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CHINESE JOURNAL OF MECHANICAL ENGINEERING
v01．18，No．1，2005
XU Daoming
Jia Zhenyuan
Guo Dongming
Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education， Dalian University of Technology, Dalian 116024．China
DIRECT AND ADAPTIVE SLICING ON CAD MODEL OF IDEAL FUNCTIONAL MATERIAL COMPONENTS(IFMC)
Abstract：A brand new direct and adaptive slicing approach is proposed．which can apparently improve the part accuracy and reduce the building time．At 1east two stages are included in this operation：getting the crossing contour of the cutting plane with the solid part and determining the layer thickness．Apart from usual SPI algorithm，slicing of the solid mode1 has its special requirements Enabling the contour 1ine segments of the cross—section as long as possible is one of them．which 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 height．The layer thickness determination can be divided into two phases：the geometry—based thickness estimation and the material—based thickness verifying．During the former phase．the 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 necessary．In addition．a(chǎn)n example using this slicing algorithm is also illustrated．
Key words：Rapid prototyping Ideal functional material components Direct and adaptive slicing Surface／plane intersection Marching
0 INTRODUCTION
Ideal 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 IFMC．which is based on the principle of manufacturing layer by layer．Compared 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 range
Slicing of the solid part is one of the elementary steps ln the process of manufacturing IFMC．which illustrates the principle of RP process
Intuitively and can be applied to relevant stages，such as orientation，
support generation，etc．
At present，slicing is mainly processed on a myriad of triangular facets approximating the part，that is，STL file．Owing to its intrinsic disadvantages，the way of directly slicing on the part model is becoming a more active research topic．which can reach any flexibly adaptive allowable secant height．Moreover,there are also two types of slicing strategy：the uniform slicing and the adaptive slicing．Compared with the former,the latter can accomplish a higher surface accuracy with less building time．
P. Kulkarni and D．Dutta discussed an accurate slicing procedure for LM process．Based on it，V．Kumar，et al ，further described a more general slicing procedure in LM for heterogeneous models．W. Y. Ma and P. R．He 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. Sambu．et alt ，K． Mani，et al extended their earlier works，say Refs．f2，31，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 thickness．The former is mainly processed to get the contour line segments of the cross．section 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 settings．Both 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 surface．The problem of slicing the solid model of the part by cutting plane is,in fact,a SPI(surface／plane intersection)problem from viewpoint of geometry, which can be regarded as a special case of SSI(surface／surface intersection problem．Approach 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 extent．
1.1 Algorithm for computing crossing point of a line with a parametric surface
Let 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 variable．Let S(u，V)denote a surface with parametric variables u and V．From 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 equations
Assuming that
Following equations may be obtained
Let t= 0 be the initial value of variable t for function f(t) ,corresponding to point ai．Let 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 size
Assume 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 step
The practical crossing point of the step line with the surface of the part is computed by the algorithm introduced in section 1．1．However,it does not mean that the resulting secant height can satisfy pre-set requirement and it is optimized．The criterion for optimized step can be various．In this paper,we set the secant height have to be 0.9[h]
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