×

zbMATH — the first resource for mathematics

Thin shell analysis from scattered points with maximum-entropy approximants. (English) Zbl 1217.74147
Summary: We present a method to process embedded smooth manifolds using sets of points alone. This method avoids any global parameterization and hence is applicable to surfaces of any genus. It combines three ingredients: (1) the automatic detection of the local geometric structure of the manifold by statistical learning methods; (2) the local parameterization of the surface using smooth meshfree (here maximum-entropy) approximants; and (3) patching together the local representations by means of a partition of unity. Mesh-based methods can deal with surfaces of complex topology, since they rely on the element-level parameterizations, but cannot handle high-dimensional manifolds, whereas previous meshfree methods for thin shells consider a global parametric domain, which seriously limits the kinds of surfaces that can be treated. We present the implementation of the method in the context of Kirchhoff – Love shells, but it is applicable to other calculations on manifolds in any dimension. With the smooth approximants, this fourth-order partial differential equation is treated directly. We show the good performance of the method on the basis of the classical obstacle course. Additional calculations exemplify the flexibility of the proposed approach in treating surfaces of complex topology and geometry.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K25 Shells
Software:
Powercrust
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexa, VIS ’01: Proceedings of the Conference on Visualization ’01 pp 21– (2001) · doi:10.1109/VISUAL.2001.964489
[2] Pauly M Point primitives for interactive modeling and processing of 3d geometry 2003
[3] Levin, Geometric Modeling for Scientific Visualization pp 37– (2003)
[4] Levoy M Whitted T The use of points as displays primitives 1985
[5] Hoppe, SIGGRAPH ’92: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques pp 71– (1992) · doi:10.1145/133994.134011
[6] Ohtake, Multi-level partition of unity implicits, ACM Transactions on Graphics (Proceedings of SIGGRAPH 2003 ) 22 pp 463– (2003) · doi:10.1145/1201775.882293
[7] Levin, The approximation power of moving least-squares, Mathematics of Computation 67 (224) pp 1517– (1998) · Zbl 0911.41016 · doi:10.1090/S0025-5718-98-00974-0
[8] Alexa, Computing rendering point set surfaces, Transactions on Visualization and Computer Graphics 9 (1) pp 3– (2003) · Zbl 05108085 · doi:10.1109/TVCG.2003.1175093
[9] Amenta, Defining point-set surfaces, ACM Transactions on Graphics 23 (3) pp 264– (2004) · Zbl 05457185 · doi:10.1145/1015706.1015713
[10] Alexa M Gross M Pauly M Pfister H Stamminger M Zwicker M Point-based computer graphics 2004
[11] Krysl, Analysis of thin shells by the element-free Galerkin method, International Journal of Solids and Structures 33 (20-22) pp 3057– (1996) · Zbl 0929.74126 · doi:10.1016/0020-7683(95)00265-0
[12] Noguchi, Element free analyses of shell and spatial structures, International Journal for Numerical Methods in Engineering 47 (6) pp 1215– (2000) · Zbl 0970.74079 · doi:10.1002/(SICI)1097-0207(20000228)47:6<1215::AID-NME834>3.0.CO;2-M
[13] Chen, A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates, International Journal for Numerical Methods in Engineering 68 pp 151– (2006) · Zbl 1130.74055 · doi:10.1002/nme.1701
[14] Rabczuk, A meshfree thin shell method for non-linear dynamic fracture, International Journal for Numerical Methods in Engineering 72 pp 525– (2007) · Zbl 1194.74537 · doi:10.1002/nme.2013
[15] MacNeal, A proposed standard set of problems to test finite element accuracy, Finite Element in Analysis and Design 1 (1) pp 3– (1985) · doi:10.1016/0168-874X(85)90003-4
[16] Bucalem, Higher-order Mitc general shell elements, International Journal for Numerical Methods in Engineering 36 pp 3729– (1993) · Zbl 0800.73466 · doi:10.1002/nme.1620362109
[17] Simo, On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization, Computer Methods in Applied Mechanics and Engineering 72 pp 267– (1989) · Zbl 0692.73062 · doi:10.1016/0045-7825(89)90002-9
[18] Cirak, Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, International Journal for Numerical Methods in Engineering 47 pp 2039– (2000) · Zbl 0983.74063 · doi:10.1002/(SICI)1097-0207(20000430)47:12<2039::AID-NME872>3.0.CO;2-1
[19] Simo, On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects, Computer Methods in Applied Mechanics and Engineering 73 pp 53– (1989) · Zbl 0724.73138 · doi:10.1016/0045-7825(89)90098-4
[20] Yang, A survey of recent shell finite elements, International Journal for Numerical Methods in Engineering 47 (1-3) pp 101– (2000) · Zbl 0987.74001 · doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<101::AID-NME763>3.0.CO;2-C
[21] Cirak, Fully C1-conforming subdivision elements for finite deformation thin-shell analysis, International Journal for Numerical Methods in Engineering 51 (7) pp 813– (2001) · Zbl 1039.74045 · doi:10.1002/nme.182.abs
[22] Engel, 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 191 pp 3669– (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4
[23] Wells, A C0 discontinuous Galerkin formulation for Kirchhoff plates, Computer Methods in Applied Mechanics and Engineering 196 (35-36) pp 3370– (2007) · Zbl 1173.74447 · doi:10.1016/j.cma.2007.03.008
[24] Noels, A new discontinuous Galerkin method for Kirchhoff-Love shells, Computer Methods in Applied Mechanics and Engineering 197 (33-40) pp 2901– (2008) · Zbl 1194.74456 · doi:10.1016/j.cma.2008.01.018
[25] Arroyo, Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, International Journal for Numerical Methods in Engineering 65 pp 2167– (2006) · Zbl 1146.74048 · doi:10.1002/nme.1534
[26] Cyron, Smooth, second order, non-negative meshfree approximants selected by maximum entropy, International Journal for Numerical Methods in Engineering 79 (13) pp 1605– (2009) · Zbl 1176.74208 · doi:10.1002/nme.2597
[27] Belytschko, Stress projection for membrane and shear locking in shell finite-elements, Computer Methods in Applied Mechanics and Engineering 51 pp 221– (1985) · Zbl 0581.73091 · doi:10.1016/0045-7825(85)90035-0
[28] Jain, Statistical pattern recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence 22 (1) pp 4– (2000) · doi:10.1109/34.824819
[29] Zhang, Principal manifolds and nonlinear dimensionality reduction via tangent space alignment, SIAM Journal on Scientific Computing 26 (1) pp 313– (2005) · Zbl 1077.65042 · doi:10.1137/S1064827502419154
[30] Lall, Structure-preserving model reduction for mechanical systems, Physica D 184 pp 304– (2003) · Zbl 1041.70011 · doi:10.1016/S0167-2789(03)00227-6
[31] Niroomandi, Model order reduction for hyperelastic materials, International Journal for Numerical Methods in Engineering (2009) · Zbl 1183.74365 · doi:10.1002/nme.2733
[32] do Carmo, Differential Geometry of Curves and Surfaces (1976)
[33] Sukumar, Construction of polygonal interpolants: a maximum entropy approach, International Journal for Numerical Methods in Engineering 61 (12) pp 2159– (2004) · Zbl 1073.65505 · doi:10.1002/nme.1193
[34] Hughes, Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 pp 4135– (2005) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[35] González, A higher order method based on local maximum entropy approximation, International Journal for Numerical Methods in Engineering (2010) · Zbl 1197.74193 · doi:10.1002/nme.2855
[36] Rosolen, On the optimum support size in meshfree methods: a variational adaptivity approach with maximum entropy approximants, International Journal for Numerical Methods in Engineering (2009) · Zbl 1188.74086 · doi:10.1002/nme.2793
[37] Fernández-Méndez, Imposing essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering 193 pp 1257– (2004) · Zbl 1060.74665 · doi:10.1016/j.cma.2003.12.019
[38] Gupta MR An information theory approach to supervised learning 2003
[39] Stroud, Approximate Calculation of Multiple Integrals (1971) · Zbl 0379.65013
[40] Ciarlet, Mathematical Elasticity, Volume III: Theory of Shells (2000)
[41] Wang, A hermite reproducing kernel approximation for thin plate analysis with sub-domain stabilized conforming integration, International Journal for Numerical Methods in Engineering 74 pp 368– (2008) · Zbl 1159.74460 · doi:10.1002/nme.2175
[42] Green, Second-order accurate constraint formulation for subdivision finite element simulation of thin shells, International Journal for Numerical Methods in Engineering 61 (3) pp 380– (2004) · Zbl 1075.74645 · doi:10.1002/nme.1070
[43] Ohtake, Sparse surface reconstruction with adaptive partition of unity and radial basis functions, Graphical Models 68 (1) pp 15– (2006) · Zbl 1103.68924 · doi:10.1016/j.gmod.2005.08.001
[44] Timoshenko, Theory of Plates and Shells (1959)
[45] Lindberg G Olson M Cowper G New developments in the finite element analysis of shells 1969 1 38
[46] Amenta N Choi S Kolluri R The power crust 249 260
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.