基于黏弹性的微注射成型流动模拟

应用有限元方法研究了微注射成型中瞬态、可压缩、非牛顿熔体流动的黏弹性对流动前沿及流动平衡的影响。基于Phan-Thien-Tanner模型建立了熔体流动的本构方程,利用Hele-Shaw假设和简化建立了瞬态、可压缩、非牛顿熔体流动的连续性方程、动量方程、能量方程;为了有效地描述微注射成型的尺寸效应,采用了边界滑移和表面张力边界条件。通过分部积分和待定系数法导出了带有边界信息的变分方程和求解应力分量的半解析公式,构造了有限元离散求解及超松驰迭代算法。模拟结果表明:熔体的黏弹性对浇口附近的压力和后续的熔体流动前沿有重要影响;与黏性模型相比,黏弹性模型可以控制模拟压力的快速增长,减少不同型腔之间的充填差异,与短射实验结果也更吻合。     

Modeling and simulation of the viscoelastic fluid mold filling process by level set method

A model for unifying a viscoelastic fluid and a Newtonian fluid is established, in which the governing equations for the viscoelastic fluid and the Newtonian fluid are successfully united into a system of generalized Navier–Stokes equations. A level set method is set up to solve the model for capturing the moving interface in the mold filling process. The physical governing equations are solved by the finite volume method on a non-staggered grid and the interpolation technique on the collocated grid is used for the pressure-velocity and the stress-velocity decoupling problems. The level set and its reinitialization equation are solved by the finite difference method, in which the spatial derivatives are discretized by the 5th-order Weighted Essentially Non-Oscillatory (WENO) scheme, and the temporal derivatives are discretized by the 3rd-order Total Variation Diminishing Runge–Kutta (TVD-R–K) scheme. The validity of the method is verified by some benchmark problems. Then a simulation of viscoelastic fluid mold filling process is pursued with the method. The moving interface and all the information of the physical quantities during the injection process are captured. The die swelling phenomenon is found in the simulation. The influences of elasticity and viscosity on the physical quantities such as stresses etc. in the mold filling process are analyzed. Numerical results show that elastic characteristics such as the stretch and die swelling etc. reinforce accordingly as Weissenberg number increases. Pressures increase continuously in the mold filling process and the pressure maintains the maximum value at the inlet. Injection velocity is proportional to injection pressure. A higher viscosity leads to a higher pressure distribution, that is, the pressure decreases as Reynolds number increases.     

基于ALE方法的3D充填流动模拟

基于任意拉格朗日-欧拉方法发展了三维充填流动的数值模拟方案.该方案采用ALE方法准确地追踪移动自由面的位置并避免了网格扭曲;基于移动最小二乘曲面拟合方法提出了移动自由面上网格节点重定位方法,将充填流动的网格更新过程简化为自由面附近的局部网格重划分过程,并通过分级多面体三角剖分实现,减小了网格划分的计算量,实现了实时网格生成.给出的数值算例结果表明了该数值模型对三维充填流动模拟的有效性.     

A simple method for simulating viscoelastic fluid flows in slit channel

A low-cost semi-analysis finite element technique, named the finite piece method (FPM) is presented in this article. It aims to solve three-dimensional (3D) viscoelastic slit flows. The viscoelastic stress of the fluid is modelled using an K-BKZ integral constitutive equation of the Wagner type. Picard iteration is used to solve non-linear equations. The FPM is tested on flow problems in both planar and contraction channels. The accuracy of the method is assessed by comparing flow distributions and pressure with results obtained by 3D finite element method (FEM). It shows that the solution accuracy is excellent and a substantial amount of computing time and memory requirement can be saved.     

A corrected symmetric SPH method to simulate viscoelastic free surface flows based on the PTT model

In this work, a corrected symmetric and periodic density reinitialized SPH (CSPDR‐SPH) method is proposed and extended to simulate the viscoelastic free surface flows based on the Phan–Thien–Tanner model. The improvements mainly lie in deriving a corrected symmetric kernel gradient, and combining it with a periodic density reinitialization procedure. In addition, a simple artificial viscosity and a simple artificial stress form are adopted. Thus, the CSPDR‐SPH method has higher accuracy and better stability than the SPH method, and conserves both linear and angular momentums. The consistency and convergence of the CSPDR‐SPH method are justified by approximating a function in one and two dimensions. The merits of CSPDR‐SPH method are demonstrated by several benchmarks. The simple flow in a two‐dimensional channel is investigated to show the capability of the CSPDR‐SPH method to simulate the viscoelastic free surface flow. Then the CSPDR‐SPH method is extended to simulate the impacting drop problem. Numerical results show that the CSPDR‐SPH method can precisely capture the viscoelastic free surface. The Reynolds number, Weissenberg number and elongation parameter have remarkable effect on the flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element analysis of transient fluid flow with free surface using VOF (volume-of-fluid) method and adaptive grid

The VOF method is adopted for the finite element analysis of transient fluid flow with a free surface. In particular, an adaptation technique for generating an adaptive grid is incorporated to capture a higher resolution of the free surface configuration. An adaptive grid is created through the refinement and mergence of elements. In this domain the elements in the surface region are made finer than those in the remaining regions for more efficient computation. Also, three techniques based on the VOF method are newly developed to increase the accuracy of the analysis, namely the filling pattern, advection treatment and free surface smoothing techniques. Using the proposed numerical techniques, radial flow with a point source and the collapse of a dam are analysed. The numerical results agree well with the theoretical solutions as well as with the experimental results. Through comparisons with the numerical results of several cases using different grids, the efficiency of the proposed technique is verified. © 1998 by John Wiley & Sons, Ltd.     

Modeling and simulation of three‐dimensional extrusion swelling of viscoelastic fluids with PTT,Giesekus and FENE‐P constitutive models

The investigation of the extrusion swelling mechanism of viscoelastic fluids has both scientific and industrial interest. However, it has been traditionally difficult to afford theoretical and experimental researches to this problem. The numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three‐dimensional extrusion swelling of viscoelastic fluids flowing through out of a circular die. The rheological responses of viscoelastic fluids are described by using three kinds of differential constitutive models including the Phan‐Thien Tanner model, the Giesekus model, and the finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation model. A streamface‐streamline method is introduced to adjust the swelling free surface. The calculation stability is improved by using the discrete elastic‐viscous split stress algorithm with the inconsistent streamline‐upwind scheme. The essential flow characteristics of viscoelastic fluids are predicted by using the proposed numerical method, and the mechanism of swelling phenomenon is further discussed.Copyright © 2013 John Wiley & Sons, Ltd.     

A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow

In this paper, a corrected particle method based on the smoothed particle hydrodynamics (SPH) method with high-order Taylor expansion (CSPH-HT) for solving the vis-coelastic flow is proposed and investigated. The validity and merits of the CSPH-HT method are first tested by solv-ing the nonlinear high order Kuramoto-Sivishinsky equation and simulating the drop stretching, respectively. Then the flow behaviors behind two stationary tangential cylinders of polymer melt, which have been received little attention, are investigated by the CSPH-HT method. Finally, the CSPH-HT method is extended to the simulation of the filling process of the viscoelastic fluid. The numerical results show that the CSPH-HT method possesses higher accuracy and stability than other corrected SPH methods and is more reliable than other corrected SPH methods.     

Three‐dimensional simulation of planar contraction viscoelastic flow by penalty finite element method

The planar contraction flow is a benchmark problem for the numerical investigation of viscoelastic flow. The mathematical model of three‐dimensional viscoelastic fluids flow is established and the numerical simulation of its planar contraction flow is conducted by using the penalty finite element method with a differential Phan‐Thien–Tanner constitutive model. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The distributions of velocity and stress obtained by simulation are compared with that of Quinzani's experimental results detected by laser–doppler velocimetry and flow‐induced birefringence technologies. It shows that the numerical results agree well with the experimental results. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids. Copyright © 2009 John Wiley & Sons, Ltd.     

聚合物注射成型流动残余应力的数值分析

建立了可压缩黏弹性聚合物熔体在薄壁型腔中充模/保压过程中非等温、非稳态流动 的数学模型,用数值方法实现了注射成型过程中流动应力和取向建立及松弛过程的模拟,研 究了熔体温度、模具温度和注射速率等工艺条件对分子冻结取向的影响,取得了与实验相符 的结果.     

A hybrid method for computing the flow of viscoelastic fluids

A hybrid method for computing the flow of viscoelastic and second-order fluids is presented. It combines the features of the finite difference technique and the shooting method. The method is accurate because it uses central differences. Its convergence is at least superlinear. The method is applied to obtain the solutions to three problems of flow of Walters' B' fluid: (a) flow near a stagnation point, (b) flow over a stretching sheet and (c) flow near a rotating disk. Numerical results reveal some new characteristics of flows which are not easy to demonstrate using the perturbation technique.     

Die-swell,splashing drop and a numerical technique for solving the Oldroyd B model for axisymmetric free surface flows

This work deals with the development of a numerical method for simulating viscoelastic axisymmetric free surface flow of an Oldroyd B fluid. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries and on the symmetry axis. The full free surface stress conditions are employed. The resulting governing equations are solved by finite differences on a Marker-and-cell (MAC) type grid. Validation is provided by simulating a pipe flow problem. The classical die-swell problem is solved and swelling ratios are provided. The height of the splash caused by a falling liquid drop for various Reynolds and Weissenberg numbers is then studied, and the height of the splash is shown to diminish with increasing viscoelasticity.     

变截面黏弹性直杆纵振时复频率分析的差分解法

从一维黏弹性本构方程出发,导出了黏弹性变截面直杆纵向振动微分方程的一般形式,采用了有限差分法,并以二阶矩阵表示的递推形式,建立了该问题的复特征值方程组。两种Maxwell黏弹性变截面（指数指数、线性函数）直杆的数值计算表明,该方法运算简单,计算精度高,能适用于求解任意变截面黏弹性直属的纵向自由振动问题。     

Numerical simulation of transient planar flow of a viscoelastic material with two moving free surfaces

In this study, we examine the numerical simulation of transient viscoelastic flows with two moving free surfaces. A modified Galerkin finite element method is implemented to the two-dimensional non-steady motion of the fluid of the Oldroyd-B type. The fluid is initially placed between two parallel plates and bounded by two straight free boundaries. In this Lagrangian finite element method, the spatial mesh deforms in time along with the moving free boundaries. The unknown shape of the free surfaces is determined with the flow field u, v, τ, p by the deformable finite element method, combined with a predictor-corrector scheme in an uncoupled fashion. The moving free surfaces and fluid motion of both Newtonian and non-Newtonian flows are investigated. The results include the influence of surface tension, fluid inertia and elasticity.     

GENSMAC3D: a numerical method for solving unsteady three‐dimensional free surface flows

A numerical method for solving three‐dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier–Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user‐friendly input data file is employed to allow almost any arbitrary three‐dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.     

Three‐dimensional finite element analysis of transient fluid flow with free‐surface using marker surface method and adaptive grid refinement

For three‐dimensional finite element analysis of transient fluid flow with free‐surface, a new marker surface method is proposed, in which the fluid flow is represented by the marker surface composed of marker elements instead of marker particles used in the marker particle method. This also involves an adaptive grid that is created under a new criterion of element categorization of filling states and the locations in the total region at each time step. The marker surface is used in order to represent the free‐surface accurately, as well as to decrease the memory and computation time, and to effectively display the predicted three‐dimensional free‐surface. By using the adaptive grid in which the elements, finer than those in internal and external regions, are distributed in the surface region through refinement and coarsening procedures, the analysis of three‐dimensional transient fluid flow with free‐surface is achieved more efficiently. Through three‐dimensional analysis of two kinds of problems using several grids, the efficiency of the proposed marker surface method and the adaptive grid are verified. Copyright © 1999 John Wiley & Sons, Ltd.