Topology Optimization for Steady-State Navier-Stokes Flow Based on Parameterized Level Set Based Method

In this paper,we consider solving the topology optimization for steady-state incompressibleNavier-Stokes problems via a new topology optimization method called parameterized level set method,which can maintain a relatively smooth level set function with a local optimality condition.The objective of topology optimization is to􀀀nd an optimal con􀀀guration of theuid and solid materials that minimizes power dissipation under a prescribeduid volume fraction constraint.An arti􀀀cial friction force is added to the Navier-Stokes equations to apply the no-slip boundary condition.Although a great deal of work has been carried out for topology optimization ofuidow in recent years,there are few researches on the topology optimization ofuidow with physical body forces.To simulate theuidow in reality,the constant body force(e.g.,gravity)is considered in this paper.Several 2D numerical examples are presented to discuss the relationships between the proposed method with Reynolds number and initial design,and demonstrate the feasibility and superiority of the proposed method in dealing with unstructuredmesh problems.Three 3D numerical examples demonstrate the proposedmethod is feasible in three-dimensional.

A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method

Considering droplet phenomena at low Mach numbers,large differences in the magnitude of the occurring characteristic waves are presented.As acoustic phenomena often play a minor role in such applications,classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction.In this work,a novel scheme based on a specific level set ghost fluid method and an implicit-explicit(IMEX)flux splitting is proposed to overcome this timestep restriction.A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface.In this part of the domain,the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases.It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method.Applica-tions to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.

Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

A LEVEL SET METHOD FOR STRUCTURAL TOPOLOGY OPTIMIZATION WITH MULTI-CONSTRAINTS AND MULTI-MATERIALS

Combining the vector level set model,the shape sensitivity analysis theory with the gradient projection technique,a level set method for topology optimization with multi-constraints and multi-materials is presented in this paper.The method implicitly describes structural material in- terfaces by the vector level set and achieves the optimal shape and topology through the continuous evolution of the material interfaces in the structure.In order to increase computational efficiency for a fast convergence,an appropriate nonlinear speed mapping is established in the tangential space of the active constraints.Meanwhile,in order to overcome the numerical instability of general topology opti- mization problems,the regularization with the mean curvature flow is utilized to maintain the interface smoothness during the optimization process.The numerical examples demonstrate that the approach possesses a good flexibility in handling topological changes and gives an interface representation in a high fidelity,compared with other methods based on explicit boundary variations in the literature.

Level set method for numerical simulation of a cavitation bubble,its growth, collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier-Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving inter-face is captured by the level set function, and the interface velocity is resolved by ＂one-side＂ velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the ＂shock wave＂-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating.

Structural Topology Design of Container Ship Based on Knowledge-Based Engineering and Level Set Method

Knowledge-Based Engineering （KBE） is introduced into the ship structural design in this paper. From the implementation of KBE, the design solutions for both Rules Design Method （RDM） and Interpolation Design Method （IDM） are generated. The corresponding Finite Element （FE） models are generated. Topological design of the longitudinal structures is studied where the Gaussian Process （GP） is employed to build the surrogate model for FE analysis. Multi-objective optimization methods inspired by Pareto Front are used to reduce the design tank weight and outer surface area simultaneously. Additionally, an enhanced Level Set Method （LSM） which employs implicit algorithm is applied to the topological design of typical bracket plate which is used extensively in ship structures. Two different sets of boundary conditions are considered. The proposed methods show satisfactory efficiency and accuracy.

LEVEL SET METHOD FOR TOPOLOGICAL OPTIMIZATION APPLYING TO STRUCTURE, MECHANISM AND MATERIAL DESIGNS

Based on a level set model, a topology optimization method has been suggestedrecently. It uses a level set to express the moving structural boundary, which can flexibly handlecomplex topological changes. By combining vector level set models with gradient projectiontechnology, the level set method for topological optimization is extended to a topologicaloptimization problem with multi-constraints, multi-materials and multi-load cases. Meanwhile, anappropriate nonlinear speed, mapping is established in the tangential space of the activeconstraints for a fast convergence. Then the method is applied to structure designs, mechanism andmaterial designs by a number of benchmark examples. Finally, in order to further improvecomputational efficiency and overcome the difficulty that the level set method cannot generate newmaterial interfaces during the optimization process, the topological derivative analysis isincorporated into the level set method for topological optimization, and a topological derivativeand level set algorithm for topological optimization is proposed.

New normalized nonlocal hybrid level set method for image segmentation

This article introduces a new normalized nonlocal hybrid level set method for image segmentation.Due to intensity overlapping,blurred edges with complex backgrounds,simple intensity and texture information,such kind of image segmentation is still a challenging task.The proposed method uses both the region and boundary information to achieve accurate segmentation results.The region information can help to identify rough region of interest and prevent the boundary leakage problem.It makes use of normalized nonlocal comparisons between pairs of patches in each region,and a heuristic intensity model is proposed to suppress irrelevant strong edges and constrain the segmentation.The boundary information can help to detect the precise location of the target object,it makes use of the geodesic active contour model to obtain the target boundary.The corresponding variational segmentation problem is implemented by a level set formulation.We use an internal energy term for geometric active contours to penalize the deviation of the level set function from a signed distance function.At last,experimental results on synthetic images and real images are shown in the paper with promising results.

The shape optimization of the arterial graft design by level set methods

The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is one of the important aspects in the arterial graft design problem since it affects the flow of the blood significantly. As an attractive design tool for this problem, level set methods are quite efficient for obtaining better shape of the graft. In this paper, a cubic spline level set method and a radial basis function level set method are designed to solve the arterial graft design problem. In both approaches, the shape of the arterial graft is implicitly tracked by the zero-level contour of a level set function and a high level of smoothness of the graft is achieved. Numerical results show the efficiency of the algorithms in the arterial graft design.

A LEVEL SET METHOD FOR MICROSTRUCTURE DESIGN OF COMPOSITE MATERIALS

Based on a level set model and the homogenization theory, an optimization al- gorithm for ?nding the optimal con?guration of the microstructure with speci?ed properties is proposed, which extends current research on the level set method for structure topology opti- mization. The method proposed employs a level set model to implicitly describe the material interfaces of the microstructure and a Hamilton-Jacobi equation to continuously evolve the ma- terial interfaces until an optimal design is achieved. Meanwhile, the moving velocities of level set are obtained by conducting sensitivity analysis and gradient projection. Besides, how to handle the violated constraints is also discussed in the level set method for topological optimization, and a return-mapping algorithm is constructed. Numerical examples show that the method exhibits outstanding ?exibility of handling topological changes and ?delity of material interface represen- tation as compared with other conventional methods in literatures.

An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.

Topology and Shape Optimization of 2-D and 3-D Micro-ArchitecturedThermoelastic Metamaterials Using a Parametric Level Set Method

2-D and 3-D micro-architectured multiphase thermoelastic metamaterials are designed and analyzed using a parametric level set method for topology optimization and the finite element method.An asymptotic homogenization approach is employed to obtain the effective thermoelastic properties of the multiphase metamaterials.Theε-constraint multi-objective optimization method is adopted in the formulation.The coefficient of thermal expansion(CTE)and Poisson’s ratio(PR)are chosen as two objective functions,with the CTE optimized and the PR treated as a constraint.The optimization problems are solved by using the method of moving asymptotes.Effective isotropic and anisotropic CTEs and stiffness constants are obtained for the topologically optimized metamaterials with prescribed values of PR under the constraints of specified effective bulk modulus,volume fractions and material symmetry.Two solid materials along with one additional void phase are involved in each of the 2-D and 3-D optimal design examples.The numerical results reveal that the newly proposed approach can integrate shape and topology optimizations and lead to optimal microstructures with distinct topological boundaries.The current method can topologically optimize metamaterials with a positive,negative or zero CTE and a positive,negative or zero Poisson’s ratio.

INTERACTION OF A SOLITARY WAVE AND A FRONT STEP SIMULATED BY LEVEL SET METHOD

As a new method, the Level Set method had been developed to compute the interface of two-phase flow. The basic mathematical theory and the detailed method to solve the free surface hydrodynamic problem had been investigated. By using the Level Set method, the transformation of a solitary wave over a front step was simulated. The results were in good agreement with laboratory experiments.

A Parameter-Free Approach to Determine the Lagrange Multiplier in the Level Set Method by Using the BESO

A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method.It is inspired by the procedure of determining the threshold of sensitivity number in the BESO method.It first computes the difference between the volume of current design and the upper bound of volume.Then,the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material.Numerical examples proved that this approach is effective to constrain the volume.More importantly,there is no parameter in the proposed approach,which makes it convenient to use.In addition,the convergence is stable,and there is no big oscillation.

LEVEL SET METHODS BASED ON DISTANCE FUNCTION

Some basic problems on the level set methods were discussed, such as the method used to preserve the distance junction , the existence and uniqueness of solution for the level set equations. The main contribution is to prove that in a neighborhood of the initial zero level set, the level set equations with the restriction of the distance function have a unique solution, which must be the signed distance function with respect to the evolving surface. Some skillful approaches were used: Noticing that any solution for the original equation was a distance function, the original level set equations were transformed into a simpler alternative form. Moreover, since the new system was not a classical one, the system was transformed into an ordinary one, for which the implicit function method was adopted.

Liquid-Gas Two-Phase Flow Simulation in Mold Filling Process with Level Set Method

Air entrapped in liquid metal during the mold filling process seriously affects the casting quality, thus it is important to track its behavior in the mold cavity. A liquid-gas two-phase flow model is developed to describe the mold filling process and predict the air entrapment defect. The model is based on the combination of SOLA and Level Set Method. The pressure and velocity fields are calculated by SOLA,and the interface movement is simulated by Level Set method as the most common interface tracking method in recent years.In order to validate the feasibility of the model,the liquid-gas two-phase simulation results were tested by the broken dam problem and the S-shaped experiment. Comparison between the experiments and simulation results show that Level Set method might be a very promising tool in two-phase flow simulation during the mold filling process.

Modeling of the Saltwater Intrusion Using the Level Set Method. Application to Henry’s Problem

The salt intrusion phenomenon is caused by overexploitation of aquifers in coastal areas. This physical phenomenon has been the subject of numerous studies and numerous methods have been proposed, with the aim of protecting the quality of the water in these aquifers. This work proposes a two-dimensional saline intrusion model using the sharp interface approach and the level set method. It consists of a parabolic equation modeling the underground flow and a hyperbolic Equation (the level set equation) which makes it possible to track the evolution of the interface. High-order numerical schemes such as the space scheme WENO5 and the third-order time scheme TVD-RK were used for the numerical resolution of the hyperbolic equation. To limit the tightening of the contour curves of the level set function, the redistanciation or reinitialization algorithm proposed by Sussma et al. (1994) was used. To ensure the effectiveness and reliability of the proposed method, two tests relating to the standard Henry problem and the modified Henry problem were performed. Recall that Henry’s problem uses the variable density modeling approach in a confined and homogeneous aquifer. By comparing the results obtained by the level set method with reinitialization (LSMR) and those obtained by Henry (1964), and by Simpson and Clement (2004), we see in the two test cases that the level set method reproduces well the toe, the tip and the behaviour of the interface. These results correspond to the results obtained by Abarca for Henry’s problem with constant dispersion coefficients. The results obtained with LSMR, reproduced the interface with a slight spacing compared to those obtained by Henry. According to Abarca (2006), this spacing is due to the absence of the longitudinal and transversal dispersion coefficients in the model.

Buckling optimization of curvilinear fiber-reinforced composite structures using a parametric level set method

Owing to their excellent performance and large design space,curvilinear fiber-reinforced composite structures have gained considerable attention in engineering fields such as aerospace and automobile.In addition to the stiffness and strength of such structures,their stability also needs to be taken into account in the design.This study proposes a level-set-based optimization framework for maximizing the buckling load of curvilinear fiber-reinforced composite structures.In the proposed method,the contours of the level set function are used to represent fiber paths.For a composite laminate with a certain number of layers,one level set function is defined by radial basis functions and expansion coefficients for each layer.Furthermore,the fiber angle at an arbitrary point is the tangent orientation of the contour through this point.In the finite element of buckling,the stiffness and geometry matrices of an element are related to the fiber angle at the element centroid.This study considers the parallelism constraint for fiber paths.With the sensitivity calculation of the objective and constraint functions,the method of moving asymptotes is utilized to iteratively update all the expansion coefficients regarded as design variables.Two numerical examples under different boundary conditions are given to validate the proposed approach.Results show that the optimized curved fiber paths tend to be parallel and equidistant regardless of whether the composite laminates contain holes or not.Meanwhile,the buckling resistance of the final design is significantly improved.

A two-dimensional numerical wave flume based on SA-MPLS method

A spatially adaptive (SA) two-dimensional (2-D) numerical wave flume is presented based on the quadtree mesh system,in which a new multiple particle level set (MPLS) method is proposed to solve the problem of interface tracking,in which common intersection may be traversed by multiple interfaces.By using the adaptive mesh technique and the MPLS method,mesh resolution is updated automatically with time according to flow characteristics in the modeling process with higher resolution around the free surface and the solid boundary and lower resolution in less important area.The model has good performance in saving computer memory and CPU time and is validated by computational examples of small amplitude wave,second-order Stokes wave and cnoidal wave.Computational results also indicate that standing wave and wave overtopping are also reasonably simulated by the model.

AN IMPROVED LEVEL-SET RE-INITIALIZATION SOLVER

Re-initialization procedure in level-set interface capturing method were investigated. The algorithm accomplishes the re-initialization step through locking the interface positions. Better accuracy was obtained both on the interface positions and the total fluid volume keeping. Though one more step of the interpolations is added in the procedure, there is no significant increase in total machine time spent.