In this blog post, the basics of finite element method will be presented. This post is far from academic and aims to describe the basic idea behind FEM calculations in a digestable manner.

Content:

Introduction

How does it work?

Meshing

Material properties

Stiffness

External forces and boundary conditions

Convergence

Time steps and increments

Results

Evaluation

**Introduction**

Finite Element Method (FEM) is a numerical technique used to solve complex engineering problems by dividing them into smaller, more manageable parts called finite elements. FEM has become an integral part of modern engineering design and analysis, enabling engineers to simulate real-world scenarios with a high degree of accuracy. It is widely used in a variety of industries, such as aerospace, automotive, and civil engineering, to analyze the behavior of structures, fluids, and other physical phenomena. Through FEM, engineers can predict the performance of a design, optimize it, and identify potential issues before constructing physical prototypes. With its versatility and practicality, FEM has revolutionized the field of engineering, allowing for more efficient and cost-effective designs that meet the demands of modern society.

Simulations can be used among others for early design concepting, design optimizations and root-cause failure analysis.

Keep it in mind that the introduction below is a pure introduction and is written so that everyone independent of profession should understand the basic idea behind FEM, but entire academic books exist just about the individual chapters below!

**How does it work?**

I have got all kinds of comments that nice animation, beautiful rainbow, nice drawing but what are they actually?

As mentioned earlier, bodies are discretized into small elements and nodes. When an external force is applied to a part of the body the neighboring nodes start to move and the rest of the nodes will react accordingly based on the boundary conditions and material properties. Displacements, strains and stresses are calculated across nodes and elements throughout the entire component where one affects the other.

**Meshing**

The software splits the component into small elements, so called finite elements which is called discretization. These elements are connected at "nodes" and one element can contain several nodes depending on the type. Elements can be 1-2 and 3 dimensional depending on the simulated problem. One thing is common: nodes have degrees of freedom, and its a question of the dimension and type of the mesh to determine in how many directions and rotations the element is capable for movements. Typically, a 2D solid element can be a triangles or quadrilaterals in shape with either flat or curved surfaces. Each node of the element will have 2 translational degrees of freedom.

Deep-dive into the meshing techniques here: Guideline to discretization (simulaxengineering.com)

**Material properties**

Material properties define how a structure responds to external loadings and deformation. These properties include parameters such as Elastic modulus, poisonâ€™s ratio, yield strength and many others. These properties are acquired through physical testing, where a small piece of material is subjected to displacement by pulling it apart while measuring the reaction force.

Based on these parameters, the software can calculate the stiffness of the component. In combination with loads, the software can predict how the component will deform, how stresses will be distributed and whether the component will fail or not under the given loading condition.

**Stiffness**

Stiffness is the ability of an object to resist deformation when a stress is applied. It is a measure of how difficult it is to change the shape or size of an object. A highly stiff material or object will undergo minimal deformation, while a less stiff material will deform more under the same force. Stiffness is typically quantified by the stiffness coefficient or modulus of elasticity. In the field of materials science and engineering, the modulus of elasticity is a measure of a material's resistance to deformation and is expressed as the ratio of stress to strain. It represents the slope of the stress-strain curve within the elastic region of the material and is indicated with capital E. (see material graph)

It is a common approach to think about elements as if they were springs. The applied force {F} will cause displacement {x} which is contingent upon the stiffness of the material. 1D bar elements consist of two nodes, and each node is restricted to movement only in the lateral directions. This characteristic makes them the optimal choice for representation purposes.

The 2 schematic above aim to represent the connection between stiffness; elastic modulus, elements and springs. As the equation shows, stiffness [K] is a function of the elastic modulus (E) which is determined using material tests. Hopefully, establishing the connection between stiffness and material characterization made it more clear why materials are the heart of the simulation.

This is the simplest introduction to stiffness and equations become slightly more complicated for 2 or 3D elements:

**External forces and boundary conditions**

In the context of FEM, "loads" refer to external forces and boundary conditions. Material properties define the stiffness of the component and loads determine the behavior and response of the structure under different conditions. External forces and displacements can be applied to edges, surfaces or even volumes. Boundary conditions are generally used to support structures in any directions or fixing areas completely. Deep-dive into boundary conditions here: Understanding loads and Boundary conditions (simulaxengineering.com)

**Convergence**

Results are obtained and the simulation has been successful when the result is converged. Convergence is reached when the equations are satisfied in the background on every single node. The equation is satisfied, when the external and internal forces are in equilibrium or with other words, as long as the matrix [K] (stiffness matrix) can support the applied forces. As mentioned, this equation must be satisfied at every single node which is why the simulation iterates across all the nodes until it finds equilibrium everywhere.

Naturally, this simplistic explanation serves as a basic introduction. However, when dealing with non-linear iterative methods, simulations involving large deformations, or fast impact scenarios, the straightforward calculation approach quickly becomes cumbersome.

**Time steps and increments**

Simulations are successful when convergence is reached which can only be achieved if the loads are applied slow enough that equilibrium is found on each node. Once the load is applied fast, the structure is not able to accommodate to the sudden shape change and simulation will terminate. In order to slow down the process, the load is split into steps and increments and the software iterates through them.

**Results **

Once the simulation is said to be converged, it spits out a colorful plot. Colors are just used to visualize the result on the component, but all of these colors are actually assigned to a value. The colors can also represent many different physical phenomena such as displacement, temperature, stresses and strain and it only depends on what the engineer is evaluating at the moment. The more pronounced the color is (dark red/dark blue) the more extreme the values are in the positive or negative direction. Colors in between the extremes are distributed based on the status of the picture we are looking at.

**Evaluation **

The evaluation of the simulation result is the last step, where the resulting stress and strain values are compared to the material's limit to see whether it can survive the applied load or not. Does the stress or strain go beyond the limits of the material? The component might undergo unwanted permanent deformation (yielding) or even cracking, and the component will fail. Cracking is only visible when materials are characterized accordingly using advanced techniques. Otherwise, engineers have to rely on the colorful plots and numbers.

I like to say that the engineer's job practically starts at the evaluation as thatâ€™s where all his knowledge and experience is combined. Simulations don't provide a green or red light to indicate good-to-go or failure. The values that the colors represent are used to make decisions based on the engineer's best knowledge meanwhile the values depend on factors like mesh type, the chosen material characterization method, and the correct boundary conditions. Engineers therefore must be able to judge the correctness of the data and convert the plots into answers.

Now that you have an idea about what the simulation does, letâ€™s go into the sub-topics where you can get a more in-depth description of each.

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