Mastering Plane Finite Elements: Unleash the Power of Two-Dimensional Problem Solving

When it comes to solving complex engineering problems, plane finite elements play a crucial role. These numerical techniques have revolutionized the analysis and simulation of two-dimensional problems across various fields of engineering.

Understanding Finite Element Method

The Finite Element Method (FEM) is a powerful numerical analysis technique that discretizes complex problems into smaller, simpler and manageable elements. Each element behaves as a representative of the whole system and allows engineers to study the behavior of the system under different conditions.

In plane finite element analysis, the problem domain is divided into smaller triangular or quadrilateral elements. These elements are interconnected, forming a mesh that accurately represents the geometry of the problem. Each element possesses defined material properties and boundary conditions, enabling the simulation of various real-life scenarios.

Plane Finite Elements for Two-Dimensional Problems: Application of the Computer Algebra System Maxima

by Andreas Öchsner(Kindle Edition)

4.8 out of 5

Language	:	English
File size	:	136965 KB
Text-to-Speech	:	Enabled
Enhanced typesetting	:	Enabled
Print length	:	307 pages
Screen Reader	:	Supported

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Advantages of Plane Finite Elements

One of the key advantages of using plane finite elements is their ability to solve complex two-dimensional problems accurately. They provide a systematic approach to obtaining approximate solution methods for a wide range of engineering scenarios.

Additionally, plane finite elements facilitate efficient computation by minimizing the number of variables compared to three-dimensional problems. This reduction in complexity not only saves computational resources but also speeds up calculation times, allowing engineers to perform analyses in a timely manner.

Moreover, the versatility of plane finite elements enables engineers to tackle problems in different branches of engineering. From structural analysis and heat transfer to fluid dynamics and electromagnetics, these elements have been extensively utilized to simulate and optimize designs.

Working Principles of Plane Finite Elements

Plane finite elements work on the principle of discretization, where a complex problem is divided into smaller elements. The shape functions of these elements help to interpolate unknown values within each element based on known values at the element nodes.

The governing equations for each element are derived using variational principles or energy methods. These equations are then assembled to form a global stiffness matrix, representing the behavior of the entire system.

Types of Plane Finite Elements

There are various types of plane finite elements, each suited for specific types of problems. Some of the commonly used elements include:

• Triangle Elements: These elements are particularly useful for analyzing irregular geometries and stress concentration problems.
• Quadrilateral Elements: Quadrilateral elements provide higher accuracy and numerical stability, making them ideal for structural and heat transfer analysis.
• Isoparametric Elements: These elements use the same shape functions for both geometry and interpolation of unknowns, offering better accuracy in simulations.

Applications of Plane Finite Elements

Plane finite elements find applications in an array of engineering disciplines. Some of the common applications include:

• Structural Analysis: Plane finite elements are used to determine stress, strain, and deformation in structural components.
• Fluid Dynamics: These elements simulate fluid flow and aid in optimizing designs for better performance and efficiency.
• Heat Transfer: Plane finite elements help analyze heat transfer phenomenon in various systems, such as heat exchangers and electronic devices.
• Electromagnetics: These elements assist in studying electromagnetic fields and their interaction with materials.

Plane finite elements have revolutionized the way engineers approach two-dimensional problem solving. Their versatility, accuracy, and efficiency make them an essential tool for analyzing and optimizing designs across various branches of engineering. By harnessing the power of plane finite elements, engineers can push the boundaries of innovation and pave the way for groundbreaking advancements.

Plane Finite Elements for Two-Dimensional Problems: Application of the Computer Algebra System Maxima

by Andreas Öchsner(Kindle Edition)

4.8 out of 5

Language	:	English
File size	:	136965 KB
Text-to-Speech	:	Enabled
Enhanced typesetting	:	Enabled
Print length	:	307 pages
Screen Reader	:	Supported

FREE

This book is intended as a study aid for the finite element method. Based on the free computer algebra system Maxima, we offer routines to symbolically or numerically solve problems from the context of two-dimensional problems. For this rather advanced topic, classical ‘hand calculations’ are difficult to perform and the incorporation of a computer algebra system is a convenient approach to handle, for example, larger matrix operations. The mechanical theories focus on the classical two-dimensional structural elements, i.e., plane elements, thin or classical plates, and thick or shear deformable plate elements. The use of a computer algebra system and the incorporated functions, e.g., for matrix operations, allows to focus more on the methodology of the finite element method and not on standard procedures. Furthermore, we offer a graphical user interface (GUI) to facilitate the model definition. Thus, the user may enter the required definitions in a source code manner directly in wxMaxima or use the GUI which is able to execute wxMaxime to perform the calculations.