任意拉格朗日—欧拉方法及其在二维数值计算中的初步应用
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摘要
本文简要介绍了ALE方法的研究历史、现状和应用前景，详细阐述了ALE方法的基本理论。并在SALE(Simplified ALE，一种简化的ALE方法，网格可以任意运动，但物质界面处的网格仍然以物质速度运动，物质界面始终为Lagrange网格线以保证网格中只有单物质出现)方法框架下，推导并给出了适用于弹塑性流计算的ALE线积分差分格式。采用Laplace反演方法和速度松弛法计算网格速度，实现了网格的自动重分，采用贡献网格法和线性插值完成输运计算。
借助于ALE方法的早期程序SHALE的基本框架，开发研制了二维弹塑性流ALE方法计算程序HEPALE。程序包含用弹塑性本构模型和Grūnrisen状态方程来描述固体材料的行为；采用JWL状态方程来描述爆轰产物的性质；采用程序燃烧法来模拟爆轰波阵面的传播。
应用HEPALE程序对平面碰撞、铜棒碰撞刚性壁(Taylor杆问题)、爆轰波的传播、炸药驱动金属平板和柱壳进行了数值模拟，并与有关理论解析结果或者实验结果以及LS-DYNA程序、Lagrange程序的计算结果进行了比较，符合程度较好。表明本文的计算方法和程序能够用于爆炸力学诸多有关问题的数值计算。与纯Lagrange程序计算的结果相比较，ALE方法在处理大变形问题时有较明显的优势。
The history, development and application of Arbitrary Lagrange-Euler (ALE) are in introduced, and the basic theory of ALE method is also elaborated in detail in this paper. Based on SALE method (Simplified ALE, in which the mesh may move with arbitrarily prescribed velocity with respected to the fluid, and Lagrange interfaces are maintained between cells containing different material.), the line loop integral difference scheme is derived which can be used to calculate two-dimensional elastic-plastic flow. The grid velocity is obtained by using both of so-called Laplace and velocity relaxation methods, and rezone is automatically done. The remap of state variables is calculated with both of donor cell and linear interpolation method.
Based on the SHALE code, the early code of ALE, the two dimensional finite difference elastic-plastic flow code HEPALE has been developed, in which the ideal elastic-plastic model and equation of state of Gruneisen are used to describe the solid material, the programmed burn technique is used to simulate the propagation of detonation, and the equation of state of JWL is used to describe the detonation products.
With the HEPALE code, the simulated results are reported about problems of plate impact, the copper bar impacted the rigid well (Taylor Bar), the propagation of detonation, and the motion of a plane and a cylinder shell driven by explosives, and they are basically in agreement with experiment, as well as one obtained by LS-DYNA. It is shown that HEPALE code is reasonable and can be applied to simulate dynamic phenomena. It is also shown that the HEPALE code has distinct advantage with the pure Lagrange method in simulating the large distortion problems.
借助于ALE方法的早期程序SHALE的基本框架，开发研制了二维弹塑性流ALE方法计算程序HEPALE。程序包含用弹塑性本构模型和Grūnrisen状态方程来描述固体材料的行为；采用JWL状态方程来描述爆轰产物的性质；采用程序燃烧法来模拟爆轰波阵面的传播。
应用HEPALE程序对平面碰撞、铜棒碰撞刚性壁(Taylor杆问题)、爆轰波的传播、炸药驱动金属平板和柱壳进行了数值模拟，并与有关理论解析结果或者实验结果以及LS-DYNA程序、Lagrange程序的计算结果进行了比较，符合程度较好。表明本文的计算方法和程序能够用于爆炸力学诸多有关问题的数值计算。与纯Lagrange程序计算的结果相比较，ALE方法在处理大变形问题时有较明显的优势。
The history, development and application of Arbitrary Lagrange-Euler (ALE) are in introduced, and the basic theory of ALE method is also elaborated in detail in this paper. Based on SALE method (Simplified ALE, in which the mesh may move with arbitrarily prescribed velocity with respected to the fluid, and Lagrange interfaces are maintained between cells containing different material.), the line loop integral difference scheme is derived which can be used to calculate two-dimensional elastic-plastic flow. The grid velocity is obtained by using both of so-called Laplace and velocity relaxation methods, and rezone is automatically done. The remap of state variables is calculated with both of donor cell and linear interpolation method.
Based on the SHALE code, the early code of ALE, the two dimensional finite difference elastic-plastic flow code HEPALE has been developed, in which the ideal elastic-plastic model and equation of state of Gruneisen are used to describe the solid material, the programmed burn technique is used to simulate the propagation of detonation, and the equation of state of JWL is used to describe the detonation products.
With the HEPALE code, the simulated results are reported about problems of plate impact, the copper bar impacted the rigid well (Taylor Bar), the propagation of detonation, and the motion of a plane and a cylinder shell driven by explosives, and they are basically in agreement with experiment, as well as one obtained by LS-DYNA. It is shown that HEPALE code is reasonable and can be applied to simulate dynamic phenomena. It is also shown that the HEPALE code has distinct advantage with the pure Lagrange method in simulating the large distortion problems.
引文
[1] Browne P. L., Rezone, A proposal for accomplishing rezoning in two-dimensional Lagrangian hydrodynamics problems, Report LA-3455-MS, Los Alamos Sci. Lab
[2] 恽寿榕，涂侯杰，梁德寿等，爆炸力学计算方法，北京理工大学出版社，1995
[3] 李德元，徐国荣，水鸿寿等，二维非定常流体力学数值模拟，科学出版社，1987
[4] Noh W. F., CEL: A time-dependent two-space-dimensional coupled Eulerian-Lagrangian code, in: B. Alder, S. Fernbach and M. Rotenberg, Eds. Methods in computational physics 3(Academic press, New York, 1964)
[5] Hirt C. W., Amsden A. A and Cook J. L., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, 1974 14: 227-253
[6] Amsden A. A. and Hirt C.W., YAQUI: an Arbitrary Lagrangian-Eulerian computer program for fluid flow at all speed, Los Alamos Scientific Laboratory report, LA-8095(June 1980)
[7] Amsden A.A., Ruppel H. M. and Hirt C. W., SALE: A simplified ALE computer program for fluid flow at all Speed, LA-8095, June 1980
[8] Amsden A. A., Ruppel H M et al, SALE-3D: A simplified ALE computer program for fluid flow at all speed, Los Alamos Scientic Laboratory, Los Alamos, NM, 1980
[9] Amsden A. A., Ruppel H. M. et al, SALE-3D: A simplified ALE computer program for calculating three-dimensional fluid flow at all speed, Los Alamos Scientic Laboratory, Los Alamos, NM, 1981
[10] Demuth R. B., Margolin L. G., Nichols B. D. and Smith B. W., SHALE: A computer program for solid dynamics, Los Alamos National Laboratory, New Mexico 87545, LA-10236
[11] Hughes T. J. R., Liu W. k. and Zinnerman T. K., Lagrangian-Eulerian finite element method for incompressible viscous flows, Comput. Meths. Appl Mech. Engrg., 1981; 29: 329～349
[12] Ramaswamy B., Kawahara M., Arbitrary Lagrangiano-Eulerian finite element method for unsteady, convective incompressible viscous free surface fluid flow, Int. J. Num. Methods, 1987; 7: 1053～1075
[13] Huerta A. and Liu W. K., Viscous flow with large free surface motion, Comput. Meths. Appl. Mech. Engrg. 1988; 69: 277～324
[14] Donea J., Fasoli-Stellap et al, Lagrangian and Eulerian finite elememt techniques for transient fluid-structure interaction problems, in: Trans. SmiRT-4, paper B1/2, USA, 1977; 15～19
[15] Belytschko T. and Kennedy J. M., Computer models for subassembly simulation, Nucl.
Engrg. Design, 1978; 49: 17～38
[16] Belytschko T., Kennedy J. M. et al, Quasi-Eulerian finite element formulation for fluid-structure interaction, J. Pressure Vessel Techno. ASME, 102(1980) 62-69
[17] Haber R. B. and Abel J., Contact slip analysis using mixed displacements, J. Engrg. Mech. ASCE, 1983; 109(2): 411～429
[18] Haber R. B. and Koh H. M., Explicit expressions for energy release rates using virtual extension, Int. J. Num. Meth. Engrg., 1985; 21: 301～315
[19] Haber R. B. and Hariandja B H, An arbitrary Lagrangian-Eulerian finite element approach to large deformation frictional contact, Computers & Structure, 1985; 20: 193～201
[20] Haber R. B., A mixed Eulerian-Lagrangian displacement model for large-deformation analysis in solid mechanics, Comput. Methods Appl. Mech. Engeg., 1984; 43: 277～292
[21] Huetink J., Analysis of metal forming processes based on a combined Euerian-Lagrangian finite element formulation, in: J. F. T. Pittman et al. eds. Numerical Methods in Industrial Forming Processes, Pineridge Press, Swamsea, U. K., 1982; 501～509
[22] Huetink J., Vreede P. T. et al, processes in mixed Eulerian-Lagrangian finite element simulation of forming process, Int. J. Num. Meth. Engrg., 1990; 30: 1441～1457
[23] Liu W. K., Chen J. S. et al, Adaptive ALE finite element with particular reference to external work rate on fractional interface, Comput. Meth. Appl. Mech. Engeg., 1991; 93: 189～216
[24] Schreurs P. J. G., Veldpaus F. E., et al, Simulation of forming processes, using the arbitrary Eulerian-Lagrangian formulation, Comput. Meth. Appl. Mech. Engeg., 1986; 58: 19～36
[25] Huetink van J., der Lugt, et al, A mixed Eulerian-Lagrangian contact element to describe boundary and interface behavior in the forming, Proc. NUMETA, 1987, 1, Nijhop, To the Netherlands
[26] Huetink J., Vreede P. T. et al, The simulation of contact problems in the forming processed using Arbitrary Eulerian-Lagrangian finite element methods, in: E. G. Thompson et al, eds. Numerical Methods in Industrial Forming Processes, Balkema, Roterdam, 1989; 549～554
[27] Liu W. K., Chen H., et al, Arbitrary Lagrangian-Eulerian stress updated for forming simulation, Proc. Symp. Recent Advances in Inelastic Analysis, ASME, Boston, USA, 1987
[28] Liu W. K., Cheng H., et al, Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua, Comput. Meth. Appl. Mech. Engeg., 1988; 68: 259～310
[29] Benson D. J., Adding ALE capability to DYNA2d: experiences and conclusions,
post-conference on IMPACT 9th Int. Conf. On Structural Mech. in Reactor Technology, Switzerland, 1987
[30] Benson D. J., An efficient, accurate, simple ALE method for nonlinear finite element programs, Comput. Meth. Appl. Mech. Engeg., 1989: 72:305～350
[31] James S. P. and Daniel E. C., MultiMaterial ALE methods in unstructured Grids, Sandia National Laboratories Albuquerque, New Mexico 87185-0819
[32] James S. P. and Kent G. B., RHALE: A MMALE shock physics code for arbitrary meshes, Sandia National Laboratories, Albuquerque, NM 87185, USA
[33] Chhabildas L.C., Konrad C.H., Mosher D.A. et al, A methodology to validate 3D arbitrary Langrangian-Eulerian codes with applications to ALEGRA, Sandia National Laboratories, Shock Physics Applications Department, Albuquerque, New Mexico 87185-1181
[34] Liu W. K., Belytschko T., et al, Arbitrary Lagrangian-Eulerian finite element method for path-dependent materials, Comput. Meth. Appl. Mech. Engeg., 1988; 58: 227～245
[35] Harlow F. H., Numerical methods for fluid dynamics, an annotated bibliography, Report LA-4281, Los Alamos Sci. Lab
[36] Hertel E. S. Jr., A survey of numerical methods for shock physics applications, Sandia National Laboratories, Albuquerque, New Mexico 87185-0819 SAND97-1015C
[37] Christiaan Stoker, Developments of the arbitrary Lagrangian-Eulerian method in non-linear solid mechanics applications to forming processes, Copyright 1999 by H.C.Stoker, Enschede, The Netherlands, Printed by FEBO druk, Enschede
[38] Ponthot, J., Efficient mesh management in Eulerian-Lagrangian method for large deformation analysis, Proceedings of the 3rd international conference on numerical methods in industrial forming processes 1989; 203～210
[39] Brackbill J. U. and Saltzman J. S., Adaptive zoning for singular problems in two dimensions, Journal of computational Physics, 1982; 46:342～368
[40] Dwyer H. A., Kee R. J. and Sanders S. R., An adaptive grid method for problems in fluid dynamics and heat transfer, AIAA 79; 1464
[41] Winslow A. M., Numerical solution of the quasi-linear Poisson equation in a non-uniform triangular mesh, J. Phys., Vol. 1, P.149
[42] Chu W. H., Development of a general finite difference approximation for a general domain, part Ⅰ. machine transformation, J. Comp. Phy., 1971; 8: 392
[43] Winslow A. M., Journal of computational Physics, 1966; 1: 149
[44] Wilkins M. L., Calculation of elastic-plastic flow, eds. Methods in Computational physics, Volume 3
[45] 王继海，多项式形式物态方程及其等熵线，爆炸与冲击，第12卷第一期，1992年1月
[46] Robert R. K., Warhead simulation techniques: hydrocodes, tactical missile warheads,Edited by Joseph Carleone, pp272
[47] Michael E. B., Detonation of high explosive in Lagrangian hydrodynamic codes using the programmed burn technique, Los Alamos Scientific Laboratory report LA-6097-MS,December 1975
[48] Mcqueen R. G., et al, Equation of state of solids form shock wave study, High-Velocity Impact Phenomena, Edited by Ray Kinslow 1970, 293～417
[49] Charles E.M., Las Alamos Shock wave Profile Data, University of California press, Los Angeles, 1982
[50] 庄仕明，刘文翰，孙承纬，点引爆圆板装药驱动飞板的实验及二维数值模拟，爆炸与冲击，爆轰论文集，第二卷：202～209
[51] Margolin, L. g., and B. D. Nichols, Momentum Control Volumes for finite difference codes, Proc. Of the third International conference on Numerical Methods in Laminar andturbulent Flow, Seattle, Washington, 1985
[2] 恽寿榕，涂侯杰，梁德寿等，爆炸力学计算方法，北京理工大学出版社，1995
[3] 李德元，徐国荣，水鸿寿等，二维非定常流体力学数值模拟，科学出版社，1987
[4] Noh W. F., CEL: A time-dependent two-space-dimensional coupled Eulerian-Lagrangian code, in: B. Alder, S. Fernbach and M. Rotenberg, Eds. Methods in computational physics 3(Academic press, New York, 1964)
[5] Hirt C. W., Amsden A. A and Cook J. L., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, 1974 14: 227-253
[6] Amsden A. A. and Hirt C.W., YAQUI: an Arbitrary Lagrangian-Eulerian computer program for fluid flow at all speed, Los Alamos Scientific Laboratory report, LA-8095(June 1980)
[7] Amsden A.A., Ruppel H. M. and Hirt C. W., SALE: A simplified ALE computer program for fluid flow at all Speed, LA-8095, June 1980
[8] Amsden A. A., Ruppel H M et al, SALE-3D: A simplified ALE computer program for fluid flow at all speed, Los Alamos Scientic Laboratory, Los Alamos, NM, 1980
[9] Amsden A. A., Ruppel H. M. et al, SALE-3D: A simplified ALE computer program for calculating three-dimensional fluid flow at all speed, Los Alamos Scientic Laboratory, Los Alamos, NM, 1981
[10] Demuth R. B., Margolin L. G., Nichols B. D. and Smith B. W., SHALE: A computer program for solid dynamics, Los Alamos National Laboratory, New Mexico 87545, LA-10236
[11] Hughes T. J. R., Liu W. k. and Zinnerman T. K., Lagrangian-Eulerian finite element method for incompressible viscous flows, Comput. Meths. Appl Mech. Engrg., 1981; 29: 329～349
[12] Ramaswamy B., Kawahara M., Arbitrary Lagrangiano-Eulerian finite element method for unsteady, convective incompressible viscous free surface fluid flow, Int. J. Num. Methods, 1987; 7: 1053～1075
[13] Huerta A. and Liu W. K., Viscous flow with large free surface motion, Comput. Meths. Appl. Mech. Engrg. 1988; 69: 277～324
[14] Donea J., Fasoli-Stellap et al, Lagrangian and Eulerian finite elememt techniques for transient fluid-structure interaction problems, in: Trans. SmiRT-4, paper B1/2, USA, 1977; 15～19
[15] Belytschko T. and Kennedy J. M., Computer models for subassembly simulation, Nucl.
Engrg. Design, 1978; 49: 17～38
[16] Belytschko T., Kennedy J. M. et al, Quasi-Eulerian finite element formulation for fluid-structure interaction, J. Pressure Vessel Techno. ASME, 102(1980) 62-69
[17] Haber R. B. and Abel J., Contact slip analysis using mixed displacements, J. Engrg. Mech. ASCE, 1983; 109(2): 411～429
[18] Haber R. B. and Koh H. M., Explicit expressions for energy release rates using virtual extension, Int. J. Num. Meth. Engrg., 1985; 21: 301～315
[19] Haber R. B. and Hariandja B H, An arbitrary Lagrangian-Eulerian finite element approach to large deformation frictional contact, Computers & Structure, 1985; 20: 193～201
[20] Haber R. B., A mixed Eulerian-Lagrangian displacement model for large-deformation analysis in solid mechanics, Comput. Methods Appl. Mech. Engeg., 1984; 43: 277～292
[21] Huetink J., Analysis of metal forming processes based on a combined Euerian-Lagrangian finite element formulation, in: J. F. T. Pittman et al. eds. Numerical Methods in Industrial Forming Processes, Pineridge Press, Swamsea, U. K., 1982; 501～509
[22] Huetink J., Vreede P. T. et al, processes in mixed Eulerian-Lagrangian finite element simulation of forming process, Int. J. Num. Meth. Engrg., 1990; 30: 1441～1457
[23] Liu W. K., Chen J. S. et al, Adaptive ALE finite element with particular reference to external work rate on fractional interface, Comput. Meth. Appl. Mech. Engeg., 1991; 93: 189～216
[24] Schreurs P. J. G., Veldpaus F. E., et al, Simulation of forming processes, using the arbitrary Eulerian-Lagrangian formulation, Comput. Meth. Appl. Mech. Engeg., 1986; 58: 19～36
[25] Huetink van J., der Lugt, et al, A mixed Eulerian-Lagrangian contact element to describe boundary and interface behavior in the forming, Proc. NUMETA, 1987, 1, Nijhop, To the Netherlands
[26] Huetink J., Vreede P. T. et al, The simulation of contact problems in the forming processed using Arbitrary Eulerian-Lagrangian finite element methods, in: E. G. Thompson et al, eds. Numerical Methods in Industrial Forming Processes, Balkema, Roterdam, 1989; 549～554
[27] Liu W. K., Chen H., et al, Arbitrary Lagrangian-Eulerian stress updated for forming simulation, Proc. Symp. Recent Advances in Inelastic Analysis, ASME, Boston, USA, 1987
[28] Liu W. K., Cheng H., et al, Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua, Comput. Meth. Appl. Mech. Engeg., 1988; 68: 259～310
[29] Benson D. J., Adding ALE capability to DYNA2d: experiences and conclusions,
post-conference on IMPACT 9th Int. Conf. On Structural Mech. in Reactor Technology, Switzerland, 1987
[30] Benson D. J., An efficient, accurate, simple ALE method for nonlinear finite element programs, Comput. Meth. Appl. Mech. Engeg., 1989: 72:305～350
[31] James S. P. and Daniel E. C., MultiMaterial ALE methods in unstructured Grids, Sandia National Laboratories Albuquerque, New Mexico 87185-0819
[32] James S. P. and Kent G. B., RHALE: A MMALE shock physics code for arbitrary meshes, Sandia National Laboratories, Albuquerque, NM 87185, USA
[33] Chhabildas L.C., Konrad C.H., Mosher D.A. et al, A methodology to validate 3D arbitrary Langrangian-Eulerian codes with applications to ALEGRA, Sandia National Laboratories, Shock Physics Applications Department, Albuquerque, New Mexico 87185-1181
[34] Liu W. K., Belytschko T., et al, Arbitrary Lagrangian-Eulerian finite element method for path-dependent materials, Comput. Meth. Appl. Mech. Engeg., 1988; 58: 227～245
[35] Harlow F. H., Numerical methods for fluid dynamics, an annotated bibliography, Report LA-4281, Los Alamos Sci. Lab
[36] Hertel E. S. Jr., A survey of numerical methods for shock physics applications, Sandia National Laboratories, Albuquerque, New Mexico 87185-0819 SAND97-1015C
[37] Christiaan Stoker, Developments of the arbitrary Lagrangian-Eulerian method in non-linear solid mechanics applications to forming processes, Copyright 1999 by H.C.Stoker, Enschede, The Netherlands, Printed by FEBO druk, Enschede
[38] Ponthot, J., Efficient mesh management in Eulerian-Lagrangian method for large deformation analysis, Proceedings of the 3rd international conference on numerical methods in industrial forming processes 1989; 203～210
[39] Brackbill J. U. and Saltzman J. S., Adaptive zoning for singular problems in two dimensions, Journal of computational Physics, 1982; 46:342～368
[40] Dwyer H. A., Kee R. J. and Sanders S. R., An adaptive grid method for problems in fluid dynamics and heat transfer, AIAA 79; 1464
[41] Winslow A. M., Numerical solution of the quasi-linear Poisson equation in a non-uniform triangular mesh, J. Phys., Vol. 1, P.149
[42] Chu W. H., Development of a general finite difference approximation for a general domain, part Ⅰ. machine transformation, J. Comp. Phy., 1971; 8: 392
[43] Winslow A. M., Journal of computational Physics, 1966; 1: 149
[44] Wilkins M. L., Calculation of elastic-plastic flow, eds. Methods in Computational physics, Volume 3
[45] 王继海，多项式形式物态方程及其等熵线，爆炸与冲击，第12卷第一期，1992年1月
[46] Robert R. K., Warhead simulation techniques: hydrocodes, tactical missile warheads,Edited by Joseph Carleone, pp272
[47] Michael E. B., Detonation of high explosive in Lagrangian hydrodynamic codes using the programmed burn technique, Los Alamos Scientific Laboratory report LA-6097-MS,December 1975
[48] Mcqueen R. G., et al, Equation of state of solids form shock wave study, High-Velocity Impact Phenomena, Edited by Ray Kinslow 1970, 293～417
[49] Charles E.M., Las Alamos Shock wave Profile Data, University of California press, Los Angeles, 1982
[50] 庄仕明，刘文翰，孙承纬，点引爆圆板装药驱动飞板的实验及二维数值模拟，爆炸与冲击，爆轰论文集，第二卷：202～209
[51] Margolin, L. g., and B. D. Nichols, Momentum Control Volumes for finite difference codes, Proc. Of the third International conference on Numerical Methods in Laminar andturbulent Flow, Seattle, Washington, 1985