Abstract
In isogeometric analysis (IGA), the boundary representation of computer-aided design (CAD) and the tensor-product non-uniform rational B-spline structure make the analysis of three-dimensional (3D) problems with irregular geometries difficult. In this paper, an IGA method for complex models is presented by reconstructing analysis-suitable models. The CAD model is represented by boundary polygons or point cloud and is embedded into a regular background grid, and a model reconstruction method is proposed to obtain the level set function of the approximate model, which can be directly used in IGA. Three 3D examples are used to test the proposed method, and the results demonstrate that the proposed method can deal with complex engineering parts reconstructed by boundary polygons or point clouds.
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Abbreviations
- B :
-
Strain-displacement matrix
- B i,p(ξ):
-
ith B-spline basis function of p-degree
- C p−k :
-
Continuity between knot spans
- d 1, d 2 :
-
SMDs of the two vertices of an edge
- D :
-
Stress-strain matrix
- Dis :
-
Distance from a point to a point on a triangle
- E :
-
Young’s modulus
- F :
-
Force vector
- J :
-
Jacobian matrix
- J 1, J 2 :
-
Transformation relation from the NURBS parametric space to the physical space and the integration parametric space to the NURBS parametric space, respectively
- k :
-
Multiplicity of the knots
- K e :
-
Element stiffness matrix
- m :
-
Number of polygons
- n :
-
Number of basis functions
- N :
-
Shape function vector
- N i :
-
Shape function
- N i,p(ξ):
-
ith NURBS basis function of p-degree
- p :
-
Order of the B-spline
- v 1 :
-
Vector from the vertex to the polygon plane
- v 2 :
-
Outward vector of the polygon
- V Ωe, \(V_{\hat{\Omega}_{\rm{e}}}\) and \(V_{\bar{\Omega}_{\rm{e}}}\) :
-
Volumes of the element in physical, NURBS parametric, and integration domains, respectively
- w i :
-
Positive weight
- w i,j :
-
Weight value corresponding to the tensor product Bi, p(ξ)Bj, q(η)
- x :
-
Physical value (e.g., coordinate, displacement, and force)
- x :
-
Physical value vector
- v :
-
Poisson’s ratio
- τ :
-
Traction of the loading region Ω
- ξ, η :
-
Variables for the coordinates in parameter space
- Ξ :
-
Knot vector
- φ :
-
Level set function value
- φ i :
-
LSF value of the ith vertex of the regular embedded element
- Ωe, \(\hat{\Omega}_{\rm{e}}\), and \(\bar{\Omega}_{\rm{e}}\) :
-
Physical, NURBS parametric, and integration domains of the element, respectively
- 2D:
-
Two-dimensional
- 3D:
-
Three-dimensional
- B-rep:
-
Boundary representation
- CAD:
-
Computer-aided design
- CAE:
-
Computer-aided engineering
- CSG:
-
Constructive solid geometry
- FCM:
-
Finite cell method
- FEA:
-
Finite element analysis
- FSI:
-
Fluid-structure interaction
- IGA:
-
Isogoemetric analysis
- LSF:
-
Level set function
- MC:
-
Marching cube
- MD:
-
Minimum distance
- NURBS:
-
Non-uniform rational B-spline
- SMD:
-
Sign minimum distance
- STEP:
-
Standard for the exchange of product model data
References
Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195
Cottrell J A, Reali A, Bazilevs Y, et al. Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering, 2006, 195(41–43): 5257–5296
Weeger O, Wever U, Simeon B. Isogeometric analysis of nonlinear Euler-Bernoulli beam vibrations. Nonlinear Dynamics, 2013, 72(4): 813–835
Benson D J, Bazilevs Y, Hsu M C, et al. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13–16): 1367–1378
Deng X, Korobenko A, Yan J, et al. Isogeometric analysis of continuum damage in rotation-free composite shells. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 349–372
Bazilevs Y, Deng X, Korobenko A, et al. Isogeometric fatigue damage prediction in large-scale composite structures driven by dynamic sensor data. Journal of Applied Mechanics, 2015, 82(9): 091008
Bazilevs Y, Korobenko A, Deng X, et al. Fluid-structure interaction modeling for fatigue-damage prediction in full-scale wind-turbine blades. Journal of Applied Mechanics, 2016, 83(6): 061010
Peng X, Atroshchenko E, Kerfriden P, et al. Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 151–185
Bazilevs Y, Calo V M, Hughes T J R, et al. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Computational Mechanics, 2008, 43(1): 3–37
Hsu M C, Akkerman I, Bazilevs Y. High-performance computing of wind turbine aerodynamics using isogeometric analysis. Computers & Fluids, 2011, 49(1): 93–100
Yan J, Deng X, Korobenko A, et al. Free-surface flow modeling and simulation of horizontal-axis tidal-stream turbines. Computers & Fluids, 2017, 158: 157–166
Bazilevs Y, Yan J, Deng X, et al. Computer modeling of wind turbines: 2. Free-surface FSI and fatigue-damage. Archives of Computational Methods in Engineering, 2019, 26(4): 1101–1115
Wang Y, Benson D J. Isogeometric analysis for parameterized LSM-based structural topology optimization. Computational Mechanics, 2016, 57(1): 19–35
Wang Y, Xu H, Pasini D. Multiscale isogeometric topology optimization for lattice materials. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 568–585
Wang Z P, Poh L H, Dirrenberger J, et al. Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 250–271
Xie X, Wang S, Xu M, et al. A new isogeometric topology optimization using moving morphable components based on R-functions and collocation schemes. Computer Methods in Applied Mechanics and Engineering, 2018, 339: 61–90
Deng X W, Wu N, Yang K, et al. Integrated design framework of next-generation 85-m wind turbine blade: modelling, aeroelasticity and optimization. Composites Part B, Engineering, 2019, 159: 53–61
Wang Z P, Poh L H. Optimal form and size characterization of planar isotropic petal-shaped auxetics with tunable effective properties using IGA. Composite Structures, 2018, 201: 486–502
Wang Z P, Poh L H, Zhu Y, et al. Systematic design of tetra-petals auxetic structures with stiffness constraint. Materials & Design, 2019, 170: 107669
Babic B, Nesic N, Miljkovic Z. A review of automated feature recognition with rule-based pattern recognition. Computers in Industry, 2008, 59(4): 321–337
Zhu X F, Hu P, Ma Z D, et al. A new surface parameterization method based on one-step inverse forming for isogeometric analysis-suited geometry. International Journal of Advanced Manufacturing Technology, 2013, 65(9–12): 1215–1227
Jaxon N, Qian X. Isogeometric analysis on triangulations. Computer-Aided Design, 2014, 46: 45–57
Martin T, Cohen E, Kirby R M. Volumetric parameterization and trivariate B-spline fitting using harmonic functions. Computer-Aided Geometric Design, 2009, 26(6): 648–664
Xu G, Mourrain B, Duvigneau R, et al. Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications. Computer-Aided Design, 2013, 45(2): 395–404
Zuo B Q, Huang Z D, Wang Y W, et al. Isogeometric analysis for CSG models. Computer Methods in Applied Mechanics and Engineering, 2015, 285: 102–124
Bazilevs Y, Calo V M, Cottrell J A, et al. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 229–263
Scott M A, Li X, Sederberg T W, et al. Local refinement of analysis-suitable T-splines. Computer Methods in Applied Mechanics and Engineering, 2012, 213–216: 206–222
Whetten C, Sederberg M, Scott M. Isogeometric analysis using the *IGA_INCLUDE_BEZIER keyword in LS-DYNA. 2019
Kim H J, Seo Y D, Youn S K. Isogeometric analysis for trimmed CAD surfaces. Computer Methods in Applied Mechanics and Engineering, 2009, 198(37–40): 2982–2995
Wang Y W, Huang Z D, Zheng Y, et al. Isogeometric analysis for compound B-spline surfaces. Computer Methods in Applied Mechanics and Engineering, 2013, 261–262: 1–15
Nagy A P, Benson D J. On the numerical integration of trimmed isogeometric elements. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 165–185
Wang Y, Benson D J, Nagy A P. A multi-patch nonsingular isogeometric boundary element method using trimmed elements. Computational Mechanics, 2015, 56(1): 173–191
Ruess M, Schillinger D, Bazilevs Y, et al. Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. International Journal for Numerical Methods in Engineering, 2013, 95(10): 811–846
Marussig B, Zechner J, Beer G, et al. Stable isogeometric analysis of trimmed geometries. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 497–521
Guo Y, Heller J, Hughes T J R, et al. Variationally consistent isogeometric analysis of trimmed thin shells at finite deformations, based on the STEP exchange format. Computer Methods in Applied Mechanics and Engineering, 2018, 336: 39–79
Marussig B, Hughes T J. A review of trimming in isogeometric analysis: challenges, data exchange and simulation aspects. Archives of Computational Methods in Engineering, 2018, 25(4): 1059–1127
Parvizian J, Düster A, Rank E. Finite cell method. Computational Mechanics, 2007, 41(1): 121–133
Düster A, Parvizian J, Yang Z, et al. The finite cell method for three-dimensional problems of solid mechanics. Computer Methods in Applied Mechanics and Engineering, 2008, 197(45–48): 3768–3782
Schillinger D, Ruess M. The finite cell method: a review in the context of higher-order structural analysis of CAD and image-based geometric models. Archives of Computational Methods in Engineering, 2015, 22(3): 391–455
Kamensky D, Hsu M C, Schillinger D, et al. An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 1005–1053
Xu F, Schillinger D, Kamensky D, et al. The tetrahedral finite cell method for fluids: immersogeometric analysis of turbulent flow around complex geometries. Computers & Fluids, 2016, 141: 135–154
Huang J, Menq C H. Automatic CAD model reconstruction from multiple point clouds for reverse engineering. Journal of Computing and Information Science in Engineering, 2002, 2(3): 160–170
Manmadhachary A, Kumar R, Krishnanand L. Improve the accuracy, surface smoothing and material adaption in STL file for RP medical models. Journal of Manufacturing Processes, 2016, 21: 46–55
Wang J, Gu D, Yu Z, et al. A framework for 3D model reconstruction in reverse engineering. Computers & Industrial Engineering, 2012, 63(4): 1189–1200
Rajon D A, Bolch W E. Marching cube algorithm: review and trilinear interpolation adaptation for image-based dosimetric models. Computerized Medical Imaging and Graphics, 2003, 27(5): 411–435
Montani C, Scateni R, Scopigno R. A modified look-up table for implicit disambiguation of marching cubes. The Visual Computer, 1994, 10(6): 353–355
Newman T S, Yi H. A survey of the marching cubes algorithm. Computers & Graphics, 2006, 30(5): 854–879
Chang M, Oh J W, Chang D S, et al. Interactive marching cubes algorithm for intraoral scanners. International Journal of Advanced Manufacturing Technology, 2017, 89(5–8): 2053–2062
Doi A, Koide A. An efficient method of triangulating equi-valued surfaces by using tetrahedral cells. IEICE Transactions on Information and Systems, 1991, 74(1): 214–224
Treece G M, Prager R W, Gee A H. Regularised marching tetrahedra: Improved iso-surface extraction. Computers & Graphics, 1999, 23(4): 583–598
Guo D, Li C, Wu L, et al. Improved marching tetrahedra algorithm based on hierarchical signed distance field and multi-scale depth map fusion for 3D reconstruction. Journal of Visual Communication and Image Representation, 2017, 48: 491–501
Zander N, Bog T, Elhaddad M, et al. FCMLab: a finite cell research toolbox for MATLAB. Advances in Engineering Software, 2014, 74: 49–63
Verhoosel C V, Van Zwieten G J, Van Rietbergen B, et al. Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 138–164
Kudela L, Kollmannsberger S, Almac U, et al. Direct structural analysis of domains defined by point clouds. Computer Methods in Applied Mechanics and Engineering, 2020, 358: 112581
Piegl L, Tiller W. The NURBS Book. New York: Springer, 1997
de Boor C. On calculating with B-splines. Journal of Approximation Theory, 1972, 6(1): 50–62
Xia Z, Wang Q, Wang Y, et al. A CAD/CAE incorporate software framework using a unified representation architecture. Advances in Engineering Software, 2015, 87: 68–85
Requicha A A G, Voelcker H B. Boolean operations in solid modeling: boundary evaluation and merging algorithms. Proceedings of the IEEE, 1985, 73(1): 30–44
Hales T C. The Jordan curve theorem, formally and informally. American Mathematical Monthly, 2007, 114(10): 882–894
Wang Y, Benson D J. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers of Mechanical Engineering, 2016, 11(4): 328–343
Eberly D. Distance between point and triangle in 3D. Geometric Tools, 2008
Wang Y, Liao Z, Ye M, et al. An efficient isogeometric topology optimization using multilevel mesh, MGCG and local-update strategy. Advances in Engineering Software, 2020, 139: 102733
Zhu B, Zhang X, Zhang H, et al. Design of compliant mechanisms using continuum topology optimization: a review. Mechanism and Machine Theory, 2020, 143: 103622
Acknowledgements
This work has been supported by the National Key R&D Program of China (Grant No. 2020YFB1708300), the National Natural Science Foundation of China (Grant Nos. 52075184 and 51705158), and the Research Grants Council via Early Career Scheme, Hong Kong, China (RGC Ref. No. 27209817). These supports are gratefully acknowledged.
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Wang, Y., Gao, L., Qu, J. et al. Isogeometric analysis based on geometric reconstruction models. Front. Mech. Eng. 16, 782–797 (2021). https://doi.org/10.1007/s11465-021-0648-0
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DOI: https://doi.org/10.1007/s11465-021-0648-0