Quadratic Basis Research Articles

Quadratic Spline Wavelets with Short Support for Fourth-Order Problems

In the paper, we propose constructions of new quadratic spline-wavelet bases on the interval and the unit square satisfying homogeneous Dirichlet boundary conditions of the second order. The basis functions have small supports and wavelets have one vanishing moment. We show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small.

Periodized radial basis functions, part I: Theory

We extend the theory of periodized RBFs. We show that the imbricate series that define the Periodic Gaussian (PGA) and Sech (PSech) basis functions are Jacobian theta functions and elliptic functions “dn”, respectively. The naive periodization fails for the Multiquadric and Inverse Multiquadric RBFs, but we are able to define periodic generalizations of these, too, by proving and exploiting a generalization of the Poisson Summation Theorem. Although applications of periodic RBFs are mostly left for another day, we do illustrate the flaws and potential by solving the Mathieu eigenproblem on both uniform and highly-adapted grids. The terms of a Fourier basis can be grouped into four classes, depending upon parity with respect to both the origin and x=π/2, and so, too, the Mathieu eigenfunctions. We show how to construct symmetrized periodic RBFs and illustrate these by solving the Mathieu problem using only the periodic RBFs of the same symmetry class as the targeted eigenfunctions. We also discuss the relationship between periodic RBFs and trigonometric polynomials with the aid of an explicit formula for the nonpolynomial part of the Periodic Inverse Quadratic (PIQ) basis functions. We prove that the rate of convergence for periodic RBFs is geometric, that is, the error can be bounded by exp⁡(−Nμ) for some positive constant μ. Lastly, we prove a new theorem that gives the periodic RBF interpolation error in Fourier coefficient space. This is applied to the “spectral-plus” question. We find that periodic RBFs are indeed sometimes orders of magnitude more accurate than trigonometric interpolation even though it has long been known that RBFs (periodic or not) reduce to the corresponding classical spectral method as the RBF shape parameter goes to 0. However, periodic RBFs are “spectral-plus” only when the shape parameter α is adaptively tuned to the particular f(x) being approximated and even then, only when f(x) satisfies a symmetry condition.

K-Block parallel addition versus 1-block parallel addition in non-standard numeration systems

Parallel addition in integer base is used for speeding up multiplication and division algorithms. k-block parallel addition has been introduced by Kornerup in [14]: instead of manipulating single digits, one works with blocks of fixed length k. The aim of this paper is to investigate how such notion influences the relationship between the base and the cardinality of the alphabet allowing block parallel addition. In this paper, we mainly focus on a certain class of real bases — the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the d-bonacci base, which satisfies the equation Xd=Xd−1+Xd−2+⋯+X+1. If in a base being a d-bonacci number 1-block parallel addition is possible on an alphabet A, then #A⩾d+1; on the other hand, there exists a k∈N such that k-block parallel addition in this base is possible on the alphabet {0,1,2}, which cannot be reduced. In particular, addition in the Tribonacci base is 14-block parallel on alphabet {0,1,2}.

A New Extension of Quadratic B\'ezier Curves with Multiple Shape Parameters

A class of polynomial basis functions with multiple shape parameters is presented, which is an extension to quadratic Bernstein basis functions. The corresponding extended curve is also constructed and the properties are studied. The shape of the curve can be adjusted by altering the values of shape parameters

Solving leaky modes on a dielectric slab waveguide involving materials of arbitrary dielectric anisotropy with a finite-element formulation

A planar optical waveguide mode solver is established based on a finite-element (FE) formulation for determining the guided and leaky modes that exist on waveguides made of anisotropic materials with an arbitrary permittivity tensor, for example, with arbitrary optic-axis orientation in the uniaxially anisotropic material case. Correct numerical determination of the complex effective index, especially its imaginary part that gives the modal leakage, is particularly emphasized referring to available analytical solutions. For the situation when the optic axis changes its direction only in the plane parallel to the waveguide interface planes, analytical characteristic equations for solving purely guided and leaky modes are separately derived in a more systematic manner compared with prior analytical formulae reported more than three decades ago, with the obtained complex effective indices agreeing excellently with FE solutions. It is found that in the FE analysis of leaky modes, the thickness of the perfectly matched layer (PML) and the PML theoretical reflection coefficient should be properly chosen. The FE formulation is based on either the three electric-field components or the three magnetic-field components using quadratic nodal bases, resulting in a quadratic eigenvalue equation that is then solved by the shift-and-invert Arnoldi method.

B\'{e}zier-like Curve and B-spline-like Curve of Order Two

In this paper, a set of basis functions with three parameters m,α and β is constructed. It contains quadratic Bernstein basis and quadratic uniform B-spline basis as special cases. Based on the bases with parameters α = 1 and α = 1 2 , we define two kinds of curves, called Bezier-like curve and B-spline-like curve, respectively. They enjoy many properties of the usual Bezier curve and B-spline curve, and has similar endpoint behavior with the usual quadratic Bezier curve and quadratic uniform B-spline curve, respectively. The advantage of the Bezier-like curve in that as long as the control points of the adjacent curve segments satisfy the G1 continuity conditions, they will Gm−2 continuous when β ∈ (0, 1], and will Gm−1 continuous when β = 0. The advantage of the B-spline-like curve in that its shape and continuity can be adjusted integrally or locally. The Bezier-like surface and B-spline-like surface are also defined and some examples are given.

Protonated water clusters

A complete set of protonated water clusters conformers up to n = 8 have been evaluated by ab-initio Hartree-Fock method in 6-311G** molecular orbitals basis set and with quadratic complete basis set method. Configurations of ground-state conformers are in a good accord with those available in literature but present dissociation energies for the light clusters agree better with experimental values. A tentative analysis of experimental conditions for observation of certain clusters is presented in comparison to selected formation paths of clusters. Dissociation energies show a broad range of values.

Analysis of torsional barrier height of HSNO as the simplest S-nitrosothiol

Torsional barrier height of thionitrous acid is analysed with Gaussian-2(G2), quadratic complete basis set (CBS-Q) and DFT-B3LYP/CBS-Q (CBS-QB3) methods. In agreement with purely intuitive arguments, it was determined that the cis to trans barrier height is nearly 5.7–6.3 kcal mol − 1. In addition, the stability of rotation as function of competing dissociation pathways and also result of natural bond orbital analysis are discussed. Results of G2, CBS-Q and CBS-QB3 calculations on the torsional dissociation and rotation barrier of HSNO are presented. It was determined that the cis to trans barrier height is nearly 5.7-6.3 kcal mol − 1. HSNO torsional potential correlated very well with a potential function, represented by nine terms truncated Fourier series.

Multilevel augmentation method with wavelet bases for singularly perturbed problem

Multilevel augmentation method with wavelet bases is demonstrated to show as the fast technique for solving singularly perturbed problems. Linear and quadratic wavelet bases are employed for constructing the full form of matrix system. To reduce the size of matrix coefficients, the multilevel augmented technique is applied at each current basis level. It is found that the multilevel augmentation method is faster than the standard multilevel method at the same order of accuracy. Convergent rates for linear and quadratic bases are 2 and 4 respectively. By the application of wavelet bases, numerical accuracy can be easily improved by increasing just desired levels in the multilevel augmentation process.

Hexagonal honeycomb cell optimisation by way of meta-model techniques

This paper presents the result of an optimisation study by linear, quadratic, Kriging and radial basis meta-models in order to augment the crashworthiness characteristic of cellular structures. Thin-walled cellular structures (honeycomb) have the ability to absorb impact energy during crashing, thus it is important to enhance the crushing efficiency and optimise the structural reliability. The optimisation carried out in this study is aimed at maximising the energy absorption characteristics using meta-models while considering some limitation on the maximum force as a constraint. Achieving these characteristics is an important factor in crashworthiness analysis, which minimises the damage in dynamic performance. The objective of using various meta-models is to qualify the meta-model in crashworthiness analysis using different point selection schemes and different number of points. The optimisation is performed in two stages; through experimental design methods in which a set of sampling points is selected from design space and polynomial fitting in order to optimise the objective. It is concluded that D-optimal best suits response surface method for model approximation. Kriging performed best by space filling and the best point selection scheme for radial basis surrogate is latin hypercube design. The results show that for optimising the crashworthiness characteristics of honeycomb, Kriging and quadratic response surface (RS) are best, in terms of accuracy and robustness point of view and the radial basis neural network would be the second best. During this optimisation, the RS was combined with detailed geometrically simplified finite element model of honeycomb cell using ANSYS/LS-DYNA, LS-DYNA and LS-opt packages. Approximated functions combined with the finite element analysis were an effective tool to optimise highly non-linear impact problems. This has led to the development of validated algorithm that enabled the development of the optimised solutions.

DRBEM solution of mixed convection flow of nanofluids in enclosures with moving walls

This paper presents the results of a numerical study on unsteady mixed convection flow of nanofluids in lid-driven enclosures filled with aluminum oxide and copper–water based nanofluids. The governing equations are solved by the Dual Reciprocity Boundary Element Method (DRBEM), and the time derivatives are discretized using the implicit central difference scheme. All the convective terms and the vorticity boundary conditions are evaluated in terms of the DRBEM coordinate matrix. Linear boundary elements and quadratic radial basis functions are used for the discretization of the boundary and approximation of inhomogeneity, respectively. Solutions are obtained for several values of volume fraction (φ), the Richardson number (Ri), heat source length (B), and the Reynolds number (Re). It is disclosed that the average Nusselt number increases with the increase in volume fraction, and decreases with an increase in both the Richardson number and heat source length.

High-order B-splines based finite elements for delamination analysis of laminated composites

Finite element analysis of the delamination process in delaminated composites is addressed using high order Bézier elements relied on the Bernstein polynomials which are commonly used in Computer Aided Design. Laminae of a composite laminate are modeled with two dimensional Bézier elements. Zero-thickness cohesive elements, which are inserted along the interface in between the laminae to model the interface cracking, employ the high order univariate Bernstein basis functions. Quadratic, cubic and quartic Bernstein basis functions are examined. An open source pre-processing code is presented to generate Bézier meshes and interface elements. Implementation of a simple yet highly efficient arc-length solver which is able to trace equilibrium paths having multiple snap-backs is given. Numerical examples are presented for the analyses of benchmark composite delamination problems which include the double cantilever beam, the end notch flexure and the mixed mode bending tests. Additionally multiple delamination analysis of a composite specimen is also presented. Additionally multiple delamination analysis of a composite specimen is also investigated. For all given examples, the highly efficient performance of the proposed formulation is found.

Finite element method solution of the simplified P3 equations for general geometry applications

The method and performance of a finite element method based simplified P3 code developed for generalized geometry applications are presented. The code takes the mesh data produced by an external general-purpose mesh generator. The weighted residual method with Galerkin weighting is directly applied to the two SP3 equations which were slightly modified to attain symmetry. The linear system for the global problem involving two unknown flux moments per node is obtained by summing up the elementwise weighted residual equations. The solution accuracy is examined for a suite of benchmark problems consisting of typical thermal and fast reactor problems together with the comparisons against the corresponding diffusion theory solutions. The effectiveness of the quadratic basis function is assessed as well against the linear function. It is demonstrated that the SP3 formulation is particularly beneficial for fast neutron systems.

The finite element simulation for the shallow impurity in quantum dots

The finite element method (FEM) has been implemented in this paper to investigate the electronic state of shallow hydrogenic impurities in spherical GaAs (Al,As) quantum dots (SQDs) by taking into account the finite value of realistic potential barrier height. The nonlinear partial differential equations have been discretized by means of Galerkin's weighted residue method assuming a uniform partition and using a quadratic Lagrange basis function for each finite element of the physical domain considered. The impurity binding energies have been calculated numerically by solving the governing equations and compared with the variational method. The results we have obtained are in excellent agreement with those published in the literature.

Analysis of width- w non-adjacent forms to imaginary quadratic bases

We consider digital expansions to the base of $\tau$, where $\tau$ is an algebraic integer. For a $w \geq 2$, the set of admissible digits consists of 0 and one representative of every residue class modulo $\tau^w$ which is not divisible by $\tau$. The resulting redundancy is avoided by imposing the width $w$-NAF condition, i.e., in an expansion every block of $w$ consecutive digits contains at most one non-zero digit. Such constructs can be efficiently used in elliptic curve cryptography in conjunction with Koblitz curves. The present work deals with analysing the number of occurrences of a fixed non-zero digit. In the general setting, we study all $w$-NAFs of given length of the expansion. We give an explicit expression for the expectation and the variance of the occurrence of such a digit in all expansions. Further a central limit theorem is proved. In the case of an imaginary quadratic $\tau$ and the digit set of minimal norm representatives, the analysis is much more refined: We give an asymptotic formula for the number of occurrence of a digit in the $w$-NAFs of all elements of $\Z[\tau]$ in some region (e.g. a disc). The main term coincides with the full block length analysis, but a periodic fluctuation in the second order term is also exhibited. The proof follows Delange's method. We also show that in the case of imaginary quadratic $\tau$ and $w \geq 2$, the digit set of minimal norm representatives leads to $w$-NAFs for \emph{all} elements of $\Z[\tau]$. Additionally some properties of the fundamental domain are stated.

Optimality of the Width-w Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width\nbd-$w$ non-adjacent form (\wNAF): each block of $w$ consecutive digits contains at most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the \wNAF{}-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of \wNAF{}s can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal \wNAF{}-expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $\tau=\pm1\pm i$, optimality depends on the parity of $w$. Computational results for small trace are given.

NURBS-Based Isogeometric Analysis of Beams and Plates Using High Order Shear Deformation Theory

Isogeometric analysis (IGA) based on nonuniform rational B-splines (NURBS) is developed for static analysis of beams and plates using the third-order shear deformation theory (TSDT). TSDT requires C1-continuity of generalized displacements; quadratic, cubic, and quartic NURBS basis functions are utilized to satisfy this requirement. Due to the noninterpolatory nature of NURBS basis functions, a penalty method is presented to enforce the essential boundary conditions. A series of numerical examples of thick and thin beams and plates with different boundary conditions are presented. Results are compared with analytical solutions and other numerical methods. First-order shear deformation theory (FSDT) is also employed and compared with TSDT in the stress analysis. The effects of aspect ratios and boundary conditions are discussed as well.

Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Mesh quality plays an essential role in finite element applications, since it affects the efficiency of the simulation with respect to solution accuracy and computational effort. Therefore, mesh smoothing techniques are often applied for improving mesh quality while preserving mesh topology. One of these methods is the recently proposed geometric element transformation method (GETMe), which is based on regularizing element transformations. It will be shown numerically that this smoothing method is particularly suitable, from an applicational point of view, since it leads to a significant reduction of discretization errors within the first few smoothing steps requiring only little computational effort. Furthermore, due to reduced condition numbers of the stiffness matrices the performance of iterative solvers of the resulting finite element equations is improved. This is demonstrated for the Poisson equation with a number of meshes of different complexity and type as well as for piecewise linear and quadratic finite element basis functions. Results are compared to those obtained by two variants of Laplacian smoothing and a state of the art global optimization-based approach.

Finite element solution of coupled-partial differential and ordinary equations in multicomponent polymeric coatings

Mass transport equations in multicomponent polymeric coatings are nonlinear coupled partial differential equations. These equations were solved using Galerkin's method of finite elements which converts them to ordinary differential equations. Residuals were made orthogonal by using quadratic basis functions. Non-uniform elements were used to capture steep concentration gradient near the top of the coating. Finite element formulation has been solved using ode15s of MATLAB. Results are in very good agreement with the earlier results using different solution techniques.

Purely periodic expansions in systems with negative base

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama’s result for positive Pisot unit base β, we find a sufficient condition so that there exist an interval J containing the origin such that the (−β)-expansion of every rational number from J is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property ( $\mathop {\mathrm {Fin}}\nolimits \, (\beta)= {\mathbb {Z}}[\beta]$ ) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that $\mathop {\mathrm {Fin}}\nolimits \, (-\beta)= {\mathbb {Z}}[\beta]$ is not necessary in the case of negative bases.