Topology and shape optimization with explicit geometric constraints using a spline-based representation and a fixed grid

Abstract Topology optimization is a computational method for finding the distribution of material such that an objective function is minimized subject to a set of constraints. In the context of structures, topology optimization aims to find the layout by changing the shape of the boundary and the number and shape of holes. Such optimized designs ultimately lead to energy savings, efficient usage of materials, and to faster and sustainable manufacturing. In this paper, we present an optimization approach that is based on explicit B-spline representation of the design, conforming with CAD standards. This parametrization enables to incorporate explicit constraints on minimum and maximum areas of holes and on curvatures of boundaries. Therefore practical design considerations such as avoiding stress concentrations in sharp corners and flexibility with respect to locations and sizes of holes can be embedded into the optimization problem. Furthermore, control of curvature can simplify machining processes leading to more efficient and sustainable manufacturing.

[1]  Martin Reimers,et al.  An unconditionally convergent method for computing zeros of splines and polynomials , 2007, Math. Comput..

[2]  Ramana V. Grandhi,et al.  A survey of structural and multidisciplinary continuum topology optimization: post 2000 , 2014 .

[3]  Cláudio A. C. Silva,et al.  Velocity fields using NURBS with distortion control for structural shape optimization , 2006 .

[4]  D. Tortorelli,et al.  A geometry projection method for continuum-based topology optimization with discrete elements , 2015 .

[5]  Nam H. Kim,et al.  Eulerian shape design sensitivity analysis and optimization with a fixed grid , 2005 .

[6]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[7]  H. A. Kim,et al.  Smooth Boundary Based Optimisation Using Fixed Grid , 2007 .

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

[9]  Osvaldo M. Querin,et al.  Topology design of two-dimensional continuum structures using isolines , 2009 .

[10]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[11]  Yi Min Xie,et al.  Introduction of fixed grid in evolutionary structural optimisation , 2000 .

[12]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[13]  Anders Clausen,et al.  Topology optimization with flexible void area , 2014 .

[14]  Mu Zhu,et al.  Casting and Milling Restrictions in Topology Optimization via Projection-Based Algorithms , 2012, DAC 2012.

[15]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[16]  Byung Man Kwak,et al.  Smooth Boundary Topology Optimization Using B-spline and Hole Generation , 2007 .

[17]  O. Sigmund.  Morphology-based black and white filters for topology optimization , 2007 .

[18]  O. Sigmund,et al.  Topology optimization approaches , 2013, Structural and Multidisciplinary Optimization.

[19]  Jian Zhang,et al.  Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons , 2016 .

[20]  Ole Sigmund,et al.  Design of materials with extreme thermal expansion using a three-phase topology optimization method , 1997, Smart Structures.

[21]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[22]  Takayuki Yamada,et al.  World Congress on Structural and Multidisciplinary Optimization , 2013 .

[23]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[24]  Rida T. Farouki,et al.  Analytic properties of plane offset curves , 1990, Comput. Aided Geom. Des..