外文翻譯--直接自適應(yīng)切片在理想材料零件的CAD模型（IFMC）

1 中國(guó)機(jī)械工程學(xué)報(bào) v01 18， No 1， 2005 徐道明 鎮(zhèn)沅家 郭東明 重點(diǎn)實(shí)驗(yàn)室精密和非傳統(tǒng)加工技術(shù)應(yīng)用教育部 ， 大連理工大學(xué) ， 116024 中國(guó)大連。 直接自適應(yīng)切片在理想材料零件的 CAD 模型（ IFMC） 摘要：一個(gè)全新的直接自適應(yīng)分層的方法，可明顯提高零件精度和減少建立時(shí)間。班至少有兩個(gè)階段都包含在這個(gè)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(xiàn)段盡可能是其中之一 這是提高制造效率，通過(guò)自適應(yīng)地調(diào)整方向的一步， 在每個(gè)交叉點(diǎn)的步驟的大小來(lái)獲得優(yōu)化的咬合高度達(dá)到。層厚度的測(cè)定可分為兩個(gè)階段：基于幾何厚度和厚度估計(jì)基于驗(yàn)證材料。前一階段的幾何公差過(guò)程分為兩個(gè)部分：各種曲線(xiàn)由圓弧近似，引入了第一部分，和 LM 過(guò)程的輪廓線(xiàn)之間的偏差和圓弧生成第二部分后一階段主要是驗(yàn)證估計(jì)在前一階段的層的厚度和確定一個(gè)新的必要的話(huà)。 關(guān)鍵詞：快速原型 理想材料零件 直接自適應(yīng)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(xué)技術(shù)發(fā)展所需的類(lèi)。 快速原型制造（ RP&M）技術(shù)，或者叫 SFF（固體無(wú) 模成形）技術(shù)，是制造的理想材料零件的基本技術(shù)。 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(shù)目前是耗時(shí)的部分依賴(lài)，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(guò)程。 闡述了 RP 工藝原理直觀(guān)，可應(yīng)用于相關(guān)的階段， 如方向，支持生成，等。 目前，切片是主要處理無(wú)數(shù)的三角面片逼近的部分，那就是， STL 文件。由于其固有的缺點(diǎn)，這樣直接切片的部分模型更是成為一個(gè)活躍的研究都可以達(dá)到任何靈活的自適應(yīng)允許割線(xiàn)的高度。此外，也有兩種類(lèi)型的分層策略：均勻分層 自適應(yīng)切片。與前者相比，后者能用較少的時(shí)間完成建設(shè)較高的表面精度。 2 P.卡尼和 D. Dutta 討論一個(gè)準(zhǔn)確的切片程序 LM過(guò)程。 在此基礎(chǔ)上， v.kumar，等人，進(jìn)一步描述了一種更一般的切片過(guò)程中的 LM非均質(zhì)模型。 W. Y.馬和 P. R.他提出了一個(gè)算法，即自適應(yīng)切片孵化戰(zhàn)略選擇。 一種新的方法，稱(chēng)為局部自適應(yīng)切片技術(shù)進(jìn)行了簡(jiǎn)要的介紹了賈斯廷 tyberg，等。 一種自適應(yīng)分層方法在二語(yǔ)習(xí)得過(guò)程西方公司旗下，三富。 ET ALT， K.瑪尼，等人擴(kuò)展他們的早期作品，說(shuō)裁判。 2,3自適應(yīng)的 CAD 模型切片 。 另一個(gè)全新的直接自適應(yīng)分層策略提出了由至少兩個(gè)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(xiàn)段盡可能根據(jù)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(chǔ)上獲得的幾何特性和材料設(shè)置綜合分析的厚度。兩者交替進(jìn)行直至切層在預(yù)方向到達(dá)的最后部分定義的取向。 1跟蹤沿交叉曲線(xiàn) 一般來(lái)說(shuō)，在 CAD 模型的表面是由平面，圓錐曲線(xiàn)和曲面。 切割零件的實(shí)體模型的切割平面問(wèn)題，事實(shí)上，一個(gè) SPI（表面平面交叉口）從幾何問(wèn)題，這可以被視為一個(gè)特殊的情況下（ SSI 表面曲面求交問(wèn)題。 SSI 問(wèn)題的方法通常分為兩類(lèi)：解析法和數(shù)值方法（主要是推進(jìn)基于或細(xì)分算法）。此外，基于微分幾何原理的算法是近年來(lái)迅速發(fā)展起來(lái)的。 平面交叉口之間 和一個(gè)參數(shù)的表面可以被視為一個(gè)擴(kuò)展 和特殊情況下的交叉參數(shù)化的表面和表面之間。 行進(jìn)中的基礎(chǔ)算法計(jì)算一個(gè)切割平面與一個(gè)理想材料零件的 CAD 模型的參數(shù)曲面求交的輪廓，其中一個(gè)突出的特點(diǎn)是允許充分利用咬合高度。 1.1 對(duì)于具有參數(shù)曲面的線(xiàn)交叉點(diǎn)算法計(jì)算 讓 代表一條直線(xiàn)，在 AI 上表面附近的點(diǎn)線(xiàn)，是本線(xiàn)和 T為參變量的方向矢量。讓我們（ U， V）表示一個(gè)曲面的參數(shù)變量 u 和 v從某一初始點(diǎn)在直線(xiàn)和平面，一個(gè)迭代過(guò)程可以進(jìn)行，得到一個(gè)真正的交叉點(diǎn)，以滿(mǎn)足表達(dá) 擴(kuò)大這種表達(dá)，我們可以得到 3 牛頓迭代法求解這組方程 假設(shè) 可以得到以下方程 讓 T = 0的函數(shù) f的變量的初始值（ T），對(duì)應(yīng)點(diǎn)的 AI。讓我們（ U， V）被認(rèn)為是最接近的表面上的點(diǎn)，即， Bz 和雙值（ U， V）的變量對(duì)初始值（ U， V）表達(dá)的（ U， V）。 毫無(wú)疑問(wèn)，迭代過(guò) 程將持續(xù)到下 是滿(mǎn)意的，其中 是一個(gè)預(yù)先設(shè)定的允許誤差，和作為一個(gè)結(jié)果，真正的交叉點(diǎn) 1.2 該步驟的方向和步長(zhǎng)的初始估計(jì) 假定曲率點(diǎn)的 Pi表面上是 Ki。那里的步進(jìn)方向和步長(zhǎng)的初步評(píng)估是根據(jù)曲率 KI測(cè)定。 在這種情況下，割線(xiàn)的高度不能滿(mǎn)足要求的優(yōu)化步驟，中間值定理和線(xiàn) 4 性插值的方法將聯(lián)合應(yīng)用，得到優(yōu)化的步進(jìn)方向和步長(zhǎng)。方向的一步，對(duì)于點(diǎn)Pt下點(diǎn) 9月的大?。ㄒ?jiàn)圖。 1）是由方程 4決定 其中一個(gè)是切向量之間的夾角，在點(diǎn) PI 和步進(jìn)方向向量 ，即，估計(jì)步長(zhǎng)方向；我是估計(jì) 的步長(zhǎng)； R 對(duì)應(yīng)估計(jì)曲率 KI 圓半徑； H 是預(yù)先設(shè)定的容許咬合高度。 圖 1 選擇下一步 1.3 優(yōu)化的步驟 實(shí)際的交叉點(diǎn)的部分的表面的步驟是在 1.1節(jié)中介紹的算法來(lái)計(jì)算的。然而，這并不意味著得到滿(mǎn)足預(yù)先設(shè)定的要求和咬合高度進(jìn)行優(yōu)化。優(yōu)化的步驟的標(biāo)準(zhǔn)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(xiàn)高度設(shè)定值。 讓 H1是計(jì)算正割高度有一定夾角的 A1對(duì)應(yīng)，這是小于 H ，而 Hg大于 H對(duì)應(yīng)的夾角銀。我們可以構(gòu)建一個(gè)變小時(shí)，即功能， = F（ H）。擴(kuò)大，我們 5 根據(jù)表面的連續(xù)性假設(shè)和中值定理，我們可以通過(guò)線(xiàn)性插值的方法獲得估計(jì)的如 步長(zhǎng)可以計(jì)算由方程（ 4）與 這個(gè)周期將被重復(fù)直到咬合高度滿(mǎn)足優(yōu)化咬合高度要求。 2 階梯效應(yīng)和遏制的問(wèn)題 兩個(gè)主要因素影響幾何計(jì)算的基礎(chǔ)層的厚度和表面加工精度是階梯效應(yīng)和遏制的問(wèn)題。換句話(huà)說(shuō)，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(yīng)是由 LM 過(guò) 程的特點(diǎn)而形成的。它是由物理參數(shù)表示：牙尖高度，如圖 2所示。 6 圖 2 階梯效應(yīng)和遏制風(fēng)格 （ 2） 安全問(wèn)題是指包含關(guān)系的部分原始 CAD模型的輪廓和沉積在 LM過(guò)程后的實(shí)際，這是通過(guò)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(xiàn) Sc 的 LM 的形成過(guò)程。 它可以從圖的情況下看到（一）正公差和案例（ B）是負(fù)公差而案例（ c）和（ d）混合公差。 3 基于幾何 層厚度估計(jì) 對(duì)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(shè)備的確定。 7 圖 3 層厚度的確定算法流程圖 幾何基礎(chǔ)層厚度計(jì)算在任何點(diǎn)上的輪廓線(xiàn)的切片平面是馬的最低層的厚度對(duì)切片輪廓各點(diǎn)的基礎(chǔ)上。 通常，一個(gè)逃離曲線(xiàn)由圓弧和直線(xiàn)近似可以被視為一個(gè)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(duì)切片平面的層位于同一縱截面的兩個(gè)點(diǎn)作為一個(gè)自由曲線(xiàn)或圓弧的終點(diǎn)。 3.1 誤差準(zhǔn)則 在某點(diǎn) 的誤差準(zhǔn)則被定義為偏離所建立的輪廓線(xiàn)的層在 LM 從正常的曲線(xiàn)在某點(diǎn)上下分層平面。一般說(shuō)來(lái)，誤差值是通過(guò)允許尖高度代表。 8 偏差的一個(gè)綜合性的概念，一般可以分為兩個(gè)部分：（ 1）的圓弧曲線(xiàn)或直線(xiàn)，逼近誤差說(shuō) 。從該層的輪廓線(xiàn)，圓弧的錯(cuò)誤，說(shuō) 。從而，允許的牙尖高度，說(shuō) ，由用戶(hù)，可以全面的價(jià)值。它們之間的關(guān)系如下圖所示 3.2 誤差分析 3.2.1逼近誤差 原來(lái)的曲線(xiàn)和逼近圓弧之間的誤差是由 ，作為顯示在圖 4A。假設(shè)在兩個(gè)端點(diǎn)曲率， Q1和 Q2，正常曲線(xiàn) K1和 K2。因此，對(duì)圓弧 C1曲率估計(jì)的定義是 從中心點(diǎn)曲線(xiàn) C2端點(diǎn)之間，說(shuō)第三季度，沿垂直方向的線(xiàn)段 q1q2，高度誤差之間的正常曲線(xiàn) C2和 C1有圓弧割線(xiàn) H2和 = | H2 |。在特殊情況下，例如，正常曲線(xiàn) C2降低到一條直線(xiàn)，圓弧的曲率為零的 C1和 =0。 3.2.2 偏差 錯(cuò)誤的定義是 F 層的輪廓線(xiàn)的偏差距離逼近圓弧，這是相對(duì)于 G有兩種情況計(jì)算誤差 F復(fù)雜一點(diǎn)：一是圓弧的謊言在一季度的圓，如圖 4b；另一個(gè)是圓弧跨越一個(gè)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(xiàn) 與取向方向兩端點(diǎn)的切矢角可以得到，如 A3的角度 圖 4c。簽署產(chǎn)品積極結(jié)果是相應(yīng)的案例（ B）而相反的是相應(yīng)的案例（ C）和（ D） （ 1） 在一個(gè)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(guò)象限 在 圖 4c，圓弧是在用過(guò)量的沉積策略的凸函數(shù)。 假設(shè)在 A3點(diǎn)四比一點(diǎn)第四季度，我們 。 在圖 4d,圓弧是缺乏沉積策略的凸函數(shù)。 (a)圓弧逼近自由曲線(xiàn) （ b)在一個(gè)單一的 象限圓弧過(guò)度沉積 （ c）在圓弧過(guò)象限過(guò)量沉積 （ d）對(duì)電弧在一個(gè)象限缺乏沉積 圖 4 逼近誤差和偏差 假設(shè)在 A3點(diǎn) Q4大于一點(diǎn) Q3， 我們有 。 在這種情況下，電弧是在一個(gè)缺乏或過(guò)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(shù)。 10 3.3 錯(cuò)誤和層厚度 如果當(dāng)前層厚度不能滿(mǎn)足牙尖高度的要求，降低層的厚度進(jìn)行估計(jì)的一種新的周期。在本文中。當(dāng)前層厚度的 DG除以 n = 100和價(jià)值的 DG / N作為層厚度遞減。 在某一層的厚度估計(jì)將被視為在該層的厚度估計(jì)過(guò)程的下一點(diǎn)的當(dāng)前層厚度的初始值。 4 基于材料 層厚度的檢驗(yàn) 目的驗(yàn)證的材料是檢查是否當(dāng)前層厚度符合要求，材料制造，如果當(dāng)前沒(méi)有獲得一個(gè)新的層厚度值。具體而言，一個(gè)隨機(jī)選擇的空間點(diǎn)上的可用區(qū)域低的切片平面某一種物質(zhì)的區(qū)域是用來(lái)驗(yàn)證當(dāng)前層的厚度，而這個(gè)過(guò)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(dāng)前層厚度不符合材料的要求，該層的厚度逐漸減小直至滿(mǎn)足要求；得到的層的厚度在這一點(diǎn)上，然后作為下一次驗(yàn)證過(guò)程的初始值；這個(gè)周 期將持續(xù)到一個(gè)預(yù)先設(shè)定的總數(shù) N點(diǎn)進(jìn)行了驗(yàn)證。 在本文中，驗(yàn)證過(guò)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(xiàn)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(shù)。 Z軸的簡(jiǎn)化，即， P1 = F（ Z1）。 把材料的第一優(yōu)先為例。從某一點(diǎn)上下分層 Q1的平面層高度 Z1，延長(zhǎng)距離當(dāng)前層的厚度，我們?cè)诟叨?22一點(diǎn) Q2。材料的體積百分比 P1， P2和 P3點(diǎn) Q1， Q2和 Q3的中間點(diǎn)，分別說(shuō)在高度 23。 結(jié)合三體積 百分比， P1， P2和 P3，我們可以從點(diǎn)七的距離與當(dāng)前層厚度沿導(dǎo)向軸構(gòu)造一個(gè)近似圓弧的體積百分比曲線(xiàn)，如圖 5所示。 11 （ a） 圓弧是單調(diào)的 （ b）圓弧是非單調(diào) 圖 5 逼近圓弧曲線(xiàn)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(yīng)的單調(diào)而相反的是定義為非單調(diào)對(duì)應(yīng)圖 5b。每種情況都有不同的解決方式。 4.2 誤差 的 分析 一個(gè)必要但不充分的條件下，本文提出驗(yàn)證當(dāng)前層厚度。三 個(gè)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(diǎn)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(shè)備可以存放在實(shí)踐中較低的切平面的層達(dá)到一定高度取向軸。在圖 5A 的情況下，下面的關(guān)系需要進(jìn)行測(cè)試，驗(yàn)證層厚度 代表這個(gè) LM 機(jī)材料分辨率。 這個(gè)方程的一個(gè)充要條件。實(shí)際上，它可以簡(jiǎn)化驗(yàn)證層厚度。從式（ 10），我們有 如圖 5b，這些變量測(cè)試的關(guān)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(yīng)用于驗(yàn)證層的厚度。在這兩個(gè)方程，考慮三個(gè)主要因素。 不滿(mǎn)足這些條件，該層的厚度必須逐漸減少執(zhí)行另一個(gè)周期的驗(yàn)證。 5 例 如圖所示，有一個(gè)自由曲面主要由兩個(gè)裁剪曲面的 NURBS 表示 ISO 10303協(xié)議如下。在正常的方向?yàn)?Y 軸相對(duì)的斜面（不繪制在圖 6）作為一個(gè)切割平面相交的表面。 圖 6 自由曲面在笛卡爾 坐標(biāo)系統(tǒng) 13 有兩種主要的誤差與切削過(guò)程一致的表面上的點(diǎn)對(duì)點(diǎn)線(xiàn)距誤差和誤差咬合高度相關(guān)的線(xiàn)段相交的曲線(xiàn)逼近。這兩個(gè)錯(cuò)誤分別是 l0-4毫米和 10-1毫米。 三種不同的算法結(jié)果的比較見(jiàn)表 l 上市，其中算法的 L是指分半步長(zhǎng)折半查找法，算法 2表示二進(jìn)制搜索法和分半步的方向； 3代表算法自適應(yīng)方法，根據(jù)中值定理和線(xiàn)性插值相結(jié)合的方法旋轉(zhuǎn)角的變化。 從表 1，它是已知的兩個(gè)算法，算法 2我很難獲得更好的結(jié)果。 3使用線(xiàn)性插值算法具有更好的綜合效果比算法 1和 2。 材料的設(shè)置屬性附加到了這部分內(nèi)容如下。 最低層厚度： 0.01mm 最大層厚度： 0。 1毫米 材料認(rèn)證面積密度： 0。 l mm2 材料沉積策略：多余的材料類(lèi)型： FGM 外表面的公差： 0.02毫米 內(nèi)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(shù) 14 其中 R 是遠(yuǎn)處的一個(gè)空間點(diǎn)遠(yuǎn)離方向軸， Z 是該點(diǎn)坐標(biāo)分量；符號(hào)“ ABS”意味著“絕對(duì)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(gè)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(jì)層厚度； DM代表材料為基礎(chǔ)的驗(yàn)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(guò)自適應(yīng)分層產(chǎn)生的層的厚度可以在一個(gè)相對(duì)較大的范圍根據(jù)綜合因素包括曲面的幾何特征和零件的材料屬性，這無(wú)疑可以與均勻切片技術(shù)相比減少建造時(shí)間。 6 結(jié)論 所描述的工作重點(diǎn)是分層制造過(guò)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(dǎo)致改進(jìn)的精度。 SPI 本文提 出的算法具有一個(gè)突出的特點(diǎn)是充分利用允許的咬合高度。自適應(yīng)切片也可以改善切削精度和減少建筑時(shí)間比較均勻的切片。幾何信息是用于預(yù)測(cè)層的厚度和材料的信息是用來(lái)驗(yàn)證層的厚度和確定一個(gè)新的必要的話(huà)。 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