Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

Gross, B. J.; Trask, N.; Kuberry, P.; Atzberger, P. J.

doi:10.1016/j.jcp.2020.109340

Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

Journal Article · 19 February 2020 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2020.109340· OSTI ID:1619207

Gross, B. J. ^[1]; Trask, N. ^[2]; ^[2]; ^[1]

Univ. of California, Santa Barbara, CA (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of California, Santa Barbara, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC04-94AL85000; SC0019246

OSTI ID:: 1619207

Alternate ID(s):: OSTI ID: 1602294; OSTI ID: 1633912; OSTI ID: 2331459

Report Number(s):: SAND-2019-5752J; 675748; TRN: US2200936

Journal Information:: Journal of Computational Physics, Vol. 409, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (67)

A Simple Mesh Generator in MATLAB Persson, Per-Olof; Strang, Gilbert SIAM Review, Vol. 46, Issue 2 https://doi.org/10.1137/S0036144503429121	journal	January 2004
Lung surfactants and different contributions to thin film stability Hermans, Eline; Saad Bhamla, M.; Kao, Peter Soft Matter, Vol. 11, Issue 41 https://doi.org/10.1039/C5SM01603G	journal	January 2015
Level Set Methods and Dynamic Implicit Surfaces Osher, Stanley; Fedkiw, Ronald Applied Mathematical Sciences https://doi.org/10.1007/b98879	book	January 2003
An RBF-FD closest point method for solving PDEs on surfaces Petras, A.; Ling, L.; Ruuth, S. J. Journal of Computational Physics, Vol. 370 https://doi.org/10.1016/j.jcp.2018.05.022	journal	October 2018
Solving Partial Differential Equations on Point Clouds Liang, Jian; Zhao, Hongkai SIAM Journal on Scientific Computing, Vol. 35, Issue 3 https://doi.org/10.1137/120869730	journal	January 2013
Intracellular Fluid Mechanics: Coupling Cytoplasmic Flow with Active Cytoskeletal Gel Mogilner, Alex; Manhart, Angelika Annual Review of Fluid Mechanics, Vol. 50, Issue 1 https://doi.org/10.1146/annurev-fluid-010816-060238	journal	January 2018
Convergence and stabilization of stress-point integration in mesh-free and particle methods Fries, Thomas-Peter; Belytschko, Ted International Journal for Numerical Methods in Engineering, Vol. 74, Issue 7 https://doi.org/10.1002/nme.2198	journal	January 2008
The effect of curvature and topology on membrane hydrodynamics Henle, M. L.; McGorty, R.; Schofield, A. B. EPL (Europhysics Letters), Vol. 84, Issue 4 https://doi.org/10.1209/0295-5075/84/48001	journal	November 2008
On generalized moving least squares and diffuse derivatives Mirzaei, D.; Schaback, R.; Dehghan, M. IMA Journal of Numerical Analysis, Vol. 32, Issue 3 https://doi.org/10.1093/imanum/drr030	journal	September 2011
Transport schemes on a sphere using radial basis functions Flyer, Natasha; Wright, Grady B. Journal of Computational Physics, Vol. 226, Issue 1 https://doi.org/10.1016/j.jcp.2007.05.009	journal	September 2007
Concentration fluctuations in binary fluid membranes Seki, K.; Komura, S.; Imai, M. Journal of Physics: Condensed Matter, Vol. 19, Issue 7 https://doi.org/10.1088/0953-8984/19/7/072101	journal	February 2007
Modeling Fractures and Barriers as Interfaces for Flow in Porous Media Martin, Vincent; Jaffré, Jérôme; Roberts, Jean E. SIAM Journal on Scientific Computing, Vol. 26, Issue 5 https://doi.org/10.1137/S1064827503429363	journal	January 2005
A scalable consistent second-order SPH solver for unsteady low Reynolds number flows Trask, Nathaniel; Maxey, Martin; Kim, Kyungjoo Computer Methods in Applied Mechanics and Engineering, Vol. 289 https://doi.org/10.1016/j.cma.2014.12.027	journal	June 2015
Relaxation dynamics of fluid membranes Arroyo, Marino; DeSimone, Antonio Physical Review E, Vol. 79, Issue 3 https://doi.org/10.1103/PhysRevE.79.031915	journal	March 2009
Transition Behaviors of Configurations of Colloidal Particles at a Curved Oil-Water Interface Lee, Mina; Xia, Ming; Park, Bum Materials, Vol. 9, Issue 3 https://doi.org/10.3390/ma9030138	journal	February 2016
Stabilization of RBF-generated finite difference methods for convective PDEs Fornberg, Bengt; Lehto, Erik Journal of Computational Physics, Vol. 230, Issue 6 https://doi.org/10.1016/j.jcp.2010.12.014	journal	March 2011
Comparison of discrete Hodge star operators for surfaces Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi Computer-Aided Design, Vol. 78 https://doi.org/10.1016/j.cad.2016.05.002	journal	September 2016
A High-Order Staggered Meshless Method for Elliptic Problems Trask, Nathaniel; Perego, Mauro; Bochev, Pavel SIAM Journal on Scientific Computing, Vol. 39, Issue 2 https://doi.org/10.1137/16M1055992	journal	January 2017
Theoretical Limitations of Discrete Exterior Calculus in the Context of Computational Electromagnetics Kotiuga, P. R. IEEE Transactions on Magnetics, Vol. 44, Issue 6 https://doi.org/10.1109/TMAG.2007.915998	journal	June 2008
Pressure-dependent surface viscosity and its surprising consequences in interfacial lubrication flows Manikantan, Harishankar; Squires, Todd M. Physical Review Fluids, Vol. 2, Issue 2 https://doi.org/10.1103/PhysRevFluids.2.023301	journal	February 2017
3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data Frank, Tobias; Tertois, Anne-Laure; Mallet, Jean-Laurent Computers & Geosciences, Vol. 33, Issue 7 https://doi.org/10.1016/j.cageo.2006.11.014	journal	July 2007
Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions Shankar, Varun; Wright, Grady B. Journal of Computational Physics, Vol. 366 https://doi.org/10.1016/j.jcp.2018.04.007	journal	August 2018
Complex Fluid-Fluid Interfaces: Rheology and Structure Fuller, Gerald G.; Vermant, Jan Annual Review of Chemical and Biomolecular Engineering, Vol. 3, Issue 1 https://doi.org/10.1146/annurev-chembioeng-061010-114202	journal	July 2012
Configurations of fluid membranes and vesicles Seifert, Udo Advances in Physics, Vol. 46, Issue 1 https://doi.org/10.1080/00018739700101488	journal	February 1997
An overview of the Trilinos project Heroux, Michael A.; Phipps, Eric T.; Salinger, Andrew G. ACM Transactions on Mathematical Software, Vol. 31, Issue 3 https://doi.org/10.1145/1089014.1089021	journal	September 2005
Compatible meshfree discretization of surface PDEs Trask, Nathaniel; Kuberry, Paul Computational Particle Mechanics, Vol. 7, Issue 2 https://doi.org/10.1007/s40571-019-00251-2	journal	June 2019
The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces Piret, Cécile Journal of Computational Physics, Vol. 231, Issue 14 https://doi.org/10.1016/j.jcp.2012.03.007	journal	May 2012
Multiscale Modeling of Membrane Rearrangement, Drainage, and Rupture in Evolving Foams Saye, R. I.; Sethian, J. A. Science, Vol. 340, Issue 6133 https://doi.org/10.1126/science.1230623	journal	May 2013
SymPy: symbolic computing in Python Meurer, Aaron; Smith, Christopher P.; Paprocki, Mateusz PeerJ Computer Science, Vol. 3 https://doi.org/10.7717/peerj-cs.103	journal	January 2017
Brownian motion in thin sheets of viscous fluid Saffman, P. G. Journal of Fluid Mechanics, Vol. 73, Issue 4 https://doi.org/10.1017/S0022112076001511	journal	February 1976
Surfaces generated by moving least squares methods Lancaster, P.; Salkauskas, K. Mathematics of Computation, Vol. 37, Issue 155 https://doi.org/10.1090/S0025-5718-1981-0616367-1	journal	September 1981
Fourth order partial differential equations on general geometries Greer, John B.; Bertozzi, Andrea L.; Sapiro, Guillermo Journal of Computational Physics, Vol. 216, Issue 1 https://doi.org/10.1016/j.jcp.2005.11.031	journal	July 2006
A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion Leung, Shingyu; Lowengrub, John; Zhao, Hongkai Journal of Computational Physics, Vol. 230, Issue 7 https://doi.org/10.1016/j.jcp.2010.12.029	journal	April 2011
Models of dynamic extraction of lipid tethers from cell membranes Nowak, Sarah A.; Chou, Tom Physical Biology, Vol. 7, Issue 2 https://doi.org/10.1088/1478-3975/7/2/026002	journal	May 2010
Generalizing the finite element method: Diffuse approximation and diffuse elements Nayroles, B.; Touzot, G.; Villon, P. Computational Mechanics, Vol. 10, Issue 5 https://doi.org/10.1007/BF00364252	journal	January 1992
A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction–Diffusion Equations on Surfaces Shankar, Varun; Wright, Grady B.; Kirby, Robert M. Journal of Scientific Computing, Vol. 63, Issue 3 https://doi.org/10.1007/s10915-014-9914-1	journal	September 2014
Smoothed particle hydrodynamics: theory and application to non-spherical stars Gingold, R. A.; Monaghan, J. J. Monthly Notices of the Royal Astronomical Society, Vol. 181, Issue 3 https://doi.org/10.1093/mnras/181.3.375	journal	December 1977
Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi Journal of Computational Physics, Vol. 312 https://doi.org/10.1016/j.jcp.2016.02.028	journal	May 2016
A parametric finite element method for fourth order geometric evolution equations Barrett, John W.; Garcke, Harald; Nürnberg, Robert Journal of Computational Physics, Vol. 222, Issue 1 https://doi.org/10.1016/j.jcp.2006.07.026	journal	March 2007
RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces Shankar, Varun; Narayan, Akil; Kirby, Robert M. Journal of Computational Physics, Vol. 373 https://doi.org/10.1016/j.jcp.2018.07.015	journal	November 2018
Finite element methods for surface PDEs Dziuk, Gerhard; Elliott, Charles M. Acta Numerica, Vol. 22 https://doi.org/10.1017/S0962492913000056	journal	April 2013
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces Fuselier, Edward J.; Wright, Grady B. Journal of Scientific Computing, Vol. 56, Issue 3 https://doi.org/10.1007/s10915-013-9688-x	journal	February 2013
A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere Flyer, Natasha; Lehto, Erik; Blaise, Sébastien Journal of Computational Physics, Vol. 231, Issue 11 https://doi.org/10.1016/j.jcp.2012.01.028	journal	June 2012
Capillarity-induced ordering of spherical colloids on an interface with anisotropic curvature Ershov, Dmitry; Sprakel, Joris; Appel, Jeroen Proceedings of the National Academy of Sciences, Vol. 110, Issue 23 https://doi.org/10.1073/pnas.1222196110	journal	May 2013
Hydrodynamics experiments with soap films and soap bubbles: A short review of recent experiments Kellay, H. Physics of Fluids, Vol. 29, Issue 11 https://doi.org/10.1063/1.4986003	journal	November 2017
A Stable Algorithm for Flat Radial Basis Functions on a Sphere Fornberg, Bengt; Piret, Cécile SIAM Journal on Scientific Computing, Vol. 30, Issue 1 https://doi.org/10.1137/060671991	journal	January 2008
Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes Sigurdsson, Jon Karl; Atzberger, Paul J. Soft Matter, Vol. 12, Issue 32 https://doi.org/10.1039/C6SM00194G	journal	January 2016
Mobility in geometrically confined membranes Domanov, Y. A.; Aimon, S.; Toombes, G. E. S. Proceedings of the National Academy of Sciences, Vol. 108, Issue 31 https://doi.org/10.1073/pnas.1102646108	journal	July 2011
L EVEL S ET M ETHODS FOR F LUID I NTERFACES Sethian, J. A.; Smereka, Peter Annual Review of Fluid Mechanics, Vol. 35, Issue 1 https://doi.org/10.1146/annurev.fluid.35.101101.161105	journal	January 2003
Surfactant Effects on Bubble Motion and Bubbly Flows Takagi, Shu; Matsumoto, Yoichiro Annual Review of Fluid Mechanics, Vol. 43, Issue 1 https://doi.org/10.1146/annurev-fluid-122109-160756	journal	January 2011
Finite element exterior calculus, homological techniques, and applications Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar Acta Numerica, Vol. 15 https://doi.org/10.1017/S0962492906210018	journal	May 2006
A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces Lehto, Erik; Shankar, Varun; Wright, Grady B. SIAM Journal on Scientific Computing, Vol. 39, Issue 5 https://doi.org/10.1137/16M1095457	journal	January 2017
Thin film dynamics on a prolate spheroid with application to the cornea Braun, R. J.; Usha, R.; McFadden, G. B. Journal of Engineering Mathematics, Vol. 73, Issue 1 https://doi.org/10.1007/s10665-011-9482-4	journal	June 2011
Surface reconstruction from unorganized points Hoppe, Hugues; DeRose, Tony; Duchamp, Tom ACM SIGGRAPH Computer Graphics, Vol. 26, Issue 2 https://doi.org/10.1145/142920.134011	journal	July 1992
Fluid-membrane tethers: Minimal surfaces and elastic boundary layers Powers, Thomas R.; Huber, Greg; Goldstein, Raymond E. Physical Review E, Vol. 65, Issue 4 https://doi.org/10.1103/PhysRevE.65.041901	journal	March 2002
Error bounds for GMLS derivatives approximations of Sobolev functions Mirzaei, Davoud Journal of Computational and Applied Mathematics, Vol. 294 https://doi.org/10.1016/j.cam.2015.08.003	journal	March 2016
Curvature-driven capillary migration and assembly of rod-like particles Cavallaro, M.; Botto, L.; Lewandowski, E. P. Proceedings of the National Academy of Sciences, Vol. 108, Issue 52 https://doi.org/10.1073/pnas.1116344108	journal	December 2011
Numerical convergence of discrete exterior calculus on arbitrary surface meshes Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 19, Issue 3 https://doi.org/10.1080/15502287.2018.1446196	journal	April 2018
Finite Element Exterior Calculus for Evolution Problems Gillette, Andrew Journal of Computational Mathematics, Vol. 35, Issue 2 https://doi.org/10.4208/jcm.1610-m2015-0319	journal	June 2017
A level-set method for interfacial flows with surfactant Xu, Jian-Jun; Li, Zhilin; Lowengrub, John Journal of Computational Physics, Vol. 212, Issue 2 https://doi.org/10.1016/j.jcp.2005.07.016	journal	March 2006
Flows on surfaces of arbitrary topology Stam, Jos ACM Transactions on Graphics, Vol. 22, Issue 3 https://doi.org/10.1145/882262.882338	journal	July 2003
Finite element approximation of the Cahn–Hilliard equation on surfaces Du, Qiang; Ju, Lili; Tian, Li Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 29-32 https://doi.org/10.1016/j.cma.2011.04.018	journal	July 2011
Simple computation of reaction–diffusion processes on point clouds Macdonald, Colin B.; Merriman, Barry; Ruuth, Steven J. Proceedings of the National Academy of Sciences, Vol. 110, Issue 23 https://doi.org/10.1073/pnas.1221408110	journal	May 2013
A compatible high-order meshless method for the Stokes equations with applications to suspension flows Trask, Nathaniel; Maxey, Martin; Hu, Xiaozhe Journal of Computational Physics, Vol. 355 https://doi.org/10.1016/j.jcp.2017.10.039	journal	February 2018
Hydrodynamic flows on curved surfaces: Spectral numerical methods for radial manifold shapes Gross, B. J.; Atzberger, P. J. Journal of Computational Physics, Vol. 371 https://doi.org/10.1016/j.jcp.2018.06.013	journal	October 2018
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds Gross, B.; Atzberger, P. J. Journal of Scientific Computing, Vol. 76, Issue 1 https://doi.org/10.1007/s10915-017-0617-2	journal	December 2017
Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes Gross, Ben J.; Atzberger, Paul J. arXiv https://doi.org/10.48550/arxiv.1803.07594	text	January 2018

Cited By (3)

Recycling augmented Lagrangian preconditioner in an incompressible fluid solver Olshanskii, Maxim A.; Zhiliakov, Alexander Numerical Linear Algebra with Applications, Vol. 29, Issue 2 https://doi.org/10.1002/nla.2415	journal	October 2021
Graph Convolutional Neural Networks for Body Force Prediction Ogoke, Francis; Meidani, Kazem; Hashemi, Amirreza arXiv https://doi.org/10.48550/arxiv.2012.02232	text	January 2020
Variational Autoencoders for Learning Nonlinear Dynamics of Physical Systems Lopez, Ryan; Atzberger, Paul J. arXiv https://doi.org/10.48550/arxiv.2012.03448	preprint	January 2020

Similar Records

First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)

Journal Article · 2022 · Journal of Computational Physics · OSTI ID:1882875

Gross, B. J.; Kuberry, P.; Atzberger, P. J.

Related Subjects

97 MATHEMATICS AND COMPUTING
Meshfree methods
High order numerical methods
Generalized Moving Least Squares
Manifolds
Exterior calculus
Curved fluid interfaces
Fluid mechanics

Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

Citation Formats

References (67)

Cited By (3)

Similar Records

Related Subjects