Unlocking the Power of Bem Based Finite Element Approaches on Polytopal Meshes - Lecture Notes In

In the fields of engineering, physics, and computer science, the finite element method revolutionized the way we solve complex problems involving partial differential equations. One of the newest and most promising developments in this domain is the BEM (Boundary Element Method) based finite element approaches on polytopal meshes.

What is BEM?

The Boundary Element Method (BEM) is a numerical technique used to solve boundary value problems in various areas of science and engineering. Instead of dividing the domain into smaller elements like traditional FEM (Finite Element Method), BEM focuses on solving problems on the boundary of the domain.

By formulating the problem in terms of the boundary conditions, BEM reduces the computational complexity and allows for more accurate solutions in certain scenarios. This approach has gained significant traction in the scientific community due to its advantages in terms of memory usage, computational speed, and accuracy.

BEM-based Finite Element Approaches on Polytopal Meshes (Lecture Notes in Computational Science and Engineering Book 130)

by A.J. Best (1st ed. 2019 Edition, Kindle Edition)

4 out of 5

Language	:	English
File size	:	7498 KB
Print length	:	263 pages
Screen Reader	:	Supported
Hardcover	:	318 pages
Item Weight	:	1.37 pounds
Dimensions	:	6 x 0.75 x 9 inches

FREE

Understanding Polytopal Meshes

Now that we have a basic understanding of BEM, let's delve into another crucial concept - polytopal meshes. In the realm of computational geometry and mesh generation, polytopal meshes refer to meshes composed of various types of cells, such as triangles, quadrilaterals, tetrahedra, and more.

These meshes provide a versatile framework for simulating complex structures and phenomena with irregular boundaries. While traditional meshing techniques struggle with handling curved domains, polytopal meshes offer a more flexible and efficient solution.

The Integration of BEM and Polytopal Meshes

Now, let's explore the synergy between the BEM and polytopal meshes. By combining these two powerful approaches, researchers and engineers can benefit from the best of both worlds.

Instead of relying solely on the FEM or BEM, which have their limitations, researchers have developed innovative techniques that exploit the advantages of polytopal meshes while leveraging the accuracy and efficiency of the BEM.

Benefits and Applications

The integration of BEM with polytopal meshes opens up a wide range of benefits and applications across various fields:

1. Computational Physics and Engineering

Simulating and analyzing physical phenomena like heat transfer, fluid flow, and electromagnetic fields become more accurate and efficient with the BEM-based finite element approaches on polytopal meshes. Researchers can now explore complex geometries with irregular boundaries while minimizing computational resources.

2. Seismic Analysis and Geotechnical Engineering

The behavior of soil and rock structures during earthquakes can be better understood and predicted using this powerful combination of methodologies. Engineers can analyze the response of complex underground structures subjected to seismic forces, leading to improved safety in construction and infrastructure projects.

3. Biomechanics

Studying the mechanics of biological structures, such as bones and soft tissues, requires advanced numerical simulations. BEM-based finite element approaches on polytopal meshes provide a robust framework for analyzing the stress distribution, deformation, and other important factors affecting biological systems.

4. Computational Electromagnetics

For analyzing electromagnetic fields and interactions, the BEM-based finite element approaches on polytopal meshes offer higher accuracy and computational efficiency. This is particularly relevant in the design and optimization of antennas, sensors, and other electromagnetic devices.

Future Directions

As the field of computational science continues to evolve, the integration of BEM approaches on polytopal meshes will witness further advancements. Researchers are working on refining the techniques, developing new algorithms, and exploring novel applications.

With the help of machine learning and artificial intelligence, the accuracy and efficiency of these approaches are expected to improve even further. This will undoubtedly lead to new discoveries, optimized design processes, and enhanced problem-solving capabilities in diverse scientific and engineering domains.

The BEM-based finite element approaches on polytopal meshes represent a breakthrough in computational science and engineering. By combining the strengths of BEM and polytopal meshes, researchers and engineers can tackle complex problems and simulations more accurately, efficiently, and intuitively than ever before.

As this field continues to advance, we can expect to see more innovative applications, improved algorithms, and game-changing solutions to the challenges we face in various industries. The future of BEM-based finite element approaches on polytopal meshes is undoubtedly promising.

BEM-based Finite Element Approaches on Polytopal Meshes (Lecture Notes in Computational Science and Engineering Book 130)

by A.J. Best (1st ed. 2019 Edition, Kindle Edition)

4 out of 5

Language	:	English
File size	:	7498 KB
Print length	:	263 pages
Screen Reader	:	Supported
Hardcover	:	318 pages
Item Weight	:	1.37 pounds
Dimensions	:	6 x 0.75 x 9 inches

FREE

This book introduces readers to one of the first methods developed for the numerical treatment of boundary value problems on polygonal and polyhedral meshes, which it subsequently analyzes and applies in various scenarios. The BEM-based finite element approaches employs implicitly defined trial functions, which are treated locally by means of boundary integral equations. A detailed construction of high-order approximation spaces is discussed and applied to uniform, adaptive and anisotropic polytopal meshes.

The main benefits of these general discretizations are the flexible handling they offer for meshes, and their natural incorporation of hanging nodes. This can especially be seen in adaptive finite element strategies and when anisotropic meshes are used. Moreover, this approach allows for problem-adapted approximation spaces as presented for convection-dominated diffusion equations. All theoretical results and considerations discussed in the book are verified and illustrated by several numerical examples and experiments.  

Given its scope, the book will be of interest to mathematicians in the field of boundary value problems, engineers with a (mathematical) background in finite element methods, and advanced graduate students.