scholarly journals A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis

2016 ◽  
Vol 18 (3) ◽  
pp. 317-353 ◽  
Author(s):  
Daniel Kraft
Keyword(s):  
Sensitivity Analysis ◽  
Time Evolution ◽  
Level Set ◽  
Shape Sensitivity ◽  
Shape Sensitivity Analysis ◽  
New Approach

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.

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.

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