On the correspondence between the three- and four-dimensional parameters of the three-dimensional rotation group

A fundamental kinematic theorem due to Euler permits synthesizing a series of three- and four-dimensional orientation parameters that correspond to each other in spaces of the same dimension. We use the theorem about the homeomorphism of two topological spaces (the three-dimensional sphere S ³ ? R ⁴ with a single punctured (removed) point and the three-dimensional space R ³) to establish a one-to-one mutually continuous correspondence between the four- and three-dimensional kinematic parameters prescribed in these spaces. The latter can be proved using the stereographic projection of points of the sphere S ³ onto the hyperplane R ³. For the normalized (Hamiltonian) Rodrigues-Hamilton parameters, we present a method of stereographic projection of a point belonging to the three-dimensional sphere S ³ onto the oriented space R ³. We present a family of local kinematic parameters obtained by the method of mapping four symmetric kinematic parameters of the space R ⁴ onto the oriented real space R ³. In contrast to the well-known four symmetric global parameters of the Rodrigues-Hamilton orientation, the synthesized three-dimensional orientation parameters are local (have two singular points ±360°). The differential equations of rotation in the three-dimensional orientation parameters are obtained by the projection method. We present the three-dimensional parameters corresponding to the classical Hamiltonian quaternions defined in the four-dimensional vector space R ⁴.     

A Variational Approach to Dynamics of Flexible Multibody Systems

Abstract

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body reference frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed     

Action-Minimizing Orbits in the Parallelogram Four-Body Problem with Equal Masses

In this paper we prove the existence of a new periodic solution for the planar Newtonian four-body problem with equal masses. On this orbit two mass points travel on one star-shaped closed curve while the other two travel on another and have the opposite orientation. The configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time. Our proof is based on a variational approach inspired by a recent work of Chenciner &; Montgomery [5]. By choosing an appropriate subspace of the Sobolev space H ¹([0,T],V), where V is the configuration space, we show that the action functional restricted to this subspace attains its infimum and any minimizer solves the Newtonian four-body problem. The orbits we found are indeed extensions of these minimizers. By studying the behavior of minimizers in reduced configuration space and comparing their action with rhomboid motions, we show that these minimizers do not experience any collision.     

Maximal Regularity of the Time - Periodic Linearized Navier–Stokes System

Time-periodic solutions to the linearized Navier–Stokes system in the n-dimensional whole-space are investigated. For time-periodic data in L ^q -spaces, maximal regularity and corresponding a priori estimates for the associated time-periodic solutions are established. More specifically, a Banach space of time-periodic vector fields is identified with the property that the linearized Navier–Stokes operator maps this space homeomorphically onto the L ^q -space of time-periodic data.     

A consistent numerical scheme for the von Mises mixed-hardening constitutive equations

Analytical solutions are derived for the von Mises mixed-hardening elastoplastic model under rectilinear strain paths, and the concept of response subspace is introduced such that the original five-dimensional problem in deviatoric stress space is reduced to a more economic two-dimensional problem, of which two coordinates (x,y) suffice to determine normalized active stress. Furthermore, in this subspace a Minkowski spacetime can be endowed, on which the group action is found to be a proper orthochronous Lorentz group SO_o(2,1). The existence of a fixed point attractor in the normalized active stress space is demonstrated by the long-term behavior deduced from the analytical solutions, which together with the response stability is further verified by Lyapunov's direct method. Two numerical schemes based on a nonlinear Volterra integral equation and on a group symmetry are derived, the latter of which exactly preserves the consistency condition for every time step. The consistent scheme is stable, accurate and efficient, because it updates the stress point automatically on the yield surface at each time step without any iteration. For the purpose of comparison and contrast, numerical results calculated by the above two schemes as well as by the radial return method were displayed for several loading examples.     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.     

Oscillating radial flow between parallel plates

An analysis is presented for laminar radial flow due to an oscillating source between parallel plates. The source strength varies according to Q=Q ₀ cos ωt, and the solution is in the form of an infinite series in terms of a reduced Reynolds number, R _a ^* =Q ₀/4πνa/(r/a)². (Q ₀ = amplitude of source strength, ω = frequency, a = half distance between plates, r = radial coordinate, t = time, and ν = kinematic viscosity.) The results are valid for small values of R _a ^* and all values of the frequency Reynolds number, α=ωa ²/ν. The effects of the parameters R _a ^* and α are discussed.     

Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C ¹ traveling wave solutions, provided that the C ¹ norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space-time R ¹⁺ⁿ , are also given.     

Mapping similarity between parallel and serial architecture kinematics

First the principles of mapping spatial points to surfaces is introduced in the context of the inverse kinematics of a general six revolute serial wrist partitioned robot. Then the advantage of choosing ideal frames is illustrated by showing that in the case of some architectures an image space formulation, though possible, may be an impediment to clear geometric insight and a satisfactory and much simpler solution. After showing how the general point mapping transformation is reduced to classical Blaschke-Grünwald planar mapping a novel three legged planar robot??s direct kinematics is solved in image space and then using conventional ??distance?? constraints. The purpose is to show why the latter approach yields spurious solutions and how the displacement pole rotation performed with kinematic mapping reliably avoids this problem. In conclusion certain other new and/or interesting reduced mobility parallel robots are discussed briefly to point out some advantages and insights gained with an image space approach. Particular effort is made to expose in detail how mapping simplifies and extends the solution of direct kinematics pertaining to Calvel??s ??Delta?? 3D translational robot.     

Global Solution to the Generalized Riemann Problem in the Presence of a Boundary and Contact Discontinuities

This work is a continuation of our previous work. In the present paper we study the global structure stability of the Riemann solution $u=U(\frac{x}{t})$ containing only contact discontinuities for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the existence and uniqueness of a global piecewise C ¹ solution containing only contact discontinuities to a class of the generalized Riemann problems for general n×n quasilinear hyperbolic systems of conservation laws in a half space. Our result indicates that this kind of Riemann solution $u=U(\frac{x}{t})$ mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary possesses a global nonlinear structure stability. Some applications to quasilinear hyperbolic systems of conservation laws occurring in physics and other disciplines, particularly to the system describing the motion of the relativistic string in Minkowski space R ^1?+?n , are also given.     

Studying the Laws of the Thermoviscoplastic Deformation of a Solid Under Nonisothermal Complex Loading. Part 2

The studies made at the thermoplasticity department of the S. P. Timoshenko Institute of Mechanics are analyzed. These studies involve experimental validations of the kinematic equation of creep damage and the constitutive equations describing simple thermoviscoelastoplastic loading, with history, of isotropic and transversally isotropic bodies, for elastoviscoplastic deformation of bodies along slightly curved paths, for complex loading along arbitrary paths lying either in a plane arbitrarily oriented in the five-dimensional space of stresses or in one coordinate plane, and for elastoplastic deformation of a body's elements along paths of moderate curvature and small torsion     

Two-Dimensional Elastic Herglotz Functions and Their Applications in Inverse Scattering

In this work solutions of the spectral Navier equation that satisfy the Herglotz boundedness condition in two-dimensional linear elasticity are presented. Navier eigenvectors in polar coordinates are introduced and it is established that they form a linearly independent and complete set in the L ²-sense on every smooth curve. It is also proved that the classical solutions of the spectral Navier equation are expressed via Navier eigenvectors, and this expansion converges uniformly over compact subsets of R ². Two far-field patterns, the longitudinal and the transverse one corresponding to the displacement field are introduced, and the Herglotz norm is expressed as the sum of the L ²-norms of these patterns over the unit circle. It is also established that the space of elastic Herglotz functions is dense in the space of the classical solutions of the spectral Navier equation. Finally, connection to inverse elasticity scattering is established and reconstructions of rigid bodies are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Delay in response of turbulent heat transfer against acceleration or deceleration of flow in a pipe

To predict the heat transfer enhancements that result from the application of a pulsating flow in a pipe, we experimentally investigated the turbulent heat transfer variations produced in response to sudden accelerations or decelerations to flows within a pipe. To accomplish this, the Reynolds numbers with the valve open (Re₁) and close (Re₀) were systematically varied in the range of 8,000 ≤ Re₁ ≤ 34,000 and 700 ≤ Re₀ ≤ 23,000, respectively, and in-pipe spatiotemporal heat transfer variations were measured using infrared thermography simultaneously with temporal variations to the in-pipe flow properties. Based on the experimental results, it was found that the heat transfer delays that occur in response to accelerations or decelerations can be characterized using the corresponding time lag Δt and first-order time constant τ. The values of Δt and τ can be expressed as non-dimensional forms of Δt/(ν/u_τ²) and τ/(R/u_τ), respectively, where u_τ is the pipe wall friction velocity, ν is the kinematic viscosity of the fluid, and R is the pipe radius.     

A Priori Assessment of Prediction Confidence for Data-Driven Turbulence Modeling

Although Reynolds-Averaged Navier–Stokes (RANS) equations are still the dominant tool for engineering design and analysis applications involving turbulent flows, standard RANS models are known to be unreliable in many flows of engineering relevance, including flows with separation, strong pressure gradients or mean flow curvature. With increasing amounts of 3-dimensional experimental data and high fidelity simulation data from Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), data-driven turbulence modeling has become a promising approach to increase the predictive capability of RANS simulations. However, the prediction performance of data-driven models inevitably depends on the choices of training flows. This work aims to identify a quantitative measure for a priori estimation of prediction confidence in data-driven turbulence modeling. This measure represents the distance in feature space between the training flows and the flow to be predicted. Specifically, the Mahalanobis distance and the kernel density estimation (KDE) technique are used as metrics to quantify the distance between flow data sets in feature space. To examine the relationship between these two extrapolation metrics and the machine learning model prediction performance, the flow over periodic hills at Re = 10595 is used as test set and seven flows with different configurations are individually used as training sets. The results show that the prediction error of the Reynolds stress anisotropy is positively correlated with Mahalanobis distance and KDE distance, demonstrating that both extrapolation metrics can be used to estimate the prediction confidence a priori. A quantitative comparison using correlation coefficients shows that the Mahalanobis distance is less accurate in estimating the prediction confidence than KDE distance. The extrapolation metrics introduced in this work and the corresponding analysis provide an approach to aid in the choice of data source and to assess the prediction performance for data-driven turbulence modeling.     

Identifying Symmetry Classes of Elasticity Tensors Using Monoclinic Distance Function

We present a method to identify the symmetry class of an elasticity tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the concept of distance in the space of tensors, and relies on the monoclinic or transversely isotropic distance function. Since the orientation of a monoclinic or transversely isotropic tensor depends on two Euler angles only, we can plot the corresponding distance functions on the unit sphere in ℝ³ and observe the symmetry pattern of the plot. In particular, the monoclinic distance function vanishes in the directions of the normals of the mirror planes, so the number and location of the zeros allows us to identify the symmetry class and the orientation of the natural coordinate system. Observing the approximate locations of the zeros on the plot, we can constrain a numerical algorithm for finding the exact orientation of the natural coordinate system.     

N-space collision model for rotation-translation energy exchange in DSMC collisions

A new discrete simulation Monte Carlo (DSMC) collision model for molecules possessing an integer number of classical degrees of freedom for molecular structure energy is proposed. The total molecular energy (translation plus molecular structure) is represented by a velocity in five-dimensional space. Each collision is an elastic N-sphere collision in N-space, where N= 3, 4, or 5. For N=5, there is a maximum chance of exchange of energy between the two components of velocity, which represent the rotation energy and the three components that represent the translational velocity. For N=3, there is no change in the rotation energy of each molecule, and for N=4, there is an intermediate chance that rotation and translation energy will be exchanged. The exchange probability ϕ can be set to give the desired rotational relaxation rate. To achieve any realistic viscosity μ(T), the N-space model must be coupled with a modified collision procedure known as ν-DSMC. The new model is shown to match the results of molecular dynamics calculations for the internal structure of a Mach 7 shock, with a saving of about 20% in CPU time compared to standard DSMC using the standard Borgnakke-Larsen exchange model.     

Direction of Vorticity and Regularity up to the Boundary: On the Lipschitz-Continuous Case

In their famous 1993 paper, Constantin and Fefferman consider the evolution Navier–Stokes equations in the whole space R ³ and prove, essentially, that if the direction of the vorticity is Lipschitz continuous in the space variables, during a given time-interval, then the corresponding solution is regular. Since Lipschitz-continuity is a very natural, basic, property, it looks interesting to go further in this particular direction. In this paper, we consider the initial-boundary value problem for the Navier–Stokes equations in a regular, bounded, domain under a slip boundary condition, and prove regularity of the solution, up to the boundary, under a weakened Lipschitz-continuity assumption on the direction of the vorticity. The interest of our result highly relies on the fact that the Lipschitz-continuity coefficient g(x, t) is sharp. This means, in a sense, that our finding possesses the same level of accuracy as that of the classical “Prodi-Serrin” type conditions; see the introductory section. It should be remarked that a similar result was already obtained in the 2009 paper by Beirão da Veiga and Berselli. In the latter, the proof of an analogous sharp result was shown under the assumption of ${\frac12}$ -H?lder continuity on the direction of vorticity. The authors also claimed, correctly, that by the same ideas the proof of such a result could be extended to H?lder exponents ${\beta \in\,]\,0,\,1\,]}$ . However the proofs would be extremely involved. On the contrary, the proof followed in this paper treat the Lipschitz case is definitely more elementary than any other proof, even if restricted to the whole space case.     

Mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux

A model is developed for the study of mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux. The model combined natural convection dominated and forced convection dominated film condensation, including effects of pressure gradient and interfacial vapor shear drag has been investigated and solved numerically. The separation angle of the condensate film layer, φ _s is also obtained for various pressure gradient parameters, P ^* and their corresponding dimensionless Grashof?'s parameters, Gr ^*. Besides, the effect of P ^* on the dimensionless mean heat transfer, will remain almost uniform with increasing P ^* until for various corresponding available values of Gr ^*. Meanwhile, the dimensionless mean heat transfer, is increasing significantly with Gr ^* for its corresponding available values of P ^*. For pure natural-convection film condensation, is obtained.     

An isoparametric piecewise representation of the anisotropic strength of polycrystalline solids

A methodology for representing the anisotropic strength of polycrystalline solids using an isoparametric mapping is presented. A four-dimensional hypersphere in five-dimensional (deviatoric) deformation rate space is defined using Lagrange elements for a suite of deformation rate directions with a specified (reference) magnitude. Onto this surface are mapped values of the flow stress required to induce plastic flow for the corresponding values of the reference deformation rate. The flow stress values are evaluated from a combination of experimental data and computational results using polycrystal plasticity. The methodology is demonstrated for three example systems: Ti–6Al–4V plate, AA5182 hot rolled plate, and AL6XN hot rolled plate.     

Investigation of shear damage considering the evolution of anisotropy

The damage that occurs in shear deformations in view of anisotropy evolution is investigated. It is widely believed in the mechanics research community that damage (or porosity) does not evolve (increase) in shear deformations since the hydrostatic stress in shear is zero. This paper proves that the above statement can be false in large deformations of simple shear. The simulation using the proposed anisotropic ductile fracture model (macro-scale) in this study indicates that hydrostatic stress becomes nonzero and (thus) porosity evolves (increases or decreases) in the simple shear deformation of anisotropic (orthotropic) materials. The simple shear simulation using a crystal plasticity based damage model (meso-scale) shows the same physics as manifested in the above macro-scale model that porosity evolves due to the grain-to-grain interaction, i.e., due to the evolution of anisotropy. Through a series of simple shear simulations, this study investigates the effect of the evolution of anisotropy, i.e., the rotation of the orthotropic axes onto the damage (porosity) evolution. The effect of the evolutions of void orientation and void shape onto the damage (porosity) evolution is investigated as well. It is found out that the interaction among porosity, the matrix anisotropy and void orientation/shape plays a crucial role in the ductile damage of porous materials.