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Multi-level \(hp\)-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes. (English) Zbl 1311.74133

Summary: The implementation of \(hp\)-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most \(hp\)-code frameworks restrict themselves to \(1\)-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local \(hp\)-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the \(hp\)-\(d\)-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level \(hp\)-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence – both in terms of degrees of freedom and in run-time – for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic \(hp\)-adaptivity.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Demkowicz L (2007) Computing with hp-adaptive finite elements, Vol. 1: one and two dimensional elliptic and maxwell problems., Applied mathematics and nonlinear science seriesChapman & Hall/CRC, Boca Raton
[2] Solin P (2004) Higher-order finite element methods., Studies in advanced mathematicsChapman & Hall/CRC, Boca Raton · Zbl 1032.65132
[3] Demkowicz L, Oden JT, Rachowicz W, Hardy O (1989) Toward a universal h-p adaptive finite element strategy, part 1. Constrained approximation and data structure. Comput Methods Appl Mech Eng 77(1-2):79-112 · Zbl 0723.73074 · doi:10.1016/0045-7825(89)90129-1
[4] Rachowicz W, Oden JT, Demkowicz L (1989) Toward a universal h-p adaptive finite element strategy part 3. Design of h-p meshes. Comput Methods Appl Mech Eng 77(1-2):181-212 · Zbl 0723.73076 · doi:10.1016/0045-7825(89)90131-X
[5] Demkowicz L, Gerdes K, Schwab C, Bajer A, Walsh T (1998) HP90: a general and flexible Fortran 90 hp-FE code. Comput Vis Sci 1(3):145-163 · Zbl 0912.68014 · doi:10.1007/s007910050014
[6] Demkowicz L, Bajer A, Rachowicz W, Gerdes K (1999) 3D hp-adaptive finite element package Fortran 90 implementation (3Dhp90), TICAM Report 99-29. The University of Texas at Austin, Texas Institute for Computational and Applied Mathematics · Zbl 0926.65046
[7] Rachowicz W, Demkowicz L (2000) An hp-adaptive finite element method for electromagnetics: part 1: data structure and constrained approximation. Comput Methods Appl Mech Eng 187(1-2): 307-335 · Zbl 0979.78031 · doi:10.1016/S0045-7825(99)00137-1
[8] Rachowicz W, Demkowicz L (2002) An hp-adaptive finite element method for electromagnetics—part II: a 3D implementation. Int J Numer Methods Eng 53(1):147-180 · Zbl 0994.78012 · doi:10.1002/nme.396
[9] Paszyński M, Demkowicz L (2006) Parallel, fully automatic hp-adaptive 3D finite element package. Eng Comput 22(3-4):255-276 · Zbl 1093.65113 · doi:10.1007/s00366-006-0036-8
[10] Solin P, Cerveny J (2006) Automatic hp-adaptivity with arbitrary-level hanging nodes, Tech Rep Research Report No. 2006-07, The University of Texas at El Paso, Department of Mathematical Sciences · Zbl 1135.65394
[11] Solin P, Cerveny J, Dolezel I (2008) Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM. Math Comput Simul 77(1):117-132 · Zbl 1135.65394 · doi:10.1016/j.matcom.2007.02.011
[12] Kus P (2011) Automatic hp-adaptivity on meshes with arbitrary-level hanging nodes in 3D. Phd thesis, Charles University, Institute of Mathematics, Prague
[13] Schröder, A.; Hesthaven, JS (ed.); Ronquist, EM (ed.), Subdivisions and multi-level hanging nodes, 317-325 (2011), Heidelberg · Zbl 1216.65161 · doi:10.1007/978-3-642-15337-2_29
[14] Rank E (1992) Adaptive remeshing and h-p domain decomposition. Comput Methods Appl Mech Eng 101(1-3):299-313 · Zbl 0782.65145 · doi:10.1016/0045-7825(92)90027-H
[15] Schillinger D, Rank E (2011) An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry. Comput Methods Appl Mech Eng 200(47-48):3358-3380 · Zbl 1230.74197 · doi:10.1016/j.cma.2011.08.002
[16] Schillinger D (2012) The p- and B-spline versions of the geometrically nonlinear finite cell method and hierarchical refinement strategies for adaptive isogeometric and embedded domain analysis. Doctoral thesis, Technische Universität München, Chair for Computation in Engineering
[17] Schillinger D, Düster A, Rank E (2012) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng 89(9):1171-1202 · Zbl 1242.74161 · doi:10.1002/nme.3289
[18] Schillinger D, Dedè L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJ (2012) An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput Methods Appl Mech Eng 249-252:116-150 · Zbl 1348.65055 · doi:10.1016/j.cma.2012.03.017
[19] Joulaian M, Düster A (2013) Local enrichment of the finite cell method for problems with material interfaces. Comput Mech 52(4):741-762 · Zbl 1311.74123 · doi:10.1007/s00466-013-0853-8
[20] Zienkiewicz O, Taylor R, Zhu J (2005) The finite element method: its basis and fundamentals, 6th edn. Butterworth-heinemann, Oxford · Zbl 1307.74005
[21] Franke DC, Düster A, Nübel V, Rank E (2010) A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2D Hertzian contact problem. Computat Mech 45(5):513-522 · Zbl 1398.74330 · doi:10.1007/s00466-009-0464-6
[22] Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola
[23] Bathe KJ (2007) Finite element procedures. Prentice Hall, Englewood Cliffs
[24] Strang G (1973) An analysis of the finite element method. Prentice-Hall, Englewood Cliffs · Zbl 0356.65096
[25] Felippa CA (2013) Introduction to Finite Element Methods · Zbl 1093.65113
[26] Ainsworth M, Senior B (1997) Aspects of an adaptive hp-finite element method: adaptive strategy, conforming approximation and efficient solvers. Comput Methods Appl Mech Eng 150(1-4): 65-87 · Zbl 0906.73057 · doi:10.1016/S0045-7825(97)00101-1
[27] Rivara M (1984) Mesh refinement processes based on the generalized bisection of simplices. SIAM J Numer Anal 21(3):604-613 · Zbl 0574.65133 · doi:10.1137/0721042
[28] Mitchell WF (1989) A comparison of adaptive refinement techniques for elliptic problems. ACM Trans Math Softw 15(4): 326-347 · Zbl 0900.65306 · doi:10.1145/76909.76912
[29] Babuška I, Aziz A (1976) On the angle condition in the finite element method. SIAM J Numer Anal 13(2):214-226 · Zbl 0324.65046 · doi:10.1137/0713021
[30] Solin P, Dubcova L, Dolezel I (2010) Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell’s equations. Adv Appl Math Mech 2(4):518-532
[31] Schneiders R (2000) Algorithms for quadrilateral and hexahedral mesh generation. Proceedings of the VKI Lecture Series on Computational Fluid Dynamic, VKI-LS, vol. 4 · Zbl 1074.65512
[32] Niekamp R, Stein E (2002) An object-oriented approach for parallel two- and three-dimensional adaptive finite element computations. Comput Struct 80(3-4):317-328 · doi:10.1016/S0045-7949(02)00004-4
[33] Fries T-P, Byfut A, Alizada A, Cheng KW, Schröder A (2011) Hanging nodes and XFEM. Int J Numer Methods Eng 86(4-5): 404-430 · Zbl 1216.74020 · doi:10.1002/nme.3024
[34] Cheng K-W, Fries T-P (2012) XFEM with hanging nodes for two-phase incompressible flow. Comput Methods Appl Mech Eng 245-246:290-312 · Zbl 1354.76099 · doi:10.1016/j.cma.2012.07.011
[35] Szymczak A, Paszyńska A, Paszyński M, Pardo D (2013) Preventing deadlock during anisotropic 2D mesh adaptation in hp-adaptive FEM. J Comput Sci 4(3):170-179 · doi:10.1016/j.jocs.2011.09.001
[36] Mote CD (1971) Global-local finite element. Int J Numer Methods Eng 3(4):565-574 · Zbl 0248.65062 · doi:10.1002/nme.1620030410
[37] Noor AK (1986) Global-local methodologies and their application to nonlinear analysis. Finite Elem Anal Design 2(4):333-346 · doi:10.1016/0168-874X(86)90020-X
[38] Zienkiewicz, OC; Craig, A.; Babuska, I. (ed.); Zienkiewicz, OC (ed.); Gago, J. (ed.); Oliviera, ER (ed.), Adaptive refinement, error estimates, multigrid solution and hierarchic finite element method concepts, 25-55 (1986), New York
[39] Belytschko T, Fish J, Engelmann BE (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59-89 · Zbl 0653.73032 · doi:10.1016/0045-7825(88)90180-6
[40] Fish J, Belytschko T (1988) Elements with embedded localization zones for large deformation problems. Comput Struct 30(1-2):247-256 · Zbl 0667.73033 · doi:10.1016/0045-7949(88)90230-1
[41] Fish J, Belytschko T (1990) A finite element with a unidirectionally enriched strain field for localization analysis. Comput Methods Appl Mech Eng 78(2):181-200 · Zbl 0708.73069 · doi:10.1016/0045-7825(90)90100-Z
[42] Belytschko T, Fish J, Bayliss A (1990) The spectral overlay on finite elements for problems with high gradients. Comput Methods Appl Mech Eng 81(1):71-89 · Zbl 0729.73200 · doi:10.1016/0045-7825(90)90142-9
[43] Rank E, Krause R (1997) A multiscale finite-element method. Comput Struct 64(1):139-144 · Zbl 0918.73222 · doi:10.1016/S0045-7949(96)00149-6
[44] Fish J (1992) The s-version of the finite element method. Comput Struct 43(3):539-547 · Zbl 0775.73247 · doi:10.1016/0045-7949(92)90287-A
[45] Kim, YH; Levit, I.; Stanley, G.; Parsons, I. (ed.); Nour-Omid, B. (ed.), A finite element adaptive mesh refinement technique that avoids multipoint constraints and transition zones, No. CED-4, 27-35 (1991), New York
[46] Fish J (1992) Hierarchical modelling of discontinuous fields. Commun Appl Numer Methods 8(7):443-453 · Zbl 0757.73054 · doi:10.1002/cnm.1630080704
[47] Fish J, Markolefas S (1993) Adaptive s-method for linear elastostatics. Comput Methods Appl Mech Eng 104(3):363-396 · Zbl 0775.73248 · doi:10.1016/0045-7825(93)90032-S
[48] Fish J, Markolefas S, Guttal R, Nayak P (1994) On adaptive multilevel superposition of finite element meshes for linear elastostatics. Appl Numer Math 14(1-3):135-164 · Zbl 0801.73068 · doi:10.1016/0168-9274(94)90023-X
[49] Moore PK, Flaherty JE (1992) Adaptive local overlapping grid methods for parabolic systems in two space dimensions. J Comput Phys 98(1):54-63 · Zbl 0753.65079 · doi:10.1016/0021-9991(92)90172-U
[50] Babuška I, Melenk JM (1997) The partition of Unity Method. Int J Numer Methods Eng 40(4):727-758 · Zbl 0949.65117 · doi:10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
[51] Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181(1-3):43-69 · Zbl 0983.65127 · doi:10.1016/S0045-7825(99)00072-9
[52] Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253-304 · Zbl 1202.74169
[53] Krause R, Rank E (2003) Multiscale computations with a combination of the h- and p-versions of the finite-element method. Comput Methods Appl Mech Eng 192(35-36):3959-3983 · Zbl 1037.74047 · doi:10.1016/S0045-7825(03)00395-5
[54] Düster A, Niggl A, Rank E (2007) Applying the hp-d version of the FEM to locally enhance dimensionally reduced models. Comput Methods Appl Mech Eng 196(37-40):3524-3533 · Zbl 1173.74413 · doi:10.1016/j.cma.2006.10.018
[55] Schillinger D, Evans JA, Reali A, Scott MA, Hughes TJR (2013) Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput Methods Appl Mech Eng 267:170-232 · Zbl 1286.65174 · doi:10.1016/j.cma.2013.07.017
[56] Szabó BA, Babuska I (1991) Finite element analysis. Wiley & Sons, New York · Zbl 0792.73003
[57] Rank E, Zienkiewicz OC (1987) A simple error estimator in the finite element method. Commun Appl Numer Methods 3(3):243-249 · Zbl 0623.65120 · doi:10.1002/cnm.1630030311
[58] Szabó, BA; Düster, A.; Rank, E.; Stein, E. (ed.), The p-version of the finite element method (2004), Chichester
[59] Shewchuk JR (1994) An introduction to the conjugate gradient method without the agonizing pain. Tech Rep · Zbl 1230.74197
[60] Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41(1):121-133 · Zbl 1162.74506 · doi:10.1007/s00466-007-0173-y
[61] Düster A, Parvizian J, Yang Z, Rank E (2008) The finite cell method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197(45-48):3768-3782 · Zbl 1194.74517 · doi:10.1016/j.cma.2008.02.036
[62] Zander N, Bog T, Elhaddad M, Espinoza R, Hu H, Joly A, Wu C, Zerbe P, Düster A, Kollmannsberger S, Parvizian J, Ruess M, Schillinger D, Rank E (2014) FCMLab: a finite cell research toolbox for MATLAB. Adv Eng Softw 74:49-63 · doi:10.1016/j.advengsoft.2014.04.004
[63] Parvizian J, Düster A, Rank E (2011) Topology optimization using the finite cell method. Optim Eng 13(1):57-78 · Zbl 1293.74357 · doi:10.1007/s11081-011-9159-x
[64] Schillinger D, Ruess M, Zander N, Bazilevs Y, Düster A, Rank E (2012) Small and large deformation analysis with the p- and B-spline versions of the finite cell method. Comput Mech 50(4):445-478 · Zbl 1398.74401 · doi:10.1007/s00466-012-0684-z
[65] Yang Z, Ruess M, Kollmannsberger S, Düster A, Rank E (2012) An efficient integration technique for the voxel-based finite cell method. Int J Numer Methods Eng 91(5):457-471 · doi:10.1002/nme.4269
[66] Yang Z, Kollmannsberger S, Düster A, Ruess M, Garcia EG, Burgkart R, Rank E (2012) Non-standard bone simulation: interactive numerical analysis by computational steering. Comput Vis Sci 14(5):207-216 · doi:10.1007/s00791-012-0175-y
[67] Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11(3-4):425-437 · doi:10.1007/s10237-011-0322-2
[68] Abedian A, Parvizian J, Düster A, Rank E (2013) The finite cell method for the J2 flow theory of plasticity. Finite Elem Anal Design 69:37-47 · doi:10.1016/j.finel.2013.01.006
[69] Abedian A, Parvizian J, Düster A, Rank E (2014) Finite cell method compared to h-version finite element method for elasto-plastic problems. Appl Math Mech 35(10):1239-1248 · doi:10.1007/s10483-014-1861-9
[70] Duczek S, Joulaian M, Düster A, Gabbert U (2013) Simulation of Lamb waves using the spectral cell method. pp. 86951U-86951U-11 · Zbl 1398.74330
[71] Cai Q (2013) Finite Cell Method for Transport Problems in Porous Media. Doctoral thesis, Technische Universität München, Munich · Zbl 1398.74401
[72] Rank E, Kollmannsberger S, Sorger C, Düster A (2011) Shell finite cell method: a high order fictitious domain approach for thin-walled structures. Comput Methods Appl Mech Eng 200(45-46):3200-3209 · Zbl 1230.74232 · doi:10.1016/j.cma.2011.06.005
[73] Rank E, Ruess M, Kollmannsberger S, Schillinger D, Düster A (2012) Geometric modeling, isogeometric analysis and the finite cell method. Comput Methods Appl Mech Eng 249-252:104-115 · Zbl 1348.74340 · doi:10.1016/j.cma.2012.05.022
[74] Ruess M, Bazilevs Y, Schillinger D, Zander N, Rank E (2012) Weakly enforced boundary conditions for the NURBS-based Finite Cell Method. In: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS). Austria, Vienna · Zbl 1352.65643
[75] Ruess M, Schillinger D, Bazilevs Y, Varduhn V, Rank E (2013) Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Int J Numer Methods Eng 95(10):811-846 · Zbl 1352.65643 · doi:10.1002/nme.4522
[76] Ruess M, Schillinger D, Özcan AI, Rank E (2014) Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Comput Methods Appl Mech Eng 269:46-71 · Zbl 1296.74013 · doi:10.1016/j.cma.2013.10.009
[77] Zander N, Kollmannsberger S, Ruess M, Yosibash Z, Rank E (2012) The finite cell method for linear thermoelasticity. Comput Math Appl 64(11):3527-3541 · Zbl 1268.74020 · doi:10.1016/j.camwa.2012.09.002
[78] Dauge M, Düster A, Rank E (2014) Theoretical and numerical investigation of the finite cell method. Tech Rep hal-00850602, version 2, Université de Rennes · Zbl 1331.65160
[79] Schillinger D, Ruess M (2014) The Finite Cell Method: a review in the context of higher-order structural analysis of CAD and image-based geometric models. Arch Comput Methods Eng, pp. 1-65 · Zbl 1348.65056
[80] Sadd MH (2009) Elasticity theory, applications and numerics. Elsevier Butterworth-Heinemann, Burlington
[81] Mitchell WF (2010) The hp-multigrid method applied to hp-adaptive refinement of triangular grids. Numer Linear Algebr Appl 17(2-3):211-228 · Zbl 1240.65356
[82] Hu N, Guo X-Z, Katz IN (1997) Multi-p preconditioners. SIAM J Sci Comput 18(6):1676-1697 · Zbl 0926.65046 · doi:10.1137/S1064827595279368
[83] Wilson EL (1974) The static condensation algorithm. Int J Numer Methods Eng 8(1):198-203 · doi:10.1002/nme.1620080115
[84] Szabó, BA; Babuska, I. (ed.); Zienkiewicz, OC (ed.); Gago, J. (ed.); Oliviera, ER (ed.), Estimation and control of error based on p convergence, 25-55 (1986), New York
[85] Niekamp R, Stein E (2001) The hierarchically graded multilevel finite element method. Comput Mech 27(4):302-304 · Zbl 0985.65144 · doi:10.1007/s004660100242
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