What is Finite Element Analysis (FEA)?

Introduction

The Finite Element Analysis, also called the Finite Element Method (FEM), is a numericaltechnique to find numerical solutions to partial differential or integral equations of field problems.In stress analysis this field is the displacement field whereas in thermal analysis it is thetemperature and in fluid flow it is the velocity potential function, and so on [1].

The FEM discretizes the physical domain into subdomains, the so­called finite elements, definedby a set of nodes; the arrangement of elements is called mesh. The unknown field is interpolatedby shape functions, usually polynomial functions, depending on a set of discrete variables (i.e.field values at the nodes). By connecting elements the field becomes interpolated over thestructure. The unknown field quantity is calculated via minimizing a function such as the potentialenergy in stress analysis; this procedure leads to a matrix of algebraic equations which are solvedwith a numerical scheme. As a result, the field values at the nodes are determined and extrapolatedover the finite elements [1], [2].

In simple words, the FEM is a numerical method to solve partial differential equations bydiscretizing the domain into a finite mesh. Numerically speaking,a set of partial differentialequations are converted into a set of algebraic equations to be solved for unknown at the nodesof the mesh. Bear in mind thatFEA is an approximate solution of a mathematicalrepresentation of a physical problem. This is to say that, best case scenario FEA will replicatean analytical solution, whether the mathematical model is according to the real physical situationor not is another matter!

A simple cantilever beam under shear force is discretrized into a set of finite elements.

Versatility and applications

The power of FEA is its versatility. Over other numerical methods many advantages arise:

applicable to many field problems: structural analysis, heat transfer, electrical/magneticalanalysis, fluid and acoustic analysis, multi­physics, etc.no geometric restrictions,boundary and load conditions are easily applied as well as non­linear characteristics,many different materials, models and mathematical representations can be implemented,combined and coupled.

Although FEA has its inherent disadvantages, it has become widely used in structural analysis dueto this set of unique features.

Solid mechanics

As I said, FEA is extensively applied to structural analysis, also called stress analysis, which is mymajor area of expertise and hence the focus of this blog. Typically, the input and output variablesinvolved in linear static stress analysis are the following:

Unknowns: displacements, strains, stresses.

A finite element solver always solves for displacements at the nodes. Once the displacements areknown, other variables can be obtained. Strains are then derived via the compatibility equationsfrom the displacement field, and stresses are subsequently derived from strains via the materialconstitutive matrix.

Solid Mechanics equations:

Equilibrium eqs. Typically in solid mechanics, any elementunder static equilibrium satisfies these set of equations.

Constitutive eqs. This is the relationship between strain andstresses. E.g. For an isotropic material, the Young´s modulus and Poisson’s ratio suffices todescribe Hooke’s Law.

Compatibility eqs. These define how the strains are related to thedisplacements. Strains are just the derivatives of the displacements.

In total, we need to solve a problem consisting of 15 equations in which 9 are partial differentialequations (good luck with that!). For simple cases, analytical solutions are available using theElasticity theory, but for complex scenarios FEA is required [3].

Input: Geometry, loading, restraints, material data.

The solid mechanics equations are fine…, but how does FEA solve for displacements?

Principle of Minimum Potential Energy (MPE)

The FEM uses a variational approach. Let’s consider an elastic body under the action of volume,surface and point loads. The MPE asserts that the structure shall deform to a position whichminimizes the total potential energy of the system, this principle will allow us to calculate thedisplacements that satisfy this condition. The Potential energy (PE) of a structural system isdefined as the sum of the strain energy (SE) and the work potential (WP)

MPE applied to a spring­mass system:

Single degree of freedom mass­spring system

The SE of a mass(m) – spring(k) system is equal to

the WP of a force (f) acting in the X direction is

and therefore the PE is as follows:

If we apply the MPE principle, the PE minimized in the deformed configuration. In order to findthis minimum let’s find the first derivative of the PE with respect to x and equate it to zero:

Which gives the position of equilibrium of the mass­spring system under the action of a singleforce – neglecting the work done by gravity:

MPE applied to a general elastic body:

The total potential energy of an elastic body under volume, surface and point forces can be

expressed as shown below:

, where epsilon is strain vector and sigma is stress vector, D is the constitutive material matrixrelating strain and stress.

, where u is the displacement vector, b is the volume force vector, t is the surface force vector andP is the nodal force vector. Hence, PE is as follows:

We can now apply the compatibility equations, those relating strains and displacements.mentioned before. In doing so, the strain vector can be regarded as a matrix derivative operator,denoted L, times the displacement vector, u. Finally, the PE is written as shown below:

We got the total potential energy of a general elastic body!, but… where are the finite elementsin all this? Nowhere to be seen, yet… When the FEM is applied to the MPE the volume andtherefore the above equation is discretized into finite elements, within each element theunknown displacements are interpolated using polynomial functions. In doing so we areassuming that the displacements vary in a polynomial fashion within each element – which maynot be correct – this is related to the so­called discretization error. Then, making use of the MPEwe will be able to calculate the derivative of the PE with respect to the interpolated displacementfield and get the desired solution to our problem.

Firsts things first. For each single element of the discretized finite mesh of our elastic body theMPE applies, and the previous equation ca be re­written as shown below. Note that thedisplacements have been interpolated using the shape functions and the super­index “e” denoteselement variables of our discretized domain.

Derivation of the algebraic equations governing the deformation of an elastic body. The FEM isused to interpolate the displacement field and discretize the domain.

Now that the potential energy has been assessed on an element by element basis, the total potentialenergy of the system simply adds up to their sum. This is typically called the assembling process.The element stiffness matrix, element force and displacement vectors refer to the local elementand have to be assembled into a global and much bigger matrix capturing the global stiffnessmatrix. For example, elements sharing nodes will contribute to the global stiffness matrix together.The stiffness terms coming from each different element is added together to get the total stiffnessof the sharing node.

Global assembling process and application of the MPE principle using the FEM.

Similarly to the single mass­spring system where the MPE principle results in an algebraicequation, the variational approach produces a set of algebraic equations. The stiffness matrix Kand the applied loads vector F will be used then to obtain the unknown nodal displacements.

The process is not yet finished, the restraints have to applied to the global stiffness matrix. As it is,this matrix is singular until the nodal restraints avoiding rigid body motions of the elastic body areset in place. In other words, no static equilibrium of the body can be found until the body isproperly restraint. The restraint nodes, whose displacements is known are then removed from theglobal stiffness matrix. This final reduced stiffness matrix is, somehow, inverted and multiplied by

the global force vector. And then, and not before, we will get the displacement vector under thedeformed configuration :).

References:

Personally, I strongly recommend having a read through the following references which I findvery useful to introduce yourself to FEA. Cook’s book very well deserves an honorable mentionas it is regarded as the FEA bible. It is a must have on your shelf!

[1] Cook, R. D. (1995), Finite element modeling for stress analysis, 1st ed, John Wiley & sons,INC., University of Wisconsin – Madison.

[2] Fuenmayor, J., Ródenas, J.J., Tarancón, J.E., Tur, M. and Vercher, A., (2009),Cálculoestructural – Método de los elementos finitos, Departamento de Ingeniería Mecánica y deMateriales (DIMM), Universidad Politécnica de Valencia.

[3] Qi, H. (2006), What is Finite Element Analysis (FEA)?, University of Colorado, availableat: http://www.colorado.edu/MCEN/MCEN4173/chap_01.pdf (accessed 20 May 2015).