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Frontiers of Mechanical Engineering

Front. Mech. Eng.    2016, Vol. 11 Issue (4) : 328-343     https://doi.org/10.1007/s11465-016-0403-0
RESEARCH ARTICLE |
Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements
Yingjun WANG1(),David J. BENSON2
1. School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, China; Department of Mechanical Engineering, McGill University, Montreal H3A0C3, Canada
2. Department of Structural Engineering, University of California, San Diego 92093, USA
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Abstract

In this paper, an approach based on the fast point-in-polygon (PIP) algorithm and trimmed elements is proposed for isogeometric topology optimization (TO) with arbitrary geometric constraints. The isogeometric parameterized level-set-based TO method, which directly uses the non-uniform rational basis splines (NURBS) for both level set function (LSF) parameterization and objective function calculation, provides higher accuracy and efficiency than previous methods. The integration of trimmed elements is completed by the efficient quadrature rule that can design the quadrature points and weights for arbitrary geometric shape. Numerical examples demonstrate the efficiency and flexibility of the method.

Keywords isogeometric analysis      topology optimization      level set method      arbitrary geometric constraint      trimmed element     
Corresponding Authors: Yingjun WANG   
Online First Date: 19 October 2016    Issue Date: 29 November 2016
 Cite this article:   
Yingjun WANG,David J. BENSON. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements[J]. Front. Mech. Eng., 2016, 11(4): 328-343.
 URL:  
http://journal.hep.com.cn/fme/EN/10.1007/s11465-016-0403-0
http://journal.hep.com.cn/fme/EN/Y2016/V11/I4/328
Fig.1  An example of geometric constraint in TO problems
Fig.2  Two mesh schemes for the example in Fig. 1. (a) Meshes by conventional finite element methods; (b) regular mesh
Fig.3  A 2D design domain and LSM
Fig.4  An example of the Greville abscissae collocation for the surface formed from knot vectors
Fig.5  Ray crossing test: One crossing denotes P1 is inside and two crossings denote P2 is outside
Fig.6  An example of element classification
Fig.7  An illustration for the approximation representation of the trimmed domain: Domain with (a) initial and (b) refined trimming control polygon
Fig.8  The element filling density computation for a boundary-crossing element: (a) Polyline approximation of the boundary and (b) the integration scheme
Fig.9  The Michell type structure with a geometric constraint: (a) Design domain and (b) distribution of initial holes
Fig.10  Optimization stages of the Michell type structure with geometric constraint: (a) Initial design, (b) Stage 2, (c) Stage 4, (d) Stage 10, (e) Stage 20 and (f) Stage 25 (final result)
Fig.11  Convergence histories of the Michell type structure with the geometric constraint
Fig.12  Optimization result of the Michell type structure without the geometric constraint [36]
Fig.13  The spanner structure with profile geometric constraint: (a) Design domain and (b) distribution of initial holes
Fig.14  Optimization stages of the spanner structure with geometric constraint: (a) Initial design, (b) Stage 2, (c) Stage 5, (d) Stage 10, (e) Stage 20, (f) Stage 30, (g) Stage 40 and (h) Stage 55 (final result)
Fig.15  Convergent histories of the spanner type structure with geometric constrain
Fig.16  Construction of geometrically constrained domain of the spanner model by R-functions. The operator

represents the R-conjunction operation [66] which can be also regarded as an intersection operation

Fig.17  The R-function LSF of the spanner domain

Ω15

in Fig. 16

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