A Moving Morphable Voids Approach for Topology Optimization With Closed B-Splines

2019 ◽  
Vol 141 (8) ◽  
Author(s):  
Bingxiao Du ◽  
Wen Yao ◽  
Yong Zhao ◽  
Xiaoqian Chen
Keyword(s):  
Topology Optimization ◽  
B Spline ◽  
B Splines ◽  
Spline Curves ◽  
Beam Design ◽  
Merging Process ◽  
Order Degree ◽  
Structural Boundary ◽  
Ks Function ◽  
Selection Of

Topology optimization with moving morphable voids (MMVs) is studied in this paper. B-spline curves are used to represent the boundaries of MMVs in the structure. Kreisselmeier–Steinhauser (KS)-function is also implemented to preserve the smoothness of the structural boundary in case the intersection of the curves happen. In order to study the influence of continuity, we propose pseudo-periodic closed B-splines (PCBSs) to construct curves with an arbitrary degree. The selection of PCBS parameters, especially the degree of B-spline, is studied and discussed. The classic Messerschmitt–Bolkow–Blohm (MBB) case is taken as an example in the numerical experiment. Results show that with the proper choice of B-spline degrees and number of control points, PCBSs have enough flexibility and stability to represent the optimized material distribution. We further reveal the mechanism of the merging process of holes and find that high-order degree PCBS could preserve more separated voids. A support beam design problem of microsatellite is also studied as an example to demonstrate the capability of the proposed method.

Structural Topology Optimization Through Explicit Boundary Evolution

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
Weisheng Zhang ◽  
Wanying Yang ◽  
Jianhua Zhou ◽  
Dong Li ◽  
Xu Guo
Keyword(s):  
Topology Optimization ◽  
Optimal Designs ◽  
Structural Topology ◽  
B Spline ◽  
Numerical Examples ◽  
Optimization Framework ◽  
And Topology ◽  
Spline Curves ◽  
Structural Boundary ◽  
Moving Morphable Components

Traditional topology optimization is usually carried out with approaches where structural boundaries are represented in an implicit way. The aim of the present paper is to develop a topology optimization framework where both the shape and topology of a structure can be obtained simultaneously through an explicit boundary description and evolution. To this end, B-spline curves are used to describe the boundaries of moving morphable components (MMCs) or moving morphable voids (MMVs) in the structure and some special techniques are developed to preserve the smoothness of the structural boundary when topological change occurs. Numerical examples show that optimal designs with smooth structural boundaries can be obtained successfully with the use of the proposed approach.

BEZIER AND B-SPLINE CURVES WITH KNOTS IN THE COMPLEX PLANE

Fractals ◽  
2011 ◽  
Vol 19 (01) ◽  
pp. 67-86 ◽  
Author(s):  
KONSTANTINOS I. TSIANOS ◽  
RON GOLDMAN
Keyword(s):  
Complex Domain ◽  
Polynomial Algorithms ◽  
Control Points ◽  
Knot Insertion ◽  
Real Numbers ◽  
Koch Curve ◽  
Natural Manner ◽  
B Spline ◽  
B Splines ◽  
Spline Curves

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

B-Spline Based Robust Topology Optimization

2015 ◽  
Author(s):  
Yu Gu ◽  
Xiaoping Qian
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
Compliant Mechanism ◽  
Analytical Description ◽  
Minimal Length ◽  
Optimization Approach ◽  
Material Density ◽  
B Spline ◽  
B Splines ◽  
Robust Formulation

In this paper, we present an extension of the B-spline based density representation to a robust formulation of topology optimization. In our B-spline based topology optimization approach, we use separate representations for material density distribution and analysis. B-splines are used as a representation of density and the usual finite elements are used for analysis. The density undergoes a Heaviside projection to reduce the grayness in the optimized structures. To ensure minimal length control so the resulting designs are robust with respect to manufacturing imprecision, we adopt a three-structure formulation during the optimization. That is, dilated, intermediate and eroded designs are used in the optimization formulation. We give an analytical description of minimal length of features in optimized designs. Numerical examples have been implemented on three common topology optimization problems: minimal compliance, heat conduction and compliant mechanism. They demonstrate that the proposed approach is effective in generating designs with crisp black/white transition and is accurate in minimal length control.

B-Splines Geometric Modeling for Dynamic Analysis Behaviour in Offshore Systems

2006 ◽  
Author(s):  
Fa´bio G. T. de Menezes ◽  
Prota´sio Dutra Martins
Keyword(s):  
Geometric Modeling ◽  
Floating Body ◽  
Design Work ◽  
Oil Drilling ◽  
B Spline ◽  
B Splines ◽  
Hydrodynamic Behaviour ◽  
Spline Curves ◽  
Definition Of ◽  
Floating Systems

This work reports a study of B-Spline curves and surfaces applied to the geometric definition of hulls of ships and oil drilling and production platforms. The research aims at defining mathematically the floating body surface in suitable formats for the analysis of functional behaviour of the design object with sophisticated methods and tools. The WAMIT system was chosen as a reference in the research due to its reliability as a professional tool for hydrodynamic behaviour of floating systems in practice. The B-Spline model is input to the WAMIT system in the required format for the analysis of hull motion response to waves. The quality of the results obtained with B-Splines modeling was compared the ones obtained with flat panels. B-Splines have shown to be an effective approach, more efficient in computing terms when compared with the flat panels approach and suitable to optimization scripts. It revealed itself as a more adequate procedure to the design work as it simplifies the hull form mathematical definition of floating systems.

Efficient Filtering in Topology Optimization via B-Splines1

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Mingming Wang ◽  
Xiaoping Qian
Keyword(s):  
Topology Optimization ◽  
Three Dimensional ◽  
Processing Unit ◽  
Optimization Scheme ◽  
B Spline ◽  
B Splines ◽  
Central Processing ◽  
Filter Size ◽  
Density Filter ◽  
3D Topology

This paper presents a B-spline based approach for topology optimization of three-dimensional (3D) problems where the density representation is based on B-splines. Compared with the usual density filter in topology optimization, the new B-spline based density representation approach is advantageous in both memory usage and central processing unit (CPU) time. This is achieved through the use of tensor-product form of B-splines. As such, the storage of the filtered density variables is linear with respect to the effective filter size instead of the cubic order as in the usual density filter. Numerical examples of 3D topology optimization of minimal compliance and heat conduction problems are demonstrated. We further reveal that our B-spline based density representation resolves the bottleneck challenge in multiple density per element optimization scheme where the storage of filtering weights had been prohibitively expensive.

Efficient Filtering in Topology Optimization via B-Splines

2014 ◽  
Author(s):  
Mingming Wang ◽  
Xiaoping Qian
Keyword(s):  
Topology Optimization ◽  
Three Dimensional ◽  
Optimization Scheme ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
Cpu Time ◽  
Filter Size ◽  
Density Filter ◽  
3D Topology

This paper presents a B-spline based approach for topology optimization of three-dimensional (3D) problems where the density representation is based on B-splines. Compared with the usual density filter in topology optimization, the new B-spline based density representation approach is advantageous in both memory usage and CPU time. This is achieved through the use of tensor-product form of B-splines. As such, the storage of the filtered density variables is linear with respect to the effective filter size instead of the cubic order as in the usual density filter. Numerical examples of 3D topology optimization of minimal compliance and heat conduction problems are demonstrated. We further reveal that our B-spline based density representation resolves the bottleneck challenge in multiple density per element optimization scheme where the storage of filtering weights had been prohibitively expensive.

Analyzing Specular Reflectivities with Parametric B-Splines

1994 ◽  
Vol 376 ◽  
Author(s):  
N. F. Berk ◽  
C. F. Majkrzak
Keyword(s):  
Scattering Length ◽  
Density Profiles ◽  
B Spline ◽  
X Ray ◽  
B Splines ◽  
Fitting Procedure ◽  
Spline Curves ◽  
Reflectivity Spectra ◽  
Specular Reflectivity ◽  
Low Dimensional

ABSTRACTA method of using parametric B-spline curves to interpret neutron and x-ray specular reflectivity spectra is described. The introduction of parametric curves for scattering length density profiles greatly expands the function space accessible to low-dimensional representations but also requires means to restrict the space to physically acceptable functions. A practical fitting procedure is outlined, and two examples are shown.

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