Topological shape optimization design of continuum structures via an effective level set method

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. This paper proposes a new level set method for topological shape optimization of continuum structure using radial basis function (RBF) and discrete wavelet transform (DWT). The boundary of the structure is implicitly represented as the zero level set of a higher-dimensional level set function. The interpolation of the implicit surface using RBF is introduced to decouple the spatial and temporal dependence of the level set function. In doing so, the Hamilton-Jacobi partial differential equation (PDE) that defines the motion of the level set function is transformed into an explicit parametric form, without requiring the direct solution of the complicated PDE using the finite difference method. Therefore, many more efficient gradient-based optimization algorithms can be applied to solve the optimization problem, via updating the expansion coefficients of the interpolant and then evolving the level set function and the boundary. Furthermore, the DWT is employed to handle the full matrix arising from the globally supported RBF interpolation. Several high stiffness but lightweight designs with smooth and clear structural boundaries are optimized and presented. The numerical results show that the proposed method can remarkably increase the efficiency in the topology optimization design of both the 2D and 3D structures

A Method of Topology Optimization for Curvature Continuous Designs

Recently, there have been many developments made in the field of topology optimization. Specifically, the structural dynamics community has been the leader of the engineering disciplines in using these methods to improve the designs of various structures, ranging from bridges to motor vehicle frames, as well as aerospace structures like the ribs and spars of an airplane. The representation of these designs, however, are usually stair-stepped or faceted throughout the optimization process and require post-process smoothing in the final design stages. Designs with these low-order representations are insufficient for use in higher-order computational fluid dynamics methods, which are becoming more and more popular. With the push for the development of higher-order infrastructures, including higher-order grid generation methods, there exists a need for techniques that handle curvature continuous boundary representations throughout an optimization process. Herein a method has been developed for topology optimization for high-Reynolds number flows that represents smooth bodies, that is, bodies that have continuous curvature. The specific objective function used herein is to match specified x-rays, which are a surrogate for the wake profile of a body in cross-flow. The parameterized level-set method is combined with a boundary extraction technique that incorporates a modified adaptive 4th-order Runge-Kutta algorithm, together with a classical cubic spline curve-fitting method, to produce curvature-continuous boundaries throughout the optimization process. The level-set function is parameterized by the locations and coefficients of Wendland C2 radial basis functions. Topology optimization is achieved by implementing a conjugate gradient optimization algorithm that simultaneously changes the locations of the radial basis function centers and their respective coefficients. To demonstrate the method several test cases are shown where the objective is to generate a smooth representation of a body or bodies that match specified x-rays. First, multiple examples of shape optimization are presented for different topologies. Then topology optimization is demonstrated with an example of two bodies merging and several examples of a single body splitting into separate bodies

Introducing the sequential linear programming level-set method for topology optimization

The authors would like to thank Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk/). Dr H Alicia Kim acknowledges the support from Engineering and Physical Sciences Research Council, grant number EP/M002322/1Peer reviewedPublisher PD

Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography

State-of-the-art in aerodynamic shape optimisation methods

Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners

Minimum length-scale constraints for parameterized implicit function based topology optimization

Open access via Springer Compact Agreement The author would like to thank the Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk/). The author also would like to acknowledge the support of the Maxwell compute cluster funded by the University of Aberdeen. Finally, the author thanks the anonymous reviewers for their helpful comments and suggestions that improved this paper.Peer reviewedPublisher PD

Sizing and Topology Design of an Aeroelastic Wingbox Under Uncertainty

The goals of this work are to use a nested optimizer to conduct simultaneous sizing (inner level) and topology (outer level) design of a wingbox, considering uncertainties in the safety factors used to define the aeroelastic constraints. These uncertainties, propagated via sampling-driven polynomial chaos, are explicitly introduced at the inner level of the method, during gradient-based sizing optimization, resulting in a stochastic optimal sizing distribution. Measures of robustness in the total structural mass are then passed to the outer level, where a global optimizer evolves the topology parameters. The results demonstrate design choices needed to improve robustness in the face of uncertain safety factors, and the various physical mechanisms driving this process