Gauss Lobatto Research Articles

Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

A highly accurate spectral algorithm for (2 + 1) fractional percolation equations with variable order (VO‐FPEs) is considered. We propose a shifted Legendre–Gauss–Lobatto collocation (SL‐GL‐C) method in conjunction with shifted Chebyshev–Gauss–Radau collocation (SC‐GR‐C) method to solve the two‐dimensional VO‐FPEs. A complete theoretical formulation is presented, and numerical results are given to illustrate the performance and efficiency of the algorithm. The superiority of the scheme to tackle VO‐FPEs is revealed, even when dealing with non‐smooth time solutions.

Space–time spectral collocation method for Klein–Gordon equation

By using the Legendre–Laguerre collocation method, we can construct a spectral collocation scheme to solve the Klein–Gordon equation on the half-line. The Laguerre function collocation method (based on the Lagrange interpolation) in space and the Legendre–Gauss–Lobatto collocation method in time are used. A Newton iterative algorithm is provided. The numerical results demonstrate the high efficiency and accuracy of suggested algorithms.

Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem

The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.

Weak form quadrature elements for non-classical Kirchhoff plate theory

In this paper, two novel versions of weak form quadrature elements are developed for bending, free vibration and stability analysis of non-classical first strain gradient Kirchhoff plate theory. In the first version, Lagrange interpolations are assumed in orthogonal directions to approximate the field variables and in the second, mixed interpolations are used, with Lagrange in one direction and Hermite in another. The elements are formulated with the aid of variational principles and the Gauss–Lobatto–Legendre quadrature points are considered as element nodes and also used for numerical integration of element matrices. The multi-degrees degrees of freedom associated with the non-classical plate theories are accounted into the formulation in a simplified way, which facilitates the application of classical and non-classical boundary conditions. The procedure to compute the weighting coefficients by incorporating the classical and non-classical degrees of freedom are explained for both element versions. Detailed mathematical formulation of the elements and their numerical implementation is presented. The efficiency of the proposed elements is demonstrated by solving numerical examples and comparing the results with the exact solutions available in the literature. Further, new results are presented for strain gradient plates with different support conditions, which can serve as reference for other researchers in this field.

Application of quadrature-based moment methods to the conditional moment closure

Two solution algorithms are developed for the conditional moment closure (CMC) using quadrature-based moment methods (QBMM). Their primary purpose is to eliminate the necessity of the additional grid for the conditioning variable (e.g., mixture fraction). As in almost every probability-density-function (PDF)-based description, the main technical hurdle is the solution of the molecular-mixing term. Here the focus is on solving this term coupled with chemistry, as opposed to spatial transport. In the first algorithm (referred to as SA-CMC), this is done by semi-analytically solving an equation for the deviation variable of the conditional scalar mean in terms of Jacobi polynomials. The mixture-fraction space is represented by the Gauss–Lobatto quadrature rule and a β-PDF. In the second algorithm (referred to as QBMM-CMC), a closed set of differential equations is written for the joint moments of a scalar and the mixture fraction, thereby eliminating the need to assume a form for the mixture-fraction PDF. Both solution algorithms are tested for multi-step H2 combustion, and the expected results are achieved with at most six quadrature nodes. Since in the proposed algorithms the scalar mean is ensured to be constant during molecular mixing, it is further concluded that the proposed algorithms are accurate, computationally cost-effective, and straightforward alternatives for traditional CMC solution methods.

An MHD Stokes eigenvalue problem and its approximation by a spectral collocation method

An eigenvalue problem is introduced for the magnetohydrodynamic (MHD) Stokes equations describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. The solution of the eigenproblem is approximated by using a spectral collocation method that is based on vanishing the residual equation at the collocation points on the physical domain which are chosen to be the Chebyshev–Gauss–Lobatto points. As the solutions are sought in the physical space, the approximations to the derivatives of the unknowns are directly evaluated. The equations are formulated in the primitive variables, and hence with inclusion of the continuity equation, the discretization of the operator results in a generalized eigenproblem with zero diagonal entries. Therefore, a penalty method is applied to circumvent the degeneracy where a perturbed form of the problem is considered, and a zero mean pressure value is introduced. The numerical prospects of the algorithm are investigated and demonstrated by a number of characteristic tests. The key features of interest are the effects of introducing a magnetic field on the eigenspectrum focusing mainly on the change of the fundamental eigenpairs, and the consequential variation of the eigenstructure with the magnetic field. The mechanisms that underlie these effects are examined by the numerical model proposed, the implications of these effects are presented, and it is shown that the flow field is considerably affected with the introduction of a magnetic field into the physical model.

Bending and free vibration analysis of orthotropic in-plane functionally graded plates using a Chebyshev spectral approach

This paper presents a Chebyshev spectral approach for the bending and free vibration analysis of orthotropic in-plane functionally graded (IPFG) plates with general boundary conditions. Material properties and distributed loadings are assumed to be varying in the plane of the plates arbitrarily. Both of them are approximated by high-order Chebyshev expansions combining with Gauss–Lobatto sampling. Then, based on the Mindline-Reissner first-order shear deformation theory, the governing equations for the bending and free vibration are derived for orthotropic IPFG plates respectively by employing the principle of minimum total potential energy and the Lagrange’s equation, which are consistent for any kind of material and loading distribution. Several numerical examples, involving various boundary conditions, types of loadings, and thickness-length ratios, are presented to demonstrate the accuracy and efficiency of the proposed method. The results are compared with those reported in the literature and those obtained by commercial finite element software ABAQUS. Excellent agreements are observed, demonstrating the effectiveness of the Chebyshev spectral method for the static and dynamic analysis of orthotropic IPFG plates.

Feasibility analysis of using the hp-adaptive Radau pseudospectral method for minimum-effort collision-free docking operations of AUV

This paper continues the previous effort on the development of a trajectory generation platform that assures minimum-control expenditure and collision-free manoeuvre of a torpedo-shaped autonomous underwater vehicle (AUV) into a funnel-shaped stationary docking station (DS). The earlier-developed guidance system was based on the Inverse Dynamics in the Virtual Domain (IDVD) method accounting for AUV’s dynamics and producing a smooth trackable trajectory, thus guaranteeing safe arrival to DS. The optimality of the real-time generated solutions has been assessed via comparing them with the Legendre–Gauss–Lobatto pseudo-spectral (PS) method solutions that could only be obtained off-line. This paper explores a possibility of employing a more advanced hp-adaptive Radau (hp-AR) PS method for the same Hamiltonian two-point boundary-value problem. The considered approach explicitly encapsulates all realistic vehicular and environmental constraints such as the AUV’s dynamics, ocean current disturbances, no-fly zones, and DS pose while minimizing the vehicle’s controls expenditure and permitting precise manoeuvring into DS. The performance evaluation of the hp-AR PS based optimization routine is carried out through extensive software-in-the-loop simulations. For completeness, computational effectiveness and solution optimality of the trajectory generator engine based on the hp-AR method is compared with two other well-known PS methods based on Legendre and Chebyshev polynomial approximation. The results of this study show the superb performance of the hp-AR method-based trajectory generator among all other PS methods and a possibility of using it along with IDVD in the real-time implementation.

Chebyshev–Gauss–Lobatto collocation method for variable-order time fractional generalized Hirota–Satsuma coupled KdV system

In this paper, the Chebyshev–Gauss–Lobatto collocation method is developed for studying the variable-order (VO) time fractional model of the generalized Hirota–Satsuma coupled KdV system arising in interaction of long waves. To define this new system, the Atangana–Baleanu fractional operator is implemented. The operational matrix of VO fractional differentiation for the shifted Chebyshev polynomials is extracted and then, a collocation scheme is established to reduce the original VO fractional problem to a system of nonlinear algebraic equations. The validity of the presented method is investigated on two numerical examples.

Time-domain Spectral Finite Element Method for Wave Propagation Analysis in Structures with Breathing Cracks

Guided waves are generally considered as a powerful approach for crack detection in structures, which are commonly investigated using the finite element method (FEM). However, the traditional FEM has many disadvantages in solving wave propagation due to the strict requirement of mesh density. To tackle this issue, this paper proposes an efficient time-domain spectral finite element method (SFEM) to analyze wave propagation in cracked structures, in which the breathing crack is modeled by defining the spectral gap element. Moreover, novel orthogonal polynomials and Gauss–Lobatto–Legendre quadrature rules are adopted to construct the spectral element. Meanwhile, a separable hard contact is utilized to simulate the breathing behavior. Finally, a comparison of the numerical results between the FEM and the SFEM is conducted to demonstrate the high efficiency and accuracy of the proposed method. Based on the developed SFEM, the nonlinear features of waves and influence of the incident mode are also studied in detail, which provides a helpful guide for a physical understanding of the wave propagation behavior in structures with breathing cracks.

A Legendre wavelet–based stability prediction method for high-speed milling processes

Induced by the regeneration mechanism, chatter vibration that has adverse effects on surface finish and productivity could occur in most cutting processes. Therefore, accurately predicting chatter and taking timely suppression measures are desperately needed. This work develops a semi-analytical Legendre wavelet–based stability prediction method in high-speed milling, of which dynamics model is commonly modeled by time-periodic delay differential equations. To begin with, the milling dynamics model is transformed into state equation. Based on analysis of the vibration modes for milling dynamic system, the period for coefficient matrix could be divided into two subintervals. Thereafter, variable transformation is introduced to convert the forced vibration interval into the standard definition interval of Legendre wavelets, which can be discretized non-uniformly into the Chebyshev–Gauss–Lobatto sampling points. On this basis, finite series Legendre wavelets are utilized to accurately approximate the state term, while the derivative of state term can be obtained simultaneously via a new Legendre wavelet operational matrix. By employing the Kronecker product and Floquet theory, the transition matrix is eventually acquired for semi-analytically predicting stability. The superiority and versatility for the presented approach are validated through comparisons with existing algorithms. The results demonstrate that it obtains high accuracy, calculation speed, and applicability.

Optimal Motion Planning for Differential Drive Mobile Robots based on Multiple-Interval Chebyshev Pseudospectral Methods

SUMMARYThis paper presents a Chebyshev Pseudospectral (PS) method for solving the motion planning problem of nonholonomic mobile robots with kinematic and dynamic constraints. The state and control variables are expanded in the Chebyshev polynomial of orderN, and Chebyshev–Gauss–Lobatto (CGL) nodes are provided for approximating the system dynamics, boundary conditions, and performance index. For the lack of enough nodes nearby the obstacles, the interpolation of trajectory may violate the obstacles and the multiple-interval strategy is proposed to deal with the violation. Numerical examples demonstrate that multiple-interval strategy yields more accurate results than the single-interval Chebyshev PS method.

Stability analysis and highly accurate numerical approximation of Fisher’s equations using pseudospectral method

In this article, we present a new pseudospectral method to approximate the solution of one- and two-dimensional Fisher’s equations, which is a prototype of nonlinear reaction–diffusion equations. The proposed method is based on Chebyshev–Gauss–Lobatto points and employed collocation in both spatial and time direction to study of several travelling wave solutions which evolves into a shock like wavefront and change the shape to some wave pattern. The stability analysis of the proposed method for Fisher’s equation is presented. The numerical results show highly accurate and stable for different values of the reaction rate coefficients. Some model examples of one- and two-dimensional Fisher’s equations are tested and detailed comparison of the proposed method with various other methods are also given.

An $h$-$p$ version of the Chebyshev spectral collocation method for Volterra integro-differential equations with vanishing delays

We propose a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for linear Volterra integro-differential equations with vanishing delays. We derive an h-p version a-priori error estimate in the H1-norm that is completely explicit in the local time steps and the local polynomial degrees. Numerical experiments are provided to confirm the theoretical results.

Analysis of the lumped mass model for the cantilever beam subject to Grob’s swelling pressure

The lumped mass model is derived from a one-mode Galerkin discretization with the Gauss–Lobatto quadrature applied to the non-linear swelling pressure term. Our reduced-order model of the problem is then analyzed to study the essential dynamics of an elastic cantilever Euler–Bernoulli beam subject to the swelling pressure described by Grob’s law. The solutions to the initial value problem for the resulting nonlinear ODE are proved to be always periodic. The numerical solution to the derived lumped mass model satisfactorily matches the finite difference solution of the dynamic beam problem. Including the effect of oscillations at the base of the beam, we show that the model exhibits resonances that may crucially influence its dynamical behavior.

Legendre-tau-Galerkin and spectral collocation method for nonlinear evolution equations

A Legendre-tau-Galerkin method is developed for nonlinear evolution problems and its multiple interval form is also considered. The Legendre tau method is applied in time and the Legendre/Chebyshev–Gauss–Lobatto points are adopted to deal with the nonlinear term. By taking appropriate basis functions, it leads to a simple discrete equation. The proposed method enables us to derive optimal error estimates in L2-norm for the Legendre collocation under the two kinds of Lipschitz conditions, respectively. Our method is also applied to the numerical solutions of some nonlinear partial differential equations by using the Legendre Galerkin and Chebyshev collocation in spatial discretization. Numerical examples are given to show the efficiency of the methods.

Superconvergent Functional Estimates from Tensor-Product Generalized Summation-by-Parts Discretizations in Curvilinear Coordinates

We investigate superconvergent functional estimates in curvilinear coordinates for diagonal-norm tensor-product generalized summation-by-parts operators. We show that interpolation/extrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summation-by-parts operator that is used to approximate the flux terms and (2) the degree of the generalized summation-by-parts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendre–Gauss–Lobatto and Legendre–Gauss element-type operators are considered. When the aforementioned condition (2) is violated for the Legendre–Gauss operators, there is an even–odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolation/extrapolation operators that correspond to the odd-degree Legendre–Gauss operators. The theory developed is confirmed through numerical examples with a steady one-dimensional problem and the unsteady two-dimensional linear convection equation.

Comparison of the Legendre–Gauss pseudospectral and Hermite–Legendre–Gauss–Lobatto methods for low-thrust spacecraft trajectory optimization

Low-thrust spacecraft propulsion systems enable fuel-efficient trajectories through space but the resulting trajectory optimization problems can be challenging. Such problems often use direct collocation methods for transcribing the optimal control problem into a nonlinear programming problem. In this work, the Hermite–Legendre–Gauss–Lobatto (HLGL) and the Legendre–Gauss pseudospectral (PS) direct collocation methods have been compared for a minimum-time low-thrust Earth-to-Mars transfer problem. It is demonstrated that trajectories with different time domains can be effectively compared using points that correspond to the same normalized times. Various metrics that describe the computational cost as well as the errors with respect to a reference trajectory have been used. For the HLGL method, the number of subintervals (m) and the order of the interpolating polynomial (n), can be varied such that the number of nodes (N) stays constant. The distribution of the number of such (m, n) pairs is analysed using the concept of divisor functions from number theory and an algorithm for determining all the possible combinations for any N value is developed. In addition, a method is presented by which the Edelbaum trajectory is used to set bounds, scale the problem, and generate a suitable initial guess. The primary contribution of this paper involves a detailed comparison of the performance of all the HLGL pairs with the PS method for nodes in the range $$N = 6$$ to $$N = 40$$ for the minimum-time low-thrust Earth-to-Mars transfer.

Space–time FEM with block-iterative algorithm for nonlinear dynamic fracture analysis of concrete gravity dam

In this paper, space–time finite element method is developed for the dynamic fracture analysis of dam–reservoir (DR) system which is supported on a perfectly rigid foundation. In this method, an auxiliary variable q representing the first-order time derivative of hydrodynamic pressure is treated as the primary unknown for the reservoir domain. Similarly, velocity v is the primary unknown for the solid domain. A three point Gauss–Lobatto quadrature rule is employed for computing time integral of those terms which contain stress term. Further, a partitioned method based on a block-iterative scheme is employed to incorporate the material nonlinearity of the concrete and to enforce the coupling between the two domains in a single iteration loop. A co-axially rotating crack model with exponential strain softening rule is employed to model the fracture of the concrete. Afterwards, numerical performance of the proposed scheme is demonstrated by analyzing two situations of dynamic fracture of Koyna concrete gravity dam.