# Three-dimensional finite-element simulation of the dynamic Brazilian tests on concrete cylinders

код для вставкиСкачатьINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 48:1089}1106 Mismatching re"nement with domain decomposition for the analysis of steady-state metal forming process K. Park and D. Y. Yang*R Department of Mechanical Engineering, Korea Advanced Institute of Science and ¹echnology, ME3214, Kusung-dong, 171-1, >ousung-gu, ¹aejon, 305-701, Korea SUMMARY The "nite element analysis of three-dimensional metal forming processes is generally subject to large computational burden due to its non-linearity. For economic computation, the mismatching re"nement, an e$cient domain decomposition method with di!erent mesh density for each subdomain, is developed in the present study. A modi"ed velocity alternating scheme for the interface treatment is proposed in order to obtain good convergence and accuracy in the mismatching re"nement. As a numerical example, the analysis of the axisymmetric extrusion processes is carried out. The results are discussed for the various velocity update schemes and for the variation of the length of overlapped region. The three-dimensional extrusion processes for a rectangular section and an E-section are analysed in order to verify the e!ectiveness of the proposed method. Comparing the results with those of the conventional method of full region analysis, the accuracy and the computational e$ciency of the proposed method are then discussed. Copyright 2000 John Wiley & Sons, Ltd. KEY WORDS: mesh re"nement; domain decomposition; iterative calculation; rigid-plastic "nite element method; extrusion process 1. INTRODUCTION In order to solve large-scale "nite element problems with good accuracy and high e$ciency, there have been various studies on adaptive mesh re"nement schemes. Much of the focus of the studies has been put on the development of error estimators and following re"nement schemes to reduce error [1}8]. The basic constraint of the conventional mesh re"nement methods is the continuity of nodes between re"ned regions and unre"ned regions. For the re"nement using triangular mesh or tetrahedral mesh [5}7], it has been known that the continuity is fully maintained. For the re"nement of quadrilateral mesh or hexahedral mesh, however, special techniques are required to control the mesh density. * Correspondence to: Dong-Yol Yang, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, ME3214, Kusung-dong, 373, Yousung-gu, Taejon 305-701, Korea R E-mail: dyyang@mail.kaist.ac.kr Copyright 2000 John Wiley & Sons, Ltd. Received 24 August 1998 Revised 23 September 1999 1090 K. PARK AND D. Y. YANG There have been various studies on the re"nement using quadrilateral mesh; Babuska and Rheinboldt [1}3] introduced a bisectionally re"ned mesh structure by imposing some geometric constraints on the irregular nodes. Choi et al. [8, 9] used transition elements for the interface region between re"ned mesh and unre"ned mesh so as to remove the arti"cial constraints. In spite of e!ective interface treatment, this method has a disadvantage that the bandwidth is highly increased for hexahedral transition elements which can possess 26 nodes at most [9]. It is not adequate for the analysis of large-scale problems, such as the analysis of three-dimensional metal forming processes. Domain decomposition has become an important subject to obtain parallelism when solving a partial di!erential equation (PDE). It is useful for reducing the problem size, and distributed processing enhances the reduction of computation time. The basic idea is to subdivide the domain of the de"nition of the PDE into a set of subdomains and then to solve the PDE problems on each subdomain. These solutions for all the subdomains are then assembled to obtain the solution for the whole domain. Schwarz [10] equated the alternating procedure which solves each subdomain concurrently with data exchange between two adjacent overlapped subdomains. The convergence of the procedure was established by Sobolev [11]. Rodrigue and Shah [12] imposed pseudoboundary conditions to accelerate the parallel Schwarz alternating procedure. Fahrat and Wilson [13] implemented recursive substructuring method which does not require overlapped region. Funaro et al. [14] introduced an iterative solving scheme between subdomains as imposing the continuity of the solution and the continuity of the normal derivative. In the "nite element analysis of metal forming processes, Doltsinis and NoK lting [15] implemented parallel substructuring method to analyse the rolling processes. The domain decomposition method is e!ective in reducing computation time by reducing the problem size as well as to ensure parallelism of the problem. Adaptive mesh re"nement can be easily implemented by imposing di!erent mesh density to each subdomain. In this study, mismatching re"nement, a mesh re"nement scheme combined with the domain decomposition method, is developed. Iterative calculation with an error minimization scheme is carried out between the two subdomains which have di!erent mesh densities. As numerical examples, the extrusion processes are analysed in order to verify the e!ectiveness of the proposed method. Many numerical parameters which a!ect the accuracy and the convergence of the solution are discussed. 2. THE RIGID-PLASTIC FINITE ELEMENT METHOD 2.1. Basic formulation In this work, the rigid-plastic "nite element method [16] is employed to simulate metal forming processes. The rigid-plastic analysis requires considerably less computation time than the elastoplastic analysis. The equilibrium demands the following relations: Copyright 2000 John Wiley & Sons, Ltd. pGHH"0 in ) (1) pGH nH"fG on !D (2) Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS 1091 where nH is the unit normal vector to a given surface, and fG is the force vector. The incompressibility condition for a rigid-plastic material is given as eR 4"eR GG"0 (3) Since the rigid-body rotation and elastic deformation of material are neglected in the rigid-plastic "nite element method, the constitutive equation is given as follows: 2 pN p " eR GH 3 eNQ GH (4) where pN "( p p , GH GH eNQ "( eR eR GH GH (5) and p and eR are components of the deviatoric stress and strain rate, respectively. From the GH GH variational principle, the functional for a rigid-plastic material can be written as follows pN deNQ d)#K* eR 4 deR 4 d)! ! D f du d!"0 G G (6) where u denotes the velocity component, and K* is the penalty constant for the incompressibility G condition. Since discretization of Equation (6) is a non-linear system of equations with respect to u , the Newton}Raphson iteration is utilized with an initial guess by assuming a linear viscous G material [17]. For the computation, bilinear quadrilateral elements are used for two-dimensional analysis, and trilinear hexahedral elements are utilized for three-dimensional analysis. In order to avoid locking due to incompressibility, selective reduced integration is employed for numerical integration. That is, the second term in Equation (6) is integrated with a lower-order rule while the other terms are integrated with a normal Gauss quadrature rule. 2.2. Error estimation In the iterative solving method with domain decomposition, there occurs some numerical error for connection between subdomains. In the present study, the error estimation is considered in view of that error caused by the iterative calculation. The result of the iterative calculation is compared with the solution which is obtained from the fully re"ned mesh. That is, the solution of the fully re"ned mesh is taken as the reference solution. For the comparison of accuracy of the iterative calculation, two error measures, such as velocity error and strain energy error, are considered. The velocity error at a speci"c point is de"ned as follows: #u!uL # #u*!uL # e" : S #u# #u*# Copyright 2000 John Wiley & Sons, Ltd. (7) Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1092 K. PARK AND D. Y. YANG where superscripts * and denote the results of the full domain analysis and the results of the iterative calculation, respectively. In the rigid}plastic formulation, the energy dissipation norm is given as follows: p2 : eR d) #;#" (8) where p and eR denote deviatoric stress and strain rate, respectively. Thus, the energy error norm is given by the following forms: #E#" (p!pL )2 : (eR !eRL ) d) : (p*!pL )2 : (eR *!eRL ) d) (9) The error measure for strain energy then can be written as the ratio of the energy dissipation norm to the energy error norm [5, 6]. #E# e " # # ;# (10) 3. ITERATIVE CALCULATION WITH DOMAIN DECOMPOSITION Let us consider the matrix equation after "nite element discretization of Equation (6) Ku"f, in ) (11) u"g, on ! (12) where K and f are sti!ness matrix and load vector, respectively, and ! is the boundary of domain ). Domain ) is subdivided into two subdomains, ) and ) , as shown in Figure 1. For the subdomain ) , by introducing the concept of Schwarz alternating procedure, [10] Equations (11) and (12) can be rewritten as follows: K uI" f , in ) (13) Figure 1. Schematic description of domain decomposition. Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS uI"g, on ! uI"uI\, on ! !! 1093 (14) (15) where subscript 1 and 2 denote the subdomain 1 and the subdomain 2, respectively, and superscript (k) means the current iteration step. In order to improve the convergence of the iterative calculation, the continuity condition of the normal derivative is taken as follows: *uI *uI\ " , on ! !! *n *n (16) where n denotes the outward normal direction. For the subdomain ) , the equations are given in the same manner K uI"f , in ) (17) uI"g, on ! (18) uI "uI, on ! !! (19) *uI *uI " , on ! !! *n *n (20) As shown above, velocity components of the overlapped interface region are obtained using the results of the neighboring subdomain. The iteration is carried out until the following criterion is satis"ed: #uI!uI\# eI" )d # uI# (21) where eI is an amount of error at the current step, # z # is a Euclidean vector norm, and d is a very small constant called as a limit of the fractional norm. 4. MISMATCHING REFINEMENT WITH DOMAIN DECOMPOSITION The main idea of the mismatching re"nement is that the solution is obtained by iterative calculation between several subdomains which have di!erent mesh densities. At the interfaces between subdomains, there exist mismatching overlapped regions where the continuity of nodes is not guaranteed due to the di!erence of mesh density or mesh con"guration. A domain for axisymmetric extrusion is divided into two subdomains, a coarse subdomain and a "ne subdomain. In the overlapped region, a projection operator from the coarse mesh to the "ne mesh is de"ned as P : RL P RL Copyright 2000 John Wiley & Sons, Ltd. (22) Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1094 K. PARK AND D. Y. YANG where n and n mean the numbers of DOFs in the overlapped region for the coarse subdomain and for the "ne subdomain, respectively. The reverse operation can be de"ned in the same manner Q : RL P RL (23) The velocity alternating procedure (Equations (15), (16), (19), and (20)) can thus be rewritten by introducing the projection operators uI"QI\uI\, on ! !! (24) *uI *uI\ "QI\ , *n *n on ! !! (25) uI"PIuI, on ! !! (26) *uI *uI "PI , *n *n on ! !! (27) where subscripts c and f denote the coarse subdomain and the "ne subdomain, respectively. The iterative alternating procedure has been known to render no problem of convergence [11]. However, it should be modi"ed so as to consider the di!erence of mesh density for the mismatching re"nement. For the consideration, let eI and eI be the velocity error measures for the coarse subdomain and the "ne subdomain, respectively #u!QI\uI\ # #u!uI # " eI" # u!uI\ # #u!uI\ # (28) # u!uI # # u!PI uI # eI" " #u!uI\ # # u!uI\ # (29) Since uI\ ensures better accuracy than uI\, the convergence criterion is ful"lled for the alternating procedure from "ne subdomain to coarse subdomain (Equations (24) and (25)), that is, eI(1. The reverse procedure (Equations (26) and (27)), however, may cause some convergence problem since uI is not always more accurate than uI\ . In the present work, for the characteristics of mismatching re"nement, a modi"ed velocity update scheme is proposed considering the solution of the previous step as well as that of the other subdomain. For both subdomains, Equations (24)}(27) can be rewritten as follows: uI"aI QI\ uI\#(1!aI) uI\, on ! !! (30) *uI *uI\ *uI\ "aIQI\ #(1!aI) , *n *n *n on ! !! (31) Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS 1095 uI"bI PI uI#(1!bI) uI\, A on ! !! (32) *uI *uI\ *uI "bIPI #(1!bI) , *n *n *n on ! !! (33) where a and b are the weight factors to consider solution of the previous step. In the modi"ed alternating scheme, eI can be revised as follows: # u!uI # # u!aI QI\ uI\!(1!aI) uI\ # eI" " # u!uI\ # #u!uI\ # (34) if aI'(1!aI), that is, 0.5(aI(1, the weighting factor for uI\ becomes higher than the weighting factor for uI\. For such a condition, eI can be rewritten as follows, which satis"es the convergence criterion eI: # u!aI QI\ uI\ # (1 #u!uI\ # (35) On the other hand, eI can be expressed in the same manner as follows: # u!uI # #u!bI PI uI!(1!bI) uI\ # eI" " #u!uI\ # # u!uI\ # (36) if bI((1!bI), that is, 0(bI(0.5, the weighting factor for uI\ becomes higher than the weighting factor for uI\. For such a condition, the convergence criterion is also met where eI can be rewritten as follows: #u!(1!bI) uI\ # (1 eI: #u!uI\ # (37) In order to ful"l the convergence criterion, aI and bI are de"ned as a function of the rate of convergence g aI"C [gI ]K, on ! !! (38) bI"C [gI ]K, on ! !! (39) In this study, C and m are taken as 1.0 and 0.5, respectively. The rate of convergence in the overlapped region at the kth step is de"ned by # eI # G gI" , in ) 5 ) G # eI\ # G Copyright 2000 John Wiley & Sons, Ltd. (40) Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1096 K. PARK AND D. Y. YANG where subscript i denotes the subdomain number. The rate of convergence is a value between 0 and 1 unless the solution diverges. If it diverges, the rate of convergence is set to 1. As the solution converges, the rate of convergence decreases continuously and the weighting factors are updated re#ecting the state of convergence. It is expected that the proposed scheme improves convergence as well as accuracy of the solution by utilizing the results of the re"ned region for all the velocity alternating procedures. 5. NUMERICAL EXAMPLES 5.1. Comparison of the velocity alternating schemes As a numerical example, an analysis of the axisymmetric extrusion process with the mismatching re"nement is carried out (Figure 2). The workpiece material used in simulation is Al6061 of which the constitutive relation is given as follows at 400 3C [18]: pN "53.60 e (MPa) (41) The subdomain 1, with a low mesh density, consists of 143 nodes and 120 elements (Figure 2(b)) while the subdomain 2, with a high mesh density, consists of 155 nodes and 120 elements (Figure 2(c)). The analyses of the subdomain 1 and subdomain 2 are carried out iteratively and the result is compared with that of a full domain analysis which consists of 575 nodes and 520 elements Figure 2(a)). Analyses are carried out in order to compare two velocity alternating procedures (Equations (24)}(27) and (30)}(33)). For each case, the variation of axial velocity components is compared. The axial velocity components are compared at the center point of the outlet as the iterative calculation is carried out (Figure 3). The dot line in Figure 3(b) means the velocity of the full domain analysis without domain decomposition, which is regarded as a reference solution for the comparison. Comparing the results, the proposed velocity alternating scheme, Equations (30)}(33), shows more accurate and stable solution than the Schwarz's alternating scheme, Figure 2. Basic concept of the nonconforming re"nement: (a) full mesh; (b) mesh for subdomain 1; (c) mesh for subdomain 2. Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS 1097 Figure 3. Axial velocity components at the reference point: (a) position of the reference point; (b) axial velocity components w.r.t. the number of iterations. Figure 4. Variation of error measures. Equations (24)}(27). It appears that the proposed velocity update scheme improves stability and accuracy of the solution in the case of the mismatching re"nement. In order to check the convergence of iterative calculation for each subdomain, the variation of error measure, which is de"ned in Equation (21), is shown in Figure 4 with respect to the number of iterations. It is concluded that convergence and stability of mismatching re"nement are assured with the Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1098 K. PARK AND D. Y. YANG proposed velocity alternating scheme. The proposed velocity alternating scheme is thus utilized for the analysis of further numerical examples. 5.2. Analysis of the axisymmetric extrusion process In order to verify the accuracy and e$ciency of mismatching re"nement, the analysis of the axisymmetric extrusion process is carried out with a steady-state isothermal assumption at 4003C. The workpiece material is Al6061 of which the constitutive relation is given in Equation (41). Three mesh structures for the analysis are considered as shown in Figure 5. The mesh 2 (Figure 5(b)) is decomposed into two subdomains with the same mesh density as mesh 1. For the mismatching re"nement, mesh 3 (Figure 5(c)), the mesh density of subdomain 1 of mesh 3 is reduced. The ram speed is set as 10 mm/s and the friction factor is assumed to be 0.3. The iterative calculation is carried out until the error measure is converged below 10\. The velocity error at the reference point and the strain energy error in the overlapped region are compared in Table I. All error measures are less than 1.0 per cent in the case of mesh 2 while a little increase, still less than a 2.0 per cent, occurs in mesh 3. The speed-up ratio relative to the analysis of the mesh 1 goes up to 1.43 times for mesh 2, and 2.62 times for mesh 3, respectively. Figure 5. Mesh systems for axisymmetric extrusion: (a) mesh 1; (b) mesh 2; (c) mesh 3. Table I. Comparison of the results of axisymmetric extrusion. Subdomain No. of nodes Velocity error (%) Energy error (%) No. of iterations Elapsed time (sec) Speed-up ratio 1 * 575 * * * 218.2 1.00 2 1 2 525 155 0.39 0.90 4 151.7 1.43 3 1 2 143 155 1.96 1.24 18 82.9 2.62 Mesh Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS 1099 5.3. E+ect of the length of overlapped region In this section, the e!ect of the length of overlapped region is investigated. Determination of the length of overlapped region is an important procedure since it may a!ect the convergence as well as the accuracy of solution. Three mesh structures are considered as shown in Figure 6. Each mesh has the same structure for subdomain 1, although the length of the overlapped region (l) is di!erent for subdomain 2. The workpiece material and process parameters are the same as those of the previous section. The results with a variation of l are presented in Table II. The convergence limit is 10\, and the velocity error measure and the speed-up ratio are de"ned with respect to the results of the full domain analysis. In the case of l"15 and 20 mm, the results show that a better convergence and an improved speed-up ratio are obtained. The accuracy of solution of both cases is still better than the result of l"10 mm. Figure 7 shows the state of convergence with the variation of l. In order to analyse the e!ect of the length of overlapped region, axial velocity component along the centerline is considered in Figure 8. In this "gure, the result of the subdomain 1 means the solution after the "rst iteration. The result of subdomain 1 shows almost the same result as that of the full analysis until the distance from the punch is 45 mm (i.e. l"15 mm), while it shows a numerical error de"ned as the di!erence from the distance 50 mm (i.e. l"10 mm). It thus appears that the di!erence of the convergence as shown in Table II is caused by the numerical Figure 6. Mesh systems for various length of overlapped region: (a) l"10 mm; (b) l"15 mm; (c) l"20 mm. Table II. Comparison of the results for various length of overlapped region. l (mm) No. of iterations Velocity error (%) Elapsed time (s) Speed-up ratio Copyright 2000 John Wiley & Sons, Ltd. 10.0 18 1.96 82.9 2.62 15.0 12 1.89 62.2 3.51 20.0 10 2.03 49.4 4.42 Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1100 K. PARK AND D. Y. YANG Figure 7. Convergence status with the variation of l. Figure 8. Axial velocity component along the centerline: (a) de"nition of l and distance; (b) variation of axial velocity component. error of the velocity. As a consequence, it is recommended that the length of overlapped region is determined as considering the result of subdomain 1 at the "rst iteration. 5.4. Analysis for extrusion of a rectangular section Let us consider three-dimensional extrusion of a rectangular section (Figure 9) in order to magnify the merit of the proposed method. The ram speed is set as 10 mm/s and the friction factor Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS 1101 Figure 9. Three-dimensional extrusion process with a rectangular section. Figure 10. Mesh systems for the rectangular section extrusion: (a) mesh 1; (b) mesh 2; (c) mesh 3. is taken to be 0.3. The workpiece material used in simulation is Al6061 of which the constitutive relation is given as follows at 4003C [18]: pN "53.60 e (MPa) (42) Analyses for three mesh systems as shown in Figure 10 are carried out with a steady-state isothermal assumption at 4003C. The computations are performed on HP 780 workstation with 64MB memory. The axial velocity components and related error measures at the four reference points (Figure 11) and the strain energy error in the overlapped region are compared in Table III. Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1102 K. PARK AND D. Y. YANG Figure 11. Position of four reference points. Figure 12. Schematic description of an &E'-section extrusion. Table III. Comparison of velocity components and error measures. Reference point Mesh 1 Mesh 2 Mesh 3 1 Velocity (mm/sec) Error (%) 158.64 * 160.62 1.24 160.61 1.24 2 Velocity (mm/s) Error (%) 153.43 * 153.96 0.35 154.51 0.70 3 Velocity (mm/s) Error (%) 135.94 * 133.41 1.89 134.10 1.35 4 Velocity (mm/s) Error (%) 145.57 * 144.73 0.58 144.97 0.41 0.69 0.91 Energy error (%) * All velocity error measures are around 1.0 per cent with a small variation. The problem size and the computing performance for the three mesh systems are compared in Table IV. The speed-up for mesh 3 is about twenty times of mesh 1 while about six times for mesh 2. It is concluded that the proposed mismatching re"nement with domain decomposition guarantees highly e$cient solution with an allowable accuracy in the "nite element analysis of three-dimensional extrusion process. 5.5. Analysis for extrusion of an *E+-section Let us consider three-dimensional extrusion of an &E'- section (Figure 12) with a high extrusion ratio, i.e., thin-walled section extrusion. The ram speed is set as 10 mm/s and the friction factor is Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1103 STEADY-STATE METAL FORMING PROCESS Table IV. Comparison of problem size and performance. Subdomain No. of nodes No. of elements No. of iterations Elapsed time (h : min) Speed-up ratio 1 * 2740 2148 * 37 : 13 1.00 2 1 2 2220 1186 1764 776 4 6 : 04 6.13 3 1 2 711 1186 512 776 8 1 : 53 19.76 Mesh Figure 13. Full mesh structure (10 886 nodes, 9240 elements). taken to be 0.3. The workpiece material used in simulation is Al6061 of which the constitutive relation is given as follows at 5003C [18]: pN "26.87 e (MPa) (43) Numerical analysis is carried out with a steady-state isothermal assumption at 5003C. As considering the symmetric condition of the geometry, only section of the workpiece is analysed. The computations are performed on Cray T3E computer with a single CPU and 128MB memory. Figure 13 shows the full mesh structure, which contains 32658 DOFs. The full analysis, however, is not possible due to the limit of memory capacity. Figure 14 shows mesh con"guration decomposed into two subdomains with mismatching re"nement. The number of DOFs is e!ectively reduced as 3630 for subdomain 1 and as 10 167 for the subdomain 2, respectively. Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 1104 K. PARK AND D. Y. YANG Figure 14. Mesh con"guration for the mismatching re"nement: (a) mesh for subdomain 1 (1210 nodes, 940 elements); (b) mesh for subdomain 2 (3389 nodes, 2352 elements); (c) mesh for the full domain. Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106 STEADY-STATE METAL FORMING PROCESS 1105 The number of iterations between subdomains is 8 with a convergence limit of 10\. It takes 2947 CPU seconds for the analysis with a given computational condition. It is concluded that the computation of a large problem size, which is even unable to carry out due to the memory capacity, can be accomplished within 1 h by employing mismatching re"nement. 6. CONCLUSION In the present work, mismatching re"nement, a new mesh re"nement scheme, has been proposed in the analysis of three-dimensional extrusion processes. The domain decomposition method with overlapping has been utilized in combination with the iterative calculation between several subdomains. For the convergence of the iterative calculation between subdomains with di!erent mesh densities, a modi"ed velocity alternating scheme has been introduced and has been compared with Schwarz alternating procedure. The proposed alternating scheme has shown better convergence than Schwarz's scheme for the mismatching re"nement. Through several numerical examples of the extrusion processes, the validity of the proposed method has been demonstrated. From the results it has been shown that computational e$ciency is highly increased, especially for the three-dimensional analysis. It is expected that the proposed method enables a three-dimensional analysis of the industrial extrusion processes with a less computational overhead. In addition, it can be easily implemented on parallel computation which still more accelerates the computational e$ciency. REFERENCES 1. Babuska I, Rheinboldt WC. Error estimates for adaptive "nite element method computation. SIAM Journal of Numerical Analysis 1978; 15:736}754. 2. Babuska I, Rheinboldt WC. A posteriori error estimates for the "nite element analysis. International Journal for Numerical Methods in Engineering 1978; 12:1597}1625. 3. Rheinboldt WC. Adaptive mesh re"nement processes for "nite element solutions. International Journal for Numerical Methods in Engineering 1981; 17:649}662. 4. Demkowicz L, Devloo Ph, Oden JT. 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St. Doltsinis I, NoK lting S. Generation and decomposition of "nite element models for parallel computations. Computer Systems in Engineering 1991; 2:427}449. 16. Lee CH, Kobayashi S. New solution to rigid plastic deformation using a matrix method. ASME, Journal of Engineering for Industry 1973; 95:865}873. 17. Yoon JH, Yang DY. Rigid-plastic "nite element analysis of three dimensional forging by considering friction on continuous curved dies with initial guess generation. International Journal of Mechanical Sciences 1988; 30:887}898. 18. Knight JW, Aluminum Extrusion Alloys. KAISER Aluminum and Chemical Sales, Inc.: Houston, 1964. Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1089}1106

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