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.

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.

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.

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.

Kantorovich-Rubinstein misfit for inverting gravity-gradient data by the level-set method

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. G55-G73
Author(s):  
Guanghui Huang ◽  
Xinming Zhang ◽  
Jianliang Qian
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Random Noise ◽  
Magnetic Data ◽  
Gravity Gradient ◽  
Local Minima ◽  
Target Model ◽  
Level Set Function ◽  
Level Set Evolution ◽  
Set Function

We have developed a novel Kantorovich-Rubinstein (KR) norm-based misfit function to measure the mismatch between gravity-gradient data for the inverse gradiometry problem. Under the assumption that an anomalous mass body has an unknown compact support with a prescribed constant value of density contrast, we implicitly parameterize the unknown mass body by a level-set function. Because the geometry of an underlying anomalous mass body may experience various changes during inversion in terms of level-set evolution, the classic least-squares ([Formula: see text]-norm-based) and the [Formula: see text]-norm-based misfit functions for governing the level-set evolution may potentially induce local minima if an initial guess of the level-set function is far from that of the target model. The KR norm from the optimal transport theory computes the data misfit by comparing the modeled data and the measured data in a global manner, leading to better resolution of the differences between the inverted model and the target model. Combining the KR norm with the level-set method yields a new effective methodology that is not only able to mitigate local minima but is also robust against random noise for the inverse gradiometry problem. Numerical experiments further demonstrate that the new KR norm-based misfit function is able to recover deep dipping flanks of SEG/EAGE salt models even at extremely low signal-to-noise ratios. The new methodology can be readily applied to gravity and magnetic data as well.

Numerical Simulation of Breaking Waves Using a Level Set Method

2002 ◽  
Author(s):  
Pablo Go´mez ◽  
Julio Herna´ndez ◽  
Joaqui´n Lo´pez ◽  
Fe´lix Faura
Keyword(s):  
Free Surface ◽  
Level Set Method ◽  
Level Set ◽  
Stokes Equations ◽  
Numerical Study ◽  
Operating Conditions ◽  
Breaking Wave ◽  
Level Set Function ◽  
Set Function ◽  
The Impact

A numerical study of the initial stages of wave breaking processes in shallow water is presented. The waves considered are assumed to be generated by moving a piston in a two-dimensional channel, and may appear, for example, in the injection chamber of a high-pressure die casting machine under operating conditions far from the optimal. A numerical model based on a finite-difference discretization of the Navier-Stokes equations in a Cartesian grid and a second-order approximate projection method has been developed and used to carry out the simulations. The evolution of the free surface is described using a level set method, with a reinitialization procedure of the level set function which uses a local grid refinement near the free surface. The ability of different algorithms to improve mass conservation in the reinitialization step of the level set function has been tested in a time-reversed single vortex flow. The results for the breaking wave profiles show the flow characteristics after the impact of the first plunging jet onto the wave’s forward face and during the subsequent splash-up.

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 local level-set method for 3D inversion of gravity-gradient data

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. G35-G51 ◽  
Author(s):  
Wangtao Lu ◽  
Jianliang Qian
Keyword(s):  
Inverse Problem ◽  
Level Set Method ◽  
Level Set ◽  
Local Level ◽  
Density Contrast ◽  
Gravity Gradient ◽  
Gravity Center ◽  
Level Set Function ◽  
Local Level Set ◽  
Set Function

We have developed a local level-set method for inverting 3D gravity-gradient data. To alleviate the inherent nonuniqueness of the inverse gradiometry problem, we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain [Formula: see text] so that we may convert the original inverse problem into a domain inverse problem. Because the unknown domain [Formula: see text] may take a variety of shapes, we parametrized the domain [Formula: see text] by a level-set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level-set function. Because the convergence of the level-set algorithm relied heavily on initializing the level-set function to enclose the gravity center of a source body, we applied a weighted [Formula: see text]-regularization method to locate such a gravity center so that the level-set function can be properly initialized. To rapidly compute the gradient of the nonlinear functional arising in the level-set formulation, we made use of the fact that the Laplacian kernel in the gravity force relation decayed rapidly off the diagonal so that matrix-vector multiplications for evaluating the gradient can be accelerated significantly. We conducted extensive numerical experiments to test the performance and effectiveness of the new method.

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