A Hybrid Level Set Optimization Design Method of Functionally Graded Cellular Structures Considering Connectivity

With the continuous advancement in topology optimization and additive manufacturing(AM)technology,the capability to fabricate functionally graded materials and intricate cellular structures with spatially varying microstructures has grown significantly.However,a critical challenge is encountered in the design of these structures–the absence of robust interface connections between adjacent microstructures,potentially resulting in diminished efficiency or macroscopic failure.A Hybrid Level Set Method(HLSM)is proposed,specifically designed to enhance connectivity among non-uniform microstructures,contributing to the design of functionally graded cellular structures.The HLSM introduces a pioneering algorithm for effectively blending heterogeneous microstructure interfaces.Initially,an interpolation algorithm is presented to construct transition microstructures seamlessly connected on both sides.Subsequently,the algorithm enables the morphing of non-uniform unit cells to seamlessly adapt to interconnected adjacent microstructures.The method,seamlessly integrated into a multi-scale topology optimization framework using the level set method,exhibits its efficacy through numerical examples,showcasing its prowess in optimizing 2D and 3D functionally graded materials(FGM)and multi-scale topology optimization.In essence,the pressing issue of interface connections in complex structure design is not only addressed but also a robust methodology is introduced,substantiated by numerical evidence,advancing optimization capabilities in the realm of functionally graded materials and cellular structures.

基于梯度增量level-set方法的复杂物体入水砰击数值模拟

复杂结构物自由表面砰击问题,如空投鱼雷、水空无人航行器潜水和高速船出入水,一直是海洋工程领域的研究热点。本文采用梯度增量level-set和虚拟网格法数值模拟复杂结构物的入水砰击,采用时间半隐式有限差分法求解不可压缩Navier-Stokes方程,虚拟网格法可重构复杂结构物界面插值模块并施加无滑移边界条件,梯度增量level-set法可捕捉非线性自由表面翻卷、射流等物理现象。基于以上方法模拟二维圆柱的常速入水砰击,并与试验和数值结果对比,验证本文数值方法的正确性。最后,数值模拟二维船型剖面和飞机剖面的入水砰击,分析了冲击压力、运动响应、压力分布和自由表面运动随入水角度的变化规律。同时,通过模拟可观察到复杂结构浸入水中后的流体分离、射流和小角度空气腔等典型砰击物理现象。

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 Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem onImplicit Surfaces

We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

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 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.

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.

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.

Simulation of Non-isothermal Injection Molding for a Non-Newtonian Fluid by Level Set Method

A non-isothermal injection molding process for a non-Newtonian viscous pseudoplastic fluid is simulated.A conservative interface capturing technique and the flow field solving method are coupled to perform a dynamic simulation.The validity of the numerical method is verified by a benchmark problem.The melt interface evolution versus time is captured and the physical quantities such as temperature,velocity and pressure at each time step are obtained with corresponding analysis.A"frozen skin"layer with the thickness increasing versus time during the injection process is found.The fact that the"frozen skin"layer can be reduced by increasing the injection velocity is numerically verified.The fountain flow phenomenon near the melt interface is also captured.Moreover,comparisons with the non-isothermal Newtonian case show that the curvatures of the interface arcs and the pressure contours near the horizontal mid-line of the cavity for the non-Newtonian pseudoplastic case is larger than that for the Newtonian case.The velocity profiles are different at different positions for the non-Newtonian pseudoplastic case,while in the case of Newtonian flow the velocity profiles are parabolic and almost the same at different positions.

Numerical simulation of single bubble rising in shear-thinning fluids by level set method

The behavior of single bubble rising in quiescent shear-thinning tlmds was lnvestlgateO numerically by level set metnoa. number of bubbles in a large range of Reynolds number and Eotvos number were investigated including spherical, oblate and spherical. The bubble shape and drag coefficient were compared with experimental results. It is observed that the simulated results show good conformity to experimental results over a wide range of Reynolds number. In addition, the detailed flow field based on the reference coordinate system moving with the bubble is obtained, and the relationship among flow field, bubble shape and velocity is discussed.

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

为快分割并且降噪，提高 penalization 学期的古典 Mumford-Shah (MS ) 模型需要，即增加 penalization 参数，它导致目标的渐渐的消失。在这份报纸，我们建议一个改进 Mumford-Shah (IMS ) 模型避免现象，并且采用 piecewise 常数水平集合方法(PCLSM ) 和坡度降下方法解决最小化问题。数字实验被给显示出新模型和算法的效率和优点。

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.

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.

New approach to develop a 3D non-isothermal computational framework for injection molding process based on level set method

The simulation of three-dimensional （3D） non-isothermal, non-Newtonian fluid filling process is an extremely difficult task and remains a challenging problem, which includes polymer melt flow with free surface coupled with transient heat transfer. This paper presents a full 3D non-isothermal two-phase flow model to predict the complex flow in melt filling process, where the Cross-WLF model is applied to characterize the rheological behav- ior of polymer melt. The governing equations are solved using finite volume method with SIMPLEC algorithm on collocated grids and the melt front is accurately captured by a high resolution level set method. A domain exten- sion technique is adopted to deal with the complex cavities, which greatly reduces the computational burden. To verify the validity of the developed 3D approach, the melts filling processes in two thin rectangular cavities （one of them with a cylindrical insert） are simulated. The predicted melt front interfaces are in good agreement with the experiment and commercial software prediction. For a case with a rather complex cavity, the dynamic filling process in a hemispherical shell is successfully simulated. All of the numerical results show that the developed numerical procedure can provide a reasonable orediction for injection molding process.

基于Level-Set方法的小孔及熔池动态形成数值模拟

构建了激光深熔焊接三维混合态连续模型,模拟了固液界面过渡层的固、液态共存和熔池高于焊接表面的特征,并采用Level-Set方法追踪了气液界面的移动,计算得到了小孔和熔池的动态形成过程.结果表明,熔池在小孔前沿薄后沿厚,温度梯度在小孔前沿大后沿小,小孔及熔池前沿和后沿存在明显的非对称性;孔壁上蒸发的金属蒸气由孔壁流向小孔中轴线,且向孔外喷射;孔底吸收的激光功率密度最大,最高温度3 700 K位于孔底,高于汽化温度567 K;小孔形成的初期阶段孔深的变化较快,但随着小孔深度的增加,孔深变化速率逐渐下降.

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.

近水面水下爆炸二维Level-set数值模拟

水下爆炸是一个多介质、动边界、瞬态非线性过程,捕捉介质之间交界面的变化是数值模拟的一个难点。本文采用Level-set方法来描述水下爆炸气体产物和流体交界面以及自由表面变化,应用TVD(Total Variation Diminishing)计算技术求解Level-set流场,首先建立了近水面水下爆炸的二维数值方法,然后,将该方法推广到流场内有刚性结构的情况,考虑了冲击波和刚性结构物的相互作用。为了模拟无限边界流场问题,引进了Thompson建立的无反射边界条件。

基于Level Set法铸造充型过程气-液两相流数值模拟

针对目前铸造充型流动模拟最为常见VOF方法的不足,建立了基于Level Set方法的气-液两相流数学模型及求解方法。应用所建立的模型,模拟了气泡的运动过程,并通过水模拟试验进行了验证,模拟结果表明,Level Set方法可以有效地模拟铸造充型过程中的气-液两相流动,是铸造模拟技术发展的新方向。

求解不可压缩两相流的复合Level Set-VOF方法

目前,不可压缩两相流问题的数值求解方法主要有VOF方法和Level Set方法等.根据CLSVOF方法基本思想,通过将VOF和Level Set两种方法结合,实现了复合Level Set-VOF方法追踪不可压缩两相流相界面的模拟计算;对比分析了VOF、Level Set和复合Level Set-VOF方法计算经典两相流问题的结果.研究结果表明,复合Level Set-VOF方法继承了VOF方法精确构造相界面和Level Set方法精确计算表面张力的优点,能够有效削弱两相流求解过程中出现的虚假流动问题,因此可以更加精确地求解不可压缩两相流问题.

Level-set法在液体晃荡研究中的应用

Level-set法是一种简便、通用、高效的计算分析空间运动界面的数值方法,本文尝试性地将Level-set方法应用于液体晃荡的研究,详细论述了应用该方法模拟液体晃荡的数学模型及数值解法。为了验证该方法处理晃荡问题的正确合理性,首先在势流假定下模拟了释放初始液面引起的自由晃荡,并将液面波动的数值结果与理论解析解进行了比较;然后模拟了二维矩形液箱做简谐横摇运动引起的液体受迫晃荡,计算结果以箱壁不同高度处的压强时间历经给出,并与实验结果进行了定量比较。计算结果初步表明采用Level-set方法研究液体晃荡现象是正确可行的。