Gauss Lobatto Research Articles

A connection between Szegő–Lobatto and quasi Gauss-type quadrature formulas

In this paper we obtain new results on positive quadrature formulas with prescribed nodes for the approximation of integrals with respect to a positive measure supported on the unit circle.We revise Szegő–Lobatto rules and we present a characterization of their existence. In particular, when the measure on the unit circle is symmetric, this characterization can be used to recover, in a more elementary way, a recent characterization result for the existence of positive quasi Gauss, quasi Radau and quasi Lobatto rules (quasi Gauss-type), due to B. Beckermann et. al. Some illustrative numerical examples are finally carried out in order to show the powerfulness of our results.

On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects

Abstract On the basis of an efficient numerical solution methodology, the free vibration response of third-order shear deformable nanobeams made of functionally graded materials (FGMs) around the postbuckling domain is investigated incorporating the effects of surface free energy. Gurtin–Murdoch elasticity theory in conjunction with von Karman geometric nonlinearity is implemented into the classical third-order shear deformation beam theory. In order to consider balance conditions on the surfaces of nanobeam, it is assumed that the bulk normal stress is distributed cubically through the thickness. The material properties are assumed to be graded in the thickness direction based on power-law distribution. Using Hamilton’s principle, the non-classical nonlinear governing differential equations of motion and associated boundary conditions are derived. Then generalized differential quadrature method is employed to discretize the governing equations and solution domain according to the Chebyshev–Gauss–Lobatto grid points. Subsequently, the pseudo-arc length continuation technique is utilized to solve the nonlinear problem. It is demonstrated that in contrast to the prebuckling domain, in the postbuckling domain, the natural frequency of FGM nanobeam decreases by increasing the value of material property gradient index. Also, it is revealed that the surface effect plays more important role on the vibration characteristics of the buckled FGM nanobeams with lower thicknesses.

Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects

Abstract. The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge–Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma–pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia–gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.

A time-domain high-order spectral finite element for the simulation of symmetric and anti-symmetric guided waves in laminated composite strips

A time domain spectral finite element is developed for improving the efficiency of numerical simulations of guided waves in laminated composite strips. The finite element relies on a new generalized laminate mechanics model formulated to represent symmetric and anti-symmetric Lamb waves. The laminate mechanics incorporate third-order polynomial terms for the approximation of axial and transverse displacement fields through the thickness and consider the displacements of the upper and lower surfaces as degrees of freedom. The laminate theory formulation is easily expanded to a high-order layerwise model. Based on the resultant governing equations of the laminate section, a new finite element with 8 nodal degrees of freedom is formulated; its nodes are collocated with Gauss–Lobatto–Legendre integration points in order to improve computational efficiency. Stiffness and mass matrices are assembled and the transient response is predicted using the explicit central differences time integration scheme. The transient response of Aluminum, Carbon Fiber Reinforced Polymer laminated and sandwich strips is investigated. Numerical results are validated against a semi-analytical solution. The accuracy and computational efficiency of the introduced element regarding the prediction of symmetric and anti-symmetric wave propagation is also quantified.

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

SummaryWe demonstrate the potential of collocation methods for efficient higher‐order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra‐weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher‐order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher‐order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. Copyright © 2014 John Wiley & Sons, Ltd.

Surface effects on the free vibration behavior of postbuckled circular higher-order shear deformable nanoplates including geometrical nonlinearity

This investigation deals with the free vibration characteristics of circular higher-order shear deformable nanoplates around the postbuckling configuration incorporating surface effects. Using the Gurtin–Murdoch elasticity theory, a size-dependent higher-order shear deformable plate model is developed which takes account all surface effects including surface elasticity, surface stress and surface density. Geometrical nonlinearity is considered based on the von Karman type nonlinear strain–displacement relationships. Also, in order to satisfy the balance conditions between bulk and surfaces of nanoplate, it is assumed that the normal stress is distributed cubically through the thickness of nanoplate. Hamilton׳s principle is utilized to derive non-classical governing differential equations of motion and related boundary conditions. Afterwards, an efficient numerical methodology based on a generalized differential quadrature (GDQ) method is employed to solve numerically the problem so as to discretize the governing partial differential equations along various edge supports using Chebyshev–Gauss–Lobatto grid points and pseudo arc-length continuation technique. A comparison between the results of present non-classical model and those of the classical plate theory is conducted. It is demonstrated that in contrast to the prebuckling domain, for a specified value of axial load in the postbuckling domain, increasing the plate thickness leads to higher frequencies.

Pseudo-spectral methods in one-dimensional magnetostriction

In this paper a pseudo-spectral method is proposed to solve a one-dimensional model of a saturated hard ferromagnetic thin-film structure within the Euler–Bernoulli kinematics. The model accounts for the non-local nature of the magneto-elastic coupling and interaction is in the form of a logarithmic potential. The proposed solution method adopts global polynomial interpolation at a main grid, given by the Gauss–Lobatto points, and it employs a secondary grid, consisting of the Gauss points, to perform the Gaussian quadrature. The two grids are non-overlapping to avoid the singularity. Interpolation relates the unknowns, evaluated at the secondary grid, to their values at the collocation grid. Furthermore, the integration interval is parted about the singularity point. The procedure is assessed through the relative equilibrium residual for different values of the approximating polynomial degree and of the quadrature order. Maximum, average and standard deviation of the error are presented. An asymptotic analysis yields the Boundary Solution to the problem and results are compared when the latter is introduced in the numerical scheme. It is shown that its contribution is important in reducing the overall error. The equilibrium residual is plotted and its behavior discussed. It is further shown that numerical precision significantly affects the results at midspan, owing to the self-equilibrium of the system, thereby a limit exists to the best accuracy which may be gained through a more accurate interpolation.

Energy conserving discontinuous Galerkin spectral element method for the Vlasov–Poisson system

We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov–Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss–Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge–Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).

An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

In this paper, we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized Zakharov system (GZS) subject to initial-boundary conditions. The Jacobi pseudospectral approximation is investigated for spatial approximation of the GZS. It possesses the spectral accuracy in space. The Jacobi–Gauss–Lobatto quadrature rule is established to treat the boundary conditions, and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable. This scheme has the advantage of allowing us to obtain the spectral solution in terms of the Jacobi parameters α and β, which therefore means that the algorithm holds a number of collocation methods as special cases. Finally, two illustrative examples are implemented to assess the efficiency and high accuracy of the Jacobi pseudo-spectral scheme.

Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

Abstract A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form ∫-11 f′(x)(1-x)α dx, α > -1, is derived and applied to approximation of the usual fractional derivative. A software implementation of such quadratures was done by the recent Mathematica package OrthogonalPolynomials (cf. [A.S. Cvetković, G.V. Milovanović, Facta Univ. Ser. Math. Inform. 19 (2004), 17–36] and [G.V. Milovanović, A.S. Cvetković, Math. Balkanica 26 (2012), 169–184]). Several numerical examples are presented and they show the effectiveness of the proposed approach.

Numerical solution of the coupled viscous Burgers equations by Chebyshev–Legendre Pseudo-Spectral method

In this paper, we consider Chebyshev–Legendre Pseudo-Spectral (CLPS) method for solving coupled viscous Burgers (VB) equations. A leapfrog scheme is used in time direction, while CLPS method is used for space direction. Chebyshev–Gauss–Lobatto (CGL) nodes are used for practical computation. The error estimates of semi-discrete and fully-discrete of CLPS method for coupled VB equations are obtained by energy estimation method. The numerical results of the present method are compared with the exact solution for two test problems.

Surface effects on the nonlinear forced vibration response of third-order shear deformable nanobeams

Given the high surface to volume ratio, the nonlinear forced vibration behavior of third-order shear deformable nanobeams in the presence of the both effects of surface stress and that of surface inertia is investigated. Gurtin–Murdoch elasticity theory is utilized within the framework of third-order shear deformation beam theory to develop a novel non-classical beam model to incorporate surface effects into the forced vibration analysis of nanobeams. A cubic variation through the thickness of nanobeam is considered for the normal stress component of the bulk in order to satisfy the surface equilibrium equations. Hamilton’s principle is used to derive size-dependent nonlinear governing differential equations of motion. The equations are solved numerically using generalized differential quadrature method with an iterative algorithm on the basis of shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, based on the Galerkin’s technique, the set of nonlinear partial differential equations are reduced into a time-varying set of ordinary differential equations of Duffing type. At the end, the pseudo arc-length method is employed to solve the set of nonlinear equations of the time domain. It is observed that by increasing the beam thickness, surface effects on the nonlinear forced vibration behavior of nanobeam diminish which leads to increasing the deviation from the linear response.

Static analysis of higher order sandwich beams by weak form quadrature element method

Based on the extended high order sandwich panel theory (EHSAPT) and differential quadrature rule, an N-node novel weak form quadrature sandwich beam element is established. Gauss Lobatto Legendre (GLL) points are utilized as element nodes and GLL quadrature is used to obtain the element stiffness matrix and work equivalent load. Detailed formulations are given. Convergence study is performed. Numerical results are presented for sandwich beams with different boundary conditions subjected to distributed loadings. Different materials for the face sheets and core of the beam structure are considered. It is shown that the proposed beam element can yield very accurate displacements and stresses as compared to theoretical solutions.

Surface energy effects on the free vibration characteristics of postbuckled third-order shear deformable nanobeams

The prime aim of the current study is to predict the free vibration behavior of third-order shear deformable nanobeams in the vicinity of postbuckling configuration and in the presence of surface effects which includes surface elasticity, residual surface stress and surface inertia. To accomplish this end, Gurtin–Murdoch elasticity theory within the framework of third-order shear deformation beam theory is employed. In order to satisfy the balance conditions between the bulk and surfaces of nanobeam, a cubic distribution is considered for the normal stress through the thickness. By using Hamilton’s principle, the non-classical governing differential equations of motion including von Karman geometric nonlinearity are derived. After using generalized differential quadrature (GDQ) method to discretize the governing equations on the basis of Chebyshev–Gauss–Lobatto grid points, the pseudo-arc length continuation technique is utilized to solve the eigenvalue problem. The natural frequencies of nanobeam corresponding to the both prebuckling and postbuckling domains are obtained for various buckling mode shapes based on the numerical solution strategy. It is demonstrated that in the prebuckling domain of the first vibration mode shape, increasing of beam thickness leads to lower natural frequency for all types of boundary conditions, but this behavior becomes reverse in the postbuckling domain.

A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation

We consider an initial-boundary-value problem for a time-fractional diffusion equation with initial condition u0(x) and homogeneous Dirichlet boundary conditions in a bounded interval [0, L]. We study a semidiscrete approximation scheme based on the pseudo-spectral method on Chebyshev–Gauss–Lobatto nodes. In order to preserve the high accuracy of the spectral approximation we use an approach based on the evaluation of the Mittag-Leffler function on matrix arguments for the integration along the time variable. Some examples are presented and numerical experiments illustrate the effectiveness of the proposed approach.

Space–time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains

The spectral method is proposed for the partial integro-differential equations with a weakly singular kernel on irregular domains. The space discretization is based on the nodal spectral element method using the Lagrange polynomials basis associated with the Gauss–Lobatto–Legendre quadrature nodes. Also the model is discretized in time with the Legendre spectral Galerkin method. The discretization leads to conversion of the problem to a Sylvester matrix equation which can be solved efficiently by the QZ algorithm (Gardiner et al., 1992). The convergence of the method is proven by providing a priori L2-error estimate. Numerical results illustrate the efficiency and spectral accuracy of the proposed method.

Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis

We explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis. The rules studied encompass tensor-product Gauss and Gauss–Lobatto rules, and certain so-called monomial rules that do no possess a tensor-product structure. The objective of the study is to determine quadrature strategies, which enjoy the same accuracy and stability behavior as full Gauss quadrature, but with significantly fewer quadrature points. Several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard C0-continuous finite elements of the same order.

Spectral-Prism Element for Multi-Scale Layered Package-Chip Co-Simulations Using the Discontinuous Galerkin Time-Domain Method

A new kind of prism element with a triangular base is presented for discretization of multi-scale layered structures within the discontinuous Galerkin time-domain framework. Mixed-order curl-conforming vector basis functions are used in the triangular bases of the prismatic element. The height of the prism adopts spectral basis functions based on Gauss–Lobatto–Legendre polynomials, with an arbitrary order of interpolation. This method combines the flexibility of triangles with the accuracy of spectral elements for layered structures. Eigenvalues obtained show better results than traditional finite elements using tetrahedrons and hexahedrons. For transient analysis, the implicit Crank–Nicholson method is implemented for sequential sub-domains. This kind of arrangement of sub-domains produces a block tridiagonal linear system, thus allowing a block Thomas algorithm to solve the system efficiently. A package-to-chip example shows the efficacy of this method.

A spectral-element discontinuous Galerkin lattice Boltzmann method for simulating natural convection heat transfer in a horizontal concentric annulus

We present a spectral-element discontinuous Galerkin lattice Boltzmann method to solve incompressible natural convection flows based on the Bousinessq approximation. A passive-scalar thermal lattice Boltzmann model is used to resolve flows for variable Prandtl number. In our model, we solve the lattice Boltzmann equation for the velocity field and the advection–diffusion equation for the temperature field. As a result, we reduce the degrees of freedom when compared with the passive-scalar double-distribution model, which requires the solution of several equations to resolve the temperature field. Our numerical solution is represented by the tensor product basis of the one-dimensional Legendre–Lagrange interpolation polynomials. A high-order discretization is employed on body-conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Within the discontinuous Galerkin framework, we weakly impose boundary and element-interface conditions through the numerical flux. A fourth-order Runge–Kutta scheme is used for time integration with no additional cost for mass matrix inversion due to fully diagonal mass matrices. We study natural convection fluid flows in a square cavity and a horizontal concentric annulus for Rayleigh numbers in the range of Ra=103–108. We validate our numerical approach by comparing it with finite-difference, finite-volume, multiple-relaxation-time lattice Boltzmann, and spectral-element methods. Our computational results show good agreement in temperature profiles and Nusselt numbers using relatively coarse resolutions.

A spectral method for the triangular cavity flow

We describe the use of a spectral collocation method to compute the characteristics of incompressible, viscous flow in a lid driven right triangular cavity with wall motion away from the right angle. First, the 2-D Gauss–Lobatto grid on the square [-1,1]×[-1,1] is mapped to a triangular domain by means of a singular mapping. A singularity subtraction technique is used to account for the most singular terms of the asymptotic expansions near the top corners of the flow. The unsteady Navier–Stokes equations in the vorticity–streamfunction formulation are then solved in this transformed grid using an influence matrix technique. A third order Adams Bashforth/Backward differentiation method appears to provide excellent numerical stability for the scheme and also permits a larger critical time step. Detailed characteristics of the flow and comparisons with published results are presented.