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What is the Finite Element Method? FEM and FEA Explained

Finite Element Analysis (FEA) is the analysis of any given physical phenomenon using the numerical technique called Finite Element Method (FEM).

It is necessary to use mathematics to comprehensively understand and quantify any physical phenomena, like structural or fluid behavior, thermal transport, wave propagation, the growth of biological cells etc. Most of these processes are described using Partial Differential Equations (PDEs). However, for a computer to solve these PDEs, numerical techniques have been developed over the last few decades and one of the prominent ones, today, is the Finite Element Method.

Applications of the Finite Element Method

Finite Element Method started with significant promise in modeling several mechanical applications related to aerospace and civil engineering. The applications of Finite Element Method are just starting to reach its potential. One of the most exciting prospects is its application to coupled problems like fluid-structure interaction; thermo-mechanical, thermo-chemical, thermo-chemo-mechanical problems; bio-mechanics & bio-medical engineering; piezoelectric, ferroelectric, electromagnetics etc.

There have been many alternative methods proposed in the recent decades. But their commercial applicability is yet to be proved. In short, FEM has just made a blip on the radar!

Partial Differential Equations

Before proceeding into FEM itself, it is firstly important to understand the different genre of PDEs and their suitability to use FEM. Understanding this is particularly important to everyone, irrespective of one’s motivation in using finite element analysis. One should constantly remind oneself that FEM is a tool and any tool is as good as its user.

Laplace equation solved on an annulus - what is Finite Element Analysis

Fig 01: Laplace equation on an annulus (Source)

PDEs can be categorized as elliptic, hyperbolic, and parabolic. When solving these differential equations boundary and / or initial conditions need to be provided. Based on the type of PDE, the necessary inputs can be evaluated. Examples for PDEs in each category include Poisson equation (Elliptic), Wave equation (Hyperbolic), Fourier law (Parabolic).

There are two main approaches to solving elliptic PDEs – Finite Difference Methods (FDM) and Variational (or Energy) Methods. FEM falls into the second category of variational methods. Variational approaches are primarily based on the philosophy of energy minimization.

Hyperbolic PDEs are commonly associated with jumps in solutions. For example, the wave equation is a hyperbolic PDE. Owing to the existence of discontinuities (or jumps) in solutions, the original FEM technology (or Bubnov-Galerkin Method) was believed to be unsuitable for solving hyperbolic PDEs. However, over the years, modifications have been developed to extend the applicability of FEM technology.

Before concluding this discussion, it is important to consider the consequence of using a numerical framework that is unsuitable for the type of PDE. Such usage leads to solutions that are known as “improperly posed.” This could mean that small changes in the domain parameters leads to large oscillations in the solutions or the solutions exist only on a certain part of the domain or time etc. Such solutions are not reliable. Well-posed solutions are defined as that where a unique solution exists continuously for the defined data. Hence, considering reliability, it is extremely important to obtain well-posed solutions.

Principle of Minimization of Energy

So how does FEM work? What is the primary driving force? The principle of minimization of energy forms the primary backbone of the Finite Element Method. In other words, when a particular boundary condition is applied on a body, this can lead to several configurations but yet only one particular configuration is realistically possible or achieved. Even when the simulation is performed multiple times, same results prevail. Why is this so?

Depiction of Virtual Work: Foundation of Finite Element Method

Fig 02: Depiction of principle of virtual work

This is governed by the principle of minimization of energy. It says that when a boundary condition (like displacement, force etc) are applied, of the numerous possible configurations that the body can take, only that configuration where the total energy is minimum is the one that is chosen.

History of the Finite Element Method

Technically, depending on one’s perspective, FEM can be said to have it’s origins in the work of Euler, as early as the 16th century. However, the earliest mathematical papers on FEM can be said be in the works of Schellback [1851] and Courant [1943].

FEM was independently developed by engineers to address structural mechanics problems related to aerospace and civil engineering. The developments started around the mid-1950s with papers of Turner, Clough, Martin and Topp [1956], Argyris [1957] and Babuska and Aziz [1972]. The books by Zienkiewicz [1971] and Strang and Fix [1973] also laid the foundations for future developments in FEM.

An interesting review on the historical development can be found in Oden [1991]. A review of FEM developments in the last 75 years can also be found on our blog at “75 years of Finite Element Method (FEM)”.

Technical Overview of Finite Element Method

Finite Element Method is in itself a semester course. Here in this article, a concise description of the working of FEM is described. Here, we will consider a simple 1-D problem to depict the various stages involved in FEA.

Weak Form

One of the first steps in FEM is to identify the related identification of the PDE associated with the physical phenomenon. The PDE (or differential form) is known as the strong form and the integral form is known as the weak form. Let us consider a simple PDE as shown below. The equation is multiplied by a trial function v(x) on both sides and integrated with the domain [0,1].

Equation 01

Now, using integration by parts, the LHS of the above equation can be reduced to

Equation 02

As it can be seen the order of continuity required for the unknown function u(x) is reduced by one. The earlier differential equation required u(x) to be differentiable at least twice while the integral equation requires it to be differentiable only once. The same can be shown for multi-dimensional functions too but the derivatives are replaced by gradients and divergence.

Without going into the mathematics here, one can prove using Riesz representation theorem that there exists a unique solution for u(x) for the integral and hence the differential form. In  addition, if f(x) is smooth, it also ensures that u(x) is also smooth.

Discretization

Once the integral or weak form has been set up, the next step is the discretization of the weak form. The integral form needs to be solved numerically and hence the integration is converted to a summation that can be calculated numerically. In addition, one of the primary goals of discretization is also to convert the integral form to a set of matrix equations that can be solved using well-known theories of matrix algebra.

Meshing of gears

Fig 03: Meshing of gears in contact

As shown in Fig. 03, the domain is divided into small pieces known as “Elements” and the corner points of each element is known as a “Node”. The unknown functional u(x) are calculated at the nodal points. Interpolation functions are defined for each element to interpolate, for values inside the element, using nodal values. These interpolation functions are also often called as shape or ansatz functions. Thus the unknown functional u(x) can be reduced to

Equation 03

where nen is the number of nodes in the element, Ni and ui are the interpolation function and unknowns associated with node i. Similarly, interpolation can be used for the other functions v(x) and f(x) present in the weak form such that the weak form can be re-written asWeak form to summation

The summation schemes can be transformed into matrix products and can be re-written as

Summation as matrix product

The weak form can now be reduced to a matrix form [K]{u} = {f}

In matrix form

Note above that the earlier trial function v(x) that had been multiplied does not exist anymore in the resulting matrix equation. Also here [K] is known as the stiffness matrix, {u} is the vector of nodal unknowns and {R} is the residual vector. Further on, using numerical integration schemes, like Gauss or Newton-Cotes quadrature, the integrations in the weak form that form the tangent stiffness and residual vector are also handled easily.

There definitely exists a lot of mathematics going into the decision in choosing interpolation functions that require a knowledge of functional spaces like Hilbert and Sobolev spaces. For more details in this regard, the references listed in the article “How can I learn FEA?” are recommended.

Solvers

Once the matrix equations have been set up, the equations are passed on to a solver to solve the system of equations. Depending on the type of problem, direct or iterative solvers are generally used. A more detailed treatment on the solvers and their working, criterion to choose between direct and iterative solvers etc is available in the blog article “How to Choose Solvers for FEM Problems: Direct or Iterative?

Different Types of Finite Element Method

As discussed earlier in the section on PDEs, traditional FEM technology has demonstrated shortcomings in modeling problems related to fluid mechanics, wave propagation etc. Several improvements have been made over the last two decades to improve the solution process and extend the applicability of finite element analysis to a wide genre of problems. Some of the important ones still being used include

Extended Finite Element Method (XFEM)

Bubnov-Galerkin method requires continuity of displacements across elements. However problems like contact, fracture and damage involve discontinuities and jumps that cannot be directly handled by Finite Element Method. To overcome this shortcoming, XFEM was born in 1990’s. XFEM works through expansion of the shape functions with Heaviside step functions. Extra degrees-of-freedom are assigned to the nodes around the point of discontinuity such that the jumps can be considered.

Generalized Finite Element Method (GFEM)

GFEM was introduced around the same time as XFEM in the 90’s. It combines the features of traditional FEM and meshless methods. Shape functions are primarily defined in the global coordinates and further multiplied by partition-of-unity to create local elemental shape functions. One of the advantages of GFEM is the prevention of re-meshing around singularities.

Mixed Finite Element Method

In several problems, like contact or incompressibility, constraints are imposed using Lagrange multipliers. These extra degrees of freedom arising from Lagrange multipliers is solved independently. The system of equations is solved like a coupled system of equations.

hp-Finite Element Method

hp-FEM is a combination of using automatic mesh refinement (h-refinement) and increase in order of polynomial (p-refinement). This is not the same as doing h- and p- refinements separately. When automatic hp-refinement is used, and an element is divided into smaller elements (h-refinement), each element can have different polynomial orders as well.

Discontinuous Galerkin Finite Element Method (DG-FEM)

DG-FEM has shown significant promise for using the idea of Finite Elements for solving hyperbolic equations where traditional Finite element methods have been weak. In addition, it has also shown promise in bending and incompressible problems which are commonly observed in most material processes. Here additional constraints are added to the weak form that includes a penalty parameter (to prevent interpenetration) and terms for other equilibrium of stresses between the elements.


References

  • Schnellback, “Probleme der Variationsrechnung,” Journal für die reine und Angewandte Mathematik , v. 41, pp. 293 – 363 (1851)
  • R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bulletin of American Mathematical Society, v. 49, pp. 1-23 (1943)
  • M. J. Turner, R. M. Clough, H. C. Martin and L. J. Topp, “Stiffness and deflection analysis of complex structures,” Journal of Aeronautical Science, v. 23, pp. 805-823 (1956)
  • J. H. Argyris, “Die matritzentheorie der Statik,” Ingenieur-Archiv XXV, pp. 174-194 (1957)
  • O. C. Zienkiewicz, “The Finite Element Method in Structural and Continuum Mechanics,” McGraw-Hill, London (1971)
  • I. Babuska and A. K. Aziz, “Survey lectures on the mathematical foundations of the finite element method,” In The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, pp. 3-636 (1972)
  • G. Strang and G. J. Fix, “An analysis of the Finite Element Method,” Prentice-Hall, Englewood Cliffs, New Jersey (1973)
  • J. T. Oden, “Finite elements: An introduction,” in Handbook of Numerical Analysis II, Finite element methods (Part I), North-Holland, Amsterdam, pp. 3-12 (1991)

    This case study shows a stress analysis of a wheel loader arm performed with the SimScale simulation platform. 

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