Principle of finite element method

Introduction to Finite Element Method

The Finite Element Method, often abbreviated as FEM, is a powerful tool used in engineering and mathematical modeling to solve complex problems.
It provides a way to transform partial differential equations, which describe various physical phenomena, into algebraic equations that are easier to solve.
In essence, the FEM approximates a continuous system by dividing it into smaller, simpler parts known as finite elements.

What is the Finite Element Method?

The Finite Element Method is a numerical method for solving problems related to fields like structural analysis, heat transfer, fluid dynamics, and electromagnetics.
It works by breaking down a large, complicated problem into smaller, more manageable pieces called elements.
These elements can be of various shapes like triangles or quadrilaterals in 2D or tetrahedrals and hexahedrals in 3D.

The Role of Nodes and Elements

In the Finite Element Method, a crucial aspect is the division of the domain into elements.
Within each element, there are specific points called nodes.
These nodes serve as a foundation for approximating the behavior of the element.
By using simple mathematical functions, known as shape functions, the distribution of variables, such as temperature or stress, is estimated across the element.

Discretization Process

Discretization is a significant step in the FEM process.
It involves dividing the problem domain into a finite number of elements and nodes.
This process transforms the continuous problem into a set of discrete problems that can be handled computationally.
Each element is an approximation of the real-world system and collectively, these elements approximate the entire system.

Historical Background

The roots of the Finite Element Method trace back to the 1940s and 1950s with the advancing needs of aerospace engineering.
The complex structures and the behaviors of these structures necessitated a method that could handle large and complicated mathematical models.
The method gained momentum with the contributions from engineers like Alexander Hrennikoff and Richard Courant who initially applied it to elastic structures.

Key Principles of Finite Element Method

The Finite Element Method relies on several key principles that make its application successful in solving diverse problems:

1. Approximation of Real-World Features

The FEM uses mathematical functions to approximate real-world features.
This reduces complex shapes into a series of simpler ones, making the analysis feasible.
The elements are designed to closely resemble portions of the actual physical model.

2. Application of Boundary Conditions

Boundary conditions are vital in FEM as they define how the model interacts with its surroundings.
Applying accurate boundary conditions ensures that the model behaves similarly to the real system.

3. Assembly of Global System

Once individual elements are analyzed, the results are compiled into a global system.
This combines all the individual element equations into one comprehensive equation.
Mathematically, this is represented as a system of algebraic equations which then needs to be solved.

4. Iterative Solution Methods

After assembling the global system, iterative methods are used to solve the algebraic equations.
These methods aim to find solutions that converge close to the exact answer.
This step is crucial in ensuring the accuracy of the results.

Applications of the Finite Element Method

The versatility of the Finite Element Method allows it to be applied across a wide range of disciplines.
Some prominent applications include:

Mechanical Engineering

In mechanical engineering, FEM is primarily used for structural analysis.
Engineers use it to analyze whether structures can withstand applied forces.
It helps in designing components that are both efficient and safe.

Aerospace Engineering

In aerospace engineering, FEM assists in assessing stress points and potential failure regions in aircraft components.
The method supports the optimization of material use, ensuring stability and performance during flight.

Civil Engineering

Civil engineers utilize FEM for structural analysis in bridge and building design.
By simulating various load scenarios, they ensure the infrastructure can endure everyday use and extreme conditions.

Biomedical Engineering

Biomechanics is another area benefiting from FEM.
It helps in simulating human tissues and organs, leading to better understanding and innovations in medical devices and treatments.

Advantages and Limitations of FEM

While Finite Element Method has numerous advantages, it also comes with a few limitations.

Advantages

– **Flexibility:** FEM can simulate complex geometries and diverse materials.
– **Accuracy:** When used correctly, it provides highly accurate results.
– **Comprehensiveness:** The method accommodates various types of analyses, from thermal to electromagnetic.

Limitations

– **Computational Demand:** Large problems can require significant computational resources.
– **Skill Requirement:** Proper use requires specialized knowledge and expertise.
– **Approximation Risk:** Errors can occur if the approximation of elements isn’t suitably accurate.

Conclusion

The Finite Element Method is an indispensable part of modern engineering analysis.
It provides a way to solve complex problems by breaking them down into simpler, manageable pieces.
Through a combination of discretization, approximation, and computation, FEM allows engineers and scientists to predict the behavior of systems accurately.
While it has a few limitations, its advantages in solving difficult engineering problems make it an invaluable tool across various disciplines.