Numerical methods for solid mechanics on overlapping grids: Linear elasticity

This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids.     

光折变介质的相位共轭波强度的近似表达式

以光折变介质中非线性耦合波微分方程的一种精确解为基础，对在推导这种精确解过程中起重要作用的一个守恒量进行讨论，通过初等变换，在不同的特殊条件下得到相位共轭波强度的几处近似表达式。与报道的其它近似方法相比，本近似方法不需要建立及求解简化的微分方程因而具有普适，简单的特点。     

Application of He's homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate

In this Letter, the problem of forced convection over a horizontal flat plate is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

NON-LINEAR MODELLING OF ROTOR DYNAMIC SYSTEMS WITH SQUEEZE FILM DAMPERS—AN EFFICIENT INTEGRATED APPROACH

Squeeze film dampers used in rotor assemblies such as aero-engines introduce non-linear damping forces into an otherwise linear rotor dynamic system. The steady state periodic response of such rotor dynamic systems to rotating out-of-balance excitation can be efficiently determined by using periodic solution techniques. Such techniques are essentially faster than time marching techniques. However, the computed periodic solutions need to be tested for stability and recourse to time marching is necessary if no periodic attractor exists. Hence, an efficient integrated approach, as presented in this paper, is necessary. Various techniques have been put forward in order to determine the periodic solutions, each with its own advantages and disadvantages. In this paper, a receptance harmonic balance method is proposed for such a purpose. In this method, the receptance functions of the rotating linear part of the system are used in the non-linear analysis of the complete system. The advantages of this method over current periodic solution techniques are two-fold: it results in a compact model, and the receptance formulation gives the designer the widest possible choice of modelling techniques for the linear part. Stability of these periodic solutions is efficiently tested by applying Floquet Theory to the modal equations of the system and time marching carried out on these equations, when necessary. The application of this integrated approach is illustrated with simulations and an experiment on a test rig. Excellent correlation was achieved between the periodic solution approach and time marching. Good correlation was also achieved with the experiment.     

An iterative multiscale finite volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).     

Maintaining the stability of nonlinear differential equations by the enhancement of HPM

Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained. In this Letter, the need for stability verification is shown through some examples. Consequently, HPM is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic, even the selected linear part is not.     

Spinor strong interaction model for baryon spectra

The spinor strong interaction model recently proposed by the author to account for meson spectra is applied to baryons. Quark-quark strong interaction is of massless scalar type. Harmonic confinement arises as naturally as linear confinement for mesons. No approximation is needed in order to derive, from the proposed covariant spinor baryon equations, coupled nonlinear radial equations for the ground-state spin-1/2 and spin-3/2 baryons in the rest frame. These equations are effectively of sixth order and call for a particle classification other than the usual unrelativistic one. Simplified analytical solutions are given. Internal functions and mass operators are analogously introduced. With these and the above simplified space-time solution, baryon data yield bare quark masses that agree approximately with those analogously obtained earlier from meson data.     

Gravitational Field of Gyratons

A gyraton is an object moving with the speed of light and having finite energy and internal angular momentum (spin). First, we derive the gravitational field of a gyraton in the linear approximation. After this we study solutions of the Einstein equations for gyratons. We demonstrate that these solutions in 4 and higher dimensions reduce to two linear problems in a Euclidean space. We obtain the exact solutions for relativistic gyratons, discuss their properties, and consider special examples.     

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.     

Nonlinear theory of the compressible gas flows over a nonuniform boundary in the gravitational field in the shallow-water approximation

A set of equations is derived for the motion of a compressible ideal gas over a nonuniform boundary in the gravitational field in the shallow-water approximation. Classical simple waves are shown not to be the solutions to this set of equations. Generalized simple waves are found to exist only in the case of a linear underlying-surface profile. All continuous and discontinuous solutions are obtained in an explicit form for the case of the boundary in the form of an inclined plane, and an analytical solution is found for the problem of the decay of an arbitrary discontinuity. This solution consists of four wave configurations. Necessary and sufficient conditions are determined for the existence of each configuration.     

Instantaneous Bethe-Salpeter Equation and Its Exact Solution

We present an approach to solve Bethe-Salpeter (BS) equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and well-determined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium, the exact solutions precisely present corrections to those of the corresponding Schrödinger equation in order v¹ (v is the relative velocity) for eigenfunctions, in order v² for eigenvalues, and the mixing between S and D components in J^PC=1^-- states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the O(v) corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states.     

Mean field approximation versus exact treatment of collisions in few-body systems

A variational principle for calculating matrix elements of the full resolvent operator for a many-body system is studied. Its mean field approximation results in nonlinear equations of Hartree (-Fock) type, with initial and final channel wave functions as driving terms. The mean field equations will in general have many solutions whereas the exact problem being linear, has a unique solution. In a schematic model with separable forces the mean field equations are analytically soluble, and for the exact problem the resulting integral equations are solved numerically. Comparing exact and mean field results over a wide range of system parameters, the mean field approach proves to be a very reliable approximation, which is not plagued by the notorious problem of defining asymptotic channels in the time-dependent mean field method.     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

Approximate solutions for half-dark solitons in spinor non-equilibrium Polariton condensates

In this work I generalize and apply an analytical approximation to analyze 1D states of non-equilibrium spinor polariton Bose–Einstein condensates (BEC). Solutions for the condensate wave functions carrying black solitons and half-dark solitons are presented. The derivation is based on the non-conservative Lagrangian formalism for complex Ginzburg–Landau type equations (cGLE), which provides ordinary differential equations for the parameters of the dark soliton solutions in their dynamic environment. Explicit expressions for the stationary dark soliton solution are stated. Subsequently the method is extended to spin sensitive polariton condensates, which yields ordinary differential equations for the parameters of half-dark solitons. Finally a stationary case with explicit expressions for half-dark solitons is presented.     

Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation

The mixed boundary value problem of the Laplace equation is considered. The method of fundamental solutions (MFS) approximates the exact solution to the Laplace equation by a linear combination of independent fundamental solutions with different source points. The accuracy of the numerical solution depends on the distribution of source points. In this paper, a weighted greedy QR decomposition (GQRD) is proposed to choose significant source points by introducing a weighting parameter. An index called an average degree of approximation is defined to show the efficiency of the proposed method. From numerical experiments, it is concluded that the numerical solution tends to be more accurate when the average degree of approximation is larger, and that the proposed method can yield more accurate solutions with a less number of source points than the conventional GQRD.     

Initial-boundary value problems for a class of nonlinear thermoelastic plate equations

This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions, it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations, by means of the Galerkin method. Moreover, it also proves the existence of strong and classical solutions.     

An Analytical Technique,Based on Natural Transform to Solve Fractional-Order Parabolic Equations

This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.     

On iterative solutions of the LTE model atmosphere problem

We discuss iterative methods for solving the coupled radiative-transfer and energy-balance equations in the LTE model atmospheres problem including isotropic coherent scattering. We show that iterative solution (e.g. by SOR techniques) of the grand matrix encountered in such problems is vastly more efficient than a direct solution, and is easily vectorized. The final computational effort is linear in the number of depths and frequencies considered, and thus this approach opens the door for the computation of both static and dynamic line-blanketed models using large numbers of depth-points and huge numbers of frequencies.The iterative methods discussed here can be applied to line-formation problems with complete redistribution and to certain classes of problems with partial redistribution (e.g. Compton scattering problems in the Fokker-Planck approximation).     

非线性迭代解析节块方法的数值稳定方法

针对非线性迭代解析节块方法在特定条件下出现的数值不稳定问题,提出了一种解决方法,并给出了数值结果。     

Light modulation by acoustic signals for high diffraction efficiency

A theoretical discussion is presented on strong Bragg acoustooptic interaction AOI of light beams in the dynamic field of an acoustic signal. A system of integrodifferential equations is formulated to describe the evolution of the angular and frequency spectra of the beams in the AOI region for a high level of acoustooptic coupling. The third-order approximation in the perturbation method is used to obtain an analytic solution. Calculations are presented on the modulation of monochromatic beams by acoustic pulses having rectangular envelopes and propagating in a lithium niobate crystal, and the same for a signal having linear frequency modulation LFM in a paratellurite crystal, which demonstrate the broadening of the beam spectrum as the depth of the acoustooptic coupling increases, together with the occurrence of an asymmetry specific to strong AOI in the response of the light field to the symmetrical acoustic signal. Tomsk State University for Control Systems and Electronics, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 42, No. 1, pp. 99–106, January, 1999.