A cell based particle method for modeling dynamic interfaces

Abstract We propose several modifications to the grid based particle method (GBPM) [21] for moving interface modeling. There are several nice features of the proposed algorithm. The new method can significantly improve the distribution of sampling particles on the evolving interface. Unlike the original GBPM where footpoints (sampling points) tend to cluster to each other, the sampling points in the new method tend to be better separated on the interface. Moreover, by replacing the grid-based discretization using the cell-based discretization, we naturally decompose the interface into segments so that we can easily approximate surface integrals. As a possible alternative to the local polynomial least square approximation, we also study a geometric basis for local reconstruction in the resampling step. We will show that such modification can simplify the overall implementations. Numerical examples in two- and three-dimensions will show that the algorithm is computationally efficient and accurate.

[1]  Shingyu Leung,et al.  Eulerian Gaussian Beams for High Frequency Wave Propagation , 2007 .

[2]  Hongkai Zhao,et al.  A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion , 2011, J. Comput. Phys..

[3]  Ian M. Mitchell,et al.  A hybrid particle level set method for improved interface capturing , 2002 .

[4]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[5]  Hongkai Zhao,et al.  A grid based particle method for moving interface problems , 2009, J. Comput. Phys..

[6]  D. Juric,et al.  A front-tracking method for the computations of multiphase flow , 2001 .

[7]  David Salac,et al.  A level set projection model of lipid vesicles in general flows , 2011, J. Comput. Phys..

[8]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[9]  Axel Voigt,et al.  Dynamics of multicomponent vesicles in a viscous fluid , 2010, J. Comput. Phys..

[10]  Yves Nievergelt,et al.  Hyperspheres and hyperplanes fitted seamlessly by algebraic constrained total least-squares , 2001 .

[11]  Riccardo Rosso,et al.  Evolution of vesicles subject to adhesion , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Andrew W. Fitzgibbon,et al.  Direct Least Square Fitting of Ellipses , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Jianliang Qian,et al.  Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime , 2009, J. Comput. Phys..

[14]  Frédéric Gibou,et al.  A second order accurate level set method on non-graded adaptive cartesian grids , 2007, J. Comput. Phys..

[15]  Said I. Abdel-Khalik,et al.  Accurate representation of surface tension using the level contour reconstruction method , 2005 .

[16]  Xiaolin Li,et al.  Simple front tracking , 1999 .

[17]  Ted Belytschko,et al.  A vector level set method and new discontinuity approximations for crack growth by EFG , 2002 .

[18]  J. Steinhoff,et al.  A New Eulerian Method for the Computation of Propagating Short Acoustic and Electromagnetic Pulses , 2000 .

[19]  Seungwon Shin,et al.  Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity , 2002 .

[20]  Hongkai Zhao,et al.  Gaussian Beam Summation for Diffraction in Inhomogeneous Media Based on the Grid Based Particle Method , 2010 .

[21]  W. Gander,et al.  Least-squares fitting of circles and ellipses , 1994 .

[22]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[23]  David E. Womble,et al.  A front-tracking method for multiphase free boundary problems , 1989 .

[24]  P. Koumoutsakos,et al.  A Lagrangian particle level set method. , 2005 .

[25]  Sabine Van Huffel,et al.  Consistent least squares fitting of ellipsoids , 2004, Numerische Mathematik.

[26]  Kaleem Siddiqi,et al.  Geometric Shock-Capturing ENO Schemes for Subpixel Interpolation, Computation and Curve Evolution , 1997, CVGIP Graph. Model. Image Process..

[27]  Jun Liu,et al.  A Splitting Algorithm for Image Segmentation on Manifolds Represented by the Grid Based Particle Method , 2013, J. Sci. Comput..

[28]  Carlos Alberto Brebbia.  Boundary element method , 1979 .

[29]  Stanley Osher,et al.  A Fixed Grid Method for Capturing the Motion of Self-Intersecting Wavefronts and Related PDEs , 2000 .

[30]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[31]  George Biros,et al.  A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D , 2009, J. Comput. Phys..

[32]  Hongkai Zhao,et al.  A grid based particle method for evolution of open curves and surfaces , 2009, J. Comput. Phys..

[33]  Kenichi Kanatani,et al.  Statistical Bias of Conic Fitting and Renormalization , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[34]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[35]  Peter Meer,et al.  Unbiased Estimation of Ellipses by Bootstrapping , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[36]  Xiaolin Li,et al.  Robust Computational Algorithms for Dynamic Interface Tracking in Three Dimensions , 1999, SIAM J. Sci. Comput..

[37]  C. Pozrikidis,et al.  Interfacial dynamics for Stokes flow , 2001 .

[38]  Frank C. Park,et al.  Evolution of inelastic plane curves , 1999 .

[39]  Eftychios Sifakis,et al.  ' s personal copy A second order virtual node method for elliptic problems with interfaces and irregular domains , 2010 .

[40]  Thomas Y. Hou,et al.  Boundary integral methods for multicomponent fluids and multiphase materials , 2001 .