Hybrid Spectral Method Research Articles

Lifting Directional Fields to Minimal Sections

Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement.

Non-Separable Covariance Kernels for Spatiotemporal Gaussian Processes Based on a Hybrid Spectral Method and the Harmonic Oscillator

Non-Separable Covariance Kernels for Spatiotemporal Gaussian Processes Based on a Hybrid Spectral Method and the Harmonic Oscillator

Hybrid spectral algorithm under a convex constrained to solve nonlinear equations

The hybrid spectral technique is the main and important part of optimization and has many numerical applications to solve nonlinear equations with convex constrained. This paper combines a new hybrid spectral method in addition to the projection technique with a new line search under convex constraints. This work proved the global convergence of the new suggested technique under some lemmas and assumptions. Also, this new technique holds many important results such as the sufficient descent property and the new search direction. The numerical experiments show the efficiency and competitiveness of the new technique compared with the other two famous methods.

A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels

A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels

A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays

In this paper, we develop a hybrid spectral method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel and vanishing delays. Our main strategy is to divide the original interval into subintervals, to employ the shifted generalized log orthogonal functions (GLOFs) as the basis on the first interval, to take the classical shifted Legendre polynomials as the basis on other intervals. We analyze the existence and uniqueness of the numerical scheme, and derive the corresponding error estimates. A series of examples demonstrate the effectiveness of the proposed method.

An Adaptive Hybrid Spectral Method for Stochastic Helmholtz Problems

The implementation of an adaptive hybrid spectral method for Helmholtz equations with random parameters is addressed in this work. New error indicators for generalized polynomial chaos for stochastic approximations and spectral element methods for physical approximations are developed, and systematic adaptive strategies are proposed associated with these error indicators. Numerical results show that these error indicators provide effective estimates for the approximation errors, and the overall adaptive procedure results in efficient approximation method for the stochastic Helmholtz equations.

Stochastic analysis of dead-time systems using a hybrid spectral method

Control systems often operate in the presence of dead-time. However, in most works, these dead-time systems are studied in a deterministic manner, which have low precision and reliability. Many natural systems often suffer stochastic noise that causes fluctuations in their behavior, making their responses deviate from nominal models. Therefore, it is important to investigate such statistical characteristic of states (mean, variance, etc.) for those stochastic systems. This problem is often called statistical analysis of a system. A hybrid spectral method represents a powerful numerical tool for statistical analysis of stochastic linear system. Thus, a hybrid spectral technique is proposed for statistical analysis of the time delay system under affections of random parameters and inputs. Numerical examples are considered to demonstrate the validity of the proposed method. Comparison with the traditional Monte-Carlo and the polynomial chaos methods is made to demonstrate the computationally lessdemanding feature of the proposed method.

Optimal Design of Stochastic Distributed Order Linear SISO Systems Using Hybrid Spectral Method

The distributed order concept, which is a parallel connection of fractional order integrals and derivatives taken to the infinitesimal limit in delta order, has been the main focus in many engineering areas recently. On the other hand, there are few numerical methods available for analyzing distributed order systems, particularly under stochastic forcing. This paper proposes a novel numerical scheme for analyzing the behavior of a distributed order linear single input single output control system under random forcing. The method is based on the operational matrix technique to handle stochastic distributed order systems. The existing Monte Carlo, polynomial chaos, and frequency methods were first adapted to the stochastic distributed order system for comparison. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method for the analysis of stochastic distributed order systems. The stability of the systems under stochastic perturbations can also be inferred easily from the moment of random output obtained using the proposed method. Based on the hybrid spectral framework, the optimal design was elaborated on by minimizing the suitably defined constrained-optimization problem.

Optimal design of fractional order linear system with stochastic inputs/parametric uncertainties by hybrid spectral method

This paper reports the design of a fractional linear system under stochastic inputs/uncertainties. The design methods were based on the hybrid spectral method for expanding the system signals over orthogonal functions. The use of the hybrid spectral method led to algebraic relationships between the first and second order stochastic moments of the input and output of a system. The spectral method could obtain a highly accurate solution with less computational demand than the traditional Monte Carlo method. Based on the hybrid spectral framework, the optimal design was elaborated by minimizing the suitably defined constrained-optimization problem.

Uncertainty propagation in stochastic fractional order processes using spectral methods: A hybrid approach

Stochastic spectral methods are widely used in uncertainty propagation thanks to its ability to obtain highly accurate solution with less computational demand. A novel hybrid spectral method is proposed here that combines generalized polynomial chaos (gPC) and operational matrix approaches. The hybrid method takes advantage of gPC’s efficient handling of large parameter uncertainties and overcomes its limited applicability to systems with relatively highly correlated inputs. The hybrid method’s use of operational matrices allows analyses of systems with low input correlations without suffering its restriction to small parameter uncertainties. The hybrid method is aimed to propagate uncertainties in fractional order systems with random parameters and random inputs with low correlation lengths. It is validated through several examples with different stochastic uncertainties. Comparison with Monte Carlo and gPC demonstrates the superior computational efficiency of the proposed method.

A hybrid radial basis function–pseudospectral method for thermal convection in a 3‐D spherical shell

A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospectral methods in a “2 + 1” approach is presented for numerically simulating thermal convection in a 3‐D spherical shell. This is the first study to apply RBFs to a full 3‐D physical model in spherical geometry. In addition to being spectrally accurate, RBFs are not defined in terms of any surface‐based coordinate system such as spherical coordinates. As a result, when used in the lateral directions, as in this study, they completely circumvent the pole issue with the further advantage that nodes can be “scattered” over the surface of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurate and provide the necessary clustering near the boundaries to resolve boundary layers. Applications of this new hybrid methodology are given to the problem of convection in the Earth's mantle, which is modeled by a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants further investigation, the study limits itself to an isoviscous mantle. Benchmark comparisons are presented with other currently used mantle convection codes for Rayleigh number (Ra) 7 × 103 and 105. Results from a Ra = 106 simulation are also given. The algorithmic simplicity of the code (mostly due to RBFs) allows it to be written in less than 400 lines of MATLAB and run on a single workstation. We find that our method is very competitive with those currently used in the literature.

Efficient conceptual framework to quantify flow uncertainty in large-scale, highly nonstationary groundwater systems

This study presents a hybrid spectral method (HSM) to estimate flow uncertainty in large-scale highly nonstationary groundwater systems. Taking advantages of spectral theories in solving unmodeled small-scale variability in hydraulic conductivity, the proposed HSM integrates analytical and numerical spectral solutions in the calculation procedures to estimate flow uncertainty. More specifically, the HSM involves two major computational steps after the mean flow equation is solved. The first step is to apply an analytical-based approximate spectral method (ASM) to predict nonstationary flow variances for entire modeling area. The perturbation-based numerical method, nonstationary spectral method (NSM), is then employed in the second step to correct the regional solution in local areas, where the variance dynamics is considered to be highly nonstationary (e.g., around inner boundaries or strong sources/sinks). The boundary conditions for the localized numerical solutions are based on the ASM closed form solutions at boundary nodes. Since the regional closed form solution is instantaneous and the more expensive perturbation-based numerical analysis is only applied locally around the strong stresses, the proposed HSM can be very efficient, making it possible to model strongly nonstationary variance dynamics with complex flow situations in large-scale groundwater systems. In this study the analytical-based ASM solutions was first assessed to quantify the solution accuracy under transient and inner boundary flow conditions. This study then illustrated the HSM accuracy and effectiveness with two synthetic examples. The HSM solutions were systematically compared with the corresponding numerical solutions of NSM and Monte Carlo simulation (MCS), and the analytical-based solutions of ASM. The simulation results have revealed that the HSM is computationally efficient and can provide accurate variance estimations for highly nonstationary large-scale groundwater flow problems.

Transient thermal convection in a spherical enclosure containing a fluid core and a porous shell

Transient natural convection in a spherical enclosure containing a central core fluid and a porous shell fully saturated with the same fluid is investigated numerically. Simulations are based on a mathematical model consisting of the Navier-Stokes equations for the fluid region, the Brinkman equation for flow through porous media, a convective diffusion equation for energy transport, and the Boussinesq approximation for buoyancy. Solutions are obtained by a hybrid spectral method which combines the concepts of Galerkin and collocation methods with Legendre and Chebyshev polynomials employed as basis functions, respectively. Time advancement is accomplished by a combined Adams-Bashforth and backward Euler schemes. The numerical results exhibit remarkable effects along the porous-fluids interface; however, the overall heat flux is only sensitive to the thermal conductivity ratio of the solid matrix to the fluid.

Three-Dimensional Viscous Flows Between Concentric Cylinders Executing Axially Variable Oscillations: A Hybrid Spectral/Finite Difference Solution

A hybrid spectral/finite difference method is developed in this paper for the analysis of three-dimensional unsteady viscous flows between concentric cylinders subjected to fully developed laminar flow and executing transverse oscillations. This method uses a partial spectral collocation approach, based on spectral expansions of the flow parameters in the transverse coordinates and time, in conjunction with a finite difference discretization of the axial derivatives. The finite difference discretization uses central differencing for the diffusion derivatives and a mixed central-upwind differencing for the convective derivatives, in terms of the local mesh Reynolds number. This mixed scheme can be used with coarser as well as finer axial mesh spacings, enhancing the computational efficiency. The hybrid spectral/finite difference method efficiently reduces the problem to a block-tridiagonal matrix inversion, avoiding the numerical difficulties otherwise encountered in a complete three-dimensional spectral-collocation approach. This method is used to compute the unsteady fluid-dynamic forces, the real and imaginary parts of which are related, respectively, to the added-mass and viscous-damping coefficients. A parametric investigation is conducted to determine the influence of the Reynolds and oscillatory Reynolds (or Stokes) numbers on the axial variation of the real and imaginary components of the unsteady forces. A semi-analytical method is also developed for the validation of the hybrid spectral method, in the absence of previous accurate solutions or experimental results for this problem. Good agreement is found between these very different methods, within the applicability domain of the semi-analytical method.

Three-Dimensional Unsteady Viscous Flows Between Oscillating Eccentric Cylinders by an Enhanced Hybrid Spectral Method

An enhanced hybrid spectral method is developed for the analysis of three-dimensional (3-D) unsteady viscous flows in an annular channel between two eccentric cylinders, when one of them executes forced axially variable transverse oscillations of specified frequency. A partial spectral formulation is used for the primitive flow variables, based on Chebyshev polynomials and Fourier expansions of the transverse coordinates and on complex exponential functions of the oscillation frequency and time. The Navier-Stokes equations are discretized in this hybrid method by using a collocation approach in a quasi-radial direction, obtained by a coordinate transformation, in conjunction with an efficient Fourier identification procedure in the azimuthal direction and a mixed-type central-upwind finite-difference scheme in the axial direction, which accommodates both finer and coarser grids. This method, developed for small amplitude oscillations, is first validated for several limiting cases, for which the solutions were found in good agreement with previous results. The method is then used to solve the 3-D unsteady viscous flows between eccentric cylinders executing axially variable transverse oscillations, for which no previous solutions are known. The new numerical solutions, obtained for transverse oscillations following the first flexural beam mode, are illustrated by the axial and azimuthal variations of the real and imaginary components of the reduced unsteady pressure, circumferential velocity component and unsteady fluid-dynamic forces.

A hybrid spectral method for the three-dimensional numerical modelling of nonlinear flows in shallow seas

A spectral method employing eddy-viscosity eigenfunctions is used to solve the full three-dimensional nonlinear hydrodynamic equations for the numerical computation of flows caused by tides or meteorological forcing. An explicit finite element method is developed to compute the nonlinear advective terms and an explicit treatment of bottom friction is used. This leads to a rapidly convergent expansion and relatively few eigenfunctions are required to obtain accurate solutions. An Arakawa B-grid is used in the horizontal directions and leapfrog time-stepping. The eigenfunctions are computed at the beginning of the program, for an arbitrary spatial dependence of eddy viscosity, using the SLEIGN subroutine. Several model problems have been used to test the accuracy, stability, and computational efficiency of the method.