【正文】 . Obviously, the flow deflection is caused by the distortion of the space time function due to the narrowness of the flow channel.For an unbounded cavity, the flow path of a melt particle will be a straight line which is a geodesic curve in the flat Euclidean space. This is analogous to the line of sight from the source (or gate).If the cavity is bounded, and with the assumption that the boundary is absorbent o?ering no reflection, a secondary source arises at the point where the line of sight hits the boundary. In this case, the cavity is partitioned by two sources, a gate and a secondary source. The secondary source is not inhibited for the melt from the gate stopped by the boundary. The space time geometry of the gate is trimmed o? due to the boundary of the cavity and the line of sight.The source characteristics for primary source (that is the gate) is vj (assume constant filling velocity) and tj (= 0, zero time delay) while the characteristics for the secondary source k is vj and tk188。djkvj
, where djkis the geodesic distance between j and k. Fig. 10(a) shows the space time function. The secondary source is flared out due to the line of sight jk. The flow fronts are obtained by sectioning the distorted space time geometry as illustrated in Fig. 10(b) and projecting on to the (2D) space as shown in Fig. 10(c). A similar Moldflow plot is shown in Fig. 10(d).. Merging Certain circumstances, two virtual sources exist and cause interference (for wave propagation) or weld/meld line (for plastic flow). Mould and die cavity is partitioned according to the line of sights from the sources. Interference arises when there is more than one source in the partition. In this case, a weld line is formed when two flow fronts meet each other.The cavity is partitioned by the line of sight because of the insert. These bounded partitions distort the space time function. The distorted space time function due to the di?ractions is shown in Fig. 11(a). The flow fronts are obtained by sectioning the distorted space time function and projecting onto the space as depicted in Fig. 11(b). The flow fronts are shown in Fig. 11(c). TheModlflow plot is shown in Fig. 11(d) for parison. 6. Flow front propagation in a cavity The flow deflection is mainly caused by two features in the cavity: inserts and change in wall these features exist in the cavity, the space timeFig. 12. A cavity with di?erent flow velocity. (a) Cavity with secondary sources。 (b) flow pattern。 (c) Moldflow plot.function is distorted. The flow paths of the melt particles in the cavity are the projection of the geodesics on the space time function onto the space.Fig. 12(a) shows a cavity with two di?erent wall thickness so that the flow velocity ratio in region J and region Kis11:5. An insert is in the region K. Meltenters the cavity from the source j. Refraction occursat the interface betweenregion J and region K, hencethe lines of sight from the primary source j enteringregion kis not straight lines. Four secondary sources k1, k2, k3 and constant filling velocity in each region is assumed. The flow front due to the distorted space time is shown in Fig. 12(b).Fig. 12(c) shows the Moldflow plot for parison. There is a discontinuity in the space time function which accounts for the existence of a weld line after the insert. Fig. 13. Filling pattern of an injection moulding with various wall thicknesses. (a) Moulding。 (b) space time function of source e12。 (c) space time function of source e23。 (d) space time function of source e13。 (e) resultant space time function in domain D3。 (f) flow front and weld line.Fig. 14. Flow simulation by MoldFlow.Interfaces between these domains. The gate location is at g in domain D1, hence domain D1 is filled by the single primary source g. As the melt goes from domain D1 to D2 through edge e12, e12 is induced to be a secondary line source. The space time function of this line source is obtained by sweeping the inverted cone with time delay along the edge e12 as shown in Fig. 13(b).Similarly, domains D3 possesses two line sources e13 and e23. Fig. 13(c) and (d) show the space time functions in domain D3 from source e13 and e23, respectively. Fig. 13(e) gives the resultant space time function in D3. The flow front propagation is obtained by sectioning this function. A weld line is formed at the intersection of two space time functions as shown in Fig. 13(f).Fig. 14 shows the simulation plotted by injection mould flow analysis software Moldflow for . 15(a) shows the flow fronts generated in a television case moulding of constant wall thickness. The part is moulded by a twogate injection mould. The image shown in Fig. 15(b) is generated by Moldflow software for parison. The two filling patterns basically conform to each other. Since this moulding consists of two gates located in the big opening and one small opening (for control buttons), at least three weld/meld lines are expected. Actually, four additionalmeld lines are shown on the two sides (two on each side) of the moulding explicitly since the part is only symmetric about one axis. Although these meld lines are not so obvious in Fig. 15(b), they still can be identified due to the change in curvatures of the flow fronts in the corresponding regions. Manufacturing operations such as moulding, casting, heat treatment, rolling, forming, and forging involve shape changes. Quantitative assessment and prediction of the result have always required the solution to partial di?erential equations from fluid dynamics and thermodynamics,the modeling of which require insight into the particular process under consideration. Then,Fig. 15. Flow fronts in a twogate moulding. (a) Flow fronts generated by the geometric approach。 (b) flow fronts generated by Moldflow , the solution to the partial di?erential equation sinvolving integration under boundary conditions isknown to be velocity of flow is assumed to be proportional to the part wal