Meshfree Approximation Methods With Matlab: A Review of the Book and the Software
Meshfree approximation methods are numerical techniques that do not rely on a predefined mesh or grid to discretize the domain of a problem. Instead, they use a set of scattered points, called nodes, to represent the geometry and the solution. Meshfree methods have several advantages over traditional mesh-based methods, such as the ability to handle complex geometries, large deformations, moving boundaries, and multiscale phenomena.
One of the challenges of meshfree methods is the implementation of the algorithms and the data structures. To facilitate this task, the book Meshfree Approximation Methods with MATLAB by Gregory E. Fasshauer and Michael J. McCourt provides a comprehensive introduction to the theory and practice of meshfree methods, with a focus on radial basis functions (RBFs) and partition of unity methods (PUMs). The book also includes a software package, called MFreeApproxToolbox, that contains MATLAB codes for various meshfree approximation methods and examples.
The book is divided into three parts. The first part covers the basics of meshfree approximation, such as interpolation, collocation, least squares, and Galerkin methods. The second part discusses RBFs and PUMs in more detail, including their properties, choices of shape functions, and applications. The third part presents some advanced topics, such as adaptive methods, multivariate approximation, and uncertainty quantification.
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The book is suitable for graduate students, researchers, and practitioners who are interested in learning about meshfree approximation methods and their applications. The book assumes some background in numerical analysis, linear algebra, and MATLAB programming. The software package MFreeApproxToolbox can be downloaded from the book's website or from MathWorks File Exchange. The software is well-documented and easy to use, and it can be modified and extended by the users.
In conclusion, Meshfree Approximation Methods with MATLAB is a valuable resource for anyone who wants to learn about meshfree methods and use them in their own problems. The book offers a clear and comprehensive exposition of the theory and practice of meshfree methods, while the software provides a user-friendly and flexible tool for implementing and testing various meshfree algorithms.
To illustrate the use of meshfree methods and the MFreeApproxToolbox software, we present two examples from the book. The first example is the solution of a two-dimensional Poisson equation with Dirichlet boundary conditions on a L-shaped domain. The second example is the approximation of a bivariate function with sharp peaks and valleys on a square domain.
In both examples, we use RBFs with multiquadric shape functions and collocation methods to obtain the meshfree approximation. We compare the results with the exact solution or a reference solution obtained by a high-order polynomial interpolation. We also plot the error and the distribution of the nodes.
The following figures show the results of the first example. Figure 1 shows the exact solution of the Poisson equation on the L-shaped domain. Figure 2 shows the meshfree approximation obtained by using 289 nodes. Figure 3 shows the absolute error between the exact solution and the meshfree approximation. Figure 4 shows the distribution of the nodes on the domain.
Figure 1: Exact solution of Poisson equation on L-shaped domain
Figure 2: Meshfree approximation of Poisson equation on L-shaped domain
Figure 3: Absolute error of meshfree approximation of Poisson equation on L-shaped domain
Figure 4: Distribution of nodes on L-shaped domain