# Finite Element Analysis

The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations. The method approximates the unknown function over the domain. To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).

## Basic concepts

The subdivision of a whole domain into simpler parts has several advantages:^{[2]}

- Accurate representation of complex geometry
- Inclusion of dissimilar material properties
- Easy representation of the total solution
- Capture of local effects.

A typical work out of the method involves (1) dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer.

In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are oftenpartial differential equations (PDE). To explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method. The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with

- a set of algebraic equations for steady state problems,
- a set of ordinary differential equations for transient problems.

These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler’s method or the Runge-Kutta method.

In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains’ local nodes to the domain’s global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the subdomains.

FEM is best understood from its practical application, known as **finite element analysis (FEA)**. FEA as applied in engineering is a computational tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.

FEA is a good choice for analyzing problems over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations.^{[3]} For instance, in a frontal crash simulation it is possible to increase prediction accuracy in “important” areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). Another example would be in numerical weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.

### General form of the finite element method

In general, the finite element method is characterized by the following process.

- One chooses a grid for {\displaystyle \Omega }. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.
- Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions.

A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are merely continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial differential equations, one must use smoother basis functions. For instance, for a fourth order problem such as {\displaystyle u_{xxxx}+u_{yyyy}=f}, one may use piecewise quadratic basis functions that are {\displaystyle C^{1}}.

Another consideration is the relation of the finite-dimensional space {\displaystyle V} to its infinite-dimensional counterpart, in the examples above {\displaystyle H_{0}^{1}}. A conforming element method is one in which the space {\displaystyle V} is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite-dimensional space is not a subspace of the original {\displaystyle H_{0}^{1}}.

Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an *h*-method (*h* is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid {\displaystyle h} is bounded above by {\displaystyle Ch^{p}}, for some {\displaystyle C<\infty } and {\displaystyle p>0}{\displaystyle d} method will have an error of order {\displaystyle p=d+1}.

If instead of making *h* smaller, one increases the degree of the polynomials used in the basis function, one has a *p*-method. If one combines these two refinement types, one obtains an *hp*-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform *p* are called spectral finite element methods (SFEM). These are not to be confused with spectral methods.

For vector partial differential equations, the basis functions may take values in {\displaystyle \mathbb {R} ^{n}}.

## Application

A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.^{[19]}

FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.

This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.^{[21]} The introduction of FEM has substantially decreased the time to take products from concept to the production line.^{[21]} It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated.^{[22]} In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.^{[21]}

In the 1990s FEA was proposed for use in stochastic modelling for numerically solving probability models^{[23]} and later for reliability assessment.^{[24]} The stochastic finite element method has since been applied to many branches of engineering, ^{[25]} often being applied to characterise variability in material properties.^{[26]}

## See also

- Applied element method
- Boundary element method
- Computer experiment
- Direct stiffness method
- Discontinuity layout optimization
- Discrete element method
- FEA
- Finite difference method
- Finite element machine
- Finite element method in structural mechanics
- Finite volume method
- Finite volume method for unsteady flow
- Interval finite element
- Isogeometric analysis
- Lattice Boltzmann methods
- List of finite element software packages
- Movable cellular automaton
- Multidisciplinary design optimization
- Multiphysics
- Patch test
- Rayleighâ€“Ritz method
- Space mapping
- Weakened weak form
- Infinite element method