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.

Application of Gridless Method to Simulation of Compressible Multi-Material Flows

The least-square gridless method was extended to simulate the compressible multi-material flows. The algorithm was accomplished to solve the Arbitrary Lagrange-Euler( ALE) formulation. The local least-square curve fits was adopted to approximate the spatial derivatives of a point on the base of the points in its circular support domain,and the basis function was linear. The HLLC( Harten-Lax-van Leer-Contact) scheme was used to calculate the inviscid flux. On the material interfaces,the gridless points were endued with a dual definition corresponding to different materials. The moving velocity of the interface points was updated by solving the Riemann problem. The interface boundary condition was built by using the Ghost Fluid Method( GFM).Computations were performed for several one and two dimensional typical examples. The numerical results show that the interface and the shock wave are well captured,which proves the effectiveness of gridless method in dealing with multi-material flow problems.

A hybrid subcell-remapping algorithm for staggered multi-material arbitrary Lagrangian-Eulerian methods

A new flux-based hybrid subcell-remapping algorithm for staggered multimaterial arbitrary Lagrangian-Eulerian (MMALE) methods is presented. This new method is an effective generalization of the original subcell-remapping method to the multi-material regime (LOUBERE, R. and SHASHKOV,M. A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. Journal of Computational Physics, 209, 105–138 (2005)). A complete remapping procedure of all fluid quantities is described detailedly in this paper. In the pure material regions, remapping of mass and internal energy is performed by using the original subcell-remapping method. In the regions near the material interfaces, remapping of mass and internal energy is performed with the intersection-based fluxes where intersections are performed between the swept regions and pure material polygons in the Lagrangian mesh, and an approximate approach is then introduced for constructing the subcell mass fluxes. In remapping of the subcell momentum, the mass fluxes are used to construct the momentum fluxes by multiplying a reconstructed velocity in the swept region. The nodal velocity is then conservatively recovered. Some numerical examples simulated in the full MMALE regime and several purely cyclic remapping examples are presented to prove the properties of the remapping method.

A robust algorithm for moving interface of multi-material fluids based on Riemann solutions

In the paper, the numerical simulation of interface problems for multiple material fluids is studied. The level set function is designed to capture the location of the material interface. For multi-dimensional and multi-material fluids, the modified ghost fluid method needs a Riemann solution to renew the variable states near the interface. Here we present a new convenient and effective algorithm for solving the Riemann problem in the normal direction. The extrapolated variables are populated by Taylor series expansions in the direction. The anti-diffusive high order WENO difference scheme with the limiter is adopted for the numerical simulation. Finally we implement a series of numerical experiments of multi-material flows. The obtained results are satisfying, compared to those by other methods.

Robust Topology Optimization of Periodic Multi-Material Functionally Graded Structures under Loading Uncertainties

This paper presents a robust topology optimization design approach for multi-material functional graded structures under periodic constraint with load uncertainties.To characterize the random-field uncertainties with a reduced set of random variables,the Karhunen-Lo`eve(K-L)expansion is adopted.The sparse grid numerical integration method is employed to transform the robust topology optimization into a weighted summation of series of deterministic topology optimization.Under dividing the design domain,the volume fraction of each preset gradient layer is extracted.Based on the ordered solid isotropic microstructure with penalization(Ordered-SIMP),a functionally graded multi-material interpolation model is formulated by individually optimizing each preset gradient layer.The periodic constraint setting of the gradient layer is achieved by redistributing the average element compliance in sub-regions.Then,the method of moving asymptotes(MMA)is introduced to iteratively update the design variables.Several numerical examples are presented to verify the validity and applicability of the proposed method.The results demonstrate that the periodic functionally graded multi-material topology can be obtained under different numbers of sub-regions,and robust design structures are more stable than that indicated by the deterministic results.

ALE Fractional Step Finite Element Method for Fluid-Structure Nonlinear Interaction Problem

A computational procedure is developed to solve the problems of coupled motion of a structure and a viscous incompressible fluid. In order to incorporate the effect of the moving surface of the structure as well as the free surface motion, the arbitrary Lagrangian-Eulerian formulation is employed as the basis of the finite element spatial discretization. For numerical integration in time, the fraction,step method is used. This method is useful because one can use the same linear interpolation function for both velocity and pressure. The method is applied to the nonlinear interaction of a structure and a tuned liquid damper. All computations are performed with a personal computer.

A COMPARATIVE STUDY OF THE OPENING AND CLOSING PROCESS OF TWO TYPES OF MECHANICAL HEART VALVES USING ALE FINITE ELEMENT METHOD

Using arbitrary Lagrangian-Eulerian(ALE)finite element method,this paper made a comparative study of the opening and closing behaviour of a downstream directional valve(DDM)and a St.Jude medical valve(SJM)through a two dimensional model of mechanical valve-blood interaction in which the valve is considered as a rigid body rotating around a fixed point,and the blood is simplified as viscous incompressible fluid It's concluded that:(1)Compared with SJM valve, DDM valve opens faster and closes the more gently.(2)The peak badk-flow-flow of DDM is smaller than that of SJM.The present investigation shows that being a better analogue of natural valve,DDM has a brighter potential on its durability than SJM.

耦合拉格朗日-欧拉方法及其在海洋工程中的应用

Combining the strengths of Lagrangian and Eulerian descriptions,the coupled Lagrangian–Eulerian methods play an increasingly important role in various subjects.This work reviews their development and application in ocean engineering.Initially,we briefly outline the advantages and disadvantages of the Lagrangian and Eulerian descriptions and the main characteristics of the coupled Lagrangian–Eulerian approach.Then,following the developmental trajectory of these methods,the fundamental formulations and the frameworks of various approaches,including the arbitrary Lagrangian–Eulerian finite element method,the particle-in-cell method,the material point method,and the recently developed Lagrangian–Eulerian stabilized collocation method,are detailedly reviewed.In addition,the article reviews the research progress of these methods with applications in ocean hydrodynamics,focusing on free surface flows,numerical wave generation,wave overturning and breaking,interactions between waves and coastal structures,fluid–rigid body interactions,fluid–elastic body interactions,multiphase flow problems and visualization of ocean flows,etc.Furthermore,the latest research advancements in the numerical stability,accuracy,efficiency,and consistency of the coupled Lagrangian–Eulerian particle methods are reviewed;these advancements enable efficient and highly accurate simulation of complicated multiphysics problems in ocean and coastal engineering.By building on these works,the current challenges and future directions of the hybrid Lagrangian–Eulerian particle methods are summarized.

ALE FINITE ELEMENT ANALYSIS OF THE OPENING AND CLOSING PROCESS OF THE ARTIFICIAL MECHANICAL VALVE

Employing arbitrary Lagrangian-Eulerian (ALE) finite element method, this poper studies the opening and closing process of a St. Jude medical valve through a two-dimensional model of the mechanical valve-blood interaction in which the valve is regarded as a rigid body rotating around a fixed point, and foe blood is simplified as viscous incompressible Newtonian fluid. The numerical analysis of the opening and closing behaviour of as St. Jude valve suggested that: 1. The whole opening and closing process of an artificial mechanical valve is consisted of four phases: (1) Opening phase; (2) Opening maintenance phase; (3) Closing phase; (4) Closing maintenance phase. 2. The St. Jude medical valve closes with prominent regurgitat which results in water-hammer effect. 3. During the opening and closing process of the St. Jude valve,high shear stresses occur in the middle region of the two leaflets and on the valve ring. The present model has made a breakthrough on the coupling computational analysis considering the interactive movement of the valve and blood.

Digital high-efficiency print forming method and device for multi-material casting molds

Sand mold 3D printing technology based on the principle of droplet ejection has undergone rapid development in recent years and has elicited increasing attention from engineers and technicians.However,current sand mold 3D printing technology exhibits several problems,such as single-material printing molds,low manufacturing efficiency,and necessary post-process drying and heating for the manufacture of sand molds.This study proposes a novel high-efficiency print forming method and device for multi-material casting molds.The proposed method is specifically related to the integrated forming of two-way coating and printing and the shortflow manufacture of roller compaction and layered heating.These processes can realize the high-efficiency print forming of high-performance sand molds.Experimental results demonstrate that the efficiency of sand mold fabrication can be increased by 200%using the proposed two-way coating and printing method.The integrated forming method for layered heating and roller compaction presented in this study effectively shortens the manufacturing process for 3D-printed sand molds,increases sand mold strength by 63.8%,and reduces resin usage by approximately 30%.The manufacture of multi-material casting molds is demonstrated on typical wheeled cast-iron parts.This research provides theoretical guidance for the engineering application of sand mold 3D printing.

A 2D Staggered Multi-Material ALE Code Using MOF Interface Reconstruction

Hydrocodes are necessary numerical tools in the fields of implosion and high-velocity impact,which often involve large deformations with changing-topology interfaces.It is very difficult for Lagrangian or Simplified Arbitrary Lagrangian-Eulerian(SALE)codes to tackle these kinds of large-deformation problems,so a stag-gered Multi-Material ALE(MMALE)code is developed in this paper,which is the explicit time-marching Lagrange plus remap type.We use the Moment Of Fluid(MOF)method to reconstruct the interfaces of multi-material cells and present an adaptive bisection method to search for the global minimum value of the nonlinear objective function.To keep the Lagrangian computations as long as possible,we develop a ro-bust rezoning method named as Combined Rezoning Method(CRM)to generate the convex,smooth grids for the large-deformation domain.Regarding the staggered remap phase,we use two methods to remap the variables of Lagrangian mesh to the rezoned one.One is the first-order intersection-based remapping method that doesn’t limit the distances between the rezoned and Lagrangian meshes,so it can be used in the applications of wide scope.The other one is the conservative second-order flux-based remapping method developed by Kucharika and Shashkov[22]that requires the rezoned element to locate in its adjacent old elements.Numerical results of triple point problem show that the result of first-order remapping method using ALE computations is gradually convergent to that of second-order remapping method using Eulerian computations with the decrease of rezoning,thereby telling us that MMALE computations should be performed as few as possible to reduce the errors of the interface reconstruction and the remapping.Numerical results provide a clear evidence of the robustness and the accuracy of this MMALE scheme,and that our MMALE code is powerful for the large-deformation problems.

SIMULATION OF FLUID-SOLID INTERACTION ON WATER DITCHING OF AN AIRPLANE BY ALE METHOD

Ditching is considered as one of the important aspects of safety performances of airplanes. It is related primarily with the fluid-solid interaction, whose studies mainly depend on experiments at the present time. Numerical and analytical methods for fluid-solid interaction by using 3-D full scale airplane＇s model will reduce the dependence on the expensive model tests. Numerical studies can be used to estimate the safety of ditching and provide a reference for the crashworthiness design. This article proposes a 3-D dynamical structural model after the real shape of an airplane and an Arbitrary Lagrange-Euler （ALE） fluid-field model, to simulate the fluid-solid interactions caused by low speed ditching. The simulation is based on interaction computational methods, within LS-DYNA nonlinear finite-element code. The results of pressure distributions and accelerating time histories of the airplane＇s subfloor are discussed in the context of the safety of ditching, and the simulation results and the analytical methods are verified.

AN ALE METHOD AND DDM WITH HIGH ACCURATE COMPACT SCHEMES FOR VORTEX-INDUCED VIBRATIONS OF AN ELASTIC CIRCULAR CYLINDER

A numerical study was conducted for the vortex-induced vibrations of anelastic circular cylinder at low Reynolds numbers. An Arbitrary Lagrangian-Eulerian (ALE) method wasemployed to deal with the fluid-structure interaction with an H-O type of non-staggered gridsincorporating the domain decomposition method (DDM), which could save the computational CPU time dueto re-meshing. The computational domain was divided into nine sub-domains including one ALEsub-domain and eight Eulerian sub-domains. The convection term and dissipation term in the N-Sequations were discretized using the third-order upwind compact scheme and the fourth-order centralcompact scheme, respectively. The motion of the cylinder was modeled by a spring-damper-mass systemand solved using the Runge-Kutta method. By simulating the non-linear fluid-structure interaction,the ''lock-in'', ''beating'' and ''phase switch'' phenomena were successfully captured, and the resultsagree with experimental data Furthermore, the vortex structure, the unsteady lift and drag on thecylinder, and the cylinder displacement at various natural frequency of the cylinder for Re = 200were discussed in detail, by which a jump transition of the wake structure was captured.

An Optimization-Based Rezoning for ALE Methods

Based on the theory of optimization,we use edges and angles of cells to represent the geometric quality of computational grids,employ the local gradients of the flow variables to describe the variation of flow field,and construct a multi-objective programming model.The solution of this optimization problem gives appropriate balance between the geometric quality and adaptation of grids.By solving the optimization problem,we propose a new grid rezoning method,which not only keeps good geometric quality of grids,but also can track rapid changes in the flow field.In particular,it performs well for some complex concave domains with corners.We also incorporate the rezoningmethod into anArbitrary Lagrangian-Eulerian(ALE)method which is widely used in the simulation of high-speed multi-material flows.The proposed rezoning and ALE methods of this paper are tested by a number of numerical examples with complex concave domains and compared with some other rezoning methods.The numerical results validate the robustness of the proposed methods.

Prediction of shear-related defect locations in semi-solid casting using numerical flow models

Contaminated surfaces of the feedstock materials in aluminum alloy casting processes often produce various types of defects which can affect the tensile properties of the final products as well as their fatigue reliabilities.Semi-solid processing takes advantage of a much higher apparent viscosity of the die cast materials by limiting the risk of oxides formed at the free surfaces to become incorporated into the casting when the material is injected into the die.Most of existing semi-solid processes that use billets as feedstock material are however tied up with a different type of contaminated surface.During the injection phase,the external-skin on the periphery of the billet,which has been in contact with air and lubricant during the transfer in the shot sleeve,can be incorporated into the casting.When subjected to a heat treatment,the lubricant is decomposed and produces lens shape porosities.This might be a cause of reject for most structural parts.To avoid this kind of defects,the paths along which the billet skin evolves must be controlled during filling.In order to investigate the possibility of skin inclusion into cast parts during injection of the billet,a two-phase finite element mixture model is employed to model the metal flow.The formation of a skin on the periphery of the billet is modeled by setting an initial solid phase concentration profile in the radial direction.Microscopic observations of the real castings show that the approach is able to model the shear layers and to predict the paths along which the"lens porosity"defects could be formed.An Arbitrary Eulerian-Lagangian(ALE) method is also investigated and appears to be very promising to follow the skin movement in the casting.

Numerical simulations for gas-structure interaction in inflated deployment of folded membrane boom

It is very important for gas-structure interaction between compressible ideal gas and elastic structure of space folded membrane booms during the inflatable deployment. In order to study this gas-structure interaction problem, Arbitrary Lagrangian-Eulerian （ALE） finite element method was employed. Gas-structure interaction equation was built based on equilibrium integration relationship, and solved by operator split method. In addition, numerical analysis of V-shape folded membrane booms inflated by gas was given, the variation of inner pressure as well as deployment velocities of inflatable boom at different stage were simulated. Moreover, these results are consistent with the experiment of the same boom~ which shows that both ALE method and operator split method are feasible and reliable methods to study gas-structure interaction problem.

UNITIVE ANALYSIS SCHEMES ON PROBLEMS OF MULTIPLE MOVING BOUNDARIES WITH 3-D LIQUID-SOLID MULTIPLE NONLINEAR COUPLING FOR UPLIFT OF ANCHOREDLIQUID STORAGE TANKS

In the paper, 3-D analysis method with unitive schemes is set up, which is used to resolve the uplift with multiple moving boundaries and multiple nonlinear coupling for anchored liquid storage tanks. hi it, an algorithm of quasi-harmonious finite elements for arbitrary quadrilateral of thin plates and shells is built up to analyze the multiple coupling problems of general thin plates and shells structures with three dimensions, the complementary equations for analyzing uplifting moving boundary problems are deduced. The axial symmetry and presumption of beam type mode are not used. In it, an algorithm is put forward for analyzing the Navier-Stokes problems of unsteady, three-dimensional, and viscous liquid with sloshing of moving boundary surfaces in large amplitude under ALE frame by scheme of time-split-steps to which linear potential theory is not applied. The algorithms can be used to analyze the solid-liquid multiple nonlinear coupling problems with 3-D moving boundary with friction in multiple places.

Fuzzy interface treatment in Eulerian method

In this paper, we incorporate fuzzy mathematics approach into the Eulerian method to simulate three dimensional multi-material interfaces. In particular, we propose a fuzzy interface treatment for describing interfaces, designing transport plans, and computing transport quantities. Using a set of three-dimensional inviscid isothermal elastic-plastic hydrodynamic equations, we simulate shaped charge jet in different filled dynamite structures. Strain and stress have been under consideration in simulations. Numerical results demonstrate that the fuzzy interface treatment is correct and efficient for three-dimensional multi-material problems.

A NEW FREE SURFACE MESH TRACING METHOD

Free surface flow problems involving large free motions are analysed using finite element techniques. In solving these problems an Arbitrary Lagrangian-Eulerian(ALE)kinematical description of the fluid domain is adopted, in which the nodal points can be displaced independently of the fluid motion. A new mesh tracing method is proposed in this paper. To confirm the effectiveness of the new method, solitary wave propagation is analysed and the numerical results are compared with the analytical results. The behaviour of the viscous fluid flow with a free surface is expressed by the unsteady Navier-Stokes equation. For numerical integration in time the velocity correction fractional step method is used.

Numerical Simulation of Macrosegregation Caused by Thermal–Solutal Convection and Solidification Shrinkage Using ALE Model

Solidifi cation shrinkage has been recognized as an important factor for macrosegregation formation. An arbitrary Lagrangian–Eulerian(ALE) model is constructed to predict the macrosegregation caused by thermal–solutal convection and solidi-fi cation shrinkage. A novel mesh update algorithm is developed to account for the domain change induced by solidifi cation shrinkage. The velocity–pressure coupling between the non-homogenous mass conservation equation and momentum equation is addressed by a modifi ed pressure correction method. The governing equations are solved by the streamline-upwind/Petrov–Galerkin-stabilized fi nite element algorithm. The application of the model to the Pb-19.2 wt%Sn alloy solidifi cation problem is considered. The inverse segregation is successfully predicted, and reasonable agreement with the literature results is obtained. Thus, the ALE model is established to be a highly effective tool for predicting the macrosegregation caused by solidifi cation shrinkage and thermal–solutal convection. Finally, the effect of solidifi cation shrinkage is analyzed. The results demonstrate that solidifi cation shrinkage delays the advance of the solidifi cation front and intensifi es the segregation.