A simple introduction to the Finite Element Method (FEM), how a Finite Element Analysis (FEA) workflow looks like and how it is used in the industry.
If you havenβt been hiding under a stone during your engineering studies, you should have heard about the Finite Element Method (FEM). And maybe you even learned some of its practical aspects by playing around with software packages such as ANSYS, ABAQUS or SimScale.
In this blog, I am going to introduce the Finite Element Method without any heavy mathematical explanations and it is by no means a course on FEM - this is reserved for future posts.
Generally speaking, the finite element method (FEM) is a numerical method used to perform a finite element analysis (FEA) of any given physical phenomenon to predict the behaviour of a structure.
The method was originally developed for engineering analysis to model and analyse complex systems in mechanical, civil, and aeronautical engineering. The basics of the method can be derived from Newton's laws of motion, conservation of mass and energy, and the laws of thermodynamics.
FEM can be used, for example, to determine the structural mechanics of different parts of a car under different loading conditions, the heat flow through engine part, or the distribution of electromagnetic radiation from an antenna.
An important aspect of FEM is how the Computer-aided design (CAD) model is prepared for the analysis and is being subdivided during meshing (discretisation into smaller elements).
CAD software such as Creo can be used to define 3D shapes of an object and then imported into a separate FEA tool which subdivides the object into appropriately sized elements according to the desired boundary conditions or mesh.
The description of nature and the laws of physics for space- and time-dependent problems are usually expressed with partial differential equations (PDEs). These equations are solved in an approximate manner by the finite element method (FEM) which is based on equations of classical methods such as the Theory of Elasticity.
PDEs are equations for an unknown function of two or more independent variables that involves partial derivatives. Below is an example of a PDE, namely the three-dimensional Laplace equation where \[ \phi \] is the dependent variable, and x, y, and z are the spatial independent variables.
In order to solve this equation, it must be subjected to so-called initial conditions and boundary conditions.
An Initial Boundary Value Problem (IBVP) consists of the Partial Differential Equation the Initial Conditions as well as the Boundary Conditions
π Initial Boundary Value Problem = PDE + ICs + BCs
Other examples of famous PDEs are:
Generally speaking, boundary conditions (BCs) are constraints necessary for the solution of a boundary value problem.
A boundary value problem is a differential equation (or system of differential equations) to be solved in a domain on whose boundary a set of conditions is known. Compared to the "initial value problem", in which only the conditions on one extreme of the interval are known. Boundary value problems are extremely important as they model a vast amount of phenomena and applications, from solid mechanics to heat transfer, from fluid mechanics to acoustic diffusion. They arise naturally in every problem based on a differential equation to be solved in space, while initial value problems usually refer to problems to be solved in time.
Generally speaking, boundary conditions for continuous systems are classified as being of two types.
Those BCs also known as Kinematic Boundary Conditions must be satisfied according to geometric constraints. For example:
\[ u(x=0) = 0 \quad \text \quad u'(x=0) = 0 \]
at a fixed end of a cantilever.
Those BCs also known as Static Boundary Conditions must be satisfied as a result of free and moment balances. For example:
is a moment boundary condition at the free end of a cantilever. So, it is prescribed by forces and moments.
From a mathematical point of view, the following five steps are essential to know in order to understand how the FEM is working behind the scenes.
Step #1
Pre-Processing, also called model preparation, is usually the most labor-intensive step of FEA.
Step #2
Element formulation π development of equations for elements
Set up the partial differential equation in its weak form
Step #3
Assembly: Set up global problem π obtaining equations for the entire system from the equations for one element
Step #4
Solving system of linear equations
Step #5
Post-Processing: determining quantities of interest, such as stresses and strains and obtaining visualisations of the response
Hint: The solver may produce impressive, colourful results which may look convincing but can be completely useless. Try to compare your solution with analytical results (if available), perform a proper convergence study and/or compare your results with existing papers / literature.
There are obviously more points to check when validating and correlating your FEA results which will be covered in another post. For now you only have to know not to trust your results and you have to employ methods to go through a meticulous verification and validation process for every analysis.
In the section on FEA Workflow, we will have a look at a more granular level of how FEA works. In practise, you are neither really busy thinking about how the matrix is assembled, nor do you care how the system of equations are solved which is just a bunch of number crunching behind the scenes.
A characteristic feature of the finite element method is that instead of seeking the approximation over the entire region, the region is divided into smaller parts, so called finite elements and the approximation is then carried out over each element.
The collection of all small parts is called the finite elements.
When the type of approximation has been chosen (is to be applied over each element), the corresponding behaviour of each element can then be determined. Having determined the behaviour of all single elements, the elements can then be patched together (Matrix Assembly), which enables us to get an approximate solution of the entire body over the entire domain.
As we mentioned before, we need to approximate a solution over our elements in order to determine the behaviour. This approximation is usually a polynomial and is, in fact, some interpolation over the element. This means we know some values at certain points within the element but not at every point. These βcertain pointsβ are called nodal points and are often located at the boundary of the element. The accuracy with which the variable changes is expressed by the approximation, which can be linear, quadratic, cubic et cetera.
In order to get a better understanding of approximation techniques, we will look at a one-dimensional bar. Consider the true temperature distribution T(x) along the bar in the picture below.
Equidistant grids can waste many nodes in areas where the solution is not important or changes in quantities (gradients) are insignificant.
The nodal points we chose do not need to be equally spaced! We now divide our bar into four elements over which the temperature is assumed to vary in a linear manner between each nodal point!
Well we see linear approximation is quite good, but we can certainly do better! If we chose a quadratic approximation the temperature distribution along the bar is way more smooth.
Nevertheless, we see that irrespective of the polynomial degree the distribution over the fin is known once we know the values at the nodal points, simple as that. If we just have one bar we would have an infinite amount of unknowns (Degree of Freedom (DoF)). But in this case we have a problem with a finite number of unknowns.
A system with a finite number of unknowns is called a discrete system
A system with an infinite number of unknowns is called a continuous system
In our case we would have a system consisting of five equations which can be solved by hand but in general we have thousands and thousands of unknowns. Luckily we have powerful machines at our fingertips allowing us to solve such problems for us which makes life way easier.
In general, the FEA workflow can be broken down into
Pre-processing is performed to create the model, generate an appropriate finite element grid, apply the appropriate boundary conditions, and view the total model.
Processing refers to the number crunching part behind the scenes.
Post-processing provides visualisation of the computed results and gives meaning to the hundreds of numbers that could otherwise not be interpreted by any human being.
This is the part where you import your CAD model. Ideally, all redundant parts of the part that do not contribute to the analysis or require excessive model setup have been removed
CAD modeling is used by many designers to create elaborate computerized models of objects before they are physically produced.
In general, there are two ways on how to create or import CAD models, depending on which FEA software you're using.
The definition of material and boundary conditions (BCs) can usually be captured at once .
Material nonlinearity involves the nonlinear behaviours of material based on current deformation, temperature, pressure and so on. Some nonlinear material models are large strain, stress-strain relationship, elastoplasticity, plasticity, creep and hyperelasticity.
As a beginner, you can be quite sure that you will mess up boundary conditions very often as you not only have to understand how BCs can be used in your model but also how the software is treating each BC.
Contact is one type of nonlinearity of the system. An abrupt change in stiffness may occur when bodies come into or out of contact with each other. This is a result of the changing nature of the contact between components in the analysis during motion.
The following questions have to be answered when meshing:
Often, critical stresses are limited to small areas of a model. When a global mesh refinement is used, the resulting mesh is often too fine in areas that are not relevant for the evaluation process.
A more efficient approach is to refine the mesh locally where critical stress regions are located. FEA tools often allow you to use Local Mesh Refinements that provide a targeted mesh refinement.
The solver is now doing a bunch of number crunching on a system of equations behind the scenes of the FE model. Systems of equations with more than 10 million equations are not uncommon but also not unsolvable considering how fast and efficient modern codes and machines are.
The first result of includes the displacements of the individual nodes. In subsequent steps, distortions, stresses as well as nodal forces can be calculated. In addition to mechanical quantities, thermal, electrical or magnetic quantities can also be calculated if a thermo-mechanical or electromagnetic solver is used.
An equation that you should know about is
which is the standard equation for the FEM where
You will see in another blog post, that if we only have a look at a linear spring element to derive the fundamental equation for the FEM, the rows of the stiffness matrix are linearly dependent which means that the system is statically underdetermined!
We have to integrate boundary conditions into the system, otherwise our solver will end up with an error.
Whenever we apply external forces and calculate displacements, this is called a Displacement-based procedure
Whenever we apply displacements and calculate the forces, this is called a Force-based procedure
Post-Processing is the part where the numbers from the solver are taken and made interpretable for the engineers - otherwise we would have a hard time decoding what all those numbers actually mean.
It is important to know it is not sufficient to simply accept the beautiful images that the machine throws at you. You need to take your engineering knowledge and intuition and really think hard if the results actually make sense and are physically accurate.
GIGO = Garbage In equals Garbage Out
Simple doing one analysis and looking at the plots won't do the job. FEA is an iterative process which requires an engineer to make adaptations to the model, fix boundary conditions and potentially perform several meshes to see if there are no significant changes in stresses or other relevant output quantities the finer we discretise our domain.
If no convergence can be seen, this can be an issue of a so-called singularity.
A finite element model will sometimes contain a so-called singularities which means that there are points in your model where values (such as stresses) tend towards an infinite value.
In the worst case, you have to go back to Step 1 and adapt your CAD model.
FEA can be applied to different types of single or multiphysics problems involving heat transfer, fluid dynamics, electric fields and more. Three main types of problems are:
Over the years, more "exotic" topics in FEA have emerged:
Although FEM is an incredibly powerful method, there are however some drawbacks.
Various applications of the FEM prove that this technique is of great value as it provides a means with which engineers can create efficient designs for large and complex problems, which cannot be solved using analytical methods.
The FEA process incorporates many complex steps and involves numerous parameters which can be tweaked by the user. Thanks to the democratisation of FEA, software is much more streamlined, easy to integrate into engineering R&D processes and much more affordable to every engineer, designer and smaller enterprises.
In the future, the use and availability of cloud-enabled services such as SimScale will be intensified during design projects. The past and present of mathematical modelling and FEA software is a success story which will be enhanced using Artificial Intelligence (AI) marking a significant step in FEA history.
Not only will engineers have to do less effort, but the analyses will be more accurate and faster augmenting the whole product development process from conception to prototyping. This allows engineers to be more creative in the design process and focus less on tuning sensitive FE parameters.
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