Modeling and simulation of injection molding process of polymer melt by a robust SPH method

In the present study, the injection molding process of polymer melt based on the generalized Newtonian fluid model is investigated by a robust smoothed particle hydrodynamics (SPH) method. The numerical method is proposed by introducing a Rusanov flux into the continuity equation to improve the prediction of the pressure distribution and employing a corrected kernel gradient to improve the computational accuracy. In addition, a robust treatment of solid boundary is presented and verified by the spin-down problem. The merits of the robust SPH method are firstly illustrated by 2D dam breaking flow. Then the numerical method is extended to deal with the flow phenomena related to injection molding process of polymer melt. A number of numerical examples including 2D injection moldings of a thin plate mold, a circular disc with core, a ring-shaped channel, and a S-shaped cavity, and 3D injection moldings of a Z-shaped cavity and a four-legged fork are conducted. The numerical results are in agreement with the experiments, which demonstrate that the SPH method proposed here is capable of handling with injection molding process of polymer melt in a robust manner. Moreover, the robust SPH method allows to recover the fluctuations-free pressure and velocity fields which in most cases cannot be easily obtained by the traditional SPH method.     

A corrected smoothed particle hydrodynamics method for solving transient viscoelastic fluid flows

In this work, a corrected smoothed particle hydrodynamics (CSPH) method is proposed and extended to the numerical simulation of transient viscoelastic fluid flows due to that its approximation accuracy in solving the Navier–Stokes equations is higher than that of the smoothed particle hydrodynamics (SPH) method, especially near the boundary of the domain. The CSPH approach comes with the idea of combining the SPH approximation for the interior particles with the modified smoothed particle hydrodynamics (MSPH) method for the exterior particles, this is because that the later method has higher accuracy than the SPH method although it also needs more computational cost. In order to show the validity of CSPH method to simulate unsteady viscoelastic flows problems, the planar shear flow problems, including transient Poiseuille, Couette flow and transient combined Poiseuille and Couette flow for the Oldroyd-B fluid are solved and compared with the analytical and SPH results. Subsequently, the general viscoelastic fluid based on the eXtended Pom–Pom (XPP) model is numerically investigated and the viscoelastic free surface phenomena of impacting drop are simulated by the CSPH for its extended application and the purpose of illustrating the ability of the proposed method. The numerical results are presented and compared with available solutions, which shows a very good agreement. All the numerical results show the higher accuracy and better stability of the CSPH than the SPH, especially for larger Weissenberg numbers.     

A decoupled finite particle method for modeling incompressible flows with free surfaces

Smoothed particle hydrodynamics (SPH) is a meshfree Lagrangian particle method, and it has been applied to different areas in engineering and sciences. One concern of the conventional SPH is its low accuracy due to particle inconsistency, which hinders the further methodology development. The finite particle method (FPM) restores the particle consistency in the conventional SPH and thus significantly improves the computational accuracy. However, as pointwise corrective matrix inversion is necessary, FPM may encounter instability problems for highly disordered particle distribution. In this paper, through Taylor series analyses with integration approximation and assuming diagonal dominance of the resultant corrective matrix, a new meshfree particle approximation method, decoupled FPM (DFPM), is developed. DFPM is a corrective SPH method, and is flexible, cost-effective and easy in coding with better computational accuracy. It is very attractive for modeling problems with extremely disordered particle distribution as no matrix inversion is required. One- and two-dimensional numerical tests with different kernel functions, smoothing lengths and particle distributions are conducted. It is demonstrated that DFPM has much better accuracy than conventional SPH, while particle distribution and the selection of smoothing function and smoothing length have little influence on DFPM simulation results. DFPM is further applied to model incompressible flows including Poiseuille flow, Couette flow, shear cavity and liquid sloshing. It is shown that DFPM is as accurate as FPM while as flexible as SPH, and it is very attractive in modeling incompressible flows with possible free surfaces.     

Numerical simulation of the filling behavior of a Non-Newtonian Fluid

A numerical model for free surface flows of non-newtonian liquids which are injected into a cavity is presented. These flows are regarded as a basic model of injection molding. Model experiments of the injection process are performed with a water-based gel. The flow equations are integrated according to the finite-volume-method. The volume of fluid method (VoF) is employed in order to describe the free surface flow of two incompressible phases, the phase interface is resolved by the method of geometric reconstruction. The Herschel-Bulkley model is used in order to describe shear-thinning behavior of the molding material and the effects of a yielding point. Different patterns of the filling flow depending on the injection parameters are evident in the experiment and the simulation. They are characterized and arranged with respect to the similarity parameters of the flow. Again, the results of the simulation are found to agree well with the experimental observations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of process parameters on injection compression molding of pickup lens

The residual stresses and shrinkages of pickup lens in injection compression molding are investigated in this study. It was realized that the behavior of residual stresses in injection compression molding parts was affected by different process conditions such as melt temperature, mold temperature, compression pressure and time. Moldings under different conditions were numerically investigated to study the effects of the process conditions on the residual stresses and shrinkage of a pickup lens with large thickness variations. The mold temperature and compression were found to be the most important factors that affect the shrinkage of lens in the thickness direction, resulting in surface profile deviation. The effect of heat transfer coefficient of the mold wall used in the molding simulation was also discussed.     

A conservative SPH scheme using exact projection with semi-analytical boundary method for free-surface flows

This paper presents a novel SPH scheme for modelling incompressible and divergence-free flow with a free surface (IDFSPH) associated with semi-analytical wall boundary conditions. In line with the projection method, the velocity field is decoupled from the pressure field in the momentum equation. A Poisson equation, serving as the pressure solver, is obtained by which pressure field is decoupled completely from the velocity field. In particular, an exact projection scheme is deployed to fulfil the requirement of the divergence-free velocity field. The condition of incompressibility is satisfied by iteratively updating the density field till the convergence. The two-equation k–ε model is employed to describe the turbulence effects in Newtonian flows. It is shown that the discretised SPH schemes have the feature of both linear and angular momentum conservations. The semi-analytical wall method implements the appropriate integrals to evaluate the boundary contributions to the mass and momentum equations. In comparison to the boundary particle methods, it can greatly enhance the feasibility and efficiency with the complex geometries. The algorithm presented within this paper is applied to several academic test cases for which either analytical results or simulations with other methods are available. The comparisons verify that this scheme is provided with convincing efficiency and extensive applicability.     

Modeling unsteady flow characteristics using smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) method has been extensively used to simulate unsteady free surface flows. The works dedicated to simulation of unsteady internal flows have been generally performed to study the transient start up of steady flows under constant driving forces and for low Reynolds number regimes. However, most of the fluid flow phenomena are unsteady by nature and at moderate to high Reynolds numbers. In this study, first a benchmark case (transient Poiseuille flow) is simulated to evaluate the ability of SPH to simulate internal transient flows at low and moderate Reynolds numbers (Re = 0.05, 500 and 1500). For this benchmark case, the performance of the two most commonly used formulations for viscous term modeling is investigated, as well as the effect of using the XSPH variant. Some points regarding using the symmetric form for pressure gradient modeling are also briefly discussed. Then, the application of SPH is extended to oscillating flows imposed by oscillating body force (Womersley type flow) and oscillating moving boundary (Stokes’ second problem) at different frequencies and amplitudes. There is a very good agreement between SPH results and exact solution even if there is a large phase lag between the oscillating pressure difference and moving boundary and the movement of the SPH particles generated. Finally, a modified formulation for wall shear stress calculations is suggested and verified against exact solutions. In all presented cases, the spatial convergence analysis is performed.     

Application of Smoothed Particle Hydrodynamics for modelling gated spillway flows

Computational models of spillways are important for evaluating and improving dam safety, optimising spillway design and updating operating conditions. Traditionally, scaled down physical models have been used for validation and to collect hydraulic data. Computational fluid dynamics (CFD) models however provide advantages in time, cost and resource reduction. CFD models also provide greater efficiency when evaluating a range of spillway designs or operating conditions. Within the present literature, most studies of computational spillway models utilise a mesh-based method. In this work we use the particle based method of Smoothed Particle Hydrodynamics (SPH) to model weir flow through a four bay, gated, spillway system. Advantages of SPH for such modelling include automatic representation of the free surface flow behaviour due to the Lagrangian nature of the method, and the ability to incorporate complex and dynamic boundary objects such as gate structures or debris. To validate the SPH model, the reservoir water depth simulated is compared with a related physical study. The effect of SPH resolution on the predicted water depth is evaluated. The change in reservoir water level with discharge rates for weir flow conditions is also investigated, with the difference in simulated and experimental water depths found to range from 0.16% to 11.48%. These results are the first quantitative validation of the SPH method to capture spillway flow in three dimensions. The agreement achieved demonstrates the capability of the SPH method for modelling spillway flows.     

Analysis of the filling capability to the microstructures in micro-injection molding

Micro-injection molding is an important fabrication process for polymer plastics with micro-features. In micro-injection molding of products with microstructures, the ability for the polymer melt to flow into the microstructures is a crucial factor for successful molding. An analytical model in micro-injection molding is constructed in this research. It has been reported that most of the filling in microstructure is done during the packing pressure. In this analytical model, the temperature of the polymer melt near the entrance of the microstructure at the end of mold filling is estimated first. With the temperature, we can calculate the injection distance into the microstructures of the mold insert during the packing stage. The model is compared with experimental results, and shows the feasibility. The experiment uses the LIGA-like lithography process to define the micro-feature and a micro-electroforming method to form the mold insert with the replicated micro-feature. The injection distance into the microstructures predicted by this analytical model shows reasonable result as compared to the experimental measurement.     

粉末注射成形填充过程的数值模拟

本文将粉末注射成形喂料在薄壁模腔中的流动视为二维流动，以流变学的基本方程为基础，建立了从动量方程、连续方程和热传递方程得到的描述PIM喂料充模二维流动的数学模型。在无滑移边界的条件下，推导了喂料熔体流导率的计算公式和压力场的控制方程，得到的压力场控制方程是一非线性椭圆偏微分方程．从而可用Galerkin方法进行数值求解，使模型的数值求解成为可能，为进一步对粉末注射成形进行计算机模拟和数值分析奠定了数学基础。     

Modeling incompressible flows using a finite particle method

This paper describes the applications of a finite particle method (FPM) to modeling incompressible flow problems. FPM is a meshfree particle method in which the approximation of a field variable and its derivatives can be simultaneously obtained through solving a pointwise matrix equation. A set of basis functions is employed to obtain the coefficient matrix through a sequence of transformations. The finite particle method can be used to discretize the Navier–Stokes equation that governs fluid flows. The incompressible flows are modeled as slightly compressible via specially selected equations of state. Four numerical examples including the classic Poiseuille flow, Couette flow, shear driven cavity and a dam collapsing problem are presented with comparisons to other sources. The numerical examples demonstrate that FPM is a very attractive alternative for simulating incompressible flows, especially those with free surfaces, moving interfaces or deformable boundaries.     

3D SPH flow predictions and validation for high pressure die casting of automotive components

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three-dimensional fluid flow with significant free surface fragmentation and splashing. A Lagrangian simulation technique that is particularly well suited to modelling HPDC is smoothed particle hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling the accurate prediction of fluid flows involving complex free surface motion.     

A corrected smooth particle hydrodynamics method for the simulation of debris flows

This article presents the development and application of a corrected smooth particle hydrodynamics (CSPH) code to the simulation of debris flow and avalanches. The advantages of a mesh‐free method over other traditional numerical methods such as the finite element method are discussed. A new frictional approach for the boundary conditions and modified constitutive equations are introduced in the SPH method. The resulting technique is then applied for the simulation of debris flows, comparing the results with those obtained from experiments reported by other researchers. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 140–163, 2004.     

A new kernel function for SPH with applications to free surface flows

Smoothed particle hydrodynamics (SPH) is a popular meshfree Lagrangian particle method, which uses a kernel function for numerical approximations. The kernel function is closely related to the computational accuracy and stability of the SPH method. In this paper, a new kernel function is proposed, which consists of two cosine functions and is referred to as double cosine kernel function. The newly proposed double cosine kernel function is sufficiently smooth, and is associated with an adjustable support domain. It also has smaller second order momentum, and therefore it can have better accuracy in terms of kernel approximation. SPH method with this double cosine kernel function is applied to simulate a dam-break flow and water entry of a horizontal circular cylinder. The obtained SPH results agree very well with the experimental results. The double cosine kernel function is also comparatively studied with two frequently used SPH kernel functions, Gaussian and cubic spline kernel functions.     

Constitutive equations of non-isothermal polymer melt

Constitutive equations of non-isothermal polymer melt are presented by the analysis of entropic free energy contribution of the macromolecular chains, which are treated as elastic dumbbell models. With describing non-isothermal dumbbell spring, as the function of temperature, the non-linear elastic coefficient expression causes the appearance of temperature gradient in stress constitutive equations. Following the constitutive equation of Hookean dumbbell model, non-isothermal stress constitutive equations of FENE and FENE-P models are derived. In deriving process of constitutive equations, the second moment approximation is used to closure FENE model. Using the non-isothermal constitutive equations, numerical simulations of polymer flow through shear cavity and planar contraction cavity are presented. And the distributions of correlative stress functions and the effects of different temperatures on stress functions are discussed. The present results are shown to explore the non-isothermal constitutive equations of elastic dumbbell models, and to search more accurately describing way of non-isothermal polymer melt.     

曲率流的参数化有限元逼近

许多物理现象可以在数学上描述为受曲率驱动的自由界面运动,例如薄膜和泡沫的演变、晶体生长,等等.这些薄膜和界面的运动常依赖于其表面曲率,从而可以用相应的曲率流来描述,其相关自由界面问题的数值计算和误差分析一直是计算数学领域中的难点.参数化有限元法是曲率流的一类有效计算方法,已经能够成功模拟一些曲面在几类基本的曲率流下的演化过程.本文重点讨论曲率流的参数化有限元逼近,它的产生、发展和当前的一些挑战.     

Optimization of the injection molding process for short-fiber-reinforced composites

A fast and effective methodology integrating the finite-element and Taguchi methods is presented to determine the optimal design conditions of the injection molding process for short-fiber-reinforced polycarbonate composites. The finite-element-based flow simulation software, M-flow, was employed to simulate the molding process to obtain the fiber orientation distributions required. The Taguchi optimization technique was used to identify the optimal settings of injection molding parameters to maximize the shear layer thickness. The effects of four main parameters — the filling time, melt temperature, mold temperature, and injection speed — on the fiber orientation or the shear layer thickness were investigated and discussed. It is found that the dominant parameter is the filling time. The best levels of the four parameters to acquire the thickest shear layer are also identified.     

Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three dimensional fluid flow with significant free surface fragmentation and splashing. A simulation method that has proved particularly suited to modelling HPDC is Smoothed Particle Hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling more accurate prediction of fluid flows involving complex free surface motion. Three practical industrial case studies of SPH simulated HPDC flows are presented; aluminium casting of a differential cover (automotive), an electronic housing and zinc casting of a door lock plate. These show significant detail in the fragmented fluid free surfaces and allow us to understand the predisposition to create defects such as porosity in the castings. The validation of flow predictions coupled with heat transfer and solidification is an important area for such modelling. One powerful approach is to use short shots, where insufficient metal is used in the casting or the casting shot is halted part way through, to leave the die cavity only partially filled. The frozen partial castings capture significant detail about the order of fill and the flow structures occurring during different stages of filling. Validation can occur by matching experimental and simulated short shots. Here we explore the effect of die temperature, metal super-heat and volume fill on the short shots for the casting of a simple coaster. The bulk features of the final solid castings are found to be in good agreement with the predictions, but the fine details appear to depend on surface behaviour of the solidifying metals. This potentially has significant implications for modelling HPDC.     

Comparative estimate of the molding capacities of injection molding machines with plunger and screw plastication

An analysis of the mold filling process, based on the equation of nonisothermal flow of an amorphous polymer melt, shows that for thin moldings the molding capacity of the machine, estimated as the maximum shot length, has a limit that does not depend on the pressure at the mold inlet and is determined by the flow rate of the polymer in the mold. As shown by an analysis of the process of compaction of the polymer in the system preceding the mold, this flow rate is many times less for plunger plastication, which is responsible for the reduced molding capacity of plunger machines. The effect can be eliminated by compressing the polymer before injecting it into the mold.Moscow Institute of Chemical Machine Building. Translated from Mekhanika Polimerov, No. 2, pp. 367–372, March–April, 1970.     

三维PTT黏弹性液滴撞击固壁面问题的改进SPH模拟

基于光滑粒子动力学(smoothed particle hydrodynamics, SPH)方法,对三维Phan-Thien Tanner(PTT)黏弹性液滴撞击固壁面问题进行了数值模拟.为了有效地防止粒子穿透固壁,且缩减三维数值模拟所消耗的计算时间,提出了一种适合三维数值模拟的改进固壁边界处理方法.为了消除张力不稳定性问题,采用一种简化的人工应力技术.应用改进SPH方法对三维PTT黏弹性液滴撞击固壁面问题进行了数值模拟,精细地捕捉了液滴在不同时刻的自由面,讨论了PTT黏弹性液滴不同于Newton(牛顿)液滴的流动特征,分析了PTT拉伸参数对液滴宽度、高度和弹性收缩比等的影响.模拟结果表明,改进SPH方法能够有效而准确地描述三维PTT黏弹性液滴撞击固壁面问题的复杂流变特性和自由面变化特征.