Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations
SIAM Journal on Numerical Analysis
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
A level set approach for computing solutions to incompressible two-phase flow
Journal of Computational Physics
A variational level set approach to multiphase motion
Journal of Computational Physics
Sequential and Parallel Splitting Methods for Bilinear Control Problems in Hilbert Spaces
SIAM Journal on Numerical Analysis
On the Topological Derivative in Shape Optimization
SIAM Journal on Control and Optimization
Maximizing band gaps in two-dimensional photonic crystals
SIAM Journal on Applied Mathematics
Band structure optimization of two-dimensional photonic crystals in H-polarization
Journal of Computational Physics
Structural boundary design via level set and immersed interface methods
Journal of Computational Physics
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Level set methods: an overview and some recent results
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Scientific Computing
Journal of Computational Physics
Incorporating topological derivatives into level set methods
Journal of Computational Physics
Structural optimization using sensitivity analysis and a level-set method
Journal of Computational Physics
Analysis of iterative algorithms of Uzawa type for saddle point problems
Applied Numerical Mathematics
A multilevel, level-set method for optimizing eigenvalues in shape design problems
Journal of Computational Physics
An extended level set method for shape and topology optimization
Journal of Computational Physics
Image Segmentation Using Some Piecewise Constant Level Set Methods with MBO Type of Projection
International Journal of Computer Vision
A piecewise constant level set method for elliptic inverse problems
Applied Numerical Mathematics
Incorporating topological derivatives into shape derivatives based level set methods
Journal of Computational Physics
Level set method with topological derivatives in shape optimization
International Journal of Computer Mathematics - INNOVATIVE ALGORITHMS IN SCIENCE AND ENGINEERING
A variational level set method for the topology optimization of steady-state Navier-Stokes flow
Journal of Computational Physics
Journal of Scientific Computing
Design of piezoelectric actuators using a multiphase level set method of piecewise constants
Journal of Computational Physics
Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design
Journal of Computational Physics
A binary level set model and some applications to Mumford-Shah image segmentation
IEEE Transactions on Image Processing
Level set method for the inverse elliptic problem in nonlinear electromagnetism
Journal of Computational Physics
Applied Numerical Mathematics
Journal of Mathematical Imaging and Vision
Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems
Journal of Scientific Computing
Hi-index | 31.45 |
We apply the piecewise constant level set method to a class of eigenvalue related two-phase shape optimization problems. Based on the augmented Lagrangian method and the Lagrange multiplier approach, we propose three effective variational methods for the constrained optimization problem. The corresponding gradient-type algorithms are detailed. The first Uzawa-type algorithm having applied to shape optimization in the literature is proven to be effective for our model, but it lacks stability and accuracy in satisfying the geometry constraint during the iteration. The two other novel algorithms we propose can overcome this limitation and satisfy the geometry constraint very accurately at each iteration. Moreover, they are both highly initial independent and more robust than the first algorithm. Without penalty parameters, the last projection Lagrangian algorithm has less severe restriction on the time step than the first two algorithms. Numerical results for various instances are presented and compared with those obtained by level set methods. The comparisons show effectiveness, efficiency and robustness of our methods. We expect our promising algorithms to be applied to other shape optimization and multiphase problems.