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Numerical algorithms of subdivision-based IGA-EIEQ method for the molecular beam epitaxial growth models on complex surfaces

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Abstract

In this paper, we develop fully discrete methods for two molecular beam epitaxial models on surfaces, where one has a fourth-order Ginzburg–Landau double-well potential, and the other has a logarithmic nonlinear potential. The main challenge is how to establish high-order spatiotemporal accurate schemes with the property of unconditional energy stability. The spatial numerical approach adopted in this paper is the recently developed isogeometric analysis framework based on loop subdivision technology, where it can provide an exact and elegant description for the surfaces with arbitrary topology, and subdivision basis functions can serve to represent the numerical solutions. The temporal discrete scheme is based on the “explicit-invariant energy quadratization” strategy. Its novelty is to introduce two auxiliary variables (one local and the other nonlocal) and then develop two specific ordinary differential equations that convert the original equations into their equivalent forms. The use of the nonlocal variable helps us avoid dealing with variable coefficients and solve only equations with constant coefficients, and the use of the local variable allows us to obtain linear schemes with energy-stable structures numerically. We further perform various numerical experiments on some well-known complex surfaces, such as the sphere, bubble, and splayed surface to verify the stability and accuracy of the developed method.

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References

  1. Schneider M, Schuller IK, Rahman A (1987) Epitaxial growth of silicon: a molecular-dynamics simulation. Phys Rev B 46:1340–1343

    Article  Google Scholar 

  2. Clarke S, Vvedensky DD (1987) Origin of reflection high-energy electron-diffraction intensity oscillations during molecular-beam epitaxy: a computational modeling approach. Phys Rev Lett 58:2235–2238

    Article  Google Scholar 

  3. Gyure MF, Ratsch C, Merriman B, Caflisch RE, Osher S, Zinck JJ, Vvedensky DD (1998) Level-set methods for the simulation of epitaxial phenomena. Phys Rev E 58(3):6927–6930

    Article  MATH  Google Scholar 

  4. Shen J, Tang T, Yang J (2016) On the maximum principle preserving schemes for the generalized Allen–Cahn equation. Commun Math Sci 14:1517–1534

    Article  MathSciNet  MATH  Google Scholar 

  5. Shen J, Yang X (2010) Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin Dyn Syst 228:1669–1691

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhao J, Wang Q, Yang X (2016) Numerical approximations to a new phase field model for immiscible mixtures of nematic liquid crystals and viscous fluids. Comput Methods Appl Mech Eng 310:77–97

    Article  Google Scholar 

  7. Eyre DJ (1998) Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: Computational and mathematical models of microstructural evolution. San Francisco, CA, 1998, Materials research society symposium proceedings, vol 529. MRS, Warrendale, PA, pp 39–46

  8. Wang C, Wang X, Wise SM (2010) Unconditionally stable schemes for equations of thin film epitaxy. Discrete Contin Dyn Syst 28(1):405–423

    Article  MathSciNet  MATH  Google Scholar 

  9. Shen J, Wang C, Wang X, Wise SM (2012) Second-order convex splitting schemes for gradient flows with Ehrlich–Schwoebel type energy: application to thin film epitaxy. SIAM J Numer Anal 50(1):105–125

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen W, Conde S, Wang C, Wang X, Wise S (2012) A linear energy stable scheme for a thin film model without slope selection. J Sci Comput 52:546–562

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang X, Zhao J, Wang Q (2017) Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J Comput Phys 333:104–127

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen L, Zhao J, Yang X (2018) Regularized linear schemes for the molecular beam epitaxy model with slope selection. Appl Numer Math 128:139–156

    Article  MathSciNet  MATH  Google Scholar 

  13. Shen J, Yang X (2020) The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems. Contemp Math 754:217–245

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang X (2021) A novel fully-decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier–Stokes equations coupled with mass-conserved Allen–Cahn phase-field model of two-phase incompressible flow. Int J Numer Methods Eng 122:283–1306

    MathSciNet  Google Scholar 

  15. Yang X, Zhang G-D (2020) Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn–Hilliard and Allen–Cahn equations with general nonlinear potential. J Sci Comput 82:55

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang X (2021) Numerical approximations of the Navier-Stokes equation coupled with volume-conserved multi-phase-field vesicles system: fully-decoupled, linear, unconditionally energy stable and second-order time-accurate numerical scheme. Comput Methods Appl Mech Eng 375:113600

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang X (2016) Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J Comput Phys 302:509–523

    MathSciNet  Google Scholar 

  18. Yang X, Yu H (2018) Efficient second order unconditionally stable schemes for a phase-field moving contact line model using an invariant energy quadratization approach. SIAM J Sci Comput 40:889–914

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang X, Zhao J, Wang Q, Shen J (2017) Numerical for a three components Cahn–Hilliard phase-field model based on the invariant energy quadratization method. Math Models Methods Appl Sci 27(11):1993–2030

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao J, Yang X, Gong Y, Wang Q (2017) A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals. Comput Methods Appl Mech Eng 318:803–825

    Article  MathSciNet  MATH  Google Scholar 

  21. Furihata D (2001) A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer Math 87:675–699

    Article  MathSciNet  MATH  Google Scholar 

  22. Lee HG, Lowengrub JS, Goodman J (2002) Modeling pinchoff and reconnection in a Hele–Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys Fluids 14:514–545

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu C, Shen J (2003) A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179:211–228

    Article  MathSciNet  MATH  Google Scholar 

  24. Dziuk G, Elliot CM (2013) Finite element methods for surface PDEs. Acta Numer 22:289–396

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao S, Xiao X, Feng X (2020) An efficient time adaptively based on chemical potential for surface Cahn–Hilliard equation using finite element approximation. Appl Math Comput 369:124901

    Article  MathSciNet  MATH  Google Scholar 

  26. Du Q, Ju L, Tian L (2011) Finite element approximation for the Cahn–Hilliard equation on surface. Comput Methods Appl Mech Eng 200:2458–2470

    Article  MathSciNet  MATH  Google Scholar 

  27. Marenduzzo D, Orlandini E (2013) Phase separation dynamics on curved surfaces. Soft Matter 9:1178–1187

    Article  Google Scholar 

  28. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  29. Bazilevs Y, Beirão da Veiga L, Cottrell JA, Hughes TJR, Sangalli G (2006) Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math Models Methods Appl Sci 16(7):1031–1090

    Article  MathSciNet  MATH  Google Scholar 

  30. Piegl LA, Tiller W (1997) The Nurbs book (monographs in visual communication), 2nd edn. Spinger, New York

    MATH  Google Scholar 

  31. Sederberg TW, Finnigan GT, Li X, Lin H, Ipson H (2008) Watertight trimmed NURBS. ACM Trans Graph 27(3):1–8

    Article  Google Scholar 

  32. Zhang Y, Bazilevs Y, Goswami S, Bajaj C, Hughes TJR (2007) Patient-Specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput Methods Appl Mech Eng 196(29–30):2943–2959

    Article  MathSciNet  MATH  Google Scholar 

  33. Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCs. ACM Trans Graph 22(3):477–484

    Article  Google Scholar 

  34. Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 5–8:229–263

    Article  MathSciNet  MATH  Google Scholar 

  35. Catmull E, Clark J (1978) Recursively generated B-spline surfaces on arbitrary topological meshes. Comput Aided Des 10(6):350–355

    Article  Google Scholar 

  36. Loop C (1978) Smooth subdivision surfaces based on triangles. Master’s thesis. Department of Mathematics, University of Utah

  37. Stam J (1998) Fast evaluation of Catmull–Clark subdivision surfaces at arbitrary parameter values. In: SIGGRAPH ’98 proceedings, pp 395–404

  38. Stam J (1998) Fast evaluation of Loop triangular subdivision surfaces at arbitrary parameter values. In: SIGGRAPH ’98, CD-ROM supplement

  39. Cirak F, Ortiz M, Schröder P (2000) Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Int J Numer Methods Eng 47:2039–2072

    Article  MATH  Google Scholar 

  40. Cirak F, Scott MJ, Antonsson EK, Ortiz M, Schröder P (2002) Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Comput Aided Des 34(2):137–148

    Article  Google Scholar 

  41. Pan Q, Xu G, Xu G, Zhang Y (2015) Isogeometric analysis based on extended Loop’s subdivision. J Comput Phys 299(15):731–746

    Article  MathSciNet  MATH  Google Scholar 

  42. Pan Q, Xu G, Xu G, Zhang Y (2016) Isogeometric analysis based on extended Catmull–Clark subdivision. Comput Math Appl 71:105–119

  43. Pan Q, Chen C, Xu G (2017) Isogeometric finite element approximation of minimal surfaces based on extended Loop subdivision. J Comput Phys 343:324–339

    Article  MathSciNet  MATH  Google Scholar 

  44. Pan Q, Rabczuk T, Xu G, Chen C (2018) Isogeometric analysis of minimal surfaces on the basis of extended Catmull–Clark subdivisions. Comput Methods Appl Mech Eng 337:128–149

    Article  MathSciNet  MATH  Google Scholar 

  45. Pan Q, Rabczuk T, Xu G, Chen C (2019) Isogeometric analysis for surface PDEs with extended Loop subdivision. J Comput Phys 398:108892

    Article  MathSciNet  MATH  Google Scholar 

  46. Pan Q, Rabczuk T, Yang X (2021) Subdivision-based isogeometric analysis for second order partial differential equations on surfaces. Comput Mech 68:1205–1221

    Article  MathSciNet  MATH  Google Scholar 

  47. Pan Q, Rabczuk T, Chen C (2022) Subdivision based isogeometric analysis for geometric flows. Int J Numer Methods Eng 123:610–633

    Article  MathSciNet  Google Scholar 

  48. Do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  49. Spivak MA (1999) A comprehensive introduction to differential geometry, vol 3, 3rd edn. Perish, Boston

    MATH  Google Scholar 

  50. Moldovan D, Golubovic L (2000) Interfacial coarsening dynamics in epitaxial growth with slope selection. Phys Rev E 61:6190–6214

    Article  Google Scholar 

  51. Ehrlich G, Hudda FG (1966) Atomic view of surface diffusion: tungsten on tungsten. J Chem Phys 44:1036

    Article  Google Scholar 

  52. Schwoebel RL (1969) Step motion on crystal surfaces: II. J Appl Phys 40:614

  53. Kobbelt L, Hesse T, Prautzsch H, Schweizerhof K (1997) Iterative mesh generation for FE-computation on free form surfaces. Eng Comput 14:806–820

    Article  MATH  Google Scholar 

  54. Dyn N, Levin D, Gregory JA (1990) A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans Graph 9(2):160–169

    Article  MATH  Google Scholar 

  55. Zorin D, Schröder P, Sweldens W (1996) Subdivision for meshes with arbitrary topology. In: SIGGRAPH ’96 proceedings, pp 71–78

  56. Doo D, Sabin M (1978) Behaviours of recursive division surfaces near extraordinary points. Comput Aided Des 10(6):356–360

    Article  Google Scholar 

  57. Peters J, Reif U (1997) The simplest subdivision scheme for smoothing polyhedra. ACM Trans Graph 16(4):420–431

    Article  Google Scholar 

  58. Zhang H, Zhang J, Yang X (2021) Stabilized invariant energy quadratization (S-IEQ) method for the molecular beam epitaxial model without slope section. Int J Numer Anal Model 18:642–655

    MathSciNet  MATH  Google Scholar 

  59. Cheng Q, Shen J, Yang X (2019) Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach. J Sci Comput 78:1467–1487

    Article  MathSciNet  MATH  Google Scholar 

  60. Kohn RV, Yan X (2003) Upper bound on the coarsening rate for an epitaxial growth model. Commun Pure Appl Math 56:1549–1564

    Article  MathSciNet  MATH  Google Scholar 

  61. Li B, Liu JG (2003) Thin film epitaxy with or without slope selection. Eur J Appl Math 14:713–743

    Article  MathSciNet  MATH  Google Scholar 

  62. Wang C, Wang X, Wise SM (2010) Unconditionally stable schemes for equations of thin film epitaxy. Discrete Contin Dyn Syst 28(1):405–423

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of Q. Pan was partially supported by National Natural Science Foundation of China (Nos. 12171147 and 12261017). The work of J. Zhang was partially supported by Hunan Provincial Natural Science Foundation of China (No. 2021JJ30456). The work of X. Yang was partially supported by National Science Foundation of USA with Grant Number DMS-2012490.

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Authors and Affiliations

  1. School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha, 410114, China

    Qing Pan & Jin Zhang

  2. Institute of Structural Mechanics, Bauhaus Universität-Weimar, 99423, Weimar, Germany

    Timon Rabczuk

  3. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Chong Chen

  4. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

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  1. Qing Pan
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Correspondence to Xiaofeng Yang.

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Pan, Q., Zhang, J., Rabczuk, T. et al. Numerical algorithms of subdivision-based IGA-EIEQ method for the molecular beam epitaxial growth models on complex surfaces. Comput Mech 72, 927–948 (2023). https://doi.org/10.1007/s00466-023-02319-6

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