June 20, 2023

# What is finite element method?

The finite element method (FEM) is a widely used mathematical technique for analyzing complex physical systems. It is applied in various fields, including engineering, physics, and computer graphics. FEM involves breaking down a system into smaller, simpler pieces called finite elements, which are then analyzed using mathematical equations to simulate the behavior of the entire system. This article discusses the history, basics, process, and applications of finite element method.

## A Brief History of Finite Element Method

The origins of Finite Element Method (FEM) can be traced back to Isaac Newton's work in the 17th century, where he used calculus to analyze and predict the motion of physical objects. However, the modern concept of FEM was not developed until the mid-20th century, when engineers and mathematicians began to explore ways to solve complex structural problems.

During the early days of FEM, engineers used math to predict the behavior of beams, plates, and shells under various loads and boundary conditions. This led to the development of methods and techniques for solving simple problems. The results were compared with experimental data, which helped validate the approach and paved the way for more complex analyses.

### Origins and Early Development

One of the earliest applications of FEM was in the aerospace industry. Engineers used the technique to analyze the behavior of aircraft structures under different loading conditions. The method proved to be extremely useful in designing lightweight and efficient structures that could withstand the stresses of flight.

Another important early development in FEM was the creation of the stiffness method. This method involves dividing a structure into smaller, simpler elements, and then analyzing the behavior of each element individually. The results are then combined to give an overall picture of the behavior of the entire structure.

As computers became more powerful, engineers were able to use FEM to solve more complex problems. The technique was used to analyze the behavior of structures under dynamic loading conditions, such as earthquakes and wind gusts. It was also used to study the behavior of materials under extreme conditions, such as high temperatures and pressures.

### Evolution and Modern Applications

Advancements in computing technology, especially the development of digital computers, revolutionized the field of finite element analysis. Today, FEM is used in a wide range of applications, including structural analysis, fluid dynamics, heat transfer, and electromagnetic analysis. The technique has become increasingly popular in various fields as it provides a valuable tool for analyzing complex physical systems.

One of the most important modern applications of FEM is in the field of biomedical engineering. The technique is used to analyze the behavior of biological tissues and organs under different conditions. This has led to the development of new medical devices and treatments, such as artificial joints and prosthetic limbs.

In conclusion, the history of FEM is a story of innovation and progress. From its humble beginnings in the 17th century, the technique has evolved into a powerful tool for solving complex engineering problems. As technology continues to advance, it is likely that FEM will continue to play an important role in the development of new technologies and solutions.

## Understanding the Basics of Finite Element Method

Before diving into finite element analysis, itâ€™s essential to understand the key concepts and terminology that are involved. Finite element method (FEM) is a numerical technique used to solve complex engineering problems. It is widely used in the aerospace, automotive, and civil engineering industries, among others. FEM involves dividing the physical system into smaller, simpler parts called finite elements, which are then analyzed to understand the behavior of the overall system.

### Key Concepts and Terminology

At the heart of FEM is the concept of approximation. Finite elements are used to approximate the behavior of the physical system being analyzed. The accuracy of the analysis depends on the number of finite elements used and the quality of the approximation. Other fundamental concepts in FEM include nodes, which are the points where finite elements are connected to each other, and degrees of freedom, which are the variables used to describe the behavior of each node.

Nodes are essential components of FEM as they represent the points where the physical system is discretized. The behavior of the system is then approximated by solving the governing equations at each node. Degrees of freedom are the variables that are used to describe the behavior of each node. These variables can include displacement, temperature, and stress, among others.

### The Role of Meshing

Meshing is the process of dividing the physical system into finite elements. This step is critical to the accuracy and efficiency of the analysis. The mesh should be fine enough to capture the detail of the physical system but not so fine as to require excessive computation time or memory. Meshing can be done using various techniques, such as structured, unstructured, and adaptive meshing.

Structured meshing involves dividing the physical system into a grid-like structure, where each element has the same shape and size. Unstructured meshing, on the other hand, involves dividing the physical system into elements of varying shapes and sizes. Adaptive meshing involves refining the mesh in areas where there is a high gradient or a change in the behavior of the physical system.

### Types of Elements

FEM utilizes various types of finite elements to approximate the behavior of the physical system. The type of element used depends on the characteristics of the system being analyzed. Common types of elements include beams, trusses, plates, and shells.

Beam elements are used to model structures that are long and slender, such as bridges and towers. Truss elements are used to model structures that are made up of interconnected bars, such as roof trusses and cranes. Plate elements are used to model thin structures, such as aircraft wings and car bodies. Shell elements are used to model curved structures, such as pressure vessels and pipes.

In conclusion, understanding the basics of finite element method is crucial when it comes to solving complex engineering problems. The concepts of approximation, nodes, degrees of freedom, meshing, and types of elements are fundamental to FEM and play a significant role in the accuracy and efficiency of the analysis.

## The Process of Finite Element Analysis

The process of finite element analysis can be broken down into three main steps: preprocessing, solving, and postprocessing.

### Preprocessing: Geometry and Material Properties

The first step in any FEM analysis is to create a finite element model of the physical system being analyzed. This typically involves defining the geometry of the system, specifying material properties, and applying boundary conditions and loads.

When defining the geometry of the system, it is important to consider all of the components that make up the system. This includes both solid and fluid components, as well as any moving parts. Material properties must also be carefully considered, as they can greatly affect the behavior of the system under different loads and conditions.

Boundary conditions and loads must also be carefully defined. This includes specifying the types of forces or pressures that will be applied to the system, as well as any constraints or restrictions that will be placed on the system.

### Solving: Applying Loads and Boundary Conditions

Once the finite element model is defined, the next step is to apply loads and boundary conditions to simulate the physical environment in which the system operates. This typically involves applying forces, pressures, temperatures, or other types of loads to the system.

When applying loads and boundary conditions, it is important to consider all of the possible scenarios that the system may encounter in its operating environment. This includes both static and dynamic loads, as well as different types of materials and environmental conditions.

In addition to applying loads and boundary conditions, the solving step also involves running the finite element analysis software to generate results. This can be a time-consuming process, especially for complex systems with many components and loads.

### Postprocessing: Analyzing and Interpreting Results

After the analysis is complete, the results must be analyzed and interpreted to generate a meaningful understanding of the system. Visualization tools, such as graphs, contour plots, and animations, are often used to help understand the results.

When analyzing and interpreting results, it is important to consider all of the factors that may have affected the behavior of the system. This includes any uncertainties or assumptions made during the modeling and analysis process, as well as any external factors that may have influenced the results.

Overall, the process of finite element analysis is a complex and iterative process that requires careful consideration of many factors. However, when done correctly, it can provide valuable insights into the behavior of physical systems and help engineers and designers make informed decisions about how to improve their designs.

## Applications of Finite Element Method

FEM can be applied to a wide range of physical systems and phenomena. Here are some of the most common applications:

### Structural Analysis

One of the most common applications of FEM is in structural analysis. Engineers use FEM to analyze the behavior of structures such as buildings, bridges, and airplanes under various loads and boundary conditions.

For example, in the design of a new building, FEM can be used to analyze the stresses and strains that will be placed on the structure by wind, earthquakes, and other external forces. This information can then be used to optimize the design and ensure that the building is safe and structurally sound.

In the aerospace industry, FEM is used to analyze the behavior of aircraft under various flight conditions. This is essential in ensuring that the aircraft is safe and reliable, and that it can withstand the stresses and strains of flight.

### Fluid Dynamics

FEM is also widely used in fluid dynamics to simulate the behavior of fluids under various conditions. This is particularly important in designing systems such as pipes, turbines, and pumps.

For example, in the design of a new pump, FEM can be used to analyze the flow of fluid through the pump and identify areas of high pressure or turbulence. This information can then be used to optimize the design and ensure that the pump is efficient and reliable.

In the oil and gas industry, FEM is used to simulate the behavior of fluids in reservoirs and pipelines. This information is essential in optimizing the production of oil and gas and ensuring that the pipelines are safe and reliable.

### Heat Transfer and Thermal Analysis

FEM is used to simulate heat transfer and thermal analysis of physical systems. This application is essential in designing systems such as engines, power plants, and electronic devices.

For example, in the design of a new engine, FEM can be used to analyze the flow of heat through the engine and identify areas of high temperature or thermal stress. This information can then be used to optimize the design and ensure that the engine is efficient and reliable.

In the electronics industry, FEM is used to simulate the behavior of electronic devices under various thermal conditions. This information is essential in ensuring that the devices are safe and reliable, and that they can withstand the stresses of operation.

### Electromagnetic Analysis

FEM is also used in electromagnetic analysis, which has applications in designing and analyzing electrical devices such as motors, transformers, and antennas.

For example, in the design of a new motor, FEM can be used to analyze the behavior of the electromagnetic fields within the motor and identify areas of high magnetic flux or eddy currents. This information can then be used to optimize the design and ensure that the motor is efficient and reliable.

In the telecommunications industry, FEM is used to simulate the behavior of antennas and other electromagnetic devices. This information is essential in ensuring that the devices are efficient and reliable, and that they can transmit and receive signals effectively.

## Conclusion

The finite element method is a powerful tool for analyzing complex physical systems. It has revolutionized the way engineers and scientists approach problem-solving by providing a way to simulate the behavior of physical systems in a virtual environment. By breaking down a physical system into smaller, simpler pieces, FEM enables accurate and efficient analysis of a wide range of phenomena.