Interactive separating streak surfaces

Streak surfaces are among the most important features to support 3D unsteady flow exploration, but they are also among the computationally most demanding. Furthermore, to enable a feature driven analysis of the flow, one is mainly interested in streak surfaces that show separation profiles and thus detect unstable manifolds in the flow. The computation of such separation surfaces requires to place seeding structures at the separation locations and to let the structures move correspondingly to these locations in the unsteady flow. Since only little knowledge exists about the time evolution of separating streak surfaces, at this time, an automated exploration of 3D unsteady flows using such surfaces is not feasible. Therefore, in this paper we present an interactive approach for the visual analysis of separating streak surfaces. Our method draws upon recent work on the extraction of Lagrangian coherent structures (LCS) and the real-time visualization of streak surfaces on the GPU. We propose an interactive technique for computing ridges in the finite time Lyapunov exponent (FTLE) field at each time step, and we use these ridges as seeding structures to track streak surfaces in the time-varying flow. By showing separation surfaces in combination with particle trajectories, and by letting the user interactively change seeding parameters such as particle density and position, visually guided exploration of separation profiles in 3D is provided. To the best of our knowledge, this is the first time that the reconstruction and display of semantic separable surfaces in 3D unsteady flows can be performed interactively, giving rise to new possibilities for gaining insight into complex flow phenomena.     

MCFTLE: Monte Carlo Rendering of Finite‐Time Lyapunov Exponent Fields

Traditionally, Lagrangian fields such as finite‐time Lyapunov exponents (FTLE) are precomputed on a discrete grid and are ray casted afterwards. This, however, introduces both grid discretization errors and sampling errors during ray marching. In this work, we apply a progressive, view‐dependent Monte Carlo‐based approach for the visualization of such Lagrangian fields in time‐dependent flows. Our approach avoids grid discretization and ray marching errors completely, is consistent, and has a low memory consumption. The system provides noisy previews that converge over time to an accurate high‐quality visualization. Compared to traditional approaches, the proposed system avoids explicitly predefined fieldline seeding structures, and uses a Monte Carlo sampling strategy named Woodcock tracking to distribute samples along the view ray. An acceleration of this sampling strategy requires local upper bounds for the FTLE values, which we progressively acquire during the rendering. Our approach is tailored for high‐quality visualizations of complex FTLE fields and is guaranteed to faithfully represent detailed ridge surface structures as indicators for Lagrangian coherent structures (LCS). We demonstrate the effectiveness of our approach by using a set of analytic test cases and real‐world numerical simulations.     

Lagrangian Visualization of Flow‐Embedded Surface Structures

The notions of Finite‐Time Lyapunov Exponent (FTLE) and Lagrangian Coherent Structures provide a strong framework for the analysis and visualization of complex technical flows. Their definition is simple and intuitive, and they are built on a deep theoretical foundation. We apply these concepts to enable the analysis of flows in the immediate vicinity of the boundaries of flow‐embedded objects by limiting the Lagrangian analysis to surfaces closely neighboring these boundaries. To this purpose, we present an approach to approximate FTLE fields over such surfaces. Furthermore, we achieve an effective depiction of boundary‐related flow structures such as separation and attachment over object boundaries and specific insight into the surrounding flow using several specifically chosen visualization techniques. We document the applicability of our methods by presenting a number of application examples.     

An Improved Eulerian Approach for the Finite Time Lyapunov Exponent

We propose a new Eulerian numerical approach to compute the Jacobian of flow maps in continuous dynamical systems and subsequently the so-called finite time Lyapunov exponent (FTLE) for Lagrangian coherent structure extraction. The original approach computes the flow map and then numerically determines the Jacobian of the map using finite differences. The new algorithm improves the original Eulerian formulation so that we first obtain partial differential equations for each component of the Jacobian and then solve these equations to obtain the required Jacobian. For periodic dynamical systems, based on the time doubling technique developed for computing the longtime flow map, we also propose a new efficient iterative method to compute the Jacobian of the longtime flow map. Numerical examples will demonstrate that our new proposed approach is more accurate than the original one in computing the Jacobian and thus the FTLE field, especially near the FTLE ridges.     

基于视频粒子流和FTLE场的人群运动分割算法

针对复杂视频监控场景中不同运动行为的人群分割,提出了将视频粒子流和有限时间李雅普诺夫指数(FTLE)场相结合的人群运动分割算法。首先利用视频粒子流来表示长周期的粒子运动估计,通过最小化包含粒子外观匹配一致性和粒子间形变的能量函数,来优化每个粒子的轨迹;接着求解粒子流图的空间梯度,并构造FTLE场;最后利用FTLE场中的拉格朗日相干结构把流图分割成运动特性不同的区域。实验结构表明,算法能从拥挤复杂的视频监控场景中有效地分割出不同运动特性的群体,且具有较好的鲁棒性。     

Interactive computation and rendering of Finite-time Lyapunov Exponent fields

In this paper, we present a novel technique that allows for the coupled computation and visualization of salient flow structures at interactive frame rates. Our approach is built upon a hierarchical representation of the Finite-time Lyapunov Exponent (FTLE) field, which is adaptively sampled and rendered to meet the need of the current visual setting. The performance of our method allows the user to explore large and complex data sets across scales and to inspect their features at arbitrary resolution. The paper discusses an efficient implementation of this strategy on graphics hardware and provides results for an analytical flow and several CFD simulation data sets.     

Truncated Integration for Simultaneous Simulation of Sintering Using a Separated Representation

Recent developments of multidimensional solvers using separated representation make it possible to account for the multidimensionality of mechanical models in materials science when doing numerical simulations. This paper aims to extend the separated representation to inseparable equations using an efficient integration scheme. It focuses on the dependence of constitutive equations on material coefficients. Although these coefficients can be optimized using few experimental results, they are not very well known because of the natural variability of material properties. Therefore, the mechanical state can be viewed as a function depending not only on time and space variables but also on material coefficients. This is illustrated in this paper by a sensitivity analysis of the response of a sintering model with respect to variations of material coefficients. The considered variations are defined around an optimized value of coefficients adjusted by experimental results. The proposed method is an incremental method using an extension of the integration scheme developed for the Hyper Reduction method. During the incremental solution, before the adaptation of the representation, an assumed separation representation is used as a reduced-order model. We claim that a truncated integration scheme enables to forecast the reduced-state variables related to the assumed separated representation. The fact that the integrals involved in the formulation can not be written as a sum of products of one-dimensional integrals, this approach reduces the extent of the integration domain.     

Structural schema integration with full and partial correspondence using the dual model

The integration of views and schemas is an important part of database design and evolution and permits the sharing of data across complex applications. The view and schema integration methodologies used to date are driven purely by semantic considerations, and allow integration of objects only if that is valid from both semantic and structural view points. We discuss a new integration method called structural integration that has the advantage of being able to integrate objects that have structural similarities, even if they differ semantically. This is possible by using the object-oriented Dual Model which allows separate representation of structure and semantics. Structural integration has several advantages, including the identification of shared common structures that is important for sharing of data and methods.     

Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.     

On Triangulating Planar Graphs under the Four-Connectivity Constraint

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included. Received August 1, 1995; revised July 8, 1996, and August 23, 1996.     

Time Line Cell Tracking for the Approximation of Lagrangian Coherent Structures with Subgrid Accuracy

Lagrangian coherent structures (LCSs) have become a widespread and powerful method to describe dynamic motion patterns in time‐dependent flow fields. The standard way to extract LCS is to compute height ridges in the finite‐time Lyapunov exponent field. In this work, we present an alternative method to approximate Lagrangian features for 2D unsteady flow fields that achieve subgrid accuracy without additional particle sampling. We obtain this by a geometric reconstruction of the flow map using additional material constraints for the available samples. In comparison to the standard method, this allows for a more accurate global approximation of LCS on sparse grids and for long integration intervals. The proposed algorithm works directly on a set of given particle trajectories and without additional flow map derivatives. We demonstrate its application for a set of computational fluid dynamic examples, as well as trajectories acquired by Lagrangian methods, and discuss its benefits and limitations.     

Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme

We focus on a simple implicit time integration scheme for the transient response solution of structures when large deformations and long time durations are considered. Our aim is to have a practical method of implicit time integration for analyses in which the widely used Newmark time integration procedure is not conserving energy and momentum, and is unstable. The method of time integration discussed in this paper is performing well and is a good candidate for practical analyses.     

Analysis of time-dependent flow-sensitive PC-MRI data

Many flow visualization techniques, especially integration-based methods, are problematic when the measured data exhibit noise and discretization issues. Particularly, this is the case for flow-sensitive phase-contrast magnetic resonance imaging (PC-MRI) data sets which not only record anatomic information, but also time-varying flow information. We propose a novel approach for the visualization of such data sets using integration-based methods. Our ideas are based upon finite-time Lyapunov exponents (FTLE) and enable identification of vessel boundaries in the data as high regions of separation. This allows us to correctly restrict integration-based visualization to blood vessels. We validate our technique by comparing our approach to existing anatomy-based methods as well as addressing the benefits and limitations of using FTLE to restrict flow. We also discuss the importance of parameters, i.e., advection length and data resolution, in establishing a well-defined vessel boundary. We extract appropriate flow lines and surfaces that enable the visualization of blood flow within the vessels. We further enhance the visualization by analyzing flow behavior in the seeded region and generating simplified depictions.     

A straightforward method of measuring MPRT using LC test cells

Abstract— A simple method based on integration over time to measure motion‐picture response time (MPRT) has been developed. Liquid‐crystal response time (LCRT) alone cannot express the motion blur perceived by the observer; one must also take into account the transition between intermediate gray levels and the sample‐and‐hold effect. The method shows similar results to other previously reported methods based on temporal integration, while being simpler and more straightforward. Indeed, just monopixel test cells are required for measuring MPRT. This method can be used for comparison between different materials, alignments surfaces, or other manufacturing details with no need of fabricating the whole LCD structure. Nevertheless, the method could also be used for characterization of commercial displays with major changes. A comparison of MPRT values for three different liquid‐crystal materials is presented in this work. The behavior of the MPRT parameter in the case of an ideal liquid‐crystal material with an LCRT equal to zero has also been studied. The results obtained for this material have been used as a reference to establish comparisons with real materials.     

A fast level set framework for large three-dimensional topography simulations

We present fast methods to describe the surface evolution of large three-dimensional structures. Based on the sparse field level set method and the hierarchical run-length encoding level set data structure optimal figures for the computation time and for the memory consumption are achieved. Furthermore, we introduce a new multi-level-set technique, which is able to incorporate multiple material regions, and which can also handle material specific surface speeds accurately. We also describe an optimal algorithm for the visibility check for unidirectional etching. The presented techniques are demonstrated on various examples.     

Source Inversion by Forward Integration in Inertial Flows

Inertial particles are finite‐sized objects traveling with a certain velocity that differs from the underlying carrying flow, i.e., they are mass‐dependent and subject to inertia. Their backward integration is in practice infeasible, since a slight change in the initial velocity causes extreme changes in the recovered position. Thus, if an inertial particle is observed, it is difficult to recover where it came from. This is known as the source inversion problem, which has many practical applications in recovering the source of airborne or waterborne pollutions. Inertial trajectories live in a higher dimensional spatio‐velocity space. In this paper, we show that this space is only sparsely populated. Assuming that inertial particles are released with a given initial velocity (e.g., from rest), particles may reach a certain location only with a limited set of possible velocities. In fact, with increasing integration duration and dependent on the particle response time, inertial particles converge to a terminal velocity. We show that the set of initial positions that lead to the same location form a curve. We extract these curves by devising a derived vector field in which they appear as tangent curves. Most importantly, the derived vector field only involves forward integrated flow map gradients, which are much more stable to compute than backward trajectories. After extraction, we interactively visualize the curves in the domain and display the reached velocities using glyphs. In addition, we encode the rate of change of the terminal velocity along the curves, which gives a notion for the convergence to the terminal velocity. With this, we present the first solution to the source inversion problem that considers actual inertial trajectories. We apply the method to steady and unsteady flows in both 2D and 3D domains.     

Structural integration: Concepts and case study

When integrating the views of a large telecomunications application database at Bellcore, it was found that some pairs of view objects had significant structural similarities but differed semantically.⁺ This observation motivated the design of the structural integration methodology described in this article. Currently existing view and schema integration methodologies are based on semantic considerations. They allow integration only if two objects agree in their semantic and structural aspects. Structural integration permits the integration of objects even if they differ semantically. This article introduces structural integration for the case of full structural correspondence. We further develop an important special case, namely structural integration for classes with attribute partial correspondence. We use a subschema of the telecommunications application to demonstrate the applicability of structural integration to situations involving the complexities of real-world databases and applications. Algorithms for checking full structural correspondence of classes and databases are presented. Structural integration has several advantages, including the identification of shared common structures that are important for sharing of data and methods.This research has been supported partially by grants from the Center for Manufacturing Systems at New Jersey Institute of Technology and from Bellcore     

Maintaining large time steps in explicit finite element simulations using shape matching

We present a novel hybrid method to allow large time steps in explicit integrations for the simulation of deformable objects. In explicit integration schemes, the time step is typically limited by the size and the shape of the discretization elements as well as by the material parameters. We propose a two-step strategy to enable large time steps for meshes with elements potentially destabilizing the integration. First, the necessary time step for a stable computation is identified per element using modal analysis. This allows determining which elements have to be handled specially given a desired simulation time step. The identified critical elements are treated by a geometric deformation model, while the remaining ones are simulated with a standard deformation model (in our case, a corotational linear Finite Element Method). In order to achieve a valid deformation behavior, we propose a strategy to determine appropriate parameters for the geometric model. Our hybrid method allows taking much larger time steps than using an explicit Finite Element Method alone. The total computational costs per second are significantly lowered. The proposed scheme is especially useful for simulations requiring interactive mesh updates, such as for instance cutting in surgical simulations.     

Analysis of shell structures under transient loading using adaptivity in time and space

The adaptive analysis of structures under transient loading leads to the question, which time integration scheme, finite elements or finite differences, is most favorably combined with an adaptive spatial FE discretization. In order to judge this, the properties of different discontinuous Galerkin (DG) and the standard Newmark method are investigated first, also concerning efficiency. In particular, the damping and dispersion effects are discussed in detail for various types of problems. It must be noted that the type of problem has to be carefully checked in order to apply the most appropriate and efficient time integration scheme. It is shown that e.g. for the wave propagation problems the DG method with linear approximations (DG P1-P1) has to be favored when adaptivity in space is applied. Finally, an adaptive time step modification scheme is presented and applied to various problems.     

Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials

In this paper, two different methods for modelling of functionally graded material (FGM) beam with continuous spatially varying material properties will be presented and compared, namely the multilayering method and the direct integration method. Both the methods are related to homogenization of spatially varying material properties of real FGM beam and to calculation of the secondary variables of the FGM beams. The multilayering method is based on the laminate theory, which is very often used by modelling of the multilayer composite beams. The direct integration method transform spatial continuous varying material properties to the effective ones by direct integration of derived homogenization rules. In next part of the paper, new multiphysical beam finite element will be presented, which in conjunction with the proposed homogenization methods can be used for very effective analysis of the FGM beam structures. The numerical experiment will be presented concerning the multiphysical (electro–thermal–structural) analysis of the chosen FGM beams with spatial continuous variation of material properties.