Applications of Finite Element Methods (FEM)

In this article, we will explore various applications of FEM in the field of Mechanical Engineering.

Structural Analysis

In structural analysis, FEM is used to compute displacement, strain and stresses in the structure subject to external loads. A structural problem can categories based on structural geometry, material properties used and involvement of time.

• 1D, 2D & 3D

Based on the geometry, there can be 1D, 2D and 3D structural analysis problems. All structures are three dimensional in real world. However, we model them as 1D if its 1 dimension is relatively larger than other two dimension. For example, bar or truss structures are often modelled as 1D structures. Similarly, the plates and shell structures subjected to plane stress and plane strain conditions can be conveniently model as 2D structures. When, all three dimensions of a structure are comparable to each other, then, we model them as 3D structures. Also, we investigates the complex 3D stress or strain state in a structure model by modelling it as 3D structural problem. The computational cost increases from 1D to 3D structural analysis.

• Linear vs Nonlinear

Based on the material properties used, a structural analysis can be considered as linear or nonlinear one. If the material properties are within the proportional limit, it will be considered as a linear structural problem or nonlinear otherwise. Generally, we design our structure to withstand loads below its yield strength. In other words, such structures are designed to operate within the proportional limit of the materials, consequently, requiring linear structural analysis.

• Static vs Dynamic

A structural problem can be static if the state variables do not change with time, otherwise, it will be a dynamic problem. For static structure problems, only spatial discretization of the domain is required, however, for solving the dynamic problems, there is need of both spatial and temporal discretization. The temporal (time) discretization is needed to evaluate how the response of structure evolves over the passage of time. The dynamic problems are frequently solved using direct integration methods e.g. central difference method, Newmark method etc. Another method for solving linear dynamic problems is the modal method. It should be noted that the dynamic problems involves the terms of inertial force, damping force, elastic force and applied force. In case of static problems, the inertial and damping force terms become zero, thus, the static problems becomes a special case of dynamic problems that involve elastic force and applied force terms only.

For dynamic problems, the global stiffness matrix is obtained by combining the element stiffness matrices. In a similar fashion, the global mass matrix is built from the element mass matrices. However, the construction of global damping matrix is done differently. Rather than building it, we often express the global damping matrix in terms of global stiffness matrix and global mass matrix. This is one way of driving the global damping matrix, and it is named ‘proportional damping model’. It is frequently used because it is convenient as compared to other damping models (i.e. viscous damping model etc.).

Here is an example of how to drive the global mass matrix from the element mass matrix shown below. Similarly, how to express the global damping matrix in terms of global stiffness matrix and global mass matrix and how to find out the relevant constants from the vibration graph representing the relationship between the excitation frequency and the damping ratio of the structure.

Modal Analysis

This analysis is carried out to compute the natural frequencies and the mode shapes of the structure, which plays very important role to design the structures. If the frequency of an external source becomes close or equal the natural frequency of a structure, then, resonance can occur. The resonance phenomenon causes high amplitude vibration that may lead to catastrophic failure.

The modal analysis can be performed by setting assumptions (i.e. free vibration and zero damping) on the equation of motion. Free vibration means that we want computed the natural frequencies of structure by neglecting the excited force term. The damping term is set zero to conveniently solve the problem using the special mathematics technique i.e. Eigenvalue problem. It should be noted the damping force just reduces the amplitude of vibration and effects on the structure frequency can be neglected.

Thermal Analysis

It is used to compute the heat transfer problems. Heat transfer can be through conduction, convection and/or radiation. The thermal problems are solved in a similar fashion as that of the structure problems. For example, for 1D heat conduction problem can be expressed by Fourier equations, which then can be solved using FEM by building a set of simultaneous algebraic equations expressed in matrix form as shown below. The thermal problems can be of steady-state or dynamic nature. For steady-state problems, only spatial discretization is needed, however, for dynamic problems, both spatial as well as temporal discretization is carried out.

Coupled Problem Analysis

All physical problems are coupled in nature. We often decoupled or simplify those using appropriate assumptions so that we can conveniently solve them. However, when and where required, the FEM can also be applied to solve coupled problems. Coupled problems are generally complicated and require relative more computational resource. For example, coupled structural-thermal analysis is often carried to analyze the structures subjected to mechanical as well as temperature loads. For this kind of coupled problem requires both mechanical and thermal properties of the structure materials as inputs. Similarly, the FEM can also be deployed to solve fluid-structure interaction (FSI) problems. Investigation of the interaction between the structural dynamics and fluid mechanics at the interface is not only intricate to solve but also consumes a lot of computation resource. 

The coupled problem, in nutshell, require multidisciplinary knowledge and their solution can be challenging due to the presence of strong nonlinearity.

Developing Finite Element Mathematical Model (Formal Approach) - Part 2

In this Part-2 of previous article, we will be covering new concepts of Jacobian matrix and Gauss-Legendre Quadrature numerical integration method.

In Part-1, we learn how to derive the element mathematical model starting from the choice of shape functions expressed in the natural coordinate system (CS). In case of structure problems, we apply the minimization of potential energy principle to drive the element stiffness matrix presented in the integral form. For 1D bar element, it was easy to solve this integral manually, however, for complex problems we often make use of numerical integration methods.

One of the various ways to solve the integral form of the element stiffness matrix is to deploy Gauss-Legendre numerical integration method, as discuss in the following:

Gauss-Legendre (GL) numerical integration

GL numerical integration method is accurate and widely used in FEM numerical computation. The general form of this GL method is shown below. In this method, the integral form of a function is computed by summing up all terms, each obtained by multiplying a weight with function value, estimated at various integration points within the domain. Since, the method requires computation at few pre-determined integration points within the domain, rather than the integration over the entire domain, therefore, its computation is fast. The accuracy is dependent on the selected integration points selected from a Table. The more are number of integration points, higher will be the accuracy of the integration results.

It should be noted that the GL method can be applied to the functions spanning over the domain ranging [-1, 1] only. In other words, we need to express every function in the natural CS which are normalized, in order to numerical integrate them using this GL method.

This can further explained with the following example where a comparison between manual integration and GL numerical integration methods. In case of GL method, 02 integration points are selected with in the normalized domain [1, -1], but still the computed results are very close to that obtained from manual integration.

Jacobian Matrix

The Jacobin matrix can be considered as a transformation matrix which translate the given information between the physical spaces expressed in local/element coordinate system, and the natural space which is normalized space expressed in natural coordinate system. Below Figure below shows the Jacobian matrix linking the physical and natural spaces for a 2D element.

In FEM, the use of Jacobian matrix is linked the GL integration method. Whenever, we need to integrate a function expressed in local/element coordinate system, first we need to transform it in the natural CS so that we can make use of GL integration method. For transformation between the local and natural CSs, the Jacobin matrix comes in action. It should be well noted that for complex engineering problems, the element geometry in local CS is often irregular which makes it difficult to perform the integration in the local CS. That is the reason, we map the physical geometry space to an abstract/mathematical space defined using natural CS which is not only normalized but also regular, making it convenient to perform numerical integration. Here for 2D element, we can see how a Jacobian matrix looks like.

Now further expand the integration of stiffness matrix of a bar element from 1D to 3D. It will look like this:

Let’s summarize the all concepts related to developing of element mathematical model. We begin with shape functions using natural CS to express the state variables with the element domain. The natural CS is used because we need to perform Gauss-Legendre numerical integration in the later staged. After executing various mathematical steps, and applying the minimization of potential energy principle, we derive an integral form of the element stiffness matrix which we developed previously using simplified direct method. The Jacobian matrix comes in action, when we need to map state variables in physical space to the natural space. This transformation makes it convenient for us to evaluate state variable which might be challenging to compute otherwise in the physical space due to its complex geometry. For details, see below Figure.

One last thing that we must keep in mind, that the element stiffness matrix is expressed in the local/element coordinates. The evaluation of the integral form of the stiffness matrix is done in the natural coordinate system is carried out for computation convenience only. For mapping between local/element CS and natural CS is carried out using Jacobian matrix. Once, the element stiffness matrix is computed in the local CS, then, it needs to be transformed according global/system CS, before the element assembly process begins. For transformation from local CS to global CS, we require a transformation matrix which we already covered in our previous articles. Here is the complete table describing the summary of transformations/mappings.

Developing Finite Element Mathematical Model (Formal Approach) - Part 1

 In this article, we will be covering how to drive the element mathematical model using formal approach that involves the concepts of shape functions, natural coordinates, minimization of potential energy and numerical integration using Gauss-Legendre quadrature method. The article has 2 parts. Part-1 covers the how to drive element stiffness matrix. Part-2, coming next, will be focused on how to make use of numerical integration to evaluate the entries of element stiffness matrix.

Let’s dig out the formal approach by considering a simple 1D bar element with 2 displacement dofs, as show in Figure 1. There are 03 types of coordinate systems (CS) to define the element mathematical model: global CS, local/element CS and natural CS. The global CS is used for element assembly process. The local CS is used to define the element stiffness matrix, and the natural CS is used to conveniently solve for the entries of element stiffness matrix etc. The concept of using natural CS will be further explained in the following text.

In case of formal approach, we begin with choosing with a displacement function that is expressed in terms of nodal values. Such functions are named as shape functions as they define the distribution of displacement field within the element domain by interpolating nodal values. The number of shape functions depends on the number of element nodes and associated dofs. For example, 2 shape functions are needed for 1D bar element with 2 nodes, each having 1 displacement dof.

Pay attention to the fact that:

·      Shape function are expressed in terms of natural coordinate system (CS), which are normalized in nature as they cover the domain space within range [-1, 1]. Why is there need of natural CS? It is needed to compute the entries of element stiffness matrix conveniently using numerical integration method.

·    Shape functions are not arbitrary selected. They must fulfill the 02 criteria which are: (1) the value of shape function at a given node is 1 and 0 at other node(s), and (2) the sum of shape functions at any point with the element domain must be equal to 1.

Here is demonstration of how to choose shape functions and recheck them.

One the choice of shape functions is made, then, we express the displacement and strain fields within elements in terms of shape functions and natural coordinates, present them in matrix form as following.

Now, to drive the element stiffness matrix, we can deploy various methods i.e. weighted residual, minimization of potential energy etc. The minimization of potential energy principle is very commonly used in the field of engineering structures, and the same we will be using here. Using this principle, we equate the work done on the structure with the corresponding strain energy stored in the structure. After doing tedious mathematics, we can easily derive the element stiffness matrix as following.

For 1D bar element with 2 nodes, we solve this integral to get the element stiffness matrix. Also, we can nodal force and displacement vectors, as following.

Developing Finite Element Mathematical Model (Direct Method)

 In this article, we will be discussing how to develop element mathematical model from scratch using the direct method. The development of an element mathematical model is a crucial task, and it belongs to step # 2 of the procedure, requiring to perform 6 steps in total, for solving problems using FEM. For further details on FEM procedure, see previous articles on this topic.

When we are using a commercial FEM software package, we choose an appropriate element type from the element library of the software depending on the nature of our problem. Being end users, we do not need to derive and implement the element mathematical model. We just need to use these ready-made elements. However, using there elements without knowing how they are developed, is not that interesting. Is not it?

Let assume that we may not have access to costly commercial software packages, or sometimes we want to solve a problem and there is no suitable element available in the element library of an open-source or commercial packages, then, what will we do? The answer is that we simply drive the element mathematical model, implement it and solve our problem.

The development of element mathematical modelling can be broadly done either by direct formulation method or using formal approach. The direct method is based on intuition and is suitable for solving simple problems. On the contrary, the formal approach requires intensive involvement of mathematics and is suitable for complex problems. Moreover, the formal approach is attributed to various methods like the weighted residuals methods (e.g. Galerkin method etc.), and the variational methods involving the calculus of variation and the minimization of potential energy (e.g. the Rayleigh-Ritz method etc.).

Here we will be deriving the mathematical model of 1D bar/spring element having 02 nodes and 1 dof per node, using direct stiffness method. It should be kept in mind that for understanding of FEM, the development of element mathematical model using direct formulation is a good starting point.

Consider an axially loaded elastic bar member as shown in Figure 1a. The governing differential equation for the bar element is as following:

In case of FEM, we need to discretize the continuous do

main of axially load bar structure member. The discretized element having 2 nodes, where each node represent 1 nodal displacement dof, subject to axial nodal forces, is shown in Figure 1b.

Let’s express continuous displacement in terms of nodal displacements of discretized element. Applying force equilibrium at each node, we can drive force equation, as shown below. Finally, expressing the force equations in the matrix form, which is the element stiffness matrix we are looking for. How simple force equation is.

The above element stiffness matrix will be assigned to the discretized domain, followed by the element assembly process and other steps involved in performing FEA to solve the problem.