Abstract
In the present work, an axi-symmetric cold forging problem is analyzed using radial basis function collocation method. The material is assumed to be rigid-plastic strain hardening. At each increment of the punch displacement, the problem is solved using an Eulerian control volume approach. The mixed pressure-velocity formulation is adopted, in which the hydrostatic stress and velocities are approximated by linear combinations of multiquadrics radial basis functions, the coefficients of which are obtained by satisfying the continuity and equilibrium equations at certain points called collocation points. The resulting non-linear equations are solved using a trust region method available in MATLAB, which is based on interior-reflective Newton method. Because of the nature of the equations, hydrostatic stress values contain spurious terms. To eliminate them, boundary conditions on hydrostatic stress are required, which are not known initially. Therefore the problem is solved in two stages. In the first stage, the problem is solved without any boundary condition for the hydrostatic stress and the forging load is computed by dividing the total power by the punch velocity. The hydrostatic stress at the punch-workpiece interface is obtained from the known forging load. In the second stage, the problem is solved again by putting the additional hydrostatic stress boundary conditions. Computational performance of the proposed method is studied by carrying out parametric study.
Similar content being viewed by others
References
Bathe KJ, Ramm E, Wilson EL (1975) Finite element formulations for large deformation dynamic analysis. Int J Numer Methods Eng 9:353–386
Park JJ, Kobayashi S (1984) Three-dimensional finite element analysis of block compression. Int J Mech Sci 26:515–525
Surdon G, Chenot JL (1987) Finite element calculation of three-dimensional hot forging. Int J Numer Methods Eng 24:2107–2117
Chitkara NR, Kim Y (2002) Simulation of upset heading, application to rigid-plastic finite element analysis and some experiments. Int J Adv Manuf Technol 20:589–597
Mungi MP, Rasane SD, Dixit PM (2003) Residual stresses in cold axisymmetric forging. J Mater Process Technol 142:256–266
Chandra A, Srivastava RA (1991) boundary element analysis of axisymmetric upsetting. Math Comput Model 15:81–92
Guo YM (1998) Analyses of forging processes by a rigid-plastic finite-boundary element method. J Mater Process Technol 84:13–19
Guo YM, Nakanishi K (1999) Axisymmetric forging analyses by the rigid plastic finite-boundary element method. J Mater Process Technol 86:208–215
Guo YM, Nakanishi K (2002) A rigid-plastic hybrid element method for simulations of metal forming. J Mater Process Technol 130–131:444–449
Zienkiewicz OC, Zhu JZ (1992) Part I: The super convergence patch recovery and a posteriori error estimates, Part II: error estimates and adaptivity. Int J Numer Methods Eng 33:1365–1382
Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: Theory and application to non-spherical Stars. Mon Not R Astron Soc 181:375–389
Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318
Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196
Belytschko T, Lu YY, Gu L (1994) Element free Galerkin methods. Int J Numer Methods Eng 37:229–256
Duarte CA, Oden JT (1996) An h-p adaptive method using clouds. Comput Methods Appl Mech Eng 139:237–262
Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38:1655–1679
Melenk JM, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314
Onate E, Idelsohn S, Zienkiewicz OC, Toylor RL, Sacco C (1996) A stabilized finite point method for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng 139:315–346
Zhu T, Zhang J, Atluri SN (1998) A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach. Comput Mech 21:223–235
Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9:987–1000
Lagaris IE, Likas A, Fotiadis DI (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11:1041–1049
Kansa EJ (1990) Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics II: solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19(8–9):147–161
Mai-Duy N, Tran-Cong T (2002) Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations. Eng Anal Bound Elem 26:133–156
Jianyu L, Siwei L, Yingjian Q, Yaping H (2003) Numerical solution of elliptical partial differential equation using radial basis function neural networks. Neural Netw 16:729–734
Leitao VMA (2001) A Meshless method for Kirchhoff plate bending problems. Int J Numer Methods Eng 52:1107–1130
Ferreira AJM, Roque CMC, Martins PALS (2003) Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Compos Part B Eng 34:627–636
Liew KM, Chen XL, Reddy JN (2004) Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates. Comput Methods Appl Mech Eng 193:205–224
Leitao VMA (2004) RBF-based meshless methods for 2D elastostatic problems. Eng Anal Bound Elem 28:1271–1281
Bathe KJ (1996) Finite element procedures. Prentice-Hall, New Delhi
Dixit US, Dixit PM (1997) A study on residual stresses in rolling. Int J Mach Tools Manuf 37:837–853
Chakrabarty J (1998) Theory of plasticity. McGraw Hill Book Company, Singapore
Franke R (1982) Scattered data interpolation: tests of some methods. Math Comput 38:181–200
Haykin S (1999) Neural networks: a comprehensive foundation. Prentice-Hall, New Jersey
Moody J, Darken CJ (1989) Fast learning in networks of locally tuned processing units. Neural Comput 1:281–294
Mai-Duy N, Tran-Cong T (2001) Numerical solution of differential equations using multiquadric radial basis function networks. Neural Netw 14:185–199
Coleman TF, Li Y (1996) An Interior, trust region approach for nonlinear minimization Subject to Bounds. SIAM J Optim 6:418–445
Johnson W, Mellor PB (1972) Engineering plasticity. Von Nostrand Reinhold Company, London
Shima S, Mori K, Osakada K (1978) Analysis of metal forming by rigid-plastic finite element method based on plasticity theory for porous material. In: Lippman H (ed) Metal forming plasticity. Springer, Berlin Heidelberg New York pp 305–317
Park JW, Kim YH, Bae WB (1997) An upper-bound analysis of metal forming processes by nodal velocity fields using a shape function. J Mater Process Techn 72:94–101
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mahadevan, P., Dixit, U.S. & Robi, P.S. Analysis of cold rigid-plastic axisymmetric forging problem by radial basis function collocation method. Int J Adv Manuf Technol 34, 464–473 (2007). https://doi.org/10.1007/s00170-006-0633-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00170-006-0633-0