Доступ предоставлен для: Guest
Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
International Journal for Multiscale Computational Engineering
Импакт фактор: 1.016 5-летний Импакт фактор: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Печать: 1543-1649
ISSN Онлайн: 1940-4352

Том 17, 2019 Том 16, 2018 Том 15, 2017 Том 14, 2016 Том 13, 2015 Том 12, 2014 Том 11, 2013 Том 10, 2012 Том 9, 2011 Том 8, 2010 Том 7, 2009 Том 6, 2008 Том 5, 2007 Том 4, 2006 Том 3, 2005 Том 2, 2004 Том 1, 2003

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v8.i4.40
pages 397-409

Adaptive Multiwavelet-Hierarchical Method for Multiscale Computation

Youming Wang
State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, People’s Republic of China, 710049
Xuefeng Chen
State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, People’s Republic of China, 710049
Zhengjia He
State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, People’s Republic of China, 710049

Краткое описание

An adaptive multiwavelet-hierarchical method characterized by high convergent rate and flexible adaptive strategy is proposed for multiscale computation of field problems. According to the Strang--Fix condition, the convergence rate of the finite element multiwavelet method is determined by the approximation order of scaling functions in the same level of multiwavelet refinement. To raise the approximation order of scaling functions, finite element multiwavelets are combined with hierarchical bases to construct a new multilevel multiwavelet-hierarchical space. An adaptive strategy for multiwavelet-hierarchical refinement is presented based on new error estimation in the form of multiwavelets and hierarchical bases, which leaves much freedom for the problem-oriented selection of multiwavelets or hierarchical functions. Numerical examples demonstrate that the proposed method is an accurate and efficient tool in solving the field problems with singularities or changes in high gradients.


  1. Abraham, F. F., Broughton, J. Q., and Bernstein, N., Spanning the length scale in dynamic simulation. DOI: 10.1063/1.168756

  2. Amaratunga, K. and Williams, J. R., Wavelet-Galerkin solutions for one-dimensional partial differential equations. DOI: 10.1002/nme.1620371602

  3. Amaratunga, K. and Sudarshan, R., Multiresolution modeling with operator-customized wavelets derived from finite elements. DOI: 10.1016/j.cma.2005.05.012

  4. Bathe, K. J., Finite Element Procedures.

  5. Bensoussan, A. ., Lions, J. L., and Papanicolaou, G., Asymptotic analysis for periodic structures, Studies in mathematics and its applications.

  6. Bertoluzza, S. and Naldi, G., A wavelet collocation method for the numerical solution of partial differential equations. DOI: 10.1006/acha.1996.0001

  7. Carnicer, J., Dahmen, W., and Pena, J., Local decompositions of refinable spaces. DOI: 10.1006/acha.1996.0012

  8. Castrillon-Candas, J. and Amaratunga, K., Spatially adapted multiwavelets and sparse representation of integral equations on general geometries. DOI: 10.1137/S1064827501371238

  9. Chen, J. S., Teng, H., and Nakano, A., Wavelet projection method for coarse graining of DNA molecules.

  10. Chen, X. F. and Yang, S. J., The construction of wavelet finite element and its application. DOI: 10.1016/S0168-874X(03)00077-5

  11. Dahmen, W. and Stevenson, R., Element-by-element construction of wavelets satisfying stability and moment conditions. DOI: 10.1137/S0036142997330949

  12. D’Heedene, S., Amaratunga, K., and Castrillon-Candas, J., Generalized hierarchical bases: A Wavelet–Ritz–Galerkin framework for Lagrangian FEM. DOI: 10.1108/02644400510572398

  13. Efendiev, Y. and Hou, T. Y., Multiscale Finite Element Methods: Theory and Applications.

  14. Fish, J., Multiscale Methods: Bridging the Scales in Science and Engineering.

  15. Fish, J. and Fan, R., Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading. DOI: 10.1002/nme.2355

  16. Fish, J. and Yuan, Z., Multiscale enrichment based on partition of unity for nonperiodic fields and nonlinear problems. DOI: 10.1007/s00466-006-0095-0

  17. Hu, W. and Chen, J. S., Multiscale method for quantum mechanics.

  18. Liu, W. K., Karpov, E. G., and Park, H. S., Nano Mechanics and Materials: Theory, Multiscale Methods and Applications.

  19. Liu, W. K., Karpov, E. G., Zhang, S., and Park, H. S., An introduction to computational nanomechanics and materials. DOI: 10.1016/j.cma.2003.12.008

  20. McVeigh, C. and Liu, W. K., Multiresolution modeling of ductile reinforced brittle composites. DOI: 10.1016/j.jmps.2008.10.015

  21. Mehraeen, S. and Chen, J. S., Wavelet Galerkin method in multiscale homogenization of heterogeneous materials. DOI: 10.1002/nme.1554

  22. Mehra, M. and Kevlahan, N. K.-R., An adaptive wavelet collocation method for the solution of partial differential equations on the sphere. DOI: 10.1016/j.jcp.2008.02.004

  23. Nuggehally, M. A., Shephard, M. S., Picu, C. R., et al., Adaptive model selection procedure for concurrent multiscale problems. DOI: 10.1615/IntJMultCompEng.v5.i5.20

  24. Pinho, P., Ferreira, P. J. S. G., and Pereira, J. R., Multiresolution analysis using biorthogonal and interpolating wavelets. DOI: 10.1109/APS.2004.1330469

  25. Reddy, J. N., An Introduction to the Finite Element Method.

  26. Rudd, R. E. and Broughton, J. Q., Coarse-grained molecular dynamics and atomic limit of finite elements. DOI: 10.1103/PhysRevB.58.R5893

  27. Shenoy, V. B., Miller, R., Tadmor, E. B., et al., Quasicontinuum models of interfacial structure and deformation. DOI: 10.1103/PhysRevLett.80.742

  28. Solin, P., Segeth, K., and Dolezel, I., Higher-Order Finite Element Methods.

  29. Strang, G., Wavelets and dilation equations: A brief introduction. DOI: 10.1137/1031128

  30. Sweldens, W., The lifting scheme: A custom-design construction of biorthogonal wavelets. DOI: 10.1006/acha.1996.0015

  31. Sweldens, W., The lifting scheme: A construction of secondgeneration wavelets. DOI: 10.1137/S0036141095289051

  32. Unser, M., Approximation power of biorthogonal wavelet expansions. DOI: 10.1109/78.489025

  33. Vasilyev, O. V., Paolucci, S., and Sen, M., A multilevel wavelet collocation method for solving partial differential equations in a finite domain. DOI: 10.1006/jcph.1995.1147

  34. Vasilyev, O. V. and Bowman, C., Second-generation wavelet collocation method for the solution of partial differential equations. DOI: 10.1006/jcph.2000.6638

  35. Vasilyev, O. V. and Kevlahan, N. K-R., An adaptive multilevel wavelet collocation method for elliptic problems. DOI: 10.1016/j.jcp.2004.12.013

  36. Xiang, J. W. and Chen, X. F., The construction of 1D wavelet finite elements for structural analysis. DOI: 10.1007/s00466-006-0102-5

  37. Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method.