Conventional Level Set Method Research Articles

Topology optimization of coated structures with layer-wise graded lattice infill for maximizing the fundamental eigenfrequency

Combining the advantages of lattice infill and coating, coated structures with lattice infill are widely used in engineering to achieve light-weight structural characteristic or certain functionalities. This paper presents a novel concurrent multiscale topology optimization (TO) framework to maximize the fundamental eigenfrequency of structures with a uniform outer coating and layer-wise graded lattice infill microstructures. The macroscale topology optimization is conducted by using the velocity field-based level set method, which inherits the implicit geometrical representation and signed distance property of the conventional level set method (smooth and clear boundaries and well-maintenance of the uniform thickness of the coating). Besides, the employment of general mathematical programming algorithms enables the velocity field-based level set method to handle multiple constraints in an easier way. At microscale, the popular density-based method is used to design layer-wise graded lattice structures. The effective material properties of microstructures are computed by using the asymptotic homogenization method, which bridges macroscale and microscale designs. With higher design flexibility, it is found that coated structures with layer-wise graded lattice infill have higher eigenfrequencies than those with periodic and uniform lattice infill microstructures. The influence of layers of lattice infill and coating thickness on the concurrent optimization of macrostructures and microstructures are carefully studied by showcasing several numerical examples, and the effectiveness of the proposed method is also confirmed, as well as the manufacturability of optimization results.

A high order flux reconstruction interface capturing method with a phase field preconditioning procedure

This paper presents a simple and highly accurate method for capturing sharp interfaces moving in divergence-free velocity fields using the high-order Flux Reconstruction approach on unstructured grids. A well-known limitation of high-order methods is their susceptibility to the Gibbs phenomenon; the appearance of spurious oscillations in the vicinity of discontinuities and steep gradients makes it difficult to accurately resolve shocks or sharp interfaces. In order to address this issue in the context of sharp interface capturing, a novel, preconditioned and localized phase-field method is developed in this work. The numerical accuracy of interface normal vectors is improved by utilizing a preconditioning procedure based on the level set method with localized artificial viscosity stabilization. The developed method was implemented in the framework of the multi-platform Flux Reconstruction open-source code PyFR [1]. Kinematic numerical tests in 2D and 3D conducted on different mesh types showed that the preconditioning procedure significantly improves accuracy with little added computational effort. The results demonstrate the conservativeness of the proposed method and its ability to capture highly distorted interfaces with superior accuracy when compared to conventional and high-order VOF and level set methods. Dynamic interface capturing validation tests were also carried out by coupling the proposed method to the entropically damped artificial compressibility variant of the incompressible Navier–Stokes equations. The high accuracy and locality of the proposed method offer a promising route to carrying out massively-parallel, high accuracy simulations of multi-phase, incompressible phenomena.

Weakly supervised serous retinal detachment segmentation in SD-OCT images by two-stage learning.

Automated lesion segmentation is one of the important tasks for the quantitative assessment of retinal diseases in SD-OCT images. Recently, deep convolutional neural networks (CNN) have shown promising advancements in the field of automated image segmentation, whereas they always benefit from large-scale datasets with high-quality pixel-wise annotations. Unfortunately, obtaining accurate annotations is expensive in both human effort and finance. In this paper, we propose a weakly supervised two-stage learning architecture to detect and further segment central serous chorioretinopathy (CSC) retinal detachment with only image-level annotations. Specifically, in the first stage, a Located-CNN is designed to detect the location of lesion regions in the whole SD-OCT retinal images, and highlight the distinguishing regions. To generate available a pseudo pixel-level label, the conventional level set method is employed to refine the distinguishing regions. In the second stage, we customize the active-contour loss function in deep networks to achieve the effective segmentation of the lesion area. A challenging dataset is used to evaluate our proposed method, and the results demonstrate that the proposed method consistently outperforms some current models trained with a different level of supervision, and is even as competitive as those relying on stronger supervision. To our best knowledge, we are the first to achieve CSC segmentation in SD-OCT images using weakly supervised learning, which can greatly reduce the labeling efforts.

A Parametric Level Set Method for Topology Optimization Based on Deep Neural Network

Abstract This paper proposes a new parametric level set method for topology optimization based on deep neural network (DNN). In this method, the fully connected DNN is incorporated into the conventional level set methods to construct an effective approach for structural topology optimization. The implicit function of level set is described by fully connected DNNs. A DNN-based level set optimization method is proposed, where the Hamilton–Jacobi partial differential equations (PDEs) are transformed into parametrized ordinary differential equations (ODEs). The zero-level set of implicit function is updated through updating the weights and biases of networks. The parametrized reinitialization is applied periodically to prevent the implicit function from being too steep or too flat in the vicinity of its zero-level set. The proposed method is implemented in the framework of minimum compliance, which is a well-known benchmark for topology optimization. In practice, designers desire to have multiple design options, where they can choose a better conceptual design base on their design experience. One of the major advantages of the DNN-based level set method is capable to generate diverse and competitive designs with different network architectures. Several numerical examples are presented to verify the effectiveness of the proposed DNN-based level set method.

Structural topology optimization using a level set method with finite difference updating scheme

In conventional level set methods for topology optimization, the evolution of level set function is solved by upwind scheme and needs to be reinitialized frequently throughout the optimization. The upwind scheme is complicated in numerical implementation and the reinitialization is time-consuming and slows down the optimization efficiency. This paper presents a level set method updated with finite difference scheme for structural topology optimization. The velocity field used to update the level set function is interpolated by piecewise basis function in order to incorporate the universal mathematical programming algorithm into the optimization. The diffusion is introduced into the Hamilton-Jacobi equation and the three-step splitting method is adopted to solve the equation. By applying the proposed method, the evolution of the level set can be updated by simple central difference scheme and the procedure of reinitialization is avoided without sacrificing the numerical stability. The classical mean compliance minimization problem with several numerical examples is used to test the validity of the proposed method.

An isogeometrical level set topology optimization for plate structures

This study presents topology optimization of plate structures by employing isogeometrical level set method. For structural analysis of plates, the IsoGeometric Analysis (IGA) approach is applied and Non-Uniform Rational B-Splines (NURBS) basis functions are used for approximation of the design domain geometry as well as the unknown deformation field. In this paper, the level set function is parametrized with Radial Basis Functions (RBFs), which is more efficient than the conventional level set method. This approach along with an approximate re-initialization scheme can maintain a smooth level set function during the optimization process and has less dependency on initial designs because of its ability to nucleate new holes inside the design domain. Due to capability of IGA method in modeling complex design domains while maintaining high accuracy in analysis, combination of IGA with RBFs level set method provides a very useful and effective technique for topology optimization problems. Several numerical examples are prepared to demonstrate the efficiency and accuracy of the method and obtained optimum topologies are compared with the results of other methods in literature.

Derivative-free level-set-based multi-objective topology optimization of flow channel designs using lattice Boltzmann method

A derivative-free level-set-based topology optimization method (DFLS-TO) was proposed and was used for channel structure design. Unlike in a conventional level-set method, quantum-behaved particle swarm optimization (QPSO) was used in DFLS-TO to optimize the level-set values at the knot points. The values at the non-knot points were interpolated by solving the Laplace equation. QPSO was combined with the Tchebycheff decomposition method to search for optimal level-set values in multi-objective problems. The channel boundary was represented by an iso-contour of the level-set function, and the B-spline method was used to convert the zigzag-like boundary into a smooth boundary. The lattice Boltzmann method (LBM) was integrated with the immersed boundary method to simulate the fluid flow. The Spalart–Allmaras model was incorporated with the LBM during the simulation of turbulent flows. To demonstrate the applicability of DFLS-TO, optimization of a pipe bend and a fluid distributor were performed.

Detailed Voxel-Based Implicit Modeling With Local Boolean Composition of Discrete Level Sets

Compared with the functionally-based implicit models, the level set models are usually defined by discretely sampling the implicit function values over the volume grid to represent the shapes, which are suitable for scientific computing and engineering. However, the voxel-based level set methods usually lose some features and details to represent the solids due to the finite grid resolutions. In this paper, we propose a more accurate voxel-based implicit modeling method based on discrete level sets. Compared with the traditional level set methods, more than one level set function can be defined in an implicit model with our approach by classifying the cells of the volume grid. The local Boolean compositions conducted over the compound cells are used to depict the interactions of different implicit surfaces represented by corresponding level set functions, and the sharp and thin details caused by Boolean operations are preserved owing to the definition of the compound cells. We show in our experiments that our method is robust to provide the more detailed implicit description of the shapes with less loss of the features than the conventional voxel-based level set methods.

The parameterized level set method for structural topology optimization with shape sensitivity constraint factor

In recent years, the parameterized level set method (PLSM) has attracted widespread attention for its good stability, high efficiency and the smooth result of topology optimization compared with the conventional level set method. In the PLSM, the radial basis functions (RBFs) are often used to perform interpolation fitting for the conventional level set equation, thereby transforming the iteratively updating partial differential equation (PDE) into ordinary differential equations (ODEs). Hence, the RBFs play a key role in improving efficiency, accuracy and stability of the numerical computation in the PLSM for structural topology optimization, which can describe the structural topology and its change in the optimization process. In particular, the compactly supported radial basis function (CS-RBF) has been widely used in the PLSM for structural topology optimization because it enjoys considerable advantages. In this work, based on the CS-RBF, we propose a PLSM for structural topology optimization by adding the shape sensitivity constraint factor to control the step length in the iterations while updating the design variables with the method of moving asymptote (MMA). With the shape sensitivity constraint factor, the updating step length is changeable and controllable in the iterative process of MMA algorithm so as to increase the optimization speed. Therefore, the efficiency and stability of structural topology optimization can be improved by this method. The feasibility and effectiveness of this method are demonstrated by several typical numerical examples involving topology optimization of single-material and multi-material structures.

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distanceregularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.

Extended finite element simulation of step-path brittle failure in rock slopes with non-persistent en-echelon joints

The aim of this paper is to investigate the brittle failure mechanism of rock slopes with non-persistent en-echelon joints using the extended finite element method (XFEM). In the research, a novel geometrical method distinguished from the conventional level set method is proposed to describe the cracks in the XFEM. The numerical form of the J-integral considering the effect of the bode force and the frictional contact force is provided. Moreover, a refined coalescence method is proposed. Then, the state function employed to define the state of the crack and the method used to define the crack growth increment are introduced. Next, some numerical tests are carried out to validate the feasibility of the XFEM. Finally, two typical rock slope modes are provided to illustrate the mechanism of the step-path brittle failure of rock slopes.

An improved parametric level set method for structural frequency response optimization problems

In conventional parametric level set methods, the compactly supported radial basis functions (CSRBF) are used to approximate the level set function due to their unique properties, such as the sparsity of the interpolation matrix. The CSRBFs only consider the contributions of knots within a narrow sub-region, which sacrifices accuracy for efficiency in the interpolation. However, the accuracy loss in the CSRBF-based method may prolong the iteration and gradually lead the topology optimization towards a worse local optimum or even an unfeasible design, especially when the allowable material usage in the design domain is relatively low. This will significantly affect the performance of the optimization method. This paper proposes an improved parametric level set method (iPLSM), which is more efficient and effective in topology optimization designs. In this method, the Gaussian radial basis function with global support is used to parameterize the level set surface, to ensure a high numerical accuracy due to the consideration of all interpolation knots in the global domain. Then, a discrete wavelet transform scheme is incorporated into the parametric form to compress the full interpolation matrix and save the computational cost. The proposed method is applied to both the global and local frequency response optimization problems under wide excitation frequency ranges, to validate its efficiency and effectiveness.

Topology optimization of conformal structures on manifolds using extended level set methods (X-LSM) and conformal geometry theory

In this paper, we propose a new method to systematically address the issue of structural shape and topology optimization on free-form surfaces. A free-form surface, also termed manifold, is conformally mapped onto a 2D rectangle domain where the level set function is defined. With the conformal mapping, the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar. Keeping this intrinsic relation in mind, we derive the Euclidean form for the Riemannian Hamilton–Jacobi equation governing the boundary evolution on the manifold, which can be solved on a 2D plane using classical level set methods, such as the upwind finite difference or fast marching method. By reducing the dimension of the problem, the topology optimization problem on the manifold embedded in the 3D space can be recast as a 2D topology optimization problem in the Euclidean space. Compared with other approaches which need project the Euclidean differential operators to the manifold, the proposed method can not only reduce the computational cost but also preserve all the advantages of conventional level set methods. The proposed method reveals the fundamental relation between topology optimization on manifolds and Euclidean planes. It provides a unified level-set-based computational framework for the generative design of conformal structures with increasing applications in different fields of interests.

A level set image segmentation method based on a cloud model as the priori contour

A novel image segmentation method combining a cloud model and a level set (CM-LS) is proposed in this article. At present, the cloud model can only obtain the rough segmentation result of an image, but the level set method is sensitive to the initial contour. The core idea of this method is to use the rough segmentation result of cloud model as the initial contour of the level set and then obtain the final result by the contour evolution. In this method, the cloud model is used to decompose the boundary of the image, which reduces the occurrence probability and occurrence degree of the instability problem caused by artificial intervention; at the same time, the convergence of the level set function is accelerated, and the initializing operation of the level set function that uses the cloud model algorithm can also effectively reduce the noise sensitivity of the function itself. Compared with the conventional level set method, the proposed method is general and accurate. The experimental data set in this article includes natural images of the Berkeley database, medical images and synthetic noise images. The experimental results show that the method is effective.

A velocity field level set method for shape and topology optimization

SummaryIn this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions.

Parameterized level-set based topology optimization method considering symmetry and pattern repetition constraints

Symmetry and repetitive patterns are very common in both natural and artificial structures. Design with symmetry and pattern repetition (SPR) can be very useful in practical engineering. In this paper, we discuss a new optimal structural design method considering SPR constraints. The basic idea consists in combining a parameterized level-set method together with radial basis function (RBF) interpolation strategy, which allows for decoupling RBF knots from the finite element (FE) analysis mesh. The merits of the decoupled manner are that RBF knots can be mapped independently on regular or irregular design domain, and SPR constraints can be imposed by simply building a bijective relationship between RBF knots regardless of analysis mesh. Furthermore, due to parameterization of the level-set function, there are fewer numerical manipulations that need to be implemented than the conventional level-set method during the optimization process. Several numerical examples are presented to show feasibility and effectiveness of the proposed method.

Topology optimization of magnetic shielding using level-set function combined with element-based topological derivatives

PurposeShape optimization using the level-set method is one of the most effective automatic design tools for electromagnetic machines. While level-set method has the advantage of being able to suppress unfeasible shape, it has a weakness of being unable to handle complex topology changes such as perforate at material region. With this method, it is only possible to define simple connected topology, and it is difficult to determine the optimal shape which has holes. Therefore, it is important to efficiently expand the searching area in the optimization process with level-set method.Design/methodology/approachIn this paper, the authors introduce the newly defined hole sensitivity which is based on concept of topological derivatives, and combine it with level-set method to effectively create holes in the search process. Furthermore, they consider a variable bandwidth of gray scale, which indicates the transition width between air and magnetic body and combine it with the hole creation method described above. With these methods, the authors aim to expand the searching area in comparison with the conventional level-set method.FindingsAs a result of applying the proposed methods to a magnetic shielding problem, the multi-layered shielding which effectively reduces the magnetic flux in the target area, is successfully produced.Originality/valueThe proposed methods enable us to effectively create a hole and to expand the searching area in the topology optimization process unlike in the case of conventional level-set method.

Parametric shape and topology optimization: A new level set approach based on cardinal basis functions

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

Convergence Acceleration of Topology Optimization Based on Constrained Level Set Function Using Method of Moving Asymptotes in 3-D Nonlinear Magnetic Field System

Topology optimization (TO) has advantages over shape optimization; for example, the design space is wider and the degrees of the shape freedom is larger. The convergence characteristic in the conventional level set (LS) method is slow because of the limitation of the time step size that is used for solving the simplified Hamilton-Jacobi equation, which is normally defined as the width of one finite element in the design domain. To overcome this difficulty, convergence acceleration using the method of moving asymptotes is investigated. The performance of the proposed method is compared with the conventional LS method for a 3-D TO problem regarding the magnetic shielding and an interior permanent magnet motor in a 3-D nonlinear magnetostatic field.

Image Segmentation with Depth Information via Simplified Variational Level Set Formulation

Image segmentation with depth information can be modeled as a minimization problem with Nitzberg–Mumford–Shiota functional, which can be transformed into a tractable variational level set formulation. However, such formulation leads to a series of complicated high-order nonlinear partial differential equations which are difficult to solve efficiently. In this paper, we first propose an equivalently reduced variational level set formulation without using curvatures by taking level set functions as signed distance functions. Then, an alternating direction method of multipliers (ADMM) based on this simplified variational level set formulation is designed by introducing some auxiliary variables, Lagrange multipliers via using alternating optimization strategy. With the proposed ADMM method, the minimization problem for this simplified variational level set formulation is transformed into a series of sub-problems, which can be solved easily via using the Gauss–Seidel iterations, fast Fourier transform and soft thresholding formulas. The level set functions are treated as signed distance functions during computation process via implementing a simple algebraic projection method, which avoids the traditional re-initialization process for conventional variational level set methods. Extensive experiments have been conducted on both synthetic and real images, which validate the proposed approach, and show advantages of the proposed ADMM projection over algorithms based on traditional gradient descent method in terms of computational efficiency.