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Meshfree Methods for Partial Differential Equations VII: Advanced Meshfree Methods

Herbert Cox

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Published in Meshfree Methods For Partial Differential Equations VII (Lecture Notes In Computational Science And Engineering 100)

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Meshfree methods are a class of numerical methods for solving partial differential equations (PDEs) that do not require a mesh to discretize the solution domain. This makes them particularly well-suited for problems with complex geometries, moving boundaries, and other situations where a traditional mesh-based method would be difficult or impossible to use.

Meshfree methods have been developed over the past few decades, and a number of different methods now exist. Some of the most popular meshfree methods include:

hp-Clouds
Meshfree finite element methods
Partition of unity methods
Radial point interpolation methods
Smoothed particle hydrodynamics methods

Each of these methods has its own advantages and disadvantages, and the best choice for a particular application will depend on the specific requirements of the problem.

Meshfree Methods for Partial Differential Equations VII (Lecture Notes in Computational Science and Engineering 100)

Meshfree Methods for Partial Differential Equations VII (Lecture Notes in Computational Science and Engineering Book 100)

by Javier Mallo

4.5 out of 5

Language	:	English
File size	:	12107 KB
Screen Reader	:	Supported
Print length	:	332 pages

Meshfree methods are based on the idea of using a set of nodes to represent the solution domain. These nodes can be located anywhere in the domain, and they do not need to be connected to each other. The solution is then approximated using a function that is defined at the nodes.

The function used to approximate the solution can be any type of function, but it is typically a polynomial or a radial basis function. Polynomial functions are easy to evaluate and differentiate, but they can become inaccurate if the solution is not smooth. Radial basis functions are more accurate than polynomial functions, but they are more expensive to evaluate and differentiate.

The choice of function used to approximate the solution will depend on the specific requirements of the problem. For problems with smooth solutions, a polynomial function may be sufficient. For problems with non-smooth solutions, a radial basis function may be necessary.

In recent years, there have been a number of significant developments in meshfree methods. These developments have made meshfree methods more accurate, efficient, and versatile.

One of the most important developments in meshfree methods has been the development of hp-clouds. Hp-clouds are a type of meshfree method that uses a hierarchical set of nodes to represent the solution domain. This allows for a more accurate approximation of the solution, especially for problems with complex geometries.

Another important development in meshfree methods has been the development of meshfree finite element methods. Meshfree finite element methods combine the advantages of meshfree methods with the advantages of finite element methods. This results in a method that is accurate, efficient, and versatile.

Partition of unity methods, radial point interpolation methods, and smoothed particle hydrodynamics methods have also seen significant development in recent years. These methods have been improved in terms of accuracy, efficiency, and versatility.

Meshfree methods are used in a wide variety of applications, including:

Solid mechanics
Fluid mechanics
Heat transfer
Electromagnetics

Meshfree methods are particularly well-suited for problems with complex geometries, moving boundaries, and other situations where a traditional mesh-based method would be difficult or impossible to use.

Meshfree methods are used to solve a wide range of problems in solid mechanics, including:

Linear and nonlinear elasticity
Plasticity
Fracture mechanics
Contact mechanics

Meshfree methods are used to solve a wide range of problems in fluid mechanics, including:

Incompressible and compressible flows
Laminar and turbulent flows
Heat transfer
Multiphase flows

Meshfree methods are used to solve a wide range of problems in heat transfer, including:

Conduction
Convection
Radiation
Phase change

Meshfree methods are used to solve a wide range of problems in electromagnetics, including:

Maxwell's equations
Wave propagation
Antenna design
Electromagnetic compatibility

Meshfree methods are a powerful tool for solving partial differential equations. They are particularly well-suited for problems with complex geometries, moving boundaries, and other situations where a traditional mesh-based method would be difficult or impossible to use.

In recent years, there have been a number of significant developments in meshfree methods. These developments have made meshfree methods more accurate, efficient, and versatile. As a result, meshfree methods are now used in a wide variety of applications, including solid mechanics, fluid mechanics, heat transfer, and electromagnetics.

Meshfree Methods for Partial Differential Equations VII: Advanced Meshfree Methods by Michael Griebel and Michael A. Schweitzer
hp-Clouds: A Method for High-Free Download Accurate Meshfree Approximation by Michael Griebel and Michael A. Schweitzer
Meshfree Finite Element Methods by Ted Belytschko, Yiyong Lu, and Lihua Gu
Partition of Unity Methods by John M. Melenk and Ivar Babuška
Radial Point Interpolation Methods by George R. Johnson and Stephen R. Cottrell
Smoothed Particle Hydrodynamics Methods by Monaghan, J. J.

Meshfree Methods for Partial Differential Equations VII (Lecture Notes in Computational Science and Engineering Book 100)

by Javier Mallo

4.5 out of 5