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.

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 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.

Risk assessment of coastal flooding disaster by storm surge based on Elevation-Area method and hydrodynamic models:Taking Bohai Bay as an example

The future inundation by storm surge on coastal areas are currently ill-defined.With increasing global sealevel due to climate change,the coastal flooding by storm surge is more and more frequently,especially in coastal lowland with land subsidence.Therefore,the risk assessment of such inundation for these areas is of great significance for the sustainable socio-economic development.In this paper,the authors use Elevation-Area method and Regional Ocean Model System(ROMS)model to assess the risk of the inundation of Bohai Bay by storm surge.The simulation results of Elevation-Area method show that either a 50-year or 100-year storm surge can inundate coastal areas exceeding 8000 km^(2);the numerical simulation results based on hydrodynamics,considering ground friction and duration of the storm surge high water,show that a 50-year or 100-year storm surge can only inundate an area of over 2000 km^(2),which is far less than 8000 km^(2);while,when taking into account the land subsidence and sea level rise,the very inundation range will rapidly increase by 2050 and 2100.The storm surge will greatly impact the coastal area within about 10-30 km of the Bohai Bay,in where almost all major coastal projects are located.The prompt response to flood disaster due to storm surge is urgently needed,for which five suggestions have been proposed based on the geological background of Bohai Bay.This study may offer insight into the development of the response and adaptive plans for flooding disasters caused by storm surge.

A Simple Egg Membrane Patch Bridging Method for Repairing Tympanic Membrane Perforation

Objective:To analyze the clinical effect of a simple egg membrane patch bridging method in repairing tympanic membrane perforation.Methods:A total of 93 tympanic membrane perforation patients admitted to the hospital between September 2022 and October 2023 were selected and divided into two groups according to the random number table method.The control group implemented the conventional treatment(n=46 cases),and the patch group adopted the simple egg membrane patch bridging method(n=47 cases).The healing rate of the tympanic membrane,the air-bone gap,the air conduction hearing threshold,the dry ear rate,and the incidence of complications in both groups were compared before and after treatment.Results:The healing rate of the tympanic membrane in the patch group was significantly higher than that of the control group(95.75%vs.76.09%),with P<0.05;there was no difference in the air-bone gap and air conduction hearing threshold levels between the two groups before treatment(P>0.05),and the hearing indexes of the patch group were significantly lower than those of the control group 3 months after treatment(P<0.05);the dry ear rate in the patch group was significantly higher than that of the control group after treatment(85.11%vs.67.39%),and the total incidence of complications was also significantly lower than that of the control group(6.38%vs.21.74%),with P<0.05.Conclusion:The simple egg membrane patch bridging method is effective in repairing tympanic membrane perforation,which can effectively improve patients’hearing levels and reduce the occurrence of post-treatment complications.Thus,it is worth popularizing and applying in the clinic.

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。

Improved Calculation of Vibrational Energy Levels in F2 Molecule using the RKR Method

The potential energy curves of the ground state X2∑＋g of the fluorine molecule have been accurately reconstructed employing the Ryderg-Klein-Rees （RKR） method extrapolated by a Hulburt and Hirschfeler potential function for longer internuclear distances. Solving the corresponding radial one-dimensional Schr？dinger equation of nuclear motion yields 22 bound vibrational levels above v=0. The comparison of these theoretical levels with the experimental data yields a mean absolute deviation of about 7.6 cm^-1 over the 23 levels. The highest vibrational level energy obtained using this method is 13308.16 cm？1 and the relative deviation compared with the experimental datum of 13408.49 cm^-1 is only 0.74%. The value from our method is much closer and more accurate than the value obtained by the quantum mechanical ab initio method by Bytautas. The reported agreement of the vibrational levels and dissociation energy with experiment is contingent upon the potential energy curve of the F2 ground state.

Topological and Shape Optimization of Flexure Hinges for Designing Compliant Mechanisms Using the Level Set Method

A flexure hinge is a major component in designing compliant mechanisms that o ers unique possibilities in a wide range of application fields in which high positioning accuracy is required. Although various flexure hinges with di erent configurations have been successively proposed, they are often designed based on designers' experiences and inspirations. This study presents a systematic method for topological optimization of flexure hinges by using the level set method. Optimization formulations are developed by considering the functional requirements and geometrical constraints of flexure hinges. The functional requirements are first constructed by maximizing the compliance in the desired direction while minimizing the compliances in the other directions. The weighting sum method is used to construct an objective function in which a self-adjust method is used to set the weighting factors. A constraint on the symmetry of the obtained configuration is developed. Several numerical examples are presented to demonstrate the validity of the proposed method. The obtained results reveal that the design of a flexure hinge starting from the topology level can yield more choices for compliant mechanism design and obtain better designs that achieve higher performance.

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.

Model for the Whole Roller Leveling Process of Plates with Random Curvature Distribution Based on the Curvature Integration Method

A model based on the curvature integration method has been applied in an online plate leveling system. However, there are some shortcomings in the current leveling models. On the one hand, the models cannot deal with the leveling process of plates with a random curvature distribution. On the other hand, the current models are suitable only for stable leveling processes and ignore the biting in and tailing out stages. This study presents a new plate-leveling model based on the curvature integration method, which can describe the leveling process of plates with random curvature distribution. Further, the model is solved in two cases in order to take the biting in and tailing out stages into consideration. The proposed model is evaluated by comparing with a plate leveling experiment. Finally, the leveling process of a plate with a wave bent is studied using the proposed model. It is found that the contact angles vary greatly during the biting in and tailing out stages. However, they are relatively steady during the 5 roller leveling stage. In addition, the contact angle of roller No. 2 is the smallest, which is close to 0. Roller leveling can effectively eliminate bending in the plate, but there are regions in the head and tail of the plate, where roller leveling is not effective. The non-leveling region length is about 2 times that of the roller space. This study proposes a quasi-static plate-leveling model, which makes it possible to analyze the dynamic straightening process using a curvature integration method. It also makes it possible to analyze the straightening process of a plate with random curvature distribution.

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.

New Method to Measure the Fill Level of the Ball Mill I——Theoretical Analysis and DEM Simulation

The accurate measurement of the fill level in the ball mill has not been resolved because of the interplay of many variable factors, which led the mill to be operated under the uneconomical condition and lost a lot of energy. At present, some methods, such as vibration method and acoustic method, have been applied for measuring the fill level by the researchers. Aiming at the problem of the traditional methods for measuring the fill level, that is, the feature variables of the fill level suffer the influences of the ball load and the water content of the coal, a novel method to measure the fill level is proposed and a possible relation between the fill level and the angular position of the maximum vibration point on the mill shell is investigated. The angular positions of the maximum vibration point on the mill shell for different fill level cases are calculated theoretically under two assumptions, respectively. Meanwhile the charge motions of the mill for different fill level cases are simulated with the discrete element method （DEM）. And the simulation results are verified by comparing the motion trajectories of steel balls and power draft of the mill. The simulated movement trajectories of the outmost layer steel balls in the mill are monitored and analyzed to obtain the angular positions of the maximum vibration point on the mill shell. Both the results of the theoretical calculation and the 3D DEM simulation show that the position of the maximum vibration point on the mill shell moves to a lower angular positions as the fill level decreasing, which provides a new idea for measuring the filllevel accurately.

A method to calculate design tide levels on the basis of numerical model of tidal current and its application

In order to determine the design tide levels in the areas without measured tide level data, especially in the areas where it is difficult to measure tidal levels, a calculation method based on a numerical model of tidal current is proposed. The essentials of the method are described, and its application is illustrated with an example. The results of the application show that the design tide levels calculated by the method are close to those determined by long-time measured tide level data, and its calculation precision is high, so it is feasible to use the method to determine the design tide levels in the areas.

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.

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.

A Simple Weighted Integration Method for Calculating Surface Tension Force to Suppress Parasitic Flow in the Level Set Approach

Parasitic flows may occur in the numerical simulation of incompressible multiphase flow due to errors in the calculation of surface tension terms, specifically for the curvature and unit normal vector. An improved method for calculating the surface tension based on the level set approach is proposed, in which the contribution of not only the center node but also the rest area of a control volume to the calculation of surface tension is considered in a balanced manner. The weighted integration method (WIM) is more consistent with the concept of a banded interface in the level set method. It is applied to the temporal evolution of a two-dimensional neutrally buoyant liquid drop and a buoyancy driven deformable bubble in an immiscible fluid for the validation of WIM. The results show that the parasitic flows are evidently suppressed by the weighted integration method. The weight factors for WIM in 3-D cases are also suggested.

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.