What is this article about?
The finite element method (FEM) is one of the most important tools for assessing and optimizing components and entire structures today. Its importance ranges from the automotive industry to aerospace engineering and mechanical engineering, where it has become indispensable in the dimensioning and analysis of components. With its help, even highly complex structures can be calculated efficiently and precisely.
Despite its performance, FEM also brings challenges. Its application requires in-depth specialist knowledge, as the quality of the results depends largely on the correct definition of the boundary conditions, the material properties and the model assumptions. In practice, however, the method is often used as a “black box” because modern software greatly simplifies the execution of calculations. This means that
the necessary expert knowledge is not always fully available – a risk that should not be underestimated.
This article offers a compact insight into the basics of the finite element method and aims to make this fascinating technique understandable for those with a technical interest. From the principles of discretization to modeling and analysis of the results, it shows how the FEM works and why it is indispensable for modern engineering practice. Part 2 provides further information on the possible errors in the analysis.
A brief summary of the procedure
What does FEM / Finite Element Method mean?
Figure 1 shows a structure that had failed in reality and Figure 1 shows the pictorial representation of the principal stresses in the affected cross-section determined using the Finite Element Method (FEM).

When comparing Figure 1 and Figure 1, it is immediately apparent that significant stress increases were also identified in the analysis in the fractured area of the structure. With correct application of the FEM, known external loads, expert model creation and subsequent analysis and evaluation, the FEM allows such problems to be identified in advance.

Review / History of the FEM / Finite Element Method
The history of the finite element method was first introduced and implemented by NASA in the 1960s. In the 1970s, the first analyses began with very simplified calculation models in the aerospace and automotive industries. Until the 1990s, only mainframe computers were required for this and the method was used almost exclusively in the automotive and aerospace industries.
For other industries, calculations on mainframe computers were simply too expensive and specialists were not available on the open market. It was therefore necessary to check very carefully what was to be analyzed, because it cost money.
Since the mid-1990s, affordable and increasingly faster computers have made FEM applicable to many areas of industry.
Software manufacturers took advantage of this boom by promising that almost everything would be predictable and that anyone could use the programs – a message that continues to shape the industry to this day.
Aim and purpose of FEM simulation
The FEM (Finite Element Method) is a numerical method that plays a central role in mechanical engineering. Its aim is to solve complex physical problems that occur in technical components and systems using computer simulations.
The finite element method is an indispensable tool for mechanical engineers. It enables detailed and precise analysis of components and systems, which leads to more efficient product development and higher quality when this method is applied correctly.
FEM calculation in mechanical engineering in four steps
The FEM calculation in mechanical engineering follows four central steps that systematically provide numerical approximate solutions:
- Model preparation: The component is broken down into small “finite elements”, e.g. hexahedrons or tetrahedrons. The mesh fineness has a decisive influence on accuracy and computing time.
- Parameter definition: Material properties, external loads and boundary conditions are specified. These determine the behavior of the model under load.
- Calculation: The FEM software solves a complex system of equations that describes all interactions between nodes and elements. Iterative methods are used for nonlinear problems.
- Postprocessing: Results such as stress distributions, deformations or temperature fields are evaluated graphically and numerically. Critical areas are analyzed and plausibility checked.
model creation
The structure to be assessed is divided into a finite number of small and simple geometric elements (“finite elements”) that are connected to each other at their nodes (Figure 2). This is based on a complex mathematical system of equations. Once “boundary conditions” have been defined, the deformations and, from this, the strains and stresses can be determined.

What is essential is:
Any deformable body is broken down into individual, finite parts, which are called “finite elements”.
These finite elements are connected to each other at the so-called nodes (Figure 4). The decisive, continuously changing quantities of the problem are represented by their values at the nodes. Continuous functions are replaced by a system of a finite number of node parameters. The solution of the individual parts ultimately results in the approximate solution for the idealized overall structure.

For different problems, there are different elements, for whose node displacement mathematical approach functions are defined that influence the result. Anyone who is not familiar with this as a “software user” cannot even
not objective
evaluate.
influence of the elements
A distinction is made between one-dimensional, two-dimensional and three-dimensional elements.
One-dimensional elements are beams and rods, which can often only be used for very simplified questions.
“ Thin ” structures are represented with two-dimensional shell elements. We speak of “ structurally thin ” when one dimension is much smaller than the other two dimensions. The hood of a car, for example, is a classic structurally “thin” structure.
Structures that must be captured using volume elements include bearing blocks, rollers, engine housings, etc.

Figure 4 shows common elements for thin structures (2D) and for volumes (3D).
All elements in Figure 4 are linear elements. There are also elements with higher approaches. The differences are discussed in Part 2 of the topic.
As can be seen in particular in Figure 2 ” FEM mesh of a washing drum “, in reality it is by no means the case that the elements used have an identical regular boundary. The elements are adapted to the contours of the structure of the component to be assessed (“construction”). As a result, flat quadrilaterals sometimes degenerate into a quadrilateral with different edge lengths, a rhombus or a curved structure if all four nodes do not lie exactly in one plane.
This affects the analysis result and the accuracy of the numerical solution.
Networking and its influence
With a finer discretization (mesh fineness), the approximate solution theoretically converges to the (exact) solution, which, however, is based (only) on analytical approaches, which also represent approximate solutions.

However, a finer mesh with lower-quality tetrahedral elements does not provide “more accurate” results than a coarser mesh with higher-quality elements.
The precise knowledge of the elements and, in addition, a technically high-quality discretization (networking) provide the necessary basis for an analysis result with good information quality.

In principle, it is possible to visualize stresses and displacements in a structure while taking into account bearing boundary conditions and external loads. This is shown in Figure 6 for the model in Figure 5. It should never be forgotten that these are approximate solutions.
Important:
Often an optimal element division can only be achieved through appropriate experience!
It is precisely this fact that creates great potential for error when users work with this finite element method without having detailed knowledge of the underlying technical mechanics, mathematics, physics and materials science.
Determining the stress on components and structures using FEM
At this point, we would like to briefly explain to laypeople how the risk of mechanical damage to components can be checked.

Three
necessary criteria
are important for the assessment (Figure 7):
- First of all, any form of mechanical damage requires that external loads (forces, pressures, temperatures, etc.) act on the component.
- The external load results in stresses in the structure. These must be known or determined. The finite element method ( FEM ) is suitable for this. This is marked with the number 2 in Figure 7.
- If these internal stresses are known, they can be compared with the mechanical properties of the material used. If the corresponding stresses occurring in the material are greater than or equal to the mechanical properties of the material used, the component can fail due to different types of failure depending on the type of load (static, dynamic).
What is essential, however, is that the external load is known and, above all, what results from it in the component. This in turn requires that the other boundary conditions are also known for an analysis. With the finite element method, bearing boundary conditions must be selected in order to obtain a result.
Important:
The external loads and the support boundary conditions also influence the result of the finite element method.
areas of application in mechanical engineering
The finite element method (FEM) is indispensable in mechanical engineering and is used in numerous classic application areas, including:
- structural mechanics
- Investigation of stresses, strains and deformations in components and assemblies.
- Examples: supporting structures, frames, housings, connecting elements.
- Structural dynamics
- Analysis of vibrations, natural frequencies and resonances.
- Examples: crankshafts, turbines, gearboxes, machine foundations.
- Thermal analysis
- Investigation of temperature distributions and heat flows.
- Examples: heat sinks, heat exchangers, cylinder heads.
- fluid mechanics (CFD coupled with FEM)
- Analysis of fluid-structure interactions.
- Examples: pumps, valves, rotor blades, aerodynamics.
- crash analyses
- Simulation of shock and impact scenarios.
- Examples: protective devices, dampers, safety components.
- life and fatigue analyses
- Calculation of component lifetime under cyclic loading.
- Examples: gears, bearings, welds.
- optimization of components
- Topology and shape optimization to reduce weight or improve mechanical properties.
- Examples: lightweight components, supporting structures.
This diversity makes FEM an indispensable tool in mechanical engineering, used from product development to fault diagnosis.
However, these advantages can only be exploited if the calculation engineer has extensive experience and, above all, detailed knowledge of the finite element method.
Example of the influence of tetrahedral elements and hexahedral elements in modeling
The base plate of an industrial press was chosen as an example. This simple structure can be represented using hexahedral elements as well as tetrahedral elements.
The load acts centrally on a circular surface. Only a quarter model was considered. With symmetrical loads and geometric symmetry, identical results are achieved with reduced models and thus less analysis effort.

Figure 8 shows the comparison of the determined displacements for one and the same structure under identical loading for tetrahedral elements (3D “right” in Figure 4) and for the hexahedral elements (3D “left” in Figure 4) with identical scaling of the results. The difference in the determined displacements is 22.9%. The structure is simply calculated to be “stiffer” using the tetrahedral elements.

Figure 9 shows the same comparison, but for the determined Von Mises equivalent stresses with identical scaling.
As expected, the differences in the voltages are larger. This is due to the follow-up calculation. Further possible errors are discussed in part 2 of the article.
Hexahedral elements calculate larger displacements (22.9%) and also higher stresses (25.0%) under otherwise identical conditions.
These are crucial differences.
This must be kept in mind when conducting analyses and/or receiving results.
Some error possibilities as (negative) multipliers
Facts and possible errors discussed in advance include:
- Only numerical approximate solutions are determined.
- The approach functions refer to analytical model approaches of (linear) engineering mechanics.
- The task influences the elements to be applied.
- The element type influences the quality of the results.
- Network quality and network fineness are reflected in the result of the analysis.
- Linear calculation across Hooke’s line is technically incorrect.
- Stress results are always less accurate than the determined displacements.
- Results at storage boundary conditions are never correct.
These potential errors influence the final result in multiple ways and are therefore negative multipliers. And that is precisely why you have to be very critical when confronted with analysis results.
What is strange, however, is that errors in the analyses lead in most cases to “high” stresses. If something like this is determined incorrectly, measures are taken to reduce the “high” stresses. If everything goes “well”, you end up with an oversized structure. In reality, nothing breaks, despite the incorrect result.
However, things become critical when precise dimensioning becomes necessary.
This often leads to failure in reality. When computational analyses are increasingly presented in legal disputes, the results are often “calculated well”. It is extremely difficult to check the plausibility of such analyses if the analysis data is not made available. However, as an expert, you can then use the result plots to identify initial indications as to whether there are any indications of errors. But only experts can recognize this. There are few of them, and in the field of expert witnesses, perhaps a proverbial handful.
basic requirements
An engineering degree with in-depth knowledge of mathematics, technical mechanics and physics is necessary. In my opinion, the focus should be on technical mechanics, strength of materials and vibration technology. It is also helpful and self-evident,
Lectures in the Finite Element Method
visited and understood. This in turn requires
sound knowledge of technical mechanics
Unfortunately, technical mechanics is a subject that is not very popular among students because it is considered “difficult”.
If you meet the above requirements, you also need
practical experience.
Results must always be critically questioned, ideally discussed with other true experts and research carried out. The exchange of information is essential.
The technical skills of a calculation engineer include modeling, analysis and evaluation, documentation, presentation and critical discussion of the results. Experience is gained in this process, which over the years leads to an ever-increasing wealth of experience, so that at the end of the day, the expert is actually there. Calculation engineers are engineers who require high technical and personal demands. Unfortunately, this is often not appreciated, because it is not usually the technical specialist who makes a career in industry.
Conclusion
The finite element method (FEM) is one of the most important tools in mechanical engineering and many other engineering disciplines today. Its ability to simulate highly complex physical phenomena precisely and efficiently has revolutionized the way engineers develop, analyze and optimize components today. From aerospace to automotive to mechanical engineering, FEM has helped to drive innovation, increase safety and use resources more efficiently.
But the power of FEM also brings with it great responsibility. Although modern software solutions have made the method accessible to a wider group of users, the
Danger that essential fundamentals are neglected.
Correct application requires deep expertise in areas such as engineering mechanics, physics, materials science and numerical mathematics. Errors in the definition of boundary conditions, material modelling or discretisation of the structure can significantly affect the quality of the results and, in the worst case, lead to wrong decisions.
The FEM is by no means a sure-fire success. Its results are
always approximations that must be carefully checked and interpreted.
This means that calculation engineers not only have sound technical knowledge, but must also be able to question critically and communicate their results clearly and comprehensibly. This requirement makes FEM more than just a technical tool – it requires a
high level of responsibility and expertise.
The future of FEM lies in its further integration into interdisciplinary development processes. Advances in computing power and the linking with other simulation methods, such as CFD (Computational Fluid Dynamics), will further expand the possible applications. Nevertheless, the quality of the results always remains linked to the skills of the users.
In summary, it can be said that FEM is an indispensable tool for modern engineering practice. However, only through critical and well-founded use of the method can its possibilities be fully exploited and errors avoided. It is not just a means to an end, but a key to innovation and progress.
provided it is applied with the necessary care and expertise.
outlook
Part 2 of this article will discuss the possible errors in application and use in more detail.
Recommendation:
Only have technical calculations using the finite element method (FEM) carried out by professionals. If you have suffered damage due to an incorrect calculation, hire an expert experienced in FEM to carry out the investigation.