On the Use of the Restitution Condition in Flexible Body Dynamics

The difference between the classical treatment offlexible body impact and the treatment of impact in flexiblemultibody dynamics is due to several fundamental reasons. Inthe classical impact theory, simple structures such as beamsand plates are used. Infinite dimensional models can bedeveloped for these simple structural elements to study theimpact dynamics and the wave propagation problem. Flexiblemultibody impact problems, on the other hand, involve bodieswith complex geometry that cannot be modeled using infinitenumber of degrees of freedom. Furthermore, the classicalimpact theory has been mainly concerned with the impactbetween a rigid mass that moves without constraints beforeit impacts a simple flexible structure. This is not amultibody simulation scenario in which the impact occursbetween kinematically constrained bodies that are subjectedto impulsive constraint forces in addition to the impactforces. These constraint forces can influence the motion ofthe two bodies immediately after impact, and as aconsequence, the simple classical theory scenario of impactdoes not apply. It is the objective of this paper to discussthe use of the restitution condition in flexible multibodyimpact problems and demonstrate that the use of thisapproach does not exclude the classical formulation.Nonetheless, the impulse momentum balance approach can serveas an effective and efficient procedure for solving theimpact problem in finite dimensional models that do not obeythe classical wave theory. Energy results of simplestructural elements are presented in order to demonstratethe consistency of using the impulse momentum balanceapproach in solving impact problems in finite dimensionalflexible body applications.     

Study on the packed volume and the void ratio of idealized human red blood cells using a ?nite-discrete element method

Numerical simulations are performed to examine the packing behavior of human red blood cells(RBCs). A combined ?nite-discrete element method(FDEM) is utilized, in which the RBCs are modeled as no-friction and no-adhesion solid bodies. The packed volume and the void ratio of a large number of randomly packed RBCs are clari?ed,and the effects of the RBC shape, the mesh size, the cell number, and the container size are investigated. The results show that the packed human RBCs with normal shape have a void ratio of 28.45%, which is slightly higher than that of the ?at or thick cells used in this study. Such information is bene?cial to the further understanding on the geometric features of human RBCs and the research on RBC simulations.     

Analytical approach in torsional multi-mass discrete-continuous systems with variable inertia

In the paper the systems torsionally deformed consisting of an arbitrary number of elastic elements connected by rigid bodies using the perturbation approach are investigated. The first rigid body represents a motor working with a constant speed, and the last one has a mass moment of inertia depending on the angular displacement. The problem is nonlinear and it is linearized after appropriate transformations. Exemplary numerical results for angular displacements for a two-mass and a three-mass systems are presented.     

Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media I: Theories

This study developed generalized mathematical models to describe the motion of fluids in porous media, and applied these models to harmonic excitation applications. The problem of fluid flow in small channels of a periodic elastic solid matrix was studied at the pore scale, and the homogenization technique was applied to predict the macroscopic behavior of reservoirs. Based on the homogenization study, five separate characteristic macroscopic models were identified according to the relation between a length scale parameter and a property contrast number. These five models can be used to interpret the corresponding responses of a saturated porous medium. The relation to existing theories, such as Darcy's law, the Telegrapher's equation and Biot's theory, was investigated. The numerical results and applications are presented in Part II of the study.     

Flexible Multibody Simulation and Choice of Shape Functions

The approach most widely used for the modelling of flexible bodies in multibody systems has been called the floating frame of reference formulation. In this methodology the flexible body motion is subdivided into a reference motion and deformation. The displacement field due to deformation is approximated by the Ritz method as a product of known shape functions and unknown coordinates depending on time only. The shape functions may be obtained using finite-element-models of flexible bodies in multibody systems, resulting in a detailed system representation and a high number of system equations. The number of system equations of such a nodal approach can be reduced considerably using a modal representation of deformation. This modal approach, however, leads to the fundamental problem of selecting the shape functions.The floating frame of reference formulation is reviewed here using a generic flexible body model, from which the various body models used in multibody simulations may be derived by formulation of specific constraint equations. Special attention is given in this investigation to the following subjects: The separation of flexible body motion into a reference motion and deformation requires the definition of a body reference frame, which in turn affects the choice of shape functions. Some alternatives will be outlined together with their advantages and disadvantages. Assuming the body deformation to be small, the system equations can be linearized. This may require considering geometric stiffening terms. The problem of how to compute these terms has been solved in literature on the instability of structures under critical loads. For finite element models the geometric stiffening terms are obtained from the tangential stiffness matrix. The generality of the flexible body model allows the definition of an object oriented data base to describe the system bodies. Such a data base includes a general interface between multibody- and finite-element-codes. By combining eigenfunctions and static deformation modes to represent body deformation one obtains a set of so-called quasi-comparison functions. When selected properly these functions can be shown to improve the representation of stresses significantly.     

Towards high-fidelity erosion prediction: On time-accurate particle tracking in turbomachinery

Erosion and fouling caused by ingested particles causes performance degradation and safety issues in turbomachinery components. Simulating these processes is a complex multiphysics and multiscale problem which has not reached a satisfactory level of maturity yet. The current state of the art approach is based on RANS solutions, which provide an averaged carrier phase on which the particles are advanced in an a posteriori manner. Upon wall impact, the particle quantities are then fed to the erosion and rebound models. In this work, we present as an alternative to this approach an Eulerian/Lagrangian simulation framework of high-order accuracy in space and time for the time-resolved prediction of particle motion in complex flows. We apply it to the LES of a particle-laden flow over a T106C low pressure turbine linear cascade. We then contrast time-averaged and time-accurate flow fields as carrier phases for the particles and highlight how the associated modeling assumptions influence the solution. Based on the particle Stokes number, we identify characteristic regimes and their interaction with the flow phase. By a detailed comparison of the particle statistics, we highlight the effects of turbulent small scale behavior and define the modeling challenges associated with finding accurate particle closure models for time-averaged simulations. This framework constitutes a first step towards high-fidelity erosion prediction for turbomachinery applications.     

Large eddy simulation of polydispersed inertial particles using two-way coupled PDF-PBE

This paper presents large eddy simulation (LES) of polydispersed turbulent recirculating flows using a two-way coupled probability density function of the population balance equation (PDF-PBE). A stochastic Monte Carlo method is adopted to solve the PDF-PBE on an ensemble of notional Lagrangian particles and the method of Stokes binning, that was recently developed by Salehi et al. (2017) is employed to explicitly treat effects of inertia. The PDF-PBE is applied to the experiment of Boŕee et al. (2001) which studied dispersion of polysized inertial particles in a bluff body configuration. The particle mass loading is 22% where the dispersed elements affect the carrier phase velocity. The simulations are performed using both the standard Smagorinsky and the Wall-Adapting Local Eddy-viscosity (WALE) subgird turbulent models. It is found that the subgrid model has a significant impact on the results, particularly on the carrier velocity. The WALE model shows a better agreement with the measurements. Different boundary conditions are tested for injection of notional particles. It is demonstrated both dispersed and carrier phase velocities are initially sensitive to the particle boundary condition whereas the difference between tested conditions becomes marginal further downstream. The best results are obtained for the particle boundary condition that accounts for effects of inertia at the inlet. Finally, the sensitivity of the PDF-PBE simulations to the number of Stokes bins is studied. It is found that eight Stokes bins are enough to accurately model the polysized particles dispersion in a bluff body configuration. The results indicate that particle dispersion is notably sensitive to the number of Stokes bins but the particle velocity predictions are much less sensitive with small variation in velocity results over the range of Stokes bins that were tested.     

An investigation into electromagnetic force models: differences in global and local effects demonstrated by selected problems

This study investigates the implications of various electromagnetic force models in macroscopic situations. There is an ongoing academic discussion which model is “correct,” i.e., generally applicable. Often, gedankenexperiments with light waves or photons are used in order to motivate certain models. In this work, three problems with bodies at the macroscopic scale are used for computing theoretical model-dependent predictions. Two aspects are considered, total forces between bodies and local deformations. By comparing with experimental data, insight is gained regarding the applicability of the models. First, the total force between two cylindrical magnets is computed. Then a spherical magnetostriction problem is considered to show different deformation predictions. As a third example focusing on local deformations, a droplet of silicone oil in castor oil is considered, placed in a homogeneous electric field. By using experimental data, some conclusions are drawn and further work is motivated.     

有限弹簧法在钢筋混凝土细观断裂分析中的应用

叙述了依据计算几何学的分割理论划分多边形单元的计算前处理系统，及其有限弹簧法在钢筋混凝土构件细观断裂数值模拟中的应用。该模拟系统的主要特色为：一是利用有限弹簧法的单元形状任意性来考虑混凝土断裂模拟时单元形状的影响；二是利用其变位不连续性模拟构件从微细裂缝到断裂的全过程；三是把钢筋和混凝土分别看成具有不同性质的微小构造单元来考虑其界面性质。数值模拟钢筋混凝土构件在拉伸作用下的断裂全过程，考察钢筋的螺纹抵抗作用及其保护层的约束机理。     

Differentially weighted direct simulation Monte Carlo method for particle collision in gas-solid flows

In gas-solid flows,particle-particle interaction(typical,particle collision) is highly significant,despite the small particles fractional volume.Widely distributed polydisperse particle population is a typical characteristic during dynamic evolution of particles(e.g.,agglomeration and fragmentation) in spite of their initial monodisperse particle distribution.The conventional direct simulation Monte Carlo(DSMC)method for particle collision tracks equally weighted simulation particles,which results in high statistical noise for particle fields if there are insufficient simulation particles in less-populated regions.In this study,a new differentially weighted DSMC(DW-DSMC) method for collisions of particles with different number weight is proposed within the framework of the general Eulerian-Lagrangian models for hydrodynamics.Three schemes(mass,momentum and energy conservation) were developed to restore the numbers of simulation particle while keeping total mass,momentum or energy of the whole system unchanged respectively.A limiting case of high-inertia particle flow was numerically simulated to validate the DW-DSMC method in terms of computational precision and efficiency.The momentum conservation scheme which leads to little fluctuation around the mass and energy of the whole system performed best.Improved resolution in particle fields and dynamic behavior could be attained simultaneously using DW-DSMC,compared with the equally weighted DSMC.Meanwhile,computational cost can be largely reduced in contrast with direct numerical simulation.     

Nonstructural geometric discontinuities in finite element/multibody system analysis

Existing multibody system (MBS) algorithms treat articulated system components that are not rigidly connected as separate bodies connected by joints that are governed by nonlinear algebraic equations. As a consequence, these MBS algorithms lead to a highly nonlinear system of coupled differential and algebraic equations. Existing finite element (FE) algorithms, on the other hand, do not lead to a constant mesh inertia matrix in the case of arbitrarily large relative rigid body rotations. In this paper, new FE/MBS meshes that employ linear connectivity conditions and allow for arbitrarily large rigid body displacements between the finite elements are introduced. The large displacement FE absolute nodal coordinate formulation (ANCF) is used to obtain linear element connectivity conditions in the case of large relative rotations between the finite elements of a mesh. It is shown in this paper that a linear formulation of pin (revolute) joints that allow for finite relative rotations between two elements connected by the joint can be systematically obtained using ANCF finite elements. The algebraic joint constraint equations, which can be introduced at a preprocessing stage to efficiently eliminate redundant position coordinates, allow for deformation modes at the pin joint definition point, and therefore, this new joint formulation can be considered as a generalization of the pin joint formulation used in rigid MBS analysis. The new pin joint deformation modes that are the result of C ⁰ continuity conditions, allow for the calculations of the pin joint strains which can be discontinuous as the result of the finite relative rotation between the elements. This type of discontinuity is referred to in this paper as nonstructural discontinuity in order to distinguish it from the case of structural discontinuity in which the elements are rigidly connected. Because ANCF finite elements lead to a constant mass matrix, an identity generalized mass matrix can be obtained for the FE mesh despite the fact that the finite elements of the mesh are not rigidly connected. The relationship between the nonrational ANCF finite elements and the B-spline representation is used to shed light on the potential of using ANCF as the basis for the integration of computer aided design and analysis (I-CAD-A). When cubic interpolation is used in the FE/ANCF representation, C ⁰ continuity is equivalent to a knot multiplicity of three when computational geometry methods such as B-splines are used. C ² ANCF models which ensure the continuity of the curvature and correspond to B-spline knot multiplicity of one can also be obtained. Nonetheless, B-spline and NURBS representations cannot be used to effectively model T-junctions that can be systematically modeled using ANCF finite elements which employ gradient coordinates that can be conveniently used to define element orientations in the reference configuration. Numerical results are presented in order to demonstrate the use of the new formulation in developing new chain models.     

Differentially weighted direct simulation Monte Carlo method for particle collision in gas–solid flows

In gas–solid flows, particle–particle interaction (typical, particle collision) is highly significant, despite the small particles fractional volume. Widely distributed polydisperse particle population is a typical characteristic during dynamic evolution of particles (e.g., agglomeration and fragmentation) in spite of their initial monodisperse particle distribution. The conventional direct simulation Monte Carlo (DSMC) method for particle collision tracks equally weighted simulation particles, which results in high statistical noise for particle fields if there are insufficient simulation particles in less-populated regions. In this study, a new differentially weighted DSMC (DW-DSMC) method for collisions of particles with different number weight is proposed within the framework of the general Eulerian–Lagrangian models for hydrodynamics. Three schemes (mass, momentum and energy conservation) were developed to restore the numbers of simulation particle while keeping total mass, momentum or energy of the whole system unchanged respectively. A limiting case of high-inertia particle flow was numerically simulated to validate the DW-DSMC method in terms of computational precision and efficiency. The momentum conservation scheme which leads to little fluctuation around the mass and energy of the whole system performed best. Improved resolution in particle fields and dynamic behavior could be attained simultaneously using DW-DSMC, compared with the equally weighted DSMC. Meanwhile, computational cost can be largely reduced in contrast with direct numerical simulation.     

断层破碎带中的透镜体及透镜体力学效应初步研究

断层破碎带中的透镜体及透镜体力学效应研究在文献中不多见。但是许多断层尤其是扭性断层中的透镜体非常发育,这些透镜体往往控制断层破碎带的强度大小和变形破坏机理,是工程岩体中必须重视的工程地质力学问题。作者经研究认为:(1)透镜体类型多种;(2)透镜体周边一般具断层泥线或泥膜分布;(3)透镜体内具有节理裂隙系统;(4)透镜体周边可以当作结构面或软弱结构面看待;(5)透镜体本身可以作为一个复合结构体看待。在原位力学试验中也揭示了断层泥透镜体在原位剪切试验和单轴压缩试验中的若干力学行为,其表现是:(1)透镜体周边具有结构面力学效应;(2)断层泥透镜体控制着强度大小、变形破坏机理。     

A reciprocity theorem in linear gradient elasticity and the corresponding Saint-Venant principle

In many practical applications of nanotechnology and in microelectromechanical devices, typical structural components are in the form of beams, plates, shells and membranes. When the scale of such components is very small, the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling. In addition, small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity. It is then useful to be able to apply approximate solutions for the extension, shear and flexure of slender bodies. Such approach requires the existence of some form of the Saint-Venant principle. The present work presents a statement of the Saint-Venant principle in the context of linear strain gradient elasticity. A reciprocity theorem analogous to Betti’s theorem in classic elasticity is provided first, together with necessary restrictions on the constitutive equations and the body forces. It is shown that the order of magnitude of displacements are in accord with the Sternberg’s statement of the Saint-Venant principle. The cases of stretching, shearing and bending of a beam were examined in detail, using two-dimensional finite elements. The numerical examples confirmed the theoretical results.     

刚体单元及其在多体系统动力学中的应用

多体系统动力学分析软件要求人工输入形状复杂物体的质量、质心位置和转动惯量,而实际上这些参量并不容易获得。本文探索了一种以组成物体的刚体单元为基本要素的新方法,并结合实际需要具体构造了刚性四面体和刚性梁单元。以刚体单元为基础并内嵌网格剖分模块的分析软件能够自动获得这些参数,从而具备处理任何复杂系统的能力。仿真结果的对比分...     

On the dependence of the threshold energy of small erodent particles on their geometry in erosion fracture

The structural-temporal approach and the incubation time criterion are used to study the threshold energy necessary to initiate erosion fracture of a material surface. The behavior of the energy threshold values depending on the indentor geometry (ball, cylinder, and body of revolution) is analyzed. The graphs of threshold energy versus impact pulse duration and radius are drawn. The difference in the behavior of energy for small particles in these cases is established.