### From Mathematics to Computer Science

Finite Element Analysis (FEA) is a discipline where mathematics, physics, engineering and computer science meet. The Finite Element Method (FEM) used to solve a FEA problem is always based on the three following steps:

*Pre-processing*where the finite element model is defined: meshing, properties, boundary conditions, loading, solution setting*Solving*where the numerical solution is computed*Post-processing*where the user reads the results and check the validity of the solution

Phases 1 & 3 are generally performed using an interface software that visualizes the mathematical models. In phase 2, a solver composed of several algorithms compute the solution corresponding to the problem submitted by the analyst. This step consists in solving matrix equations for unknown nodal values which approximate the continuous solution.

### The Magic of Discretization

The continuous solution fields such as stress, displacement, temperature, pressure… are approximated using a discrete model composed of a set of piecewise continuous functions defined over a finite number of elements.

## The Key Concept of Meshing

If the elements are small enough and are spread intelligently across the analyzed part, the numerical solution can closely approximate reality

However, even if the higher number of elements and nodes gives a higher accuracy, it is not a reason to always create a high-density mesh with a maximum possible number of nodes and elements. The reason is because by increasing the number of nodes, you will increase the solution time. The FE engineer should find the balance between a desired level of accuracy and the mesh density (the number of nodes and elements) using the available hardware resources. It is better to have a FEM with a reasonable mesh size giving an error of 10% within 1 day rather than a FEM giving a 0% error after 1 month of computation. In industry, the absolute accuracy is not the goal of a Finite Element Analysis.

### The General FEA Process

### The FEA Solution Process

### How FEA Can Help You?

- Stress & Structural Engineers
- Dynamics Engineers
- Loads Engineers
- Designers

- Will it break?
- Will it deform to much?
- Will it hold together?
- Can it be lighter?
- Can it be stronger and durable?

### How Accurate is FEA?

The structural engineer performs a FEM to predict an approximate response of the structure under a given excitation. So, a legitimate question that can be asked is about the accuracy of the prediction.

The accuracy of the prediction will depend on the differences between the real structure and the Finite Element Model. Obviously, a perfect match is impossible because many assumptions are done when the structural model is built:

- The CAD (Computer Aided Design) geometry of the real structure is simplified by removing unnecessary details
- The CAD is discretized by creating meshing of the different parts which compose the structure
- Assumptions are done for the modeling of the joints
- Material models are selected
- The real loadings are discretized to make them understandable by the FEM
- Assumptions about boundary conditions are made
- The analyst selects the behaviors that the FEM will capture

Each of the above assumptions will produce a deviation between the real structure and the FE model.

FEA can yield to spectacularly accurate results compare to test results. However, in a complex FEM, it is not possible to obtain an error less than 1% in the whole model. Generally, a target of ±10% global error permits to make very good predictions. This target does not prevent to have errors less than 5% in local regions of the FEM.

So, in order to have a highly accurate FEM, the FE analyst will have to pay close attention to all details. All the above points will have to be perfect, especially if the model is composed of several parts assembled together. A single component model will match very easily the real life while complex assemblies will require more efforts.

Finally, you should know all of the inputs used to the problem: the material properties, the loading conditions, the boundary conditions and any elements that can affect the results. These inputs may have uncertainties in them. For example, the material properties and loading may not be defined precisely, but most of the time, you will have to do with the available inputs. It is important to keep this in mind during the modeling process because there is no advantage in trying to solve a model for greater accuracy than the input data admits.

### Why Do Finite Element Analysis?

The structural problems are described by governing equations that can be solved on a paper to predict how the structure will behave. However, it can be done only for simple part shapes. But the reality is never simple and most of the time it is too complicated to solve the structural problems using closed form solutions. The Finite Element Method permits the engineers to model very complex structures by doing realistic assumptions which allow predicting the accurate behaviors of the structure.

The most common advantages of doing FEA for the development of a product are:

- Capacity to model complex systems
- Reduction of the development time
- Analyze different configurations of a design
- Optimization of the performance and cost
- Reduction of testing
- Safety improvement
- Faster achievement of the required quality
- Improvement of the data available to engineers in the decision making
- A better understanding of all components of a system allowing more rational design

In the modern industries, Finite Element Analysis is a key activity in developing high-performance products. No one questions the usefulness and the power of such a tool. So, let me answer the question “Why do FEA?”

Before the intensive usage of Computer Aided Engineering (CAE), and particularly FEA, the development of products in the mechanical structural industry was based on prototyping. The prototypes were built at different key stages of the development process. So, it was common to build dozens of prototypes before to get the final product. Today, prototyping is still used, but less intensively since usually a prototype is built only at the end of the development process, to validate the concept.

So, the idea is not to completely remove the prototyping, but rather to minimize it. FEA must not be seen as another task in the product development process. It becomes integrated with the entire product development process. It is the reason why the last FEA packages offer multiphysics capability. This capability permits engineers to improve the life and service of complex products by simulating various phenomena which occur at the same time and which influence the behavior and the performance of the product.

Today, the fastest and more powerful hardware the engineers can access with massive parallel processing capabilities allow simulating most of the conditions the product will experiment during its life. For example, it is possible to simulate all the potential failure conditions a structure will face to estimate all the consequences of these failures to develop safer and more durable products.

Finally, the economic aspect plays a major role in favor to FEA: physical prototyping cost is huge and continues to increase every year. On the other hand, the cost of the CAE engineers, even if it increases every year, is less that prototyping. And what happens to the costs of the simulation systems (FEA licenses + workstations + servers)? They dramatically decrease over the years. Today, the hourly cost of a complete simulation system is much lower than the hourly cost of a CAE engineer’s team and even more widely below the hourly cost of a prototyping process.

When FEA is implemented early in the design cycle, the development process is guided in the right direction early, when mistakes are costliest. Once a FEM has been correlated to physical tests, the engineers can make modifications and easily simulate these modifications, without having to physically build and test each new design.

In summary, the finite element method is a very mature method for structural analysis. The costs of this technology to everyday design tasks have been dropping while the capabilities delivered by the method expand constantly. The powerful and the accuracy of the commercial FEA packages increase so exponentially that the question “Why Do Finite Element Analysis?” becomes “Why Not?”

With FEA, it is now possible to deliver higher quality products in a shorter design cycle. However, the question of the ability of analysts to use this method arises. Any engineer in mechanics cannot use this method without an advanced education in Finite Element Analysis.