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 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.

Design of Compliant Thermal Actuators Using Structural Optimization Based on the Level Set Method

2011 ◽  
Vol 11 (1) ◽  
Author(s):  
Takayuki Yamada ◽  
Shintaro Yamasaki ◽  
Shinji Nishiwaki ◽  
Kazuhiro Izui ◽  
Masataka Yoshimura
Keyword(s):  
Structural Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Compliant Mechanisms ◽  
Optimization Method ◽  
Equilibrium Equations ◽  
Thermal Actuators ◽  
Level Set Function ◽  
Initialization Scheme ◽  
Set Function

Compliant mechanisms are designed to be flexible to achieve a specified motion as a mechanism. Such mechanisms can function as compliant thermal actuators in micro-electromechanical systems by intentionally designing configurations that exploit thermal expansion effects in elastic material when appropriate portions of the mechanism structure are heated or are subjected to an electric potential. This paper presents a new structural optimization method for the design of compliant thermal actuators based on the level set method and the finite element method (FEM). First, an optimization problem is formulated that addresses the design of compliant thermal actuators considering the magnitude of the displacement at the output location. Next, the topological derivatives that are used when introducing holes during the optimization process are derived. Based on the optimization formulation, a new structural optimization algorithm is constructed that employs the FEM when solving the equilibrium equations and updating the level set function. The re-initialization of the level set function is performed using a newly developed geometry-based re-initialization scheme. Finally, several design examples are provided to confirm the usefulness of the proposed structural optimization method.

A Meshless Level Set Method for Shape and Topology Optimization

2011 ◽  
Vol 308-310 ◽  
pp. 1046-1049 ◽  
Author(s):  
Yu Wang ◽  
Zhen Luo
Keyword(s):  
Topology Optimization ◽  
Free Boundary ◽  
Level Set Method ◽  
Level Set ◽  
Level Set Methods ◽  
Discrete Level ◽  
Shape And Topology Optimization ◽  
Level Set Function ◽  
And Topology ◽  
Set Function

This paper proposes a meshless Galerkin level set method for structural shape and topology optimization of continua. To taking advantage of the implicit free boundary representation scheme, structural design boundary is represented through the introduction of a scalar level set function as its zero level set, to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface. Compactly supported radial basis functions (CSRBFs) are used to parameterize the level set function and also to construct the shape functions for mesh free function approximation. The meshless Galerkin global weak formulation is employed to implement the discretization of the state equations. This provides a pathway to simplify two numerical procedures involved in most conventional level set methods in propagating the discrete level set functions and in approximating the discrete equations, by unifying the two different stages at two sets of grids just in terms of one set of scattered nodes. The proposed level set method has the capability of describing the implicit moving boundaries without remeshing for discontinuities. The motion of the free boundary is just a question of advancing the discrete level set function by finding the design variables of the size optimization in time. One benchmark example is used to demonstrate the effectiveness of the proposed method. The numerical results showcase that this method has the ability to simplify numerical procedures and to avoid numerical difficulties happened in most conventional level set methods. It is straightforward to apply the present method to more advanced shape and topology optimization problems.

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.

Topology Optimization of Total Femur Structure: Application of Parameterized Level Set Method Under Geometric Constraints

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Xiaowei Deng ◽  
Yingjun Wang ◽  
Jinhui Yan ◽  
Tao Liu ◽  
Shuting Wang
Keyword(s):  
Topology Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Narrow Band ◽  
Geometric Constraints ◽  
Contour Method ◽  
Viable Option ◽  
Level Set Function ◽  
Set Function ◽  
Constraint Level

Optimization of the femur prosthesis is a key issue in femur replacement surgeries that provide a viable option for limb salvage rather than amputation. To overcome the drawback of the conventional techniques that do not support topology optimization of the prosthesis design, a parameterized level set method (LSM) topology optimization with arbitrary geometric constraints is presented. A predefined narrow band along the complex profile of the original femur is preserved by applying the contour method to construct the level set function, while the topology optimization is carried out inside the cavity. The Boolean R-function is adopted to combine the free boundary and geometric constraint level set functions to describe the composite level set function of the design domain. Based on the minimum compliance goal, three different designs of 2D femur prostheses subject to the target cavity fill ratios 34%, 54%, and 74%, respectively, are illustrated.

Design of Compliant Thermal Actuators Using Structural Optimization Based on the Level Set Method

2008 ◽  
Author(s):  
Takayuki Yamada ◽  
Shintaro Yamasaki ◽  
Shinji Nishiwaki ◽  
Kazuhiro Izui ◽  
Masataka Yoshimura
Keyword(s):  
Structural Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Compliant Mechanisms ◽  
Optimization Method ◽  
Equilibrium Equations ◽  
Micro Electro Mechanical Systems ◽  
Thermal Actuators ◽  
Level Set Function ◽  
Set Function

Compliant mechanisms are a new type of mechanism, designed to be flexible to achieve a specified motion as a mechanism. Such mechanisms can function as compliant thermal actuators in Micro-Electro Mechanical Systems (MEMS) by intentionally designing configurations that exploit thermal expansion effects in elastic material when appropriate portions of the mechanism structure are heated. This paper presents a new structural optimization method for the design of compliant thermal actuators based on the level set method and the Finite Element Method (FEM). First, an optimization problem is formulated that addresses the design of compliant thermal actuators considering the magnitude of the displacement at the output location. Next, the topological derivatives that are used when introducing holes during the optimization process are derived. Based on the optimization formulation and the level set method, a new structural optimization algorithm is constructed that employs the FEM when solving the equilibrium equations and updating the level set function. The re-initialization of the level set function is performed using a newly developed geometry-based re-initialization scheme. Finally, several design examples are provided to confirm the usefulness of the proposed structural optimization method.

Multi-Stage Design Method for Practical Compliant Mechanisms by Topology and Shape Optimizations and Shape Conversion Method Utilizing Level Set Method

2008 ◽  
Author(s):  
Masakazu Kobayashi ◽  
Shinji Nishiwaki ◽  
Masatake Higashi
Keyword(s):  
Topology Optimization ◽  
Shape Optimization ◽  
Level Set Method ◽  
Level Set ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Design Method ◽  
Stage Design ◽  
Conversion Method ◽  
Multi Stage

This paper proposes a multi-stage design method for a design of practical compliant mechanisms. The proposed method consists of topology and shape optimizations and a shape conversion method that incorporates two optimizations. In the 1st stage, an initial and conceptual compliant mechanism is created by topology optimization. In the 2nd stage, an initial model of shape optimization is created from the result of topology optimization by the shape conversion method based on the level set method. In the 3rd stage, the shape optimization yields a detailed shape of the compliant mechanism by considering non-linear deformation and stress concentration. Execution of the shape optimization after the topology optimization enables evaluation of stress concentration and large deformation effect that are normally difficult for the traditional topology optimization. On the other side, the precise conversion from the model by topology optimization to the one for the shape optimization becomes possible by the shape conversion method that is utilizing the level set method. Using the proposed multi-stage method, a practical compliant mechanism can be designed with the designer’s minimum efforts that are indications of design conditions of the topology and shape optimizations and several parameters and threshold values of the shape conversion method.

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.

scholarly journals Level set-based topology optimisation of a compliant mechanism design using mathematical programming

2011 ◽  
Vol 2 (1) ◽  
pp. 91-98 ◽  
Author(s):  
M. Otomori ◽  
T. Yamada ◽  
K. Izui ◽  
S. Nishiwaki
Keyword(s):  
Mathematical Programming ◽  
Mechanism Design ◽  
Level Set ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Topology Optimisation ◽  
Structural Optimisation ◽  
Level Set Function ◽  
Set Function ◽  
Compliant Mechanism Design

Abstract. We propose a structural optimisation method, based on the level set method and using mathematical programming such as the method of moving asymptotes (MMA), which we apply to the design of compliant mechanisms. A compliant mechanism is a monolithic joint-free mechanism designed to be flexible to obtain a specified motion. In the design of compliant mechanisms, several requirements such as the direction of the deformation and stress concentrations must be considered to obtain the specified mechanical function. Topology optimisation, the most flexible type of structural optimisation, has been successfully used as a design optimisation method for compliant mechanisms, but the utility of topology optimisation results is often spoiled by a plethora of impractical designs such as structures containing grayscale areas. Level set-based topology optimisation methods are immune to the problem of grayscales since the boundaries of the optimal configuration are implicitly represented using the level set function. The proposed method updates the level set function using mathematical programming to facilitate the treatment of constraint functionals. To verify its capability, we apply our method to compliant mechanism design problems that include displacement constraints and stress constraints.

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.

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