Fractal self-transform functions

The concept of fractal (self-similar) self-transform functions is examined. A general method to prove existence of these functions is introduced, and necessary conditions for this existence are derived. The results are general and apply to all transforms with product-type kernels.     

A model of groundwater contaminant transport using the CVBEM

The Complex Variable Boundary Element Method or CVBEM has been used to develop a simple but powerful numerical analog of contaminant transport in a saturated, confined groundwater aquifer. The presented numerical technique is based upon a mean-square fit of the boundary conditions which includes the effects of sources and sinks defined within the problem domain. The numerical analogue produces locations of streamlines and the time-evolution of the contaminant front location.     

A collocation CVBEM using program Mathematica

The well-known complex variable boundary element method (CVBEM) is extended for using collocation points not located at the usual boundary nodal point locations. In this work, several advancements to the implementation of the CVBEM are presented. The first advancement is enabling the CVBEM nodes to vary in location, impacting the modeling accuracy depending on chosen node locations. A second advancement is determining values of the CVBEM basis function complex coefficients by collocation at evaluation points defined on the problem boundary but separate and distinct from nodal point locations (if some or all nodes are located on the problem boundary). A third advancement is the implementation of these CVBEM modeling features on computer program Mathematica, in order to reduce programming requirements and to take advantage of Mathematica’s library of mathematical capabilities and graphics features.     

Wavefront expansion basis functions and their relationships

Wavefront expansion basis functions are important in representing ocular aberrations and phase perturbations due to atmospheric turbulence. A general discussion is presented for the conversions of the coefficients between any two sets of basis functions. Several popular sets of basis functions, namely, Zernike polynomials, Fourier series, and Taylor monomials, are discussed and the conversion matrices between any two of these basis functions are derived. Some analytical and numerical examples are given to demonstrate conversion of coefficients of different basis function sets.     

An algorithm for optimizing CVBEM and BEM nodal point locations

The Complex Variable Boundary Element Method or CVBEM is a numerical technique for approximating particular partial differential equations such as the Laplace or Poisson equations (which frequently occur in physics and engineering problems, among many other fields of study). The advantage in using the CVBEM over traditional domain methods such as finite difference or finite element based methods includes the properties that the resulting CVBEM approximation is a function: (i) defined throughout the entire plane, (ii) that is analytic throughout the problem domain and almost everywhere on the problem boundary and exterior of the problem domain union boundary; (iii) is composed of conjugate two-dimensional real variable functions that are both solutions to the Laplace equation and are orthogonal such as to provide the “flow net” of potential and stream functions, among many other features. In this paper, a procedure is advanced that locates CVBEM nodal point locations on and exterior of the problem boundary such that error in matching problem boundary conditions is reduced. That is, locating the nodal points is part of modeling optimization process, where nodes are not restricted to be located on the problem boundary (as is the typical case) but instead locations are optimized throughout the exterior of the problem domain as part of the modeling procedure. The presented procedure results in nodal locations that achieve considerable error reduction over the usual methods of placing nodes on the problem boundary such as at equally spaced locations or other such procedures. Because of the significant error reduction observed, the number of nodes needed in the model is significantly reduced. It is noted that similar results occur with the real variable boundary element method (or BEM).The CVBEM and relevant nodal location optimization algorithm is programmed to run on program Mathematica, which provides extensive internal modeling and output graphing capabilities, and considerable levels of computational accuracy. The Mathematica source code is provided.     

Fractal measures for meromorphic functions of finite order

We prove several essential fractal properties, such as positivity, finiteness or local infinity, of Hausdorff and packing measures of radial Julia sets for large subclasses of entire and meromorphic functions considered in our previous work: çGeometric thermodynamical formalism and real analyticity for meromorphic functions of finite order'. Most of the results proven are shown to be optimal.     

Higher-order basis functions for MoM calculations

The extension of Rao, Wilton and Glisson basis functions on flat-faceted triangular elements to different element shapes, higher-order geometry and higher-order function support is outlined. A curvilinear coordinate formulation is used to obtain a family of finite elements for more accurate method of moments (MoM) computations. Results for the first two orders of geometry and function support in 1, 2 and 3 dimensions are presented. The basis functions are hierarchical, in that mixed orders of geometry and of function support can be used together in a single calculation to allow efficient local refinement. Practical issues of element assembly, treatment of singular and non-singular MoM integrals and of loop basis functions are addressed.     

Consideration of a CVBEM approximation for plane hydrodynamics

The complex variable boundary element method (CVBEM) represents a very useful working tool for applied mathematics in general and for hydrodynamics in particular. The goal of the present work is to give a brief survey on some approximation procedures given the data on the working domain contour in view of getting a rapid convergence of the whole process within a CVBEM. In the first section of this work we will see how some important mixed boundary value problems, for simply connected domains and for the Laplace operator, are transformed into equivalent Dirichlet problems and consequently these Dirichlet problems are the only boundary value problems we will deal with for a CVBEM. In the second section of the present work, some convergent algorithms of a CVBEM, for a Dirichlet boundary problem are built up in view of fluid dynamics (but not only!) applications.     

Global response approximation with radial basis functions

In this article, a study is performed on the accuracy of radial basis functions (RBFs) in creating global metamodels for both low- and high-order nonlinear responses. The response surface methodology (RSM), which typically uses linear or quadratic polynomials, is inappropriate for creating global models for highly nonlinear responses. The RBF, on the other hand, has been shown to be accurate for highly nonlinear responses when the sample size is large. However, for most complex engineering applications only limited numbers of samples can be afforded; it is desirable to know whether the RBF is appropriate in this situation, especially when the augmented RBF has to be used. Because the types of true responses are typically unknown a priori, it is essential for high-fidelity metamodeling to have an RBF or RBFs that are appropriate for linear, quadratic, and higher-order responses. To this end, this study compares a variety of existing basis functions in both non-augmented and augmented forms with various types of responses and limited numbers of samples. This article shows that the augmented RBF models created by Wu’s compactly supported functions are the most accurate for the various test functions used in this study.     

Weakly equilibrated basis functions for elasticity problems

In this paper weakly equilibrated basis functions (EqBFs) are introduced for the development of a boundary point method. This study is the extension of the one in (Int. J. Numer. Methods Engng. 81 (2010) 971–1018) using exponential basis functions (EBFs) which are available just for partial differential equations (PDEs) with constant coefficients. Here the EqBFs are evaluated numerically to solve more general PDEs with non-constant coefficients. The EqBFs are found through weighted residual integrals defined over a fictitious domain embedding the main domain. A series of Chebyshev polynomials are used for the construction of the basis functions. By properly choosing the weight functions as the product of two unidirectional functions, here with Gaussian distribution, the main 2D integrals are written as the product of the simpler 1D ones. The results of the integrals can be stored for further use; however in some particular cases the EqBFs may be stored as a set of library functions. The results may also be found useful for those who are interested in residual-free functions in other numerical methods. For the verification, we discuss on the validity of the solution through an essential and comprehensive test procedure followed by several numerical examples.     

CVBEM for solving De Saint-Venant solid under shear forces

Evaluation of shear stresses distribution due to external shear forces applied to De Saint-Venant beams has been solved through Complex Variable Boundary Element Method properly extended, to benefit from advantages of this method, so far widely used for twisted solids. Extending the above method, further simplifications have been introduced such as those of performing line integrals only, instead of domain integrals. Numerical applications confirm accuracy and efficiency of the proposed extended version of the method, since the good agreement with results proposed in literature.     

Analysis of thick plates by radial basis functions

In this paper, we use a higher-order shear deformation theory and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thick plates. Through numerical experiments, the capability and efficiency of this collocation technique for static and vibration problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.     

The Hermite collocation method using radial basis functions

In this note, numerical experiments are carried out to study the convergence of the Hermite collocation method using high order polyharmonic splines and Wendland's RBFs.     

An orthonormal basis functions method for moment problems

In this paper, a new computational method is developed to recover an unknown function from its moments with respect to general kernel functions. By using the Gram–Schmidt orthonormalization technique, our method is shown to be efficient and can be interpreted as a generalization of the Talenti method. Convergence and error estimates are also discussed. For the purposes of verification and application, the method is applied to solve both Cauchy problem for Laplace equation and a Fredholm integral equation of the first kind.     

Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions

A two‐stage numerical procedure using Chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation. The purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations. Our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously. Numerical results on the approximation of eight two‐dimensional test functions and their derivatives are given. To demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations, we solve a modified Helmholtz equation. Near machine precision can be achieved for all these test problems. Copyright © 2006 John Wiley & Sons, Ltd.     

Radial basis functions methods for solving Fokker-Planck equation

In this paper two numerical meshless methods for solving the Fokker-Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker-Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and N_e errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.     

The null space and non-conventional basis functions in the mixed finite element method

The two matrix equations obtained from the mixed finite element method are uncoupled through the use of the QR decomposition. The procedure obtains the basis for the null space which is composed of solutions to the homogeneous flux-equilibrium equation. The result is that, even for statically indeterminate problems, stresses can be obtained without solving for the displacement field. Furthermore, the stress field can be decomposed directly into contributions from applied forces and from prescribed displacements. The appearance of kinetic modes can easily be monitored. The possibility of utilizing non-conventional basis functions for use in conjunction with the QR decomposition is explored briefly. Simple examples are given to illustrate the theoretical results.