Bulk-metal forming processes from computational modelling via sensitivity analysis to tool shape optimization

• Autorzy: Antunez H. J.
• Języki publikacji: EN

Abstrakty

EN

Within the proceding framework, the present work is a collection of developments performed at the Institute of Fundamental Technological Research of the Polish Academy of Science. All of them are concerning one specific way of modelling metal forming processes; this is especially suited for hot forming condition, and has conceived for steady-state processes, although it can be applied in transient ones as well. The naturally suited application of this model is extrusion; however, it can be used also in free forging, cutting and rolling (including seamless tube rolling), for which results are shown as well. Stationary and transient processes are separately presented here; not only for the sake of clarity of the exposition and for historical reasons, but to show the analysis problem, sensitivity analysis and shape optimization as successive steps within a logical line of thinking. Essentially this process was followed in the computer simulation of metal forming, feeding into each step all the results obtained in the proceding ones. In this work, the steady state is presented first, starting by the flow approach. Following, the descretization by finite elemtens is given. Here the different features accounted for in the model are explained. One the analysis model is complete, the sensitivity is ready to be introduced. First, parametric sensitivity is discussed, what is incidentally useful to show the available methods for sensitivity analysis. Shape sensitivity is considered next, followed by the optimization algorithm which finds, according to given criteria and design restrictions, the optimum design. Afterwards transient processes are considered. A full transient formulation is concidered and used to obtain an incremental method that makes use of linear elements due to a proper time-step splitting. Attention is focused on the full explicit version. Further, sensitivity analysis within such model and discretization is shown. In addition, the pseudo-concentration method is briefly revisited and used in connection to Fourier series expansion of the problem of seamless tube rolling. Some additional -but significant- topics are discussed in the course of the main presentation. The simulation of almost perfect plasticity poses the problem of uniqueness (or its lack), and this is discussed in the context of cutting simulation. Pressure stabilization is necessary to apply a friction model based on a Coulomb-type law. In this context a bilinear interpolation is introduced, which takes advantages of a method for similar pressure stabilization used in fluid mechanics. Sensitivity analysis suggests that its results can have an additional application in evaluating the effect on the solution of numerical parameters needed in some models. This is the case of the upwind parameters in coupled thermo-mechanical problems and the time step in time integration of transient processes. The introduction of shape sensitivity analysis of fomring processes gives the occasion to consider the extension to large displacements of the two available methods. In the shape optimization part, the problem of shape parameterization requires special attention. Two different techniques of interpolating points in the discretized domain in terms of the design parameters are proposed.

Słowa kluczowe

EN

computer science; metallurgy; product engineering

PL

informatyka; hutnictwo; technologia mechaniczna

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Prace Instytutu Podstawowych Problemów Techniki PAN
• Rocznik: 2001
• Tom: nr 1
• Strony: 3--207
• Opis fizyczny: Bibliogr. 123 poz.

Twórcy

autor

Antunez H. J.

• Instytut Podstawowych Problemów Techniki PAN

Bibliografia

• [1] H.J. Antunez. Andlisis por elementos finitos del conformado de metales, eon ori- entación a la laminación de tubos sin costura. PhD thesis, Universidad Nacional de Córdoba, Argentina, 1990.
• [2] H.J. Antunez. Non-unique numerical Solutions in visco-plasticity. Eng. Trans., 40:525-536, 1992.
• [3] H.J. Antunez. Thermo-mechanical modelling and sensitivity analysis for metal- forming operations. Comput. Meth. Appl. Mech. Engrg., 161:113-125, 1998.
• [4] H.J. Antunez. Linear elements for metal forming problems within the flow ap- proach. Comput. Meth. Appl. Mech. Engrg., 190(5-7):783-801, 2000.
• [5] H.J. Antunez. Sensitivity analysis of transient metal forming problems with incompressible linear elements. Comput. Assisted Mech. Eng. Sci., 7(4):449-460, 2000.
• [6] H.J. Antunez and S. Idelsohn. Topics in numerical solution of isothermal and thermal-coupled forming processes. Latm Am. Appl. Res., 20:69-83, 1990.
• [7] H.J. Antńnez and S. Idelsohn. Using pseudo-concentrations in the analysis of transient metal forming processes. Eng. Comput., 9(5):547-559, 1992.
• [8] H.J. Antunez and S. Idelsohn. Transient analysis of tube rolling processes by a semi-analytical formulation. Int. j. numer, methods eng., 37:3621-3632, 1994.
• [9] H.J. Antunez, S. Idelsohn and N. Dvorkin. Metal forming analysis by fourier series expansion and further uaes of pseudo-concentrations. Comput. Struci., 44(l-2):435 451, 1992.
• [10] H.J. Antunez and M. Kleiber. Parameter sensitivity of metal forming processes. Comput. Assisted Mech. Eng. Sci., 3:263-282, 1996.
• [11] H.J. Antunez and M. Kleiber. Sensitivity analysis of metal forming processes involving frictional contact in steady State. J. Mater. Proc. Technology, 60:485 491, 1996.
• [12] H.J. Antunez and M. Kleiber. Sensitivity of forming processes to shape param- eters. Comput. Meth. Appl. Mech. Engrg., 137:189-206, 1996.
• [13] H.J. Antunez, M. Kleiber and W. Sosnowski. Shape and non-shape sensitivity and optimization of metal forming processes. In E. Ońate, D.R.J. Owen and E. Hinton, editors, Computational Plasticity, Proc. COMPLAS VI, pages 783-791. Barcelona, Pineridge Press, 1997.
• [14] H.J. Antunez, O.C. Zienkiewicz, R.L. Taylor and J. Rojek. Explicit formulation and incompressible linear elements for metal forming problems. In S.R. Idelsohn, E. Ońate and E.N. Dvorkin, editors, Computational Mechanice - New trends and applications, WCCM IV, Buenos Aires, 29 June 2July 1998, Barcelona, 1988. CIMNE - IACM.
• [15] J.S. Arora. An exposition of the materiał derivative approach for structural shape sensitivity analysis. Comput. Meth. Appl. Mech. Engrg., 105:41-62, 1993.
• [16] J.S. Arora and J.B. Cardoso. Variational principle for shape sensitivity analysis. Al A A J., 30:538-547, 1992.
• [17] J.S. Arora, T.H. Lee and J.B. Cardoso. Structural shape sensitivity analysis: relationship between materiał derivative and control volume approaches. Al A A J., 30:1638-1648, 1992.
• [18] J.S. Badrinaraganan and N.B. Zabaras. A sensitivity analysis for the optimal design for metal-forming processes. Comput. Meth. Appl. Mech. Engrg., 129:319-348, 1996.
• [19] T. Belytchko, D.P. Flanagan and J.M. Kennedy. Finite element methods with user controlled mesh for fluid structure interaction. Comput. Meth. Appl. Mech. Engrg., 33:669-688, 1982.
• [20] J.A. Bennett and M.E. Botkin. Shape optimization with geometrie structural description and adaptive refinement. AIAA J., 23:458-464, 1985.
• [21] J.A. Bennett and M.E. Botkin, editors. The optimum shape: automated structural design. Plenum Press, New York, 1986.
• [22] J. Bonet and A.J. Burton. A simple average nodal pressure tetrahedral ele¬ment for incompressible and nearly incompressible dynamie explicit applications. Communications in Numerical Methods in Engtneering, 14:437-449, 1998.
• [23] V. Braibant and C. Fleury. Shape optimal design: a performing cad oriented formulation. In AIAA Papers 84-0857, 1984.
• [24] A.N. Brooks and T.J.R. Hughes. Streamline upwind Petrov-Galerkin formula- tions for conyection dominated flows with particular emphasis on the incompress¬ible Navier-Stokes eąuations. Comput. Meth. Appl. Mech. Engrg., 32:199-259, 1982.
• [25] S.M. Byon and S.M. Hwang. Process optimal design in non-isothermal, steady-state metal forming by the finite element method. Int. j. numer, methods eng., 46(7): 1075-1100, 1999.
• [26] A. Chandra and S. Mukherjee. A finite element analysis of metal-forming problems with an elastic-viscoplastic materiał. Int. j. numer, methods eng., 20:1613-1628, 1984.
• [27] J-H. Cheng and N. Kikuchi. An analysis of metal forming processes using large deformation elastic-plastic formulations. Comput. Meth. Appl. Mech. Engrg., 49:71-108, 1985.
• [28] J.L. Chenot and M. Bellet. The viscoplastic approach for the finite element modelling of metal forming processes. In P. Hartley, I. Pillinger and C.E.N. Sturges, editors, Numencal modelling of materiał deformation processes, pages 179-224, London-Berlin, 1992. Springer-Verlag.
• [29] K.K. Choi. Shape design aensitivity analysis of displacement and stress constraints. Struct. Mech., 13(1):27—41, 1985.
• [30] K.K. Choi and E.J. Haug. Shape design sensitivity analysis of elastic structures. J. Struć. Mech., 11:231-269, 1983.
• [31] K.K. Choi and H.G. Seong. A domain method for shape design sensitivity analysis of built-up structures. Comput. Meth. Appl. Mech. Engrg., 57:1-15, 1986.
• [32] A.J. Chorin. A numerical method for solving incompressible viscous problems. J. Comput. Phys., 2:12-26, 1967.
• [33] A.J. Chorin. On the convergence of discrete approximation to the Navier-Stokes equations. Math. Comput., 23, 1969.
• [34] S.H. Chung and S.M. Hwang. Optimal process design in non-isothermal, non-steady metal forming by the finite element method. Int. j. numer, methods eng., 42(8):1343-1390, 1998.
• [35] G.C. Cornfield and R.H. Johnson. Theoretical prediction of plastic flow in hot rolling including the effect of various temperaturę distributions. J. Iron and Steel Inst., 211:567-573, 1973.
• [36] J.B. Dalin and J.L. Chenot. Finite element computation of bulging in continu- ously cast Steel with a viscoplastic model. Int. j. numer, methods eng., 25:147-163, 1988.
• [37] P.R. Dawson and E G. Thompson. Finite element analysis of steady state elasto- visco-plastic flow by the initial stress rate method. Int. j. numer, methods eng., 12:47-57, 1978.
• [38] K. Denis and R.T. Haftka. Two approaches to sensitmty analysis for shape variation of structures. J. Mech. Struć. Machines, 16:501-522, 1989.
• [39] K. Dems and Z. Mróz. Variational approach by means of adjoint Systems to structural optimization and sensitmty analysis. II. Structure shape variation. Solids Struct., 20(6):527-552, 1984.
• [40] K. DemB and Z. Mr6z. Shape sensitivity analysis and optimal design of physically nonlinear plates. Arch. Mech., 41:481-501, 1989.
• [41] J. Van der Lugt and J. Huetink. Thermal mechanically coupled finite element analysis in metal forming processes. Comput. Meth. Appl. Mech. Engrg., 54:154-160, 1986.
• [42] J. Donea, S. Giuliani and J.I. Halleux. The arbitrary lagrangian eulerian finite element method for transient dynamic fluid structure interaction. Comput. Meth. Appl. Mech. Engrg., 33, 1982.
• [43] R. Fletcher and C.M. Reeves. Function minimization by conjugate gradients. Computer Journal, 7:149-154, 1964.
• [44] L. Fourment, T. Balan and J. Chenot. Shape optimization for the forging process. In D.R. J. Owen, E. Onate and E. Hinton, editors, Computational Plasticity, Proc. COMPLAS IV, pages 1369-1381. Barcelona, Pineridge Press, 1995.
• [45] L. Fourment, T. Balan and J.L. Chenot. Optimal design for non-steady-state metal forming processes-ii. application of shape optimization in forging. Int. j. numer. methods eng., 39(l):51-65, 1996.
• [46] L. Fourment and J.L. Chenot. Optimal design for non-steady-state metal forming processes-i. shape optimization method. Int. j. numer. methods eng., 39(1):33-50, 1996.
• [47] L.P. Franca, T.J.R. Hughes and M. Ballestra. Two classes of mixed finite element methods. Comput. Meth. Appl. Mech. Engrg., 69:89-129, 1988.
• [48] M. Gadala, G. Oravas and M. Dokainish. A consistent eulerian formulation of large deformation problems in statics and dynamics. Int. J. Non-Linear Mechanics, 18(1):21—35, 1983.
• [49] J.C. Gelin and B.V. Nguegang. Modelling of thermomechanical coupling during hot forming of viscoplastic materials. J. Mater. Proc. Technology, 60:441-446, 1996.
• [50] V. Girault and P.A. Raviart. Finite element approximation of the Navier-Stokes equations. Springer Verlag, 1980.
• [51] G.Y. Goon, P.I. Poluchin, W.P. Poluchin and B.A. Prudcowsky. The plastic deformation of metals. Metallurgica (in Russian), 1968.
• [52] A.M. Habraken and S. Cescotto. An automatic remeshing technique for finite element simulation of forming processes. Int. j. numer. methods eng., 30:1503-1525, 1990.
• [53] R.T. Haftka and R.K. Grandhi. Structural shape optimization - a survey. Comput. Meth. Appl. Mech. Engrg., 57:91-106, 1986.
• [54] C.S. Han, R.V. Grandhi and R. Srinivasan. Optimum design of forging die shapes using nonlinear finite element analysis. AIAA J., 31:774-781, 1993.
• [55] E.J. Haug, K.K. Choi and V. Komkov. Design Sensitivity Analysis of Structural Systems. Series in Math. Sci. Eng. Academic Press, Orlando, 1986.
• [56] J.C. Heinrich, P.S. Huyakorn, A.R. Mitchell and O.C. Zienkiewicz. An ’upwind’ finite element scheme for two-dimensional convective transport equation. Int. j. numer. methods eng., 11:131-143, 1977.
• [57] J.C. Heinrich and O.C. Zienkiewicz. Quadratic finite element schemes for two di-mensional convective transport problems. Int. j. numer. methods eng., 11:1831—1844, 1977.
• [58] H.R. Hill. The mathematical theory of plasticity. Clarendon Press, Oxford, 1950.
• [59] R. Hill. On the state of stress in a plastic-rigid body at the yield point. Phil. Mag., 42:868-875, 1951.
• [60] N.J. Hoff. Approximate analysis of structures in the presence of moderately large deformation. Quart. Appl. Math., 2(1) :49, 1954.
• [61] J.W. Hou, J.L. Chen and J.S. Sheen. A computational method for shape optimization. In AIAA Papers 85-0713, 1985.
• [62] J. Huetink. Analysis of metal forming processes based on a combined eulerian- lagrangian finite element formulation. In J.F. Pittman, R.D. Wood, J.M. Alexander and O.C. Zienkiewicz, editors, Numerical Methods in Industrial Forming Processes, pages 501-509. Pineridge Press, 1983.
• [63] T. J.R. Hughes, L.P. Franca and M. Balestra. A new finite element formulation for computational fluid dynamic: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation for the Stokes problems accommodating equal-order interpolations. Comput. Meth. Appl. Mech. Engrg., 59:85-99, 1986.
• [64] S. Idelsohn, M. Storti and N. Nigro. Stability analysis of mixed finite element formulations with special mention of equal order interpolations. Int. J. Num. Methd. Fluids, 20:1003-1022, 1985.
• [65] B.M. Irons. A frontal solution program for finite elements. Int. j. numer. methods eng., 2:6-32, 1970.
• [66] M.S. Joun and S.M. Hwang. Die shape optimal design in three-dimensional shape metal extrusion by the finite element method. Int. j. numer. methods eng., 41:311-335, 1998.
• [67] A. Karagiannis and H. Mavridis. A finite element convergence study for shearthinning flow problems. Int. J. Num. Meth. Fluids, 8:128-138, 1988.
• [68] M. Kawahara and K. Ohmiya. Finite element analysis of density flow using the velocity correction method. Int. J. Num. Methd. Fluids, 5:981-993, 1985.
• [69] P. Kelly, A. Nakazawa, O.C. Zienkiewicz and J.C. Heinrich. A note on upwinding and anisotropic balancing dissipation in finite elements approximations to convective diffusion problems. Int. j. numer. methods eng., 15:1705-1711, 1980.
• [70] M. Kleiber, H.J. Antunez, T.D. Hien and P. Kowalczyk. Parameter Sensitivity in Nonlinear Mechanics. Wiley, Chichester, 1997.
• [71] M. Kleiber and T.D. Hien. Nonlinear dynamics of complex axisymmetric structures under arbitrary loads. Comput. Meth. Appl. Mech. Engrg., 37:93-107, 1983.
• [72] M. Kleiber, T.D. Hien, H.J. Antunez and P. Kowalczyk. Parameter sensitivity of elasto-plastic response. Engng. Computations, 12:263 280, 1995.
• [73] M. Kleiber and W. Sosnowski. Parameter sensitivity analysis in frictional contact problems of sheet metal forming. Comput. Mech., 16:297-306, 1995.
• [74] S. Kobayashi, S. Oh and T. Altan. Metal Forming and the Finite Element Method. Oxford University Press, 1989.
• [75] J. Kusiak. Optimization techniques in computer simulation oj metal forming processes (in polish). Wydawdnictwo AGH, Krakow, 1995.
• [76] J. Kusiak and E.G. Thompson. Optimization techniques for extrusion die shape design. In E.G. Thompson, R.D. Wood, O.C. Zienkiewicz and A. Samuelsson, editors, KUMIF0RM ’ 86, Numerical Methods in Industrial Forming Processes, pages 569 574. Balkema, 1989
• [77] K. Kuzman, E.Pfeifer, N. Bay and J. Hunding. Control of material flow in a combined backward can-forward rod extrusion. J. Mater. Proc. Technology, 60:141-147, 1996.
• [78] J.Y. Lamant, M. Larrecq, J.P. Birat, J.L. Hengsen, J.D. Weber and J.C. Dhuyvetter. Study of slab bulging in a continuous caster. In Metals Society Conference, London, 1985.
• [79] C.H. Lee and S. Kobayashi. New solutions to rigid plastic deformation problems using a matrix method. J. Eng. for Ind., Thins. ASME, 95:865-873, 1973.
• [80] T.H. Lee, J.S. Arora and V. Kumar. Shape design sensitivity analysis of viscoplastic structures. Comput. Meth. Appl. Mech. Engrg., 108:237-259, 1993.
• [81] L. Malvern. Introduction to the mechanics of a continuous medium. Prentice Hall, Englewood Cliffe, 1969.
• [82] T.D. Marusich and M. Ortiz. Modelling and simulation of high-speed machining. Int. j. numer. methods eng., 38(21):3675-3694, 1995.
• [83] Z. Mrdz. Variational methods in sensitivity analysis and optimal design. Eur. J. Mech. A/Solids, 13:115-147, 1994.
• [84] Z. Mr6z, M.P. Kamat and R.H. Plant. Sensitivity analysis and optimal design of nonlinear beams and plates. J. Struct. Mech., 13:245-266, 1985.
• [85] T. De Mulder. The role of bulk viscosity in stabilized finite element formulations for incompressible flow: a review. Comput. Meth. Appl. Mech. Engrg., 163:1-10, 1998.
• [86] E. Onate, M. Kleiber and C. Agelet de Saracibar. Plastic and viscoplastic flow of void containing metals: applications to axisymmetric sheet forming problems. Int. j. numer. methods eng., 25:225-251, 1988.
• [87] R. Michalowski and Z. Mr6z. Associated and non-associated sliding rules in contact friction problems. Arch. Mech., 30:259-276, 1978.
• [88] P. Perzyna. The constitutive equation for rate-sensitive plastic materials. Quart. Appl. Math., 20:321-332, 1963.
• [89] P. Perzyna. Fundamental problems in viscoplasticity, chapter 9. Academic Press, New York, 1966.
• [90] H. Petryk. Non-unique slip-line field solutions for the wedge indentation problem. J. Mechanique Appliquie, 4(3) :255—282, 1980.
• [91] H. Petryk. On the stability of non-uniquely defined processes of plastic deformations. J. Mechanique thiorique et appliqude, spec. nr.:187-202, 1982.
• [92] H. Petryk. Slip line field solutions for sliding contact. In Proc.Inst.Mech.Engrs., Int. Con/. Tribology - Friction, Lubrication and Wear, Fifty Years On, pages 987-994, London, 1987.
• [93] H. Petryk and Z. Mr6z. Time derivatives of integrals and functionals defined on varying volume and surface domains. Arch. Mech., 38:697-724, 1986.
• [94] P.G. Phelan, C.A. Vidal and R.B. Haber. Explicit sensitivity analysis of nonlinear elastic systems. In Proc. 1st Int. Conf. Computer Aided Optimal Design of Structures, Southampton, U.K., 1989.
• [95] K. Schittkowski. NLPQL: a fortran subroutine solving constrained nonlinear programming problems. Annals oj Operations Research, 5:485-500, 1986.
• [96] G.E. Schneider, G.D. Raithby and M.M. Yovanovich. Finite element analysis of incompressible fluid flow incorporating equal order pressure and velocity interpolation. In C. Taylor, K. Morgan and C.A. Brebbia, editors, Numerical Methods in Laminar and Turbulent Flow, Plymouth, 1978. Pentech Press.
• [97] P.J. Schreurs, F.F. Veldpaus and W.A. Brakalmans. An eulerian and lagrangian finite element model for the simulation of geometrical non-linear hyper elastic and elaato-plastic deformation processes. In Proc. Conf. on Industrial Forming Processes, pages 491-500. Pineridge Press, 1983.
• [98] G.S. Sekhon and J.L. Chenot. Numerical simulation of continuous chip formation during non-steady orthogonal cutting. Engng. Computations, 10:31-48, 1993.
• [99] W. Sosnowski and M. Kleiber. A study on the influence of friction evolution on thickness changes in sheet metal forming. J. Mater. Proc. Tech., 60:469-474, 1996.
• [100] J.S. Strenkowski and J.T. Carroll. A finite element model of orthogonal metal cutting. J. Engrg. Ind., 107:349-354, 1985.
• [101] A.U. Sulijoadikusomo and O.W. Dillon. Temperature distribution of the asixymmetric extrusion of ti-6al-4v. Univ. of Kentucky - Lexington, Int. rep., 1978.
• [102] W. Szczepinski. Introduction to the mechanics of plastic forming of metals. Sijthoff-Noordhoff, Alphen aan den Rijn, 1979.
• [103] E. Thompson. Use of pseudo-concentrations to follow creeping viscous flows during transient analysis. Int. J. Num. Meth. Fluids, 6:749-761, 1986.
• [104] E. Thompson and R.E. Smelser. Transient analysis of forging operations by the pseudo-concentration method. Int. j. numer. methods eng., 28:177-189, 1989.
• [105] J.J.M. Too. On numerical modelling of hot rolling of metals. Int. j. numer. methods eng., 30:1699-1718, 1990.
• [106] D.A. Tortorelli, J.A. Tomasko, T.E. Morthland and J.A. Dantzig. Optimum design of nonlinear parabolic systems. Part II: Variable spatial domain with applications to casting optimization. Comput. Meth. Appl. Mech. Engrg., 113:157-172, 1994.
• [107] D.A. Tortorelli and Z. Wang. A systematic approach to shape sensitivity analysis. Solids Struct., 30:1181-1212, 1993.
• [108] L.A. Winnicki and O.C. Zienkiewicz. Plastic (or visco-plastic) behaviour of axisymmetric bodies subjected to non-symmetric loading—semi-analytical finite element solution. Int. j. numer. methods eng., 14:1399-1412, 1979.
• [109] R.J. Yang and K.K. Choi. Accuracy of finite element based shape design sensitivity analysis. J. Struct. Mech., 13:223-239, 1985.
• [110] P. Yu and J.C. Heinrich. Petrov-Galerkin method for the time dependent convective transport equation. Int. j. numer. methods eng., 23:883-901, 1986.
• [111] O.C. Zienkiewicz. Flow formulation for numerical solution of forming processes. In J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander, editors, Numerical Analysis of Forming Processes, pages 1-44, Chichester, 1984. Wiley.
• [112] O.C. Zienkiewicz and R. Codina. A general algorithm for compressible and incompressible flow - Part I. The split, characteristic-based scheme. Int. j. numer. methods eng., 20:869-885, 1995.
• [113] O.C. Zienkiewicz and I.C. Conneau. Visco-plasticity—plasticity and creep in elastic solid — a unified approach. Int. j. numer. methods eng., 8:821-845, 1974.
• [114] O.C. Zienkiewicz and P.N. Godbole. Flow of plastic and viscoplastic solids with special reference to extrusion and forming processes. Int. j. numer. methods eng., 8:3-16, 1974.
• [115] O.C. Zienkiewicz and P.N. Godbole. Flow of plastic and viscoplastic solids with special reference to non-newtonian (plastic) fluids. In Finite elements in fluids, volume 1, pages 25-55, 1974.
• [116] O.C. Zienkiewicz, P.C. Jain and E. Onate. Flow of solids during forming and extrusion: some aspects of numerical solutions. Solids Struct., 14:15-38, 1978.
• [117] O.C. Zienkiewicz, Y.C. Liu and G.C. Huang. Error estimation and adaptivity in flow formulation for forming problems. Int. j. numer. methods eng., 25:23-42, 1988.
• [118] O.C. Zienkiewicz, K. Morgan, B.V. Satya Sai, R. Codina and M. Vasquez. A general algorithm for compressible and incompressible flow - Part II. Tests on the explicit form. Int. j. numer. methods eng., 20:887-913, 1995.
• [119] O.C. Zienkiewicz, E. Onate and J.C. Heinrich. A general formulation for coupled thermal flow of metals using finite elements. Int. j. numer. methods eng., 17:1497-1514, 1981.
• [120] O.C. Zienkiewicz, J. Rojek, R.L. Taylor and M. Pastor. Triangles and tetrahedra in explicit dynamic codes for solids. Int. j. numer. methods eng., 43:565-584, 1998.
• [121] O.C. Zienkiewicz and J. Wu. Incompressibility without tears—how to avoid restrictions of mixed formulations. Int. j. numer. methods eng., 32:1189-1209, 1991.
• [122] O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Int. j. numer. methods eng., 24:337-357, 1987.
• [123] J.P. Zolesio. The material derivative (or speed) method for shape optimization. In E.J. Haug and J. Cea, editors, Optimization of Distributed Parameter Structures. Sijthoff fc Noordhoff, 1981.