【正文】 w as the sa me as the overall area reductio n. O nce the final posi tion of a mater ial point w as assumed, a third order polyno mial w as fit betw ee n the die entrance a nd exit points. Gunase kera et al.[44] refined this me tho d to allow for reentrant geo me tries. Ponala gusa my et al. [45] proposed using Be zier curves for designi ng strea mlined e xtr usion dies. Kang a nd Ya ng[46] used finite ele me nt models to predict the opti mal bearing length for a n“ L‖ shape e xtr usion. S tudies on the design of three dime nsio nal extrusio n dies have been limi ted. T he co ntrolled strain rate co ncept has o nl y been appl ied to a xisymme tric extrusions and not to threedime nsional extrusions [19,20,26]. 3. The adaptable die design method The adap table die design method has been developed and is described in de tail in a series of papers [1–5]. The method has bee n exte nded to no na xisymme tric three di mensio nal extr usion o f a ro und bar to a recta ngular shape [6]. T he major criterion used i n developing the me thod w as to mi ni mi ze the distortion in the prod uct. T he present paper provides a brief overview of the me thod and results fro m these previous studies. Fig. 1. Schematic diagram of axisymmetric extrusion using spherical coordinate system through a die of arbitrary shape . Velocity field An upper bo und a nalysis of a metal for mi ng proble m requires a ki n ma tically ad missi ble velocity field. Fig. 1 s how s the process para meters in a sche matic diagra m w ith a sp herical coordinate s yste m (r, θ , φ ) and the three velocity zones that are used in the upper bound a nalysis of a xisymme tric extrusio n thro ugh a die with a n arbitrary die s hape. T he material is assumed to be a perfec tly plastic ma terial w ith flow strength, 0? .he friction, w hich exists be tw een the defor matio n zo ne in he w ork piece and the die, is characterized by a frictional shear tress, 0 /3mf??? , w here the co nsta nt friction fac tor, mf, can take values from 0 to 1. The ma terial starts as a cylinder of radius Ro and is extr uded into a cylindrical produc t of radius fR . Rigid bod y flow occurs in zo nes I a nd III, w ith velocities of 0v and fv , respectively. Zo ne II is the defor ma tio n re gion, w here the velocity is fairly co mplex. Tw o spherical surfaces of velocity discontinuity Γ 1 and Γ 2 separate the three velocity zones. The surface Γ 1 is located a dista nce 0r from the origin and the surface Γ2 is located a distance fr from the origin. The coordinate sys te m is centered at the converge nce point o f the die. The co nverge nce point is defined by the intersection o f the axis of symme try w ith a line at a ngle α that goes through the point where the die begins its deviation from a cylindrical shape a nd the exi t point o f the die. Fig. 1 s how s the position of the coordinate sys te m origi n. T he die s urface, w hic h is labeled ψ (r) in Fig. 1, is given in the sp herical coordinate system. ψ(r) is the a ngular position o f the die surface as a functio n of the radial dista nce fro m the origi n. T he die length for the defor matio n regio n i s give n b y the parameter L. The best velocity field to describe the flow in the de for matio n region is the si ne 1 velocity field [1,2] . This velocity field uses a base radial velocity, rv , w hich is modified by a n additional ter m prised to tw o func tions w ith eac h functio n co ntaini ng pseudoindepe ndent para me ters to deter mi ne the radial velocity ponent in zone II: rrUv???? (1) The ε function permits flexibility of flow in the radial, r, direction, and the γ function permits flexibility of flow in the a ngular, θ , direction. T he val ue o f rv is deter mined b y ass umi ng proportional distances in a cylindrical sense from the centreline: 200 2s in c o ss inr rvv r ? ?????? ???? (2) This velocity field w as fo und to be the best representatio n of the flow in the de for matio n regio n o f a n e xtr usion process for a n arbitrarily shaped die. The ε func tion is represented as a series of Le ge ndre polyno mials tha t are ortho go nal over defor ma tion zo ne. T he representation of ε is: ? ?0an iii AP x? ??? (3) Where ? ?0002 / 11/ffRR rx w it hR R r? ?????? ia being the coefficients of the Lenge ndre polyno mials Pi(x) and an being the order of the representatio n. T here is a restriction that: ? ?1 3an oddiiAA??? ?， ? ?0 2an eveniiAA??? ? (4) The re mai ning hi gher order coefficients (A2 to A an ) are the pseudoindepe ndent para meters, w ith values deter mi ned b y mi ni mi zatio n of the total pow er. Le ge ndre polyno mials are used so that higher order terms ca n be added to the function w itho ut causi ng signi ficant c hanges i n the coefficients o f the lower order polyno mials. This feature of the Le ge ndre polyno mials occurs because the y are orthogonal over a finite distance. The γ function that satisfies the boundary conditions and allow s the best description of the flow is: 0 11 c o s1 c o sb iniiBB????????? ?????? (5) where 0 1 1bn iiBB i??? ?? and the high order coefficients B1 to B bn are pseudoindepe nde nt para meters w ith val ues deter mined b y mi ni mizatio n o f the total pow er. The order of the representa tion is bn . It has b