A Level Set Method With a Bounded Diffusion for Structural Topology Optimization

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Benliang Zhu ◽  
Rixin Wang ◽  
Hai Li ◽  
Xianmin Zhang
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Jacobi Equation ◽  
Structural Topology ◽  
Signed Distance Function ◽  
Diffusion Term ◽  
Signed Distance ◽  
Level Set Function ◽  
Set Function ◽  
Hamilton Jacobi Equation

In level-set-based topology optimization methods, the spatial gradients of the level set field need to be controlled to avoid excessive flatness or steepness at the structural interfaces. One of the most commonly utilized methods is to generalize the traditional Hamilton−Jacobi equation by adding a diffusion term to control the level set function to remain close to a signed distance function near the structural boundaries. This study proposed a new diffusion term and built it into the Hamilton-Jacobi equation. This diffusion term serves two main purposes: (I) maintaining the level set function close to a signed distance function near the structural boundaries, thus avoiding periodic re-initialization, and (II) making the diffusive rate function to be a bounded function so that a relatively large time-step can be used to speed up the evolution of the level set function. A two-phase optimization algorithm is proposed to ensure the stability of the optimization process. The validity of the proposed method is numerically examined on several benchmark design problems in structural topology optimization.

Improvement of topology optimization method based on level set function in magnetic field problem

2018 ◽  
Vol 37 (2) ◽  
pp. 630-644 ◽  
Author(s):  
Yoshifumi Okamoto ◽  
Hiroshi Masuda ◽  
Yutaro Kanda ◽  
Reona Hoshino ◽  
Shinji Wakao
Keyword(s):  
Magnetic Field ◽  
Topology Optimization ◽  
Level Set ◽  
Jacobi Equation ◽  
Field Problem ◽  
Content Type ◽  
Topology Optimization Problem ◽  
Level Set Function ◽  
Set Function ◽  
Hamilton Jacobi Equation

PurposeThe purpose of this paper is the improvement of topology optimization. The scope of the paper is focused on the speedup of optimization. Design/methodology/approachTo achieve the speedup, the method of moving asymptotes (MMA) with constrained condition of level set function is applied instead of solving the Hamilton–Jacobi equation. FindingsThe acceleration of convergence of objective function is drastically improved by the implementation of MMA. Originality/valueNormally, the level set method is solved through the Hamilton–Jacobi equation. However, the possibility of introducing mathematical programming is clear by the constrained condition. Furthermore, the proposed method is suitable for efficiently solving the topology optimization problem in the magnetic field system.

Generative Design of Multi-Material Hierarchical Structures via Concurrent Topology Optimization and Conformal Geometry Method

2019 ◽  
Author(s):  
Long Jiang ◽  
Shikui Chen ◽  
Xianfeng David Gu
Keyword(s):  
Topology Optimization ◽  
Conformal Mapping ◽  
Level Set ◽  
Basis Function ◽  
Material Properties ◽  
Structural Topology Optimization ◽  
Structural Topology ◽  
Signed Distance ◽  
Level Set Function ◽  
Set Function

Abstract Topology optimization has been proved to be an automatic, efficient and powerful tool for structural designs. In recent years, the focus of structural topology optimization has evolved from mono-scale, single material structural designs to hierarchical multimaterial structural designs. In this research, the multi-material structural design is carried out in a concurrent parametric level set framework so that the structural topologies in the macroscale and the corresponding material properties in mesoscale can be optimized simultaneously. The constructed cardinal basis function (CBF) is utilized to parameterize the level set function. With CBF, the upper and lower bounds of the design variables can be identified explicitly, compared with the trial and error approach when the radial basis function (RBF) is used. In the macroscale, the ‘color’ level set is employed to model the multiple material phases, where different materials are represented using combined level set functions like mixing colors from primary colors. At the end of this optimization, the optimal material properties for different constructing materials will be identified. By using those optimal values as targets, a second structural topology optimization is carried out to determine the exact mesoscale metamaterial structural layout. In both the macroscale and the mesoscale structural topology optimization, an energy functional is utilized to regularize the level set function to be a distance-regularized level set function, where the level set function is maintained as a signed distance function along the design boundary and kept flat elsewhere. The signed distance slopes can ensure a steady and accurate material property interpolation from the level set model to the physical model. The flat surfaces can make it easier for the level set function to penetrate its zero level to create new holes. After obtaining both the macroscale structural layouts and the mesoscale metamaterial layouts, the hierarchical multimaterial structure is finalized via a local-shape-preserving conformal mapping to preserve the designed material properties. Unlike the conventional conformal mapping using the Ricci flow method where only four control points are utilized, in this research, a multi-control-point conformal mapping is utilized to be more flexible and adaptive in handling complex geometries. The conformally mapped multi-material hierarchical structure models can be directly used for additive manufacturing, concluding the entire process of designing, mapping, and manufacturing.

Topology Optimization of Compliant Mechanisms Using Level Set Method without Re-Initialization

2011 ◽  
Vol 130-134 ◽  
pp. 3076-3082 ◽  
Author(s):  
Ben Liang Zhu ◽  
Xian Min Zhang
Keyword(s):  
Topology Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Signed Distance Function ◽  
Optimal Process ◽  
Level Set Function ◽  
New Formulation ◽  
Set Function

In this paper, a new level set method for topology optimization of compliant mechanisms is presented. A new formulation is developed and built in the traditional level set method to force the level set function to be close to a signed distance function during the optimal process. The validity of the method is illustrated by topology optimization of a widely studied compliant mechanism.

Parametric Structural Shape and Topology Optimization With a Variational Distance-Regularized Level Set Method

2016 ◽  
Author(s):  
Long Jiang ◽  
Shikui Chen
Keyword(s):  
Topology Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Optimization Process ◽  
Signed Distance ◽  
Level Set Function ◽  
Distance Property ◽  
Initialization Scheme ◽  
Set Function ◽  
Distance Regularization

In conventional level set methods, the slope of the level set function needs to be well controlled to maintain the numerical stability during the topology optimization process. One common solution is to regularize the level set function to be a signed distance function, which is usually achieved by periodically implementing the so called re-initialization scheme to force the level set function to gain the desired signed distance property. However, the re-initialization scheme will bring some unwanted drawbacks to the optimization process, such as zero level set drifting, time consuming etc. In addition, re-initialization is usually implemented outside the optimization loop, which will cause convergence issues. In this paper, a distance regularization functional is introduced to the structural topology optimization objective functional to ensure the signed distance property of the level set function near the structure boundaries. This functional can also keep the level set function to be constant-value at positions far away from the structural boundaries. The radial basis function (RBF) based parameterization technique together with the mathematical programming are utilized to improve the potential capability of handling multiple constraints for the topology optimization. The combination of these two techniques makes the level set based topology optimization be capable of handling complicated multi-constrained problems with higher numerical efficiency, leaving no compromise to multiple drawbacks. To demonstrate the validity of the proposed scheme, benchmark examples on minimum compliance structural optimization are employed. This type of problem is computed by the conventional level set method with the introduced distance regularization functional, the RBF based parametric level set and at last, the distance regularized RBF based parametric level set separately to demonstrate their differences.

A topology optimization algorithm based on topological derivative and level-set function for designing phononic crystals

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rolando Yera ◽  
Luisina Forzani ◽  
Carlos Gustavo Méndez ◽  
Alfredo E. Huespe
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Optimization Algorithm ◽  
Phononic Crystal ◽  
Phononic Crystals ◽  
Lagrangian Function ◽  
Topological Derivative ◽  
Content Type ◽  
Level Set Function ◽  
Set Function

PurposeThis work presents a topology optimization methodology for designing microarchitectures of phononic crystals. The objective is to get microstructures having, as a consequence of wave propagation phenomena in these media, bandgaps between two specified bands. An additional target is to enlarge the range of frequencies of these bandgaps.Design/methodology/approachThe resulting optimization problem is solved employing an augmented Lagrangian technique based on the proximal point methods. The main primal variable of the Lagrangian function is the characteristic function determining the spatial geometrical arrangement of different phases within the unit cell of the phononic crystal. This characteristic function is defined in terms of a level-set function. Descent directions of the Lagrangian function are evaluated by using the topological derivatives of the eigenvalues obtained through the dispersion relation of the phononic crystal.FindingsThe description of the optimization algorithm is emphasized, and its intrinsic properties to attain adequate phononic crystal topologies are discussed. Particular attention is addressed to validate the analytical expressions of the topological derivative. Application examples for several cases are presented, and the numerical performance of the optimization algorithm for attaining the corresponding solutions is discussed.Originality/valueThe original contribution results in the description and numerical assessment of a topology optimization algorithm using the joint concepts of the level-set function and topological derivative to design phononic crystals.

scholarly journals A local solution approach for level-set based structural topology optimization in isogeometric analysis

2020 ◽  
Vol 7 (4) ◽  
pp. 514-526
Author(s):  
Zijun Wu ◽  
Shuting Wang ◽  
Renbin Xiao ◽  
Lianqing Yu
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Isogeometric Analysis ◽  
Iteration Process ◽  
Solid Material ◽  
Optimization Approach ◽  
Structural Topology ◽  
Zero Level ◽  
Solution Approach ◽  
Level Set Function

Abstract This paper develops a new topology optimization approach for minimal compliance problems based on the parameterized level set method in isogeometric analysis. Here, we choose the basis functions as level set functions. The design variables are obtained with Greville abscissae based on the corresponding collocation points. The zero-level set boundaries are derived from the level set function values of the interpolation points in all knot spans. In the optimization iteration process, the whole design domain is discretized into two types of subdomains around the zero-level set boundaries, undesign area with void materials and redesign domain with solid materials. To decrease the size of equations and the computational consumptions, only the solid material area is recalculated and the void material area is discarded according to the high accuracy of isogeometric analysis. Numerical examples demonstrate the validity of the proposed optimization method.

Parametric Shape and Topology Optimization: A New Level Set Approach Based on Cardinal Kernel Functions

2017 ◽  
Author(s):  
Long Jiang ◽  
Shikui Chen ◽  
Xiangmin Jiao
Keyword(s):  
Topology Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Material Property ◽  
Kernel Function ◽  
Level Set Methods ◽  
Level Set Function ◽  
Set Function ◽  
Distance Regularization ◽  
Parametric Level Set Method

The parametric level set method is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, conventional levels let methods can be easily coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Furthermore, the parametric level set scheme not only can inherit the original advantages of the conventional level set methods, such as clear boundary representation and high topological changes handling flexibility but also can alleviate some un-preferred features from the conventional level set methods, such as needing re-initialization. However, in the RBF-based parametric level set method, it was difficult to determine the range of the design variables. Moreover, with the mathematically driven optimization process, the level set function often results in significant fluctuations during the optimization process. This brings difficulties in both numerical stability control and material property interpolation. In this paper, an RBF partition of unity collocation method is implemented to create a new type of kernel function termed as the Cardinal Basis Function (CBF), which employed as the kernel function to parameterize the level set function. The advantage of using the CBF is that the range of the design variable, which was the weight factor in conventional RBF, can be explicitly specified. Additionally, a distance regularization energy functional is introduced to maintain a desired distance regularized level set function evolution. With this desired distance regularization feature, the level set evolution is stabilized against significant fluctuations. Besides, the material property interpolation from the level set function to the finite element model can be more accurate.

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