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布局优化的掩模固化快速成型工艺X. Zhang， B. Zhou，Y. Zeng，P.Gu*唐佳红 译机械制造工程系、卡尔加里大学，大学路2500，卡尔加里，加拿大艾伯塔省T2n1n4 摘要 快速成型制造技术，可直接用于CAD模型的产品开发，制造工具和制造模具以及功能部件。掩模固化(SGC)技术中的快速成型技术， 适合建立多种不同零部件的快速成型样机几何尺寸的批量生产，其成本降至最低。 然而，平面CAD模型环境的布局是费时的。 由于树脂的成本高， 一批SGC的业务在任何工业环境中的布局模式是成功的关键。 本文利用模拟退火布局优化技术。开发一个软件系统是为了协助前部分进行各种型号的CAD几何形状的布置。 STL系统接受来自任何实体造型环境的信息。下面提供几个例子来说明技术的有效性。 1. 引言全球的主导产品竞争日益需要厂商更加灵活地适应瞬息万变的市场需求。 大幅削减产品开发时间将改善企业对市场的需求，因此赢得竞争优势。 快速成型制造技术也不断提高了厂商建立三维模型和原型等几方面快速反应能力; 而具有成本效益的生产模式和复杂模具曲面1的快速成型技术可直接从CAD模型生成原型。 各种快速成型技术的出现，包括固化、选择性激光烧结(SLS)、熔融沉积制造、叠层实体制造、三维印刷和掩模固化(SGC)。它们有一个共同特点:成型制作是加入材料而不是像传统的去除材料或变形工序2。 这些技术可以填补空间不确定具体部分的概念设计。该技术也可大大提高工作效率和模具制作工艺格局。SGC的过程能在单一格局内生产多种不同形状和尺寸的零件，因此适合批量生产。 在SGC的制造过程中(图1)，使负截面的一部分产生静电收取玻璃板。这就像平时使用的激光打印机。 在此期间，一层液态固化树脂是一种分布式的到位工作。 然后放在工作空间的灯和表面之间用玻璃板挡起来，不是用激光束或者是紫外线灯注满空间和照亮整个层。剩余的液态树脂是用吸尘器抹去。 向后移动的样盘上曝露在紫外线灯下第二次凝固的液态树脂没有被吸尘器完全清理干净。 在这层充满热蜡液的空间，蜡被一个冷态金属碟子冷却到固体之后，这个树脂蜡层被刀具快速的粉碎成指定的大小。样盘从磨床回到曝光厅后，树脂的新层被应用。SGC程序能同时由一个限制简单的CAD 模型形成多个复杂部件的玻璃装料碟子。 因此，可以用来生产机器前部分分批多型号。然而，在SGC的建立过程中， 树脂不能被重复使用，因为它已经部分被恢复。 如果我们要建立一个单一的部分，并且没有其它任何地方共享区段，这部分可以很昂贵，除非占有大多数的托盘模式。 此外， 大多数其它快速成型机必须依次编织每个过渡部分。 虽然这些时间可由机器同时生产多种零部件而减少，生产时间的长短取决零件的数量和零部件的几何形状。 在SGC的过程中，每一层曝露在具有相同时间紫外线灯下的跨度为预设操作。 因此消费和制造树脂的每一层时间都是固定的， 一批独立的几何形状和数量的零部件。 因此， 当一批零部件的生产成本单纯依靠被生产层的数字时，为了最大限度地减少成本和生产率， 各地应在每一批封装时给予尽可能低的示范区，使托盘制作的一批零部件成本降到最小。 当两个或两个以上的部分是在在三维图形环境中同一时间制造时，具体操作模拟包装件应放置在它们或它们的CAD模型上。在计算机屏幕上的一个封袋内，可保证部件不相互干涉， 而且它们完全在限制的体积之内。 每批零件可用于不同的程序和不同客户。 因此零件形状和大小可以相差很大， 使得很难找到用手动解决的最佳经营模式的布局。 因此，需要找到一个电脑系统的优化，配置一批生产成本最低的布局。 图1 描述SGC的制造过程2. 相关研究工作 布局问题的模型可归纳为典型装箱问题。 应用装箱问题可用在集成电路设计、切割和股票等其它领域。典型的二维和三维装箱问题已被证明是疑难问题3。生产效率最优解的密切近似算法的开发对装箱问题具有重要现实意义是。 这些方法包括:线性规划、启发式技术，模拟退火(SA)和遗传分析(GA)。 线性规划方法已成功地广泛应用于学习、普通切割的问题中。 然而，由于它们的结构或大小，这些方法对许多真正的问题是不适当的。 在这种情况下，应用启发式手段，如动态规划搜索方法。 动态规划法是一种把单个问题转换成一系列单阶段问题的算法其难度是如何迅速确定最优决策。树-搜索的方法是将所有可能的解决方法罗列成树状结构，在同一条路径上开始和结束，当其被认为是已经找到的最佳答案或者是已经知道会造成令人不满意的解决方案。上述大部分办法要么不给最佳或接近最佳的解决方案要么不适用多种应用而且比较复杂形式的问题4。为克服线性规划、启发式方法的局限， 研究成果已使用SA和GA解决封装问题。 rao及Iyengar5适用于多样化的装箱问题。 大量仿真实验表明典型的启发式方法已经显著的改进。 cagan6摸索SA用二维和三维解决问题。适应退火时间表，多分辨率建模与动态步骤提出改进策略选择算法性能。 Han和Na 7嵌套方法提出了两个阶段:初级布局阶段，改善布局阶段。 自组织算法辅助设计制造了一种优的初步布局; 然后用SA布局作初步改善。 Corcoran 8探索GA问题在三维封装中的应用以便GA更好的解决三维封装问题。 值得一提的是，ikonen等9用GA来为机器解决三维模型的规划问题。 上述的研究大多简化了零部件的形状。 ikonen几何方法的零件不需要在封装前简化。 不过用这个方法搜索的封装过程可能非常费时 (例如15个部件需要8.5小时)。 总之，已证明它有解决GA和SA装箱问题能力。不过，GA和SA在效率和效益上很大程度取决于空间解来实施策略和目标功能。 在这项研究中， SA算法目前应用于封装的具体问题，根据客观战略SGC的搜索功能和过程。 3. 制定示范布局问题 在这项工作中，SGC的模盘代表容器的上限。 这个问题的布局模式研究一批堪称封装大小不同零件的容器(集装箱)。 封装任务具有以下三个目标: *装修进入指定型号集装箱;*避免模型重叠; *实现高密度封装，换句话说，实现最低整体水平。 在目前的应用中， 每一部分CAD模型用STL格式来表示。它由三角面坐标数据及其相应三维空间表面组成。 相对于其它格式的CAD模型，STL格式非常简单。 然而，如果STL模型是一个完整的信息编码算法，搜索会很费时，这是因为很平常的机械部分有成千上万的三角面的STL模型，一个简单操作如旋转或移动就是重新定位每一个三角面的新模式。 此外，这些动作都是用数千迭代在SA搜索过程，因此简化是必要的。 大多数的研究者只是包抄一部分，因为它是一个长方形盒子容易执行，使包装算法变得过于复杂 4。 这种方法还用于这项研究。 不同之处在于制定具体的搜索功能和战略目标。 三维模型图可以描述以下步骤(图二): 1.解析STL文件的零件制造、每一个封闭部分产生示范。2. 据具体算法封装。箱体批量的大小相同，机器的最大体积相同。 3. 计算每一部分更新STL文件。 4. 最后结果显示布局。步骤1，3，4涉及计算机图形学、可视化技术、 所不明确的问题，涉及到包装、较易执行现有图形工具。 我们的重点是在第2步。 制定这一问题的关键是树立正确的算法。 装箱问题可以作为最优化问题是方程。 (1):CAD文件包括所有生产包装信息的部件图2 典型包装的程序其中n是模型的数量、pi是封装状态模型、p是布局、 P是一批规划布局、B是轮廓边界PiPj两模型之间的交接、 PiB箱体边界和模型之间的交接、h总体规划高度与f高度目标函数映射成为集P实数。 目前这个问题，是一个单批零件的布局满足限制方程(1)的界定。 目标函数定义为整体的布局高度。 典型的封装状态是由封袋的位置和方向决定的。 每一个部分的坐标是由封袋的几何中心定义的。集装箱的左下角定义为整体的坐标原点。 封袋的几何中心确定封袋在箱中的位置。 典型的方位有六个如图(3)所示。 一个状态对另一个状态的转变可通过一个或两个旋转操作实现。旋转操作是局部调整旋转轴附近的封袋。 例如，图(3)把封袋的一种状态换成另一种状态，它可以绕X轴旋转也可以绕Y轴旋转。 每个封装袋表面总是平行或垂直于正交轴。所以每个封袋的形式pi表示为:其中xi、 yi 、zi 相对应的表示第i个封袋的顶点。 i 表示它的方向，n是用来记数的;Xi 、Yi、 Zi 表示第i个封袋沿x、 y、 z 方向的长度。目标函数f 定义为在有效方向上搜索到的合适的点。在这项研究中，目的是尽量减少总体布局的高度可以由下列公式: 图3 自由度导向示范在搜查过程中， 通过已知的模型结果搜索更加准确的解决方法。 在最后目标函数的设计中应该避免重叠。 因此目标函数有两个条件f2: 当两个平行长方形箱子重叠，重叠部分也是一个长方形。 在方程(4)中Oij(x)、 Oij(y)和Oij(z) 表示第i个箱子和第j个箱子重叠部分沿x 、y 、z 方向的长度。Oij 表示重叠部分三个方向长度的总和。重叠量化方程(4)的线性叠加最常用于多目标函数优化7，9: 不过，这个功能很少能够成功的优化这个封装的问题。 在搜索过程中， 解答f值方法是搜索目标函数值。 不同的重量物品的比较结果，可以不同。 因此这两个重量相对价值是至关重要的。f2变化范围是由各部分的数量和大小决定的，一对重量不同包装情况找不到一个比较好的方法。 即使有重叠和客观高度的值也很难找到数学方法决定自己价值。如果w2远大于W1;目标函数将更为敏感变更重叠量f2: 该模型将放置在搜索过程初期消除重叠。最后，将模型在集装箱里堆高，如果高度比意义更是值得考虑，较为敏感的全局高度和重叠量的目标函数将变化，因为目标函数值将因为高度下降而减少。 避免难以确定的重量值、 非线性目标函数和定义使用于这项研究。 下面的目标函数: 重叠高度上限是由集装箱高度决定的。当f2=1时因为f21无法成立函数值使f等于0， 否则会引起冲突。换句话说，当目标函数值不再减少，就不会得到改善。 在方程(4)重叠两个封袋重叠长度的计算总结三个方面。用数学给这两个封袋重叠部分定义合适的目标函数方程(7): 在搜查过程中利用非线性目标函数搜索重叠的客观职能是好的， 假设移动不可行组态(布局)可以导致上级可行配置(布局)4。 以此为目标函数， 算法引导搜索的走向会降低，布局重叠高度后已经很小。虽然还存在小小的重叠的型号，最后的总体布局的高度普遍较低。 SA办法来解决上述问题的最优解比较试验都是封装重叠量化使用重叠方程(4)确定。SA办法在使用第一种方法时表现较差或没有更好的表现。测试结果将在后文表明。这里指的封装算法不能保证找到一个可接受的解决方案。 消除重叠解决小SA所得的补偿算法是一个简单的程序。 SA搜索过程完成后，如果存在重叠， 补偿程序可以找出重叠模型和解决措施，显示他们最低重叠量。 由于可能产生新的重复动作，再次检查所有型号； 是否出现新的重叠，直到程序没有继续存在的重叠模式。重叠在大多数情况下，可以轻松地解决后执行这项补偿计划。在测试的情况下，整体水平SA提高不明显。4. 模拟退火方法4.1 观念SASA是基于物理概念金属热处理为了找到最低能量金属状态。数学方法是用退火冷却时间表，这是一个使功能下降的方法。SA是一个统计力学随机计算技术寻找附近所有最低成本所得的解决大型优化问题的方法。 在许多情况下， 寻找一个客观的取小价值与功能的自由度受到许多限制，是一种矛盾的疑难问题。由于有许多客观的功能将趋于局部极小。优化程序解决这类问题，应寻找样本空间，如此才能使寻找最优或接近最优解在合理时间有一个高的概率。SA是由Kirkpatrick et al 10决定的。 这一构想源自Metropolis11。 最初的起始温度T; 给予最大值，并假设系统处于初始状态开始计算目标函数值f。国家规定用p表示; 然后测试六个点的转变职能。 如果此举导致目标函数有所改善， 用新的函数代替原函数。其目标函数值也用这个新解。这可以大概计算目标函数值。 共认的概率决定如下: 温度值开始随时间下降。 每个温度系统都有几次不稳定。这反复进行温度迭代计算称为Markov链。迭代连环次数称为链长。最初，动作通过空间状态几乎随机的，造成了广泛的探索空间的客观功能。由于概率接受劣质动作下跌，它们往往很难使算法收敛到最优。 5. 封装策略51. 运动装置 SA从解(s)开始搜索的一个或多个进程，然后选择认定其预先的解。 此过程用“”表示; 这是一种新颖的操作办法p可以从旧的中得到， 移动装置是由各个动作依据模型布局组成。 当新布局由旧产生，一个动作是有移动装置中几个可能的动作中选出的。 在运动装置中定义六个动作的问题如下: *转动。 改变封袋方向一个模式不改变中心位置封袋。 共有6个方向可以选择。 然而，可能性最小的方向给予更多的高度封套。 *漫步。 随意改变的中心位置在封袋正负x 正负y 正负z方向，货厢的运动方向和移动距离取决于当前的温度。 在更高温度较长距离的运动，企图将选择一个更大的变化，实现功能的成本，并避免局部极小。 *交换。 改变两个封袋地点; 两个封袋是随机抽选调换。 *回到起点(0，0，0)。 随机挑选封袋提出一个方向，使沿线的X; Y和Z座标，封袋温度减少的那一刻将减少移动距离。*消除重叠。 计算向量之间重叠每个封袋沿线移动相应载体。 这一行动旨在减少整体重叠、 其实并不能消除重叠，因为以后排除一切可能产生重叠的新举措。 *靠墙移动。 在提出抽样封袋x-y平面(不改变其Z坐标位置)，直到它触及墙往就近封袋或其它容器壁、 然后向下直到它触及封袋或其它容器底部。 实际上有两个在移动装置中的基本操作旋转、漫步中， 其它被视为一个特殊的操作或动作顺序。 后者的主要目的是提高四个业务的算法效率。 52 重叠计算检测重叠计算评价两个或两个以上封袋重叠的程度。封袋预计二维平面坐标，这使计算方便。 一旦有重叠的两个封袋，就有重叠每个二维平面(X-Z、X-Y、Y-Z平面)。 如果任何两个平面之间没有重叠，这两个封袋之间就没有重叠 如图(4)。图4 模型布局优化算法SA6.结论 快速成型制造技术可大幅度降低产品开发时间，迅速适应市场需求。作为一个普通的快速成型技术工艺，能生产多种零部件SGC的同步进程。不过，需要做研究，以减少生产成本，提高机械效率。 SGC的SA过程用于解决在模型中布局优化、发现问题，提高了生产效率和降低成本。新的目标函数功能将于传统线性函数相比较。有效的比较办法是SA。 软件开发工具包可以从机器上减少繁琐且未必有效的工作部分。它也可以由工程师接收STL格式。封袋自动生成和更新的STL文件的封袋型号以及相应布置的办法。 最后的布置可转为VRML的格式和可视VRML。 最新日志档案可直接装入机床制造一批多种零部件。 致谢 这项研究得到了加拿大自然科学及工程研究理事会 (NSERC)战略性str192769补助。参考资料 1 Yan X， Gu P. A review of rapid prototyping technologies and systems. Computer-Aided Des 1996;28(4):30718.2 Kruth JP. Material increases manufacturing by rapid prototyping techniques. Ann CIRP 1991;40/2:60313.3 Garey MR， Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. San Fransisco: W.H. Freeman and Co.， 1979.4 Szykman S， Cagan J. A simulated annealing-based approach to three-dimensional component packing. J Mech Des 1995;117: 30814.5 Rao RL， Iyengar SS. Bin-packing by simulated annealing. Comput Math Appl 1994;27(5):7189.6 Cagan J. A shape annealing solution to the constrained geometric Knapsack problem. Computer-Aided Des 1994;28(10):7639.7 Han G-C， Na S-J. Two-stage approach for nesting in twodimensional cutting problems using neural network and simulated annealing. J Eng Manuf 1996:50919.8 Corcoran III AL， Wainwright Roger L. A genetic algorithm for packing in three dimensions. Symposium on Applied Computing(SAC 92)， Kansas City， March 13， 1992， p. 102130.9 Ikonen I， Biles WE. A genetic algorithm for optimal object packing in a selective laser sintering rapid prototyping machine. Seventh International Conference on Flexible Automation and Intelligent Manufacturing， Middlesbrough UK， 1997， p. 7519.10 Kirschman S， Gelatt II CD， et al. Optimization by simulated annealing. Science 1983;220:67180.11 Metropolis M， Rosenbluth A， et al. Equation of state calculations by fast computing machines. J Chem Phys 1953;21:108792. Model layout optimization for solid ground curing rapidprototyping processesX. Zhang, B. Zhou，Y. Zeng，P. Gu*Department of Mechanical and Manufacturing Engineering, The University of Calgary, 2500 University Drive, Calgary, Alberta, Canada T2N 1N4AbstractRapid prototyping technologies are capable of directly manufacturing physical objects from CAD models and have been increasingly used in product development, tool and die making and fabrication of functional parts. Solid ground curing (SGC) technology, one of the rapid prototyping technologies, is suitable of building multiple parts with different geometry and dimensions in batch production of rapid prototypes to minimize the cost of prototypes. However, the layout of CAD models in a graphic environment is time-consuming. Because of high cost of the resin, the layout of models in a batch is critical for the success of the SGC operations in any industrial environment. This paper presents the layout optimization using simulated annealing techniques. A software system was developed to assist Cubital operators to layout CAD models with various geometric shapes. The system accepts STL files from any solid modeling environment. Several examples are provided to illustrate the techniques and effectiveness of the approach.1. IntroductionIncreasing global competition requires product oriented manufacturing firms to become more flexible and responsive to the ever-changing market. Substantial reduction of product development time will improve firms response to market demands and therefore gain competitive advantages. Rapid prototyping and manufacturing technologies have been improving manufacturers responsiveness in several aspects such as rapid creation of 3D models and prototypes; and cost-effective production of patterns and moulds with complex surfaces 1. Rapid prototyping technologies are capable of directly generating physical objects from CAD models. A variety of rapid prototyping technologies have emerged including stereolithography, selective laser sintering (SLS), fused deposition manufacturing, laminated object manufacturing, 3D printing and solid ground curing (SGC). They have a common feature: the prototype is produced by adding materials, rather than removing or deforming materials as in traditional manufacturing processes 2. These technologies can fill the uncertainty void between the conceptual design and an actual part. The technologies can also significantly improve the efficiency of pattern and mould making processes. SGC processes can produce multiple parts with different geometries and dimensions in a single setup and therefore are suitable for batch production. In the building process of SGC (Fig. 1), a mask is generated by electrostatically charging a glass plate with a negative image of the cross-section of the part, which is similar to the process used in the laser printer. In the meantime, a thin layer of liquid photo-curable resin is spread across the surface of the work place. Then, the glass plate with the mask is placed between the lamp and the surface of the workspace. Instead of using a laser beam, an UV lamp is used to flood the chamber and expose and solidify the entire layer. After the residual liquid resin is wiped off by a vacuum cleaner, the model tray moves back under the UV lamp for a second exposure that solidifies the liquid resin that is not totally cleaned by the vacuum. The voids in this layer are filled with hot liquid wax, which acts as support for overhangs and isolated parts. After the wax is cooled down to solid by a cold metal plate, this resin/wax layer is milled with a fly cutter to a specified thickness. The new layer of resin is applied when the model tray moves from the milling station back to the exposure chamber. The SGC process can produce multiple parts at the same time by simply slicing a batch of CAD models and charging the glass plate with a negative image of multiple cross-sections. Thus, Cubital machines can be used for production of multiple models in batches. However, in the building process of SGC, the resin that does not contribute to the part and is wiped off cannot be reused because it has been partially cured during the initial exposure. If one needs to build a single part and there are no other parts to share the block, this part can be very expensive unless it occupies most of the model tray. Moreover, most of the other rapid prototyping machines have to weave the cross-sections of each part in a batch one by one. Although the lead time on these machines can be reduced by producing multiple parts, the production time does depend on the number of parts and the parts geometry. In the SGC process, the UV lamp exposes every layer with the same time span as predetermined by the operator. Thus the resin consumption and fabrication time per layer are constant, independent of the parts geometry and the number of the parts in the batch. Consequently, the time and the cost of producing a batch of parts simply depend on the number of layers produced in the fabrication. In order to maximize productivity and minimize cost, parts in every batch should be packed as low as possible within the given area of the model tray so that the fabrication layers for the batch of parts can be minimized. When more than one part is manufactured at the same time, within the 3D graphic environment, the operator simulates packing of actual parts by placing them, or their CAD models, inside an envelope on a computer screen and makes sure that parts do not intersect with each other, and that they are totally inside the building volume. Parts in each batch can be used for different applications and for different customers. Thus the shape and size of parts can vary dramatically, which makes it even more difficult for an operator to find an optimal model layout solution manually. Therefore, a computerized system is needed to find the optimal batch configuration layout for the minimum cost of production.2. Related research workThe model layout problem can be categorized into the well-known bin-packing problem. Applications of the bin-packing problem can be found in VLSI layout design, stock cutting and other fields. The classical 2D and 3D bin-packing problems have been proven to be NP-hard 3. Since the bin-packing problem is of practical importance, efficient approximation algorithms that produce close-to-optimal solutions have been developed. These approaches include linear programming, heuristic techniques, simulated annealing (SA) and genetic algorithm (GA).Linear programming methods have been extensively studied and successfully applied to a broad class of stock cutting problems. However, these methods are not appropriate for many real problems due to their structures or sizes. In such cases, heuristic methods are used, such as dynamic programming and tree-search methods. Dynamic programming is a method that converts a problem into a series of single-stage problems. The difficulty is in quickly determining the optimal decisions. The tree-search method enumerates all possible solutions in a tree Dynamic programming is a method that converts a problem into a series of single-stage problems.-like structure. Heuristics start out on one path and terminate when either an optimal solution is believed to have been found or the path is known to result in an unsatisfactory solution. Most of the above approaches either do not give an optimal or near optimal solution or are not applicable to a wide variety of applications, and the formulations of problems are rather complicated 4.To overcome the limitations of linear programming and heuristic techniques, research efforts have been made using SA and GA to solve packing problems. Rao and Iyengar 5 applied SA to a variety of the bin-packing problems. Extensive simulation experiments demonstrated that the solutions obtained by SA showed a significant improvement over those obtained by any of the well-known heuristic methods. Cagan 6 explored 2D and 3D layout problems using SA. An adaptive annealing schedule, multi-resolution modeling and a dynamic move selection strategy were proposed to improve algorithm performance. Han and Na 7 proposed a nesting approach with two stages: initial layout stage and layout improvement stage. A self organization assisted layout algorithm generates a good initial layout; then SA was used to improve the initial layout. Corcoran 8 explored GA in 3D packing problems and showed that GA yielded good solutions for 3D packing problem.It is worthwhile to mention that Ikonen et al. 9 used GA to solve a 3D model layout problem for an SLS machine.Most of the research mentioned above simplify shapes of the parts. Ikonens approach does not require geometry of parts to be simplified before packing. However, the searching process of the packing can be very time-consuming using this method (e.g. 8.5 h for 15 parts).In short, it has been demonstrated that GA and SA are capable of solving bin-packing problems. However, the performances of GA and SA in terms of efficiency and effectiveness depend greatly on the solution space, the implementation strategies and the objective functions. In this study, SA algorithm was applied to the present packing problem according to the specific objective function and searching strategies for SGC processes.3. Formulation of model layout problemIn this work, the SGC model tray is represented as a container with an upper limit. The model layout problem in this research can be described as packing a batch of parts of different sizes into the container (bin). Packing tasks are characterized by the following three objectives:* fitting models into the specified container;* avoiding any overlap between models;* achieving high packing density, in other words, achieving the minimum overall height.In the present application, the CAD model of each part is represented in STL format. It consists of coordinate information of facet triangles and their corresponding outward pointing normal in a 3D space. Compared to other formats of CAD models, STL format is very straightforward. However, the search would be very time-consuming if the complete information of an STL model is encoded in the algorithm. This is because one simple operation such as rotate or move means recalculation of the new location of everytriangle facet of a model, and it is very common for a mechanical part to have hundreds and thousands of triangle facets in an STL model. Moreover, these operations are used in thousands of iterations during the searching process of SA. Thus the simplification is necessary. Most of the researchers simply envelop a part with a rectangular box because it is easy to implement and keeps the packing algorithm from becoming too complicated 4. This method was also used in this research. The difference lies in the formulation of objective functions and specific search strategies. The 3D model layout can then be described in the following steps (Fig. 2):1. Parse STL files of the parts to be manufactured and generate an envelope for each part model.2. Pack envelopes in a bin according to a specific algorithm. The bin has the same size as that of the maximum manufacturing volume of the Cubital machine.3. Calculate the updated STL file for each part.4. Demonstrate the final result of the layout. Steps 1, 3, and 4 involve techniques in computer graphics and visualization, which are not specifically related to the packing problem and are relatively easier to implement with existing graphic tools. Our focus is on Step 2. The formulation of this problem is critical to establish a proper algorithm.The bin-packing problem can be represented as an optimization problem in Eq. (1):where n is the number of models, pi the packing state of a model, p the layout, P the set of all layouts, B the boundary of bin, pipj the overlap between any two models, piB the overlap between a model and a bin, h the overall height of a layout and f the objective function mapping the set P into the set of real numbers. In this present problem, a layout is a batch of parts satisfying the constraints defined by Eq. (1). The objective function is defined as the overall height of a layout. The packing state of a model is determined by the location (coordinates) and the orientation of models envelope. For each part a local coordinate system is defined, the origin of which is the geometric center of its envelope. The bottom-left corner of the container (bin) is defined as the global origin. The coordinates of the geometric center of an envelope determine the location of this envelope in the container. The number of state of model orientation is six as shown in Fig. 3.The change from one state to another can be realized through one or two rotational operations. The operation rotate is to spin an envelope 901 around one axis of its local coordinate system. For example, to change the envelope in Fig. 3 from state (a) to state (b), it can be rotated around its x-axis (state (d) and then rotated around its y-axis. The surfaces of every envelope are always perpendicular or parallel to the orthogonal axis. The state of each envelope pi is represented aswhere xi，yi and zi stand for the coordinates of the center of the envelope of the ith model; i is the orientation of the envelope; n is the total number of parts to be packed; Xi ，Yi and Zi are the lengths of the ith envelope along x， y and z directions.The objective function f defines a goodness value for each packing candidate to guide the search in a promising direction. In this study, the goal is to minimize the overall layout height which can be determined by the following equation:During the search process, overlap is permitted to allow models to be moved through one another and results in a more thorough search of the solution space. Since any overlap should be avoided in a final layout, the overlap is penalized in the objective function. Thus the objective function has a second term, f2:When two rectangular boxes overlap in parallel, the overlap is also a rectangular box. In Eq. (4), OijexT; OijeyT and Oij ezT are the lengths of the overlap between the envelopes of the ith and the jth models along x， y and z directions, respectively. Oij is the sum of the lengths of the overlap along the three orthogonal directions.The overlap quantification in Eq. (4) and the following linearly weighted objective function are most commonly used for multi-objective optimizations 7,9:However, this function can seldom result in successful optimization in this packing problem. In the search process, f value of one solution was compared with that of another solution so that the search always goes in the direction of reducing the value of the objective function. With different values of the weights, the comparison results can be very different. Thus the relative value of these two weights is critical. Since the range of f2 varies dramatically according to the number of parts in a batch and the dimensions of parts, there is no single good choice of a pair of weights for different packing instances. Even if there were a pair of weights reflecting the importance of the overlap and the height objectively, it would be difficult to decide their values mathematically. If the overlap f2 is given more weight than it deserves, namely, w2 being much larger than w1; the objective function would be more sensitive to the change of the overlap value f2: The models would be placed apart in the early stage of the search process to eliminate the overlap. Finally, the models would be stacked higher in the container. If the height is given more significance than it deserves, the objective function would be more sensitive to the change of the overall height and the overlap would be encouraged because the value of the objective function would be reduced with layouts of lower height.To avoid the difficulty of determining the values of weights, a nonlinear objective function and another overlap definition were proposed and used in this study. The objective function is given below:ZMAX is the upper limit of the container. It was set f2 = 1 when f21 because the value of the objective function f could not be equal to 0, otherwise it would cause premature convergence. In other words, the solution would not be improved when the objective function value could not be further minimized.In Eq. (4) the overlap between two envelopes was calculated by summing up the overlapped lengths in three dimensions. Another mathematical definition ofthe overlap between two envelopes was given to fit for the objective function in Eq. (7):The motivation for using the nonlinear objective functions is that overlap is good during the search process, assuming that moving through infeasible configuration (layout) can lead to superior feasible configuration (layout) 4. Using this objective function, the algorithm would guide the search towards lowering the layout height after the overlap is already very small. Thus the overall height in the final layout solution is generally low although tiny overlap still exists in the models.SA approach is used to solve the above optimization problem. Tests were conducted to compare the packing using the quantified overlap with those using the overlap defined in Eq. (4). SA showed worse performance or no better performance when using the first method. The results of these tests would be shown later in this paper.The packing algorithm defined here cannot guarantee to find an admissible solution. To eliminate the small overlap in the solution obtained from SA, a simple compensation algorithm was programmed. After the SA search process is completed, if overlap exists, the compensation program can be executed to identify the overlapped models and move them apart along a distance indicated by the minimum overlap. Since the movements may generate new overlaps, all models are checked again. If new overlaps occur, the procedurecontinues until no overlap exists among the models. Overlap can be easily eliminated in most cases after executing this compensation program. In the test cases, the overall height achieved through SA was increased insignificantly.4. Simulated annealing approach4.1. Concepts of SASA is loosely based on the physical idea of annealing (cooling) metals in order to find the lowest energy-state of that metal. The mathematical interpretation of annealing is a cooling schedule, which is a decreasing function of time. SA is a stochastic computational technique derived from statistical mechanics for finding near global-minimum-cost solutions to large optimization problems. In many instances, finding the global minimum value of an objective function with many degrees of freedom subject to conflicting constraints is an NP-hard problem, since the objective function will tend to have many local minima. A procedure for solving optimization problems of this type should sample the search space in such a way that it has a high probability of finding the optimal or a near-optimal solution in a reasonable time.SA was proposed by Kirkpatrick et al. 10. The idea was derived from the algorithm by Metropolis 11. Initially, the starting temperature, T; is given a high value and the system is assumed to start at an initial state p.The value of the objective function f at that point is calculated. A generation mechanism is defined, so that given a state p; a new state p0 can be obtained randomly from the neighborhood of p; which is then evaluated.=f(p)-f(p)is the change in objective function because of the move. If this move leads to an improvement in the objective function, the new design is accepted and becomes the current state. SA also accepts the new solution when its objective function value is worse than that of the old one. This can be calculated with the socalled Metropolis probability. The probability for a state to be accepted is determined as follows:The temperature starts out high and decreases with time. At each temperature, the system is perturbed several times. The set of iterations carried out at each value of the temperature is called a Markov chain. The number of iterations in a chain is sometimes referred to as the chain length. Initially, moves made through the state space are almost random, resulting in a broad exploration of the objective function space. As the probability of accepting inferior moves decreases, they tend to get rejected, allowing the algorithm to converge to an optimum.5. Packing strategies5.1. The move setThe SA search process begins with one or multiple solution (s), then finds other solutions through predefined move. A move,; is an operation through which a new layout solution p0 can be derived from the old one p:p_!p0: A move set is a set comprised of different moves through which perturbations can be made to the model layout. When a new layout is generated from the old one, a move is selected from the move set with certain possibilities. Six moves are defined in the move set for solving the present problem:* Rotate. To change the orientation of the envelope of a model without changing the location of the envelope center. There are six orientations to be selected. However, more possibilities are given to the orientation that minimizes the height of envelope.* Stroll. To randomly change the location of the center of the envelope in 7x， 7y or 7z directions. The moving distance depends on the current temperature. At a higher temperature, a longer distance of movement will be selected in an attempt to achieve a greater change in the cost function and to avoid local minimal.* Swap. To switch the locations of two envelopes; two envelopes to be swapped are selected randomly.* Move towards origin (0,0,0). To move a randomly selected envelope along a direction so that the x; y and z coordinates of that envelope will be decreased. The moving distance depends on the temperature at that moment.* Eliminate overlap. To calculate the overlapping vector between envelopes then move each envelope along the corresponding vector. This operation aims to decrease the overall overlaps, not actually eliminate all overlaps because after the move new overlaps might be generated.* Move against wall. To move a randomly selected envelope in x2y plane (without changing its z position) towards the nearest wall until it touches other envelopes or the wall of container, and then move the envelope downward until it touches other envelopes or the bottom of container. In fact there are two basic operations in the move set above, which are rotate and stroll, other operations can be considered as a special stroll operation or a sequence of stroll operations. The main purpose of the latter four operations is to improve the efficiency of the algorithm.5.2. Overlap detection and calc
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