Numerical methods for biomembranes: Conforming subdivision methods versus non-conforming PL methods

Chen, Jingmin; Yu, Thomas; Brogan, Patrick; Kusner, Robert; Yang, Yilin; Zigerelli, Andrew

doi:10.1090/mcom/3584

Numerical methods for biomembranes: Conforming subdivision methods versus non-conforming PL methods
HTML articles powered by AMS MathViewer

by Jingmin Chen, Thomas Yu, Patrick Brogan, Robert Kusner, Yilin Yang and Andrew Zigerelli HTML | PDF

Math. Comp. 90 (2021), 471-516

Abstract:

The Canham-Helfrich-Evans models of biomembranes consist of a family of geometric constrained variational problems. In this article, we compare two classes of numerical methods for these variational problems based on piecewise linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline approximation and can be viewed as higher order versions of PL methods, one may expect that the only difference between the two methods is in the accuracy order. In this paper, we prove that a numerical method based on minimizing any one of the ‘PL Willmore energies’ proposed in the literature would fail to converge to a solution of the continuous problem, whereas a method based on minimization of the bona fide Willmore energy, well-defined for SS but not PL surfaces, succeeds. Motivated by this analysis, we propose also a regularization method for the PL method based on techniques from conformal geometry. We address a number of implementation issues crucial for the efficiency of our solver. A software package called Wmincon accompanies this article, and provides parallel implementations of all the relevant geometric functionals. When combined with a standard constrained optimization solver, the geometric variational problems can then be solved numerically. To this end, we realize that some of the available optimization algorithms/solvers are capable of preserving symmetry, while others manage to break symmetry; we explore the consequences of this observation.

References

Gregory Arden, Approximation properties of subdivision surfaces, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–University of Washington. MR 2702127
John W. Barrett, Harald Garcke, and Robert Nürnberg, Finite element approximation for the dynamics of asymmetric fluidic biomembranes, Math. Comp. 86 (2017), no. 305, 1037–1069. MR 3614011, DOI 10.1090/mcom/3162
Matthias Bauer and Ernst Kuwert, Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. 10 (2003), 553–576. MR 1941840, DOI 10.1155/S1073792803208072
T. Baumgart, S. T. Hess, and W. W. Webb, Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension, Nature, 425 (Oct 2003), 821–824.
L. Bers. Riemann Surfaces. Courant Institute of Mathematical Sciences, New York University, New York, 1958.
Alexander I. Bobenko, A conformal energy for simplicial surfaces, Combinatorial and computational geometry, Math. Sci. Res. Inst. Publ., vol. 52, Cambridge Univ. Press, Cambridge, 2005, pp. 135–145. MR 2178318
A. I. Bobenko and P. Schröder, Discrete Willmore flow, Eurographics Symposium on Geometry Processing, 2005, pp. 101–110.
Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry, Graduate Studies in Mathematics, vol. 98, American Mathematical Society, Providence, RI, 2008. Integrable structure. MR 2467378, DOI 10.1007/978-3-7643-8621-4
Andrea Bonito, Ricardo H. Nochetto, and M. Sebastian Pauletti, Parametric FEM for geometric biomembranes, J. Comput. Phys. 229 (2010), no. 9, 3171–3188. MR 2601095, DOI 10.1016/j.jcp.2009.12.036
Kenneth A. Brakke, The surface evolver, Experiment. Math. 1 (1992), no. 2, 141–165. MR 1203871
K. A. Brakke, Surface Evolver manual version 2.70, Available at http://facstaff.susqu.edu/brakke/evolver/downloads/manual270.pdf (2018/04/20), August 2013.
John P. Brogan, Yilin Yang, and Thomas P.-Y. Yu, Numerical methods for biomembranes based on piecewise linear surfaces, Numerical mathematics and advanced applications—ENUMATH 2017, Lect. Notes Comput. Sci. Eng., vol. 126, Springer, Cham, 2019, pp. 809–817. MR 3977037, DOI 10.1007/978-3-319-96415-7_{7}6
A. M. Bruckstein, A. N. Netravali, and T. J. Richardson, Epi-convergence of discrete elastica, Appl. Anal. 79 (2001), no. 1-2, 137–171. MR 1880530, DOI 10.1080/00036810108840955
P. B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theoret. Biology, 26 (1970), no. 1, 61–76.
Jingmin Chen, Sara Grundel, and Thomas P. Y. Yu, A flexible $C^2$ subdivision scheme on the sphere: with application to biomembrane modelling, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 459–483. MR 3686779, DOI 10.1137/16M1076794
G. R. Cowper, Gaussian quadrature formulas for triangles, Internat. J. Numer. Methods in Engrg. 7 (1973), no. 3, 405–408.
K. Crane, Discrete Conformal Geometry, Proceedings of Symposia in Applied Mathematics, American Mathematical Society, 2020.
Frank E. Curtis, Tim Mitchell, and Michael L. Overton, A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles, Optim. Methods Softw. 32 (2017), no. 1, 148–181. MR 3579748, DOI 10.1080/10556788.2016.1208749
Klaus Deckelnick, Gerhard Dziuk, and Charles M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer. 14 (2005), 139–232. MR 2168343, DOI 10.1017/S0962492904000224
M. Desbrun, M. Meyer, P. Schröder, and A. H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’99, ACM Press/Addison-Wesley Publishing Co. New York, 1999, pp. 317–324.
Qiang Du, Chun Liu, Rolf Ryham, and Xiaoqiang Wang, A phase field formulation of the Willmore problem, Nonlinearity 18 (2005), no. 3, 1249–1267. MR 2134893, DOI 10.1088/0951-7715/18/3/016
Qiang Du, Chun Liu, and Xiaoqiang Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys. 198 (2004), no. 2, 450–468. MR 2062909, DOI 10.1016/j.jcp.2004.01.029
Qiang Du, Chun Liu, and Xiaoqiang Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, J. Comput. Phys. 212 (2006), no. 2, 757–777. MR 2187911, DOI 10.1016/j.jcp.2005.07.020
T. Duchamp, A. Certain, A. DeRose, and W. Stuetzle, Hierarchical computation of PL harmonic embeddings, Preprint, July 1997.
Gerhard Dziuk and John E. Hutchinson, The discrete Plateau problem: algorithm and numerics, Math. Comp. 68 (1999), no. 225, 1–23. MR 1613695, DOI 10.1090/S0025-5718-99-01025-X
Gerhard Dziuk and John E. Hutchinson, The discrete Plateau problem: convergence results, Math. Comp. 68 (1999), no. 226, 519–546. MR 1613699, DOI 10.1090/S0025-5718-99-01026-1
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385–524. MR 956352, DOI 10.1112/blms/20.5.385
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
E. A. Evans, Bending resistance and chemically induced moments in membrane bilayers, Biophysical J. 14 (1974), no. 12, 923–931.
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015. MR 3409135
Feng Feng and William S. Klug, Finite element modeling of lipid bilayer membranes, J. Comput. Phys. 220 (2006), no. 1, 394–408. MR 2281635, DOI 10.1016/j.jcp.2006.05.023
George Francis, John M. Sullivan, Rob B. Kusner, Ken A. Brakke, Chris Hartman, and Glenn Chappell, The minimax sphere eversion, Visualization and mathematics (Berlin-Dahlem, 1995) Springer, Berlin, 1997, pp. 3–20. MR 1607221
Philip E. Gill, Walter Murray, and Michael A. Saunders, SNOPT: an SQP algorithm for large-scale constrained optimization, SIAM Rev. 47 (2005), no. 1, 99–131. MR 2149103, DOI 10.1137/S0036144504446096
Xianfeng David Gu, Feng Luo, and Shing-Tung Yau, Recent advances in computational conformal geometry, Commun. Inf. Syst. 9 (2009), no. 2, 163–195. MR 2606088
X. Gu, Y. Wang, T. Chan, P.M. Thompson, and S.-T. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, 18th International Conference on Information Processing in Medical Imaging, (Ambleside, UK), Lecture Notes in Computer Science, vol. 2732, Springer-Verlag, Berlin and Heidelberg, 2003, pp. 172–184.
Xianfeng Gu, Ren Guo, Feng Luo, Jian Sun, and Tianqi Wu, A discrete uniformization theorem for polyhedral surfaces II, J. Differential Geom. 109 (2018), no. 3, 431–466. MR 3825607, DOI 10.4310/jdg/1531188190
Xianfeng David Gu, Feng Luo, Jian Sun, and Tianqi Wu, A discrete uniformization theorem for polyhedral surfaces, J. Differential Geom. 109 (2018), no. 2, 223–256. MR 3807319, DOI 10.4310/jdg/1527040872
W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch C, 28 (1973), no.11, 693–703.
Lucas Hsu, Rob Kusner, and John Sullivan, Minimizing the squared mean curvature integral for surfaces in space forms, Experiment. Math. 1 (1992), no. 3, 191–207. MR 1203874
M. Jarić, U. Seifert, W. Wintz, and M. Wortis, Vesicular instabilities: The prolate-to-oblate transition and other shape instabilities of fluid bilayer membranes, Phys. Rev. E, 52 (1995), 6623–6634.
Jürgen Jost, Compact Riemann surfaces, 3rd ed., Universitext, Springer-Verlag, Berlin, 2006. An introduction to contemporary mathematics. MR 2247485, DOI 10.1007/978-3-540-33067-7
F. Jülicher, U. Seifert, and R. Lipowsky, Conformal degeneracy and conformal diffusion of vesicles, Physical Rev. Lett. 71 (1993), no. 3, 452–455
O. Kahraman, N. Stoop, and M. M. Müller, Fluid membrane vesicles in confinement, New J. Phys. 14 (2012), no. 9, 095021.
O. Kahraman, N. Stoop, and M. M. Müller, Morphogenesis of membrane invaginations in spherical confinement, Europhys. Lett. 97 (2012), no. 6, 68008.
Laura Gioia Andrea Keller, Andrea Mondino, and Tristan Rivière, Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint, Arch. Ration. Mech. Anal. 212 (2014), no. 2, 645–682. MR 3176354, DOI 10.1007/s00205-013-0694-9
Rob Kusner, Comparison surfaces for the Willmore problem, Pacific J. Math. 138 (1989), no. 2, 317–345. MR 996204
Rob Kusner, Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994) A K Peters, Wellesley, MA, 1996, pp. 67–72. MR 1417949
H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
G. H. W. Lim, M. Wortis, and R. Mukhopadhyay, Stomatocyte–discocyte–echinocyte sequence of the human red blood cell: Evidence for the bilayer–couple hypothesis from membrane mechanics, PNAS, 99 (2002), no. 6, 16766–16769.
R. Lipowsky, The conformation of membranes, Nature, 349 (1991), 475–481.
C. T. Loop, Smooth subdivision surfaces based on triangles, Master’s thesis, Department of Mathematics, University of Utah, 1987.
Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing-Tung Yau, Optimization of surface registrations using Beltrami holomorphic flow, J. Sci. Comput. 50 (2012), no. 3, 557–585. MR 2886341, DOI 10.1007/s10915-011-9506-2
Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782. MR 3152944, DOI 10.4007/annals.2014.179.2.6
Mark Meyer, Mathieu Desbrun, Peter Schröder, and Alan H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and mathematics III, Math. Vis., Springer, Berlin, 2003, pp. 35–57. MR 2047000
X. Michalet and D. Bensimon, Observation of stable shapes and conformal diffusion in genus 2 vesicles, Science, 269 (1995), no. 5224, 666–668.
X. Michalet, D. Bensimon, and B. Fourcade, Fluctuating vesicles of nonspherical topology, Phys. Rev. Lett., 72 (Jan 1994), 168–171.
Jorge Nocedal and Stephen J. Wright, Numerical optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. MR 2244940
Richard S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30. MR 547524
Ulrich Reif, A degree estimate for subdivision surfaces of higher regularity, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2167–2174. MR 1327042, DOI 10.1090/S0002-9939-96-03366-7
Ulrich Reif and Peter Schröder, Curvature integrability of subdivision surfaces, Adv. Comput. Math. 14 (2001), no. 2, 157–174. MR 1843304, DOI 10.1023/A:1016685104156
H. Schumacher, On $H^2$-gradient flows for the Willmore Energy, ArXiv e-prints, March 2017.
Henrik Schumacher and Max Wardetzky, Variational convergence of discrete minimal surfaces, Numer. Math. 141 (2019), no. 1, 173–213. MR 3903206, DOI 10.1007/s00211-018-0993-z
Johannes Schygulla, Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 901–941. MR 2928137, DOI 10.1007/s00205-011-0465-4
U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys. 46 (1997), no. 1, 13–137.
Leon Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), no. 2, 281–326. MR 1243525, DOI 10.4310/CAG.1993.v1.n2.a4
J. Stam, Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values, Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’98, ACM, New York, 1998, pp. 395–404.
J. Stam, Evaluation of Loop subdivision surfaces, SIGGRAPH’99 Course Notes, 1999.
John M. Sullivan, Curvatures of smooth and discrete surfaces, Discrete differential geometry, Oberwolfach Semin., vol. 38, Birkhäuser, Basel, 2008, pp. 175–188. MR 2405666, DOI 10.1007/978-3-7643-8621-4_{9}
Max Wardetzky, Convergence of the cotangent formula: an overview, Discrete differential geometry, Oberwolfach Semin., vol. 38, Birkhäuser, Basel, 2008, pp. 275–286. MR 2405672, DOI 10.1007/978-3-7643-8621-4_{1}5
Joel L. Weiner, On a problem of Chen, Willmore, et al, Indiana Univ. Math. J. 27 (1978), no. 1, 19–35. MR 467610, DOI 10.1512/iumj.1978.27.27003
Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645–680. MR 1503303, DOI 10.2307/1968482
T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493–496 (English, with Romanian and Russian summaries). MR 202066
W. Wintz, H.-G. Döbereiner, and U. Seifert, Starfish vesicles, EPL (Europhysics Letters) 33 (1996), no. 5, 403.
G. Xie and T. P.-Y. Yu, On approximation properties of subdivision surfaces, In Preparation, 2019.
T. P.-Y. Yu and J. Chen, Uniqueness of Clifford torus with prescribed isoperimetric ratio, Preprint, March 2020.
Siming Zhao, Timothy Healey, and Qingdu Li, Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles, Comput. Methods Appl. Mech. Engrg. 314 (2017), 164–179. MR 3589870, DOI 10.1016/j.cma.2016.07.011
Denis Zorin, A method for analysis of $C^1$-continuity of subdivision surfaces, SIAM J. Numer. Anal. 37 (2000), no. 5, 1677–1708. MR 1759912, DOI 10.1137/S003614299834263X
D. Zorin, Smoothness of stationary subdivision on irregular meshes, Constr. Approx. 16 (2000), no. 3, 359–397. MR 1759894, DOI 10.1007/s003659910016