Legendre Gauss Lobatto Research Articles

A Legendre Spectral-Element Method to Incorporate Topography for 2.5D Direct-Current-Resistivity Forward Modeling

An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective approach for identifying geo-electrical anomalies. An efficient compromise, or potentially an alternative, is found in two-and-a-half-dimensional (2.5D) modeling, which employs a three-dimensional current source within a two-dimensional subsurface medium. Consequently, a Legendre spectral-element algorithm is developed specifically for 2.5D direct-current-resistivity forward modeling, taking into account the presence of topography. This numerical algorithm can combine the complex geometric flexibility of the finite-element method with the high precision of the spectral method. To solve the wavenumber-domain electrical potential variational problem, which is converted into the two-dimensional Helmholtz equation with mixed boundary conditions, the Gauss–Lobatto–Legendre (GLL) quadrature is employed in all discrete quadrilateral spectral elements, ensuring identical Legendre polynomial interpolation and quadrature points. The Legendre spectral-element method is applied to solve a two-dimensional Helmholtz equation and a resistivity half-space model. Numerical experiments demonstrate that the proposed approach yields highly accurate numerical results, even with a coarse mesh. Additionally, the Legendre spectral-element algorithm is employed to simulate the apparent resistivity distortions caused by surface topographical variations in the direct-current resistivity Wenner-alpha array. These numerical results affirm the substantial impact of topographical variations on the apparent resistivity data obtained in the field. Consequently, when interpreting field data, it is crucial to consider topographic effects to the extent they can be simulated. Moreover, our numerical method can be extended and implemented for a more accurate computation of three-dimensional direct-current-resistivity forward modeling.

An efficient meshless technique based on collocation and RBFs for solving nonlinear VIEs of third kind with proportional delays

This study provides a numerical scheme for solving nonlinear third-kind Volterra delay integral equations (VDIEs) using mesh-free approaches based on the collocation technique with radial basis functions (RBFs). The proposed approach is stable and requires less computer memory. It combines RBFs and a collocation method technique based on scattered points to represent the solution of third-kind VDIEs by interpolating RBFs using Legendre-Gauss–Lobatto nodes and weights. A description of the approach to the suggested equations is presented. Furthermore, the error analysis of the proposed scheme is investigated. A numerical example that clearly illustrates the validity of the proposed scheme is presented. This problem was also compared with the moving least squares collocation technique and other methods, demonstrating the reliability and efficiency of this approach for solving third-kind VDIEs.

Density estimation for time-dependent PDE with random input by a Legendre-based multi-element probabilistic collocation method

This paper proposed a Legendre-based multi-element probabilistic collocation method for time-dependent stochastic differential equations, used for density estimation of unknown functions. This method involves discretizing the stochastic space, and on each element, constructing Lagrange interpolation basis functions based on Legendre–Gauss–Lobatto collocation/quadrature nodes. The proposed method is applied to approximate one-dimensional/two-dimensional smooth/non-smooth functions and is tested for accuracy in approximating random function values, density estimations, and mathematical expectations. This method is applied to stochastic nonlinear Schrödinger equations and coupled stochastic nonlinear Schrödinger equations, and all numerical results are compared with Monte Carlo simulation.

Spectral solutions for fractional Klein–Gordon models of distributed order

The Klein–Gordon equation is a fundamental theoretical physics concept, governing the behavior of relativistic quantum particles with spin-zero. Its numerical solution is crucial in fields like quantum field theory, particle physics, and cosmology. The study explores numerical methodologies for solving this equation, highlighting their significance and challenges. This study uses the collocation method to approximate fractional Klein–Gordon models of distributed order based on Shifted Jacobi orthogonal polynomials and Shifted fractional order Jacobi orthogonal functions. While, the distributed term (integral term) was treat using Legendre–Gauss–Lobatto quadrature. It assesses residuals through finite expansion and yields accurate numerical results. The method is more factual and fair when initial and boundary conditions are enforced. Numerical simulations are presented to demonstrate the method’s accuracy, particularly in fractional Klein–Gordon models of distributed order. Furthermore, we offer a few numerical test scenarios to show that the method is able to maintain the non-smooth solution of the underlying issue.

A Convergent Legendre Spectral Collocation Method for the Variable-Order Fractional-Functional Optimal Control Problems

In this paper, a numerical method is applied to approximate the solution of variable-order fractional-functional optimal control problems. The variable-order fractional derivative is described in the type III Caputo sense. The technique of approximating the optimal solution of the problem using Lagrange interpolating polynomials is employed by utilizing the shifted Legendre–Gauss–Lobatto collocation points. To obtain the coefficients of these interpolating polynomials, the problem is transformed into a nonlinear programming problem. The proposed method offers a significant advantage in that it does not require the approximation of singular integral. In addition, the matrix differentiation is calculated accurately and efficiently, overcoming the difficulties posed by variable-order fractional derivatives. The convergence of the proposed method is investigated, and to validate the effectiveness of our proposed method, some examples are presented. We achieved an excellent agreement between numerical and exact solutions for different variable orders, indicating our method’s good performance.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. In addition, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients, and vortex dominated flows.

An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.

UAV Trajectory Optimization for Maximum Soaring in Windy Environment

Optimal trajectory planning in a windy environment is a complex problem when airplane performance characteristics are considered. This paper introduces a novel form of Legendre pseudospectral optimization to solve boundary value problems in UAV trajectory planning. The proposed architecture applies Legendre–Gauss–Lobatto and Hamilton–Jacobi–Bellman equations to generate candidate pieces of trajectories with respect to the UAV dynamic constraints. Analytical performance-based solutions are also developed for sample cases to achieve an optimal criterion in trajectory planning. Moreover, the notion of wind soaring is exploited to use the beneficial effects of tailwind velocities in trajectory planning. Integral cost functions are handled by Gauss-type quadrature rules. Simulations demonstrate the efficiency of the pseudospectral method compared with other solvers, in eliminating the difficulties of boundary value problems by employing the boundary points in the interpolation equation. The effectiveness of the proposed approach is demonstrated through dynamic simulations.

Three-Dimensional Forward Modeling of Transient Electromagnetic Method Considering Induced Polarization Effect Based on Spectral Element Method

The transient electromagnetic method (TEM) is widely used in the exploration of mineral, petroleum, and geothermal resources due to its sensitivity to low-resistivity bodies, limited site constraints, and strong resistance to interference. In practical applications, the TEM often uses a long wire source instead of an idealized horizontal electric dipole (HED) source as the excitation source. This is due to the complex external conditions and the relatively large distance between the receiving zone and the transmitter source. Compared to the HED, the long wire source can provide a larger excitation current, generating stronger signals to meet the requirements of a higher signal-to-noise ratio or deeper exploration. It also produces longer-duration signals, thereby providing better resolution. Additionally, for the interpretation of TEM data, three-dimensional forward modeling plays a crucial role. However, the mature traditional TEM forward method is based on a simple, sometimes inappropriate model, as it is well established that the induced polarization (IP) effect is widely present in the deep earth, especially in oil and gas reservoirs. The presence of the IP effect results in negative responses in field data that do not conform to the traditional theoretical decay law of TEM, which can significantly impact data processing and inversion results. To address this issue, a TEM forward modeling method considering the IP effect based on the spectral element method (SEM) has been developed in this study. Firstly, starting from the Helmholtz equation satisfied by the time domain electric field, we introduce the Debye model with polarization information into the forward modeling by utilizing the differential form of Ohm’s law. As a result, we derive the boundary value problem for the time domain electric field that considers the induced polarization effect. Using Gauss–Lobatto–Legendre (GLL) polynomials as the basis functions, the SEM is employed to discretize the governing equations at each time step and obtain spectral element discretization equations. Then, temporal discretization equations are derived using the second-order backward Euler formula, and the linear system of equations is solved using the Pardiso direct solver. Finally, the electromagnetic responses at any time channel are obtained via SEM interpolation and numerical integration, thereby achieving three-dimensional TEM forward modeling considering the IP effect. The results indicate that this method can effectively reflect the spatial distribution of polarizable subsurface media. It provides valuable references for studying the polarization parameters of subsurface media and performing a three-dimensional inversion of TEM data considering the induced polarization effect.

Numerical behavior of the variable-order fractional Van der Pol oscillator

In this article, we investigate the behavior of a Van der Pol oscillator based on the variable-order Caputo fractional derivatives. After variable-order fractional modeling, we discretize the obtained equations using the Legendre–Gauss–Lobatto points and employ Lagrange interpolating functions. An algebraic system is gained that approximates the variables and their fractional derivatives. Also, an approach is suggested to calculate the differentiation matrix related to the variable-order Caputo fractional derivative. Moreover, an algorithm is presented for solving the variable-order Caputo fractional Van der Pol equation on large time-interval. Numerical simulations are provided to represent the applicability of the suggested method and to see the treatment of variable-order Caputo fractional Van der Pol oscillator.

Partial Newton-Correction Method for Multiple Fixed Points of Semi-linear Differential Operators by Legendre–Gauss–Lobatto Pseudospectral Method

Partial Newton-Correction Method for Multiple Fixed Points of Semi-linear Differential Operators by Legendre–Gauss–Lobatto Pseudospectral Method

Analysis of Acoustic–Magnetic Fields Induced by Underwater Pressure Wave in a Finite-Depth Ocean

As underwater disturbances (natural or artificial) occur in the ocean, moving seawater crossing the geomagnetic fields will produce weak circular currents. These currents can induce measurable magnetic fields, which might be useful for monitoring ocean internal waves using aeromagnetic survey. In this research, a spectral-element method (SEM) based on Gauss–Lobatto–Legendre (GLL) polynomials is presented to characterize the magnetic field induced by the underwater pressure waves. A concise mathematical model is established through combining the acoustic wave equations and Maxwell’s equations. Specifically, the acoustic–magnetic coupling simulation adopts the nodal-based SEM for acoustic analysis and edge-based SEM for electromagnetic analysis. The proposed SEM has spectral accuracy, as the error exponentially decreases with the order of the basis functions. Additionally, by adopting an independent modeling and mesh scheme in two solvers, respectively, the waste of computing resources is avoided. The experimental analysis demonstrates that the induced magnetic fields mechanically propagate with the acoustic wave, producing the pseudo-radiation phenomenon. The signals of these magnetic fields may extend for tens of kilometers and exist for hours under certain circumstances, which provide a theoretical basis for underwater target identification via high-sensitivity atomic magnetometer.

Hybrid method for modelling anti-plane vibrations of layered waveguides with bonded composite joints

The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.

Pseudospectral Convex Programming for Free-Floating Space Manipulator Path Planning

To efficiently plan the point-to-point path for a 7-degrees-of-freedom (7-DOF) free-floating space manipulator system, a path planning method based on Legendre pseudospectral convex programming (LPCP) is proposed. First, the non-convex dynamics are approximated by utilizing the first-order Taylor expansion in the vicinity of the initial guess path, which results in a convex system. Next, the linearized dynamics are discretized at Legendre–Gauss–Lobatto collocation points to transcribe the differential equations to a set of equality constraints. To obtain a reliable initial guess trajectory, the auxiliary path planning problem of the 7-DOF space manipulator with a fixed base is initially resolved. Additionally, the penalty function method is introduced to enhance the convergence performance of the LPCP. Finally, simulation results show that the proposed algorithm in this paper can generate the point-to-point path and has higher computational efficiency than the general sequential convex programming method while ensuring optimality.

A SPACE-TIME SPECTRAL COLLOCATION METHOD FOR SOLVING THE VARIABLE-ORDER FRACTIONAL FOKKER-PLANCK EQUATION

A numerical approach for solving the variable-order fractional FokkerPlanck equation (VO-FFPE) is proposed. The computational scheme is based on the shifted Legendre Gauss-Lobatto and the shifted Chebyshev GaussRadau collocation methods. The VO-FFPE is written as a truncated series of shifted Legendre and shifted Chebyshev polynomials for space and time variables, respectively. The residuals of the VO-FFPE at the shifted Legendre Gauss-Lobatto and shifted Chebyshev Gauss-Radau quadrature points are estimated. The original problem is converted into a system of algebraic equations that can be solved easily. Several examples are presented to demonstrate the efficacy of the technique.

Entropy-stable Gauss collocation methods for ideal magneto-hydrodynamics

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms.Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. [1, “Efficient Entropy Stable Gauss Collocation Methods”. SIAM (2019)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system.We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.

A numerical study on fractional optimal control problems described by Caputo‐Fabrizio fractional integro‐differential equation

AbstractThis paper provides a numerical technique for evaluating the approximate solution of fractional optimal control problems with the Caputo‐Fabrizio (CF) fractional integro‐differential equation. First, due to the Gegenbauer polynomials definition and CF‐fractional derivative, we present the modified operational matrix and complement vector of integration and CF‐fractional derivative. Then, the corresponding discretization of the problem is obtained with the help of the optimization method and the Legendre‐Gauss‐Lobatto (LGL) quadrature rule. The technique of obtaining the proposed matrices and LGL‐quadrature method leads to obtaining the approximate solution with high precision. Moreover, the error of the performance index obtained by the computational method is investigated. Finally, we implement the methodology in several examples. So that, the effect of various parameters defined in the method is illustrated in tables and plots.

A time-domain POD approach based on numerical implicit and explicit Green’s functions for 3D elastodynamic analysis

This paper is concerned with the development of time integration methods based on numerical Green’s functions applied in conjunction with the proper orthogonal decomposition (POD) technique to solve three-dimensional elastodynamic problems discretized by the FEM. The convolution integral in the velocity expression of these methods is firstly carried out through integration by parts, and then approximated numerically, resulting in a simpler and more accurate convolution than that reported in the literature. To compute the solution samples, required by the POD technique, the Implicit Green’s functions Approach (ImGA)-Newmark method rewritten in terms of the ultimate spectral radius is employed. The proposed ImGA scheme is truly self-starting and easy to implement with just one free parameter. To solve the reduced-order model, a new highly accurate time integration scheme based on the Explicit Green’s functions Approach (ExGA)-Runge–Kutta method is proposed, considering the Gauss–Lobatto-Legendre (GLL) points to perform both the time integration of the Green’s functions and the approximation of the convolution integrals. To verify the accuracy and efficiency of the proposed formulation, the numerical results are compared with those from the full-order model. Moreover, undamped solution samples are utilized to build a reduced-order model that can take into account a Rayleigh damping matrix.

Performance Assessment of Convex Low-Thrust Trajectory Optimization Methods

Different discretization and trust-region methods are compared for the low-thrust fuel-optimal trajectory optimization problem using successive convex programming. In particular, the differential and integral formulations of the adaptive pseudospectral Legendre–Gauss–Radau method, an arbitrary-order Legendre–Gauss–Lobatto technique based on Hermite interpolation, and a first-order-hold discretization are considered. The number of discretization points and segments is varied. Moreover, two hard-trust-region methods and a soft-trust-region strategy are compared. It is briefly discussed whether these methods, if implemented on relevant hardware, would fulfill the general requirements for onboard guidance. A perturbed cubic interpolation and the propagation of the nonlinear dynamics are used to generate initial guesses of varying quality. Interplanetary transfers to a near-Earth asteroid, Venus, and asteroid Dionysus are chosen to assess the overall performance.

Trajectory Optimization and Cost Estimation of Laser Propulsion Systems Considering Transmission Losses

We optimized the launch trajectory of an laser-sustained-plasma-type (LSP) laser propulsion rocket at an altitude of 500 km by considering transmission efficiency. The launch cost was calculated based on the results of trajectory calculations. The transmission loss was considered to be the effect of atmospheric turbulence and focusing the performance of the lens. In the optimization problem, the Legendre–Gauss–Lobatto (LGL) method was used for discretization, and the sequential quadratic programming method was used to solve the optimization problem. The payload ratio was 18.0% with a laser power of 200 kW. In the launch calculation, the costs of the laser, electricity, propellant, and airframe were considered for the experience curve effect. The calculation indicated the cost of 267,000 launches was 29.1% of the cost of chemical propulsion.