Level Set Method for Topological Optimization of the Naiver-Stokes Fluid Flow

2012 ◽  
Vol 182-183 ◽  
pp. 1668-1672
Author(s):  
Xian Bao Duan ◽  
Fu Cai Qian ◽  
Xin Qiang Qin
Keyword(s):  
Sensitivity Analysis ◽  
Fluid Flow ◽  
Level Set Method ◽  
Level Set ◽  
Topological Optimization ◽  
Shape Sensitivity ◽  
Shape Sensitivity Analysis ◽  
Material Derivative ◽  
Level Set Function ◽  
Stokes Fluid

This paper presents a general algorithm for topological optimization of the incompressible Navier-Stokes fluid flow based on a level set method. This is a direct extension of our previous work on Stokes flow of such problems. First we obtain the shape sensitivity analysis using the material derivative concept and adjoint variable technique, and then we couple the shape sensitivity analysis result into the level set function as the advection velocity. Since the level set method is implemented in an Euleran framework, the computational cost of the proposed algorithm is moderate. A Benchmark example is provided to illustrate the efficiency and validity of this method.

RECONSTRUCTION OF SINGULAR SURFACES BY SHAPE SENSITIVITY ANALYSIS AND LEVEL SET METHOD

2006 ◽  
Vol 16 (08) ◽  
pp. 1347-1373 ◽  
Author(s):  
GUILLAUME BAL ◽  
KUI REN
Keyword(s):  
Sensitivity Analysis ◽  
Velocity Field ◽  
Level Set Method ◽  
Level Set ◽  
Diffusion Processes ◽  
Shape Sensitivity ◽  
Shape Sensitivity Analysis ◽  
Singular Surfaces ◽  
Impedance Tomography ◽  
Dimension One

We consider the reconstruction of singular surfaces from the over-determined boundary conditions of an elliptic problem. The problem arises in optical and impedance tomography, where void-like structure or cracks may be modeled as diffusion processes supported on co-dimension one surfaces. The reconstruction of such surfaces is obtained theoretically and numerically by combining a shape sensitivity analysis with a level set method. The shape sensitivity analysis is used to define a velocity field, which allows us to update the surface while decreasing a given cost function, which quantifies the error between the prediction of the forward model and the measured data. The velocity field depends on the geometry of the surface and the tangential diffusion process supported on it. The latter process is assumed to be known in this paper. The level set method is next applied to evolve the surface in the direction of the velocity field. Numerical simulations show how the surface may be reconstructed from noisy estimates of the full, or local, Neumann-to-Dirichlet map.

A Velocity Predictor–Corrector Scheme in Level Set-Based Topology Optimization to Improve Computational Efficiency

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Benliang Zhu ◽  
Xianmin Zhang ◽  
Sergej Fatikow
Keyword(s):  
Sensitivity Analysis ◽  
Topology Optimization ◽  
Velocity Field ◽  
Level Set Method ◽  
Level Set ◽  
Velocity Fields ◽  
Shape Sensitivity ◽  
Shape Sensitivity Analysis ◽  
Predictor Corrector ◽  
Mean Compliance

This paper presents an optimization method for solving level set-based topology optimization problems. A predictor–corrector scheme for constructing the velocity field is developed. In this method, after the velocity fields in the first two iterations are calculated using the shape sensitivity analysis, the subsequent velocity fields are constructed based on those obtained from the first two iterations. To ensure stability, the velocity field is renewed based on the shape sensitivity analysis after a certain number of iterations. The validity of the proposed method is tested on the mean compliance minimization problem and the compliant mechanisms synthesis problem. This method is quantitatively compared with other methods, such as the standard level set method, the solid isotropic microstructure with penalization (SIMP) method, and the discrete level set method.

Shape Sensitivity Analysis of Linear-Elastic Cracked Structures Under Mode-I Loading

2002 ◽  
Vol 124 (4) ◽  
pp. 476-482 ◽  
Author(s):  
Guofeng Chen ◽  
Sharif Rahman ◽  
Young Ho Park
Keyword(s):  
Sensitivity Analysis ◽  
Finite Difference Methods ◽  
Mode I ◽  
J Integral ◽  
Shape Sensitivity ◽  
Shape Sensitivity Analysis ◽  
Material Derivative ◽  
Linear Elastic ◽  
Mode I Loading ◽  
Element Method

A new method is presented for shape sensitivity analysis of a crack in a homogeneous, isotropic, and linear-elastic body subject to mode-I loading conditions. The method involves the material derivative concept of continuum mechanics, domain integral representation of the J-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques, such as the finite element method, boundary element method, or others. Since the J-integral is represented by domain integration, only the first-order sensitivity of displacement field is needed. Two numerical examples are presented to illustrate the proposed method. The results show that the maximum difference in calculating the sensitivity of J-integral by the proposed method and reference solutions by analytical or finite-difference methods is less than three percent.

Minimizing the quadratic mean temperature gradient for the heat-conduction problem using the level-set method

2007 ◽  
Vol 221 (2) ◽  
pp. 235-248 ◽  
Author(s):  
C G Zhuang ◽  
Z H Xiong ◽  
H Ding
Keyword(s):  
Heat Conduction ◽  
Temperature Gradient ◽  
Objective Function ◽  
Level Set Method ◽  
Level Set ◽  
Heat Conduction Problem ◽  
Shape Sensitivity ◽  
Material Derivative ◽  
Quadratic Mean ◽  
Mean Temperature

This paper presents a numerical algorithm for minimizing the quadratic mean temperature gradient for the heat-conduction problem on the basis of the shape derivative for an elliptical system and the level-set method for a propagating surface. The level-set method as an implicit boundary model is employed to represent the optimal boundaries of heat transfer material. The objective function of the optimization problem is the quadratic mean temperature gradient. The shape of physical domain is treated as the design variable. The material derivative theory of the continuum mechanics and the adjoint method are used to implement the shape sensitivity analysis of the objective function. Since the level-set approach itself cannot generate new holes in the material region, as a remedy, the topological derivative of the elliptic equations that generates new holes to suppress the topological dependence of initialization is introduced. Numerical examples demonstrate that the proposed method is an effective technique for the optimal design of the heat-conduction problem.

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