Scalar features in time-dependent fluid flow are traditionally visualized using 3D representation, and their topology changes over time are often conveyed with abstract graphs. Using such techniques, however, the structural details of feature separation and the temporal evolution of features undergoing topological changes are difficult to analyze. In this paper, we propose a novel approach for the spatio-temporal visualization of feature separation that segments feature volumes into regions with respect to their contribution to distinct features after separation. To this end, we employ particle-based feature tracking to find volumetric correspondences between features at two different instants of time. We visualize this segmentation by constructing mesh boundaries around each volume segment of a feature at the initial time that correspond to the separated features at the later time. To convey temporal evolution of the partitioning within the investigated time interval, we complement our approach with spatio-temporal separation surfaces. For the application of our approach to multiphase flow, we additionally present a feature-based corrector method to ensure phase-consistent particle trajectories. The utility of our technique is demonstrated by application to two-phase (liquid-gas) and multi-component (liquid-liquid) flows where the scalar field represents the fraction of one of the phases.
For the visualization in multiphase flow, we use a corrector method to ensure that the advected particles remain in the given phase during tracing. Related to this problem are approaches for surface tracking. Stam @cite_30 proposed a method for the computation of interface velocities to properly translate surfaces. Bojsen- @cite_31 developed a method for tracking surfaces undergoing topology changes, without prior information on the underlying physics. In the work by @cite_24 , Lagrangian transport is used for correct displacement interpolation. @cite_6 optimize the transportation in terms of Wasserstein distances, which allows for efficient shape transformation. In our work, we analyze the evolution of volumes, and computing Wasserstein distances in this case would be computationally prohibitive and also would not guarantee physically correct correspondences.
@cite_30 @cite_31 @cite_6 @cite_24
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Abstract Since the seminal work of [Sussman, M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] on coupling the level set method of [Osher S, Sethian J. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 1988;79:12–49] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both of its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton–Jacobi WENO [Jiang G-S, Peng D. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J Sci Comput 2000;21:2126–43], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [Enright D, Fedkiw R, Ferziger J, Mitchell I. A hybrid particle level set method for improved interface capturing. J Comput Phys 2002;183:83–116] and the coupled level set volume of fluid method [Sussman M, Puckett EG. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J Comput Phys 2000;162:301–37], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [Losasso F, Gibou F, Fedkiw R. Simulating water and smoke with an octree data structure, ACM Trans Graph (SIGGRAPH Proc) 2004;23:457–62]. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the ?earth-mover's metric?). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. In our context, the resulting mapping is area preserving and minimizes angle distortion in the optimal mass sense. Indeed, we first begin with an angle-preserving mapping (which may greatly distort area) and then correct it using the mass transport procedure derived via a certain gradient flow. In order to obtain fast convergence to the optimal mapping, we incorporate a multiresolution scheme into our flow. We also use ideas from discrete exterior calculus in our computations. We propose a novel approach for optical flow estimation, targeted at large displacements with significant occlusions. It consists of two steps: i) dense matching by edge-preserving interpolation from a sparse set of matches; ii) variational energy minimization initialized with the dense matches. The sparse-to-dense interpolation relies on an appropriate choice of the distance, namely an edge-aware geodesic distance. This distance is tailored to handle occlusions and motion boundaries - two common and difficult issues for optical flow computation. We also propose an approximation scheme for the geodesic distance to allow fast computation without loss of performance. Subsequent to the dense interpolation step, standard one-level variational energy minimization is carried out on the dense matches to obtain the final flow estimation. The proposed approach, called Edge-Preserving Interpolation of Correspondences (EpicFlow) is fast and robust to large displacements. It significantly outperforms the state of the art on MPI-Sintel and performs on par on Kitti and Middlebury.