A divergence-free semi-implicit finite volume scheme for ideal,viscous, and resistive magnetohydrodynamics

In this paper, we present a novel pressure-based semi-implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence-free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi-implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi-implicit pressure-based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density-based Godunov-type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi-implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.     

Radiative heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary

The heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary over a flat plate, over a 90-deg wedge, and in stagnation flow is solved by application of the Galerkin method with the particular solution boundary condition I _p (τ₀,ξ,?μ) of the equation of radiative transfer for an inhomogeneous term and the Box method. The exact integral expressions for the radiation part of this problem are developed. The coupling between convective and radiative heat transfer in boundary-layer flows is described by a set of nonlinear simultaneous equations including differential equations and integrodifferential equations. The Galerkin method and the particular solution boundary condition I _p (τ₀,ξ,?μ) are used to analyze the radiation part of the problem. The nonsimilar boundary-layer equations are solved by the Box method. The present numerical procedure solutions are compared in tables with the other exact treating results, the P-3, and P-1 approximation methods for the case of isotropically scattering boundary-layer flows. The effects of linearly anistropically scattering and reflecting surface are taken into account. It is found that the present method is a reliable and efficient numerical procedure and scattering leads to a reduction in the total heat flux. The influence of the forward-backward scattering parameter on the total heat flux decreases with the increase of the surface reflectivity.     

Symbolic Image, Hyperbolicity, and Structural Stability

The aim of the paper is substantiation of a constructive method for verification of hyperbolicity and structural stability of discrete dynamical systems. The main tool here is a symbolic image which is a directed graph constructed by a finite covering of the projective bundle. Hyperbolicity is tested by calculation of the Morse spectrum (the limit set of Lyapunov exponents of pseudo trajectories) which can be found for a given accuracy by the symbolic image [24]. If the Morse spectrum does not contain 0, then the chain recurrent set is hyperbolic and the system is Ω-stable. Thus, the symbolic image gives an opportunity to verify these properties. A diffeomorphism f is shown to be structurally stable if and only if the Morse spectrum does not contain 0 and for the complementary differential there is no connection CR ⁺→CR ^? on the protective bundle. These conditions are verified by an algorithm based on the symbolic image of the complementary differential.     

Independent research and development progress in large-scale wind turbine blade with coordinated aerodynamics,structure, and load

Abstract According to the three key elements in blade design process, i.e., aerodynamic design, structure design, and load prediction, the independent research and development (R&D) progress of blade design is summarized and analyzed. The calculational fluid dynamics (CFD) method, the vortex method, and the blade element momentum method (BEM) are described. Based on the widely used BEM method, the solutions for the blade design in low-speed wind area are obtained. A brief overview of the traditional design and analysis methods based on beam models is given. The defects of these methods used for simulating the structure of large-scale composite blade are analyzed. The application progress of the finite element method (FEM) used in the blade structure analysis is shown. The effects of load prediction on the blades and entire wind turbine are introduced. The progress in load forecasting is described. With the analysis of the relationship among these three key elements, it is concluded that developing a blade optimization design system with coordinated aerodynamics, structure, and load will truly meet the requirement of high efficiency and low cost. The main directions for further study are pointed out, e.g., high efficiency and low load airfoils, structural nonlinear finite element analysis, aerodynamic structure coupling research, and establishing different design standards. The aim is to establish a blade R&D system suitable for the conditions of wind resources in China and promote the development of wind power in the country.     

Viscosity,first normal-stress coefficient,and molecular stretching in dilute polymer solutions

Warner's numerical method for finitely extensible nonlinear elastic (“FENE”) dumbbells in a dilute suspension undergoing steady-state shear flow has been improved by assuming a form of the distribution function that removes the singularity at R = 0 and improves the behavior of the weight function in the Galerkin expression for large b. The comparison of the results of the present method to those of Christiansen and Bird's extrapolation and Warner's numerical solution indicates the success of this method. The material functions, the dumbbell elongation, and the distribution function for steady-state shear flow are given. In addition the newly obtained results are used to assess the accuracy of two approximate methods referred to as the FENE-P model and the FENE-P-B model.     

Asymptotics,Saturation Fronts,and High Resolution Reservoir Characterization

Using an asymptotic methodology we formulate a fast, accurate algorithm for the inversion of multi-phase flow data. The approach is appropriate for many common reservoir production strategies such as CO₂ and water flooding. The technique compares well to a purely numerical method with a significant reduction in computation time. In an application to fractional flow data from the North Robertson field in West Texas, 100,000 permeability and porosity parameters are determined on a workstation. Generally, higher permeability, approaching 1milli-Darcy, is found in the eastern portion of the reservoir. The permeability estimates agree with type curve analysis for material and volumetric balances and a previous numerical pilot-point inversion.     

A control-volume based finite element method for three-dimensional incompressible turbulent fluid flow,heat transfer,and related phenomena

A control-volume based finite element method of equal-order type for three-dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena is presented. The discretization equations are based mainly on the physics of the phenomena under consideration, more than on mathematical arguments. Special emphasis is devoted to the discretization of the convective terms and the continuity equation, and to the treatment of the boundary conditions imposed by the use of a high Reynolds k-?, type turbulence model. The pressure-velocity coupling in the fluid flow calculation is made from a derivative of the original SIMPLER method, without pressure correction. The discretized equations are solved in a sequential, rather than a coupled, form with significant advantage in the required computer time and storage. The method is an extension of a former version proposed by us for two-dimensional, laminar problems, and is here successfully applied to the following situations: three-dimensional deflected turbulent jet, and flows in 90° and 45° junctions of ducts with rectangular cross sections. The calculated results are in very good agreement with the experimental and numerical (obtained with the well established finite difference method) data available in the literature.     

Patterns,fabric, anisotropy,and soil elasto-plasticity

This paper proposes a new method in the theory of soil plasticity – an advance on Hill [The Mathematical Theory of Plasticity, Clarendon Press, Oxford]. The method assumes that soil fabric consists of inter-locking, inter-twining, inter-laced, juxtaposed, and superposed elementary units called “patterns”. A mechanics of patterns is developed. As well as elastic and plastic components, a third strain-increment component is deduced which helps explain non-associated flow. The proposed method leads to explanations of critical states, anisotropy, sensitivity, the Bauschinger effect, and swept-out memory. All these appear in the method as near-inescapable features of plastic solids. Results are illustrated in detail for plane strain biaxial processes.     

A note on divergence,rotation, gradient and their associated theorems

In this note, the essence and some supplements for the unified definition of divergence, rotation and gradient advanced by Tai have been presented based on the method of exterior differential form with an expression of vectors of tensors. The main purpose of this note is to introduce the useful expressions and their applications, and to simplify the proofs of many theorems in various field theories, and they are also important because of their utility for establishing a wide class of principles.     

A high-resolution laser Doppler anemometer: design, qualification, and uncertainty

A new high-resolution laser Doppler anemometer (LDA) has been developed with a working distance of 350 mm, allowing operation in lab-scale wind tunnels. The measurement volume size is 35 μm in diameter by 60 μm in length, allowing resolution of the smallest turbulence scales even at fairly high Reynolds numbers. The controversial question of velocity and validation bias in LDA data is resolved with an experimental method for measuring and removing those effects. Uncertainty estimates are also derived for all the mean and Reynolds stress measurements. Received: 27 June 1999/Accepted: 30 August 2000     

Legendre integrators, post-processing and quasiequilibrium

A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapest way to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (“maximum entropy”, MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.     

Design,manufacturing, and flight testing of a fixed wing micro air vehicle with Zimmerman planform

An optimized and comprehensive method is used to design and manufacture a fixed wing micro air vehicle (MAV) with Zimmerman planform. The design process includes four stages which are the specification of the flight mission, determination of the best aspect ratio, identification of the optimum wing loading and thrust loading values, and estimation of the weight of the structural components of the MAV. To do this, various statistical and analytical methods are utilized. Based on an aerodynamic analysis, the results show that an optimum aspect ratio that maximizes the performance of the Zimmerman MAV for a well-defined cruise speed is determined. Considering six possible flights, a constraint analysis is performed and an optimum wing loading value is determined. It is shown that the computational method is beneficial to determine the exact masses for the structural components including the wing, fuselage, and vertical tail. Using the 3D panel method, the determination of the shape of the reflexed airfoil for the MAV is successfully done by minimizing the drag force and the angle of attack to use less powerful motor and avoid any stall effect, respectively. A stability analysis is then performed to check the safe flight of the designed vehicle. During test flight, the results show that the designed Zimmerman MAV satisfies the pre-defined specification. The final characteristics of the manufactured MAV are: wingspan of 44 cm, weight of 450 g, aspect ratio of 1.51, cruise speed of 20 m/s, and flight endurance of 20 min.     

The viscous-slip, diffusion-slip, and thermal-creep problems for a binary mixture of rigid spheres described by the linearized Boltzmann equation

An analytical version of the discrete-ordinates method (the ADO method) is used with recently established analytical expressions for the rigid-sphere scattering kernels to develop concise and particularly accurate solutions to the viscous-slip, the diffusion-slip, and the half-space thermal-creep problems for a binary gas mixture described by the linearized Boltzmann equation. In addition to a computation of the viscous-slip, the diffusion-slip, and the thermal-slip coefficients, for the case of Maxwell boundary conditions for each of the two species, the velocity, heat-flow, and shear-stress profiles are established for each species of particles. Numerical results are reported for two binary mixtures (Ne–Ar and He–Xe) with various molar concentrations.     

An implicit method for frictional contact, impact and rolling

In this paper an implicit method for frictional contact, impact and rolling is suggested. A nonclassical formulation of a two-dimensional hyperelastic body unilaterally constrained to rigid supports is proposed by following the ideas of Moreau and Jean. A total Lagrangian formulation of the system is given. The elastic properties are defined by coupling the second Piola–Kirchhoff stress to the Green–Lagrange strain via the Kirchhoff–St. Venant law. The equation of motion is written in the spirit of Moreau by using the mean value impulses introduced by Jean. The mean value impulses appear explicitly in the equation of motion. In such manner the treatment of nonconstant kinematic transformation matrices becomes straightforward. The rigid supports are described by smooth functions. By utilizing these functions and the mean value impulses, new contact/impact laws of Signorini and Coulomb type are formulated. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. Three two-dimensional examples are solved by the method: a contact problem, an impact problem and a rolling contact problem.     

Fatigue crack-tip deformation,crack growth and life prediction in polycarbonate

Low-cycle fatigue fracture of polycarbonate is investigated. Local strain in the vicinity of the notch root and crack tip has been measured in real time by using a fine-grid method. The relationships among the local strain, crack initiation from the notch root and crack propagation of the crack tip are studied, and a method for more precise life estimation is suggested. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Asymptotics, structure, and integration of sound-proof atmospheric flow equations

Relative to the full compressible flow equations, sound-proof models filter acoustic waves while maintaining advection and internal waves. Two well-known sound-proof models, an anelastic model by Bannon and Durran’s pseudo-incompressible model, are shown here to be structurally very close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing ? _t ρ in the mass continuity equation and slightly modifying the gravity term, whereas the pseudo-incompressible model results from dropping ? _t p from the pressure equation. For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudo-incompressible model approaches the zero Mach number, variable density flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible flow equations, yet the pseudo-incompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphere–ocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dörnbrack (Int J Numer Meth Fluids 56:1513–1519, 2007). Despite these useful properties, neither model can be derived by a low-Mach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura and Phillips model, which assumes weak stratification of the order of the Mach number squared, can be obtained as a leading-order model from systematic low Mach number asymptotic analysis. Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudo-incompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper, we propose a second-order finite-volume projection method for the anelastic and pseudo-incompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the descent of a cold air bubble in the small-scale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work.     

Singular solitons,shock waves,and other solutions to potential KdV equation

This paper addresses the potential Korteweg–de Vries equation. The singular 1-soliton solution is obtained by the aid of ansatz method. Subsequently, the $G^{\prime }/G$ -expansion method and the exp-function approach also gives a few more interesting solutions. Finally, the Lie symmetry analysis leads to another plethora of solution to the equation. These results are going to be extremely useful and applicable in applied mathematics and theoretical physics.     

Apollonius coordinates,theN-body problem,and continuation of periodic solutions

This paper treats theN-body problem and its relation to various restricted problems. For each solution of the Kepler problem a generalization of the pulsating coordinates used to express the Hamiltonian of the elliptic restricted three-body problem is given. These coordinates are called Apollonius coordinates. The method of symplectic scaling is used to give a precise derivation of the elliptic restricted problem showing the precise asymptotic relationship between the restricted problem and the full three-body problem. This derivation obviates the proof of the fact that a nondegenerate periodic solution of the elliptic restricted three-body problem can be continued into the full three-body problem under mild nonresonance assumptions. Also, the method of symplectic scaling is used to give a precise derivation of the elliptic Hill lunar equation showing the precise relationship between the elliptic Hill lunar equation and the full three-body problem. A similar continuation theorem is established.     

Viscosity approximation with weak contractions for fixed point problem,equilibrium problem,and variational inequality problem

This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction.The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality.Under suitable conditions,some strong convergence theorems are established in the framework of Hilbert spaces.The results presented in the paper improve and extend the corresponding results of Colao et al.（Colao,V.,Acedo,G.L.,and Marino,G.An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings.Nonlinear Anal.71,2708–2715（2009））,Plubtieng and Punpaeng（Plubtieng,S.and Punpaeng,R.A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.J.Math.Anal.Appl.336,455–469（2007））,Colao et al.（Colao,V.,Marino,G.,and Xu,H.K.An iterative method for finding common solutions of equilibrium problem and fixed point problems.J.Math.Anal.Appl.344,340–352（2008））,Yao et al.（Yao,Y.,Liou,Y.C.,and Yao,J.C.Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.Fixed Point Theory Application 2007,Article ID 64363（2007）DOI 10.1155/2007/64363）,and others.     

Vehicle vibrating on a soft compacting soil half-space: Ground vibrations, terrain damage, and vehicle vibrations

The point of departure of the present work may be either an interest in vehicle vibrations themselves, or in ground vibrations and terrain damage due to vehicles traveling off-road. The vibrations of a vehicle traversing dry, soft terrain, which is either rough or undulating, may be significantly modified by the dynamic interaction of the vehicle with the soil, particularly due to losses of energy by soil compaction and as elastic waves. The present work provides a prediction methodology for both vehicle and soil vibrations, accounting for the effects mentioned above. An expedient linear method is compared to a rheologically-based non-linear method. In the linear method, the soil compaction is incorporated as a loss factor in the dynamic stiffness of the otherwise elastic half-space; the imaginary part of that dynamic stiffness already includes the effects of wave damping. The non-linear model treats the compaction using a general rheological model for soils exhibiting both viscous and thixotropic effects, and requires iterative solution. A key feature of the latter model is the hypothesis that the stress distribution may be approximately regarded as quasi-static when calculating compaction losses; that approximation is expected to hold at low frequencies, since the P-wavelength in the soil is then much greater than the dimensions of the zone in which most compaction occurs. The methods predict that the soil compaction and excited ground vibrations have maxima at the vehicle bounce and hop resonances, and at high frequencies at which the Rayleigh wavelength approaches the order of the contact patch diameter. Moreover, sufficiently soft, compactable soils, but fully realizable in nature, control the vehicle response at the hop resonance, and possibly also at the bounce resonance.