Spectral Method For Solution Research Articles

Spectral eigenfunction decomposition of a Fokker–Planck operator for relativistic heavy-ion collisions

A spectral solution method is proposed to solve a previously developed non-equilibrium statistical model describing partial thermalization of produced charged hadrons in relativistic heavy-ion collisions, thus improving the accuracy of the numerical solution. The particle’s phase-space trajectories are treated as a drift-diffusion stochastic process, leading to a Fokker–Planck equation (FPE) for the single-particle probability distribution function. The drift and diffusion coefficients are derived from the expected asymptotic states via appropriate fluctuation–dissipation relations, and the resulting FPE is then solved numerically using a spectral eigenfunction decomposition. The calculated time-dependent particle distributions are compared to Pb–Pb data from the ATLAS and ALICE collaborations at the Large Hadron Collider.

Adaptive spectral solution method for Fredholm integral equations of the second kind

Adaptive spectral solution method for Fredholm integral equations of the second kind

SIMULATION MODEL OF A QS WITH HYPEREXPONENTIAL DISTRIBUTION IN THE GPSS WORLD ENVIRONMENT

This article presents results obtained on the development of a pseudo-random sequence generator for simulation modeling of queueing system (QS) with second-order hyperexponential input distribution. In the scientific literature on GPSS WORLD, including web resources, the authors did not find any data in this subject area. The GPSS WORLD library includes many generators of various distribution laws, but there are no generators for such important composite distributions according to the queuing theory. It is known that the hyperexponential distribution concept provides a wide range of changes in the random variable variation coefficient from unity to an arbitrarily large value. Due to the fact that the variation coefficients of arrival and service intervals play an important role in estimating the delay of queue claims in queuing systems, thus this distribution concept plays a special role in modeling of QS. For a hyperexponential distribution of the second order, the authors previously obtained numerical-analytical results based on the Lindley integral equation spectral solution method. The article describes a developed algorythm and a program on GPSS WORLD to simulate the functioning of QS with hyperexponential input distribution incl. description of operation of logical switches. The difference between analytical and simulation modeling of the two-phase representation of a given distribution concept is explained on the example of the developed model. The adequacy of the obtained results was confirmed by comparing the simulation results with the results of numerical simulation in the Mathcad environment.

Continuous Grasping Force Estimation With Surface EMG Based on Huxley-Type Musculoskeletal Model.

Continuous grasping force estimation based on electromyography (EMG) signals, is very useful in practical applications including prosthetic control and human force observation. However, implementing the practical grasping force estimation usually considers a trade-off between the computational precision and resources. Specifically, the estimation based on the Huxley-type muscle model reaches detailed approximation of physiological process at a cost of larger computational resources for solving nonlinear partial differential equations while the counterpart with a traditional Hill-type muscle model. In this article, we achieve the grasping force estimation based on a reduced Huxley-type musculoskeletal model with high accuracy yet low time delay. Leveraging on a balanced truncation method, we further reduce the dimensionality of the spectral method solution in the Huxley-type musculoskeletal model for the model simplification. In addition, we introduce the Kalman filter method to process the EMG signals obtained by an armband, yielding better real-time performances and accuracy compared to the signal treatment using the traditional EMG filter method. Moreover, we also implement an effective identification of the model parameters using a particle swarm method. Finally, we trained the model on the first day and made grasping force estimation experiments involved with three participants over the course of a month. We envision that this effective and practical method would further improve the practical applications in the field of grasping force estimation.

A multigrid solver for the coupled pressure-temperature equations in an all-Mach solver with VoF

We present a generalisation of the all-Mach solver of Fuster and Popinet (2018) [1] to account for heat diffusion between two different compressible phases. By solving a two-way coupled system of equations for pressure and temperature, the current code is shown to increase the robustness and accuracy of the solver with respect to classical explicit discretization schemes. Different test cases are proposed to validate the implementation of the thermal effects: an Epstein-Plesset like problem for temperature is shown to compare well with a spectral method solution. The code also reproduces free small amplitude oscillations of a spherical bubble where analytical solutions capturing the transition between isothermal and adiabatic regimes are available. We show results of a single sonoluminescent bubble (SBSL) in standing waves, where the result of the DNS is compared with that of other methods in the literature. Moreover, the Rayleigh collapse problem is studied in order to evaluate the importance of thermal effects on the peak pressures reached during the collapse of spherical bubbles. Finally, the collapse of a bubble near a rigid boundary is studied reporting the change of heat flux as a function of the stand-off distance.

The influence of compatibility equations of non‐linear elasticity on convergence for spectral method solutions

AbstractIn structural mechanics the stresses are of a primary interest. Due to the required regularity of the approximation it is often inconvenient to obtain the stresses from the displacement methods. Mixed and hybrid methods are thus proposed to avoid this obstacle. The compatibility conditions should be fulfilled in order to obtain a unique solution.In this paper the compatibility equations of non‐linear elasticity are considered. Despite its very long history, this issue has not been satisfactorily resolved for non‐simply‐connected bodies until recently [1]. Its well known however, that the compatibility conditions are linearly dependent.The aim of the present contribution is to include the spectral methods [2] in order to improve the finite element solution procedure for the reversible elastodynamics problem [3], formulated for a large strain deformation with the deformation gradient tensor as one of the basic unknowns. In the case of the small strains the components of the deformation gradient tensor are of different magnitude. The main contribution is the derivation of the solution strategy in order to take this information into account. Applying this strategy to the implicit time integration method the exponent decrease of the condition numbers of the iteration matrices was achieved. Consequently the accuracy of the calculations should increase and the convergence for spectral method solutions should considerably improve. The theoretical findings are confirmed by a selected numerical example from structural mechanics.

Spectral method solutions of a variable aspect ratio duct flow in a uniform transverse magnetic field with two different thermal boundary conditions

Abstract A laminar electrically-conducting flow inside an electrically-insulated rectangular duct in a uniform transverse magnetic field with both uniform surface temperature and uniform heat flux boundary conditions are considered numerically. The problem with aspect ratios of the range from 1:10 to 10:1 and Hartmann number M up to 1000 is solved using a highly accurate technique which is spectral method. The flow variables are expanded in terms of linear combinations of Chebyshev polynomials chosen to satisfy the boundary conditions implicitly. The resulting equations are collocated using the Gauss points to produce a system of nonlinear algebraic equations which is solved iteratively using Gauss elimination. Convergence properties of the numerical method reveals that for aspect ratio of less than 4 to 1, the flow is well resolved with as small as 29 by 29 Chebyshev polynomials; however, as the aspect ratio increases to more than 4 to 1, the number of polynomials required for an adequate resolution can be as high as 49 by 49 polynomials. It is found when the magnetic field is turned on, the pressure drop, in general, increases with the field for different aspect ratios. However, the pressure drop increase will be slower near the aspect ratio of 10. Also, the heat transfer increases with the field for most of the cases, but for some cases the field will have adverse effect on the heat transfer, particularly, for constant surface temperature boundary condition at aspect ratio > 4 and M 4 and M

An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations

An explicit high-order semi-Lagrangian method is developed for the solution of Lagrangian transport equations in Eulerian-Lagrangian formulations. The method is consistent with an explicit, discontinuous spectral element method (DSEM) discretization of the Eulerian formulation. The semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles seeded at the Gauss quadrature points from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian-Lagrangian formulations. Numerical tests in one and two dimensions for linear and non-linear advection show that the method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator.

Adaptive spectral solution method for the Landau and Lenard-Balescu equations

We present an adaptive spectral method for solving the Landau/Fokker-Planck equation for electron-ion systems. The heart of the algorithm is an expansion in Laguerre polynomials, which has several advantages, including automatic conservation of both energy and particles without the need for any special discretization or time-stepping schemes. One drawback of such an expansion is the O(N3) memory requirement, where N is the number of polynomials used. This can impose an inconvenient limit in cases of practical interest, such as when two particle species have widely separated temperatures. The algorithm we describe here addresses this problem by periodically re-projecting the solution onto a judicious choice of new basis functions that are still Laguerre polynomials but have arguments adapted to the current physical conditions. This results in a reduction in the number of polynomials needed, at the expense of increased solution time. Because the equations are solved with little difficulty, this added time is not of much concern compared to the savings in memory. To demonstrate the algorithm, we solve several relaxation problems that could not be computed with the spectral method without re-projection. Another major advantage of this method is that it can be used for collision operators more complicated than that of the Landau equation, and we demonstrate this here by using it to solve the non-degenerate quantum Lenard-Balescu (QLB) equation for a hydrogen plasma. We conclude with some comparisons of temperature relaxation problems solved with the latter equation and the Landau equation with a Coulomb logarithm inspired by the properties of the QLB operator. We find that with this choice of Coulomb logarithm, there is little difference between using the two equations for these particular systems.

Meshless spectral method for solution of time-fractional coupled KdV equations

In this article, an efficient and accurate meshless spectral interpolation method is formulated for the numerical solution of time-fractional coupled KdV equations that govern shallow water waves. Meshless shape functions constructed via radial basis functions (RBFs) and point interpolation are used for discretization of the spatial operator. Approximation of fractional temporal derivative is obtained via finite differences of order O(τ2−α) and a quadrature formula. The formulated method is applied to various test problems available in the literature for its validation. Approximation quality and efficiency of the method is measured via discrete error norms E2, E∞ and Erms. Convergence analysis of the proposed method in space and time is numerically determined by varying nodal points M and time step-size τ respectively. Stability of the proposed method is discussed and affirmed computationally, which is an important ingredient of the current study.

Fourier-based spectral method solution to finite strain crystal plasticity with free surfaces

A plastically dilatational material model is proposed that enables to simulate the mechanical response of non-compact geometries (containing free surfaces) by means of established spectral methods without any particular adaptations, i.e. in combination with arbitrary constitutive laws describing the remainder of the simulated geometry and under mixed boundary conditions. The versatility of this material model and more accurate representation of empty space in comparison to an isotropic elastic model employing low stiffness is demonstrated for the cases of void growth under biaxial extension and grain-scale deformation behavior of an oligocrystalline dogbone-shaped aluminum sample under uniaxial tension.

A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.

An iterative spectral solution method for thin elastic plate flexure with variable rigidity

S U M M A R Y Thin plate flexure theory provides an accurate model for the response of the lithosphere to vertical loads on horizontal length scales ranging from tens to hundreds of kilometres. Examples include flexure at seamounts, fracture zones, sedimentary basins and subduction zones. When applying this theory to real world situations, most studies assume a locally uniform plate thickness to enable simple Fourier transform solutions. However, in cases where the amplitude of the flexure is prominent, such as subduction zones, or there are rapid variations in seafloor age, such as fracture zones, these models are inadequate. Here we present a computationally efficient algorithm for solving the thin plate flexure equation for non-uniform plate thickness and arbitrary vertical load. The iterative scheme takes advantage of the 2-D fast Fourier transform to perform calculations in both the spatial and spectral domains, resulting in an accurate and computationally efficient solution. We illustrate the accuracy of the method through comparisons with known analytic solutions. Finally, we present results from three simple models demonstrating the differences in trench outer rise flexure when 2-D variations in plate rigidity and applied bending moment are taken into account. Although we focus our analysis on ocean trench flexure, the method is applicable to other 2-D flexure problems having spatial rigidity variations such as seamount loading of a thermally eroded lithosphere or flexure across the continental–oceanic crust boundary.

Relativistic effects in the tidal interaction between a white dwarf and a massive black hole in Fermi normal coordinates

We consider tidal encounters between a white dwarf and an intermediate mass black hole. Both weak encounters and those at the threshold of disruption are modeled. The numerical code combines mesh-based hydrodynamics, a spectral method solution of the self-gravity, and a general relativistic Fermi normal coordinate system that follows the star and debris. Fermi normal coordinates provide an expansion of the black hole tidal field that includes quadrupole and higher multipole moments and relativistic corrections. We compute the mass loss from the white dwarf that occurs in weak tidal encounters. Secondly, we compute carefully the energy deposition onto the star, examining the effects of nonradial and radial mode excitation, surface layer heating, mass loss, and relativistic orbital motion. We find evidence of a slight relativistic suppression in tidal energy transfer. Tidal energy deposition is compared to orbital energy loss due to gravitational bremsstrahlung and the combined losses are used to estimate tidal capture orbits. Heating and partial mass stripping will lead to an expansion of the white dwarf, making it easier for the star to be tidally disrupted on the next passage. Finally, we examine angular momentum deposition. By including the octupole tide, we are able for the first time to calculate deflection of the center of mass of the star and debris. With this observed deflection, and taking into account orbital relativistic effects, we compute directly the change in orbital angular momentum and show its balance with computed spin angular momentum deposition.

A spectral method solution to crystal elasto-viscoplasticity at finite strains

A significant improvement over existing models for the prediction of the macromechanical response of structural materials can be achieved by means of a more refined treatment of the underlying micromechanics. For this, achieving the highest possible spatial resolution is advantageous, in order to capture the intricate details of complex microstructures. Spectral methods, as an efficient alternative to the widely used finite element method (FEM), have been established during the last decade and their applicability to the case of polycrystalline materials has already been demonstrated. However, until now, the existing implementations were limited to infinitesimal strain and phenomenological crystal elasto-viscoplasticity. This work presents the extension of the existing spectral formulation for polycrystals to the case of finite strains, not limited to a particular constitutive law, by considering a general material model implementation. By interfacing the exact same material model to both, the new spectral implementation as well as a FEM-based solver, a direct comparison of both numerical strategies is possible. Carrying out this comparison, and using a phenomenological constitutive law as example, we demonstrate that the spectral method solution converges much faster with mesh/grid resolution, fulfills stress equilibrium and strain compatibility much better, and is able to solve the micromechanical problem for, e.g., a 2563 grid in comparable times as required by a 643 mesh of linear finite elements.

Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide

The Lippmann–Schwinger integral equation describes the scattering of acoustic waves from an inhomogeneous medium. For scattering problems in free space, Vainikko proposed a fast spectral solution method exploiting the convolution structure of this equation's integral operator and the fast Fourier transform. Although the integral operator of the Lippmann–Schwinger integral equation for scattering in a planar three-dimensional waveguide is not a convolution, we show in this paper that the separable structure of the kernel allows to construct fast spectral collocation methods. The numerical analysis of this method requires smooth material parameters; for discontinuous materials there is no theoretical convergence statement. Therefore, we construct a Galerkin variant of Vainikko's method avoiding this drawback. For several distant scattering objects inside the three-dimensional waveguide this discretization technique would lead to a computational domain consisting of one large box containing all scatterers and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects.

Spectral method for solution of the fractional transport equation

In this paper, the Chebyshev polynomials expansion method is applied to find both an analytical solution of the fractional transport equation in the one-dimensional plane geometry and its numerical approximations. The idea of the method is in reducing of the fractional transport equation to a system of the linear fractional differential equations for the unknown coefficients of the Chebyshev polynomials expansion. The obtained system of equations is then solved by using the operational method for the Caputo fractional derivative.

A spectral solution of nonlinear mean field dynamo equations: Without inertia

In this paper we describe in detail spectral solution method for solving nonlinear mean field dynamo equations without inertial effects in a rapidly rotating spherical shell filled with electrically conducting fluid taken into account no-slip velocity boundary conditions in the core for a finitely conducting inner core and an insulating mantle. Sample results are provided for nonlinear alpha squared dynamos. The method is suitable for solving many geophysical, astrophysical and engineering problems.

Composite spectral method for solution of the diffusion equation with specification of energy

AbstractMany physical subjects are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions replacing the classic boundary conditions. In this article, we introduce a new numerical method for solving the one‐dimensional parabolic equation with nonlocal boundary conditions. The approximate proposed method is based upon the composite spectral functions. The properties of composite spectral functions consisting of terms of orthogonal functions are presented and are utilized to reduce the problem to some algebraic equations. The method is easy to implement and yields very accurate result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

An angular spectral method for solution of the heat equation in spheroidal geometries

A spectral numerical method is presented for solving the heat equation in oblate or prolate spheroids. Cartesian coordinates are scaled to transform the spheroidal geometry into a spherical geometry. The diffusion term in the transformed equation is anisotropic, being enhanced in the polar directions. The transformed equation is discretised in angle using a truncated spherical harmonic expansion of the temperature in transformed spherical polar coordinates. The anisotropic diffusion term is reduced to block tridiagonal form using recurrence relations for spherical harmonics. The radial coordinate is discretised using finite differences in scaled radius but other radial schemes are possible. Without heat sources and with a homogeneous Dirichlet boundary condition the problem reduces to an eigenproblem for the decay rate. The results are compared to the separated variables solution, which employs oblate or prolate spheroidal wave functions. The method directly extends to other scalar problems in spheroidal geometries, which have the highest derivatives in Laplacian form, including passive advection-diffusion of a scalar. The method may be extended, with difficulty, to problems with vector diffusion or ellipsoidal geometries.