABAQUS
- All of these

Correct option- d

Explanation: CAE software or computer-aided engineering software uses the FEM algorithms to perform finite element simulations. Popular software that is used extensively is, ABAQUS, ANSYS, COMSOL Multiphysics, NX-Siemens, etc.

4. Which of the following PDE is associated with discontinuities in the solution?

- Elliptic PDE
- Hyperbolic PDE
- Parabolic PDE
- None of these

Correct option- b

Explanation: Hyperbolic PDEs are generally associated with jumps in the solution or with discontinuities. The FEM is not suitable for the solution of such equations.

5. The elliptic PDEs are solved by which of the following methods?

- Variational methods
- Finite difference methods (FDM)
- Both a and b
- Only a

Correct option- c

Explanation: The elliptic type of PDEs are usually smooth PDEs and they are solved by variational methods or the finite difference method (FDM).

## Related Concepts

- Basis functions
- NAFEMS
- Mesh refinement
- The Galerkin method

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