Wavelet transforms originated in geophysics in the early 1980s for the analysis of seismic signals. Since then, significant mathematical advances in wavelet theory have enabled a suite of applications in diverse fields. In geophysics the power of wavelets for analysis of nonstationary processes that contain multiscale features, detection of singularities, analysis of transient phenom- ena, fractal and multifractal processes, and signal com- pression is now being exploited for the study of several processes including space-time precipitation, remotely sensed hydrologic fluxes, atmospheric turbulence, can- opy cover, land surface topography, seafloor bathymetry, and ocean wind waves. It is anticipated that in the near future, significant further advances in understanding and modeling geophysical processes will result from the use of wavelet analysis. In this paper we review the basic properties of wavelets that make them such an attractive and powerful tool for geophysical applications. We dis- cuss continuous, discrete, orthogonal wavelets and wavelet packets and present applications to geophysical processes.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/97RG00427

Wavelet analysis for geophysical applications

A new methodology to build discrete models of boundary-value problems is presented. The h-pcloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomialreproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and a several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater -exibility than traditional h-p finite element methods. Several numerical experiments in I-D and 2-D are also presented. @ 1996 John Wiley & Sons, Inc. In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc. Moreover, in most computer-aided design work, the generation of an appropriate mesh constitutes, by far, the costliest portion of the computer-aided analysis of products and processes. These are among the reasons that interest in so-called meshless methods has grown rapidly in recent years. Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. The analysis of problems such as crack propagation, penetration, and large deformations, can, in principle, be greatly simplified by the use of meshless methods. A growing crack, for example, can be modeled by simply extending the free surfaces that correspond to the crack [ 11. The analysis of large deformation problems by, e.g., finite element methods, may require the continuous remeshing of the domain to avoid the breakdown of the calculation due to

/pdf/h-p-clouds-an-h-p-meshless-method-55own9epaq.pdf

H‐p clouds—an h‐p meshless method

http://hogwarts.ucsd.edu/~pkrysl/publications/shells.pdf

Analysis of thin shells by the Element-Free Galerkin method

A meshless approach to the analysis of arbitrary Kirchhoff plates by the Element-Free Galerkin (EFG) method is presented. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into “finite elements”. The satisfaction of the C
1 continuity requirements are easily met by EFG since it requires only C
1 weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchhoff theory. The requirements of consistency are met by the use of a quadratic polynomial basis. A subdivision similar to finite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. It is shown, that high accuracy can be achieved for arbitrary grid geometries, for clamped and simply-supported edge conditions, and for regular and irregular grids. Numerical studies are presented which show that the optimal support is about 3.9 node spacings, and that high-order quadrature is required.

Analysis of thin plates by the element-free Galerkin method

A meshless approach to the analysis of arbitrary Kirchho plates by the Element-Free Galerkin (EFG) method is presented. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into \nite elements". The satisfaction of the C 1 continuity requirements are easily met by EFG since it requires only C 1 weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchho theory. The requirements of consistency are met by the use of a quadratic polynomial basis. A subdivision similar to nite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. It is shown, that high accuracy can be achieved for arbitrary grid geometries, for clamped and simply-supported edge conditions, and for regular and irregular grids. Numerical studies are presented which show that the optimal support is about 3:9 node spacings, and that high-order quadrature is required.

/pdf/analysis-of-thin-plates-by-the-element-free-3d9sw9g7za.pdf

Analysis of Thin Plates by the Element-Free

In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi-grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies1) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem.

In order to demonstrate the wavelet technique, we consider the one-dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund2 and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.

Sam Qian

Papers

Wavelet–Galerkin solutions for one‐dimensional partial differential equations

Wavelets and the Numerical Solution of Partial Differential Equations

Wavelets and the numerical solution of boundary value problems