Gauss Lobatto Research Articles

Numerical simulation of wave propagation using spectral finite elements

For the analysis of wave propagation at high frequencies, the spectral finite element method is under investigation. In contrast to the conventional finite element method, high-order shape functions are used. They are composed of Lagrange polynomials with nodes at the Gauss–Lobatto–Legendre points. The Gauss–Lobatto–Legendre integration scheme is applied in order to obtain a diagonal mass matrix. The resulting system equations can be solved efficiently. In the numerical examples, spectral finite elements with shape functions of different order are applied to a plane strain problem. The numerical examples cover structures without and with stiffness discontinuities. It is shown that the results agree well with analytical and experimental solutions.

Spectral formulation of finite element methods using Daubechies compactly-supported wavelets for elastic wave propagation simulation

A novel and generic formulation of the wavelet-based spectral finite element approach, which is applicable to linear transient dynamics and elastic wave propagation problems, is presented in this paper. In order to spectrally formulate the variational form of governing equations in a temporally-decoupled manner, the wavelet-Galerkin discretization based on Daubechies compactly-supported wavelets is employed. The spectral formulation reduces the problem to a set of temporally independent equations which can be solved in parallel. It is demonstrated that for the spatial discretization, any class of standard finite element method can be adopted to facilitate capturing the complex geometries and boundary conditions. It is demonstrated that the wavelet-based temporal discretization is not influenced by the finite element mesh. Also frequency-dependent damping models are straight-forward to apply. These, along with the fact that the method is directly amenable to parallel computation make the approach very promising for wave propagation problems.To attain hp-refinement capability and spectral convergence properties for the spatial discretization, a higher-order Lagrangian FEM on the Gauss–Lobatto–Legendre grid, i.e. spectral element method, is adopted. For the sake of numerical investigation, a number of 2D and 3D examples including anisotropic and non-homogeneous structures are studied. The accuracy and the convergence rate of the method are evaluated and found to be superior to the explicit Newmark time integration. Possible approaches for speeding up the method are also discussed.

An efficient pseudo‐spectral Legendre–Galerkin method for solving a nonlinear partial integro‐differential equation arising in population dynamics

The pseudo‐spectral Legendre–Galerkin method (PS‐LGM) is applied to solve a nonlinear partial integro‐differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS‐LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS‐LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS‐LGM with a semi‐implicit time integration method such as second‐order backward differentiation formula and Adams‐Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two‐dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd.

A composite Chebyshev finite difference method for nonlinear optimal control problems

In this paper, a composite Chebyshev finite difference method is introduced and is successfully employed for solving nonlinear optimal control problems. The proposed method is an extension of the Chebyshev finite difference scheme. This method can be regarded as a non-uniform finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev–Gauss–Lobatto points. The convergence of the method is established. The nice properties of hybrid functions are then used to convert the nonlinear optimal control problem into a nonlinear mathematical programming one that can be solved efficiently by a globally convergent algorithm. The validity and applicability of the proposed method are demonstrated through some numerical examples. The method is simple, easy to implement and yields very accurate results.

On the remainder term of Gauss–Radau quadrature with Chebyshev weight of the third kind for analytic functions

For analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ∓1 and a sum of semi-axes ρ>1, for Gauss–Radau quadrature formula with Chebyshev weight function of the third kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi in his paper [W. Gautschi, On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures, Rocky Mounatin J. Math. 21 (1991) 209-206]. In this way the last unproved conjecture from the mentioned paper is now verified.

A numerical investigation of a spectral‐type nodal collocation discontinuous Galerkin approximation of the Euler and Navier–Stokes equations

SUMMARYIn this paper, we focus on the applicability of spectral‐type collocation discontinuous Galerkin methods to the steady state numerical solution of the inviscid and viscous Navier–Stokes equations on meshes consisting of curved quadrilateral elements. The solution is approximated with piecewise Lagrange polynomials based on both Legendre–Gauss and Legendre–Gauss–Lobatto interpolation nodes. For the sake of computational efficiency, the interpolation nodes can be used also as quadrature points. In this case, however, the effect of the nonlinearities in the equations and/or curved elements leads to aliasing and/or commutation errors that may result in inaccurate or unstable computations. By a thorough numerical testing on a set of well known test cases available in the literature, it is here shown that the two sets of nodes behave very differently, with a clear advantage of the Legendre–Gauss nodes, which always displayed an accurate and robust behaviour in all the test cases considered.Copyright © 2012 John Wiley & Sons, Ltd.

Reissner–Mindlin Legendre spectral finite elements with mixed reduced quadrature

Legendre spectral finite elements (LSFEs) are examined through numerical experiments for static and dynamic Reissner–Mindlin plate bending and a mixed-quadrature scheme is proposed. LSFEs are high-order Lagrangian-interpolant finite elements with nodes located at the Gauss–Lobatto–Legendre quadrature points. Solutions on unstructured meshes are examined in terms of accuracy as a function of the number of model nodes and total operations. While nodal-quadrature LSFEs have been shown elsewhere to be free of shear locking on structured grids, locking is demonstrated here on unstructured grids. LSFEs with mixed quadrature are, however, locking free and are significantly more accurate than low-order finite-elements for a given model size or total computation time.

Least-Squares Spectral Method for the solution of a fractional advection–dispersion equation

Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection–dispersion equations using Caputo or Riemann–Liouville fractional derivatives.A Gauss–Lobatto–Jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition. Exponential convergence rate of the operator is verified when increasing the order of the approximation.Solutions are calculated for fractional-time and fractional-space differential equations. Comparisons with finite difference schemes are included. A significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate.

The existence and construction of rational Gauss-type quadrature rules

Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss–Radau (m=1) and Gauss–Lobatto (m=2) quadrature formulas that approximate F{f}. These are quadrature formulas with n positive weights and with the n-m remaining nodes real and distinct, so that the quadrature is exact in a (2n-m)-dimensional space of rational functions. Further, we also consider the case in which the functional is defined by a positive bounded Borel measure on an interval, for which it is required in addition that the nodes are all in the support of the measure.

A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions

In this research, a new numerical method is applied to investigate the nonlinear controlled Duffing oscillator. This method is based on the radial basis functions (RBFs) to approximate the solution of the optimal control problem by using the collocation method. We apply Legendre–Gauss–Lobatto points for RBFs center nodes in order to use the numerical integration method more easily; then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the dynamic systems, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

An optimized spectral difference scheme for CAA problems

In the implementation of spectral difference (SD) method, the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes. The errors incurred in using extrapolation and interpolation would result in instability. On the other hand, the difference between the left and right conserved variables at the edge interface will introduce dissipation to the SD method when applying a Riemann solver to compute the flux at the element interface. In this paper, an optimization of the extrapolation and interpolation schemes for the fourth order SD method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers. The optimized coefficients of the extrapolation and interpolation are presented. And the dispersion error of the original and optimized schemes is plotted and compared. An improvement of the dispersion error over the resolvable wavenumber range of SD method is obtained. The stability of the optimized fourth order SD scheme is analyzed. It is found that the stability of the 4th order scheme with Chebyshev–Gauss–Lobatto flux points, which is originally weakly unstable, has been improved through the optimization. The weak instability is eliminated completely if an additional second order filter is applied on selected flux points. One and two dimensional linear wave propagation analyses are carried out for the optimized scheme. It is found that in the resolvable wavenumber range the new SD scheme is less dispersive and less dissipative than the original scheme, and the new scheme is less anisotropic for 2D wave propagation. The optimized SD solver is validated with four computational aeroacoustics (CAA) workshop benchmark problems. The numerical results with optimized schemes agree much better with the analytical data than those with the original schemes.

On some properties of 2D spectral finite elements in problems of wave propagation

A computational analysis of plane progressive wave propagation in plane stress body is presented. The initial-boundary value problem of linear elastodynamics of Cauchy continuum is approximated spatially by specially designed multi-node C0 displacement-based isoparametric quadrilateral spectral finite elements. To integrate element matrices the Gauss–Lobatto–Legendre quadrature rule is used. The temporal discretization is carried out by Newmark type algorithm reformulated to accommodate the structure of local element matrices. The developed multi-node spectral elements with Gauss–Lobatto–Legendre nodes are validated by running some statics and dynamics tests to investigate the presence of locking effect and of spurious zero-energy modes. Dynamic tests, dedicated to wave propagation in L-shaped structure, are concentrated on energy propagation through right-hand angle of the construction.

Overlapping Schwarz preconditioned eigensolvers for spectral element discretizations

Model generalized eigenproblems associated with self-adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered. Their numerical approximation is based on Gauss–Lobatto–Legendre conforming spectral elements defined by Gordon–Hall transfinite mappings. The resulting discrete eigenproblems are solved iteratively with a Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method, accelerated by an overlapping Schwarz preconditioner. Several numerical tests show the good convergence properties of the proposed preconditioned eigensolver, such as its scalability and quasi-optimality in the discretization parameters, which are analogous to those obtained for overlapping Schwarz preconditioners for linear systems.

Solution of linear optimal control problems with time delay using a composite Chebyshev finite difference method

SUMMARYIn this paper, a composite Chebyshev finite difference method is introduced and applied for finding the solution of optimal control of time‐delay systems with a quadratic performance index. This method is an extension of the Chebyshev finite difference scheme. The proposed method can be regarded as a nonuniform finite difference scheme and is based on a hybrid of block‐pulse functions and Chebyshev polynomials using the well‐known Chebyshev–Gauss–Lobatto points. Various types of time‐delay systems are included to demonstrate the validity and the applicability of the technique. The method is easy to implement and provides very accurate results. Copyright © 2012 John Wiley & Sons, Ltd.

Least-squares spectral element solution of incompressible Navier–Stokes equations with adaptive refinement

Least-squares spectral element solution of steady, two-dimensional, incompressible flows are obtained by approximating velocity, pressure and vorticity variable set on Gauss–Lobatto–Legendre nodes. Constrained Approximation Method is used for h- and p-type nonconforming interfaces of quadrilateral elements. Adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient. Effective use of p-refinement to overcome poor mass conservation drawback of least-squares formulation and successful use of h- and p-refinement together to solve problems with geometric singularities are demonstrated. Capabilities and limitations of the developed code are presented using Kovasznay flow, flow past a circular cylinder in a channel and backward facing step flow.

Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions

A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre–Gauss–Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme.

Verification of a Spectral-Element Method Code for the Southern California Earthquake Center LOH.3 Viscoelastic Case

This article presents the verification of a spectral-element method (SEM) code using the so-called layer over half-space 3 (LOH.3) benchmark model of the Southern California Earthquake Center to 7 Hz with a minimum S-wave velocity of 2000 m/s. First, the approach of Liu and Archuleta (2006) is successfully implemented in an explicit SEM, then, it is shown that the SEM code displays excellent goodness of fits (GOFs) with five Gauss–Lobatto–Legendre (GLL) nodes per wavelength at 7 Hz. A parametric study on the influence of the number of GLL nodes per wavelength shows that more than five GLL nodes per minimum wavelength do not significantly increase the accuracy; however, a decrease of the accuracy starts to be seen from four GLL nodes. A parametric study of the influence of the number of relaxation mechanism on the accuracy of the numerical response shows that, for this specific case, six to eight relaxation mechanisms are sufficient to reproduce the reference solution, whereas below six the viscoelastic response is strongly affected and tends to an elastic response when using two relaxation mechanisms only.

Chebyshev–Legendre pseudo-spectral method for the generalised Burgers–Fisher equation

In this paper, we consider numerical approximation of generalised Burgers–Fisher equation using the pseudo-spectral method. For the time discretization we apply Crank–Nicolson /leapfrog scheme. The space discretization is based on Legendre Galerkin formulation while the Chebyshev–Gauss–Lobatto (CGL) nodes are used in practical computation, which is called “Chebyshev–Legendre” method. The stability and convergence are rigorously set up. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.

A quadrature tau method for fractional differential equations with variable coefficients

In this article, we develop a direct solution technique for solving multi-order fractional differential equations (FDEs) with variable coefficients using a quadrature shifted Legendre tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the tau method for FDEs with variable coefficients is treated using the shifted Legendre–Gauss–Lobatto quadrature. Numerical results are given to confirm the reliability of the proposed method for some FDEs with variable coefficients.

INTERACTIVE CHEBYSHEV–LEGENDRE ALGORITHM FOR LINEAR QUADRATIC OPTIMAL REGULATOR SYSTEMS

In this paper, we derive an algorithm to solve the linear quadratic (LQ) optimal regulator problems. The new approach is based on efficient Legendre and Chebyshev formulae at the Chebyshev–Gauss–Lobatto points. The technique enjoys advantages of both the Legendre and Chebyshev approximations near the end points. To show the numerical behavior of the proposed method, the simulation results of an example are presented.