Three‐dimensional numerical simulation of injection molding filling of optical lens and multiscale geometry using finite element method

This article presents the development, verification, and validation of three‐dimensional (3‐D) numerical simulation for injection molding filling of 3‐D parts and parts with microsurface features. For purpose of verification and comparison, two numerical models, the mixed model and the equal‐order model, were used to solve the Stokes equations with three different tetrahedral elements (Taylor‐Hood, MINI, and equal‐order). The control volume scheme with tetrahedral finite element mesh was used for tracking advancing melt fronts and the operator splitting method was selected to solve the energy equation. A new, simple memory management procedure was introduced to deal with the large sparse matrix system without using a huge amount of storage space. The numerical simulation was validated for mold filling of a 3‐D optical lens. The numerical simulation agreed very well with the experimental results and was useful in suggesting a better processing condition. As a new application area, a two‐step macro–micro filling approach was adopted for the filling analysis of a part with a micro‐surface feature to handle both macro and micro dimensions while avoiding an excessive number of elements. POLYM. ENG. SCI., 46:1263–1274, 2006. © 2006 Society of Plastics Engineers     

A full 3D finite element analysis of the powder injection molding filling process including slip phenomena

A full 3D finite element analysis system has been developed to simulate a Powder Injection Molding (PIM) filling process for general three‐dimensional parts. The most important features of the analysis system developed in this study are i) to incorporate the slip phenomena, the most notable rheological characteristics of PIM feedstock, into the finite element formulation based on a nonlinear penalty‐like parameter and ii) to simulate the transient flow during the filling process with a predetermined finite element mesh with the help of a volume fill factor and a melt front smoothing scheme. The treatment of the nonlinear slip boundary condition was successfully validated via a steady state pipe flow. For the purpose of comparisons, not only the numerical simulations but also experimental short‐shot experiments were performed with two 3D mold geometries using two typical materials of slip and no‐slip cases. The good agreements between the numerical and experimental results indicate that the melt front tracking scheme successfully simulates the transient filling process.     

Numerical simulation of moving free surface problems in polymer processing using volume‐of‐fluid method

We have developed a numerical algorithm based on 2D/3D finite element method for solving non‐Newtonian fluid flow with the moving free surface encountered in polymer processing. The power law model is considered as a rheological constitutive equation. The standard Galerkin finite element formulation/penalty formulation are applied to discrctize the governing equations, the volume‐of‐fluid (VOF) scheme is used to track the moving free surface, and the donor‐acceptor model introduced by Hirt and Nichols is modified and implemented on FEM. We applied the numerical scheme to simulate fountain flow and viscous buckling problems. For fountain flow, the numerical prediction of this study is in good agreement with the experimental results of other investigators. For viscous buckling, both 2D and 3D numerical simulations show that the shear thinning effect retards buckling. As this algorithm is very effective in treating moving free surface problems and requires less memory than previous algorithms, it may help solve challenging problems in polymer processing such as transient visroelastic flow simulations with moving free surfaces.     

基于P1/PO四面体宏元的三维注塑充填速度场压力场有限元数值模拟

计算效率和解的稳定性是影响三维注塑充填有限元数值模拟的关键因素.针对黏性不可压缩聚合物熔体三维充填过程的速度场和压力场,以提高计算机求解速度为出发点,分析了采用P1/P0四面体单元(速度线性,压力常数)得到的有限元方程的解不收敛的原因,提出一种采用P1/P0四面体宏元离散空间域的求解方案,从而降低了求解的自由度数量,提...     

Finite element analysis of thermoplastic melts flow through the metering and die regions of single screw extruders

This research work is devoted to the development of a mathematical model for the simulation of the flow of polymer melts through the metering and die regions of single screw extruders. The sets of the governing equations (flow and energy) are solved using the finite element method. The power‐law model is used to describe the non‐Newtonian rheological behavior of the fluid. The standard Galerkin technique is used in conjunction with the continuous penalty scheme to solve the flow equations. Due to the low thermal diffusivity of the polymer melts, a streamline upwinding Petrov–Galerkin method is used to obtain convergent and stable results for the energy equation. This method is based on the extension of a previously developed scheme. The overall solution strategy is based on the Picard iterative scheme. Simulation results are obtained for the flow of a polypropylene melt through the metering and die zones of a laboratory scale extruder. To validate the proposed model, the results of the computer simulations are compared with experimentally measured mass flow rate and pressure profile. These comparisons show that there is very good agreement between the model predictions and actual data. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 676–689, 1999     

A finite element method for flow in compression molding of thin and thick parts

Based on Barone and Caulk's model and a generalized variational functional, a finite element simulation was developed for the compression molding of thin and thick parts. For solving the u-v-p type equations, an element-based penalty method and a mixed formulation were implemented. Numerical results show that the new model gives better accuracy in velocity and velocity gradient than the Hele-Shaw formulation for cases where both models are appropriate. Predictions of velocity and its gradient by the model are compared with other FEM results and BEM solutions. Using a fixed base mesh that covers the mold cavity, a new technique was developed for tracking the moving flow front. Temporary elements and nodes are generated for the filled part of elements intersected by the flow front. This method allows a smooth representation of the flow front and has exact boundary conditions on the flow front. The scheme is demonstrated for compression molding of an elliptical and an L-shaped charge.     

Finite element modeling of the flow of a rubber compound through an axisymmetric die using the CEF viscoelastic constitutive equation

This work is devoted to the simulation of the flow of a high viscosity NR/SBR rubber compound through the die of a single screw extruder with axisymmetric geometry. An in-house developed computer code based on the use of continuous penalty finite element method was employed. Three constitutive equations including two generalized Newtonian models namely; power-law and Carreau and an explicit viscoelastic model named CEF (Criminale-Ericksen-Fillbey) were used to reflect the rheological behavior of the material. Using the parameters of the rheological models determined by a slit die rheometry technique, the flow of the compound was simulated through the die and results were compared with experimentally measured mass flow rates. It is shown that for high viscosity rubber compounds the use of generalized Newtonian models which do not take the normal stress in simple shear flow into consideration gives rise to significant errors in prediction of mass flow rates. On the other hand, comparing the simulations results using the CEF equation with experimental data revealed that this model is the best compromise between generalized Newtonian and full viscoelastic models which need high computational costs and effort. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Multi-scale finite element modelling of laminar steady flow through highly permeable porous media

The Brinkman equation is used to model the isothermal flow of Newtonian fluids in a highly permeable porous medium between two parallel plates. Due to the multi-scale nature of flow regimes represented by the Brinkman equation, the standard Galerkin method requires excessive mesh refinement to achieve stable and accurate results. To avoid this, a residual free method is used to derive appropriate bubble functions for inclusion in the finite element discretisations. It is shown that by using bubble enriched shape functions the standard Galerkin method can yield accurate and stable solutions for multi-scale problems. In this paper the performances of the third and fifth order bubble functions used in conjunction with bilinear Lagrangian elements to solve the Brinkman equation via a penalty finite element scheme, are reported.     

数值模拟黏弹流体通过椭圆环模具的挤出胀大

The numerical simulation of extrudate swell is significant in extrusion processing.Precise prediction of extrudate swell is propitious to the control of melt flow and the quality of final products.A mathematical model of three-dimensional(3D)viscoelastic flow through elliptical ring die for polymer extrusion was investigated.The penalty function formulation of viscoelastic incompressible fluid was introduced to the finite element model to analyze 3D extrusion problem.The discrete elastic viscous split stress(DEVSS)and streamline-upwind PetrovGalerkin(SUPG)technology were used to obtain stable simulation results.Free surface was updated by updating the streamlines which needs less memory space.According to numerical simulation results,the effect of zero-shear viscosity and elongation parameter on extrudate swell was slight,but with the increase of volumetric flow rate and relax time the extrudate swell ratio increased markedly.Finally,the numerical simulation of extrudate swell flow for low-density polyethylene(LDPE)melts was investigated and the results agreed well with others’work.These conclusions provided quantitative basis for the forecasting extrudate swell ratio and the controlling of extrusion productivity shape.     

Finite element analysis of switching domains using ferroelectric and ferroelastic micromechanical model for single crystal piezoceramics

Here, the domain switching in a ferroelectric and ferroelastic single crystal subjected to electrical, electromechanical, and mechanical loading condition is studied. An isoparametric 3D electromechanical hexahedral finite element introducing the micromechanical constitutive law for domain switching is proposed to investigate the non-linear response of single crystal piezoceramics. The micromechanical model considered here is based on thermodynamic approach and internal variables, accounting for the electromechanical interaction energy between the domains. The volume fractions of six distinct uni-axial variants are treated as the internal variables to describe the microscopic state of the material at any given loading level. Furthermore, the formulation includes a realistic phase transition from a cubic unit cell to tetragonal one in the single crystal piezoceramics under the application of external load. The non-linear electromechanical constitutive equations obtained are solved using an implicit integration technique employing the return-mapping algorithm. The model developed here is tested for its applicability considering variety of benchmarks, and it brings out the behaviour of piezoceramic materials as observed in experimental studies.The hexahedral finite element presented is also implemented in the commercial finite element code Abaqus via the User Element subroutine.     

Finite element modeling of curing of epoxy matrix composites

A two‐dimensional finite element model is developed to simulate and analyze the mechanisms pertaining to resin flow, heat transfer, and consolidation of laminated composites during autoclave processing. The model, which incorporates some of the best features of models already in existence, is based on Darcy's law, the convection–diffusion heat equation, and appropriate constitutive relations. By using a weighted residual method, a two‐dimensional finite element formulation for the model is presented and a finite element code is developed. Numerical examples, including a comparison of the present numerical results with one‐dimensional and two‐dimensional analytical solutions, are given to indicate the accuracy the finite element formulation. Moreover, using the finite element code, the one‐dimensional cure process of a laminate made of 228 and 380 plies of AS4/3501‐6 unidirectional tape is simulated and numerical results are compared with available experimental results. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 103: 2310–2319, 2007     

X-ray computed tomography based microstructure reconstruction and numerical estimation of thermal conductivity of 2.5D ceramic matrix composite

In this study, the X-ray computed tomography was adopted to build 3D finite element models of the 2.5D ceramic matrix composite. The threshold segmentation method was used to identify the SiC matrix. An approach was developed to identify and number the matrix regions. The geometrical features of the 2.5D microstructure such as symmetry and periodicity were utilized to identify the boundaries of warp and weft. The warp, weft and porous matrix were denoted by different colors. The finite element model of 2.5D microstructure was built based on the color of each pixel. The finite element models combined with homogenization method were then used to predict the thermal conductivity of material. An element compression method was developed to reduce the total number of elements. The in plane and out-of-plane thermal conductivities were estimated by both numerical and experimental methods. The comparisons show that the numerical results fit experiments well.     

塑料异型材挤出口模的三维罚有限元设计

讨论了现有塑料异型材挤出口模研究中的流变学分析方法,指出了前人运用流变学方法设计挤出口模的不足。本文采用三维罚有限元方法,对幂律流体在塑料异型材口模中的流动进行了计算机模拟研究,阐述了采用三维罚有限元方法进行塑料异型材挤出口模设计的重要意义     

A finite piece method for simulating polymer melt flows in extrusion sheet dies

A finite piece method is proposed to simulate three‐dimensional slit flows in extrusion sheet dies in this paper. The simulations concern incompressible fluids obeying different constitutive equations: generalized Newtonian (Carreau‐Yasuda law), and viscoelastic Phan‐Thien Tanner (PTT) models. Numerical simulations are carried out for the isothermal and nonisothermal flows of polymer melt through sheet dies. The Picard iteration method is utilized to solve nonlinear equations. The results of the finite piece method are compared with the three‐dimensional (3D) finite element method (FEM) simulation and experiments. At the die exit, the relative error of the volumetric flow between the finite piece method and the 3D FEM is below 1.2%. The discrepancy of the pressure distributions does not exceed 6%. The Maximum error of the uniformity index between the simulations and experiments is about 2.3%. It shows that the solution accuracy of the finite piece method is excellent, and a substantial amount of computing time and memory requirement can be saved. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

A hybrid 3D/2D finite element technique for polymer processing operations

A hybrid 3D/2D finite element method is proposed to simulate polymeric fluid motions for various polymer processing operations that may involve complex flow geometries. The flow regime is divided into 2D and 3D regions, and the fluid flow is assumed to follow the Hele Shaw model in the 2D region. Mathematically the 2D and the 3D regions are linked by the conservation of mass and the continuation of pressure variations. Two example calculations have been used to illustrate the accuracy and efficiency of the new numerical technique, and it was found that a substantial amount of computing time can be saved to obtain numerical solutions that are as accurate as the complete 3D calculations.     

Prediction of fiber orientation in the thickness plane during flow molding of short fiber composites

Fiber orientations caused by the flow in the thickness plane during injection molding of short fiber reinforced polymer composites has been simulated. The Lagrangian scheme was employed for the finite element analysis. Flow fields were solved by using a penalty method with Uzawa's scheme and orientation fields were also solved by using the second order orientation tensor. A generalized Newtonian fluid whose rheological behavior is independent of fiber orientation was assumed. Automatic mesh generation using an elliptic grid generator was developed for quadrilateral elements. Mold filling and orientation analyses were performed for a cavity of rectangular cross section. To determine the orientation state in other cross-sectional geometries, numerical analyses were also performed for two different typical cross sections. As the result, orientation of short fibers in the flow field was analyzed qualitatively and quantitatively. According to the state of short fiber orientation in the thickness plane, the orientation field can be classified into three regions in the flow direction and three layers in the thickness direction. Orientation of short fibers was mainly influenced by elongational and shear flows. It was observed that critical values are present for upper limits of orientation. Effects of initial orientation at the inlet on the orientation field were examined.     

Development of a discrete element model with moving realistic geometry to simulate particle motion in a Mi-Pro granulator

This paper presents the implementation of a methodology incorporating a 3D CAD geometry into a 3D discrete element method (DEM) code; discussing some of the issues which were experienced. The 3D CAD model was discretised into a finite element mesh and the finite wall method was employed for contact detection between the elements and the spherical particles. The geometry was based on a lab scale Mi-Pro granulator. Simulations were performed to represent dry particle motion in this piece of equipment. The model was validated by high speed photography of the particle motion at the surface of the Mi-Pro's clear bowl walls. The results indicated that the particle motion was dominated by the high speed impeller and that a roping regime exists. The results from this work give a greater insight into the particle motion and can be used to understand the complex interactions which occur within this equipment.     

A virtual finite element model for centered and eccentric mixer configurations

An implementation of the virtual finite element method with unstructured grids for the modeling of laminar flow in eccentric mixers is presented. The effect of the meshing strategy on the quality of the computed flow field is first carefully investigated with a centered impeller. It is shown that both the number of elements in the vicinity of the impeller and the number of kinematics constraints imposed in the virtual finite element formulation control the computational accuracy. The method is then applied to the case of an eccentric mixer provided with a Rushton turbine showing the capabilities of the proposed approach.     

Numerical study of nonisothermal polymer extrusion flow with a differential viscoelastic model

A numerical study of nonisothermal viscoelastic flow is conducted to investigate the complex flow characteristics of polymer melts in the extrusion process. A general thermodynamic model for the energy conversion related to viscoelastic fluid flow is introduced. The mathematical model for three‐dimensional nonisothermal viscoelastic flow of the polymer melts obeying a differential constitutive equation (Phan‐Thien and Tanner model) is established. A decoupled algorithm based on the penalty finite element method is performed on the calculation. The discrete elastic‐viscous split stress (DEVSS) algorithm, incorporating the streamline‐upwind Petrov‐Galerkin (SUPG) scheme is employed to improve the computation stability. Essential flow characteristics of polymer melts in the extrusion die for hollow square plastic profile is investigated based on the proposed numerical scheme with ignoring the outer thermal resource. The energy partitioning, which quantified the conversion of mechanical energy into thermal energy, is discussed. The effects of volume flow rate and die contraction angle upon the flow patterns are further investigated. POLYM. ENG. SCI., 2008. © 2007 Society of Plastics Engineers