Conventional Level Set Method Research Articles

A Meshless Level Set Method for Shape and Topology Optimization

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.

Simulation of a premixed turbulent flame by a conservative level set method

The premixed turbulent V-flame is numerically simulated in this paper. The technique used in Chan et al. [Effect of intense turbulence on turbulent premixed V-flame, Int. J. Eng. Sci. 41 (2003), pp. 903–916; Simulation of a premixed turbulent flame with the discrete vortex method, Int. J. Numer. Methods Eng. 48 (2000), pp. 613–627] was based on the random vortex method and the conventional level set method. However, this paper is an improvement of the previous technique by using a conservative level set method. The new algorithm uses a bi-cubic interpolation process together with Newton's method to reconstruct the velocity field to improve the conservation of the level set method. The numerical example in this paper indicates the efficiency of the new method in solving the problem of turbulent premixed V-flame by comparing the results with available data.

Skull‐stripping method for brain MRI using a 3D level set with a speedup operator

To extract the brain region from brain magnetic resonance (MR) images using a fast 3D level set method and a refinement process. The proposed method introduces a speedup operator to the conventional 3D level set method in order to accelerate the level set evolution. While the processing time for brain extraction is reduced by the speedup operator, the accuracy of brain extraction is also improved by adopting a refinement process. The speedup operator yielded a 75% reduction in the total iteration numbers for the synthesized volume. The proposed method was applied to several datasets and compared with previous methods, ie, BrainVisa, BET, and FreeSurfer. The proposed method provided a Jaccard index of 0.971 ± 0.0114 for the BrainWeb dataset, 0.864 ± 0.035 for the IBSR dataset, and 0.9414 ± 0.0517 for a self-produced dataset acquired with a 3T MRI system. Utilizing a speedup operator, the proposed method reduced the evolution time. Robust and accurate results for various datasets were obtained in experiments.

Moving-boundary problems solved by adaptive radial basis functions

The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach.

A level-set based topology optimization using the element connectivity parameterization method

This contribution presents a novel and versatile approach to geometrically nonlinear topology optimization by combining the level-set method with the element connectivity parameterization method or ECP. The combined advantages of both methods open up the possibility to treat a wide range of optimization problems involving complex physical and/or geometrical nonlinearities in a general and elegant manner. The level-set method features shape optimization on a fixed mesh, leading to intrinsically black-and-white designs. This approach allows a clear description of location and orientation of the interface, whereas topological changes can still be handled easily. A popular concept used in conventional level-set methods is to map the level-set function to volume-fraction design variables for every element of a finite element mesh. The resulting element density variables are then used to scale the Young’s modulus in each element using the Ersatz material approach. In this work we employ a modified material interpolation method, in which the element density variables, based on a per-element integration of a regularized Heaviside operator applied to the level-set function, are used as element connectivity design variables. The resulting crisp boundary topology optimization method exploits the advantages of ECP in the field of complex nonlinearities and eliminates the need for penalization by the implicit level-set description of the design.

Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction

Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TV-regularizers, making them difficult to compute. The previously introduced method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the concept to image segmentation problems, we build fast solvers which can out-perform more conventional schemes, such as duality based methods and graph-cuts. The convex segmentation schemes also substantially outperform conventional level set methods, such as the Chan-Vese level set-based segmentation algorithm. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions. The primary purpose of this paper is to examine the effectiveness of Split Bregman techniques for solving these problems, and to compare this scheme with more conventional methods.

Structural shape and topology optimization based on level-set modelling and the element-propagating method

This article introduces the element-propagating method to structural shape and topology optimization. Structural optimization based on the conventional level-set method needs to solve several partial differential equations. By the insertion and deletion of basic material elements around the geometric boundary, the element-propagating method can avoid solving the partial differential equations and realize the dynamic updating of the material region. This approach also places no restrictions on the signed distance function and the Courant–Friedrichs–Lewy condition for numerical stability. At the same time, in order to suppress the dependence on the design initialization for the 2D structural optimization problem, the strain energy density is taken as a criterion to generate new holes in the material region. The coupled algorithm of the element-propagating method and the method for generating new holes makes the structural optimization more robust. Numerical examples demonstrate that the proposed approach greatly improves numerical efficiency, compared with the conventional level-set method for structural topology optimization.

Design of piezoelectric actuators using a multiphase level set method of piecewise constants

This paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi-material piezoelectric actuators with in-plane motion. First, an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain. Compared with standard level set methods using n scalar functions to represent 2 n phases, each constant value in the present method denotes one material phase and 2 n phases can be represented by 2 n pre-defined constants. Thus, only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities. In the context of designing smart actuators with in-plane motions, the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints. Thus, the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi-implicit scheme with additive operator splitting (AOS) algorithm. In such a way, the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized. The compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators. The major advantage of the present method is to remove numerical difficulties associated with the solution of the Hamilton–Jacobi equations in most conventional level set methods, such as the CFL condition, the regularization procedure to retain a signed distance level set function and the non-differentiability related to the Heaviside and the Delta functions. Two widely studied examples are chosen to demonstrate the effectiveness of the present method.

A new level set method for systematic design of hinge-free compliant mechanisms

This paper presents a new level set-based method to realize shape and topology optimization of hinge-free compliant mechanisms. A quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism. A semi-implicit scheme with an additive operator splitting (AOS) algorithm is employed to solve the Hamilton–Jacobi partial differential equation (PDE) in the level set method. The design of compliant mechanisms is mathematically represented as a general non-linear programming with a new objective function augmented by the high-order energy term. The structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints. In doing so, it is expected that numerical difficulties such as the Courant–Friedrichs–Lewy (CFL) condition and periodically applied re-initialization procedures in most conventional level set methods can be eliminated. In addition, new holes can be created inside the design domain. The final mechanism configurations consist of strip-like members suitable for generating distributed compliance, and solving the de-facto hinge problem in the design of compliant mechanisms. Two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms.

Level set contour extraction method based on support value filter

Active contour method is a powerful approach for object contour extraction. This paper presents a new object contour extraction method, which combines the level set evolution with the support value filter. It analyzes image under the least squares support vector machine (LS-SVM) framework and uses the support values to represent salient features underlying image. The Gaussian filter, used in conventional level set method to compute the edge indicator, is replaced by the support value filter deduced from the mapped LS-SVM. The level set evolution method is implemented on the feature image obtained by convolving the support value filter with the original image. Experiments are undertaken on the synthetic and real images. The experimental results demonstrate that the support value filter can provide a good estimate for the optimal step size in a gradient descent algorithm and the proposed method has advantages over the direct level set evolution method in converging speed and contour extraction accuracy.

A level set method for shape and topology optimization of large‐displacement compliant mechanisms

AbstractA parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements. A level set model is established mathematically as the Hamilton–Jacobi equation to capture the motion of the free boundary of a continuum structure. The structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension. The radial basis function with compact support is then applied to interpolate the level set function, leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation. In doing so, the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients. As a result, the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method. With the concept of the shape derivative, the geometrical non‐linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms. Several numerical benchmark examples illustrate the effectiveness of the present level set method, in particular, its capability of generating new holes inside the material domain. The proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme, the reinitialization procedure, and the velocity extension algorithm in the conventional level set method. Copyright © 2008 John Wiley & Sons, Ltd.

Shape-topology optimization of stokes flow via variational level set method

In this paper, the conventional level set method is extended as an efficient approach for shape optimization. The motion of the dynamic interface is governed by Stokes equations. Although this method is not specifically designed for topology optimization, it can handle topology changes easily. Furthermore, a relatively smooth evolution can be maintained without re-initialization. The cost of this method is moderate. The accuracy and efficiency of the proposed algorithm are illustrated by several numerical examples.

A semi-implicit level set method for structural shape and topology optimization

This paper proposes a new level set method for structural shape and topology optimization using a semi-implicit scheme. Structural boundary is represented implicitly as the zero level set of a higher-dimensional scalar function and an appropriate time-marching scheme is included to enable the discrete level set processing. In the present study, the Hamilton–Jacobi partial differential equation (PDE) is solved numerically using a semi-implicit additive operator splitting (AOS) scheme rather than explicit schemes in conventional level set methods. The main feature of the present method is it does not suffer from any time step size restriction, as all terms relevant to stability are discretized in an implicit manner. The semi-implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance. Hence, the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency. Resultantly, it allows enhanced relaxation on the time step size originally limited by the Courant–Friedrichs–Lewy (CFL) condition of the explicit schemes. The stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements. Furthermore, the present method avoids additional cost to globally reinitialize the level set function for regularization purpose. It is noted that the periodically applied reinitializations are time-consuming procedures. In particular, the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating, splitting and merging processes, which makes the final design independent of initial guess, and helps reduce the probability of converging to a local minimum. The availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs.

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.

Shape and topology optimization of compliant mechanisms using a parameterization level set method

In this paper, a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms. The structural shape boundary is implicitly embedded into a higher-dimensional scalar function as its zero level set, resultantly, establishing the level set model. By applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function, the temporal and spatial Hamilton–Jacobi equation from the conventional level set method is then discretized into a series of algebraic equations. Accordingly, the original shape and topology optimization is now fully transformed into a parameterization problem, namely, size optimization with the expansion coefficients of interpolants as a limited number of design variables. Design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints. The structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method. It is highlighted that the present method can not only inherit the merits of the implicit boundary representation, but also avoid some unfavorable features of the conventional discrete level set method, such as the CFL condition restriction, the re-initialization procedure and the velocity extension algorithm. Finally, an extensively investigated example is presented to demonstrate the benefits and advantages of the present method, especially, its capability of creating new holes inside the design domain.

A level set‐based parameterization method for structural shape and topology optimization

AbstractThis paper presents an effective parametric approach by extending the conventional level set method to structural shape and topology optimization using the compactly supported radial basis functions (RBFs) and the optimality criteria (OC) method. The structural design boundary is first represented implicitly by embedding into a higher‐dimensional level set function as its zero level set, and the RBFs of a favorable smoothness are then applied to interpolate the level set function. The original initial value problem is thus converted to a parametric optimization, with the expansion coefficients of the interplant posed as the design variables.The OC method is then applied to advance the structure boundary in terms of the velocity field derived from the parametric optimization. Hence, the structural shape and topology optimization is now transformed into a process of iteratively finding coefficients to update the level set function to achieve an optimal configuration. The numerical considerations of the conventional level set method, including upwind schemes, velocity extension, and reinitialization, are eliminated. The proposed scheme is capable of addressing structural shape fidelity and topology change simultaneously and of keeping the boundary smooth during the optimization process. Furthermore, numerical convergence is expected to be improved. A widely investigated example, in the framework of structural stiffness designs, is applied to demonstrate the efficiency and accuracy of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.

An extended level set method for shape and topology optimization

In this paper, the conventional level set methods are extended as an effective approach for shape and topology optimization by the introduction of the radial basis functions (RBFs). The RBF multiquadric splines are used to construct the implicit level set function with a high level of accuracy and smoothness and to discretize the original initial value problem into an interpolation problem. The motion of the dynamic interfaces is thus governed by a system of coupled ordinary differential equations (ODEs) and a relatively smooth evolution can be maintained without reinitialization. A practical implementation of this method is further developed for solving a class of energy-based optimization problems, in which approximate solution to the original Hamilton–Jacobi equation may be justified and nucleation of new holes inside the material domain is allowed for. Furthermore, the severe constraints on the temporal and spatial discretizations can be relaxed, leading to a rapid convergence to the final design insensitive to initial guesses. The normal velocities are chosen to perform steepest gradient-based optimization by using shape sensitivity analysis and a bi-sectioning algorithm. A physically meaningful and efficient extension velocity method is also presented. The proposed method is implemented in the framework of minimum compliance design and its efficiency over the existing methods is highlighted. Numerical examples show its accuracy, convergence speed and insensitivity to initial designs in shape and topology optimization of two-dimensional (2D) problems that have been extensively investigated in the literature.

Topology description function based method for material design

ABSTRACT The purpose of this paper is to investigate the application of topology description function (TDF) in material design. Using TDF to describe the topology of the microstructure, the formulation and the solving technique of the design problem of materials with prescribed mechanical properties are presented. By presenting the TDF as the sum of a series of basis functions determined by parameters, the topology optimization of material microstructure is formulated as a size optimization problem whose design variables are parameters of TDF basis functions and independent of the mesh of the design domain. By this method, high quality topologies for describing the distribution of constituent material in design domain can be obtained and checkerboard problem often met in the variable density method is avoided. Compared with the conventional level set method, the optimization problem can be solved simply by existing optimization techniques without the process to solve the ‘Hamilton-Jacobi-type' equation by the difference method. The method proposed is illustrated with two 2D examples. One gives the unit cell with positive Poisson's ratio, the other with negative Poisson's ratio. The examples show the method based on TDF is effective for material design.

改良レベルセット法による非圧縮性気液二相流のシミュレーション

An enhanced resolution capturing method has been developed for topologically complex, incompressible gas-liquid two-phase flows. The method is based on the level set method to represent the interface combined with two recent advances in the treatment of the interface, a boundary-condition-capturing method [Kong et al., J. Sci. Comput. 15 (2000) 323] to accurately satisfy the jump condition at the gas-liquid interface, and a particle level set method [Enright et al., J. Comput. Phys. 183 (2002) 83] in which massless marker particles are used to enhance the resolution of the interface. We present direct comparisons between computations made with the improved level set method and those with the conventional level set method. Several benchmark flows in two dimensions establish that the present method has clear superiority over the classical method, especially in computations where the spatial resolution is not as high.

Radial basis functions and level set method for structural topology optimization

AbstractLevel set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.