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.

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.

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

scholarly journals Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2198
Author(s):  
Nikolaj Ezhov ◽  
Frank Neitzel ◽  
Svetozar Petrovic
Keyword(s):  
Numerical Stability ◽  
Spline Approximation ◽  
Point Of View ◽  
Direct Relation ◽  
Monomial Basis ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
Spline Representation ◽  
Lagrangian Form

In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

Topology Optimization of the Caudal Fin of the Three-Dimensional Self-Propelled Swimming Fish

2014 ◽  
Vol 6 (06) ◽  
pp. 732-763 ◽  
Author(s):  
Zhiqiang Xin ◽  
Chuijie Wu
Keyword(s):  
Topology Optimization ◽  
Swimming Speed ◽  
Moving Boundary ◽  
Swimming Performance ◽  
Three Dimensional ◽  
The Body ◽  
Caudal Fin ◽  
Swimming Efficiency ◽  
Vorticity Flux ◽  
3D Topology

AbstractBased on the boundary vorticity-flux theory, topology optimization of the caudal fin of the three-dimensional self-propelled swimming fish is investigated by combining unsteady computational fluid dynamics with moving boundary and topology optimization algorithms in this study. The objective functional of topology optimization is the function of swimming efficiency, swimming speed and motion direction control. The optimal caudal fin, whose topology is different from that of the natural fish caudal fin, make the 3D bionic fish achieve higher swimming efficiency, faster swimming speed and better maneuverability. The boundary vorticity-flux on the body surface of the 3D fish before and after optimization reveals the mechanism of high performance swimming of the topology optimization bionic fish. The comparative analysis between the swimming performance of the 3D topology optimization bionic fish and the 3D lunate tail bionic fish is also carried out, and the wake structures of two types of bionic fish show the physical nature that the swimming performance of the 3D topology optimization bionic fish is significantly better than the 3D lunate tail bionic fish.

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.

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.

scholarly journals 3D Topology Optimization and Mesh Dependency for Redesigning Locking Compression Plates Aiming to Reduce Stress Shielding

2021 ◽  
Vol 7 (3) ◽  
pp. 339
Author(s):  
A. A. Al-Tamimi
Keyword(s):  
Topology Optimization ◽  
Three Dimensional ◽  
Stress Shielding ◽  
Load Condition ◽  
Metallic Materials ◽  
Metallic Material ◽  
Combined Loads ◽  
Locking Compression Plates ◽  
Strain Energy Minimization ◽  
3D Topology

Current fixation plates for bone fracture treatments are built with biocompatible metallic materials such as stainless steel, titanium, and its alloys (e.g., Ti6Al4V). The stiffness mismatch between the metallic material of the plate and the host bone leads to stress shielding phenomena, bone loss, and healing deficiency. This paper explores the use of three dimensional topology-optimization, based on compliance (i.e., strain energy) minimization, reshaping the design domain of three locking compression plates (four-screw holes, six-screw holes, and eight-screw holes), considering different volume reductions (25, 45, and 75%) and loading conditions (bending, compression, torsion, and combined loads). A finite-element study was also conducted to measure the stiffness of each optimized plate. Thirty-six designs were obtained. Results showed that for a critical value of volume reductions, which depend on the load condition and number of screws, it is possible to obtain designs with lower stiffness, thereby reducing the risk of stress shielding.

scholarly journals A NON-GRADIENT APPROACH FOR THREE DIMENSIONAL TOPOLOGY OPTIMIZATION

2021 ◽  
Vol 59 (3) ◽  
pp. 368
Author(s):  
Minh Ngoc Nguyen ◽  
Nha Thanh Nguyen ◽  
Minh Tuan Tran
Keyword(s):  
Topology Optimization ◽  
Comparative Study ◽  
Isotropic Material ◽  
Three Dimensional ◽  
Logistic Function ◽  
Interpolation Scheme ◽  
Compliance Minimization ◽  
Material Interpolation ◽  
Density Filter ◽  
Gradient Approach

The present work is devoted to the extension of the non-gradient approach, namely Proportional Topology Optimization (PTO), for compliance minimization of three-dimensional (3D) structures. Two schemes of material interpolation within the framework of the solid isotropic material with penalization (SIMP), i.e. the power function and the logistic function are analyzed. Through a comparative study, the efficiency of the logistic-type interpolation scheme is highlighted.  Since no sensitivity is involved in the approach, a density filter is applied instead of sensitivity filter to avoid checkerboard issue

The Synthesis of Follower Motions of Camoids Using Nonparametric B-Splines

1996 ◽  
Vol 118 (1) ◽  
pp. 138-143 ◽  
Author(s):  
Der Min Tsay ◽  
Guan Shyong Hwang
Keyword(s):  
Spline Functions ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
Surface Interpolation ◽  
Spline Surface ◽  
Motion Function ◽  
Knot Sequences ◽  
Independent Parameters

This paper proposes a tool to synthesize the motion functions of the camoid-follower mechanisms. The characteristics of these kinds of motion functions are that they possess two independent parameters. To implement the work, this study applies the nonparametric B-spline surface interpolation, whose spline functions are constructed by the closed periodic B-splines and the de Boor’s knot sequences in the two parametric directions of the motion function, respectively. The rules and the restrictions needed to be noticed in the process of synthesis are established. Numerical examples are also given to verify the feasibility of the proposed method.

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.

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