On generalized donating regions: Classifying Lagrangian fluxing particles through a fixed curve in the plane

Zhang, Qinghai

doi:10.1016/j.jmaa.2014.11.043

Cited by 8 publications

(12 citation statements)

References 12 publications

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“…This geometric flexibility has been one major motivation to develop semi-Lagrangian numerical schemes for solving conservation laws, cf. [38,41].…”

Section: Illustrative Discussion Of the Main Resultsmentioning

confidence: 99%

Lagrangian Transport Through Surfaces in Volume-Preserving Flows

Karrasch¹

2016

SIAM J. Appl. Math.

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Advective transport of scalar quantities through surfaces is of fundamental importance in many scientific applications. From the Eulerian perspective of the surface it can be quantified by the well-known integral of the flux density. The recent development of highly accurate semi-Lagrangian methods for solving scalar conservation laws and of Lagrangian approaches to coherent structures in turbulent (geophysical) fluid flows necessitate a new approach to transport from the (Lagrangian) material perspective. We present a Lagrangian framework for calculating transport of conserved quantities through a given surface in n-dimensional, fully aperiodic, volume-preserving flows. Our approach does not involve any dynamical assumptions on the surface or its boundary.

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“…This geometric flexibility has been one major motivation to develop semi-Lagrangian numerical schemes for solving conservation laws, cf. [38,41].…”

Section: Illustrative Discussion Of the Main Resultsmentioning

confidence: 99%

Lagrangian Transport Through Surfaces in Volume-Preserving Flows

Karrasch¹

2016

SIAM J. Appl. Math.

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“…6.6]; cf. [12,22,23]. Generally, ∂D does not need to be the topological boundary of the donating region D, but we keep the notation ∂D by abuse of notation, and refer to it as the bounding curve in the sequel.…”

Section: Application To Two-dimensional Flowsmentioning

confidence: 99%

“…In section 2 we formulate the problem of flux computation and prove our main result, theorem 1, which solves the donating region problem posed by Zhang [21]; cf. also Zhang's prior work [22,23]. In section 3 we specify the general theory to two-dimensional flow problems and provide an algorithmic overview.…”

Section: Introductionmentioning

confidence: 99%

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Lagrangian Transport through Surfaces in Compressible Flows

Hofherr¹,

Karrasch²

2018

SIAM J. Appl. Dyn. Syst.

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A material-based, i.e., Lagrangian, methodology for exact integration of flux by volume-preserving flows through a surface has been developed recently in [Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present paper, we first generalize this framework to general compressible flows, thereby solving the donating region problem in full generality. Second, we demonstrate the efficacy of this approach on a slightly idealized version of a classic two-dimensional mixing problem: transport in a cross-channel micromixer, as considered recently in 1 We use the term surface as the short form of hypersurface, i.e., a codimension-one submanifold of space or space-time. In three dimensions, this refers to classic surfaces, in two dimensions to curves. arXiv:1707.00518v2 [physics.flu-dyn]

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“…Numerical results in [35] suggest that the PAM method is superior to VOF methods in terms of accuracy. In a later paper [29], the PAM method is connected to VOF methods via the donating region analysis [30,27].…”

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confidence: 99%

Fourth-Order Interface Tracking in Two Dimensions via an Improved Polygonal Area Mapping Method

Zhang¹,

Fogelson²

2014

SIAM J. Sci. Comput.

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We present an improved PAM (iPAM) method as the first fourth-order interface tracking method whose convergence rates are independent of C 1 (derivative) discontinuities of the interface. As an improved version of the polygonal area mapping (PAM) method [Q. Zhang and P. L.-F. Liu, J. Comput. Phys., 227 (2008), pp. 4063-4088], the accuracy of the iPAM method is achieved via (i) augmenting the abstract data structure of PAM to faithfully represent multiple components of material regions within a single cell, (ii) removing restrictive assumptions of PAM, (iii) adjusting the volume of represented cell material regions via polygon ear removal, and (iv) maintaining a relation (h L = r h h α ) between the Eulerian grid size h and the Lagrangian length scale that measures the distance between adjacent interface markers. Unlike volume-of-fluid methods and level-set methods, merging and separation of the tracked material is not automatically handled by the iPAM method. Instead, flexible subgrid resolutions are provided so that algorithmic behaviors of interface merging and separation can be determined from the specific physics of the problem under study. The iPAM method is a product of interdisciplinary research of different fields such as ordinary differential equations, computational geometry, and general topology. Its superior accuracy, efficiency, and versatility over previous interface tracking methods are demonstrated by a set of benchmark tests. In particular, for the vortex shear test on the 128 × 128 grid, the fourth-order iPAM method could be hundreds of times more efficient than state-of-the-art volume-of-fluid methods.

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On generalized donating regions: Classifying Lagrangian fluxing particles through a fixed curve in the plane

Cited by 8 publications

References 12 publications

Lagrangian Transport Through Surfaces in Volume-Preserving Flows

Lagrangian Transport Through Surfaces in Volume-Preserving Flows

Lagrangian Transport through Surfaces in Compressible Flows

Fourth-Order Interface Tracking in Two Dimensions via an Improved Polygonal Area Mapping Method

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