铝型材挤压有限体积法数值模拟技术研究与系统开发
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摘要
随着国民经济的快速发展,铝型材在各个领域得到了越来越广泛的应用。但是由于铝型材品种规格多种多样,铝型材挤压过程材料的变形量大,流动规律复杂,挤压模具承受载荷状况恶劣,使得铝型材挤压产品开发和模具的设计成为决定挤压件质量的关键环节。依赖经验设计和试模返修的传统生产模式已无法满足铝型材快速发展的需要。在效率就是生命,质量就是关键的市场经济环境下,铝型材企业最重视的就是模具设计加工成功率的提高,所以迫切需要可靠的科学理论来指导工艺及模具设计。使用数值模拟方法能够模拟铝型材挤压过程中金属流动状况,包括速度、应力、应变、压力等各种物理场量的分布及变化情况,由此可以预测可能产生的缺陷,评价工艺及模具结构参数是否合理,并对其进行及时修改,从一定程度上避免了费时、费力、高成本的试模返修过程,从而能够快速合理地进行工艺与模具设计,缩短生产周期。金属塑性成形领域应用最为广泛的数值模拟方法是拉格朗日有限元方法,但是由于铝型材挤压的变形剧烈,挤压比通常比较大,用拉格朗日有限元法模拟铝型材挤压,会遇到严重的网格畸变和频繁的网格再划分,带来CPU计算时间增加、体积损失较大、误差严重、模拟结果失真等问题,甚至当挤压比达到一定数值时,模拟计算会最终因严重的网格畸变和体积损失而难以进行下去。
相对于拉格朗日有限元法,基于欧拉网格的有限体积数值模拟方法,因其网格的欧拉描述而能够避免网格畸变和网格再划分,在大变形的体积成形过程的模拟上显示出其优越性。商品化软件Msc/SuperForge是目前唯一基于有限体积法,应用于金属塑性成形领域的数值模拟工具,国内外许多研究者利用其进行了铝型材挤压过程的模拟。但是SuperForge不是专门针对铝型材挤压的软件,并且通过对该软件的应用发现,与有限元软件相比,Superforge在大变形比的铝型材挤压过程模拟方面具有优越性,但计算的精度和模拟的效率之间的平衡问题是Msc/SuperForge软件的瓶颈。Msc/SuperForge使用类似于有限元网格的三角面片包裹金属表面的方法来描述边界,跟踪自由表面,所以当网格比较稀疏的时候,虽然计算速度较有限元方法快,但是模拟结果比较粗糙,三角面片带来的毛刺现象比较严重。要使模拟的挤压件表面质量提高,就要增加网格数量,但是由于Msc/SuperForge中欧拉背景网格尺寸是统一的,不能对用户感兴趣的关键区域进行局部加密,因此模拟质量的少许提高便会带来网格数目的三次方增长,造成计算效率严重下降。
本文针对铝型材挤压这种大变形过程,旨在根据热挤压工艺条件下铝合金材料的性质和成形过程的特点,研究铝型材挤压有限体积法模拟的关键技术,寻找适合的途径和方法,建立铝型材挤压有限体积法数值模拟模型,将适合大变形过程模拟的欧拉网格以及有限体积法灵活地引入铝型材挤压过程模拟中来,为铝型材挤压生产实际提供简便高效的理论指导工具。
本文细致研究了热挤压工艺条件下金属材料的性质,尤其是铝合金的本构关系,认为,不考虑铝型材挤压中的弹性变形,可以将铝合金视为刚塑性或粘塑性材料,并且铝型材热挤压一般是在铝合金的再结晶温度以上进行,因此金属的动力学粘度主要取决于应变速率与温度,所以将挤压中的铝合金视为非线性牛顿流体,可采用流体力学中简单高效的SIMPLE算法迭代计算铝型材挤压中的速度场、压力场,利用能量守恒方程求解温度场,继而根据铝型材挤压的本构方程,利用求解出的温度和等效应变速率对铝合金的动力学粘度进行更新。
铝型材的挤压一般根据材料性质、挤压坯料高径比的不同,分为稳态挤压和非稳态挤压。而实际的挤压过程也分为稳态挤压阶段和非稳态挤压阶段,当坯料的高径比比较大时,稳态挤压在整个挤压过程中所占比重很高。在工业生产中,铝型材的生产一般都是连续挤压,有效的挤压件都是在挤压的稳态阶段成型的,因此对铝型材挤压稳态过程或状态进行模拟非常必要。所以,本文首先进行了铝型材挤压稳态流动过程的有限体积数值模拟。结合铝型材挤压材料的本构方程及材料的特性,研究了铝型材挤压稳态流动过程有限体积法数值模拟的关键技术,推导建立了铝型材挤压过程中动量守恒方程、连续性方程、能量守恒方程的三维离散格式,得到了稳态SIMPLE算法的三维公式和求解流程,提出了铝型材挤压过程中材料粘度的迭代更新方法。给出了迭代收敛的判据,并且实现了网格的局部加密,不仅提高了计算效率,而且保证了模拟质量。在此基础上,利用Virual C++程序平台,编制了程序,建立了铝型材挤压稳态过程有限体积法数值模拟模型,利用该模型进行了稳态算例模拟,并且与有限元软件DEFORM-3D、CASFORM,有限体积软件SuperForge的模拟结果进行对比验证,证明了本文建立的铝型材挤压三维稳态过程有限体积数值模拟模型的正确性和高效性。
挤压过程进行到稳态之前的阶段从一定程度上会影响稳态阶段金属的流动,并且决定着挤压制品的前端形状、应变的积累等,而这些都是制品是否发生翘曲、开裂等缺陷的重要因素;为了使数值模拟能够真实反映铝型材挤压的实际过程,增强铝型材挤压数值模拟软件对实际生产的指导作用,有必要对铝型材挤压由非稳态到稳态整个变化的过程进行研究,得到挤压中重要物理场量随时间的变化规律。因此本文在稳态研究的基础上,拓展建立了铝型材非稳态挤压过程中动量守恒方程、连续性方程、能量守恒方程的三维离散格式,得到了三维非稳态SIMPLE算法的公式和算法流程。针对材料的非稳态流动,将计算流体领域广泛应用的VOF方法引入铝型材挤压过程中的金属流动前沿的动态捕捉中来,真实地模拟了金属前沿流动趋势,得到了良好的表面质量。基于VOF方法,提出根据自由表面位置,自动识别金属流体区域,流场计算采用局部计算方法,即:用控制方程求解金属单元变量,在自由表面单元施加动力学以及运动学条件,空气单元跳过不参与计算,提高了计算效率。提出局部采用“动网格技术”的方法,既解决了挤压垫片移动引起的计算区域变化问题,又避免了全局网格变化带来的计算时间的浪费。根据计算的稳定性条件,提出用时间步长的自动调节方法来进一步提高模拟计算的效率。利用以上关键技术,建立了铝型材挤压三维非稳态过程有限体积数值模拟模型,自主开发了铝型材挤压三维非稳态过程有限体积数值模拟软件,并且利用典型的铝型材挤压非稳态实例验证了铝型材挤压三维非稳态过程有限体积法数值模拟模型是正确、高效的。
以上稳态及非稳态模型均是在三维笛卡尔正交网格上建立的,笛卡尔正交网格虽然能够模拟任何形状区域的流动,程序适应性强,但是正交网格对复杂边界或曲线边界采用“阶梯状”近似逼近,在网格较粗的情况下会带来计算误差和边界施加的困难。为了使计算网格边界与流体区域边界更加吻合,使边界条件的施加更加方便、科学,计算更加贴合实际挤压过程,本文研究了适体网格的生成技术,在此基础上着重研究了网格的非正交性对铝型材挤压控制方程中对流项、扩散项的影响,推导建立了三维适体网格下各控制方程的离散公式,基于正交网格下的VOF方法和移动网格技术,充分考虑网格非正交性的影响,拓展建立了适体网格下的VOF方法和移动网格技术。非正交边界上采用剪切摩擦模型,并在此基础上提出了适体网格下边界条件的施加方法。以上述关键技术为依托,建立了更加贴合挤压实际,更加适应复杂区域挤压过程模拟的适体网格下的铝型材挤压有限体积法数值模拟数学模型,对三维铝型材挤压过程有限体积法数值模拟软件进行了网格适体性优化,对典型的稳态以及非稳态挤压过程实例进行了模拟、对比与分析,验证了该模型的有效性。
实验是检验数值模拟结果是否正确的最具说服力的工具。为了进一步验证本文所建立的铝型材挤压有限体积数值模拟模型的正确性,本文结合铝型材挤压的工艺和模具设计知识,针对十字花形件的挤压设计了一组挤压模具,进行了挤压实验。通过模拟结果与实验结果的比较,进一步验证了本文建立的铝型材挤压有限体积法数值模拟模型的正确性。
With the fast development of national economy, aluminum profile products are getting wider employment in various fields. For the aluminum profiles' variant specifications and extrusion dies' fierce working condition due to complex material flow status and the large deformation, the product development and the mould design are more and more important. The traditional product pattern relying on experience design and trial-correct courses can't satisfy the fast development of the aluminum alloy profile product. Under the economic entironment where the efficiency and the quality are the most important factors, the enterprises think much of the improvement of the efficiency of the die design and machining. Reliable and effective numerical simulation methods are required. They can predict the metal flow pattern, including the distribution and the development of the physical fields, such as velocity, stress, strain and pressure, etc. and then forecast the defect that possibly appears, evaluate if the process and mould conductive parameter are rational and if not. These parameters will be amended, and the trial-correct courses wasting time, manpower and money will be avoided on some degree. And then the process and die design will be completed rapidly and rationally, shortening the product cycle. The most popular numerical simulation method in the metal plastic deformation fields is the finite element method based on Lagrangian mesh. But if the Lagrangian based finite element method is used to simulate the extrusion processes with severely large deformations, serious mesh distort and frequent remeshing will happen. The remeshing will bring the waste of the CPU time, volume loss, the distortion of the results and serious error, etc. When the Reduction ratio in area is pretty large, the simulation can't continue because of the severe mesh distortion and calculation error.
Compared with Laglanrian based finite element method, Eulerian based finite volume method can avoid mesh distort and remeshing for its Euler mesh, and shows its advantages in simulating bulking forming with large deformations. Commercial software Msc/SuperForge is the only tool based on finite volume method to simulate bulking metal forming processes. But SuperForge is not a software that specially aims at aluminum alloy profile extrusion. Many researchers in the world simulate the aluminum alloy profile extrusion with SuperForge. It was found that the SuperForge has many advantages in simulating process with large deformation such as extrusion over finite element softwares, but the balance of the calculation accuracy and the simulation effective is the bottleneck of the SuperForge software. SuperForge uses triangular facets encapsulating the FVM grid to track the free surface and the wall boundary of the deforming body. When the FVM mesh is pretty sparse, the simulation speed is relative high, at the same time, the simulation results are very coarse, and the serious burr brought by the triangular facets is serious. A thick mesh is required if the improvement of the simulated surface finish is needed. But because the Eulerian background mesh is unit and can't be locally refined at the interested place, improvement of the surface property will bring the increase of the mesh number in cubic way, decreasing the simulation effective.
According to aluminum alloy extrusion with large deformation, based on the character of the aluminum alloy during the hot extrusion, this paper aims to study the key technologies of the aluminum alloy profile extrusion simulation by finite volume method, to find suitable and effective methods to build aluminum alloy profile extrusion simulation model by using finite volume method, and introduce the finite volume method and the Euler mesh suitable for large deformation simulation to the aluminum alloy profile extrusion simulation, supplying a simple and effective academic instruct tool to the real aluminum alloy extrusion product.
In this paper, the characteristics of the material in aluminum alloy profile extrusion processes are studied carefully, especially the constitutive equation of the metal. The elastic deformation of the processes is ignored and then the material is assumed to be rigid-plastic or visco-plastic material. Isotropic material properties are assumed. The dynamic viscosity is prominently influenced by the equivalent strain-rate and temperature, for the aluminum alloy profile extrusion processes always takes place above the recrystal temperature. So hot aluminum alloy material is described as a non-linear Newtonian fluid material, and the SIMPLE algorithm can be used to iteratively calculate the velocity, pressure fields, and then the temperature is calculated by the energy equation. After that, the dynamic viscosity is updated based on the constitutive equation using the calculated temperature and effective strain-rate.
Aluminum profile extrusion includes steady extrusion and unsteady extrusion. The actual extrusion processes is also divided into steady stage and unsteady stage. Because in the industrial production, aluminum profile product is always a continuous processes. The valid extrudate takes shape in the steady stage, further more, when the ratio of the height and the radius is large, the proportion of the steady stage is higher. So the steady flow pattern of the aluminum alloy profile extrusion is studied and simulated by the finite volume method. Combined with the material characters and the constitutive equation, the key technologies of the aluminum profile steady extrusion processes simulation by finite volume method are studied. The three-dimensional discretised forms of the momentum conservation equation, continuity equation and energy conservation equation are obtained. The three dimensional equations and the flow program of the SIMPLE algorithm are given. The method to update the dynamic viscosity of the material in the aluminum alloy profile extrusion is yielded. The convergence criterion is also discussed. Local refinement of the mesh is realized improving the simulation effective, and at the same time, insuring the simulation quality. On the base of above, the code is programmed on Virual C++ plat, and the steady aluminum alloy profile extrusion simulation model using finite volume method is built. Some steady cases are simulated by the model, and the results are compared with those simulated by FEM software DEF0RM-3D and CASFORM, FVM software SuperForge. The comparisons prove the accuracy and the effectiveness of the steady aluminum alloy profile extrusion simulation model using finite volume method.
The starting unsteady process before the extrusion reaches its steady state can influence the steady flow and decide the forehead shape and the accumulation of the strain etc., which are the important factors determining if the defects such as bend and craze will happen. To make the simulation model match the real processes of the aluminum extrusion, and reinforce the instruct function of the software to the actual extrusion product, it is necessary to study the change of the physical fields versus the time. On the base of the research of the steady-state flow, the three-dimensional discretised forms of the momentum conservation, continuity equation and energy conservation equation for unsteady-stat aluminum alloy profile extrusion are obtained. The equations and the flow chart of the three-dimensional unsteady SIMPLE algorithm are also got. According to the unsteady flow of the metal, the VOF method which is widely used in the Computational Fluid Dynamics is induced into the catch of the metal free surface. The flowing trend of the metal free surface is simulated factually and good surface finish is obtained. Based on the VOF method, the flowing fields are calculated by a 'part calculation method', that is, the variables on the elements full of metal fluid are calculated by the government equation, the dynamic and kinematic boundary conditions are implemented on the free surface elements, and the air elements are skipped. This 'part calculation method' can improve the calculation efficiency. The 'Moving grid system' is advised to locally used, which can not only solute the calculation domain changing problem rising by the moving of the punch, but also avoid the time waste if all the mesh is changed. The auto-control of the time step is put forward to improve the efficiency of the calculation, based on the stability command of the calculation. On the base of the key technology above, three-dimensional unsteady aluminum alloy profile extrusion finite volume method simulation model is built, and the software is programmed. Several unsteady extrusion cases are simulated to prove the accuracy and the efficiency of the aluminum alloy profile extrusion simulation model by finite volume method.
All the above steady and the unsteady extrusion models are build under Cartesian orthogonal grid which can adapted to flow caluculation with all kinds of boundaryes. But when the mesh is corse, the "Stepwise approximation" will introduce error and make the boundary condition treatment difficulty. To make the boundary of the numerical grid match the flowing domain for the convenience of the implement of the boundary condition, fitting the calculation to the actual extrusion process, the body-fitted mesh generation technology is researched, on the base of which, the influence of the grids' non-orthogonal to the conductive term and the diffusion term etc. is studied, the three dimensional discretised forms of the governing equations on body-fitted grid are built. On the base of the VOF method and the moving grid system, combining the influence of the grid's non-orthogonal, the VOF method and the moving grid system technology on the non-orthogonal grid system are built, too. Shear friction model are used on the boundary, and then the velocity boundary condition implement method is brought forward. By these key technologies, the aluminum alloy profile extrusion model by finite volume method on the body-fitted mesh system is built, and the according software is coded, too. Typical steady and unsteady extrusion cases are used to prove the efficiency of the model.
Experiment is the most powerful tool improving the accuracy of the simulation. So to prove the accuracy of the aluminum alloy profile extrusion finite volume method model, a set of extrusion dies for aα-shaped extrudate are designed on the base of the extrusion arts and the dies. The experiment is done successfully and the comparing between the experiment results and the simulation ones proves that the aluminum alloy profile extrusion finite volume method model built in this paper is accurate again.
相对于拉格朗日有限元法,基于欧拉网格的有限体积数值模拟方法,因其网格的欧拉描述而能够避免网格畸变和网格再划分,在大变形的体积成形过程的模拟上显示出其优越性。商品化软件Msc/SuperForge是目前唯一基于有限体积法,应用于金属塑性成形领域的数值模拟工具,国内外许多研究者利用其进行了铝型材挤压过程的模拟。但是SuperForge不是专门针对铝型材挤压的软件,并且通过对该软件的应用发现,与有限元软件相比,Superforge在大变形比的铝型材挤压过程模拟方面具有优越性,但计算的精度和模拟的效率之间的平衡问题是Msc/SuperForge软件的瓶颈。Msc/SuperForge使用类似于有限元网格的三角面片包裹金属表面的方法来描述边界,跟踪自由表面,所以当网格比较稀疏的时候,虽然计算速度较有限元方法快,但是模拟结果比较粗糙,三角面片带来的毛刺现象比较严重。要使模拟的挤压件表面质量提高,就要增加网格数量,但是由于Msc/SuperForge中欧拉背景网格尺寸是统一的,不能对用户感兴趣的关键区域进行局部加密,因此模拟质量的少许提高便会带来网格数目的三次方增长,造成计算效率严重下降。
本文针对铝型材挤压这种大变形过程,旨在根据热挤压工艺条件下铝合金材料的性质和成形过程的特点,研究铝型材挤压有限体积法模拟的关键技术,寻找适合的途径和方法,建立铝型材挤压有限体积法数值模拟模型,将适合大变形过程模拟的欧拉网格以及有限体积法灵活地引入铝型材挤压过程模拟中来,为铝型材挤压生产实际提供简便高效的理论指导工具。
本文细致研究了热挤压工艺条件下金属材料的性质,尤其是铝合金的本构关系,认为,不考虑铝型材挤压中的弹性变形,可以将铝合金视为刚塑性或粘塑性材料,并且铝型材热挤压一般是在铝合金的再结晶温度以上进行,因此金属的动力学粘度主要取决于应变速率与温度,所以将挤压中的铝合金视为非线性牛顿流体,可采用流体力学中简单高效的SIMPLE算法迭代计算铝型材挤压中的速度场、压力场,利用能量守恒方程求解温度场,继而根据铝型材挤压的本构方程,利用求解出的温度和等效应变速率对铝合金的动力学粘度进行更新。
铝型材的挤压一般根据材料性质、挤压坯料高径比的不同,分为稳态挤压和非稳态挤压。而实际的挤压过程也分为稳态挤压阶段和非稳态挤压阶段,当坯料的高径比比较大时,稳态挤压在整个挤压过程中所占比重很高。在工业生产中,铝型材的生产一般都是连续挤压,有效的挤压件都是在挤压的稳态阶段成型的,因此对铝型材挤压稳态过程或状态进行模拟非常必要。所以,本文首先进行了铝型材挤压稳态流动过程的有限体积数值模拟。结合铝型材挤压材料的本构方程及材料的特性,研究了铝型材挤压稳态流动过程有限体积法数值模拟的关键技术,推导建立了铝型材挤压过程中动量守恒方程、连续性方程、能量守恒方程的三维离散格式,得到了稳态SIMPLE算法的三维公式和求解流程,提出了铝型材挤压过程中材料粘度的迭代更新方法。给出了迭代收敛的判据,并且实现了网格的局部加密,不仅提高了计算效率,而且保证了模拟质量。在此基础上,利用Virual C++程序平台,编制了程序,建立了铝型材挤压稳态过程有限体积法数值模拟模型,利用该模型进行了稳态算例模拟,并且与有限元软件DEFORM-3D、CASFORM,有限体积软件SuperForge的模拟结果进行对比验证,证明了本文建立的铝型材挤压三维稳态过程有限体积数值模拟模型的正确性和高效性。
挤压过程进行到稳态之前的阶段从一定程度上会影响稳态阶段金属的流动,并且决定着挤压制品的前端形状、应变的积累等,而这些都是制品是否发生翘曲、开裂等缺陷的重要因素;为了使数值模拟能够真实反映铝型材挤压的实际过程,增强铝型材挤压数值模拟软件对实际生产的指导作用,有必要对铝型材挤压由非稳态到稳态整个变化的过程进行研究,得到挤压中重要物理场量随时间的变化规律。因此本文在稳态研究的基础上,拓展建立了铝型材非稳态挤压过程中动量守恒方程、连续性方程、能量守恒方程的三维离散格式,得到了三维非稳态SIMPLE算法的公式和算法流程。针对材料的非稳态流动,将计算流体领域广泛应用的VOF方法引入铝型材挤压过程中的金属流动前沿的动态捕捉中来,真实地模拟了金属前沿流动趋势,得到了良好的表面质量。基于VOF方法,提出根据自由表面位置,自动识别金属流体区域,流场计算采用局部计算方法,即:用控制方程求解金属单元变量,在自由表面单元施加动力学以及运动学条件,空气单元跳过不参与计算,提高了计算效率。提出局部采用“动网格技术”的方法,既解决了挤压垫片移动引起的计算区域变化问题,又避免了全局网格变化带来的计算时间的浪费。根据计算的稳定性条件,提出用时间步长的自动调节方法来进一步提高模拟计算的效率。利用以上关键技术,建立了铝型材挤压三维非稳态过程有限体积数值模拟模型,自主开发了铝型材挤压三维非稳态过程有限体积数值模拟软件,并且利用典型的铝型材挤压非稳态实例验证了铝型材挤压三维非稳态过程有限体积法数值模拟模型是正确、高效的。
以上稳态及非稳态模型均是在三维笛卡尔正交网格上建立的,笛卡尔正交网格虽然能够模拟任何形状区域的流动,程序适应性强,但是正交网格对复杂边界或曲线边界采用“阶梯状”近似逼近,在网格较粗的情况下会带来计算误差和边界施加的困难。为了使计算网格边界与流体区域边界更加吻合,使边界条件的施加更加方便、科学,计算更加贴合实际挤压过程,本文研究了适体网格的生成技术,在此基础上着重研究了网格的非正交性对铝型材挤压控制方程中对流项、扩散项的影响,推导建立了三维适体网格下各控制方程的离散公式,基于正交网格下的VOF方法和移动网格技术,充分考虑网格非正交性的影响,拓展建立了适体网格下的VOF方法和移动网格技术。非正交边界上采用剪切摩擦模型,并在此基础上提出了适体网格下边界条件的施加方法。以上述关键技术为依托,建立了更加贴合挤压实际,更加适应复杂区域挤压过程模拟的适体网格下的铝型材挤压有限体积法数值模拟数学模型,对三维铝型材挤压过程有限体积法数值模拟软件进行了网格适体性优化,对典型的稳态以及非稳态挤压过程实例进行了模拟、对比与分析,验证了该模型的有效性。
实验是检验数值模拟结果是否正确的最具说服力的工具。为了进一步验证本文所建立的铝型材挤压有限体积数值模拟模型的正确性,本文结合铝型材挤压的工艺和模具设计知识,针对十字花形件的挤压设计了一组挤压模具,进行了挤压实验。通过模拟结果与实验结果的比较,进一步验证了本文建立的铝型材挤压有限体积法数值模拟模型的正确性。
With the fast development of national economy, aluminum profile products are getting wider employment in various fields. For the aluminum profiles' variant specifications and extrusion dies' fierce working condition due to complex material flow status and the large deformation, the product development and the mould design are more and more important. The traditional product pattern relying on experience design and trial-correct courses can't satisfy the fast development of the aluminum alloy profile product. Under the economic entironment where the efficiency and the quality are the most important factors, the enterprises think much of the improvement of the efficiency of the die design and machining. Reliable and effective numerical simulation methods are required. They can predict the metal flow pattern, including the distribution and the development of the physical fields, such as velocity, stress, strain and pressure, etc. and then forecast the defect that possibly appears, evaluate if the process and mould conductive parameter are rational and if not. These parameters will be amended, and the trial-correct courses wasting time, manpower and money will be avoided on some degree. And then the process and die design will be completed rapidly and rationally, shortening the product cycle. The most popular numerical simulation method in the metal plastic deformation fields is the finite element method based on Lagrangian mesh. But if the Lagrangian based finite element method is used to simulate the extrusion processes with severely large deformations, serious mesh distort and frequent remeshing will happen. The remeshing will bring the waste of the CPU time, volume loss, the distortion of the results and serious error, etc. When the Reduction ratio in area is pretty large, the simulation can't continue because of the severe mesh distortion and calculation error.
Compared with Laglanrian based finite element method, Eulerian based finite volume method can avoid mesh distort and remeshing for its Euler mesh, and shows its advantages in simulating bulking forming with large deformations. Commercial software Msc/SuperForge is the only tool based on finite volume method to simulate bulking metal forming processes. But SuperForge is not a software that specially aims at aluminum alloy profile extrusion. Many researchers in the world simulate the aluminum alloy profile extrusion with SuperForge. It was found that the SuperForge has many advantages in simulating process with large deformation such as extrusion over finite element softwares, but the balance of the calculation accuracy and the simulation effective is the bottleneck of the SuperForge software. SuperForge uses triangular facets encapsulating the FVM grid to track the free surface and the wall boundary of the deforming body. When the FVM mesh is pretty sparse, the simulation speed is relative high, at the same time, the simulation results are very coarse, and the serious burr brought by the triangular facets is serious. A thick mesh is required if the improvement of the simulated surface finish is needed. But because the Eulerian background mesh is unit and can't be locally refined at the interested place, improvement of the surface property will bring the increase of the mesh number in cubic way, decreasing the simulation effective.
According to aluminum alloy extrusion with large deformation, based on the character of the aluminum alloy during the hot extrusion, this paper aims to study the key technologies of the aluminum alloy profile extrusion simulation by finite volume method, to find suitable and effective methods to build aluminum alloy profile extrusion simulation model by using finite volume method, and introduce the finite volume method and the Euler mesh suitable for large deformation simulation to the aluminum alloy profile extrusion simulation, supplying a simple and effective academic instruct tool to the real aluminum alloy extrusion product.
In this paper, the characteristics of the material in aluminum alloy profile extrusion processes are studied carefully, especially the constitutive equation of the metal. The elastic deformation of the processes is ignored and then the material is assumed to be rigid-plastic or visco-plastic material. Isotropic material properties are assumed. The dynamic viscosity is prominently influenced by the equivalent strain-rate and temperature, for the aluminum alloy profile extrusion processes always takes place above the recrystal temperature. So hot aluminum alloy material is described as a non-linear Newtonian fluid material, and the SIMPLE algorithm can be used to iteratively calculate the velocity, pressure fields, and then the temperature is calculated by the energy equation. After that, the dynamic viscosity is updated based on the constitutive equation using the calculated temperature and effective strain-rate.
Aluminum profile extrusion includes steady extrusion and unsteady extrusion. The actual extrusion processes is also divided into steady stage and unsteady stage. Because in the industrial production, aluminum profile product is always a continuous processes. The valid extrudate takes shape in the steady stage, further more, when the ratio of the height and the radius is large, the proportion of the steady stage is higher. So the steady flow pattern of the aluminum alloy profile extrusion is studied and simulated by the finite volume method. Combined with the material characters and the constitutive equation, the key technologies of the aluminum profile steady extrusion processes simulation by finite volume method are studied. The three-dimensional discretised forms of the momentum conservation equation, continuity equation and energy conservation equation are obtained. The three dimensional equations and the flow program of the SIMPLE algorithm are given. The method to update the dynamic viscosity of the material in the aluminum alloy profile extrusion is yielded. The convergence criterion is also discussed. Local refinement of the mesh is realized improving the simulation effective, and at the same time, insuring the simulation quality. On the base of above, the code is programmed on Virual C++ plat, and the steady aluminum alloy profile extrusion simulation model using finite volume method is built. Some steady cases are simulated by the model, and the results are compared with those simulated by FEM software DEF0RM-3D and CASFORM, FVM software SuperForge. The comparisons prove the accuracy and the effectiveness of the steady aluminum alloy profile extrusion simulation model using finite volume method.
The starting unsteady process before the extrusion reaches its steady state can influence the steady flow and decide the forehead shape and the accumulation of the strain etc., which are the important factors determining if the defects such as bend and craze will happen. To make the simulation model match the real processes of the aluminum extrusion, and reinforce the instruct function of the software to the actual extrusion product, it is necessary to study the change of the physical fields versus the time. On the base of the research of the steady-state flow, the three-dimensional discretised forms of the momentum conservation, continuity equation and energy conservation equation for unsteady-stat aluminum alloy profile extrusion are obtained. The equations and the flow chart of the three-dimensional unsteady SIMPLE algorithm are also got. According to the unsteady flow of the metal, the VOF method which is widely used in the Computational Fluid Dynamics is induced into the catch of the metal free surface. The flowing trend of the metal free surface is simulated factually and good surface finish is obtained. Based on the VOF method, the flowing fields are calculated by a 'part calculation method', that is, the variables on the elements full of metal fluid are calculated by the government equation, the dynamic and kinematic boundary conditions are implemented on the free surface elements, and the air elements are skipped. This 'part calculation method' can improve the calculation efficiency. The 'Moving grid system' is advised to locally used, which can not only solute the calculation domain changing problem rising by the moving of the punch, but also avoid the time waste if all the mesh is changed. The auto-control of the time step is put forward to improve the efficiency of the calculation, based on the stability command of the calculation. On the base of the key technology above, three-dimensional unsteady aluminum alloy profile extrusion finite volume method simulation model is built, and the software is programmed. Several unsteady extrusion cases are simulated to prove the accuracy and the efficiency of the aluminum alloy profile extrusion simulation model by finite volume method.
All the above steady and the unsteady extrusion models are build under Cartesian orthogonal grid which can adapted to flow caluculation with all kinds of boundaryes. But when the mesh is corse, the "Stepwise approximation" will introduce error and make the boundary condition treatment difficulty. To make the boundary of the numerical grid match the flowing domain for the convenience of the implement of the boundary condition, fitting the calculation to the actual extrusion process, the body-fitted mesh generation technology is researched, on the base of which, the influence of the grids' non-orthogonal to the conductive term and the diffusion term etc. is studied, the three dimensional discretised forms of the governing equations on body-fitted grid are built. On the base of the VOF method and the moving grid system, combining the influence of the grid's non-orthogonal, the VOF method and the moving grid system technology on the non-orthogonal grid system are built, too. Shear friction model are used on the boundary, and then the velocity boundary condition implement method is brought forward. By these key technologies, the aluminum alloy profile extrusion model by finite volume method on the body-fitted mesh system is built, and the according software is coded, too. Typical steady and unsteady extrusion cases are used to prove the efficiency of the model.
Experiment is the most powerful tool improving the accuracy of the simulation. So to prove the accuracy of the aluminum alloy profile extrusion finite volume method model, a set of extrusion dies for aα-shaped extrudate are designed on the base of the extrusion arts and the dies. The experiment is done successfully and the comparing between the experiment results and the simulation ones proves that the aluminum alloy profile extrusion finite volume method model built in this paper is accurate again.
引文
[1]陈泽中,包忠诩,柳和生等.铝型材挤压研究进展[J].金属成形工艺,2000,18(5):1-5
[2]高军,赵国群,李丽华.铝合金型材挤压技术现状与发展趋势[J].汽车工艺与材料,2002(6):21-24
[3]刘静安.铝型材挤压模具设计,制造,使用及维修[M].北京:冶金工业出版社,2002
[4]J.Zhou,L.Li and J.Duszczyk,3D FEM simulation of the whole cycle of aluminium extrusion throughout the transient state and the steady state using the updated Langrangian approach[J],Journal of Material Processing Technology,2003,134(3):383-397
[5]Y.B.Park,D.Y.Yang,Investigation into non-steady-state three-dimensional helical extrusion of twisted sections by the rigid-plastic finite element method[J],Engineering Computations,1997,14(6):649-668
[6]J.Lof,Elasto-viscoplastic FEM simulations of the aluminium flow in the beating area for extrusion of thin-walled sections[J],Journal of Materials Processing Technology,2001,114:174-183
[7]E Halvorsen,T.Aukrust,Studies of the mechanisms for bucking and waving in aluminum extrusion by use of a Lagrangian FEM software[J],International Journal of Plasticity,2006,22(1):158-173
[8]J.Lof,Y.Blockhuis,FEM simulations of the extrusion of the complex thin walled aluminium sections[J],Journal of Materials Processing Technology,2002,122:344-354
[9]P.Zhi,T.Sheppard,Simulation of multi-hole die extrusion[J],Materials Science and Engineering A,2004,367:329-342
[10]于沪平,彭颖红,阮雪榆.平面分流焊合模成型过程的数值模拟[J].锻压技术,1999,24(5):9-11
[11]E Chanda,J.Zhou,J.Duszczyk,FEM analysis of aluminum extrusion through square and round dies[J],Materials and Design,2000,vol.21:323-335
[12]刘汉武,丁惮,崔建忠.铝型材挤压分流组合模有限元分析和计算[J].模具工业,1999,(4):9-11
[13]X.Duan et al.Application of finite element method in the hot extrusion of aluminum alloys[J].Materials Science and Engineering A,2004,369:66-75
[14]N.Hao et al.Numerical design of the die land for shape extrusion[J].Journal of Materials Processing Technology,2000,101:81-84
[15]Z.H.Chen,C.Y.Tang,Simulation of the sheet metal extrusion process by the enhanced assumed strain finite element method[J].Journal of Materials Processing Technology,1999,91:250-256
[16]G.Li,J.T.Jinn,Recent development and applications of three-dimensional finite element modeling in bulk forming processes.Journal of Materials Processing Technology,2001,113:40-45
[17]Xinjian Duan,X.Velay,Application of finite element method in the hot extrusion of aluminium alloys[J],Materials Science and Engineering A,2004,369:66-75
[18]周飞,彭颖红,阮雪榆.铝型材挤压过程有限元数值模拟[J].中国有色金属学报,1998,8(4):637-642
[19]K.Mori,K.Osakada,H.Yamaguchi.Prediction of curvature of an extruded bar with noncircular cross-section by a 3D rigid-plastic finite element method[J].Int.J.Mech Sci.1993,35(10):879-887
[20]R.Shivpuri,S.Momin,T.Altan.Computer aided design of dies to control dimensional quality of extruded shapes[J],CirP Ann.1992,41(1):275-279
[21]Z.Jia.Three-dimensional simulations of the hollw extrusion and drawing using the finite element method.Master's Thesis,Ohio University,Athens,OH,1994
[22]闫洪等.角铝型材挤压过程的数值模拟[J].中国有色金属学报,2001,11(2):202-205
[23]J.Lof.Elasto-viscoplaxtic FEM simulation of the aluminum flow in the bearing area for extrusion thin-walled sections[J].Journal of Materials Processing Technology,2001,114:174-183
[24]赵国群.金属体积塑性成形过程数值模拟技术与仿真系统[J].金属成形工艺,2003,21(5):5-8
[25]Q.Li,C.J.Smith,Finite element investigations upon the influence of pocket die designs on metal flow in aluminum extrusion Part I.Effect of pocket angle and volume on metal flow[J],Journal of Materials Processing Technology,2003,135:189-196
[26]Q.Li,et al.Finite element modeling investigations upon the influence of pocket die designs on metal flow in aluminum extrusion Part II.Effect of pocket geometry configurations on metal flow[J],Journal of Materials Processing Technology,2003,135:197-203
[27]N.Hao,et al.Numerical design of the die land for shape extrusion[J].Journal of Materials Processing Technology,2000,101:81-84
[28]田柱平,郝南海.铝型材挤压模工作带形状的数值设计[J].模具技术,1999,1:19-21
[29]C.Tapas,et al.A comparative study on iso-speed extrusion and isothermal extrusion of 6061 A1 alloy using 3D FEM simulation[J].Journal of Materials Processing Technology,2001,114:145-153
[30]J.Zhou,et al.Computer simulated and experimentally verified isothermal extrusion of 7075 aluminum through continuous ram speed variation.Journal of Materials Processing Technology[J],2004,146:203-212
[31]X.Duan,T.Sheppard.Simulation and control of microstructure evolution during hot extrusion of hard aluminum alloys[J].Materials science and Engineering,2003,A351:282-292
[32]史翔.异型材挤压设计理论及其应用[J].轻合金加工技术,2000,28(1):21-24
[33]颜建辉,柳瑞清.正确选择铝型材挤压速度的方法[J].轻合金加工技术,2002,30(7):36-37
[34]闫洪.工艺参数对铝型材挤压变形规律的影响[J].中国有色金属学报,2002,12(6):1154-1161
[35]闫洪.基于数值模拟的铝型材挤压变形规律的研究[J].锻压机械,2000,5:29-30
[36]J.C.Malas,Effect of microstructural complexity on the hot deformation behavior of aluminum alloy 2024[J],Materials Science and Engineering A,2004,368:41-47
[37]A.H.Clausen,Sensitivity of model parameters in stretch bending of aluminium extrusions[J],International Journal of Mechanical Sciences,2001,47:427-453
[38]R.Hambli.Damage and fracture simulation during the extrusion processes[J],Computer methods in applied mechanics and engineering,2000,186:109-120
[39]T.Aukrust,Coupled FEM and texture modelling of plane strain extrusion of an aluminium alloy[J],International Journal of Plasticity,1997,13(1/2):111-125
[40]R.Couch et al,3D metal forming applications of ALE techniques.In Simulation of Materials Processing:Theory and Applications[C],Shen SF,Dawson PR(eds).Balkema:Rotterdam,1995:401-406
[41]E.H.Atzema,J.Huetink,Finite element analysis of forward/backword extrusion using ALE techniques[C].Proceedings of Iternational Conference of Numerical Methods in Industrial Forming Processes,NUMIFORM 95,Ithaca,1995:383-388
[42]D.Y.Yang,K.Park,Y.S.Kang.Integrated finite element simulation for the hot extrusion of complicated A1 alloy profiles[J],Journal of Materials Processing Technology,2001,111:25-30
[43]H.G.Mooi et al.Simulation of complex aluminum extrusion using an arbitrary Eulerian Lagrangian formulation.In Simulation of Materials Processing:Throry and Applications,Shen SF,Dawson PR(eds).Balkema:Rotterdam,1995:869-874
[44]H.N Bayomi,M.S.Gadala.Simulation of large deformation problems using the arbitrary Lagrangian-Eulerian formulation[C].Proceedings of the European Conference on Computational Mechanics,ECCM'99,Munchen,1999
[45]Y.S.Kang,D.Y.Yang.Rigid-viscoplastic finite element anlysis of hot square die extrusion of complicated profiles with flow guides and lands by arbitrary Langrangian-Eulerian formulation.In Simulation of Materials Processing:Theory and Applications,Shen SF,Dawson PR(eds).Balkema:Rotterdam,1995:841-846
[46]E.H.Atzema.Finite element analysis of forward/backward extrusion using ALE Techniques[C].Proceedings of International Conference of Numerical Methods in Industrial Forming Processes,NUMIFORM 95,Ithaca,1995:383-388
[47]A.Paine,M.Aloe,J.Walters,Simulation of hot extrusion processes[J],Aluminium Extrusion,2002,7(1):39-44
[48]A.E.Tekkaya,Current state and future developments in the simulation of forming processes[C].ICFG 97,Den Bosch,1997
[49]E.D.Vries,P.Ding,Simulation of 3D forging and extrusion problems using a finite volume method[C].Proceeding of 17~(th)MSC JAPAN Users Conference,Tokyo:Msc Software Japan Ltd.,1999,1:155-161
[50]B.V.Mehtaa.3D flow analysis inside shear and streamlined extrusion dies for feeder plate design[J],Journal of Materials Processing Technology,2001,113:93-97
[51]A.J.Williams,T.N.Croft,M.Cross.Computational modelling of metal extrusion and forging process[J],Journal of Materials Processing Technology,2002,125-126:573-582
[52]Wu Xianghong,Zhao Guoqun,Numerical simulation and die structure optimization of an aluminum rectangular hollow pipe extrusion process[J].Materials Science& Engineering A,2006,435-436
[53]Wu xianghong,Zhao Guoqun,Simulation based porthole die structure optimization of an aluminum profile extrusion process[J].Transations of Nonferrous Metals Society of China,2006,16(z3):1261-1264
[54]吴向红,赵国群,孙胜,娄淑梅.挤压速度和摩擦状态对铝型材挤压过程的影响[J],塑性工程学报,2007,(1):36-41
[55]吴向红,赵国群,孙胜,娄淑梅.使用分流组合模的铝型材挤压过程数值模拟[J],中国机械工程,2006,17:106-109
[56]黄克坚,包忠诩,陈泽中,等.有限体积法在挤压模具设计中的运用[J].稀有金属材料与工程,2004,(8):855-857
[57]黄克坚,包忠诩,周天瑞.有限体积数值模拟技术在型材挤压变形规律研究中的运用[J].轻合金加工技术,2003,31(4):29-31
[58]黄克坚,包忠诩.有限体积数值模拟技术在宽展挤压模具设计中的应用[J].塑性工程学报,2005,12(6):34-37
[59]黄光法,林高用.大挤压比铝型材挤压过程中的数值模拟[J].中国有色金属学报,2006,16(5):887-893
[60]周飞,苏丹.有限体积法模拟铝型材挤压成形过程[J].中国有色金属学报,2003,13(1):65-70
[61]周飞,苏丹,彭颖红.有限体积法仿真金属塑性成形的基本理论[J].上海交通大学学报,2002,36(7):915-919
[62]周飞.铝型材挤压有限元/有限体积复合数值模拟技术研究上海交通大学博士学位论文
[63]罗超,李大永.薄壁铝型材挤压成形的一种有效模拟方法[J].上海交通大学学报,2004,38(7):1134-1137
[64]周飞,苏丹.铝型材挤压有限元和有限体积对比模拟[J].上海交通大学学报,2003,37(7):1072-1076
[65]陈泽中,娄臻亮.复杂铝型材挤压成形有限体积仿真[J].上海交通大学学报,2005,39(1):28-40
[66]李大永,王洪俊.薄壁铝型材挤压有限体积分步模拟[J].上海交通大学学报,2005,39(1):6-9
[67]E Zhou,D.Su,FEM and FVM compound numerical simulation of aluminum extrusion processes.[J],Trans.Nonferrous Met.Soc.China,2003,13(2):381-385
[68]Z.Z Chen,Z.L.Lou,Finite volume simulation and mould optimization of aluminum profile extrusion[J],Journal of Materials Processing Technology,2007,190(1-3):382-386
[69]H.K.Versteeg,W.Malalasekera,An Introduction to Computational Fluid Dynamics[M].The Finite Volume Method Approach,Prentice Hall,1996.
[70]王福军.计算流体动力学分析-CFD软件原理与应用[M].清华大学出版社,2004
[71]陶文铨著.计算传热学的近代进展[M].北京:科学出版社,2000.06
[72]朱自强.应用计算流体力学[M].北京:北京航天航空大学出版社,1998
[73]S.V.Pantanker.Numerical Heat Transfer and Fluid Flow[M].McGraw-Hill Book Company,1984
[74](美)帕坦卡著.传热与流体流动的数值计算[M].张政译.北京:科学出版社,1989
[75]陶文铨.数值传热学[M].西安:西安交通大学出版社,2001
[76]彭颖红 金属塑性成形仿真技术[M].上海:上海交通大学出版社,1999
[77]吴向红 铝型材挤压过程有限体积数值模拟及软件开发技术的研究.山东大学博士学位论文
[78]O.C.Zienkiewicz,P.N.Godbole.Flow of Plastic and Viscoplastic Solids with Special Reference to Extrusion and Forming Process.International Journal for Numerical Methods in Engineering,1974,8:3-15
[79]M.S.Eikemo,M.S.Espedal,et al,On the numerical solution of a three dimensional extrusion model[J],Computing and Visualization in Science,1997,1(1)1-14
[80]F.Halvorsen,T.Aukrust,Studies of the mechanisms for bucking and waving in aluminum extrusion by use of a Lagrangian FEM software[J],International Journal of Plasticity,2006:22(1)158-173.
[81]C.M.Sellars,W.J.Tegart,Hot workability.Int.Met.Rev.1972,17:1-24
[82]T.Sheppard,D.Wright.Deformation of flow stress:Part 1 constitutive equations for aluminium alloys at elevated temperatures[J].Metal Technol.6215-223
[83]T.Sheppard,A.Jackson,Constitutive equations for use in prediction of flow stress during extrusion of aluminium alloys[J].Mater.Sci.Technol.,13
[84]J.van de Langkruis,W.H.Kool and S.van der Zwaag~*,Assessment of constitutive equations in modeling the hot deformability of some overaged AL-Mg-Si alloys with varying solute contents[J],Materials Science and Engineering,1999:266(1-2)135-145
[85]M.Zhou,M.P.Clode,Hot torsion tests to model the deformation behavior of aluminium alloys at hot working temperatures[J],Journal of Materials Processing Technology,1997:72(1)78-85
[86]J.Lof,FEM simulations of aluminium extrusion using an elasto-viscoplastic material model[C],Proceedings of the Seventh International Aluminium Extrusion Technology Seminar,ET2000,Chicago,USA,vol II:157-168
[87]T.ALTAN,S.I.OH,H.L.GEGEL,Metal forming:fundamentals and applications[M],American society for metals,Ohio:Carnes Publication Services,1983
[88]S.V.Patankar,Numerical Heat Transfer and Fluid Flow[M],Hemisphere fluid Publishing Corporation.Taylor& Francis Group,New York,1980
[89]I.Demirdzic,M.Peric,Space conservation law in finite volume calculations of fluid flow[J].International Journal for Numerical Methods in Fluids,1988:1037-1050
[90]J.H.Ferziger,M.Peric,Computational Methods for Fluid Dynamics[M],Springer-Vedag Berlin Heidelberg New York,2002,373-381
[91]E H.Harlow,J.E.Welsh,Numerical calculation time dependent viscous incompressible flow with free surface[J].Phys.Fluids,1965,8:2182-2189
[92]赵大勇,李维仲.VOF方法中几种界面重构技术的比较[J].热科学与技术,2003,2(4):318-322
[93]刘儒勋.数值模拟方法和运动界面追踪[M].合肥:中国科学技术大学出版社,2001:36-195
[94]C.W.Hirt,B.D.Nichols.Volume of Fluid(VOF)Method for the Dynamics of Free Boundaries[J],Journal of Computational Physics,1981,39:201-225
[95]S.Muzaferja,M.Peric.Computation of free-surface flows using finite volume method and moving grids[J],Numerical Heat Transfer Part B,1997,32:369-384
[96]胡影影.三维VOF方法--PLIC3D算法研究[J],力学季刊,2003,24(2):238-243
[97]Denis Gueyffier,J.Li.Volume-of-Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows[J],Journal for Computational Physics,1999,152:423-456
[98]何安定,鹿院卫.两相流动中自由界面的数值模拟[J],油气储运,2000,19(10):15-18
[99]N.Ashgriz,J.Y.Poo.FLAIR:Flux Line-Seglnent Model for Advection and Interface Reconstruction[J],Journal for Computational Physics,1991,93:449-468
[100]J.P.Wang.Finite-volume-type VOF method on dynamically adaptive quadtree grids[J],International Journal for Numerical Methods in Fluids,2004,45:485-508
[101]娄淑梅,赵国群.有限体积法在三维非稳态铝合金挤压过程模拟中的应用[J],机械工程学报,2007,4(32),122-126
[102]娄淑梅,赵国群.铝型材挤压过程有限体积法数值模拟技术研究[J],塑性工程学报,2006,13(4):34-37
[103]Shumei Lou,Guoqun Zhao,Rui Wang,Modeling of Aluminum Profile Extrusion Processes using Finite Volume MethodiC],Proceedings of the 1~(st)International Conference & 7~(th)AUN/SEED-net fieldwise Seminar on Manufacturing and Material Processing,Kuala Lumpur,2006:351-355
[104]娄淑梅,赵国群,王锐,铝型材非稳态挤压有限体积法模拟关键技术研究与应用[J],中国机械工程.2006,17:96-99
[105]戴挺,赵建新,适体坐标系下充型流动模拟的计算方法[J],特种铸造及有色合金,2006,26(2):83-86
[106]M.Manhart,H.Wengle,Large eddy simulation of turbulent boundary layer over a hemisphere.In P.Voke,L.Kleiser,J.P.Chollet(eds),Proc.1~(st)ERCOFTAC Workshop on Direct and Large Eddy Simulation,299-310,Kluwer Academic Publishers,Dordrecht
[107]J.C.Chai,H.S.Lee,S.V.Patankar.Treatment of irregular geometries using a Cartesian coordinates finite-volume radiation heat transfer procedure[J].Numer Heat Transfer,Part B.1994.26:1798-197
[108]J.J.Quirk.An alternative to unstructured grids for computing gas dynamic flows around arbitrary complex two-dimensional bodies[J].Comput Fluids,1994.23(1):125-142
[109]A.Nagano,N.Satofuka,N.Shimomura,A Cartesian grid approach to compressible viscous flow computations.In:Computational fuid dynamics'96,New York:John Wiley & Sons Ltd,1996.540-546
[110]Z.J.Wang,A quadtree based adaptive Cartesian grid flow solver for Navier-Stokes equations[J],Comput Fluids.1998.27(4):529-549
[111]M.Faghri,E.M.Sparrow,A.T.Prata,Finite difference solutions of convection-diffusion problems in irregular domain,using a non-orthogonal coordinate transformation[J].Numer Heat Transfer,1984.7:183-209
[112]Y.Asako,M.Faghri,Heat transfer and fluid flow analysis for an array of interrupted plates,positioned obliquely to the flow direction.In:Proceedings of the Eighth International Heat Tranfer Conference,1986.2:421-427
[113]M.Feghri,Y.Asako,Numerical determination of heat transfer and pressure drop characteristics for a converging-diverging flow channel[J].ASME J Heat Transfer,1987.109:599-605
[114]Y.Asako,M.Faghri,Finite-volume solution for laminar flow and heat transfer in a conjugated duct[J].ASME J Heat Transfer,1987.109:627-634
[115]沙曾鲁.采用适体坐标分析任意几何区域中的流动与换热问题[J],西安交通大学学报,1994,28(5):42-47
[116]揭冠周,介玉新,模拟自由面渗流的适体坐标变换方法[J],清华大学学报,2003,43(2):273-276
[117]介玉新,揭冠周,用适体坐标变换方法求解渗流[J],岩土工程学报,2004,26(1):52-56
[118]茅泽育,采用适体坐标变换方法数值模拟天然河道河冰过程[J],冰川冻土,2003,25(Suppl2):214-219
[119]侯国祥,一种适体坐标下复连通区域的流场数值计算方法[J],水动力学研究与进展,2003,18(1):86-92
[120]M.Hinatsu,J.H.Ferziger,Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique[J].Int.J.Numer.Methods Fluids,1991,13,971-997
[121]R.E.Smith,Algebraic grid generation[J].Appl Math Compt,1982.10-11:137-170
[122]J.F.Thompson,Numerical grid generation foundation and applications[M].New York:North-Holland,1985
[123]A.M.Winslow,Numerical solution of the quasilinear Poisson equation in a nonuniform triangle mesh[J].J Comput Phys,1967,2:49-172
[124]J.E Thompson,F.C.Thames,Automatic numerical geeration of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies[J],J Comput Phys,1974,15:299-319
[125]J.L.Steger,D.S.Chaussee,Generation of body-fitted coordinates using hyperbolic partial differential equations[J].SIAMJ Sci Stat Comp,1980.1(4):431-437
[126]Y.N.Jeng,Y.L.Shu,Grid generation for internal flow problems by methods using hyperbolic equation[J],Numer Heat Transfer.Part b,1995,27:43-61
[127]M.T.Nair,Orthogonal grid generation for Navier-Stokes computation[J],Int J Numer Methods Fluids,1998,28:215-224
[128]S.Nakamura,Marching grid generation using parabolic partial differential equations,In:Thompson J E ed.Numerical grid generation.New York,:Elsevier,1982:79-105
[129]J.K.hodge,Noniterative adaptive grid generation using parabolic partial differential equations.AIAAJ,1987,25(4):542-549
[130]M.Kordula,M.Vinakur,Efficient computation of volume in flow predictions,AIAAJ.,1983,21:917-918
[131]E.Ceretti,E.Taupin,T.Altan,Simulation of metal flow and fracture applications in orthogonal cutting,blanking,and cold extrusion[M],CIRP Ann.1997,46(1):187-190
[132]U.Lang,W.Michaeli,Development of a mathematical model for the calculation of the pressure drop in extrusion dies[C], in Proceedings of the 55~(th) Annual Technical Conference on Manufacturing Research, 1995,Chapter 155, pp. 315-319
[133] G E. Oosterkamp, A. N. Doyle, Improving tooling performance in aluminum extrusion[C], in: Proceedings of the 11~(th) National Conference on Manufacturing Research, 1995, Chapter 155, pp. 17-21
[2]高军,赵国群,李丽华.铝合金型材挤压技术现状与发展趋势[J].汽车工艺与材料,2002(6):21-24
[3]刘静安.铝型材挤压模具设计,制造,使用及维修[M].北京:冶金工业出版社,2002
[4]J.Zhou,L.Li and J.Duszczyk,3D FEM simulation of the whole cycle of aluminium extrusion throughout the transient state and the steady state using the updated Langrangian approach[J],Journal of Material Processing Technology,2003,134(3):383-397
[5]Y.B.Park,D.Y.Yang,Investigation into non-steady-state three-dimensional helical extrusion of twisted sections by the rigid-plastic finite element method[J],Engineering Computations,1997,14(6):649-668
[6]J.Lof,Elasto-viscoplastic FEM simulations of the aluminium flow in the beating area for extrusion of thin-walled sections[J],Journal of Materials Processing Technology,2001,114:174-183
[7]E Halvorsen,T.Aukrust,Studies of the mechanisms for bucking and waving in aluminum extrusion by use of a Lagrangian FEM software[J],International Journal of Plasticity,2006,22(1):158-173
[8]J.Lof,Y.Blockhuis,FEM simulations of the extrusion of the complex thin walled aluminium sections[J],Journal of Materials Processing Technology,2002,122:344-354
[9]P.Zhi,T.Sheppard,Simulation of multi-hole die extrusion[J],Materials Science and Engineering A,2004,367:329-342
[10]于沪平,彭颖红,阮雪榆.平面分流焊合模成型过程的数值模拟[J].锻压技术,1999,24(5):9-11
[11]E Chanda,J.Zhou,J.Duszczyk,FEM analysis of aluminum extrusion through square and round dies[J],Materials and Design,2000,vol.21:323-335
[12]刘汉武,丁惮,崔建忠.铝型材挤压分流组合模有限元分析和计算[J].模具工业,1999,(4):9-11
[13]X.Duan et al.Application of finite element method in the hot extrusion of aluminum alloys[J].Materials Science and Engineering A,2004,369:66-75
[14]N.Hao et al.Numerical design of the die land for shape extrusion[J].Journal of Materials Processing Technology,2000,101:81-84
[15]Z.H.Chen,C.Y.Tang,Simulation of the sheet metal extrusion process by the enhanced assumed strain finite element method[J].Journal of Materials Processing Technology,1999,91:250-256
[16]G.Li,J.T.Jinn,Recent development and applications of three-dimensional finite element modeling in bulk forming processes.Journal of Materials Processing Technology,2001,113:40-45
[17]Xinjian Duan,X.Velay,Application of finite element method in the hot extrusion of aluminium alloys[J],Materials Science and Engineering A,2004,369:66-75
[18]周飞,彭颖红,阮雪榆.铝型材挤压过程有限元数值模拟[J].中国有色金属学报,1998,8(4):637-642
[19]K.Mori,K.Osakada,H.Yamaguchi.Prediction of curvature of an extruded bar with noncircular cross-section by a 3D rigid-plastic finite element method[J].Int.J.Mech Sci.1993,35(10):879-887
[20]R.Shivpuri,S.Momin,T.Altan.Computer aided design of dies to control dimensional quality of extruded shapes[J],CirP Ann.1992,41(1):275-279
[21]Z.Jia.Three-dimensional simulations of the hollw extrusion and drawing using the finite element method.Master's Thesis,Ohio University,Athens,OH,1994
[22]闫洪等.角铝型材挤压过程的数值模拟[J].中国有色金属学报,2001,11(2):202-205
[23]J.Lof.Elasto-viscoplaxtic FEM simulation of the aluminum flow in the bearing area for extrusion thin-walled sections[J].Journal of Materials Processing Technology,2001,114:174-183
[24]赵国群.金属体积塑性成形过程数值模拟技术与仿真系统[J].金属成形工艺,2003,21(5):5-8
[25]Q.Li,C.J.Smith,Finite element investigations upon the influence of pocket die designs on metal flow in aluminum extrusion Part I.Effect of pocket angle and volume on metal flow[J],Journal of Materials Processing Technology,2003,135:189-196
[26]Q.Li,et al.Finite element modeling investigations upon the influence of pocket die designs on metal flow in aluminum extrusion Part II.Effect of pocket geometry configurations on metal flow[J],Journal of Materials Processing Technology,2003,135:197-203
[27]N.Hao,et al.Numerical design of the die land for shape extrusion[J].Journal of Materials Processing Technology,2000,101:81-84
[28]田柱平,郝南海.铝型材挤压模工作带形状的数值设计[J].模具技术,1999,1:19-21
[29]C.Tapas,et al.A comparative study on iso-speed extrusion and isothermal extrusion of 6061 A1 alloy using 3D FEM simulation[J].Journal of Materials Processing Technology,2001,114:145-153
[30]J.Zhou,et al.Computer simulated and experimentally verified isothermal extrusion of 7075 aluminum through continuous ram speed variation.Journal of Materials Processing Technology[J],2004,146:203-212
[31]X.Duan,T.Sheppard.Simulation and control of microstructure evolution during hot extrusion of hard aluminum alloys[J].Materials science and Engineering,2003,A351:282-292
[32]史翔.异型材挤压设计理论及其应用[J].轻合金加工技术,2000,28(1):21-24
[33]颜建辉,柳瑞清.正确选择铝型材挤压速度的方法[J].轻合金加工技术,2002,30(7):36-37
[34]闫洪.工艺参数对铝型材挤压变形规律的影响[J].中国有色金属学报,2002,12(6):1154-1161
[35]闫洪.基于数值模拟的铝型材挤压变形规律的研究[J].锻压机械,2000,5:29-30
[36]J.C.Malas,Effect of microstructural complexity on the hot deformation behavior of aluminum alloy 2024[J],Materials Science and Engineering A,2004,368:41-47
[37]A.H.Clausen,Sensitivity of model parameters in stretch bending of aluminium extrusions[J],International Journal of Mechanical Sciences,2001,47:427-453
[38]R.Hambli.Damage and fracture simulation during the extrusion processes[J],Computer methods in applied mechanics and engineering,2000,186:109-120
[39]T.Aukrust,Coupled FEM and texture modelling of plane strain extrusion of an aluminium alloy[J],International Journal of Plasticity,1997,13(1/2):111-125
[40]R.Couch et al,3D metal forming applications of ALE techniques.In Simulation of Materials Processing:Theory and Applications[C],Shen SF,Dawson PR(eds).Balkema:Rotterdam,1995:401-406
[41]E.H.Atzema,J.Huetink,Finite element analysis of forward/backword extrusion using ALE techniques[C].Proceedings of Iternational Conference of Numerical Methods in Industrial Forming Processes,NUMIFORM 95,Ithaca,1995:383-388
[42]D.Y.Yang,K.Park,Y.S.Kang.Integrated finite element simulation for the hot extrusion of complicated A1 alloy profiles[J],Journal of Materials Processing Technology,2001,111:25-30
[43]H.G.Mooi et al.Simulation of complex aluminum extrusion using an arbitrary Eulerian Lagrangian formulation.In Simulation of Materials Processing:Throry and Applications,Shen SF,Dawson PR(eds).Balkema:Rotterdam,1995:869-874
[44]H.N Bayomi,M.S.Gadala.Simulation of large deformation problems using the arbitrary Lagrangian-Eulerian formulation[C].Proceedings of the European Conference on Computational Mechanics,ECCM'99,Munchen,1999
[45]Y.S.Kang,D.Y.Yang.Rigid-viscoplastic finite element anlysis of hot square die extrusion of complicated profiles with flow guides and lands by arbitrary Langrangian-Eulerian formulation.In Simulation of Materials Processing:Theory and Applications,Shen SF,Dawson PR(eds).Balkema:Rotterdam,1995:841-846
[46]E.H.Atzema.Finite element analysis of forward/backward extrusion using ALE Techniques[C].Proceedings of International Conference of Numerical Methods in Industrial Forming Processes,NUMIFORM 95,Ithaca,1995:383-388
[47]A.Paine,M.Aloe,J.Walters,Simulation of hot extrusion processes[J],Aluminium Extrusion,2002,7(1):39-44
[48]A.E.Tekkaya,Current state and future developments in the simulation of forming processes[C].ICFG 97,Den Bosch,1997
[49]E.D.Vries,P.Ding,Simulation of 3D forging and extrusion problems using a finite volume method[C].Proceeding of 17~(th)MSC JAPAN Users Conference,Tokyo:Msc Software Japan Ltd.,1999,1:155-161
[50]B.V.Mehtaa.3D flow analysis inside shear and streamlined extrusion dies for feeder plate design[J],Journal of Materials Processing Technology,2001,113:93-97
[51]A.J.Williams,T.N.Croft,M.Cross.Computational modelling of metal extrusion and forging process[J],Journal of Materials Processing Technology,2002,125-126:573-582
[52]Wu Xianghong,Zhao Guoqun,Numerical simulation and die structure optimization of an aluminum rectangular hollow pipe extrusion process[J].Materials Science& Engineering A,2006,435-436
[53]Wu xianghong,Zhao Guoqun,Simulation based porthole die structure optimization of an aluminum profile extrusion process[J].Transations of Nonferrous Metals Society of China,2006,16(z3):1261-1264
[54]吴向红,赵国群,孙胜,娄淑梅.挤压速度和摩擦状态对铝型材挤压过程的影响[J],塑性工程学报,2007,(1):36-41
[55]吴向红,赵国群,孙胜,娄淑梅.使用分流组合模的铝型材挤压过程数值模拟[J],中国机械工程,2006,17:106-109
[56]黄克坚,包忠诩,陈泽中,等.有限体积法在挤压模具设计中的运用[J].稀有金属材料与工程,2004,(8):855-857
[57]黄克坚,包忠诩,周天瑞.有限体积数值模拟技术在型材挤压变形规律研究中的运用[J].轻合金加工技术,2003,31(4):29-31
[58]黄克坚,包忠诩.有限体积数值模拟技术在宽展挤压模具设计中的应用[J].塑性工程学报,2005,12(6):34-37
[59]黄光法,林高用.大挤压比铝型材挤压过程中的数值模拟[J].中国有色金属学报,2006,16(5):887-893
[60]周飞,苏丹.有限体积法模拟铝型材挤压成形过程[J].中国有色金属学报,2003,13(1):65-70
[61]周飞,苏丹,彭颖红.有限体积法仿真金属塑性成形的基本理论[J].上海交通大学学报,2002,36(7):915-919
[62]周飞.铝型材挤压有限元/有限体积复合数值模拟技术研究上海交通大学博士学位论文
[63]罗超,李大永.薄壁铝型材挤压成形的一种有效模拟方法[J].上海交通大学学报,2004,38(7):1134-1137
[64]周飞,苏丹.铝型材挤压有限元和有限体积对比模拟[J].上海交通大学学报,2003,37(7):1072-1076
[65]陈泽中,娄臻亮.复杂铝型材挤压成形有限体积仿真[J].上海交通大学学报,2005,39(1):28-40
[66]李大永,王洪俊.薄壁铝型材挤压有限体积分步模拟[J].上海交通大学学报,2005,39(1):6-9
[67]E Zhou,D.Su,FEM and FVM compound numerical simulation of aluminum extrusion processes.[J],Trans.Nonferrous Met.Soc.China,2003,13(2):381-385
[68]Z.Z Chen,Z.L.Lou,Finite volume simulation and mould optimization of aluminum profile extrusion[J],Journal of Materials Processing Technology,2007,190(1-3):382-386
[69]H.K.Versteeg,W.Malalasekera,An Introduction to Computational Fluid Dynamics[M].The Finite Volume Method Approach,Prentice Hall,1996.
[70]王福军.计算流体动力学分析-CFD软件原理与应用[M].清华大学出版社,2004
[71]陶文铨著.计算传热学的近代进展[M].北京:科学出版社,2000.06
[72]朱自强.应用计算流体力学[M].北京:北京航天航空大学出版社,1998
[73]S.V.Pantanker.Numerical Heat Transfer and Fluid Flow[M].McGraw-Hill Book Company,1984
[74](美)帕坦卡著.传热与流体流动的数值计算[M].张政译.北京:科学出版社,1989
[75]陶文铨.数值传热学[M].西安:西安交通大学出版社,2001
[76]彭颖红 金属塑性成形仿真技术[M].上海:上海交通大学出版社,1999
[77]吴向红 铝型材挤压过程有限体积数值模拟及软件开发技术的研究.山东大学博士学位论文
[78]O.C.Zienkiewicz,P.N.Godbole.Flow of Plastic and Viscoplastic Solids with Special Reference to Extrusion and Forming Process.International Journal for Numerical Methods in Engineering,1974,8:3-15
[79]M.S.Eikemo,M.S.Espedal,et al,On the numerical solution of a three dimensional extrusion model[J],Computing and Visualization in Science,1997,1(1)1-14
[80]F.Halvorsen,T.Aukrust,Studies of the mechanisms for bucking and waving in aluminum extrusion by use of a Lagrangian FEM software[J],International Journal of Plasticity,2006:22(1)158-173.
[81]C.M.Sellars,W.J.Tegart,Hot workability.Int.Met.Rev.1972,17:1-24
[82]T.Sheppard,D.Wright.Deformation of flow stress:Part 1 constitutive equations for aluminium alloys at elevated temperatures[J].Metal Technol.6215-223
[83]T.Sheppard,A.Jackson,Constitutive equations for use in prediction of flow stress during extrusion of aluminium alloys[J].Mater.Sci.Technol.,13
[84]J.van de Langkruis,W.H.Kool and S.van der Zwaag~*,Assessment of constitutive equations in modeling the hot deformability of some overaged AL-Mg-Si alloys with varying solute contents[J],Materials Science and Engineering,1999:266(1-2)135-145
[85]M.Zhou,M.P.Clode,Hot torsion tests to model the deformation behavior of aluminium alloys at hot working temperatures[J],Journal of Materials Processing Technology,1997:72(1)78-85
[86]J.Lof,FEM simulations of aluminium extrusion using an elasto-viscoplastic material model[C],Proceedings of the Seventh International Aluminium Extrusion Technology Seminar,ET2000,Chicago,USA,vol II:157-168
[87]T.ALTAN,S.I.OH,H.L.GEGEL,Metal forming:fundamentals and applications[M],American society for metals,Ohio:Carnes Publication Services,1983
[88]S.V.Patankar,Numerical Heat Transfer and Fluid Flow[M],Hemisphere fluid Publishing Corporation.Taylor& Francis Group,New York,1980
[89]I.Demirdzic,M.Peric,Space conservation law in finite volume calculations of fluid flow[J].International Journal for Numerical Methods in Fluids,1988:1037-1050
[90]J.H.Ferziger,M.Peric,Computational Methods for Fluid Dynamics[M],Springer-Vedag Berlin Heidelberg New York,2002,373-381
[91]E H.Harlow,J.E.Welsh,Numerical calculation time dependent viscous incompressible flow with free surface[J].Phys.Fluids,1965,8:2182-2189
[92]赵大勇,李维仲.VOF方法中几种界面重构技术的比较[J].热科学与技术,2003,2(4):318-322
[93]刘儒勋.数值模拟方法和运动界面追踪[M].合肥:中国科学技术大学出版社,2001:36-195
[94]C.W.Hirt,B.D.Nichols.Volume of Fluid(VOF)Method for the Dynamics of Free Boundaries[J],Journal of Computational Physics,1981,39:201-225
[95]S.Muzaferja,M.Peric.Computation of free-surface flows using finite volume method and moving grids[J],Numerical Heat Transfer Part B,1997,32:369-384
[96]胡影影.三维VOF方法--PLIC3D算法研究[J],力学季刊,2003,24(2):238-243
[97]Denis Gueyffier,J.Li.Volume-of-Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows[J],Journal for Computational Physics,1999,152:423-456
[98]何安定,鹿院卫.两相流动中自由界面的数值模拟[J],油气储运,2000,19(10):15-18
[99]N.Ashgriz,J.Y.Poo.FLAIR:Flux Line-Seglnent Model for Advection and Interface Reconstruction[J],Journal for Computational Physics,1991,93:449-468
[100]J.P.Wang.Finite-volume-type VOF method on dynamically adaptive quadtree grids[J],International Journal for Numerical Methods in Fluids,2004,45:485-508
[101]娄淑梅,赵国群.有限体积法在三维非稳态铝合金挤压过程模拟中的应用[J],机械工程学报,2007,4(32),122-126
[102]娄淑梅,赵国群.铝型材挤压过程有限体积法数值模拟技术研究[J],塑性工程学报,2006,13(4):34-37
[103]Shumei Lou,Guoqun Zhao,Rui Wang,Modeling of Aluminum Profile Extrusion Processes using Finite Volume MethodiC],Proceedings of the 1~(st)International Conference & 7~(th)AUN/SEED-net fieldwise Seminar on Manufacturing and Material Processing,Kuala Lumpur,2006:351-355
[104]娄淑梅,赵国群,王锐,铝型材非稳态挤压有限体积法模拟关键技术研究与应用[J],中国机械工程.2006,17:96-99
[105]戴挺,赵建新,适体坐标系下充型流动模拟的计算方法[J],特种铸造及有色合金,2006,26(2):83-86
[106]M.Manhart,H.Wengle,Large eddy simulation of turbulent boundary layer over a hemisphere.In P.Voke,L.Kleiser,J.P.Chollet(eds),Proc.1~(st)ERCOFTAC Workshop on Direct and Large Eddy Simulation,299-310,Kluwer Academic Publishers,Dordrecht
[107]J.C.Chai,H.S.Lee,S.V.Patankar.Treatment of irregular geometries using a Cartesian coordinates finite-volume radiation heat transfer procedure[J].Numer Heat Transfer,Part B.1994.26:1798-197
[108]J.J.Quirk.An alternative to unstructured grids for computing gas dynamic flows around arbitrary complex two-dimensional bodies[J].Comput Fluids,1994.23(1):125-142
[109]A.Nagano,N.Satofuka,N.Shimomura,A Cartesian grid approach to compressible viscous flow computations.In:Computational fuid dynamics'96,New York:John Wiley & Sons Ltd,1996.540-546
[110]Z.J.Wang,A quadtree based adaptive Cartesian grid flow solver for Navier-Stokes equations[J],Comput Fluids.1998.27(4):529-549
[111]M.Faghri,E.M.Sparrow,A.T.Prata,Finite difference solutions of convection-diffusion problems in irregular domain,using a non-orthogonal coordinate transformation[J].Numer Heat Transfer,1984.7:183-209
[112]Y.Asako,M.Faghri,Heat transfer and fluid flow analysis for an array of interrupted plates,positioned obliquely to the flow direction.In:Proceedings of the Eighth International Heat Tranfer Conference,1986.2:421-427
[113]M.Feghri,Y.Asako,Numerical determination of heat transfer and pressure drop characteristics for a converging-diverging flow channel[J].ASME J Heat Transfer,1987.109:599-605
[114]Y.Asako,M.Faghri,Finite-volume solution for laminar flow and heat transfer in a conjugated duct[J].ASME J Heat Transfer,1987.109:627-634
[115]沙曾鲁.采用适体坐标分析任意几何区域中的流动与换热问题[J],西安交通大学学报,1994,28(5):42-47
[116]揭冠周,介玉新,模拟自由面渗流的适体坐标变换方法[J],清华大学学报,2003,43(2):273-276
[117]介玉新,揭冠周,用适体坐标变换方法求解渗流[J],岩土工程学报,2004,26(1):52-56
[118]茅泽育,采用适体坐标变换方法数值模拟天然河道河冰过程[J],冰川冻土,2003,25(Suppl2):214-219
[119]侯国祥,一种适体坐标下复连通区域的流场数值计算方法[J],水动力学研究与进展,2003,18(1):86-92
[120]M.Hinatsu,J.H.Ferziger,Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique[J].Int.J.Numer.Methods Fluids,1991,13,971-997
[121]R.E.Smith,Algebraic grid generation[J].Appl Math Compt,1982.10-11:137-170
[122]J.F.Thompson,Numerical grid generation foundation and applications[M].New York:North-Holland,1985
[123]A.M.Winslow,Numerical solution of the quasilinear Poisson equation in a nonuniform triangle mesh[J].J Comput Phys,1967,2:49-172
[124]J.E Thompson,F.C.Thames,Automatic numerical geeration of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies[J],J Comput Phys,1974,15:299-319
[125]J.L.Steger,D.S.Chaussee,Generation of body-fitted coordinates using hyperbolic partial differential equations[J].SIAMJ Sci Stat Comp,1980.1(4):431-437
[126]Y.N.Jeng,Y.L.Shu,Grid generation for internal flow problems by methods using hyperbolic equation[J],Numer Heat Transfer.Part b,1995,27:43-61
[127]M.T.Nair,Orthogonal grid generation for Navier-Stokes computation[J],Int J Numer Methods Fluids,1998,28:215-224
[128]S.Nakamura,Marching grid generation using parabolic partial differential equations,In:Thompson J E ed.Numerical grid generation.New York,:Elsevier,1982:79-105
[129]J.K.hodge,Noniterative adaptive grid generation using parabolic partial differential equations.AIAAJ,1987,25(4):542-549
[130]M.Kordula,M.Vinakur,Efficient computation of volume in flow predictions,AIAAJ.,1983,21:917-918
[131]E.Ceretti,E.Taupin,T.Altan,Simulation of metal flow and fracture applications in orthogonal cutting,blanking,and cold extrusion[M],CIRP Ann.1997,46(1):187-190
[132]U.Lang,W.Michaeli,Development of a mathematical model for the calculation of the pressure drop in extrusion dies[C], in Proceedings of the 55~(th) Annual Technical Conference on Manufacturing Research, 1995,Chapter 155, pp. 315-319
[133] G E. Oosterkamp, A. N. Doyle, Improving tooling performance in aluminum extrusion[C], in: Proceedings of the 11~(th) National Conference on Manufacturing Research, 1995, Chapter 155, pp. 17-21