Inertial Steady 2D Vector Field Topology

Vector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite‐sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first‐order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite‐sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.     

Finite‐Time Mass Separation for Comparative Visualizations of Inertial Particles

The visual analysis of flows with inertial particle trajectories is a challenging problem because time‐dependent particle trajectories additionally depend on mass, which gives rise to an infinite number of possible trajectories passing through every point in space‐time. This paper presents an approach to a comparative visualization of the inertial particles’ separation behavior. For this, we define the Finite‐Time Mass Separation (FTMS), a scalar field that measures at each point in the domain how quickly inertial particles separate that were released from the same location but with slightly different mass. Extracting and visualizing the mass that induces the largest separation provides a simplified view on the critical masses. By using complementary coordinated views, we additionally visualize corresponding inertial particle trajectories in space‐time by integral curves and surfaces. For a quantitative analysis, we plot Euclidean and arc length‐based distances to a reference particle over time, which allows to observe the temporal evolution of separation events. We demonstrate our approach on a number of analytic and one real‐world unsteady 2D field.     

Flow‐Induced Inertial Steady Vector Field Topology

Traditionally, vector field visualization is concerned with 2D and 3D flows. Yet, many concepts can be extended to general dynamical systems, including the higher‐dimensional problem of modeling the motion of finite‐sized objects in fluids. In the steady case, the trajectories of these so‐called inertial particles appear as tangent curves of a 4D or 6D vector field. These higher‐dimensional flows are difficult to map to lower‐dimensional spaces, which makes their visualization a challenging problem. We focus on vector field topology, which allows scientists to study asymptotic particle behavior. As recent work on the 2D case has shown, both extraction and classification of isolated critical points depend on the underlying particle model. In this paper, we aim for a model‐independent classification technique, which we apply to two different particle models in not only 2D, but also 3D cases. We show that the classification can be done by performing an eigenanalysis of the spatial derivatives' velocity subspace of the higher‐dimensional 4D or 6D flow. We construct glyphs that depict not only the types of critical points, but also encode the directional information given by the eigenvectors. We show that the eigenvalues and eigenvectors of the inertial phase space have sufficient symmetries and structure so that they can be depicted in 2D or 3D, instead of 4D or 6D.     

Visualizing the Phase Space of Heterogeneous Inertial Particles in 2D Flows

In many scientific disciplines, the motion of finite‐sized objects in fluid flows plays an important role, such as in brownout engineering, sediment transport, oceanology or meteorology. These finite‐sized objects are called inertial particles and, in contrast to traditional tracer particles, their motion depends on their current position, their own particle velocity, the time and their size. Thus, the visualization of their motion becomes a high‐dimensional problem that entails computational and perceptual challenges. So far, no visualization explored and visualized the particle trajectories under variation of all seeding parameters. In this paper, we propose three coordinated views that visualize the different aspects of the high‐dimensional space in which the particles live. We visualize the evolution of particles over time, showing that particles travel different distances in the same time, depending on their size. The second view provides a clear illustration of the trajectories of different particle sizes and allows the user to easily identify differences due to particle size. Finally, we embed the trajectories in the space‐velocity domain and visualize their distance to an attracting manifold using ribbons. In all views, we support interactive linking and brushing, and provide abstraction through density volumes that are shown by direct volume rendering and isosurface slabs. Using our method, users gain deeper insights into the dynamics of inertial particles in 2D fluids, including size‐dependent separation, preferential clustering and attraction. We demonstrate the effectiveness of our method in multiple steady and unsteady 2D flows.     

Detecting particle swarm optimization

Here, we propose a detecting particle swarm optimization (DPSO). In DPSO, we define several detecting particles that are randomly selected from the population. The detecting particles use the newly proposed velocity formula to search the adjacent domains of a settled position in approximate spiral trajectories. In addition, we define the particles that use the canonical velocity updating formula as common particles. In each iteration, the common particles use the canonical velocity updating formula to update their velocities and positions, and then the detecting particles do search in approximate spiral trajectories created by the new velocity updating formula in order to find better solutions. As a whole, the detecting particles and common particles would do the high‐performance search. DPSO implements the common particles' swarm search behavior and the detecting particles' individual search behavior, thereby trying to improve PSO's performance on swarm diversity, the ability of quick convergence and jumping out the local optimum. The experimental results from several benchmark functions demonstrate good performance of DPSO. Copyright © 2008 John Wiley & Sons, Ltd.     

As‐Killing‐As‐Possible Vector Fields for Planar Deformation

Cartoon animation, image warping, and several other tasks in two‐dimensional computer graphics reduce to the formulation of a reasonable model for planar deformation. A deformation is a map from a given shape to a new one, and its quality is determined by the type of distortion it introduces. In many applications, a desirable map is as isometric as possible. Finding such deformations, however, is a nonlinear problem, and most of the existing solutions approach it by minimizing a nonlinear energy. Such methods are not guaranteed to converge to a global optimum and often suffer from robustness issues. We propose a new approach based on approximate Killing vector fields (AKVFs), first introduced in shape processing. AKVFs generate near‐isometric deformations, which can be motivated as direction fields minimizing an “as‐rigid‐as‐possible” (ARAP) energy to first order. We first solve for an AKVF on the domain given user constraints via a linear optimization problem and then use this AKVF as the initial velocity field of the deformation. In this way, we transfer the inherent nonlinearity of the deformation problem to finding trajectories for each point of the domain having the given initial velocities. We show that a specific class of trajectories — the set of logarithmic spirals — is especially suited for this task both in practice and through its relationship to linear holomorphic vector fields. We demonstrate the effectiveness of our method for planar deformation by comparing it with existing state‐of‐the‐art deformation methods.     

Inertially focused diamagnetic particle separation in ferrofluids

Continuous flow separation of target particles from a mixture is essential to many chemical and biomedical applications. There has recently been an increasing interest in the integration of active and passive particle separation techniques for enhanced sensitivity and flexibility. We demonstrate herein the proof-of-concept of a ferrofluid-based hybrid microfluidic technique that combines passive inertial focusing with active magnetic deflection to separate diamagnetic particles by size. The two operations take place in series in a continuous flow through a straight rectangular microchannel with a nearby permanent magnet. We also develop a three-dimensional numerical model to simulate the transport of diamagnetic particles during their inertial focusing and magnetic separation processes in the entire microchannel. The predicted particle trajectories are found to be approximately consistent with the experimental observations at different ferrofluid flow rates and ferrofluid concentrations.     

Terminal open-loop control for a hypervelocity unmanned flying vehicle in the atmosphere. Part 1

An algorithm for the terminal open-loop control of a hypervelocity flying vehicle with a high lift-drag (L/D) ratio is proposed; this algorithm ensures the implementation of spatial programmed hitting trajectories steering the vehicle to the given spatial domain with required high accuracy and required conditions of approaching this domain. The algorithm is based on the a priori construction of a rational hitting trajectory and the corresponding control vector; the control vector components are the attack and bank angles. At the beginning of the terminal phase, only constraints on the individual components of the vector of initial conditions are imposed: the ranges of altitudes, velocities, and flight path angle. In addition, the requirement for the minimum admissible resolution of the onboard radar that provides informational support at the terminal phase is taken into account in the trajectory construction. Examples of programmed hitting trajectories are given that confirm the fact that the proposed algorithm for the generation of such trajectories and the corresponding attainability sets is computationally efficient.     

Streamline generation code for particle dynamics description in numerical models of turbulence

Streamline Version 4 is a versatile Fortran 77 & C++ program for calculating charged test particle trajectories or field-lines for user-specified fields using the test-particle method. The user has the freedom to specify any type of field (analytical, tabulated in files, time dependent, etc.) and maintains complete control over initial conditions of trajectories/field-lines and boundary conditions of specified fields. The structure of Streamline was redesigned from previous versions in order to know not only particle or field-lines positions and velocities at each step of the simulations, but also the instantaneous field values as seen by particles. This was made to compute the instantaneous value of the particle’s magnetic moment, but other applications are possible too. Accuracy tests of the code are shown for different cases, i.e., particles moving in constant magnetic field, magnetic plus constant electric field and wave field. In addition in the last part of the paper we concentrate our discussion on the study of velocity space diffusion of charged particles in turbulent slab fields, paying attention to the discretization of the fields and the temporal discretization of the dynamical equations. The diffusion of charged particles is a very common topic in plasma physics and astrophysics since it plays an important role in many different phenomena such as stochastic particle acceleration, diffusive shock acceleration, solar energetic particle propagation, and the scattering required for the solar modulation of galactic cosmic rays.     

塑化过程螺槽中三维流场的数值模拟

采用CROSS模型表示聚氯乙烯(PVC)的黏度特征,使用POLYFLOW软件数值模拟了塑料注射成型机螺杆计量段螺槽中熔体在塑化过程的三维等温流场,求解和分析了3条参考直线、yz截面和xy截面上不同时刻螺槽中的压强场、速度场、剪切速率场和黏度场,数值计算的结果表明:在螺棱附近区域物料的剪切速率大,物料剪切稀化作用增强,物料黏度减小。并采用粒子运动轨迹示踪的方法研究了塑化过程中注塑机粒子运动轨迹。得知塑化过程中注塑机粒子运动轨迹比挤出机复杂得多,有三种典型的运动方式:一部分粒子边旋转边向负Z方向运动、另一部分粒子在旋转的同时先向负Z方向运动后向正Z方向运动,还有一部分粒子和边旋转边向正Z方向运动。     

Toward a Lagrangian Vector Field Topology

In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure for unsteadiness which describes the rate of change of the velocities in a fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio‐temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful. In the steady case the proposed criterion reduces to the classical definition of critical points. As a first step we show that finding an optimal Galilean frame of reference can be obtained implicitly by analyzing the acceleration field. In a second step we show that this can be extended by switching to the Lagrangian frame of reference. This way the criterion can detect critical points moving along intricate trajectories. We analyze the behavior of the proposed criterion based on two analytical vector fields for which a correct solution is defined by their inherent symmetries and present results for numerical vector fields.     

Optimal trajectories in brachistochrone problem with Coulomb friction

A two-parameter family of optimal curves in the brachistochrone problem in the case of Coulomb friction is found. The problem is represented in the form of the standard time minimization control problem. The normal component of the support reaction is used as control. It turned out that the formula for the optimal control, which does not include adjoint variables, has a singularity at the zero motion velocity. A system of ordinary differential equations is derived for which the solution of the Cauchy initial value problem makes it possible to obtain optimal trajectories that have a vertical tangent at the initial point. The self-similarity property of such trajectories is proved. It is shown how this property can be used to obtain by scaling all optimal trajectories from the set of optimal trajectories with fixed initial conditions and different terminal slope angles of the tangent.     

Enhanced size-dependent trapping of particles using microvortices

Inertial microfluidics has been attracting considerable interest for size-based separation of particles and cells. The inertial forces can be manipulated by expanding the microchannel geometry, leading to formation of microvortices for selective isolation and trapping of particles or cells from a mixture. In this work, we aim to enhance our understanding of particle trapping in such microvortices by developing a model of selective particle entrapment. Design and operational parameters including flow conditions, size of the trapping region, and target particle concentration are explored to elucidate their influence on trapping behavior. Our results show that the size dependence of trapping is characterized by a threshold Reynolds number, which governs the selective entry of particles into microvortices from the main flow. We show that concentration enhancement on the order of 100,000× and isolation of targets at concentrations as low as 1/mL is possible. Ultimately, the insights gained from our systematic investigation suggest optimization solutions that enhance device performance (efficiency, size selectivity, and yield) and are applicable to selective isolation and trapping of large rare cells as well as other applications.     

Equations of motion and ballistic paths of volcanic ejecta

When the equations of motion are broken down into component form their solution provides the coordinates of positions of moving particles at successive times, making it possible to plot the paths of those particles. In two dimensions the equations contain horizontal and vertical accelerations which, when added separately in each component's direction, provide the net force per unit mass acting horizontally and vertically. Repeated integration of accelerations leads to particle velocities and displacements, hence particle positions and paths. Though conceptually applicable to many kinds of motion in geomorphology, the method is applied here to paths of volcanic ejecta in equatorial latitudes. The horizontal component includes the sum of inertial, resistance and aeolian terms whereas the vertical component is the sum of inertial, gravitational, resistance, lift and total centrifugal terms. The total centrifugal effect on a body in motion on a rotating Earth is itself composed of two components. One component is the conventionally conceived centrifugal effect at right angles to Earth's polar axis. The second component is the Coriolis effect at right angles to the velocity vectors that are tangential to the path of motion and at right angles to the axis of spin of the moving body. When the axis of spin is coincident with the polar axis, as it is for eastward motion of a body on the equator, both components act in the same direction, in this case upwards vertically. Completing the scenario is the inclusion of the equatorial tropopause which impedes further vertical motion but allows horizontal motion to continue, an aspect of considerable importance with respect to transport of dust (volcanic and aeolian) and pollutants.     

Inertial effects on cylindrical particle migration in linear shear flow near a wall

Micro-/nanoparticle-based systems are regarded as one of the possible candidates due to the engineerability and multifunctionality to maximize the accumulation of the nano-/microparticle-based drug delivery system on the target. Recent advances in nanotechnology enable the fabrication of diverse particle shapes from simple spherical particles to more complex shapes. The particle dynamics in blood flow and endocytosis characteristics of non-spherical particles change as the non-sphericity effect increases. We used a numerical approach to investigate the particle dynamics in linear shear flow near a wall. We examined the dynamics of slender cylindrical particles with aspect ratio γ = 5.0 in terms of particle trajectory, velocity, and force variation for different Stokes numbers over time. We identified the rotating inertia of particle near a wall as the source of inertial migration toward the wall. The drift velocity of slender cylindrical particles is comparable to that of discoidal particles. We discuss the possibilities and limitations of using slender cylindrical particles as a drug delivery system.     

面向对象设计在捷联惯导算法仿真中的应用

针对捷联惯导系统（SINS）的算法设计,介绍了SINS的姿态、速度和位置基本微分方程组;以解捷联惯导一阶微分方程组的算法仿真为例,提出了基于面向对象程序设计中类的概念和运算符重载的功能,设计了捷联惯导算法的四阶龙格-库塔数值解法;该设计为工程实现中直接求解含标量、向量和矩阵等混合形式的一阶微分方程组的数值解提供了一定的参考价值;仿真结果表明：即使对恶劣的纯锥运动,该算法精度也很高。     

颗粒物料输送过程运动特性的离散元模拟

为揭示颗粒物料输送过程中的运动规律以及卸料轨迹的影响因素,采用离散单元 法建立胶带输送机运输模型,对颗粒物料的输送和抛射行为进行模拟。通过颗粒物料输送过程 的运动速度和抛射轨迹分析发现：颗粒在胶带输送机上的运动分为加速、抛射和碰撞3 个阶段, 颗粒在碰撞阶段与挡板和其他颗粒发生多次碰撞,使其速度和方向不断改变,是影响颗粒卸料 落点的主要因素;颗粒物料按照入料顺序分为入料初期颗粒、入料中间颗粒和入料末期颗粒, 入料中间颗粒在抛射和碰撞反弹阶段都会受到其他颗粒的干扰,其卸料轨迹和落点存在较大差 异,碰撞过程中能量损失较大,是影响输送效率的重要原因。     

Inertial focusing of microparticles in curvilinear microchannels with different curvature angles

Inertial microfluidics has become one of the emerging topics due to potential applications such as particle separation, particle enrichment, rapid detection and diagnosis of circulating tumor cells. To realize its integration to such applications, underlying physics should be well understood. This study focuses on particle dynamics in curvilinear channels with different curvature angles (280°, 230°, and 180°) and different channel heights (90, 75, and 60 µm) where the advantages of hydrodynamic forces were exploited. We presented the cruciality of the three-dimensional particle position with respect to inertial lift forces and Dean drag force by examining the focusing behavior of 20 µm (large), 15 µm (medium) and 10 µm (small) fluorescent polystyrene microparticles for a wide range of flow rates (400–2700 µL/min) and corresponding channel Reynolds numbers. Migration of the particles in lateral direction and their equilibrium positions were investigated in detail. In addition, in the light of our findings, we described two different regions: transition region, where the inner wall becomes the outer wall and vice versa, and the outlet region. The maximum distance between the tight particle stream of 20 and 15 µm particles was obtained in the 90 high channel with curvature angle of 280° at Reynolds number of 144 in the transition region (intersection of the turns), which was the optimum condition/configuration for focusing.     

Adaptive three-dimensional guidance of flying vehicle at fixed point with specified approach direction

The problem of steering a flying vehicle to a fixed point with a specified orientation of the terminal velocity is studied. The method is based on a modification of the well-known proportional navigation method in which the navigation parameter relating the angular rotation velocities of the line of sight and the velocity vector is chosen based on the current characteristics of the trajectory. The adaptivity of the method is ensured by the periodic correction of this parameter. For three-dimensional (3D) trajectories, a combination of two motions—(a) guidance in the plane formed by the given direction of the terminal velocity and the current state of the vehicle (in this plane, the specified terminal velocity direction is ensured using the modified proportional navigation) and (b) control in the orthogonal plane based on the classical proportional navigation with a fixed navigation parameter to minimize the rotation of the guidance plane. The proposed method does not require onboard trajectory prediction but forms the control using the current navigation data. Examples of various types of 3D trajectories of a gliding vehicle with a high lift-to-drag ratio are discussed.     

Volume preserving viscoelastic fluids with large deformations using position-based velocity corrections

We propose a particle-based hybrid method for simulating volume preserving viscoelastic fluids with large deformations. Our method combines smoothed particle hydrodynamics (SPH) and position-based dynamics (PBD) to approximate the dynamics of viscoelastic fluids. While preserving their volumes using SPH, we exploit an idea of PBD and correct particle velocities for viscoelastic effects not to negatively affect volume preservation of materials. To correct particle velocities and simulate viscoelastic fluids, we use connections between particles which are adaptively generated and deleted based on the positional relations of the particles. Additionally, we weaken the effect of velocity corrections to address plastic deformations of materials. For one-way and two-way fluid-solid coupling, we incorporate solid boundary particles into our algorithm. Several examples demonstrate that our hybrid method can sufficiently preserve fluid volumes and robustly and plausibly generate a variety of viscoelastic behaviors, such as splitting and merging, large deformations, and Barus effect.