Piecewise Constant Level Set Method Research Articles

Topology optimization of multimaterial distribution based on isogeometric boundary element and piecewise constant level set method

Based on isogeometric boundary element method (IGA BEM) and piecewise constant level set (PCLS) method, a new topology optimization approach is proposed to design the distribution of sound absorbing materials (SAMs) on structural surfaces. First, an efficient IGA BEM is developed with the combination of multipatch discretization and parallel CPU design. A complex submarine model comprising multiple non-uniform rational B-spline surfaces is built for engineering optimization design. Second, the PCLS method is applied to represent the multiple SAM domains without gray elements, and the sensitivities of the objective function with respect to PCLS functions are obtained using the adjoint variable method. Finally, the optimization algorithm based on PCLS method is improved by a novel penalty coefficient formulation and ordered volume constraints. The improved optimization algorithm can realize the control of multimaterial volume fractions and make the objective function steadily converge for different volume structures without the need to change any parameters. The accuracy, efficiency, and applicability of the proposed approach in engineering are demonstrated by several numerical examples.

A Non-Ersatz Material Approach for the Topology Optimization of Elastic Structures Based on Piecewise Constant Level Set Method

A Non-Ersatz Material Approach for the Topology Optimization of Elastic Structures Based on Piecewise Constant Level Set Method

A piecewise constant level set method for damage identification of continuum structures based on natural frequencies

The damage identification of continuum structures from the observation of its natural frequencies is investigated. The piecewise constant level set (PCLS) method is applied to represent the shape and topology of the damaged region. The functional derivative of the objective function with respect to PCLS function is derived with the application of the continuum sensitivity analysis. A regularized restarted conjugate gradient algorithm is proposed. Identifications of single and multiple damaged regions in 2D cantilever structures are discussed, illustrating the validity of the proposed algorithm.

An approach for topology optimization of damping layer under harmonic excitations based on piecewise constant level set method

The topology optimization of a thin plate structure with bounded damping layer patches under external harmonic excitations to suppress the vibrations of specified points in the plate is investigated. The piecewise constant level set (PCLS) method is applied to represent the region with damping material and the region only with base material. Applying Melosh-Zienkiewicz-Cheung (MZC) element, the stiffness matrix, the mass matrix and the damping matrix are expressed in detail, where the global non-proportional damping matrix is considered. The functional derivative of the objective function (the squared vibration amplitudes of specified points in a plate) with respect to the PCLS function is deduced, by introducing the adjoint problems and applying matrix sensitivity analysis. The quadratic penalty method and the total variation regularization are utilized to fulfill the volume constraint and to avoid checkerboard patterns, respectively. A penalty gradient algorithm is proposed. In numerical experiments, three rectangular plates with different boundary conditions and locations of the external excitations are investigated. Effects of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.

An approach for maximizing the smallest eigenfrequency of structure vibration based on piecewise constant level set method

Maximization of the smallest eigenfrequency of the linearized elasticity system with area constraint is investigated. The elasticity system is extended into a large background domain, but the void is vacuum and not filled with ersatz material. The piecewise constant level set (PCLS) method is applied to present two regions, the original material region and the void region. A quadratic PCLS function is proposed to represent the characteristic function. Consequently, the functional derivative of the smallest eigenfrequency with respect to PCLS function takes nonzero value in the original material region and zero in the void region. A penalty gradient algorithm is proposed, which initializes the whole background domain with the original material and decreases the area of original material region till the area constraint is satisfied. 2D and 3D numerical examples are presented, illustrating the validity of the proposed algorithm.

Topology optimization of steady Navier-Stokes flow via a piecewise constant level set method

This paper presents a piecewise constant level set method for the topology optimization of steady Navier-Stokes flow. Combining piecewise constant level set functions and artificial friction force, the optimization problem is formulated and analyzed based on a design variable. The topology sensitivities are computed by the adjoint method based on Lagrangian multipliers. In the optimization procedure, the piecewise constant level set function is updated by a new descent method, without the needing to solve the Hamilton-Jacobi equation. To achieve optimization, the piecewise constant level set method does not track the boundaries between the different materials but instead through the regional division, which can easily create small holes without topological derivatives. Furthermore, we make some attempts to avoid updating the Lagrangian multipliers and to deal with the constraints easily. The algorithm is very simple to implement, and it is possible to obtain the optimal solution by iterating a few steps. Several numerical examples for both two- and three-dimensional problems are provided, to demonstrate the validity and efficiency of the proposed method.

Piecewise constant level set method for topology optimization of unilateral contact problems

The paper deals with the structural optimization of the elastic body in unilateral contact with a rigid foundation using the level set approach. A piecewise constant level set method is used to represent the evolution of interfaces rather than the standard method. The piecewise constant level set function takes distinct constant values in each subdomain of a whole design domain. Using a two-phase approximation the original structural optimization problem is reformulated as an equivalent constrained optimization problem in terms of the piecewise constant level set function. Necessary optimality condition is formulated. Finite difference and finite element methods are applied as the approximation methods. Numerical examples are provided and discussed.

Maximizing band gaps in two-dimensional photonic crystals in square lattices

This paper is devoted to a numerical algorithm for the maximization of band gaps in two-dimensional photonic crystals in square lattices. We first apply the finite element method to solve the eigenvalue problem, then use the piecewise constant level set (PCLS) method to maximize the band gaps. The PCLS method is very powerful for representing and modeling regions of different structures. Extremely large gaps are realized with gallium arsenide material, for transverse magnetic field (TM), transverse electric field (TE), and for complete band gaps. When the mean gap frequency is below 1, the biggest gap is about 0.2922 for the TE.

Piecewise constant level set method for structural topology optimization with MBO type of projection

In this paper, we combine a Piecewise Constant Level Set (PCLS) method with a MBO scheme to solve a structural shape and topology optimization problem. The geometrical boundary of structure is represented implicitly by the discontinuities of PCLS functions. Compared with the classical level set method (LSM) for solving Hamilton---Jacobi partial differential equation (H-J PDE) we don't need to solve H-J PDE, thus it is free of the CFL condition and the reinitialization scheme. For solving optimization problem under some constraints, Additive Operator Splitting (AOS) and Multiplicative Operator Splitting (MOS) schemes will be used. To increase the convergency speed and the efficiency of PCLS method we combine this approach with MBO scheme. Advantages and disadvantages are discussed by solving some examples of 2D structural topology optimization problems.

Efficient Dual Algorithms for Image Segmentation Using TV-Allen-Cahn Type Models

AbstractVariational image segmentation based on the Mumford and Shah model [31], together with implementation by the piecewise constant level-set method (PCLSM) [26], leads to fully nonlinear Total Variation (TV)-Allen-Cahn equations. The commonly-used numerical approaches usually suffer from the difficulties not only with the non-differentiability of the TV-term, but also with directly evolving the discontinuous piecewise constant-structured solutions. In this paper, we propose efficient dual algorithms to overcome these drawbacks. The use of a splitting-penalty method results in TV-Allen-Cahn type models associated with different “double-well” potentials, which allow for the implementation of the dual algorithm of Chambolle [8]. Moreover, we present a new dual algorithm based on an edge-featured penalty of the dual variable, which only requires to solve a vectorial Allen-Cahn type equation with linear ∇(div)-diffusion rather than fully nonlinear diffusion in the Chambolle’s approach. Consequently, more efficient numerical algorithms such as time-splitting method and Fast Fourier Transform (FFT) can be implemented. Various numerical tests show that two dual algorithms are much faster and more stable than the primal gradient descent approach, and the new dual algorithm is at least as efficient as the Chambolle’s algorithm but is more accurate. We demonstrate that the new method also provides a viable alternative for image restoration.

Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.

A boundary piecewise constant level set method for boundary control of eigenvalue optimization problems

This paper investigates optimization of the least eigenvalue of −Δ with the constraint of one-dimension Hausdorff measure of Dirichlet boundary. We propose the boundary piecewise constant level set (BPCLS) method based on the regularity technique to combine two types of boundary conditions into a single Robin boundary condition. We derive the first variation of the least eigenvalue w.r.t. the BPCLS function and propose a penalty BPCLS algorithm and an augmented Lagrangian BPCLS algorithm. Numerical results are reported for experiments on ellipse and L-shape domains.

Variational piecewise constant level set methods for shape optimization of a two-density drum

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.

3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization

3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization

Piecewise constant level set method for structural topology optimization

AbstractIn this paper, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem. In the classical level set method, the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function, e.g. the signed distance function. Instead, in the PCLS approach the boundary is described by discontinuities of PCLS functions. The PCLS method is related to the phase‐field methods, and the topology optimization problem is defined as a minimization problem with piecewise constant constraints, without the need of solving the Hamilton–Jacobi equation. The result is not moving the boundaries during the iterative procedure. Thus, it offers some advantages in treating geometries, eliminating the reinitialization and naturally nucleating holes when needed. In the paper, the PCLS method is implemented with the additive operator splitting numerical scheme, and several numerical and procedural issues of the implementation are discussed. Examples of 2D structural topology optimization problem of minimum compliance design are presented, illustrating the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

An Improved Mumford-Shah Model and Its Applications to Image Processing with the Piecewise Constant Level Set Method

For quick segmentation and denoising, the classical Mumford-Shah (MS) model needs to enhance the penalization term, i.e. to increase the penalization parameter, which leads to gradual disappearance of objects. In this paper, we propose an improved Mumford-Shah (IMS) model to avoid the phenomenon, and adopt the piecewise constant level set method (PCLSM) and the gradient descent method to solve the minimization problem. Numerical experiments are given to show the efficiency and advantages of the new model and the algorithms.

A piecewise constant level set method for elliptic inverse problems

We apply a piecewise constant level set method to elliptic inverse problems. The discontinuity of the coefficients is represented implicitly by a piecewise constant level set function, which allows to use one level set function to represent multiple phases. The inverse problem is solved using a variational penalization method with the total variation regularization of the coefficients. An operator splitting scheme is used to get efficient and robust numerical schemes for solving the obtained problem. Numerical experiments show that the method can recover coefficients with rather complicated geometry of discontinuities under a moderate amount of noise in the observation data.