Numerical Solution for Schrödinger Eigenvalue Problem Using Isogeometric Analysis on Implicit Domains

Abstract

We study the accuracy and performance of isogeometric analysis on implicit domains when solving time-independent Schrödinger equation. We construct weighted extended PHT-spline basis functions for analysis, and the domain is presented with same basis functions in implicit form excluding the need for a parameterization step. Moreover, an adaptive refinement process is formulated and discussed with details. The constructed basis functions with cubic polynomials and only $C^{1}$ continuity are enough to produce a higher continuous field approximation while maintaining the computational cost for the matrices as low as possible. A numerical implementation for the adaptive method is performed on Schrödinger eigenvalue problem with double-well potential using 3 examples on different implicit domains. The convergence and performance results demonstrate the efficiency and accuracy of the approach.

Keywords

Isogeometric analysis / Finite element analysis / PHT-splines / Schrodinger equation / Eigenvalue problems

References

[1] Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW. Isogeometric analysis using t-splines. Comput. Methods Appl. Mech. Eng.. 2010, 199 5–8 229-263

[2] Bazilevs Y, Beirao da Veiga L, Cottrell JA, Hughes TJ, Sangalli G. Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci.. 2006, 16 07 1031-1090

[3] Buchegger F, Jüttler B. Planar multi-patch domain parameterization via patch adjacency graphs. Comput. Aided Des.. 2017, 82 2-12

[4] Buffa, A., Sangalli, G., Schwab, C.: Exponential convergence of the hp version of isogeometric analysis in 1D. In: Spectral and High Order Methods for Partial Differential Equations-ICOSAHOM 2012, pp. 191–203. Springer (2014)

[5] Cimrman R, Novak M, Kolman R, Tuma M, Plesek J, Vackar J. Convergence study of isogeometric analysis based on Bézier extraction in electronic structure calculations. Appl. Math. Comput.. 2018, 319 138-152

[6] Cimrman R, Novak M, Kolman R, Tuma M, Vackar J. Isogeometric analysis in electronic structure calculations. Math. Comput. Simul.. 2018, 145 125-135

[7] Cottrell JA, Hughes TJ, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. 2009 Hoboken: Wiley

[8] Davydov D, Young TD, Steinmann P. On the adaptive finite element analysis of the kohn-sham equations: methods, algorithms, and implementation. Int. J. Numer. Methods Eng.. 2016, 106 11 863-888

[9] Demmel JW. Applied Numerical Linear Algebra. 1997 Philadelphia: SIAM

[10] Deng F, Zeng C, Deng J. Boundary-mapping parametrization in isogeometric analysis. Commun. Math. Stat.. 2016, 4 2 203-216

[11] Deng J, Chen F, Feng Y. Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math.. 2006, 194 2 267-283

[12] Deng J, Chen F, Jin L. Dimensions of biquadratic spline spaces over T-meshes. J. Comput. Appl. Math.. 2013, 238 68-94

[13] Deng J, Chen F, Li X, Hu C, Tong W, Yang Z, Feng Y. Polynomial splines over hierarchical T-meshes. Graph. Models. 2008, 70 4 76-86

[14] Falini A, Jüttler B. Thb-splines multi-patch parameterization for multiply-connected planar domains via template segmentation. J. Comput. Appl. Math.. 2019, 349 390-402

[15] Harrell EM. Double wells. Commun. Math. Phys.. 1980, 75 3 239-261

[16] Hofer C, Langer U, Toulopoulos I. Discontinuous galerkin isogeometric analysis of elliptic diffusion problems on segmentations with gaps. SIAM J. Sci. Comput.. 2016, 38 6 A3430-A3460

[17] Hollig K. Finite Element Methods with B-Splines. 2003 Philadelphia: SIAM

[18] Höllig K, Apprich C, Streit A. Introduction to the web-method and its applications. Adv. Comput. Math.. 2005, 23 1–2 215-237

[19] Höllig K, Reif U, Wipper J. Weighted extended B-spline approximation of Dirichlet problems. SIAM J. Numer. Anal.. 2001, 39 2 442-462

[20] Hughes TJ, Cottrell JA, Bazilevs Y. Isogeometric analysis: CAD, finite elements, nurbs, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng.. 2005, 194 39–41 4135-4195

[21] Koç R, Haydargil D. Solution of the Schrödinger equation with one and two timensional double-well potentials. Turk. J. Phys.. 2004, 28 3 161-167

[22] Krasilnikov P. Two-dimensional model of a double-well potential: proton transfer upon hydrogen bond deformation. Biophysics. 2014, 59 2 189-198

[23] Kuang Y, Hu G. An adaptive fem with ITP approach for steady Schrödinger equation. Int. J. Comput. Math.. 2018, 95 1 187-201

[24] Li M, Huang C, Wang P. Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algorithms. 2017, 74 2 499-525

[25] Li S, Zhang Z. Computing eigenvalues and eigenfunctions of Schrödinger equations using a model reduction approach. Commun. Comput. Phys.. 2018, 24 1073-1100

[26] Milburn G, Corney J, Wright EM, Walls D. Quantum dynamics of an atomic Bose–Einstein condensate in a double-well potential. Phys. Rev. A. 1997, 55 6 4318

[27] Motamarri P, Gavini V. Subquadratic-scaling subspace projection method for large-scale Kohn–Sham density functional theory calculations using spectral finite-element discretization. Phys. Rev. B. 2014, 90 11 115127

[28] Phung-Van P, Tran LV, Ferreira A, Nguyen-Xuan H, Abdel-Wahab M. Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads. Nonlinear Dyn.. 2017, 87 2 879-894

[29] Piegl L, Tiller W. The NURBS Book. 2012 Berlin: Springer

[30] Qarariyah A, Deng F, Yang T, Liu Y, Deng J. Isogeometric analysis on implicit domains using weighted extended PHT-splines. J. Comput. Appl. Math.. 2019, 350 353-371

[31] Ram-Mohan LR. Finite Element and Boundary Element Applications in Quantum Mechanics. 2002 Oxford: Oxford University Press

[32] Schillinger D, Evans JA, Reali A, Scott MA, Hughes TJ. Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput. Methods Appl. Mech. Eng.. 2013, 267 170-232

[33] Sederberg TW, Zheng J, Bakenov A, Nasri A. T-splines and T-NURCCS. ACM Trans. Graph. (TOG). 2003, 22 3 477-484

[34] Song L, Guo Z, Chai GB, Wang Z, Li Y, Luan Y. A finite element method to investigate the elastic properties of pillared graphene sheet under different conditions. Carbon. 2018, 140 210-217

[35] Sun J, Zhou A. Finite Element Methods for Eigenvalue Problems. 2016 Boca Raton: Chapman and Hall/CRC

[36] Takacs T, Jüttler B. Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis. Comput. Methods Appl. Mech. Eng.. 2011, 200 49–52 3568-3582

[37] Wu M, Wang Y, Mourrain B, Nkonga B, Cheng C. Convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis. Comput. Aided Geom. Des.. 2017, 52 170-189

[38] Xu G, Mourrain B, Duvigneau R, Galligo A. Parameterization of computational domain in isogeometric analysis: methods and comparison. Comput. Methods Appl. Mech. Eng.. 2011, 200 23–24 2021-2031

[39] Xu J, Chen F, Deng J. Two-dimensional domain decomposition based on skeleton computation for parameterization and isogeometric analysis. Comput. Methods Appl. Mech. Eng.. 2015, 284 541-555

[40] Yan J, Lin S, Bazilevs Y, Wagner GJ. Isogeometric analysis of multi-phase flows with surface tension and with application to dynamics of rising bubbles. Comput. Fluids. 2019, 179 777-789

[41] Yu P, Anitescu C, Tomar S, Bordas SPA, Kerfriden P. Adaptive isogeometric analysis for plate vibrations: an efficient approach of local refinement based on hierarchical a posteriori error estimation. Comput. Methods Appl. Mech. Eng.. 2018, 342 251-286

[42] Zhu Y, Chen F. Modified bases of PHT-splines. Commun. Math. Stat.. 2017, 5 4 381-397

Funding

National Natural Science Foundation of China(11771420)