Nonlinear manifold learning for meshfree finite deformation thin-shell analysis

International Journal for Numerical Methods in Engineering

Volumn 93, Issue 7, 2013, Pages 685-713

  Millán, Daniel ^a   Rosolen, Adrian ^a   Arroyo, Marino ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Maximum entropy approximants; Meshfree methods; Nonlinear dimensionality reduction; Partition of unity; Point set surfaces; Shells

Indexed keywords

APPROXIMANTS; MESH-FREE METHOD; NONLINEAR DIMENSIONALITY REDUCTION; PARTITION OF UNITY; POINT-SET SURFACES; SHELLS;

BENCHMARKING; MAXIMUM ENTROPY METHODS; NONLINEAR ANALYSIS; TOPOLOGY;

TWO DIMENSIONAL;

EID: 84873084248     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.4403     Document Type: Article

References (79)

1. MacNeal R, Harder R. A proposed standard set of problems to test finite element accuracy. Finite Element in Analysis and Design 1985; 1(1):3-20.
2. Bucalem M, Bathe J. Higher-order MITC general shell elements. International Journal for Numerical Methods in Engineering 1993; 36(21):3729-3754.
3. Simo J, Fox D. On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering 1989; 72:267-304.
4. Krysl P, Belytschko T. Analysis of thin shells by the element-free Galerkin method. International Journal of Solids and Structures 1996; 33(20-22):3057-3078.
5. Cirak F, Ortiz M, Schröder P. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. International Journal for Numerical Methods in Engineering 2000; 47(12):2039-2072.
6. 1-conforming subdivision elements for finite deformation thin-shell analysis. International Journal for Numerical Methods in Engineering 2001; 51(7):813-833.
7. Hughes T, Cottrell J, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 2005; 194:4135-4195.
8. Cottrell J, Hughes T, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons, Ltd: Hoboken, NJ, 2009.
9. Cottrell J, Reali A, Bazilevs Y, Hughes T. Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering 2006; 195(41-43):5257-5296.
10. Kiendl J, Bletzinger KU, Linhard J, Wüchner R. Isogeometric shell analysis with Kirchhoff-Love elements. Computer Methods in Applied Mechanics and Engineering 2009; 198(49-52):3902-3914.
11. Bazilevs Y, Calo V, Cottrell J, Evans J, Hughes T, Lipton S, Scott M, Sederberg T. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering 2010; 199(5-8):229-263.
12. Engel G, Garikipati K, Hughes T, Larson M, Mazzei L, Taylor R. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Computer Methods in Applied Mechanics and Engineering 2002; 191:3669-3750.
13. 0 discontinuous Galerkin formulation for Kirchhoff plates. Computer Methods in Applied Mechanics and Engineering 2007; 196(35-36):3370-3380.
14. Wells G, Dung N. Geometrically nonlinear formulation for thin shells without rotation degrees of freedom. Computer Methods in Applied Mechanics and Engineering 2008; 197(35-36):3370-3380.
15. Noels L, Radovitzky R. A new discontinuous Galerkin method for Kirchhoff-Love shells. Computer Methods in Applied Mechanics and Engineering 2008; 197(33-40):2901-2929.
16. Noels L. A discontinuous Galerkin formulation of non-linear Kirchhoff-Love shells. International Journal for Numerical Methods in Engineering 2009; 78(3):296-323.
17. Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W. Surface reconstruction from unorganized points. In SIGGRAPH '92: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques. ACM: New York, NY, 1992; 71-78.
18. Alexa M, Behr J, Cohen-Or D, Fleishman S, Levin D, Silva C. Point set surfaces. In VIS '01: Proceedings of the Conference on Visualization '01. IEEE Computer Society: Washington, DC, 2001; 21-28.
19. Pauly M. Point primitives for interactive modeling and processing of 3D geometry. PhD Thesis, Federal Institute of Technology (ETH) of Zurich, 2003.
20. Levin D. Mesh-independent surface interpolation. In Geometric Modeling for Scientific Visualization, Brunnett G, Hamann B, Mueller H, Linsen L (eds). Springer-Verlag: Heidelberg, 2003; 37-49.
21. Ohtake Y, Belyaev A, Alexa M, Turk G, Seidel H. Multi-level partition of unity implicits. ACM Transactions on Graphics (Proc. SIGGRAPH 2003) 2003; 22:463-470.
22. Alexa M, Behr J, Cohen-Or D, Fleishman S, Levin D, Silva C. Computing and rendering point set surfaces. Transactions on Visualization and Computer Graphics 2003; 9(1):3-15.
23. Amenta N, Kil Y. Defining point-set surfaces. ACM Transactions on Graphics 2004; 23(3):264-270.
24. Alexa M, Gross M, Pauly M, Pfister H, Stamminger M, Zwicker M. Point-based computer graphics. SIGGRAPH 2004 Course Notes 2004.
25. Noguchi H, Kawashima T, Miyamura T. Element free analyses of shell and spatial structures. International Journal for Numerical Methods in Engineering 2000; 47(6):1215-1240.
26. Chen J, Wang D. A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates. International Journal for Numerical Methods in Engineering 2006; 68(2):151-172.
27. Rabczuk T, Areias P, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering 2007; 72(5):524-548.
28. Millán D, Rosolen A, Arroyo M. Thin shell analysis from scattered points with maximum-entropy approximants. International Journal for Numerical Methods in Engineering 2011; 85(6):723-751.
29. Belytschko T, Stolarski H, Liu W, Carpenter N, Ong J. Stress projection for membrane and shear locking in shell finite-elements. Computer Methods in Applied Mechanics and Engineering 1985; 51:221-258.
30. Arroyo M, Ortiz M. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering 2006; 65(13):2167-2202.
31. Pearson K. On lines and planes of closest fit to systems of points in space. Philosophical Magazine 1901; 2(6):559-572.
32. Hotelling H. Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 1933; 24(7):498-520.
33. Karhunen K. Zur spektraltheorie stochastischer prozesse. Annales Academiae Scientiarum Fennicae 1946; 34:1-7.
34. Loève M. Probability Theory. University series in higher mathematics, Van Nostrand: New Jersey, 1955.
35. Lorenz E. Empirical orthogonal functions and statistical weather prediction. Statistical Forecasting Project, Scientific Report 1, MIT, Department of Meteorology, Cambridge, MA, 1956.
36. Lumley JL. The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation, Yaglom AM, Tatarski VI (eds). Nauka: Moscow, 1967; 166-178.
37. Jolliffe IT. Principal Component Analysis, 2nd edn, Springer Series in Statistics, Springer: New York, NY, 2002.
38. Jain AK, Duin R, Mao J. Statistical pattern recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 2000; 22(1):4-37.
39. Zhang Z, Zha H. Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM Journal of Scientific Computing 2005; 26(1):313-338.
40. Lall S, Krysl P, Marsden J. Structure-preserving model reduction for mechanical systems. Physica D 2003; 184:304-318.
41. Niroomandi S, Alfaro I, Cueto E, Chinesta F. Model order reduction for hyperelastic materials. International Journal for Numerical Methods in Engineering 2010; 81(9):1180-1206.
42. Torgerson W. Multidimensional scaling: I. Theory and method. Psychometrika 1952; 17:401-419.
43. Cox TF, Cox MAA. Multidimensional Scaling, 2nd edn. Chapman & Hall: Boca Raton, FL, 2001.
44. Lee J, Verleysen M. Nonlinear Dimensionality Reduction, Information Science and Statistics. Springer: New York, NY, 2007.
45. Sammon JW. A nonlinear mapping for data structure analysis. IEEE Transactions on Computers 1969; 18:401-409.
46. Hérault J, Oliva A, Guérin-Dugué A. Scene categorisation by curvilinear component analysis of low frequency spectra. Proceedings of the 5th European Symposium on Artificial Neural Networks, Bruges, Belgium, 1997; 91-96.
47. Hérault J, Jausions-Picaud C, Guérin-Dugué A. Curvilinear component analysis for high-dimensional data representation: I. Theoretical aspects and practical use in the presence of noise. In Engineering Applications of Bio-Inspired Artificial Neural Networks, Vol.1607, Mira J, Sánchez-Andrés J (eds), Lecture Notes in Computer Science. Springer: Berlin/Heidelberg 1999; 625-634.
48. vanderMaaten LJP, Postma EO, vanden Herik HJ. Dimensionality reduction: a comparative review. Technical Report TiCC-TR 2009-005, Tilburg University, 2009.
49. Tenenbaum J, deSilva V, Langford J. A global geometric framework for nonlinear dimensionality reduction. Science 2000; 290(5500):2319-2323.
50. Roweis S, Saul L. Nonlinear dimensionality reduction by locally linear embedding. Science 2000; 290(5500):2323-2326.
51. Roweis S, Saul L. Think globally, fit locally: unsupervised learning of low dimensional manifolds. Journal of Machine Learning Research 2003; 4:119-155.
52. de Silva V, Tenenbaum J. Global versus local methods in nonlinear dimensionality reduction. In Advances in Neural Information Processing Systems, Vol.15. MIT Press: Cambridge, MA, 2003; 705-712.
53. Choi H, Choi S. Robust kernel Isomap. Pattern Recognition 2007; 40(3):853-862.
54. Donoho D, Grimes C. Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Sciences 2003; 100(10):5591-5596.
55. Belkin M, Niyogi P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 2003; 15:1373-1396.
56. Chang H, Yeung DY. Robust locally linear embedding. Pattern Recognition 2006; 39(6):1053-1065.
57. Zhang Z, Wang J. MLLE: modified locally linear embedding using multiple weights. In Advances in Neural Information Processing Systems, Vol.19, Schölkopf B, Platt J, Hoffman T (eds). MIT Press: Cambridge, MA, 2007; 1593-1600.
58. Gámez AJ, Zhou CS, Timmermann A, Kurths J. Nonlinear dimensionality reduction in climate data. Nonlinear Processes in Geophysics 2004; 11:393-398.
59. Das P, Moll M, Stamati H, Kavraki L, Clementi C. Low-dimensional, free-energy landscapes of protein-folding reactions by nonlinear dimensionality reduction. Proceedings of the National Academy of Sciences 2006; 103(26):9885-9890.
60. Brown WM, Martin S, Pollock SN, Coutsias EA, Watson JP. Algorithmic dimensionality reduction for molecular structure analysis. The Journal of Chemical Physics 2008; 129(6):064118.
61. Vanderplas J, Connolly A. Reducing the dimensionality of data: locally linear embedding of Sloan galaxy spectra. The Astronomical Journal 2009; 138(5):1365-1379.
62. doCarmo MP. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
63. de Ridder D, Duin R. 2002. Locally linear embedding for classification. Technical Report PH-2002-0, Department of Imaging Science & Technology, Delft University of Technology, Delft.
64. Grassberger P, Procaccia I. Measuring the strangeness of strange attractors. Physica D 1983; 9(1-2):189-208.
65. Karypis G, Kumar V. METIS: Unstructured graph partitioning and sparse matrix ordering system, version 4.0, 1998. Available from: http://www.cs.umn.edu/metis [Accessed on 2011].
66. Sha F, Saul L. Analysis and extension of spectral methods for nonlinear dimensionality reduction. In Proceedings of the 22nd International Conference on Machine Learning, ICML '05. ACM: New York, NY, 2005; 784-791.
67. Mount D, Arya S. ANN: a library for approximate nearest neighbor searching, version 1.1, 2010. Available from: http://www.cs.umd.edu/mount/ANN [Accessed on 2011].
68. Simo J, Fox D, Rifai M. On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering 1989; 73:53-92.
69. Ciarlet PG. Mathematical Elasticity: Theory of Shells, Vol.III: North-Holland, 2000.
70. Sze KY, Liu XH, Lo SH. Popular benchmark problems for geometric nonlinear analysis of shells. Journal Finite Elements in Analysis and Design 2004; 40:1551-1569.
71. Millán D. Geometrically nonlinear meshfree thin-shell analysis in the context of Kirchhoff-Love theory, 2011. Available from: http://www.youtube.com/RDanielMillan [Accessed on 2011].
72. Green S, Turkiyyah G. Second-order accurate constraint formulation for subdivision finite element simulation of thin shells. International Journal for Numerical Methods in Engineering 2004; 61(3):380-405.
73. Turk G, Levoy M. Zippered polygon meshes from range images. In Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '94. ACM: New York, NY, 1994; 311-318.
74. Rosolen A, Millán D, Arroyo M. Second order convex maximum entropy approximants with applications to high order PDE. International Journal for Numerical Methods in Engineering, under revision, 2012.
75. Bompadre A, Perotti L, Cyron C, Ortiz M. Convergent meshfree approximation schemes of arbitrary order and smoothness. Computer Methods in Applied Mechanics and Engineering 2012:221-222:83-183.
76. Cirak F, Long Q. Subdivision shells with exact boundary control and non-manifold geometry. International Journal for Numerical Methods in Engineering 2011; 88(9):897-923.
77. Rosolen A, Arroyo M. Blending isogeometric analysis and maximum entropy meshfree approximants. In preparation, 2012.
78. Nocedal J, Wright S. Numerical Optimization. Springer: New York, 1999.
79. Conn A, Gould N, Toint P. A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM Journal on Numerical Analysis 1991; 28(2):545-572.