## Integrated Modeling, Finite-Element Analysis, and Engineering Design for Thin-Shell Structures using Subdivision (2002)

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Venue: | Computer-Aided Design |

Citations: | 13 - 2 self |

### BibTeX

@ARTICLE{Cirak02integratedmodeling,,

author = {Fehmi Cirak and Michael J. Scott and Erik K. Antonsson and Michael Ortiz and Peter Schröder},

title = {Integrated Modeling, Finite-Element Analysis, and Engineering Design for Thin-Shell Structures using Subdivision},

journal = {Computer-Aided Design},

year = {2002},

volume = {34},

pages = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many engineering des98 applications require geometric modeling and mechanical s imulation of thin flexibles tructures ,s uchas thos e found in the automotive and aerosH ce indus ries Traditionally, geometric modeling, mechanical s mulation, and engineering des ign are treated as s eparate modules requiring di#erent methods and represR tations Due to the incompatibility of the involved repres9 tations the trans ition from geometric modeling to mechanicals imulation,as wellas in the oppos ite direction, requires s ubs tantial e#ort. However, for engineering des ign purpos es e#- cient trans ition between geometric modeling and mechanicalsa ulationis esio tial. We propos e the us ofs ubdivis ons5AP ces as a common foundation for modeling, s mulation, and des gn in a unified framework. Subdivis ons686 ces provide a flexible and e#cient tool for arbitrary topology free-forms urface modeling, avoiding many of the problems inherent in traditionals pline patch bas d approaches The underlying bass functions are als ideallys5 ted for a finite-element treatment of thes o-called thins hell equations , which des cribe the mechanical behavior of the modeleds tructures The res ulting s lvers are highly s alable, providing an e#cient computational foundation for des ign exploration and optimization. We demons rate our claims withs everal des5R examples s howing the versH6P64 y and high accuracy of the propos d method. Key words: Subdivis on Surfaces Finite-Elements Shells ; 1

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Citation Context ...] which generalize classic quartic box-splines, but we hasten to point out that the basic algorithms are equally applicable to other subdivision schemes, in particular the scheme of Catmull and Clark =-=[8]-=- which generalizes bi-cubic splines. Both subdivision schemes have the necessary smoothness to be ideally suited as shape functions in the finite-element treatment of thin-shell equations. Section 3 d... |

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Citation Context ...uadrilaterals and is approximating, while the scheme of Doo and Sabin is dual, uses quadrilaterals, and is approximating. Interpolating primal schemes based on triangles were introduced by Dyn et al. =-=[12]-=- and improved by Zorin et al. [53], while Kobbelt, et al., [20] constructed a quadrilateral interpolating scheme. These latter schemes are generally not smooth enough for either geometric modeling or ... |

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Citation Context ... impossible. Methods for coping with this computational difficulty at any one level of design resolution include polynomial and other approximations of the performance function (Design of Experiments =-=[28, 33, 37]-=-, Kriging [42], MARS [15], Response Surface methods [21]), local approximation of partial derivatives (sensitivity analysis [22]), directed pointwise search (classical optimization [32]) and ad hoc se... |

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Citation Context ..., while the scheme of Doo and Sabin is dual, uses quadrilaterals, and is approximating. Interpolating primal schemes based on triangles were introduced by Dyn et al. [12] and improved by Zorin et al. =-=[53]-=-, while Kobbelt, et al., [20] constructed a quadrilateral interpolating scheme. These latter schemes are generally not smooth enough for either geometric modeling or finite-element analysis of thin-sh... |

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Citation Context ... specially tailored for sparse matrices. • A classical approach to avoid the use of smooth shape functions in finite-element computations is based on the theory of thick-shells with shear deformatio=-=n [1, 4, 7, 40, 41]. -=-The related finite-element implementations require only piecewise continuous shape functions, but lead to problems such as shear locking for thin-shells – especially in the presence of severe elemen... |

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Citation Context ...ther zero or infinite stretching energies near irregular vertices. Such methods can nonetheless be useful for scattered data interpolation after suitable modifications near the extraordinary vertices =-=[27]-=- or elimination of the infinite energy modes [14]. To accurately and consistently describe the mechanical behavior of shells a formulation in terms of intrinsic surface properties is required. 3.2 Kin... |

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Citation Context ...l mesh may have valence other than six. In the original scheme Loop [25] proposed weights: w = 1 k � 5 8 − � 3 1 + 8 4 � � 2 2π cos k Other values for w also give smooth surfaces. For examp=-=le, Warren [48]-=- proposed a simpler choice for w: w = 3 8k for k>3 and w = 3 16 for k =3 Note that the weights used by the subdivision scheme depend only on the connectivity of the mesh and are independent of the nod... |

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Citation Context ...e limit surface may be obtained as follows. For each triangle in the control mesh we choose (θ 1 ,θ 2 ) as two of its barycentric coordinates within their natural range: T = {(θ 1 ,θ 2 ), s. t. θ=-= α ∈ [0, 1], 0 ≤ -=-θ 1 + θ 2 ≤ 1} The triangle T in the (θ 1 ,θ 2 )-plane may be regarded as a master or standard element domain. It should be emphasized that this parameterization is defined locally for each elem... |

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Citation Context ...ith more traditional approaches for the construction of shape functions, which are typically based on Hermite interpolation. It is well known that this leads to fifth order polynomials over triangles =-=[50]-=-. Higher order shape functions however are not suitable for practical problems with e.g. reentrant corners, jumps in the material properties, or point loads which exhibit singularities in the exact so... |

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Citation Context ... specially tailored for sparse matrices. • A classical approach to avoid the use of smooth shape functions in finite-element computations is based on the theory of thick-shells with shear deformatio=-=n [1, 4, 7, 40, 41]. -=-The related finite-element implementations require only piecewise continuous shape functions, but lead to problems such as shear locking for thin-shells – especially in the presence of severe elemen... |

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Citation Context ...ty at any one level of design resolution include polynomial and other approximations of the performance function (Design of Experiments [28, 33, 37], Kriging [42], MARS [15], Response Surface methods =-=[21]-=-), local approximation of partial derivatives (sensitivity analysis [22]), directed pointwise search (classical optimization [32]) and ad hoc selection guided by experience and intuition. For multi-at... |

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Citation Context ... specially tailored for sparse matrices. • A classical approach to avoid the use of smooth shape functions in finite-element computations is based on the theory of thick-shells with shear deformatio=-=n [1, 4, 7, 40, 41]. -=-The related finite-element implementations require only piecewise continuous shape functions, but lead to problems such as shear locking for thin-shells – especially in the presence of severe elemen... |

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Citation Context ...y on the value of the objective function and does not require function derivatives. If derivatives (sensitivities) are available more sophisticated optimization algorithms can be utilized (see, e.g., =-=[13, 30]-=- among many others). 5.1 Square Plate The first example is the design of a uniformly loaded roof over a square shaped area (Figure 8). The roof is supported at the four corners. The design objective i... |

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Citation Context ...t more than a simple optimization problem. In multi-attribute problems, trade-offs among criteria can play a determining role, and the designer is frequently interested in a Pareto frontier of points =-=[18, 26, 46]-=- rather than a single optimum. A Pareto point is a point in the set of possible designs that matches or exceeds the performance of any other possible design point on at least one attribute; if one poi... |

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Citation Context .... In a traditional framework this requires the generation of a finite-element mesh separate from the original CAD model, bringing with it the computational disadvantage of keeping two representations =-=[6, 16, 30]-=-. The subdivision based approach is computationally efficient and representationally unified way to use the subdivision control mesh to parameterize both the geometry and the finite-element model. The... |

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Citation Context .... In a traditional framework this requires the generation of a finite-element mesh separate from the original CAD model, bringing with it the computational disadvantage of keeping two representations =-=[6, 16, 30]-=-. The subdivision based approach is computationally efficient and representationally unified way to use the subdivision control mesh to parameterize both the geometry and the finite-element model. The... |

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Citation Context ...loped in different communities with limited interactions. To our knowledge few attempts have been made in the past to unify geometric modeling and simulation based on a common representation paradigm =-=[17, 20]-=-. Our contribution in this paper is a computationally simple and theoretically well-justified framework for free-form geometric modeling coupled with finite-element analysis of thin-shells for purpose... |