成型充填过程中非等温非牛顿粘性流动的ALE有限元与无网格自适应耦合模拟
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摘要
注塑成型是目前塑料成型的主要方法之一，生产的制品如磁带盒、计算机键盘等。该工艺的流程为，首先将熔融的高聚物熔体以较大的流速注射入模具型腔中，直至充满整个腔体，然后待塑料熔体充分固化后开启模具脱出塑料制品，最后再循环进入下一个制品的生产。
为提高模具的生产效率和水平，人们迫切需要发展针对该工艺过程的高效的数值分析工具来取代传统的费时费力的试模修模的过程。因而，在过去的数十年中，注塑成型过程的数值模拟引起了越来越广泛的关注。尽管如此，仍然还有很多问题有待进一步的研究，如自由面的准确追踪以及稳定、高效、精确地求解控制方程等。本论文的研究工作正是以解决这些问题为目标而展开的。
移动自由面的准确追踪对于注塑成型过程的数值模拟具有十分重要的意义。根据建立方程所采用参考系的不同，目前已有的处理方法大体上可归类为拉格朗日方法、欧拉方法以及任意拉格朗日欧拉方法(ALE)。在ALE方法中，计算网格的运动独立于物质构型的运动，结合了拉格朗日方法和欧拉方法的优点。本文第三章将给出一个基于ALE方法的自由面追踪和网格生成-重生成模型，可在准确确定移动自由面空间位置的同时尽量避免发生网格扭曲。在所发展的模型中，充填域变质量体系的实时网格生成被简化为移动自由面附近区域的多边形三角化过程，大大减少了网格处理所需的计算量。此外，还引入了局部Laplacian光顺方案来提高计算网格的质量，并发展了多种机制来处理自由面与模具边界的接触问题。
实现注塑成型模拟的另外一个关键问题是须发展能稳定求解控制方程初边值问题的数值方法。众所周知，采用标准Galerkin方法模拟不可压缩流动问题通常会导致两种类型的虚假数值振荡。其一来源于控制方程组的可约化混合特性，该特性限制了速度-压力(u-p)插值空间的自由选取。不满足所谓LBB条件的非协调的插值模式，如u-p的等低阶插值模式，将导致压力场的虚假空间振荡。第二种类型的数值振荡是由控制方程的对流特性引起的，尤其易发生在对流占优的情况下，这是由于标准的空间Galerkin离散方法只对自伴随的算子方程有效。
对于第一个问题，本文通过在有限增量微积分(FIC)的理论框架下重建质量守恒方程，并理性地引入一个辅助变量来避免空间高阶导数的计算，提出了压力稳定型分步算法(PS-FSA)，有效消除了压力场的虚假数值振荡。与经典的分步算法(C-FSA)相比，该算法具有更好的压力稳定性。此外，本文还采用这种压力稳定机制改善了经典的特征线基分裂算法(CBS)的压力稳定性。PS-FSA算法具体的推导过程以及它的数值验证将在第四章中给出。
对于第二个问题，本文发展了基于ALE描述的广义特征线Galerkin(CG)方法，有效处理了对流占优问题控制方程的空间离散。而且，与经典的CG方法相比，该方法能使用更大的时间步长。实际上，经典的CG方法以及经典的Crank-Nicolson(CN)方法能分别作为两个特例而被纳入到本文所提出的广义CG方法的框架中。此外，本文方法还将经典CG方法由欧拉描述下推广到ALE描述下，以便于和前述ALE自由面追踪技术相结合。应指出的是，若将参考系固定于空间不动，ALE描述可退化为欧拉描述，即相对于经典CG方法来说，本文方法更具一般性。将所发展的广义CG方法和分步算法以及第四章给出的压力稳定技术相结合，本文还提出了一个迭代型压力稳定的广义CBS算法，并将其应用于非等温非牛顿流的数值模拟中。该算法具体的推导过程以及它的数值验证将在第六章中给出。
此外，本文还提出了一个有限元与无网格离散区域剖分的自适应算法，可自动地将整个计算域划分成分别应用有限元法(FE)、无网格法(MF)以及它们的耦合插值进行空间离散的三个子域。基于该自适应算法以及连续掺混法(CBM)，本文提出了有限元与无网格的自适应耦合空间离散方法，并将其应用于注塑成型的数值模拟中。所发展的自适应耦合方法可在充分结合有限元法与无网格法各自优点的同时避免它们各自的缺点。该方法具体的实现细节以及展现它相对于传统有限元法和无网格法优势的数值验证将在第五章中给出。
为了论文的完整性，本文第二章还简单介绍了伽辽金型无网格法的基本概念和它所面临的问题以及相应的处理方法。实现本文算法的计算机程序及其数据结构的说明放在第七章。第八章总结全文并展望进一步的研究内容。
Injection molding is one of the most important industrial processes for the manufacturingof plastic products. Examples of such products are cassette tape boxes, computer keyboardsand so on. In the production process, molten polymer is injected with high velocity into anempty mold. Once the cavity is filled up and the polymer material is sufficiently solidified,the mold opens momentarily to eject the plastic component and the cycle repeats.
Due to the constant demand for developing efficient analysis tools to replace the costlyand time-consuming experimental trial-and-error approach, numerical simulation of injectionmolding process has attracted increasing interest over the past years. However, there are stillmany aspects that require further research. For example, the free surface tracking and thenumerical solution for initial and boundary value problems of the governing equations withacceptable levels of overall performance in stability, efficiency, accuracy and robustness arestill open subjects. The present thesis is an effort towards these objectives.
Accurately tracking the moving free surface plays an important role in the simulation ofinjection molding process. At present, the available strategies to tackle this problem can bemainly classified into Lagrangian, Eulerian and Arbitrary Lagrangian-Eulerian(ALE)methods depending on the configurations to which continuum mechanics formulations arereferred. Owing to the superiority of the ALE method which combines the respectiveadvantages of both Lagrangian and Eulerian methods by means of defining the mesh motionindependent of the material motion, a free surface tracking and mesh generation model basedon the ALE method is presented in Chapter 3 which can accurately determine locations ofadvancing free surfaces and meanwhile to minimize the distortion of the computational mesh.In this model, the real-time mesh generation of the domain with variable mass of the filledpolymer melts is simplified as a polygon's triangulation in the filled zone near the movingfront, that saves CPU time significantly. In addition, a local Laplacian smoothing technique isintroduced to improve the mesh quality and several strategies are proposed to cope with thecontact problems between the moving free surfaces and the boundaries of the mold cavity.
Another critical ingredient to achieve the simulation is the robust numerical solutionscheme for initial and boundary value problems of the governing equations. It is well knownthat numerical modeling of incompressible flows with the classical Galerkin method maysuffer from numerical instabilities due to two main sources. The first attributes to the mixedcharacter of the governing equations which restricts the choice of interpolation spaces for thevelocity and pressure (u-p) fields. The incompatible interpolations, for example, the equallow order u-p interpolations, that violate the LBB condition may induce spurious spatialoscillations in the resulting pressure field. The second is associated with the convective character of the equations which induces oscillations particularly in the convection dominatedcases, as the standard Galerkin method is only valid for self-adjoint operator equations.
As for the first problem, the contribution of the present thesis is that a Pressure StabilizedFractional Step Algorithm (PS-FSA) is developed which can effectively remove the spuriouspressure oscillations and has better pressure stability than that of the Classical Fractional StepAlgorithm (C-FSA). The proposed PS-FSA is based on re-writing the mass balance equationin the framework of the Finite Increment Calculus (FIC) theory and introducing an additionalvariable into the algorithm in a logical way to avoid the calculation of high order spatialderivatives. In addition, such pressure stabilization mechanism is also extended to theclassical Characteristic Based Split (CBS) algorithm to enhance its pressure stability. Thedetailed derivation of PS-FSA and its numerical validations are presented in Chapter 4.
As for the second problem, the contribution of the present thesis is that a generalizedversion of the Characteristic Galerkin (CG) method in ALE framework is developed whichcan effectively cope with the convection dominated problem and can use larger time step sizethan that of the classical CG method. In fact, the classical CG and the classicalCrank-Nicolson (CN) methods can be classified as two special cases of this generalized CGmethod respectively. In addition, the generation is also exhibited by the fact that the ALEdescription employed for the derivation of the proposed method can reduce to the Euleriandescription used in the classical CG method if the reference coordinates are fixed in space.Due to this generation, it is convenient to combine the proposed CG method with the ALEfree surface tracking techniques mentioned above. By combining the proposed generalizedCG method with the fractional step algorithm and the pressure stabilization techniquedeveloped in Chapter 4, an iterative pressure-stabilized generalized CBS algorithm is formedand used for non-isothermal non-Newtonian fluid flows. The detailed derivation of thealgorithm and its numerical validations are presented in Chapter 6.
Another important contribution of the present thesis is that a self-adaptive domainpartition algorithm is proposed which can automatically partition the whole computationaldomain into three sub-domains where the finite element (FE), meshfree (MF) and theircoupled approximations are employed respectively. Based on this adaptive procedure and theContinuous Blending Method (CBM), an adaptive coupled FE and MF method is alsoproposed for the simulation of injection molding process, which can adequately exploit therespective strong points of FE and MF methods and meanwhile avoid their respective weakpoints. The details of this method and the numerical results to demonstrate its superiority overthe independent FE and MF methods are presented in Chapter 5.
For the purpose of self-completeness, the fundamental concepts and issues of themeshfree method with Galerkin weak form are summarized in Chapter 2. The computer program and data structures to implement the algorithms presented in this thesis for thenumerical simulation of injection molding process are provided in Chapter 7. Conclusion andfuture developments are given in Chapter 8.
为提高模具的生产效率和水平，人们迫切需要发展针对该工艺过程的高效的数值分析工具来取代传统的费时费力的试模修模的过程。因而，在过去的数十年中，注塑成型过程的数值模拟引起了越来越广泛的关注。尽管如此，仍然还有很多问题有待进一步的研究，如自由面的准确追踪以及稳定、高效、精确地求解控制方程等。本论文的研究工作正是以解决这些问题为目标而展开的。
移动自由面的准确追踪对于注塑成型过程的数值模拟具有十分重要的意义。根据建立方程所采用参考系的不同，目前已有的处理方法大体上可归类为拉格朗日方法、欧拉方法以及任意拉格朗日欧拉方法(ALE)。在ALE方法中，计算网格的运动独立于物质构型的运动，结合了拉格朗日方法和欧拉方法的优点。本文第三章将给出一个基于ALE方法的自由面追踪和网格生成-重生成模型，可在准确确定移动自由面空间位置的同时尽量避免发生网格扭曲。在所发展的模型中，充填域变质量体系的实时网格生成被简化为移动自由面附近区域的多边形三角化过程，大大减少了网格处理所需的计算量。此外，还引入了局部Laplacian光顺方案来提高计算网格的质量，并发展了多种机制来处理自由面与模具边界的接触问题。
实现注塑成型模拟的另外一个关键问题是须发展能稳定求解控制方程初边值问题的数值方法。众所周知，采用标准Galerkin方法模拟不可压缩流动问题通常会导致两种类型的虚假数值振荡。其一来源于控制方程组的可约化混合特性，该特性限制了速度-压力(u-p)插值空间的自由选取。不满足所谓LBB条件的非协调的插值模式，如u-p的等低阶插值模式，将导致压力场的虚假空间振荡。第二种类型的数值振荡是由控制方程的对流特性引起的，尤其易发生在对流占优的情况下，这是由于标准的空间Galerkin离散方法只对自伴随的算子方程有效。
对于第一个问题，本文通过在有限增量微积分(FIC)的理论框架下重建质量守恒方程，并理性地引入一个辅助变量来避免空间高阶导数的计算，提出了压力稳定型分步算法(PS-FSA)，有效消除了压力场的虚假数值振荡。与经典的分步算法(C-FSA)相比，该算法具有更好的压力稳定性。此外，本文还采用这种压力稳定机制改善了经典的特征线基分裂算法(CBS)的压力稳定性。PS-FSA算法具体的推导过程以及它的数值验证将在第四章中给出。
对于第二个问题，本文发展了基于ALE描述的广义特征线Galerkin(CG)方法，有效处理了对流占优问题控制方程的空间离散。而且，与经典的CG方法相比，该方法能使用更大的时间步长。实际上，经典的CG方法以及经典的Crank-Nicolson(CN)方法能分别作为两个特例而被纳入到本文所提出的广义CG方法的框架中。此外，本文方法还将经典CG方法由欧拉描述下推广到ALE描述下，以便于和前述ALE自由面追踪技术相结合。应指出的是，若将参考系固定于空间不动，ALE描述可退化为欧拉描述，即相对于经典CG方法来说，本文方法更具一般性。将所发展的广义CG方法和分步算法以及第四章给出的压力稳定技术相结合，本文还提出了一个迭代型压力稳定的广义CBS算法，并将其应用于非等温非牛顿流的数值模拟中。该算法具体的推导过程以及它的数值验证将在第六章中给出。
此外，本文还提出了一个有限元与无网格离散区域剖分的自适应算法，可自动地将整个计算域划分成分别应用有限元法(FE)、无网格法(MF)以及它们的耦合插值进行空间离散的三个子域。基于该自适应算法以及连续掺混法(CBM)，本文提出了有限元与无网格的自适应耦合空间离散方法，并将其应用于注塑成型的数值模拟中。所发展的自适应耦合方法可在充分结合有限元法与无网格法各自优点的同时避免它们各自的缺点。该方法具体的实现细节以及展现它相对于传统有限元法和无网格法优势的数值验证将在第五章中给出。
为了论文的完整性，本文第二章还简单介绍了伽辽金型无网格法的基本概念和它所面临的问题以及相应的处理方法。实现本文算法的计算机程序及其数据结构的说明放在第七章。第八章总结全文并展望进一步的研究内容。
Injection molding is one of the most important industrial processes for the manufacturingof plastic products. Examples of such products are cassette tape boxes, computer keyboardsand so on. In the production process, molten polymer is injected with high velocity into anempty mold. Once the cavity is filled up and the polymer material is sufficiently solidified,the mold opens momentarily to eject the plastic component and the cycle repeats.
Due to the constant demand for developing efficient analysis tools to replace the costlyand time-consuming experimental trial-and-error approach, numerical simulation of injectionmolding process has attracted increasing interest over the past years. However, there are stillmany aspects that require further research. For example, the free surface tracking and thenumerical solution for initial and boundary value problems of the governing equations withacceptable levels of overall performance in stability, efficiency, accuracy and robustness arestill open subjects. The present thesis is an effort towards these objectives.
Accurately tracking the moving free surface plays an important role in the simulation ofinjection molding process. At present, the available strategies to tackle this problem can bemainly classified into Lagrangian, Eulerian and Arbitrary Lagrangian-Eulerian(ALE)methods depending on the configurations to which continuum mechanics formulations arereferred. Owing to the superiority of the ALE method which combines the respectiveadvantages of both Lagrangian and Eulerian methods by means of defining the mesh motionindependent of the material motion, a free surface tracking and mesh generation model basedon the ALE method is presented in Chapter 3 which can accurately determine locations ofadvancing free surfaces and meanwhile to minimize the distortion of the computational mesh.In this model, the real-time mesh generation of the domain with variable mass of the filledpolymer melts is simplified as a polygon's triangulation in the filled zone near the movingfront, that saves CPU time significantly. In addition, a local Laplacian smoothing technique isintroduced to improve the mesh quality and several strategies are proposed to cope with thecontact problems between the moving free surfaces and the boundaries of the mold cavity.
Another critical ingredient to achieve the simulation is the robust numerical solutionscheme for initial and boundary value problems of the governing equations. It is well knownthat numerical modeling of incompressible flows with the classical Galerkin method maysuffer from numerical instabilities due to two main sources. The first attributes to the mixedcharacter of the governing equations which restricts the choice of interpolation spaces for thevelocity and pressure (u-p) fields. The incompatible interpolations, for example, the equallow order u-p interpolations, that violate the LBB condition may induce spurious spatialoscillations in the resulting pressure field. The second is associated with the convective character of the equations which induces oscillations particularly in the convection dominatedcases, as the standard Galerkin method is only valid for self-adjoint operator equations.
As for the first problem, the contribution of the present thesis is that a Pressure StabilizedFractional Step Algorithm (PS-FSA) is developed which can effectively remove the spuriouspressure oscillations and has better pressure stability than that of the Classical Fractional StepAlgorithm (C-FSA). The proposed PS-FSA is based on re-writing the mass balance equationin the framework of the Finite Increment Calculus (FIC) theory and introducing an additionalvariable into the algorithm in a logical way to avoid the calculation of high order spatialderivatives. In addition, such pressure stabilization mechanism is also extended to theclassical Characteristic Based Split (CBS) algorithm to enhance its pressure stability. Thedetailed derivation of PS-FSA and its numerical validations are presented in Chapter 4.
As for the second problem, the contribution of the present thesis is that a generalizedversion of the Characteristic Galerkin (CG) method in ALE framework is developed whichcan effectively cope with the convection dominated problem and can use larger time step sizethan that of the classical CG method. In fact, the classical CG and the classicalCrank-Nicolson (CN) methods can be classified as two special cases of this generalized CGmethod respectively. In addition, the generation is also exhibited by the fact that the ALEdescription employed for the derivation of the proposed method can reduce to the Euleriandescription used in the classical CG method if the reference coordinates are fixed in space.Due to this generation, it is convenient to combine the proposed CG method with the ALEfree surface tracking techniques mentioned above. By combining the proposed generalizedCG method with the fractional step algorithm and the pressure stabilization techniquedeveloped in Chapter 4, an iterative pressure-stabilized generalized CBS algorithm is formedand used for non-isothermal non-Newtonian fluid flows. The detailed derivation of thealgorithm and its numerical validations are presented in Chapter 6.
Another important contribution of the present thesis is that a self-adaptive domainpartition algorithm is proposed which can automatically partition the whole computationaldomain into three sub-domains where the finite element (FE), meshfree (MF) and theircoupled approximations are employed respectively. Based on this adaptive procedure and theContinuous Blending Method (CBM), an adaptive coupled FE and MF method is alsoproposed for the simulation of injection molding process, which can adequately exploit therespective strong points of FE and MF methods and meanwhile avoid their respective weakpoints. The details of this method and the numerical results to demonstrate its superiority overthe independent FE and MF methods are presented in Chapter 5.
For the purpose of self-completeness, the fundamental concepts and issues of themeshfree method with Galerkin weak form are summarized in Chapter 2. The computer program and data structures to implement the algorithms presented in this thesis for thenumerical simulation of injection molding process are provided in Chapter 7. Conclusion andfuture developments are given in Chapter 8.
引文
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[1] 张雄，刘岩．无网格法．北京：清华大学出版社，2004．
[2] Huerta A, Mendez SF. Enrichment and coupling of the finite element and meshless methods. Int. J. Numer. Methods Engrg., 2000, 48: 1615-1636.
[3] Belytschko T, Organ D, Krongauz Y. A coupled finite element-element free Galerkin method. Comput. Mech., 1995, 17: 186-195.
[4] Krongauz Y, Belytschko T. Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput. Methods Appl. Mech. Engrg., 1996, 131: 133-145.
[5] Huerta A, Mendez SF, Liu WK. A comparison of two formulations to blend finite elements and mesh-free methods. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 1105-1117.
[6] Liu WK, Uras RA, Chen Y. Enrichment of the finite element method with the reproducing kernel particle method. J. Appl. Mech.,ASME, 1997, 64: 861-870.
[7] Wagner GJ, Liu WK. Hierarchical enrichment for bridging scales and mesh-free boundary conditions. Int. J. Numer. Methods Engrg., 2001, 50: 507-524.
[8] Hegen D. Element-free Galerkin methods in combination with finite element approaches. Computer Methods in Applied Mechanics and Engineering, 1996, 135: 143-166.
[9] Belytschko T, Fleming M. Smoothing, enrichment and contact in the element-free Galerkin method. Computers and Structures, 1999, 71: 173-195.
[10] Belytschko T, Krongauz Y, Fleming M et al.. Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational and Applied Mathematics, 1996, 74: 111-126.
[11] Belytschko T, Krongauz Y, Organ D et al.. Meshless methods: An overview and recent developments. Comput. Methods Appl. Mech. Engrg., 1996, 139: 3-47.
[12] Belytschko T, Lu YY, Gu L. Element free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37: 229-256.
[13] Li XK, Duan QL, Han XH et al.. Adaptive coupled arbitrary lagrangian-eulerian finite element and meshfree method for injection molding process. Int. J. Numer. Methods Engrg., accepted.
[14] 宋超．非结构化自适应有限元网格生成的AFT方法：(博士学位论文)．大连：大连理工大学，2004．
[15] Nassehi V. Practical aspects of finite element medelling of polymer processing. Chichester: John Wiley and Sons, 2001.
[16] Silliman WJ, Scriven LE. Separating flow near a static contact line: Slip at a wall and shape of a free surface. J. Comput. Phys., 1980, 34: 287-313.
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[9] Duan QL, Li XK. An ALE based iterative CBS algorithm for non-isothermal non-Newtonian flow with adaptive coupled finite element and meshfree method. Comput. Methods Appl. Mech. Engrg. submitted.
[10] 段庆林，李锡夔．不可压缩非等温非牛顿流的基于广义特征线的分步算法．大连理工大学学报，已投出．
[11] Han XianHong, Li XiKui. An iterative stabilized CNBS-CG scheme for incompressible non-isothermal non-Newtonian fluid flow. Int. J. Heat and Mass transfer, 2007, 50: 847-856.
[12] 吴其晔，巫静安．高分子材料流变学导论．北京：化学工业出版社，1994．
[13] Li Xikui, Duan Qinglin. Meshfree iterative stabilizaed Taylor-Galerkin and characteristic-based split (CBS) algorithms for incompressible N-S equations. Comput. Methods Appl. Mech. Engrg., 2006, 195: 6125-6145.
[14] Onate E.. A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation. Comput. Methods Appl. Mech. Engrg., 2000, 182: 355-370.
[15] Codina R.. Pressure stability in fractional step finite element methods for incompressible flows. J. Comput. Phys., 2001, 170: 112-140.
[16] Bell BC, Surana KS. p-version least squares finite element formulation for two-dimensional incompressible Newtonian and non-Newtonian non-isothermal fluid flow. Computers & Structures, 1995, 54: 83-96.
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