【正文】 een s how n [3] that an = 6 and bn = 2 are us ually s ufficient to provide reasonable flexibility for the flow field in the deformation region. . Die shape The die shape is described by the functionψ (r). The adaptable die shape is described by a set of Legendre polynomials: ? ?0iii c p x? ?? ? (6) where ? ?? ?0002 / 11/ffRR rx w ithrRR? ??????and ic being the coefficients o f the Le ge ndre polyno mials Pi (x). T he order of the Le ge ndre polyno mial representa tion is . The bo undar y co nditions a t the e ntrance a nd exit of the deformation region require that: At r = 0r , ψ= α At r = fr , ψ= α (7) If a strea mlined die is used then this func tion must mee t tw o additional boundary conditions: At r = 0r , 0 tanr r? ??? ? ?? At r = fr, 00 ta nfRr rR? ??? ? ?? (8) . Distortion criteria The criterion that w as found to mini mi ze the distortion i n the extr usion produc t involves mi ni mizi ng the volumetric effec tive strain rate deviation [4,5] . T he vol ume tric effective strain rate deviatio n i n the deformation zone is: Where with: (10) and ij?? are the ponents of the strain rate field. . Determining the adaptable die shape The search for the op ti mal coe fficients for the Le ge ndre polyno mials representing the die shape is no t cons trained. A nested opti mizatio n ro uti ne is used w ith the velocity field (inside loop) being mi ni mized w ith respect to the e xternally supplied pow er for the process, and the die shape (outside loop) being adap ted to mi ni mi ze the distortion criterion. T he fi nal shape is called an adap table die shape, since the shape has adapted to meet the specified criterion. Fig. 2. Streamlined adaptable die shape with no adaptatio n in the rotational directiowith red = , L/Ro= ,mf= ,Rr/Ro= and μ = . The area reduction ratio is red, Rr is the cornered radius of the rectangular product, and μ is the height to width ratio of the rectangular product. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.) . Extension to three dimensional nonaxisymmetric shapes In e xte ndi ng the adaptabl e die design method fro m a xisymme tric flow to no na xisymme tr ic threedime nsional flow in the defor matio n region requires several special considerations [6]. First, the velocity field needs to be modified to allow for rotational move me nt in the defor matio n regio n. Seco nd, the bearing re gion o n the e xit side of the die needs to be analyzed properly. Third, the functio ns used to describe the die shape need to have so me fle xibility in the rota tional direction (. ψ(r, φ ). T his fle xibility allow s die shape adap tatio n with respect to the rotational coordinate, φ . 4. Die shape to minimize distortion To illus trate the adaptable die design method a specific threedi me nsional e xa mple is presented. A n e xtrusio n upper bo und model was used to deter mi ne adaptable die sur face shapes, w hic h mi ni mi ze distortio n through mi ni mizi n g the vol ume tric effective strain rate deviation in the de for matio n zo ne for the e xtrusio n o f a cylindrical billet i nto a ro und cor nered rectangular prod uct. Tw o sche mes w ere used. In the first me thod there w as no flexibility allow ed in the rota tional direction, ψ (r), whereas in the seco nd me thod the die shape w as able to adapt in the rotational direction, ψ (r, φ ). In both me thods a strea mlined conditio n w as used at the entra nce and e xit re gions o f the die. The quarter sections (φ =0 to π /2) of both die s hapes are given i n Figs. 2 a nd 3. Fi g. 2is the die shape w ith no rotatio nal flexibility a nd i n Fi g. 3 the die w as allow ed to adapt its shape i n the rota tional direction to red uce distortion. For both of these exa mples the area reduc tion w as 60% and the rectangular product had a width to height ratio, μ, of . In Fi g. 4, the e xtr usion die surfac e s hape o n the tw o recta ngular symmetr y planes of the produc t is presented. T he adaptable die shape with rotational flexibility is different fro m the die shape obtained witho ut adap tatio n in the rota tional direction especially along the φ = π /2 symmetry plane. The adaptable die shape geo metr y along the φ = π /2 symmetry plane increases the speed of the ma terial in the defor matio n zo ne near the e xit. Fig. 5 show s t he resul ting distorted grid in the e xtr udate o n the tw o symmetr y planes. T he adaptable die shape show s a smaller differe nce in distortion as co mpared to the die shape without the rotational flexibility. Fig. 3. Streamlined adaptable die shape with adapta tion of the die shape in the rotational direction with red = , L/Ro= ,mf= ,Rr/Ro= and μ = . Fig. 4. Streamlined die shape with no adaptation as the rotational direction pared to streamlined die shape with adaptation as a function of the rotational direction—plotted along rectangular symmetry planes. Fig. 5. Extrudate distorted grid along rectangular symmetry planes for extrusion through a streamlined die shape with no adaptation as the rotational direction pared to a streamlined die shape with adaptation as a function of the rotational direction—plotted along rectangular symmetry planes. Ar is the width of he extrudate and Br is the height of the extrudate. 5. Summary This paper presented an