Isogeometric analysis and shape optimization via boundary integral

In this paper, we present a boundary integral based approach to isogeometric analysis and shape optimization.For analysis, it uses the same basis, Non-Uniform Rational B-Spline (NURBS) basis, for both representing object boundary and for approximating physical fields in analysis via a Boundary-Integral-Equation Method (BIEM). We propose the use of boundary points corresponding to Greville abscissae as collocation points. We conducted h-, p- and k-refinement study for linear elasticity and heat conduction problems. Our numerical experiments show that collocation at Greville abscissae leads to overall better convergence and robustness. Replacing rational B-splines with the linear B-Splines as shape functions for approximating solution space in analysis does not yield significant difference in convergence.For shape optimization, it uses NURBS control points to parameterize the boundary shape. A gradient based optimization approach is adopted where analytical sensitivities of how control points affect objective and constraint functions are derived. Two 3D shape optimization examples are demonstrated.Our study finds that the boundary integral based isogeometric analysis and optimization have the following advantages: (1) the NURBS based boundary integral exhibits superior computational advantages over the usual Lagrange polynomials based BIEM on a per degree-of-freedom basis; (2) it bypasses the need for domain parameterization, a bottleneck in current NURBS based volumetric isogeometric analysis and shape optimization; (3) it offers tighter integration of CAD and analysis since both the geometric models for both analysis and optimization are the same NURBS geometry. Highlights? A NURBS based boundary integral method has been developed. ? It has superior computational advantages over traditional boundary integral. ? Greville collocation leads to accurate and stable analysis results. ? A boundary integral based isogeometric shape optimization method has been developed.

[1]  Carlos Alberto Brebbia,et al.  Boundary Elements: An Introductory Course , 1989 .

[2]  A. Becker.  The Boundary Element Method in Engineering: A Complete Course , 1992 .

[3]  O. C. Zienkiewicz,et al.  The Finite Element Method for Solid and Structural Mechanics , 2013 .

[4]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[5]  Javier Jiménez,et al.  A critical evaluation of the resolution properties of B-Spline and compact finite difference methods , 2001 .

[6]  Subrata Mukherjee,et al.  Shape optimization in three-dimensional linear elasticity by the boundary contour method , 1999 .

[7]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[8]  Anders Klarbring,et al.  An Introduction to Structural Optimization , 2008 .

[9]  Seung-Hyun Ha,et al.  Isogeometric shape design optimization: exact geometry and enhanced sensitivity , 2009 .

[10]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[11]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[12]  Xiaoping Qian,et al.  NURBS based molecular force calculation , 2009, Symposium on Solid and Physical Modeling.

[13]  William C. Regli,et al.  2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling , 2009 .

[14]  R. T. Fenner,et al.  OPTIMUM SHAPE DESIGN AND POSITIONING OF FEATURES USING THE BOUNDARY INTEGRAL EQUATION METHOD , 1996 .

[15]  T. N. E. Greville,et al.  Theory and applications of spline functions , 1969 .

[16]  C. D. Boor,et al.  Collocation at Gaussian Points , 1973 .

[17]  Hongwei Lin,et al.  Watertight trimmed NURBS , 2008, ACM Trans. Graph..

[18]  Gerald Farin,et al.  Curves and surfaces for cagd , 1992 .

[19]  J. Watson,et al.  Effective numerical treatment of boundary integral equations: A formulation for three‐dimensional elastostatics , 1976 .

[20]  Martin Kögl,et al.  Boundary Element Methods for Engineers and Scientists: An Introductory Course With Advanced Topics , 2004 .

[21]  Xiaoping Qian,et al.  A general, accurate procedure for calculating molecular interaction force. , 2009, Journal of colloid and interface science.

[22]  Xiaoping Qian,et al.  A B-spline-based approach to heterogeneous objects design and analysis , 2007, Comput. Aided Des..

[23]  Panagiotis D. Kaklis,et al.  An isogeometric BEM for exterior potential-flow problems in the plane , 2009, Symposium on Solid and Physical Modeling.

[24]  Jon Trevelyan,et al.  Evolutionary structural optimisation based on boundary representation of NURBS. Part I: 2D algorithms , 2005 .

[25]  Bert Jüttler,et al.  Mathematics of Surfaces XII , 2007 .

[26]  Ole Sigmund,et al.  Isogeometric shape optimization of photonic crystals via Coons patches , 2011 .

[27]  R. Haftka,et al.  Elements of Structural Optimization , 1984 .

[28]  Tom Lyche,et al.  Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis , 2010 .

[29]  Martin Aigner,et al.  Swept Volume Parameterization for Isogeometric Analysis , 2009, IMA Conference on the Mathematics of Surfaces.

[30]  W. Wall,et al.  Isogeometric structural shape optimization , 2008 .

[31]  Elaine Cohen,et al.  Volumetric parameterization and trivariate B-spline fitting using harmonic functions , 2009, Comput. Aided Geom. Des..

[32]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[33]  Xiaoping Qian.  Full analytical sensitivities in NURBS based isogeometric shape optimization , 2010 .

[34]  J. Glinsky,et al.  The general. , 1982, Nursing.