Hermite Method Research Articles

MODELLING OF DISPLACEMENT WASHING OF PULP FIBERS USING THE HERMITE COLLOCATION METHOD

A non-linear mathematical model for displacement washing of pulp fibers with suitable boundary conditions is presented. Model equations are divided into two phases: particle phase and external fluid phase. The Hermite collocation method (HCM) is employed to solve the non-linear model equations. The validity and efficiency of the proposed model is demonstrated by comparing the numerical results with the experimental ones. Stability of the proposed method is checked by evaluating L2 and L? norms at different time intervals. Industrial parameters such as displacement ratio and wash yield have been calculated to check the validity of the model. Numerical values obtained from the model equations are presented in terms of 2D graphs.

Synthesis of phase and magnitude of images by Hermite projection method

The task of the synthesis of phase and magnitude from different images by Hermite projection method has been considered. In Hermite projection method the magnitude and the phase of Fourier transform are approximated using finite Fourier series of Hermite functions. It was shown that the synthesis results using projection approach are better than synthesis results using the discrete Fourier transform. The new algorithm of images boundary Gibbs effect suppression was introduced.

Numerical analysis of drift resistive inertial ballooning modes

Three numerical techniques employing differentiation matrices are used to investigate the linear stability of drift-resistive-inertial ballooning mode for conditions that occur in tokamak edge plasmas. Hermite and Sinc spectral methods are applied to compute the ballooning mode eigenvalues and eigenvectors. In addition, a finite difference method is utilized to construct a differentiation matrix in order to verify results obtained using the spectral methods. It is shown that the spectral methods converge more rapidly than the finite difference method. Ballooning and strongly ballooning approximations are used to calculate the eigen-spectrum. The techniques that are utilized in this paper for calculating eigenvalues are quite general and are relevant to investigate other modes that use the ballooning mode formalism.

Hermite radial basis collocation method for vibration of functionally graded plates with in-plane material inhomogeneity

A meshfree Hermite-type collocation method with radial basis functions is presented to investigate the free vibration of thin FGM plates with in-plane material inhomogeneity. For the eigenvalue analysis of thin plate vibration, conventional radial basis collocation method (RBCM) leads to an over-determined system for the eigenproblems, while Hermite radial basis collocation method (HRBCM) introducing more degrees of freedom can result in a determined system. Convergence and comparison studies with analytical solutions demonstrate the effectiveness and accuracy of the proposed method. Effects on the natural frequencies and mode shapes originated from the material inhomogeneity are numerically investigated.

Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.

Mathematical problems of calculating the multi-zone models of a cycle of internal combustion engine with spark ignition

The article describes the features of mathematical models of internal combustion engine cycles which do not allow to put ordinary differential equations (ODE) of these models in the Cauchy form. The authors proposed a stable algorithm of numerical integration of ODE systems that are not solved with respect to derivatives, using Hermite splines and methods of nonlinear programming.

Solution of non linear singular perturbation equation using Hermite collocation method

A numerical method for solving second order, transient, parabolic partial differential equation is presented. The spatial discretization is based on Hermite collocation method (HCM). It is a combination of orthogonal collocation method and piecewise cubic Hermite interpolating polynomials. The solution is obtained in terms of cubic Hermite interpolating basis. Numerical results have been plotted in terms of time space graphs to illustrate the applicability and efficiency of the HCM in terms of convergence and stability analysis. The present method has been compared with orthogonal collocation method for different value of e. Introduction The orthogonal collocation method is one of several weighted residual techniques. Orthogonal collocation method for numerical solution of partial differential equations has been followed various investigators (Ghanaei & Rahimpour (2010), Rohman et al. (2011), Wu et al. (2011) & Zugasti (2012)) due to its compailibilty, simplicity and accuracy which makes it different from other numerical methods such that Galerkin (Nadukandi et al. (2010)), Least square method (Ren et al. (2012)) and finite difference (Jha (2013)). In this method, an approximate solution is substituted into the differential equation to form the 5398 Happy Kumar, Shelly Arora and R. K. Nagaich residual. Then, this residual is set to zero at collocation points. The choice of collocation points plays an important role in collocation techniques for the convergence and efficiency. Generally, zeros of Jacobi polynomials are taken as collocation points over the normalized interval. Hermite collocation method (HCM) is combination of orthogonal collocation method and piecewise cubic Hermite interpolating polynomials. HCM can be expressed as a linear combination of Hermite basis functions. Hermite functions have great advantages that functions and its slope are continuous at junction points. Numerous investigators have used this technique in different manner to solve different type of problem such as near-singular problems (Lang and Sloan (2002)), convection-diffusion problems (Rocca et al. (2005)) and nonlinear Lane–Emden type equations (Peirce (2010)). Hermite collocation is considered for the discretization of second-order boundary value problems, the usual choice of Hermite is either quadratic or cubic at one or two collocation points. In the case of quadratic or cubic, Hermite collocation in second order problems, the computed approximations exhibit up to fourth order convergence (Prenter (1975), Sun (2000), Parand et al. (2010)). Consider the second order non-linear parabolic boundary value problem: x y x y t z t y ∂ ∂ − ∂ ∂ = ∂ ∂ − + ∂ ∂ 2 2 1 e θ θ (x,t)∈(0,1)×(0,T) (1) ) (y f z = (2) 0 2 1 = ∂ ∂ + x y q y q at x = 0 (3) 0 4 3 = ∂ ∂ + x y q y q at x = 1 (4) 1 = y at t= 0 (5) where e,Θ, q1, q2, q3, q4 are constants Cubic Hermite polynomials Hermite interpolating polynomials was first introduced by Charles Hermite (1822-1905). It is an extension of Lagrange interpolating polynomials as in Hermite interpolating polynomials, both the function and its derivative are to be assigned values at interpolating point. An nth-order Hermite polynomial in x is a polynomial of order 2n+1 and therefore, cubic Hermite interpolating polynomials particular case of general Hermite interpolating polynomials for n=1. It consists of two node points and two tangents in cubic polynomial and defined as 1 3 2 ) ( 2 3 1 + − = x x x H x x x x H + − = 2 3 2 2 ) ( Solution of non linear singular perturbation equation 5399 2 3 3 3 2 ) ( x x x H + − = 2 3 4 ) ( x x x H − = Fig.1. Cubic Hermite interpolating polynomials Figure.1. shows the behavior of the Cubic Hermite interpolating polynomials. The values of H1,H2, H3 and H4 lies within [0, 1] as x goes from 0 to 1 and their derivatives are unity or zero at the end points. The Cubic Hermite approximation is defined as: ) (u y l =∑ = 4 1 ) ( ) ( i i i u H t a l Where l = 1, 2,.......,k (6) Where, k is the number of elements and ) (t ai l ’s are the continuous functions of ‘t’ in l th element.{Hi(u)} are piecewise cubic Hermite polynomials as defined above. The first and second order discretized derivatives of the trial function l y taken at j th collocation point are defined by Aji and Bji respectively, where Aji=H’i(uj) and Bji=H”i(uj). After applying Hermite collocation method, the following set of collocation equations is obtained: ∑ ∑ = = − = ∂ ∂ − + ∂ ∂ 4

Hermite finite elements for diffusion phenomena

Two new Hermite finite elements are shown to be an advantageous alternative to well-known mixed methods in the simulation of diffusion processes in heterogeneous anisotropic media. Both are N-simplex based for N=2 and N=3 and provide flux continuity across inter-element boundaries. One of the methods denoted by P2H was introduced by the first author and collaborator for the case of homogeneous and isotropic media. Its extension to the case of heterogeneous and/or anisotropic cases is exploited here, keeping an implementation cost close to the popular Raviart–Thomas mixed finite element of the lowest order, known as RT0. The other method studied in detail in this work is a new Hermite version of the latter element denoted by RT0M. Formal results are given stating that, at least in the case of a constant diffusion, RT0M is significantly more accurate than RT0, although both elements have essentially the same implementation cost. A thorough comparative numerical study of the Hermite methods and RT0 is carried out in the framework of highly heterogeneous media among other cases. It turns out that both are globally superior all the way, and roughly equivalent to each other in most cases.

Rapid anharmonic vibrational corrections derived from partial Hessian analysis

Vibrational analysis within a partial Hessian framework can successfully describe the vibrational properties of a variety of systems where the vibrational modes of interest are localized within a specific region of the system. We have developed a new approach to calculating anharmonic frequencies based on vibrational frequencies and normal modes obtained from a partial Hessian analysis using second-order vibrational perturbation theory and the transition optimized shifted Hermite method. This allows anharmonic frequencies for vibrational modes that are spatially localized to be determined at a significantly reduced computational cost. Several molecular systems are examined in order to demonstrate the effectiveness of this method including organic molecules adsorbed on the Si(100)-2×1 surface, model peptides in solution, and the C-H stretching region of polycyclic aromatic hydrocarbons. Overall, for a range of systems, anharmonic frequencies calculated using the partial Hessian approach are found to be in close agreement with the results obtained using full anharmonic calculations while providing a significant reduction in computational cost.

Investigating the Calculation of Anharmonic Vibrational Frequencies Using Force Fields Derived from Density Functional Theory

The calculation of anharmonic vibrational frequencies for a set of small molecules has been examined to explore the merit of applying such computationally expensive approaches for large molecules with density functional theory. The performance of different hybrid and gradient-corrected exchange-correlation functionals has been assessed for the calculation of anharmonic vibrational frequencies using second-order vibrational perturbation theory with two- and four-mode couplings and compared to the recently developed transition optimized shifted Hermite method. A range of exchange-correlation functionals (B3LYP, BLYP, EDF1, EDF2, B97-1, B97-2, HCTH-93, HCTH-120, HCTH-147, and HCTH-407) have been evaluated with reference to a large experimental data set comprising 88 species and 655 modes as well as a smaller set of shifts in frequency because of anharmonicity derived from experimental data. The anharmonic frequencies calculated using hybrid functionals provide the best agreement with experiment, and are not significantly improved by frequency scaling factors, indicating an absence of significant systematic error. For the molecules studied, the B97-1 and B97-2 functionals give the closest overall agreement with experiment, although the improvement over the best case for pure harmonic frequencies is modest. Predictions of the experimental anharmonic shifts are closest for the B3LYP and EDF2 functionals, with B97-1 performing well because of a good description of the harmonic force field. Investigations using modified hybrid functionals with increased fractions of Hartree-Fock exchange indicate that approximately 20% Hartree-Fock exchange is optimal.

Simulation of compressible flows using Hermite methods

Spectral element methods based on Hermite interpolation have a number of unique properties. First of all, the stabilization inherent in the interpolation process is sufficient to suppress nonlinear instabilities observed with other discretization schemes and leads to accurate linear transport of nonsmooth solutions. Second, and most important, they allow purely local time-stepping procedures limited only by geometric domain-of-dependence requirements. Thus high-order Hermite methods maximize the computation-to-communication ratio and therefore they admit highly efficient implementations on multicore processors. In this talk we focus on the application of Hermite methods to simulate unsteady compressible flows. Examples will include the direct simulation of the aeroacoustics of a low Reynolds number subsonic jet, as well as studies of more basic sound radiating flows. The latter will illustrate the coupling of Hermite methods with more standard discontinuous Galerkin discretizations to handle physical boundaries.

P-adaptive Hermite methods for initial value problems

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems posed in 1+1 and 2+1 dimensions.

Convergence of a split-step Hermite method for the Gross-Pitaevskii equation

Convergence of a split-step Hermite method for the Gross-Pitaevskii equation

Cubic B-spline curve approximation by curve unclamping

A new approach for cubic B-spline curve approximation is presented. The method produces an approximation cubic B-spline curve tangent to a given curve at a set of selected positions, called tangent points, in a piecewise manner starting from a seed segment. A heuristic method is provided to select the tangent points. The first segment of the approximation cubic B-spline curve can be obtained using an inner point interpolation method, least-squares method or geometric Hermite method as a seed segment. The approximation curve is further extended to other tangent points one by one by curve unclamping. New tangent points can also be added, if necessary, by using the concept of the minimum shape deformation angle of an inner point for better approximation. Numerical examples show that the new method is effective in approximating a given curve and is efficient in computation.

Reprint of: Development of Arbitrary-Order Hermite Methods for Simulation and Analysis of Turbulent Jet Noise

In this short note a brief description of Hermite methods is given. Previous and ongoing development of arbitrary order Hermite methods for the simulation of turbulent jets is also presented. In addition we outline how Hermite methods can be hybridized with discontinuous Galerkin methods to handle boundary conditions in a straightforward way.

Development of arbitrary-order hermite methods for simulation and analysis of turbulent jet noise

In this short note a brief description of Hermite methods is given. Previous and ongoing development of arbitrary order Hermite methods for the simulation of turbulent jets is also presented. In addition we outline how Hermite methods can be hybridized with discontinuous Galerkin methods to handle boundary conditions in a straightforward way.

On a Steffensen–Hermite method of order three

In this paper we study a third order Steffensen type method obtained by controlling the interpolation nodes in the Hermite inverse interpolation polynomial of degree 2. We study the convergence of the iterative method and we provide new convergence conditions which lead to bilateral approximations for the solution; it is known that the bilateral approximations have the advantage of offering a posteriori bounds of the errors. The numerical examples confirm the advantage of considering these error bounds.

P-Cyclic SOR for BVPs with periodic boundary conditions

The employment of finite element or finite difference discretization schemes, for the numerical solution of Boundary Value Problems (BVPs) with periodic type Boundary Conditions (BCs), leads to a large and sparse linear system whose coefficient matrix is in normal p-cyclic form. The use of block iterative methods, for the solution of such linear systems, and the demand for fast convergence rates, require the optimal repartitioning of the coefficient matrix. In this work, we make use of the finite element Hermite collocation method to discretize the BVP and the SOR iterative method to solve the corresponding sparse linear system. The optimal repartitioning of the collocation coefficient matrix leads to SOR methods with optimal rates of convergence.

High-order time-splitting Hermite and Fourier spectral methods

In this paper, we are concerned with the numerical solution of the time-dependent Gross–Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions; on the other hand, restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration.

Frequency-domain fatigue analysis of wide-band stationary Gaussian processes using a trimodal spectral formulation

In this paper, a procedure for fatigue analysis of a general wide-band stationary Gaussian process is developed in the frequency domain using a trimodal spectral formulation, based on a generalization of the principle proposed by Jiao and Moan for predicting bimodal fatigue damage. The novel method approximates the rainflow cycles with large amplitudes as the sum of the envelopes of the process components. First, the method is derived and the corresponding accuracy is demonstrated for ideal trimodal Gaussian processes whose spectra exhibit peaks at three well separated modes, each assumed to be narrow-banded. Hermite numerical integration method is applied to evaluate the fatigue damage due to a Rayleigh sum distribution since there is no closed-form solution for a random variable which is the sum of more than two Rayleigh random variables. Then, the method is further developed and applied to general wide-band Gaussian processes by dividing the spectrum into three segments with the same variances and calculating the fatigue damage in the same way as for the ideal trimodal processes. Based on extensive time series simulated with spectra of wave- and wind-induced structural responses and with generally defined wide-band spectra, the rainflow cycle counting method has been applied to estimate the fatigue damage in the time domain in order to check the accuracy of the proposed method and other frequency-domain methods. It is found that the narrow-band assumption can be very well made for a process with a Vanmarcke’s bandwidth parameter which is less than 0.5. For most of the simulated processes with bandwidth parameters between 0.5 and 0.85, the proposed method slightly overestimates the fatigue damage on average. If the bandwidth is greater than 0.85, the fatigue damage obtained by the trimodal formulation may be significantly overestimated in some cases, however spectra with very high bandwidth parameters might be unrealistic. Anyway, it has been shown that the overestimation can be alleviated by using a formulation with more modes for these cases. Moreover, two empirical formulae for wide-band fatigue damage estimation, one derived by Dirlik and the other proposed by Benasciutti and Tovo, have also been verified to be very accurate but slightly underestimate the fatigue damage for a wide range of the bandwidth parameters considered in this paper.