Sources of Nonlinearity

We live in a complex, nonlinear world, but what are the sources of nonlinearity in finite element analysis (FEA)?

1. Contact Nonlinearity - is significant when the functioning of an assembly involves the interaction of parts.
2. Material Nonlinearity - such as time-dependent (visco) behaviours, plasticity and damage effects amongst many other characteristics commonly seen in nonlinear materials.
3. Geometric Nonlinearity - occurs when the structure experience large strain/large deformations (This is typically when strains are larger than 10%) which causes stress stiffening of structures.

Plane Stress vs. Plane Strain vs. Axisymmetric Elements

A summary of the differences between plane stress, plane strain and axisymmetric elements.

Element	Plane stress	Plane strain	Axisymmetric
Required input geometric property	Thickness.	Thickness.	None. User should find out the angle of segment assumed by the FE software.
Assumptions	• σz=0 • εz=-v/E(σx+ σy)	• εz=0 • σz=v(σx+ σy)	Axisymmetric loading and structure.
Degree of freedom per node	Translations Ux, Uy	Translations Ux, Uy	Translations Ur, Uz
Stress output	σx, σy, σxy	σx, σy, σz, σxy	σr, σz, σrz, σθ
Strain output	εx, εy, εz, εxy	εx, εy, εxy	εr, εz, εrz, εθ

Also see
Types of elements in the element library

Steps in FEA: An Overview

In the FEA vs. FEM article, we distinguished the two by introducing FEM as the numerical foundation of FEA. It is the underlying method that makes FEA works. Let us now look at the following steps involved in running a general FEA and their conceptual difference should become even clearer when compared with the Steps in FEM.

     The common steps for carrying out a general purpose FEA is summarised below:

1. Model Idealisation
2. Spatial Domain Identification
3. Element Selection
4. Mesh Discretisation
5. Material and Geometric Definition
6. Boundary Conditions
7. Pre-analysis Checks
8. Job Submission
9. Results Verification

1. Idealise Geometric Model

     The first step involves defeaturing small details of the model. Defeaturing is necessary but must always be done carefully as it is often the major source of error of an analysis. Assumptions made in this step will affect the results strongly and must be coherent throughout the analysis. For example, fillets are often omitted as they are difficult to mesh. However, defeaturing fillets in the region of interest for stress can be a hazardous decision due to stress singularity, which is common in structures with sharp re-entrant corners.

Thin Shell vs. Thick Shell

Thick shells are capable of modelling transverse shear deformation whilst thin shells do not. Thick shells are governed by the Mindlin–Reissner (thick shell) theory. As the shell thickness decreases, the problem tends to favour Kirchhoff (thin shell) theory which neglect the inclusion of transverse shear deformation. This is pretty much similar to the thin (Euler-Bernoulli) vs. thick (Timoshenko) beams comparison.

     Typical thickness for thin shell is <5% whilst thick shell theory applies within the 5-10% range. Anything significantly >10% should not be modelled using plate theories.

A comparison of the differences between thin and thick shell theories.

Theory	Thin shell Kirchoff-Love	Thick shell Mindlin-Reissner
Thickness vs. percentage of in-plane dimensions	Thickness < 5%	5% < thickness < 10%
Key assumptions	• Plane remains plane • Normal remains normal • Thickness is not affected by deformation	• Plane remains plane
Degree of freedom per node	• Translations Ux, Uy, Uz • Rotations Rx, Ry, Rz	• Translations Ux, Uy, Uz • Rotations Rx, Ry, Rz
Transverse shear deformation	No	Yes

See also
Types of elements in the element library

Membrane, Plate and Shell Elements

A comparison of the differences between membrane, plate and shell elements.

Element	Membrane	Plate	Shell
Engineering components	Pressure vessels, oil tanks, ship hull, wing skin
Analogous to	Truss	Beam with no axial stiffness	Beam
Number of nodes	Usually three: One each at the top, middle and bottom
Displacement degree of freedom	• Translations Ux, Uy	• Translations Uz • Rotations Rx, Ry	• Translations Ux, Uy, Uz • Rotations Rx, Ry, Rz
Stress output	• In-plane stresses	• Bending stresses • Additional transverse shear stresses for thick plates	• In-plane stress • Bending stress • Additional transverse shear stresses for thick shells

Also read
Types of elements in the element library

Thin Beam vs. Thick Beam

Standard beam theory (Euler-Bernoulli bending theory) assumes no deformation by shear. This can be safely applied to thin beams that are long and slender. For short and deep beams however, displacement due to shear becomes hugely important which is included in the formulation of thick beam element. Below is an exaggerated example of shear deformation:

Displacement due to shear that standard beam theory does not take into account

     This effect is the key that distinguish between the Euler-Bernoulli and Timoshenko (thick beam theory) bending theories. See below for a direct comparison between thin and thick beams.

Table illustrating the differences between thin (Euler-Bernoulli) beams and thick (Timoshenk) beams. Diagram in "look" section is adapted from Wikipedia.

Also read
Types of elements in the element library

What are Isoparametric Elements?

The truth is, you are probably using isoparametric elements without noticing it. In classes, we might have been asked to derive stiffness equations for elements of simple shapes such as rectangles or cuboids. This is to favour hand-calculations using simple equations.

In the real world however, most objects take irregular shapes. A Jacobian mapping process is therefore required to accommodate for this shape irregularity. Non-isoparametric elements can only be implemented to regular shapes and use shape functions solely for the purpose of displacement interpolation. Isoparametric elements on the other hand can be used to model irregular shapes. They use shape functions not only for displacement interpolation, but also to represent the irregular element geometry. This means shape functions are now responsible for both the displacement interpolation and element shape. This also means that modelling a curved surface within one single element is now made possible.

A mesh of isoparametric elements

In practice therefore, most elements offered by FEA programs are isoparametric elements. They are so widely used that they are not often stated in the user manuals.

Straight Beam vs. Curved Beam

Table illustrating the differences between straight and curved beams

See also

Types of elements in the element library

Truss vs. Beam

Table illustrating the differences between truss and beam elements.

See also

Types of elements in the element library

Types of Elements in FEA

One of the requirements to become a good finite element analyst is to be aware of a range of standard elements that are best for specific applications. Whilst commercial FEA programs all offer a library of elements for users to choose from, it can be difficult to decide which to use due to the large amount of elements available. Luckily, these elements can be categorised into three classes by their shape:

FEA vs. FEM

Today when we hear about finite element method (FEM), the first thing that comes across our mind would be a beautiful picture like this:

Simulation result adapted from DTE Desktop Engineering

which is not entirely true.

What are Shape Functions?

Shape functions are polynomial functions that interpolate the discrete displacements into continuous functions. They are therefore also known as interpolation functions. The order of polynomial represents the maximal capability of the shape functions to model a displacement field within each element.

Practical decisions in choosing the order of shape functions require the best balance between accuracy and computational cost. Similar to stiffness and force matrices, shape functions are first defined locally and subsequently assembled into global shape functions.

What is Jacobian Mapping?

During the assembly of the global stiffness matrix (K), local stiffnesses are numerically integrated in the parent coordinate (the polynomial is a function of s). This is better illustrated by considering the following element:

which sits in a normalised system that varies between -1 and +1. This requires to be mapped to the actual element coordinate (change to a function of x):

This mapping requires the use of Jacobian matrix (J), which is commonly used for coordinate transformation in mathematics.

What is Gaussian Quadrature?

Gaussian quadrature is a method that replaces an integral by a sum. As an example, consider the following 1D integral:

Gaussian quadrature numerically integrates f(s) using a sum,

Steps to Derive k and Assemble into K

Figure 1 is a flowchart illustrating the sequence for computing the stiffness matrix of a simple problem (the concept is similar to more complex problems). Computing the stiffness matrix involves two main steps:

1. Derive local stiffness matrices (k)
2. Assemble k into K, the global stiffness matrix

Matrix Methods: Direct vs. Iterative

Direct vs. Iterative methods
The two approaches available for solving global stiffness matrix (K) in FEM are:

Implications on FEA solver

• For linear simulation e.g. KU = f , Gaussian elimination can be applied directly
• For nonlinear simulation e.g. K(u)U=f, stiffness is dependent on displacements (u). Therefore an iterative method must be used.

See also
The role of matrix solution in the FEM process

What is Meshing?

Meshing involves:

1. discretisation of a continuum into finite number of elements
2. defining element type (determined by shape functions)
3. nodal connectivity

A finite element mesh adapted from Dr TE Kendon's research page

Traditionally, meshing is performed by human user. Automatic meshing technologies are becoming more readily available and user friendly in FEA pre-processors as the competition in the FEA software market gets fierce than ever.

Also read

The role of meshing in FEM

Virtual Work and Weak Formulation

Weak formulations (a.k.a. variational formulations) of partial differential equations (PDEs) are hugely important in the FEM as they enable the concepts of linear algebra in the analysis of PDEs. This concept transform PDEs into sets of linear equations (a matrix) that can eventually be manipulated and inverted using standard matrix methods.

Transforming an equation from strong to weak form requires the use of virtual function, and hence the name Principle of Virtual Work.

Also read
A few worked examples (external pdf)
The role of the Principle of Virtual Work in FEM

Boundary Conditions (BCs) vs. Displacement BCs in FEM

This post aims to address the question that arises when one cannot distinguish between boundary conditions (BCs) and displacement BCs in the flowchart of FEM process.

Since FEM is all about solving the FE equation in matrix form, we approach this question using the classic FE equation of a linear elastic problem, KU = f. Let us assume it expands to look like Figure 1:

Figure 1: FE Equation of a linear elastic problem, KU = f

We can now interpret the difference between BCs and displacement BCs from the physical and mathematical perspectives:

Steps in FEM: An Overview

Figure 1 is a flowchart illustrating the FEM process for a linear static problem (the concept is similar to more complex problems):

Figure 1: A finite element method process for solving linear static problems (Click to enlarge)

Brief explaination of the different stages in Figure 1:

What is the Finite Element Method (FEM)?

In mathematics, partial differential equations (PDEs) can be solved either analytically (exact, continuous) or numerically (approximation, discrete).

Illustrating the position of FEM in the overall picture for solving PDEs

Finite element method (FEM) is a numerical method for solving PDEs. FEM is usually applied to solve for PDEs in continuum mechanics for solid mechanics problems.

Two other popular methods are the finite difference and finite volume methods (FDM FVM), which are frequently used to solve problems in fluid mechanics.

Partial Differential Equations in Continuum Mechanics

Continuum mechanics studies the mechanical behaviour of a material that has been mathematically idealised using partial differential equations (PDEs). Hence, continuum mechanics solves PDEs to understand material behaviours.

Why use idealisation?
It is not economic to fully describe material behaviours over a large range of conditions. Instead, the mathematical expression is idealised such that material behaviour is only represented for a reasonable range of loading conditions. See examples.

How to mathematically idealise/ model a material?
In continuum mechanics, materials are modelled using PDEs. These are equations assembled from (1) physical laws that are universal and (2) constitutive equations that are material specific.

PDEs in continuum mechanics = physical laws + constitutive equations

Examples of PDEs

• Navier equation
• Wave equation as special case of Navier equation
• Heat equation
• Laplace's equation as special case of heat equation
• Navier-Stokes equations

What is Continuum Mechanics?

After generating a continuum (by mathematically representing a real material), the mechanical behaviour of such continuum can then be studied. This is continuum mechanics - the study of materials' mechanical behaviour using mathematical models.

     Solid mechanics and fluid mechanics are two special cases of continuum mechanics.

Physical Laws vs. Constitutive Equations

See also

What are Constitutive Equations?

Physical Laws of Continuum Mechanics

What are Constitutive Equations?

Physical laws of continuum mechanics are valid for any continuum. Constitutive equations are mathematical relationships between kinetics and kinematics quantities of a specific continuum (recall that a continuum is just a mathematical idealisation of a material). In other words, constitutive equations describe the behaviour of a material subjected to certain loading conditions.

Examples
Consider two analyses:

1. With same loading conditions but using two different materials i.e. steel vs. aluminium
2. With same material but test at two different temperatures i.e. room temperature vs. 1000ºC

In the above examples, the constitutive equations are responsible for the the followings

1. Differing response for steel and aluminium under the same loading conditions
2. In practice, a material is only described over a range of conditions it is expected to encounter. Therefore, constitutive equations are formulated to describe different response of the same material over an acceptable range i.e. two different sets of equations for 10-30ºC and 900-1200ºC temperature range for the above analysis

Reynolds' Transport Theorem

The physical laws of continuum mechanics can be generalised using Reynolds' Transport Theorem as follows:

Physical Laws of Continuum Mechanics

Physical laws are empirical laws derived by repeated observation of physical phenomena. These laws established links between kinematics (of continua) and kinetics (of deformation) without considering the physical properties (such as elasticity, density, viscosity and thermal conductivity) of the continuum (of the material) itself. This means that they are valid for any continuum (any material).

     The most important physical laws that govern the mechanics of continua are:

1. Conservation of Mass - the mass of an isolated system will remain constant over time
2. Conservation of Momentum (Newton's Second Law) - rate of change of (linear/ angular) momentum is equal to the resultant (force/ moment) acting on the system
3. Conservation of Energy (First Law of Thermodynamics) - rate of change of energy is equal to the difference between rate of heat input and rate of work output

     Note that the above laws can be generalised using Reynolds' Transport Theorem.

Figure. An experiment illustrating the law of conservation of mass (adapted from Grade 8 EAP)

What is a Continuum?

In continuum mechanics, a continuum is an idealised mathematical representation of a real material such as solid, liquid or gas.

Kinetics vs. Kinematics

A summary of the differences between a kinematics and kinetics study.

	Kinematics	Kinetics
Study of	Motion (Deformation and flow)	Forces (Surface and body)
Considers	Motion disregarding the forces and moments that cause the motion.	Motion and the forces underlying this motion.

See also

What is Kinetics?

What is Kinematics?

What is Kinetics?

Whilst kinematics deals with the motion and deformation of a body, kinetics studies the forces that produce motion. The two types of forces are:

1. Body forces (per unit mass or volume) are non-contact forces that act on the total mass of a continuum. Examples include gravitational force, electromagnetic force and inertial force.
2. Surface forces (per unit area) are contact forces that act across an internal or external surface of a body. Examples include pressure, contact forces and frictional forces.

Figure. Magnetic (non-contact) vs. frictional (contact) forces. Images adapted from Newtown High School and Joey's Blog.

What is Kinematics?

Kinematics is the study of motion, disregarding the forces that cause the motion.

     In order to define the motion of a body (i.e. the deformation and flow of a solid or fluid body under external loading), a continuum can be broken down into multiple "particles" and the relative change of position of each particle is monitored (see Figure).

Figure. The kinematic quantities of a classical particle: mass (m), position (r), velocity (v), acceleration (a). Image adapted from Wikipedia.

What is Deformation and Flow?

Materials (or continua) are generally classified as solids or fluids. Their main difference is that when subjected to external loading:

• a solid body will undergo time-independent deformation, and
• a fluid body will flow as a function of time.

     Materials that have both fluids and solids characteristics also exist. These include shear thickening materials such as custard (see Video), shear thinning materials such as blood, and other materials with time-dependent "visco" characteristics such as elastomers.

Video. The shear thickening effect of corn starch mixed with water.