article

3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

Authors:

Tobias Preusser,

Lars Ole SchwenAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 33, Issue 5

Pages 2115 - 2143

https://doi.org/10.1137/100791750

Published: 01 September 2011 Publication History

Abstract

For scalar and vector-valued elliptic boundary value problems with discontinuous coefficients across geometrically complicated interfaces, a composite finite element approach is developed. Composite basis functions are constructed, mimicking the expected jump condition for the solution at the interface in an approximate sense. The construction is based on a suitable local interpolation on the space of admissible functions. We study the order of approximation and the convergence properties of the method numerically. As applications, heat diffusion in an aluminum foam matrix filled with polymer and linear elasticity of microstructured materials, in particular, specimens of trabecular bone, are investigated. Furthermore, a numerical homogenization approach is developed for periodic structures and real material specimens which are not strictly periodic but are considered as statistical prototypes. Thereby, effective macroscopic material properties can be computed.

References

[1]

L. Adams and Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24 (2002), pp. 463-479.

[2]

G. Allaire, Shape Optimization by the Homogenization Method, Appl. Math. Sci. 146, Springer-Verlag, New York, 2002.

[3]

T. Arbogast, Numerical subgrid upscaling of two-phase flow in porous media, in Numerical Treatment of Multiphase Flows in Porous Media, Lecture Notes in Phys. 552, Springer, Berlin, 2000, pp. 35-49.

[4]

T. Arbogast, An overview of subgrid upscaling for elliptic problems in mixed form, in Current Trends in Scientific Computing, Contemp. Math. 329, AMS, Providence, RI, 2003, pp. 21-32.

[5]

T. Arbogast, S. E. Minkoff, and P. T. Keenan, An operator-based approach to upscaling the pressure equation, in Computational Methods in Water Resources XII, Volume 1: Computational Methods in Contamination and Remediation of Water Resources, Computational Mechanics Publications, Southampton, UK, 1998, pp. 405-412.

[6]

G. P. Astrakhantsev, Method of fictitious domain for a second-order elliptic equation with natural boundary conditions, Comput. Math. Math. Phys., 18 (1978), pp. 114-121.

[7]

I. Babuška, U. Banerjee, and J. E. Osborn, Survey of meshless and generalized finite element methods: A unified approach, Acta Numer., 12 (2003), pp. 1-125.

[8]

I. Babuška and J. M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 727-758.

[9]

J. W. Barrett and C. M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal., 7 (1987), pp. 283-300.

[10]

J. W. Barrett and C. M. Elliott, A practical finite element approximation of a semi-definite Neumann problem on a curved domain, Numer. Math., 51 (1987), pp. 23-36.

[11]

T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Internat. J. Numer. Methods Engrg., 45 (1999), pp. 601-620.

[12]

T. Belytschko, N. Moës, S. Usui, and C. Parimi, Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 50 (2001), pp. 993-1013.

[13]

M. Bern and D. Eppstein, Mesh generation and optimal triangulation, in Computing in Euclidean Geometry, Lecture Notes Ser. Comput. 1, World Scientific, River Edge, NJ, 1992, pp. 23-90.

[14]

A. Brandt, Methods of Systematic Upscaling, Technical report MCS06-05, Weizmann Institute of Science, Rehovot, Israel, 2006.

[15]

M. C. Chang, C. L. Liu, and T. H. Chen, Polymethylmethacrylate augmentation of pedicle screw for osteoporotic spinal surgery: A novel technique, Spine, 33 (2008), pp. E317-E324.

[16]

S.-I. Chen, R.-M. Lin, and C.-H. Chang, Biomechanical investigation of pedicle screw-vertebrae complex: A finite element approach using bonded and contact interface conditions, Med. Eng. Phys., 25 (2003), pp. 275-282.

[17]

C. Daux, N. Moës, J. Dolbow, N. Sukumar, and T. Belytschko, Arbitrary branched and intersecting cracks with the extended finite element method, Internat. J. Numer. Methods Engrg., 48 (2000), pp. 1741-1760.

[18]

C. Decolon, Analysis of Composite Structures, Taylor & Francis, New York, 2002.

[19]

C. A. Duarte, I. Babuška, and J. T. Oden, Generalized finite element methods for three-dimensional structural mechanics problems, Comput. & Structures, 77 (2000), pp. 215-232.

[20]

C. A. Duarte, O. N. Hamzeh, T. J. Liszka, and W. W. Tworzydlo, A generalized finite element method for the simulation of three-dimensional dynamic crack propagation, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 2227-2262.

[21]

C. A. Duarte, D.-J. Kim, and D. M. Quaresma, Arbitrarily smooth generalized finite element approximations, Comput. Methods Appl. Mech. Engrg., 196 (2006), pp. 33-56.

[22]

A. Düster, J. Parvizian, Z. Yang, and E. Bank, The finite cell method for three-dimensional problems of solid mechanics, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 3768-3782.

[23]

K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, in Acta Numerica, 1995, Acta Numer., Cambridge University Press, Cambridge, UK, 1995, pp. 105-158.

[24]

R. E. Ewing, R. D. Lazarov, T. F. Russell, and P. S. Vassilevski, Local refinement via domain decomposition techniques for mixed finite element methods with rectangular Raviart-Thomas elements, in Domain Decomposition Methods for Partial Differential Equations, T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., SIAM, Philadelphia, 1990, pp. 98-114.

[25]

R. D. Falgout, An introduction to algebraic multigrid, Comput. Sci. Engrg., 8 (2006), pp. 24-33.

[26]

T. Gao, W. H. Zhang, J. H. Zhu, Y. J. Xu, and D. H. Bassir, Topology optimization of heat conduction problem involving design-dependent heat load effect, Finite Elem. Anal. Des., 44 (2008), pp. 805-813.

[27]

L. J. Gibson, Biomechanics of cellular solids, J. Biomech., 38 (2005), pp. 377-399.

[28]

R. Glowinski and Y. Kuznetsov, Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1498-1506.

[29]

Y. Guéguen, M. Le Ravalec, and L. Ricard, Upscaling: Effective medium theory, numerical methods and the fractal dream, Pure Appl. Geophys., 163 (2006), pp. 1175-1192.

[30]

R. M. Gulrajani, The forward and inverse problems of electrocardiography, IEEE Eng. Med. Biol. Mag., 17 (1998), pp. 84-101.

[31]

Y. Guo, M. Oritz, T. Belytschko, and E. A. Repetto, Triangular composite finite elements, Internat. J. Numer. Methods Engrg., 47 (2000), pp. 287-316.

[32]

W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Appl. Math. Sci. 95, Springer-Verlag, New York, 1994.

[33]

W. Hackbusch and S. A. Sauter, Composite finite elements for problems containing small geometric details. Part II: Implementation and numerical results, Comput. Vis. Sci., 1 (1997), pp. 15-25.

[34]

W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math., 75 (1997), pp. 447-472.

[35]

W. Hackbusch and S. A. Sauter, A new finite element approach for problems containing small geometric details, Arch. Math. (Brno), 34 (1998), pp. 105-117.

[36]

A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5537-5552.

[37]

T. P. Harrigan, M. Jasty, R. W. Mann, and W. H. Harris, Limitations of the continuum assumption in cancellous bone, J. Biomech., 21 (1988), pp. 269-275.

[38]

K. Höllig, C. Apprich, and A. Streit, Introduction to the Web-method and its applications, Adv. Comput. Math., 23 (2005), pp. 215-237.

[39]

K. Höllig, U. Reif, and J. Wipper, Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39 (2001), pp. 442-462.

[40]

S. J. Hollister, D. P. Fyhrie, K. J. Jepsen, and S. A. Goldstein, Application of homogenization theory to the study of trabecular bone mechanics, J. Biomech., 24 (1991), pp. 825-839.

[41]

S. J. Hollister and N. Kikuchi, A comparison of homogenization and standard mechanics analyses for periodic porous composites, Comput. Mech., 10 (1992), pp. 73-95.

[42]

T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.

[43]

R. Huang, N. Sukumar, and J.-H. Prévost, Modeling quasi-static crack growth with the extended finite element method. Part II: Numerical applications, Int. J. Solids Struct., 40 (2003), pp. 7539-7552.

[44]

M. A. Hyman, Non-iterative numerical solution of boundary value problems, Appl. Sci. Res. B, 2 (1952), pp. 325-351.

[45]

T. L. Kline, M. Zamir, and E. L. Ritman, Accuracy of microvascular measurements obtained from micro-CT images, Ann. Biomed. Eng., 38 (2010), pp. 2851-2864.

[46]

V. Kosmopoulos and T. S. Keller, Damage-based finite-element vertebroplasty simulations, Eur. Spine J., 13 (2004), pp. 617-625.

[47]

Z. Li, The immersed interface method using a finite element formulation, Appl. Numer. Math., 27 (1998), pp. 253-267.

[48]

Z. Li, An overview of the immersed interface method and its applications, Taiwanese J. Math., 7 (2003), pp. 1-49.

[49]

Z. Li, T. Lin, and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 1996 (2003), pp. 61-98.

[50]

F. Liehr, T. Preusser, M. Rumpf, S. Sauter, and L. O. Schwen, Composite finite elements for $3$D image based computing, Comput. Vis. Sci., 12 (2009), pp. 171-188.

[51]

W. E. Lorensen and H. E. Cline, Marching cubes: A high resolution $3$D surface construction algorithm, SIGGRAPH Comput. Graph., 21 (1987), pp. 163-169.

[52]

A.-M. Matache and C. Schwab, Two-scale FEM for homogenization problems, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 537-572.

[53]

J. M. Melenk, On Generalized Finite Element Methods, Ph.D. thesis, University of Maryland, College Park, MD, 1995.

[54]

W. F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software, 15 (1989), pp. 326-347.

[55]

J. D. Moulton, J. E. Dendy Jr., and J. M. Hyman, The black box multigrid numerical homogenization algorithm, J. Comput. Phys., 142 (1998), pp. 80-108.

[56]

J. T. Oden, C. A. M. Duarte, and O. C. Zienkiewicz, A new cloud-based $hp$ finite element method, Comput. Methods Appl. Mech. Engrg., 153 (1998), pp. 117-126.

[57]

S. J. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.

[58]

J. Parvizian, A. Düster, and E. Rank, Finite cell method, Comput. Mech., 41 (2007), pp. 121-133.

[59]

A. Pegoretti, L. Fambri, G. Zappini, and M. Bianchetti, Finite element analysis of a glass fibre reinforced composite endodontic post, Biomaterials, 23 (2002), pp. 2667-2682.

[60]

I. Ramière, P. Angot, and M. Belliard, A fictitious domain approach with spread interface for elliptic problems with general boundary conditions, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 766-781.

[61]

M. Rech, S. Sauter, and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math., 102 (2006), pp. 681-708.

[62]

A. Rohlmann, T. Zander, and G. Bergmann, Spinal loads after osteoporotic vertebral fractures treated by vertebroplasty or kryoplasty, Eur. Spine J., 15 (2006), pp. 1255-1264.

[63]

M. Rumpf, L. O. Schwen, H.-J. Wilke, and U. Wolfram, Numerical homogenization of trabecular bone specimens using composite finite elements, in Proceedings of the Fraunhofer 1st Conference on Multiphysics Simulation—Advanced Methods for Industrial Engineering, Bonn, Germany, 2010.

[64]

V. K. Saul'ev, On solution of some boundary value problems on high performance computers by fictitious domain method, Siberian Math. J., 4 (1963), pp. 912-925.

[65]

S. A. Sauter and R. Warnke, Composite finite elements for elliptic boundary value problems with discontinuous coefficients, Computing, 77 (2006), pp. 29-55.

[66]

L. O. Schwen, Composite Finite Elements for Trabecular Bone Microstructures, Ph.D. thesis, University of Bonn, Bonn, Germany, 2010.

[67]

L. O. Schwen, T. Preusser, and M. Rumpf, Composite finite elements for $3$D elasticity with discontinuous coefficients, in Proceedings of the 16th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, Ulm, Germany, to appear.

[68]

L. O. Schwen, U. Wolfram, H.-J. Wilke, and M. Rumpf, Determining effective elasticity parameters of microstructured materials, in Proceedings of the 15th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, Ulm, Germany, 2008, pp. 41-62.

[69]

J. R. Shewchuk, What is a good linear element? Interpolation, conditioning, and quality measures, in Proceedings of the 11th International Meshing Roundtable, Ithaca, New York, Sandia National Laboratories, 2002, pp. 115-126.

[70]

G. H. Shortley and R. Weller, The numerical solution of Laplace's equation, J. Appl. Phys., 9 (1938), pp. 334-344.

[71]

M. Stolarska, D. L. Chopp, N. Moës, and T. Belytschko, Modelling crack growth by level sets in the extended finite element method, Internat. J. Numer. Methods Engrg., 51 (2001), pp. 943-960.

[72]

T. Strouboulis, K. Copps, and I. Babuška, The generalized finite element method, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 4081-4193.

[73]

K. Stüben, A review of algebraic multigrid, J. Comput. Appl. Math., 128 (2001), pp. 281-309.

[74]

L. Tartar, An introduction to the homogenization method in optimal design, in Optimal Shape Design, Lecture Notes in Math. 1740, Springer, New York, 2000, pp. 47-156.

[75]

S.-H. Teng and C. W. Wong, Unstructured mesh generation: Theory, practice and applications, Internat. J. Comput. Geom. Appl., 10 (2000), pp. 227-266.

[76]

P. Thoutireddy, J. F. Molinari, E. A. Repetto, and M. Oritz, Tetrahedral composite finite elements, Internat. J. Numer. Methods Engrg., 53 (2002), pp. 1337-1351.

[77]

G. M. Treece, R. W. Prager, and A. H. Gee, Regularised marching tetrahedra: Improved iso-surface extraction, Comput. Graph., 23 (1999), pp. 583-598.

[78]

U. Trottenberg, C. Osterlee, and A. Schüller, Multigrid, Academic Press, San Diego, London, 2001.

[79]

B. van Rietbergen, R. Müller, D. Ulrich, P. Rüegsegger, and R. Huiskes, Tissue stresses and strain in trabeculae of canine proximal femur can be quantified from computer reconstructions, J. Biomech., 32 (1999), pp. 165-173.

[80]

B. van Rietbergen, A. Odgaard, J. Kabel, and R. Huiskes, Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture, J. Biomech., 29 (1996), pp. 1653-1657.

[81]

T. Vdovina, S. E. Minkoff, and O. Korostyshevskaya, Operator upscaling for the acoustic wave equation, Multiscale Model. Simul., 4 (2005), pp. 1305-1338.

[82]

A. Wittek, R. Kikinis, S. K. Warfield, and K. Miller, Brain shift computation using a fully nonlinear biomechanical model, in Medical Image Computing and Computer-Assisted Intervention—MICCAI 2005, Lecture Notes in Comput. Sci. 3750, J. Duncan and G. Gerig, eds., Springer-Verlag, Berlin, Heidelberg, 2005, pp. 583-590.

[83]

U. Wolfram, H.-J. Wilke, and P. K. Zysset, Rehydration of vertebral trabecular bone: Influences on its anisotropy, its stiffness and the indentation work with a view to age, gender and vertebral level, Bone, 46 (2010), pp. 348-354.

[84]

J. Xu, Theory of Multilevel Methods, Ph.D. thesis, Cornell University, Ithaca, New York, 1989.

[85]

W. Zhe, M. Hui, G. Manling, and G. Dong, A simulation of the abnormal EEG morphology by the $3$-D finite element method, in Proceedings of the 2005 27th Annual Conference of the IEEE Engineering in Medicine and Biology Society, Shanghai, China, 2005, pp. 3620-3623.

[86]

P. K. Zysset, R. W. Goullet, and S. J. Hollister, A global relationship between trabecular bone morphology and homogenized elastic properties, J. Biomed. Eng., 120 (1998), pp. 640-646.

Cited By

Marcián PNarra NBorák LChamrad JWolff J(2022)Biomechanical performance of cranial implants with different thicknesses and material propertiesComputers in Biology and Medicine10.1016/j.compbiomed.2019.04.016109:C(43-52)Online publication date: 20-Apr-2022
https://dl.acm.org/doi/10.1016/j.compbiomed.2019.04.016
Han DWang PHe XLin TWang J(2016)A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma-lunar surface interactionsJournal of Computational Physics10.1016/j.jcp.2016.05.057321:C(965-980)Online publication date: 15-Sep-2016
https://dl.acm.org/doi/10.1016/j.jcp.2016.05.057

Index Terms

3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients
1. Applied computing
  1. Physical sciences and engineering
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations
    2. Numerical analysis
      1. Computations in finite fields
      2. Numerical differentiation

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 33, Issue 5

^† Special Section: 2010 Copper Mountain Conference

2011

972 pages

ISSN:1064-8275

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Society for Industrial and Applied Mathematics

United States

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Published: 01 September 2011

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Cited By

Marcián PNarra NBorák LChamrad JWolff J(2022)Biomechanical performance of cranial implants with different thicknesses and material propertiesComputers in Biology and Medicine10.1016/j.compbiomed.2019.04.016109:C(43-52)Online publication date: 20-Apr-2022
https://dl.acm.org/doi/10.1016/j.compbiomed.2019.04.016
Han DWang PHe XLin TWang J(2016)A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma-lunar surface interactionsJournal of Computational Physics10.1016/j.jcp.2016.05.057321:C(965-980)Online publication date: 15-Sep-2016
https://dl.acm.org/doi/10.1016/j.jcp.2016.05.057

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