外文翻譯--直接自適應切片在理想材料零件的CAD模型（IFMC）

1 中國機械工程學報 v01 18， No 1， 2005 徐道明 鎮(zhèn)沅家 郭東明 重點實驗室精密和非傳統(tǒng)加工技術應用教育部 ， 大連理工大學 ， 116024 中國大連。 直接自適應切片在理想材料零件的 CAD 模型（ IFMC） 摘要：一個全新的直接自適應分層的方法，可明顯提高零件精度和減少建立時間。班至少有兩個階段都包含在這個操作：得到的切削平面與固體部分和確定的層厚度的交叉輪廓。除了通常的 SPI 算法，該固體模型切片它的特殊要求，使橫截面的輪廓線段盡可能是其中之一 這是提高制造效率，通過自適應地調整方向的一步， 在每個交叉點的步驟的大小來獲得優(yōu)化的咬合高度達到。層厚度的測定可分為兩個階段：基于幾何厚度和厚度估計基于驗證材料。前一階段的幾何公差過程分為兩個部分：各種曲線由圓弧近似，引入了第一部分，和 LM 過程的輪廓線之間的偏差和圓弧生成第二部分后一階段主要是驗證估計在前一階段的層的厚度和確定一個新的必要的話。 關鍵詞：快速原型 理想材料零件 直接自適應切片 表面平面 交叉 行軍 0引言 理想材料零件（ IFMC）是一種新型的材料組分為科學技術發(fā)展所需的類。 快速原型制造（ RP&M）技術，或者叫 SFF（固體無 模成形）技術，是制造的理想材料零件的基本技術。 它是基于的原理制造層的層。與傳統(tǒng)制造工藝相比，那些使用 RP&M 技術目前是耗時的部分依賴，但在處理具有寬范圍的形狀零件 具有 柔性 固體部分的切片是一種理想材料零件的基本步驟在制造過程。 闡述了 RP 工藝原理直觀，可應用于相關的階段， 如方向，支持生成，等。 目前，切片是主要處理無數的三角面片逼近的部分，那就是， STL 文件。由于其固有的缺點，這樣直接切片的部分模型更是成為一個活躍的研究都可以達到任何靈活的自適應允許割線的高度。此外，也有兩種類型的分層策略：均勻分層 自適應切片。與前者相比，后者能用較少的時間完成建設較高的表面精度。 2 P.卡尼和 D. Dutta 討論一個準確的切片程序 LM過程。 在此基礎上， v.kumar，等人，進一步描述了一種更一般的切片過程中的 LM非均質模型。 W. Y.馬和 P. R.他提出了一個算法，即自適應切片孵化戰(zhàn)略選擇。 一種新的方法，稱為局部自適應切片技術進行了簡要的介紹了賈斯廷 tyberg，等。 一種自適應分層方法在二語習得過程西方公司旗下，三富。 ET ALT， K.瑪尼，等人擴展他們的早期作品，說裁判。 2,3自適應的 CAD 模型切片 。 另一個全新的直接自適應分層策略提出了由至少兩個階段：得到的交叉輪廓和確定層的厚度。前者主要是處理得到的斷面輪廓線段盡可能根據固體部分的幾何特征，后者試圖確定切片層由輪廓在第一階段的基礎上獲得的幾何特性和材料設置綜合分析的厚度。兩者交替進行直至切層在預方向到達的最后部分定義的取向。 1跟蹤沿交叉曲線 一般來說，在 CAD 模型的表面是由平面，圓錐曲線和曲面。 切割零件的實體模型的切割平面問題，事實上，一個 SPI（表面平面交叉口）從幾何問題，這可以被視為一個特殊的情況下（ SSI 表面曲面求交問題。 SSI 問題的方法通常分為兩類：解析法和數值方法（主要是推進基于或細分算法）。此外，基于微分幾何原理的算法是近年來迅速發(fā)展起來的。 平面交叉口之間 和一個參數的表面可以被視為一個擴展 和特殊情況下的交叉參數化的表面和表面之間。 行進中的基礎算法計算一個切割平面與一個理想材料零件的 CAD 模型的參數曲面求交的輪廓，其中一個突出的特點是允許充分利用咬合高度。 1.1 對于具有參數曲面的線交叉點算法計算 讓 代表一條直線，在 AI 上表面附近的點線，是本線和 T為參變量的方向矢量。讓我們（ U， V）表示一個曲面的參數變量 u 和 v從某一初始點在直線和平面，一個迭代過程可以進行，得到一個真正的交叉點，以滿足表達 擴大這種表達，我們可以得到 3 牛頓迭代法求解這組方程 假設 可以得到以下方程 讓 T = 0的函數 f的變量的初始值（ T），對應點的 AI。讓我們（ U， V）被認為是最接近的表面上的點，即， Bz 和雙值（ U， V）的變量對初始值（ U， V）表達的（ U， V）。 毫無疑問，迭代過 程將持續(xù)到下 是滿意的，其中 是一個預先設定的允許誤差，和作為一個結果，真正的交叉點 1.2 該步驟的方向和步長的初始估計 假定曲率點的 Pi表面上是 Ki。那里的步進方向和步長的初步評估是根據曲率 KI測定。 在這種情況下，割線的高度不能滿足要求的優(yōu)化步驟，中間值定理和線 4 性插值的方法將聯合應用，得到優(yōu)化的步進方向和步長。方向的一步，對于點Pt下點 9月的大?。ㄒ妶D。 1）是由方程 4決定 其中一個是切向量之間的夾角，在點 PI 和步進方向向量 ，即，估計步長方向；我是估計 的步長； R 對應估計曲率 KI 圓半徑； H 是預先設定的容許咬合高度。 圖 1 選擇下一步 1.3 優(yōu)化的步驟 實際的交叉點的部分的表面的步驟是在 1.1節(jié)中介紹的算法來計算的。然而，這并不意味著得到滿足預先設定的要求和咬合高度進行優(yōu)化。優(yōu)化的步驟的標準可以是多種多樣的。在本文中，我們將有咬合高度 0.9 H H H ，其中 H 為許用割線高度設定值。 讓 H1是計算正割高度有一定夾角的 A1對應，這是小于 H ，而 Hg大于 H對應的夾角銀。我們可以構建一個變小時，即功能， = F（ H）。擴大，我們 5 根據表面的連續(xù)性假設和中值定理，我們可以通過線性插值的方法獲得估計的如 步長可以計算由方程（ 4）與 這個周期將被重復直到咬合高度滿足優(yōu)化咬合高度要求。 2 階梯效應和遏制的問題 兩個主要因素影響幾何計算的基礎層的厚度和表面加工精度是階梯效應和遏制的問題。換句話說，基于幾何層厚度的允許的牙尖高度主要取決與切片平面在一定高度的原始 CAD 模型的表面形狀。 （ 1） 階梯效應是由 LM 過 程的特點而形成的。它是由物理參數表示：牙尖高度，如圖 2所示。 6 圖 2 階梯效應和遏制風格 （ 2） 安全問題是指包含關系的部分原始 CAD模型的輪廓和沉積在 LM過程后的實際，這是通過平面的輪廓的討論，在算法中沉積的策略表示，如圖 2所示。 讓 Sc的部分原始 CAD 模型的二維輪廓； S1是逼近折線 Sc 的 LM 的形成過程。 它可以從圖的情況下看到（一）正公差和案例（ B）是負公差而案例（ c）和（ d）混合公差。 3 基于幾何 層厚度估計 對某些層 的層厚度的確定算法的粗糙的流程圖如圖所示，最大層的厚度是由特定的 LM 工藝和設備的確定。 7 圖 3 層厚度的確定算法流程圖 幾何基礎層厚度計算在任何點上的輪廓線的切片平面是馬的最低層的厚度對切片輪廓各點的基礎上。 通常，一個逃離曲線由圓弧和直線近似可以被視為一個圓的曲率為零。因此我們可以集中我們的討論在圓弧誤差分析對切片平面的層位于同一縱截面的兩個點作為一個自由曲線或圓弧的終點。 3.1 誤差準則 在某點 的誤差準則被定義為偏離所建立的輪廓線的層在 LM 從正常的曲線在某點上下分層平面。一般說來，誤差值是通過允許尖高度代表。 8 偏差的一個綜合性的概念，一般可以分為兩個部分：（ 1）的圓弧曲線或直線，逼近誤差說 。從該層的輪廓線，圓弧的錯誤，說 。從而，允許的牙尖高度，說 ，由用戶，可以全面的價值。它們之間的關系如下圖所示 3.2 誤差分析 3.2.1逼近誤差 原來的曲線和逼近圓弧之間的誤差是由 ，作為顯示在圖 4A。假設在兩個端點曲率， Q1和 Q2，正常曲線 K1和 K2。因此，對圓弧 C1曲率估計的定義是 從中心點曲線 C2端點之間，說第三季度，沿垂直方向的線段 q1q2，高度誤差之間的正常曲線 C2和 C1有圓弧割線 H2和 = | H2 |。在特殊情況下，例如，正常曲線 C2降低到一條直線，圓弧的曲率為零的 C1和 =0。 3.2.2 偏差 錯誤的定義是 F 層的輪廓線的偏差距離逼近圓弧，這是相對于 G有兩種情況計算誤差 F復雜一點：一是圓弧的謊言在一季度的圓，如圖 4b；另一個是圓弧跨越一個四分之一圓，位于半圈， 在 圖 4c 和 4d 顯示。它們將分別在下面討論 。 簽署了包括交叉曲線 與取向方向兩端點的切矢角可以得到，如 A3的角度 圖 4c。簽署產品積極結果是相應的案例（ B）而相反的是相應的案例（ C）和（ D） （ 1） 在一個單一的象限圓弧 圓弧半徑 圖?；谄矫鎺缀?，我們有 這 9 （ 2）圓弧過象限 在 圖 4c，圓弧是在用過量的沉積策略的凸函數。 假設在 A3點四比一點第四季度，我們 。 在圖 4d,圓弧是缺乏沉積策略的凸函數。 (a)圓弧逼近自由曲線 （ b)在一個單一的 象限圓弧過度沉積 （ c）在圓弧過象限過量沉積 （ d）對電弧在一個象限缺乏沉積 圖 4 逼近誤差和偏差 假設在 A3點 Q4大于一點 Q3， 我們有 。 在這種情況下，電弧是在一個缺乏或過量沉積策略具有相同的處理方法如上所述的情況下 圖 .4c 或 圖 .4d 分別凹函數。 10 3.3 錯誤和層厚度 如果當前層厚度不能滿足牙尖高度的要求，降低層的厚度進行估計的一種新的周期。在本文中。當前層厚度的 DG除以 n = 100和價值的 DG / N作為層厚度遞減。 在某一層的厚度估計將被視為在該層的厚度估計過程的下一點的當前層厚度的初始值。 4 基于材料 層厚度的檢驗 目的驗證的材料是檢查是否當前層厚度符合要求，材料制造，如果當前沒有獲得一個新的層厚度值。具體而言，一個隨機選擇的空間點上的可用區(qū)域低的切片平面某一種物質的區(qū)域是用來驗證當前層的厚度，而這個過程的初始值是由材料的區(qū)域的幾何形狀確定如第 3節(jié)所提到的材料屬性；如果當前層厚度不符合材料的要求，該層的厚度逐漸減小直至滿足要求；得到的層的厚度在這一點上，然后作為下一次驗證過程的初始值；這個周 期將持續(xù)到一個預先設定的總數 N點進行了驗證。 在本文中，驗證過程主要集中在功能梯度材料（功能梯度材料）。 4.1材料的檢查 要在取向方向接近的材料的體積百分比曲線圓弧的方法不同于使用第 3節(jié)中的方法。在材料區(qū)域的某些材料的體積百分比可以被視為在取向方向的高度的函數。 Z軸的簡化，即， P1 = F（ Z1）。 把材料的第一優(yōu)先為例。從某一點上下分層 Q1的平面層高度 Z1，延長距離當前層的厚度，我們在高度 22一點 Q2。材料的體積百分比 P1， P2和 P3點 Q1， Q2和 Q3的中間點，分別說在高度 23。 結合三體積 百分比， P1， P2和 P3，我們可以從點七的距離與當前層厚度沿導向軸構造一個近似圓弧的體積百分比曲線，如圖 5所示。 11 （ a） 圓弧是單調的 （ b）圓弧是非單調 圖 5 逼近圓弧曲線的材料 讓圓弧的中心是（ Zo， PO）。如果（ Z1-Zo） x（ Z2 Zo） 0，圓弧的定義為5A 條相應的單調而相反的是定義為非單調對應圖 5b。每種情況都有不同的解決方式。 4.2 誤差 的 分析 一個必要但不充分的條件下，本文提出驗證當前層厚度。三 個主要因素，材料變異性的界限，材料分辨率在逼近圓弧的端點的材料的體積百分比，主要考慮。 在圖 5， 代表材料的體積百分比， LM 設備可以存放在實踐中較低的切平面的層達到一定高度取向軸。在圖 5A 的情況下，下面的關系需要進行測試，驗證層厚度 代表這個 LM 機材料分辨率。 這個方程的一個充要條件。實際上，它可以簡化驗證層厚度。從式（ 10），我們有 如圖 5b，這些變量測試的關系 12 或者 在 P4是圓弧的材料百分比極值。 同樣，我們有 由式表示的條件。（ 11）和（ 13）是必要但不充分的條件下，可方便地應用于驗證層的厚度。在這兩個方程，考慮三個主要因素。 不滿足這些條件，該層的厚度必須逐漸減少執(zhí)行另一個周期的驗證。 5 例 如圖所示，有一個自由曲面主要由兩個裁剪曲面的 NURBS 表示 ISO 10303協議如下。在正常的方向為 Y 軸相對的斜面（不繪制在圖 6）作為一個切割平面相交的表面。 圖 6 自由曲面在笛卡爾 坐標系統(tǒng) 13 有兩種主要的誤差與切削過程一致的表面上的點對點線距誤差和誤差咬合高度相關的線段相交的曲線逼近。這兩個錯誤分別是 l0-4毫米和 10-1毫米。 三種不同的算法結果的比較見表 l 上市，其中算法的 L是指分半步長折半查找法，算法 2表示二進制搜索法和分半步的方向； 3代表算法自適應方法，根據中值定理和線性插值相結合的方法旋轉角的變化。 從表 1，它是已知的兩個算法，算法 2我很難獲得更好的結果。 3使用線性插值算法具有更好的綜合效果比算法 1和 2。 材料的設置屬性附加到了這部分內容如下。 最低層厚度： 0.01mm 最大層厚度： 0。 1毫米 材料認證面積密度： 0。 l mm2 材料沉積策略：多余的材料類型： FGM 外表面的公差： 0.02毫米 內部表面的公差： 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 為第一優(yōu)先的成分的材料分布函數 14 其中 R 是遠處的一個空間點遠離方向軸， Z 是該點坐標分量；符號“ ABS”意味著“絕對值”。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）在這個例子。 一部分連續(xù)的層厚度從 z = 15是在表 2中列出的向上的我（我 = 1， 2， .” 10），是第 i層； DG代表的幾何特征估計層厚度； DM代表材料為基礎的驗證如上所述在層的厚度，這是第 i 層的最后一層厚度。 從表 2可以看出，通過自適應分層產生的層的厚度可以在一個相對較大的范圍根據綜合因素包括曲面的幾何特征和零件的材料屬性，這無疑可以與均勻切片技術相比減少建造時間。 6 結論 所描述的工作重點是分層制造過程的理想材料零件。直接切片方法直接切片的部分原始 CAD 模型，通常保持足夠的幾何信息，優(yōu)于 STL 文件，因此，導致改進的精度。 SPI 本文提 出的算法具有一個突出的特點是充分利用允許的咬合高度。自適應切片也可以改善切削精度和減少建筑時間比較均勻的切片。幾何信息是用于預測層的厚度和材料的信息是用來驗證層的厚度和確定一個新的必要的話。 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