Finite Element Modelling Of Non-isothermal Non-Newtonian Viscoelastic Flow In Mould Filling Process

Injection molding process is composed of three stages: filling, packing and cooling. The first stage, filling process, plays an important role to the quality of the final products, since quantitative predictions of the physical variables such as the velocities, temperature and stresses and their distributions over the filled domain and their evolutions with the filling process will bring significant influences to the successive two stages.The aim of the present thesis work is to develop finite element methods for numerical simulation of the non-isothermal non-Newtonian viscoelastic fluid flows in the mould filling process.With removing the Hele-Shaw approximation from the mathematical model developed in the present thesis, that significantly enhances its applicability to realistic problems, the partial differential equations governing the mass, momentum and energy conservations along with the constitutive equation describing the rheological behavior of filling molten polymer, the initial and boundary conditions of mould filling problems are derived and formulated.To achieve the numerical simulation of the mould filling process and the prediction of the physical quantities evolved in filling molten polymer, two critical problems should be resolved. One is the robust numerical solution scheme for initial and boundary value problems of the governing equations, and the other is to accurately determine locations of moving free surfaces. These two tasks are regarded as the objectives of the research carried out in this thesis. Aiming to the objectives of this research, the main original contributions of the present thesis can be summarized as follows:●To devise an ALE free surface tracking method and mesh generation-remeshing scheme for the injection molding process.●To develop an iterative version of stabilized fractional step algorithm for the numerical solution of the incompressible N-S equations in high viscosity fluid flows.●To present an iterative stabilized scheme I_CNBS_CG for incompressible non-isothermal non-Newtonian fluid flows, in which the Crank-Nicolson method based split (CNBS) scheme and the characteristic Galerkin (CG) method are respectively used to discretize and solve the non-Newtonian momentum-mass conservation equations and the energy conservation equation in consideration of their convective character. ●To devise and derive a mixed finite element formulation abbreviated as I_PS_DEVSS_CNBS scheme for viscoelastic flows, in which the finite increment calculus based pressure stabilization process and the discrete elastic viscous stress split (DEVSS) method using the CNBS scheme are introduced within a general framework of the iterative version of the fractional step algorithm. The streamline-upwind (SU) method is particularly chosen to deal with the convective term in the constitutive equation of viscoelastic flows. With the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables are successfully used with numerical stability and high convergence rate even for the simulation of viscoelastic flows with high Weissenberg numbers.The contents of the thesis are outlined as follows:In chapter 1, the background and motivation of this thesis are presented with the review of existing literatures related to the numerical simulation of mould filling process. The following topics related to this thesis work are briefly introduced and discussed: the significance of the thesis work; a description of the molding process; the state of arts of mathematic models for filling process and the constitutive models for polymer melt flows; the temperature effects on viscous fluid flows; advancing free surface capturing and tracking methods; crucial scientific problems encountered and to be resolved in development of the robust numerical solution schemes for the mathematical models of mould filling process; At the end of this chapter, the main research work fulfilled and presented in this thesis is summarized.Chapter 2 lists the nomenclature and symbols frequently used in the thesis, gives the fundamental equations of melt polymer flows, describes some typical rheological phenomena and explains some fundamental concepts such as the free surface, the slip/non-slip boundary condition and the LBB condition.In chapter 3, a free surface tracking and mesh generation-remeshing scheme is presented in an ALE framework for numerical simulations of the injection molding process. The additional equations determining the movement of mesh nodes on the free surface are introduced in a self-adaptive manner, which makes it possible to properly track the moving free surface in the numerical simulation of the filling process in different types of complex shaped moulds. The real-time mesh generation of the domain occupied by the filled polymer whose mass is variable with time marching, is simplified as a polygon's triangulation of the filled zone near the moving filling front at every given number of time steps, which saves CPU time significantly. In addition, a local Laplacian smoothing scheme is proposed to improve the mesh quality effectively. Different types of the wall-touching occurring between nodes of the free surface and mould walls are analyzed and the corresponding schemes to tackle them are proposed. Numerical results for modeling the filling processes in a typical mould demonstrate the capability and performance of the proposed free surface tracking and the mesh generation scheme for the injection molding process.In chapter 4, two modified versions of the fractional step algorithm using characteristic based split (CBS) and Crank-Nicolson difference method based split (CNBS) are presented to solve highly viscous flow problems. The proposed modified versions of the algorithm are based on introducing an iterative procedure into the algorithm and allow much larger time step sizes than those required to the existing explicit and semi-implicit versions. In addition, a numerical study of stability at acceptable convergence rate and accuracy as well as capability in circumventing the restriction imposed by the LBB condition for the proposed iterative versions of the algorithm is carried out with the plane Poisseuille flow problem under different Reynolds numbers ranging from low to high viscosities. Moreover, the improved performance of the proposed versions of the algorithm is validated by the plane Poisseuille flow and the lid-driven cavity flow problems, and application to numerical simulation of the polymer injection molding process is presented.In chapter 5, An iterative stabilized fractional step scheme abbreviated as I_CNBS_CG is developed, in which the CNBS method and characteristic Galerkin (CG) method are respectively used to discretize and solve the Non-Newtonian viscous momentum - mass conservation equations and the energy conservation equation in consideration of their convective character, and the CNBS and CG methods are integrated into a staggered solution frame by using the iterative procedure. The proposed I_CNBS_CG scheme particularly suits to numerically model the non-isothermal Non-Newtonian viscous fluid flows with moderate or high viscosity and low thermal conductivity, such as molten polymer flow process in a mould cavity. Numerical experiments with various Non-Newtonian viscous fluid models demonstrate the improved performances of the proposed scheme. In addition, the proposed scheme is used to simulate an injection process with a non-isothermal Carreau fluid through a typical mould.In chapter 6, a mixed finite element scheme abbreviated as I_PS_DEVSS_CNBS scheme is presented for modeling viscoelastic flow problems. The finite incremental calculus (FIC) pressure stabilization process and the discrete elastic-viscous stress-splitting method (DEVSS) are introduced into the general framework of the iterative version of the fractional step algorithm with the use of the CNBS. Inconsistent streamline upwinding method (SU) is employed to spatially discretize the constitutive equation of viscoelastic fluids. The LBB compatibility conditions which restrict the choice of finite element interpolations for the mixed variables can be circumvented by using the proposed scheme, and the finite elements with equal low order interpolation approximations for stress-velocity-pressure variables are successfully used for the simulation of viscoelastic flows. In addition, a variety of viscoelstic models, including UCM/Oldroyd-B, PTT and XPP models have been integrated into the proposed scheme to solve two benchmark problems of the viscoelasitc flow, i.e. planar Poiseuille flow and 4:1 sudden contraction flow problems. The numerical results demonstrate prominent stability and accuracy of both pressure and stress distributions over the flow domains provided by the proposed scheme within the Weissenberg number range studied in the present work, as compared with the analytical solutions or the reference solutions reported in the literatures.In chapter 7, a finite element program for simulating non-isothermal non-Newtonian viscoelastic flows in mould filling process is described, and the flow charts of three main program modules are provided, including the preprocessor module, the analysis module and the flee-surface tracking and mesh generation module.The main contributions of the thesis are presented in details and the further work is suggested in chapter 8.