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購(gòu)買設(shè)計(jì)請(qǐng)充值后下載，資源目錄下的文件所見(jiàn)即所得，都可以點(diǎn)開(kāi)預(yù)覽，資料完整，充值下載可得到資源目錄里的所有文件?！咀ⅰ浚篸wg后綴為CAD圖紙，doc,docx為WORD文檔，原稿無(wú)水印，可編輯。具體請(qǐng)見(jiàn)文件預(yù)覽，有不明白之處，可咨詢QQ:12401814 CHINESE JOURNAL OF MECHANICAL ENGINEERING v0118，No1，2005 XU DaomingJia ZhenyuanGuo DongmingKey Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education， Dalian University of Technology, Dalian 116024China DIRECT AND ADAPTIVE SLICING ON CAD MODEL OF IDEAL FUNCTIONAL MATERIAL COMPONENTS(IFMC)Abstract：A brand new direct and adaptive slicing approach is proposedwhich can apparently improve the part accuracy and reduce the building timeAt 1east two stages are included in this operation：getting the crossing contour of the cutting plane with the solid part and determining the layer thicknessApart from usual SPI algorithm，slicing of the solid mode1 has its special requirements Enabling the contour 1ine segments of the crosssection as long as possible is one of themwhich is for improving manufacturing efficiency and is reached by adaptively adjusting the step direction and the step size at every crossing point to obtain optimized secant heightThe layer thickness determination can be divided into two phases：the geometrybased thickness estimation and the materialbased thickness verifyingDuring the former phasethe geometry tolerance is divided into two parts：a variety of curves are approximated by a circular arc，which introduces the first part，and the deviation error between the contour line in LM process and the circular arc generates the second part The latter phase is mainly verifying the layer thickness estimated in the former stage and determining a new one if necessaryIn additionan example using this slicing algorithm is also illustrated Key words：Rapid prototyping Ideal functional material components Direct and adaptive slicing Surfaceplane intersection Marching0 INTRODUCTIONIdeal functional material components(IFMC)is a novel class of material component required for the development of science and technology Rapid prototyping and manufacturing(RP&M) technology,or called SFF(solid freeform fabrication) technology，is a fundamental technology for manufacturing of IFMCwhich is based on the principle of manufacturing layer by layerCompared with traditional manufacturing processes，those of applying RP&M technology currently are time-consuming with part dependence，but flexible in handling parts with shapes of wide rangeSlicing of the solid part is one of the elementary steps ln the process of manufacturing IFMCwhich illustrates the principle of RP process Intuitively and can be applied to relevant stages，such as orientation，support generation，etcAt present，slicing is mainly processed on a myriad of triangular facets approximating the part，that is，STL fileOwing to its intrinsic disadvantages，the way of directly slicing on the part model is becoming a more active research topicwhich can reach any flexibly adaptive allowable secant heightMoreover,there are also two types of slicing strategy：the uniform slicing and the adaptive slicingCompared with the former,the latter can accomplish a higher surface accuracy with less building timeP. Kulkarni and DDutta discussed an accurate slicing procedure for LM processBased on it，VKumar，et al ，further described a more general slicing procedure in LM for heterogeneous modelsW. Y. Ma and P. RHe introduced a developed algorithm，namely an adaptive slicing and selective hatching strategy A brand new approach，termed as the local adaptive slicing technique is briefly introduced by Justin Tyberg,et al .An adaptive slicing method is adopted in SLA process by A.P. West，S.P. Sambuet alt ，K Mani，et al extended their earlier works，say Refsf2，31，to adaptive slicing of CAD model Another brand new direct and adaptive slicing strategy proposed in this paper consists of at least two stages：getting the crossing contour and determining the layer thicknessThe former is mainly processed to get the contour line segments of the crosssection as long as possible according to geometry features of the solid part while the latter intends to determine the thickness of the slicing layer built from the contour obtained in the first stage based on the comprehensive analysis of both geometry features and material settingsBoth of them are conducted alternatively until the slicing layer reaches the end of the part in the direction of pre-defined orientation 1 TRACING ALONG THE CROSSING CURVE Generally,the surface in CAD model is expressed by plane,conic and parametric surfaceThe problem of slicing the solid model of the part by cutting plane is,in fact,a SPI(surfaceplane intersection)problem from viewpoint of geometry, which can be regarded as a special case of SSI(surfacesurface intersection problemApproach to SSI problem is usually classified into two categories：the analytic method and the numerical method (mainly marching-based or subdivision-based algorithms) . Moreover, algorithms based on the principle of differential geometry are developed rapidly in recent years. Intersection between a plane and a parametric surface can be regarded as an extension and a special case of the intersection between a parametric surface and a surface A marching-based algorithm is employed in this paper to compute intersection contours of a cutting plane with a parametric surface of the CAD model of IFMC,a distinguished characteristic of which is the utilization of allowable secant height to full extent1.1 Algorithm for computing crossing point of a line with a parametric surfaceLet represent a straight line，where ai is a point on the line near a surface，is the direction vector of this line and t stands for parametric variableLet S(u，V)denote a surface with parametric variables u and VFrom certain initial points at both the straight line and the surface，an iteration process can be conducted to get a true crossing point，which satisfies expression Expanding this expression，we can obtain The Newton-Raphson method is applied to solve this system of equationsAssuming that Following equations may be obtained Let t= 0 be the initial value of variable t for function f(t) ,corresponding to point aiLet S(u ，v )be the point that is closest to a on surface S ,that is，point bz and the dual value(u ，v ) are the initial values of variable pair(u，v)for expression S(u，v).It is no doubt that the iteration process will be continued until condition is satisfied，where is a preset allowable error， and as a result， the true crossing point 1.2 Initial estimation of the step direction and the step sizeAssume that the curvature at point Pi on the surface is Ki There by the initial evaluation of the step direction and the step size are determined according to curvature Ki. in the case that the secant height can not meet the requirement of optimized step , the intermediate value theorem and the linear interpolation method will be jointly applied to get the optimized step direction and step size . The step direction and the sept size for the next point of point Pt (see Fig .1) is decided by Eq . (4) where a is the separation angle between the tangent vector Vt at point pi and the step direction vector that is , estimated step direction;l is the estimated step size; r is the circle radius corresponding to estimated curvature ki ; h is pre-set allowable secant height . 1.3 Optimized stepThe practical crossing point of the step line with the surface of the part is computed by the algorithm introduced in section 11However,it does not mean that the resulting secant height can satisfy pre-set requirement and it is optimizedThe criterion for optimized step can be variousIn this paper,we set the secant height have to be 0.9hhh，whereh stands for the pre-set value of the allowable secant heightLet h1 be a calculated secant height corresponding with certain included angle a1，which is less thanh，while hg is greater than h corresponding with included angle agWe can construct a function of variable h，that is， =f(h)Expanding it，we haveAccording to surface continuity assumption and the intermediate value theorem，we can obtain an estimated by linear interpolation method as followsThe step size can be calculated by Eq.(4) with.This cycle will be repeated until the secant height satisfies optimized secant height requirement.2 STAIRCASE EFFECT AND CONTAINMENT PROBLEMTwo main factors that affect the calculation of geometry-based layer thickness and surface finish accuracy are the staircase effect and the containment problemIn other words，the geometry-based layer thickness is mainly determined by the allowable cusp height and the surface shape of the original CAD model over the slicing plane at certain height(1) Staircase effect is formed by the characteristic of LM processIt is represented by physic parameter：the cusp height as shown in Fig2 (2) Containment problem refers to the containing relationship of the contour of the original CAD model of the part and the actual one after depositing in LM process，which is discussed through planar profile and is denoted by deposition strategy in this algorithm，as shown in Fig2 Let Sc be the 2D profile of the original CAD model of the part；S1 be the approximating fold lines of Sc formed by the LM processIt can be seen from Fig2 that case (a) is positive tolerance and case (b) is negative tolerance while case (c) and (d) are mixed tolerance3 GEOMETRY-BASED LAYER THICKNESS ESTIMATIONThe rough flowchart of layer thickness determination algorithm for certain layer is illustrated in Fig.3 and the maximum layer thickness is determined by specific LM process and equipment Geometry-abased layer thickness calculation at any point on the contour line in the slicing plane is the basis for geeing the minimum layer thickness among all points on the slicing contourUsually,a flee curve can be approximated by a circular arc and a straight line can be regarded as a circle with zero curvatureTherefore we can focus our discussion on error analysis of the circular arcTwo points on both slicing planes of the layer lying in the same longitudinal section are taken as the endpoints of a free curve or a circular arc.3. 1 Error criterionThe error criterion at certain point is defined as deviation of the built up contour line of the layer in LM from the normal curve at certain point on lower slicing planeIn generalthe error value is represented by allowable cusp height.The deviation error is a comprehensive concept which can generally be divided into two parts：(1)The error of the circular arc approximating a curve or a straight line，say The error of the circular arc from the contour line of the layer，say Thereby,the allowable cusp height，say ，set by the user,can be a comprehensive value of themThe relationship between them is shown as below 3.2 Error analysis3.2.1 Approximating error The error between the original curve and the approximating circular arc is represented by , as shown in Fig.4a. Assume that the curvatures at both endpoints, q1 and q2, of normal curve are k1 and k2. Therefore, an estimate of curvature of the circular arc c1 is defined as. From a middle point between endpoints of curve C2, say q3, along the direction perpendicular to line segment q1q2, the height error between normal curve C2 and circular arc c1 has secant h2 and =|h2| . In special cases, for example, the normal curve C2 degrades to a straight line l, the curvature of circular arc c1 is zero and =0.3.2.2 Deviation error The definition of error is the deviation error of the contour line of the layer away from the approximating circular arc, which is a little complex compared with There are two cases for calculating error : one is that the circular arc lies within a quarter of circle, as shown in Fig.4b; another is that the circular arc spans over a quarter of circle and lies within one-half circle, as shown in Figs.4c and 4d. They are to be discussed in the following, respectively The signed included angle of tangent vectors at both end-points of crossing curve with the orientation direction can be obtained, such as a3 in Fig.4c. The positive consequence of the product of both signed angles is corresponding with case (b) while the opposite is corresponding with case (c) and case (d)(1) Circular arc in one single quadrant The radius of arc is in Fig.4b. Based on plane geometry, we have Where (2) Circular arc over one quadrant In Fig.4c, circular arc is in a convex function with excess deposition strategy. Assuming a3 at point q4 is greater than the one at point q4, we have . In Fig.4d, circular arc is in a convex function with deficient deposition strategy. Assuming a3 at point q4 is greater than the one at point q3, we have . In the case that arc is in a concave function with deficient or excess deposition strategy has the same tackling method as mentioned above to case of Fig.4c or Fig.4d. Respectively.3.3 Error and layer thickness If the current layer thickness can not meet the cusp height requirement, a reduced layer thickness is used to perform a new cycle of estimation. In this paper. the current layer thickness dg is divided by N=l00 and the value dg /N is taken as decrement of the layer thickness. The estimate of the layer thickness at certain point will be taken as an initial value of the current layer thickness at next point in the process of estimate of the layer thickness.4 MATERIAL-BASED LAYER THICKNESS VERIFYINGThe purpose of material verifying is to check if the current layer thickness can meet material manufacturing requirement, and to obtain a new value of the layer thickness if the current one failed. Specifically, the material attribute of a randomly selected space point on the available region of lower slicing plane of a certain material region is used to verify the current layer thickness while the initial value of this process is determined by geometry shape of the material region as mentioned in section 3; If the current layer thickness fails to meet the material requirement, the layer thickness will be reduced gradually until it satisfies the requirement; the layer thickness obtained at this point is then taken as the initial value for the next verifying process; this cycle will continue until a pre-set total number of N points are verified. In this paper, the verifying process is mainly focused on FGM (functionally gradient material).4.1 Material checkThe way to get approximating circular arc of the material volume percentage curve in the direction of orientation differs from the way used in section 3. The volume percentage of certain material in material region can be regarded as a function of height in the orientation direction. axis z for simplification, that is, p1=f(z1).Take material with No.l priority for an example. From certain point q1 on lower slicing plane of the layer with height z1, prolonging a distance of current layer thickness, we have another point q2 at height 22 . The material volume percentages are p1, p2 and p3 at point q1, q2 and the middle point of them, say q3 at height 23, respectively. Combining three volume percentages, pl, p2 and p3, we can construct an approximating circular arc of the volume percentage curve from point qi along orientation axis with a distance of current layer thickness, as shown in Fig.5. Let the center of circular arc be (zo, Po). If (z1 - zo) x (z2 - zo) 0 , the circular arc is defined as monotone corresponding with Fig.5a while the opposite is defined as non-monotone corresponding with Fig.5b. Each case has different tackling way.4.2 Error analysis A necessary but not sufficient condition is proposed in this paper to verify the current layer thickness. Three main factors, the material variation tolerance boundaries, the material resolution and the material volume percentages at endpoints of approximating arc, are mainly taken into consideration. In Fig.5, represents material volume percentage that the LM equipment may deposit in practice over lower slicing plane of the layer at certain height in the orientation axis. In case of Fig. 5a, the relationship below is needed to be tested to verify the layer thickness where stands for material resolution of this LM machine. This equation is a sufficient and necessary condition. Actually, it can be simplified to verify the layer thickness. From Eq.(10), we have In case of Fig.5b, the tested relationship of those variables is Or where p4 stands for the material percentage extremum of this circular arc. Similarly, we have The conditions represented by Eqs.(11) and (13) are necessary but not sufficient conditions, which are convenient to be applied to verify the layer thickness. In both equations, three main factors are taken into account. Without meeting those conditions, the layer thickness has to bereduced gradually to perform another cycle of verifying.5 EXAMPLE As shown in Fig.6, there is a freeform surface mainly composed of two trimmed surface patches which are represented by NURBS following IS0 10303 protocol. An oblique plane relative to axis y in normal direction (not drawn in the Fig.6) is taken as a cutting plane to intersect the surface. There are two main errors associated with the cutting process the error for coincident points on a surface apart from point on a line and the error for secant height of the line segments approxmating the intersecting curve. Both errors are l0-4 mm and 10-1 mm, respectively. Comparison of results of three different algorithms is listed in Table l, in which algorithm l refers to the binary search method with the split-half step size while algorithm 2 denotes the binary search method with the split-half step direction; algorithm 3 represents adaptive method in which the rotation angle changes according to combination of intermediate value theorem and linear interpolation method. From Table l, it is known that both algorithm I and algorithm 2 are hard to obtain better results. Algorithm 3 using linear interpolation achieves a better comprehensive result than both algorithms 1 and 2. The setting of the material attribute attached to this part is listed as below. The minimum layer thickness:0.01mm The maximum layer thickness: 0. 1 mm Material certification area density : 0. l / mm2 Material deposition strategy: Excess Material type: FGM Exterior surface tolerance: 0.02 mm Interior surface tolerance: 0.05 mm Material lower tolerance: 0.0 mm Material upper tolerance:0.1mm Material resolution: 0. 1 The material distribution function for ingredient with No.1 priority is where r is the distance of a space point away from the orientation axis while z is the coordinate component of that point; symbol abs means absolute value. The origin of the part CAD model and the origin of the material function are coincident and the orientation vector is (0.0, 1.0, 0.0) in this example. A part of successive layer thickness upward from z =15.0 is listed in Table 2 in which No.i ( i = 1, 2, . , 10) is the ith layer; dg stands for the layer thickness estimated by geometry feature; dm represents the layer thickness after material-based verifying as mentioned above, which is the final layer thickness of the ith layer. It can be seen from Table 2 that the layer thickness gen