外文翻译--直接自适应切片在理想材料零件的CAD模型（IFMC）

1 中国机械工程学报 v01 18， No 1， 2005 徐道明 镇沅家 郭东明 重点实验室精密和非传统加工技术应用教育部 ， 大连理工大学 ， 116024 中国大连。 直接自适应切片在理想材料零件的 CAD 模型（ IFMC） 摘要：一个全新的直接自适应分层的方法，可明显提高零件精度和减少建立时间。班至少有两个阶段都包含在这个操作：得到的切削平面与固体部分和确定的层厚度的交叉轮廓。除了通常的 SPI 算法，该固体模型切片它的特殊要求，使横截面的轮廓线段尽可能是其中之一 这是提高制造效率，通过自适应地调整方向的一步， 在每个交叉点的步骤的大小来获得优化的咬合高度达到。层厚度的测定可分为两个阶段：基于几何厚度和厚度估计基于验证材料。前一阶段的几何公差过程分为两个部分：各种曲线由圆弧近似，引入了第一部分，和 LM 过程的轮廓线之间的偏差和圆弧生成第二部分后一阶段主要是验证估计在前一阶段的层的厚度和确定一个新的必要的话。 关键词：快速原型 理想材料零件 直接自适应切片 表面平面 交叉 行军 0引言 理想材料零件（ IFMC）是一种新型的材料组分为科学技术发展所需的类。 快速原型制造（ RP&M）技术，或者叫 SFF（固体无 模成形）技术，是制造的理想材料零件的基本技术。 它是基于的原理制造层的层。与传统制造工艺相比，那些使用 RP&M 技术目前是耗时的部分依赖，但在处理具有宽范围的形状零件 具有 柔性 固体部分的切片是一种理想材料零件的基本步骤在制造过程。 阐述了 RP 工艺原理直观，可应用于相关的阶段， 如方向，支持生成，等。 目前，切片是主要处理无数的三角面片逼近的部分，那就是， STL 文件。由于其固有的缺点，这样直接切片的部分模型更是成为一个活跃的研究都可以达到任何灵活的自适应允许割线的高度。此外，也有两种类型的分层策略：均匀分层 自适应切片。与前者相比，后者能用较少的时间完成建设较高的表面精度。 2 P.卡尼和 D. Dutta 讨论一个准确的切片程序 LM过程。 在此基础上， v.kumar，等人，进一步描述了一种更一般的切片过程中的 LM非均质模型。 W. Y.马和 P. R.他提出了一个算法，即自适应切片孵化战略选择。 一种新的方法，称为局部自适应切片技术进行了简要的介绍了贾斯廷 tyberg，等。 一种自适应分层方法在二语习得过程西方公司旗下，三富。 ET ALT， K.玛尼，等人扩展他们的早期作品，说裁判。 2,3自适应的 CAD 模型切片 。 另一个全新的直接自适应分层策略提出了由至少两个阶段：得到的交叉轮廓和确定层的厚度。前者主要是处理得到的断面轮廓线段尽可能根据固体部分的几何特征，后者试图确定切片层由轮廓在第一阶段的基础上获得的几何特性和材料设置综合分析的厚度。两者交替进行直至切层在预方向到达的最后部分定义的取向。 1跟踪沿交叉曲线 一般来说，在 CAD 模型的表面是由平面，圆锥曲线和曲面。 切割零件的实体模型的切割平面问题，事实上，一个 SPI（表面平面交叉口）从几何问题，这可以被视为一个特殊的情况下（ SSI 表面曲面求交问题。 SSI 问题的方法通常分为两类：解析法和数值方法（主要是推进基于或细分算法）。此外，基于微分几何原理的算法是近年来迅速发展起来的。 平面交叉口之间 和一个参数的表面可以被视为一个扩展 和特殊情况下的交叉参数化的表面和表面之间。 行进中的基础算法计算一个切割平面与一个理想材料零件的 CAD 模型的参数曲面求交的轮廓，其中一个突出的特点是允许充分利用咬合高度。 1.1 对于具有参数曲面的线交叉点算法计算 让 代表一条直线，在 AI 上表面附近的点线，是本线和 T为参变量的方向矢量。让我们（ U， V）表示一个曲面的参数变量 u 和 v从某一初始点在直线和平面，一个迭代过程可以进行，得到一个真正的交叉点，以满足表达 扩大这种表达，我们可以得到 3 牛顿迭代法求解这组方程 假设 可以得到以下方程 让 T = 0的函数 f的变量的初始值（ T），对应点的 AI。让我们（ U， V）被认为是最接近的表面上的点，即， Bz 和双值（ U， V）的变量对初始值（ U， V）表达的（ U， V）。 毫无疑问，迭代过 程将持续到下 是满意的，其中 是一个预先设定的允许误差，和作为一个结果，真正的交叉点 1.2 该步骤的方向和步长的初始估计 假定曲率点的 Pi表面上是 Ki。那里的步进方向和步长的初步评估是根据曲率 KI测定。 在这种情况下，割线的高度不能满足要求的优化步骤，中间值定理和线 4 性插值的方法将联合应用，得到优化的步进方向和步长。方向的一步，对于点Pt下点 9月的大小（见图。 1）是由方程 4决定 其中一个是切向量之间的夹角，在点 PI 和步进方向向量 ，即，估计步长方向；我是估计 的步长； R 对应估计曲率 KI 圆半径； H 是预先设定的容许咬合高度。 图 1 选择下一步 1.3 优化的步骤 实际的交叉点的部分的表面的步骤是在 1.1节中介绍的算法来计算的。然而，这并不意味着得到满足预先设定的要求和咬合高度进行优化。优化的步骤的标准可以是多种多样的。在本文中，我们将有咬合高度 0.9 H H H ，其中 H 为许用割线高度设定值。 让 H1是计算正割高度有一定夹角的 A1对应，这是小于 H ，而 Hg大于 H对应的夹角银。我们可以构建一个变小时，即功能， = F（ H）。扩大，我们 5 根据表面的连续性假设和中值定理，我们可以通过线性插值的方法获得估计的如 步长可以计算由方程（ 4）与 这个周期将被重复直到咬合高度满足优化咬合高度要求。 2 阶梯效应和遏制的问题 两个主要因素影响几何计算的基础层的厚度和表面加工精度是阶梯效应和遏制的问题。换句话说，基于几何层厚度的允许的牙尖高度主要取决与切片平面在一定高度的原始 CAD 模型的表面形状。 （ 1） 阶梯效应是由 LM 过 程的特点而形成的。它是由物理参数表示：牙尖高度，如图 2所示。 6 图 2 阶梯效应和遏制风格 （ 2） 安全问题是指包含关系的部分原始 CAD模型的轮廓和沉积在 LM过程后的实际，这是通过平面的轮廓的讨论，在算法中沉积的策略表示，如图 2所示。 让 Sc的部分原始 CAD 模型的二维轮廓； S1是逼近折线 Sc 的 LM 的形成过程。 它可以从图的情况下看到（一）正公差和案例（ B）是负公差而案例（ c）和（ d）混合公差。 3 基于几何 层厚度估计 对某些层 的层厚度的确定算法的粗糙的流程图如图所示，最大层的厚度是由特定的 LM 工艺和设备的确定。 7 图 3 层厚度的确定算法流程图 几何基础层厚度计算在任何点上的轮廓线的切片平面是马的最低层的厚度对切片轮廓各点的基础上。 通常，一个逃离曲线由圆弧和直线近似可以被视为一个圆的曲率为零。因此我们可以集中我们的讨论在圆弧误差分析对切片平面的层位于同一纵截面的两个点作为一个自由曲线或圆弧的终点。 3.1 误差准则 在某点 的误差准则被定义为偏离所建立的轮廓线的层在 LM 从正常的曲线在某点上下分层平面。一般说来，误差值是通过允许尖高度代表。 8 偏差的一个综合性的概念，一般可以分为两个部分：（ 1）的圆弧曲线或直线，逼近误差说 。从该层的轮廓线，圆弧的错误，说 。从而，允许的牙尖高度，说 ，由用户，可以全面的价值。它们之间的关系如下图所示 3.2 误差分析 3.2.1逼近误差 原来的曲线和逼近圆弧之间的误差是由 ，作为显示在图 4A。假设在两个端点曲率， Q1和 Q2，正常曲线 K1和 K2。因此，对圆弧 C1曲率估计的定义是 从中心点曲线 C2端点之间，说第三季度，沿垂直方向的线段 q1q2，高度误差之间的正常曲线 C2和 C1有圆弧割线 H2和 = | H2 |。在特殊情况下，例如，正常曲线 C2降低到一条直线，圆弧的曲率为零的 C1和 =0。 3.2.2 偏差 错误的定义是 F 层的轮廓线的偏差距离逼近圆弧，这是相对于 G有两种情况计算误差 F复杂一点：一是圆弧的谎言在一季度的圆，如图 4b；另一个是圆弧跨越一个四分之一圆，位于半圈， 在 图 4c 和 4d 显示。它们将分别在下面讨论 。 签署了包括交叉曲线 与取向方向两端点的切矢角可以得到，如 A3的角度 图 4c。签署产品积极结果是相应的案例（ B）而相反的是相应的案例（ C）和（ D） （ 1） 在一个单一的象限圆弧 圆弧半径 图。基于平面几何，我们有 这 9 （ 2）圆弧过象限 在 图 4c，圆弧是在用过量的沉积策略的凸函数。 假设在 A3点四比一点第四季度，我们 。 在图 4d,圆弧是缺乏沉积策略的凸函数。 (a)圆弧逼近自由曲线 （ b)在一个单一的 象限圆弧过度沉积 （ c）在圆弧过象限过量沉积 （ d）对电弧在一个象限缺乏沉积 图 4 逼近误差和偏差 假设在 A3点 Q4大于一点 Q3， 我们有 。 在这种情况下，电弧是在一个缺乏或过量沉积策略具有相同的处理方法如上所述的情况下 图 .4c 或 图 .4d 分别凹函数。 10 3.3 错误和层厚度 如果当前层厚度不能满足牙尖高度的要求，降低层的厚度进行估计的一种新的周期。在本文中。当前层厚度的 DG除以 n = 100和价值的 DG / N作为层厚度递减。 在某一层的厚度估计将被视为在该层的厚度估计过程的下一点的当前层厚度的初始值。 4 基于材料 层厚度的检验 目的验证的材料是检查是否当前层厚度符合要求，材料制造，如果当前没有获得一个新的层厚度值。具体而言，一个随机选择的空间点上的可用区域低的切片平面某一种物质的区域是用来验证当前层的厚度，而这个过程的初始值是由材料的区域的几何形状确定如第 3节所提到的材料属性；如果当前层厚度不符合材料的要求，该层的厚度逐渐减小直至满足要求；得到的层的厚度在这一点上，然后作为下一次验证过程的初始值；这个周 期将持续到一个预先设定的总数 N点进行了验证。 在本文中，验证过程主要集中在功能梯度材料（功能梯度材料）。 4.1材料的检查 要在取向方向接近的材料的体积百分比曲线圆弧的方法不同于使用第 3节中的方法。在材料区域的某些材料的体积百分比可以被视为在取向方向的高度的函数。 Z轴的简化，即， P1 = F（ Z1）。 把材料的第一优先为例。从某一点上下分层 Q1的平面层高度 Z1，延长距离当前层的厚度，我们在高度 22一点 Q2。材料的体积百分比 P1， P2和 P3点 Q1， Q2和 Q3的中间点，分别说在高度 23。 结合三体积 百分比， P1， P2和 P3，我们可以从点七的距离与当前层厚度沿导向轴构造一个近似圆弧的体积百分比曲线，如图 5所示。 11 （ a） 圆弧是单调的 （ b）圆弧是非单调 图 5 逼近圆弧曲线的材料 让圆弧的中心是（ Zo， PO）。如果（ Z1-Zo） x（ Z2 Zo） 0，圆弧的定义为5A 条相应的单调而相反的是定义为非单调对应图 5b。每种情况都有不同的解决方式。 4.2 误差 的 分析 一个必要但不充分的条件下，本文提出验证当前层厚度。三 个主要因素，材料变异性的界限，材料分辨率在逼近圆弧的端点的材料的体积百分比，主要考虑。 在图 5， 代表材料的体积百分比， LM 设备可以存放在实践中较低的切平面的层达到一定高度取向轴。在图 5A 的情况下，下面的关系需要进行测试，验证层厚度 代表这个 LM 机材料分辨率。 这个方程的一个充要条件。实际上，它可以简化验证层厚度。从式（ 10），我们有 如图 5b，这些变量测试的关系 12 或者 在 P4是圆弧的材料百分比极值。 同样，我们有 由式表示的条件。（ 11）和（ 13）是必要但不充分的条件下，可方便地应用于验证层的厚度。在这两个方程，考虑三个主要因素。 不满足这些条件，该层的厚度必须逐渐减少执行另一个周期的验证。 5 例 如图所示，有一个自由曲面主要由两个裁剪曲面的 NURBS 表示 ISO 10303协议如下。在正常的方向为 Y 轴相对的斜面（不绘制在图 6）作为一个切割平面相交的表面。 图 6 自由曲面在笛卡尔 坐标系统 13 有两种主要的误差与切削过程一致的表面上的点对点线距误差和误差咬合高度相关的线段相交的曲线逼近。这两个错误分别是 l0-4毫米和 10-1毫米。 三种不同的算法结果的比较见表 l 上市，其中算法的 L是指分半步长折半查找法，算法 2表示二进制搜索法和分半步的方向； 3代表算法自适应方法，根据中值定理和线性插值相结合的方法旋转角的变化。 从表 1，它是已知的两个算法，算法 2我很难获得更好的结果。 3使用线性插值算法具有更好的综合效果比算法 1和 2。 材料的设置属性附加到了这部分内容如下。 最低层厚度： 0.01mm 最大层厚度： 0。 1毫米 材料认证面积密度： 0。 l mm2 材料沉积策略：多余的材料类型： FGM 外表面的公差： 0.02毫米 内部表面的公差： 0.05毫米 材料下公差： 0毫米 材料上公差： 0.1mm 材料分辨率： 0.1 为第一优先的成分的材料分布函数 14 其中 R 是远处的一个空间点远离方向轴， Z 是该点坐标分量；符号“ ABS”意味着“绝对值”。零件的 CAD 模型的起源和材料性能的起源是一致的和定向的矢量是（ 0， 1， 0）在这个例子。 一部分连续的层厚度从 z = 15是在表 2中列出的向上的我（我 = 1， 2， .” 10），是第 i层； DG代表的几何特征估计层厚度； DM代表材料为基础的验证如上所述在层的厚度，这是第 i 层的最后一层厚度。 从表 2可以看出，通过自适应分层产生的层的厚度可以在一个相对较大的范围根据综合因素包括曲面的几何特征和零件的材料属性，这无疑可以与均匀切片技术相比减少建造时间。 6 结论 所描述的工作重点是分层制造过程的理想材料零件。直接切片方法直接切片的部分原始 CAD 模型，通常保持足够的几何信息，优于 STL 文件，因此，导致改进的精度。 SPI 本文提 出的算法具有一个突出的特点是充分利用允许的咬合高度。自适应切片也可以改善切削精度和减少建筑时间比较均匀的切片。几何信息是用于预测层的厚度和材料的信息是用来验证层的厚度和确定一个新的必要的话。 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 15 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 an 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 16 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 17 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 18 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 19 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 0.9hh h， whereh stands for the pre-set value of the allowable secant height Let h1 be a calculated secant height corresponding with certain included angle a1， 20 which is less thanh， while hg is greater than h corresponding with included angle ag We can construct a function of variable h， that is， =f(h) Expanding it， we have According to surface continuity assumption and the intermediate value theorem， we can obtain an estimated by linear interpolation method as follows The 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 PROBLEM Two main factors that affect the calculation of geometry-based layer thickness and surface finish accuracy are the staircase effect and the containment problem In 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 process It is represented by physic parameter： the cusp height as shown in Fig 2 21 (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 Fig 2 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 process It can be seen from Fig 2 that case (a) is positive tolerance and case (b) is negative tolerance while case (c) and (d) are mixed tolerance 3 GEOMETRY-BASED LAYER THICKNESS ESTIMATION The 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 22 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 contour Usually,a flee curve can be approximated by a circular arc and a straight line can be 23 regarded as a circle with zero curvature Therefore we can focus our discussion on error analysis of the circular arc Two 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 criterion The 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 plane In general the 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 them The relationship between them is shown as below 3.2 Error analysis 3.2.1 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. 24 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. 25 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 26 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 VERIFYING The 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 check The 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. 27 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 28 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 be reduced 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 29 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 met