An automated functional decomposition method based on morphological changes of material flows
基于材料流形态变化的一种自动化功能分解方法

1. Introduction  1. 引言

Conceptual design is an important phase in the product design process (Ullman Citation2015). The goal of conceptual design is to analyse the design requirements of a design task, build its functional structure and find principle solutions. Generally, the overall function is a design purpose, which is defined with uncertainty and is unable to be solved directly. Thus, functional decomposition is important as a natural means of breaking the overall function into more concrete and explicit sub-functions until they can be solved by known working principles (Pahl et al. Citation2007).
概念设计是产品设计过程中的一个重要阶段（Ullman 2015）。概念设计的目的是分析设计任务的设计要求，构建其功能结构并找到基本解决方案。通常，整体功能是一个设计目标，它被定义为不确定且无法直接解决。因此，功能分解作为将整体功能分解为更具体和明确的子功能的一种自然手段非常重要，直到这些子功能可以通过已知的工作原理得到解决（Pahl 等人 2007）。

Because the functions of mechanical products are usually represented as the transformation between input and output flows (material, signal, energy) in existing methods, functional decomposition also focuses on the changes of flows. However, to the authors’ knowledge, most of these methods have paid more attention to changes of flow types and physical properties, while morphological changes (changes of shape, size and structure) in material flow have rarely been considered or have been referred to with rough description. For instance, on the functional basis, the verb ‘shape’ is used for functions that ‘To mold or form a flow’, but it is limited to analysis of the ‘morphological changes’ in detail. One reason for this absence might be that morphological information is difficult to be described and modelled formally in conceptual design stages.
因为现有方法通常将机械产品的功能表示为输入和输出流（物质、信号、能量）之间的转换，功能分解也关注流的变化。然而，据作者所知，这些方法大多更关注流类型和物理性质的变化，而材料流中的形态变化（形状、尺寸和结构的变化）很少被考虑或仅以粗略描述提及。例如，在功能基础上，动词“shape”用于表示“塑造或形成流”的功能，但它仅限于对“形态变化”进行详细分析。这种缺失的一个原因可能是形态信息在概念设计阶段难以被形式化描述和建模。

The changes of material flow's shape and structural form, such as pipe bending, are the primary functional requirements of various mechanical products. However, the requirement descriptions are usually intentional and objective, such as ‘a pipe bender accepts the user's description about the manner in which a pipe should be bent’ and ‘a pipe bender bends the pipe in a predetermined manner’; thus, it is difficult for the designers to determine principle solutions and the physical structure of the product directly.
材料流形状和结构形式的变化，如管道弯曲，是各种机械产品的主要功能要求。然而，这些要求描述通常是主观和客观的，例如“管道弯曲机接受用户关于管道应如何弯曲的描述”和“管道弯曲机以预定的方式弯曲管道”；因此，设计人员难以直接确定产品的原理方案和物理结构。

In this study, the formal representation of the morphological properties of the material flow is discussed first. Thus, designers can translate the intentional description of a material's morphological changes into a relatively formal and objective description. The designer can analyse a use case in which the morphological change requirement is specified with the original and the desired forms of the material flow. Then, the material's input and output morphological properties can be formally represented in an overall function definition. Based on the detailed morphological information, an automated sub-task decomposition process is proposed to decompose the overall morphological changes of a material flow into several sub-functions. Each of the sub-functions refers to a local shape deformation or structural change. Moreover, with the consideration of the sequential and spatial coupling relations of the material flow's morphological properties, a planning algorithm is provided to determine the implementation order of the sub-functions for local morphological changes. The morphological change sub-functions and their logical implementation order would benefit the subsequent design process for the principle determination and physical structure implementation of the product.
在本研究中，首先讨论了物质流形态特性的形式化表示。这样，设计师可以将物质形态变化的目的性描述转化为相对形式化和客观的描述。设计师可以分析一个指定了物质流原始形态和期望形态的用例，其中包含形态变化要求。然后，物质流的输入和输出形态特性可以在整体功能定义中形式化表示。基于详细的形态信息，提出了一种自动子任务分解过程，将物质流的整体形态变化分解为多个子功能。每个子功能指代一个局部形状变形或结构变化。此外，考虑到物质流形态特性的时序和空间耦合关系，提供了一种规划算法来确定局部形态变化子功能的执行顺序。 形态变化子功能及其逻辑实施顺序将有助于后续的产品原理确定和物理结构实施设计过程。

The remainder of this paper is organised as follows. Section 2 outlines the related works about functional representation and decomposition. An overview of our method is provided in Section 3. In Section 4, the morphological information modelling methods for material flow are introduced. Sections 5–7 present the decomposition principles, functional decomposition process and planning algorithm to determine the implementation order of the sub-functions, respectively. A case study is provided in Section 8 to demonstrate the proposed method. A comparative analysis of the proposed method with other methods is discussed in Section 9. Finally, the paper finishes with a conclusion and suggestions for future work in Section 10.
本文其余部分组织如下。第 2 节概述了关于功能表示和分解的相关研究。第 3 节提供了我们方法的总览。第 4 节介绍了材料流形态信息建模方法。第 5 至 7 节分别介绍了分解原则、功能分解过程以及确定子功能实施顺序的规划算法。第 8 节提供了一个案例研究以展示所提出的方法。第 9 节讨论了所提出方法与其他方法的比较分析。最后，本文在第 10 节以结论和对未来工作的建议结束。

2. Related work  2. 相关工作

Function is the fundamental concept in engineering practice because the goal of a design activity is to deliver artefacts with desired functionalities (Chandrasekaran and Josephson Citation2000). Keuneke (Citation1991), Deng (Citation2002) and Fantoni, Apreda, and Bonaccorsi (Citation2009) preferred to see ‘function’ as the action or behaviour performed on a flow, while Gero (Citation1990), Umeda and Tomiyama (Citation1995) considered function to be an intermediary linking the intentional description of a design purpose to the physical behaviour or structure of an artefact (Vermaas and Dorst Citation2007). The distinct conceptualisations of function lead to different functional representation approaches (Summers, Eckert, and Goel Citation2013). With these techniques, complex system design problems can be simplified into representations that are more readily solvable (Nagel, Bohm, and Linsey Citation2014). A function can either be described in natural language or be represented by a mathematical formula to show the input/output transformation (Chakrabarti and Bligh Citation2001). Chakrabarti and Blessing (Citation1996) divided the existing function representation methods into three categories. One popular method is the verb + noun pair. To propose a common representation of function, researchers at National Institute of Standards and Technology (NIST) (Szykman, Racz, and Sriram Citation1999) developed generic taxonomies of over 130 engineering functions and over 100 associated flows. A similar effort is the functional basis developed by Stone and Wood (Citation2000). Over 100 products are studied and compared to propose standard functional vocabulary. Hirtz et al. (Citation2002) reconciled and evolved these two previous efforts to combine them together to the final version of ‘Functional Basis’ (FB), which is widely accepted. For example, the system called FunctionCAD (Nagel et al. Citation2009; Nagel Citation2011) is developed to support the formalised modelling of functions based on the FB. Most of the other works have been based on the input/output flow transformation or the input/output flow's state transformation (Deng, Britton, and Tor Citation2000; Borgo et al. Citation2011; Sen, Summers, and Mocko Citation2013a).
功能是工程实践中的基本概念，因为设计活动的目标是为用户提供具有所需功能的产品（Chandrasekaran 和 Josephson 2000）。Keuneke（1991）、Deng（2002）以及 Fantoni、Apreda 和 Bonaccorsi（2009）倾向于将“功能”视为对流程执行的动作或行为，而 Gero（1990）、Umeda 和 Tomiyama（1995）则认为功能是连接设计目的的意图描述与产品的物理行为或结构的中间环节（Vermaas 和 Dorst 2007）。功能的独特概念化导致了不同的功能表示方法（Summers、Eckert 和 Goel 2013）。通过这些技术，复杂系统的设计问题可以简化为更易于求解的表示形式（Nagel、Bohm 和 Linsey 2014）。功能可以用自然语言描述，也可以用数学公式表示输入/输出的转换（Chakrabarti 和 Bligh 2001）。Chakrabarti 和 Blessing（1996）将现有的功能表示方法分为三类。一种流行的方法是动词+名词组合。 为了提出一种通用的功能表示方法，美国国家标准与技术研究院（NIST）（Szykman、Racz 和 Sriram 1999）的研究人员开发了一般分类法，包括超过 130 种工程功能和超过 100 种相关流程。类似的努力是 Stone 和 Wood（2000）开发的功能基础。研究并比较了超过 100 种产品，以提出标准功能词汇。Hirtz 等人（2002）协调并发展了这两项先前的努力，将它们结合起来，形成了最终版本的“功能基础”（FB），该版本被广泛接受。例如，名为 FunctionCAD（Nagel 等人 2009；Nagel 2011）的系统被开发出来，用于基于 FB 对功能进行形式化建模。其他大多数工作都是基于输入/输出流程转换或输入/输出流程的状态转换（Deng、Britton 和 Tor 2000；Borgo 等人 2011；Sen、Summers 和 Mocko 2013a）。

In these traditional function representation methods, the extremely comprehensive classification of function and flow (Stone and Wood Citation2000) exists to distinguish the diversity of functional behaviours and to demonstrate input and output flows. Although textual representation can facilitate human understanding, it does not contain the detailed semantics of flows and functions, thus hampering the functional decomposition process. Researchers have expanded the semantics of functions and flows in different manners. Some of them have used ontology to organise functional knowledge between domains and entities, such as function and behaviour representation language (Sasajima et al. Citation1995) and functional concept ontology (Kitamura et al. Citation2002). Researchers noted that flows play different roles in a function (Otto and Wood Citation2001; Kitamura and Mizoguchi Citation2003; Nagel et al. Citation2007; Sen, Summers, and Mocko Citation2013a). They divided flows into primary flow and carrier flow, and they identified the carry relations between them. Sen, Summers, and Mocko (Citation2011) proposed a function verb redefinition protocol to extend a function's semantic information. Based on this protocol, formal representation for graph-based functional models was proposed to support physical-based reasoning (Sen, Summers, and Mocko Citation2013b). The type and number of input/output flows are considered, and the mass and energy conservation are emphasised in this work, while other attributes of a flow are ignored. Chen et al. (Citation2015a) preferred to see flow affected by function as a ‘thing’. A thing is a substantial entity endowed with all of its properties, including its geometrical form (Bunge Citation1977). Based on this idea, the flow's semantics representation was extended by its type and flow properties in our previous work (Yuan et al. Citation2016). However, the representation of material flow's geometrical form remains an unsolved problem. The difficulty includes the complexity of spatial relationships that can exist between multiple objects and the variety of shape forms, and both of these issues are difficult to depict quantitatively in the conceptual design stage. To support the representation of geometric models for mechanism components, Chen et al. (Citation2009) proposed a modelling method using geometric elements and constraints to describe the shape of components and the spatial relations between them, which could be a solution for the modelling of material flows' shapes.
在这些传统的功能表示方法中，功能与流程的极其详尽的分类（Stone and Wood 2000）用于区分功能行为的多样性，并展示输入和输出流程。尽管文本表示可以促进人类理解，但它不包含流程和功能的详细语义，从而阻碍了功能分解过程。研究人员以不同方式扩展了功能和流程的语义。其中一些人使用本体来组织领域和实体之间的功能知识，例如功能与行为表示语言（Sasajima et al. 1995）和功能概念本体（Kitamura et al. 2002）。研究人员指出，流程在功能中扮演不同的角色（Otto and Wood 2001; Kitamura and Mizoguchi 2003; Nagel et al. 2007; Sen, Summers, and Mocko 2013a）。他们将流程分为主要流程和载体流程，并确定了它们之间的承载关系。Sen, Summers, and Mocko (2011) 提出了一种功能动词重新定义协议来扩展功能的语义信息。 基于此协议，提出了基于图的函数模型的正式表示方法，以支持基于物理的推理（Sen、Summers 和 Mocko 2013b）。该方法考虑了输入/输出流的类型和数量，并在本研究中强调了质量和能量守恒，而忽略了流的其他属性。Chen 等人（2015a）倾向于将受函数影响的流视为一个“事物”。事物是一个具有其所有属性（包括其几何形状）的实体（Bunge 1977）。基于这一思想，我们在先前的工作中通过流的类型和流属性扩展了流的语义表示（Yuan 等人 2016）。然而，材料流的几何形状表示仍然是一个未解决的问题。困难包括多个对象之间可能存在的空间关系的复杂性以及形状形式的多样性，这些问题在概念设计阶段都难以定量描述。为了支持机制组件的几何模型表示，Chen 等人 (2009) 提出了一种使用几何元素和约束来描述组件形状及其空间关系的建模方法，这可以作为一种解决材料流形状建模问题的方案。

Based on the information contained in a function representation, the functional decomposition can be used to capture sub-functions (van Eck, McAdams, and Vermaas Citation2007). The Design Repository ProjectFootnote¹ is an ongoing project that involved researchers from UMR, the University of Texas at Austin and NIST. It stores general design knowledge from 184 products and 6906 artefacts. With the archived function-based knowledge from existing products by reverse engineering, functional decomposition and concept generation for familiar products are supported (Bohm, Vucovich, and Stone Citation2005). However, the system design for novel products would be more challenging. To understand the tasks of a product and find principle solutions, Umeda et al. (Citation1996) divided the functional decomposition process into task decomposition and casual decomposition and stated that task decomposition is not physically derivable. Vermaas (Citation2013) proved that it was formally impossible to analyse sub-functions as parts of an overall function. Thus, task decomposition, which breaks the overall functions into independent sub-tasks, is difficult to process automatically and is usually achieved manually. At the same time, researchers have proposed diverse function models to explain the causality of the nature and function of a system, such as the Function–Behaviour–Structure (FBS) model defined by Gero (Citation1990), the FBState model (Umeda et al. Citation1990, Citation1996; Umeda and Tomiyama Citation1997) and the structure–behaviour–function (SBF) model (Goel and Chandrasekaran Citation1989; Yaner and GOEL Citation2006). The mutual interaction defined in these models can guide casual decomposition, which breaks a function into its support functions (Miles Citation2015) and finds the required physical behaviour or structure to implement them. Recently, based on these classical models, functional decomposition approaches have been proposed. Using the FBS model and the vocabulary from FB, Helms, Shea, and Hoisl (Citation2009) developed computational design synthesis based on graph-grammars. Because of the limitations of FB itself, only the types of flow are considered in the rules for decomposition, which can generate a large number of unsatisfactory solutions. Moreover, it would be difficult to decompose a function with multiple input/output flows into different types. There are also some implemented tools, such as Komoto and Tomiyama's Function Modular (Citation2011, Citation2012), the KOM (Knowledge Organising Module) developed by Russo and Montecchi (Citation2011a, Citation2011b) and the SAPPhIRE of Chakrabarti and Bligh (Citation1994, Citation2001). A common point in these decomposition tools is the exploitation of the physical principle. The physical principle can be represented by a formula; therefore, it is objective and convenient for qualitative reasoning (De Kleer and Brown Citation1984). However, the physical principle is usually corresponded directly to a certain function. Chen et al. (Citation2015b) noted that the relation between function and its principle solutions are logically unreasonable. The inflexible relation between these entities results in the automaticity of the existing decomposition tools being rare for principle searching and mapping.
基于功能表示中包含的信息，功能分解可用于捕获子功能（van Eck, McAdams, and Vermaas 2007）。设计知识库项目 ¹ 是一个正在进行中的项目，涉及来自 UMR、德克萨斯大学奥斯汀分校和 NIST 的研究人员。它存储了来自 184 种产品和 6906 个实体的通用设计知识。通过逆向工程获取现有产品的基于功能的存档知识，可以支持熟悉产品的功能分解和概念生成（Bohm, Vucovich, and Stone 2005）。然而，新产品的系统设计将更具挑战性。为了理解产品的任务并找到基本解决方案，Umeda 等人（1996）将功能分解过程分为任务分解和因果分解，并指出任务分解在物理上无法推导。Vermaas（2013）证明了将子功能作为整体功能的一部分进行分析在形式上是不可能的。因此，任务分解将整体功能分解为独立的子任务，难以自动处理，通常需要手动完成。 与此同时，研究人员提出了多种功能模型来解释系统的自然属性和功能之间的因果关系，例如 Gero（1990）定义的功能-行为-结构（FBS）模型、FBState 模型（Umeda 等人 1990、1996 年；Umeda 和 Tomiyama 1997 年）以及结构-行为-功能（SBF）模型（Goel 和 Chandrasekaran 1989 年；Yaner 和 GOEL 2006 年）。这些模型中定义的相互作用可以指导因果分解，即将功能分解为其支持功能（Miles 2015 年），并找到实现这些功能所需的物理行为或结构。最近，基于这些经典模型，提出了功能分解方法。使用 FBS 模型和来自 FB 的词汇，Helms、Shea 和 Hoisl（2009 年）开发了基于图-文法的计算设计合成方法。由于 FB 本身的局限性，分解规则中仅考虑了流的类型，这可能导致大量不令人满意的解决方案。此外，将具有多个输入/输出流的函数分解为不同类型将非常困难。 也有一些已实现的工具，例如 Komoto 和 Tomiyama 的函数模块（2011，2012），Russo 和 Montecchi 开发的 KOM（知识组织模块）（2011a，2011b）以及 Chakrabarti 和 Bligh 的 SAPPhIRE（1994，2001）。这些分解工具的共同点是利用物理原理。物理原理可以用公式表示；因此，它对定性推理是客观且方便的（De Kleer 和 Brown 1984）。然而，物理原理通常直接对应于某一特定功能。Chen 等人（2015b）指出，功能与其原理解决方案之间的关系在逻辑上是不合理的。这些实体之间僵化的关系导致现有的分解工具在原理搜索和映射方面的自动化程度较低。

To bridge the gap between function and principle solutions, the concept of functional effect (Yuan et al. Citation2016) was proposed to reflect the essence of a function, which can also be used to map a product's function to the physical effects required to implement it. Currently, there is no way to decompose the function of material flow's morphological changes. To aid designers in determining the morphologic states of a flow and to find the process for shaping or deforming a material in conceptual design, the material flow's structure and geometric form representation and morphological decomposition were followed with interest in our previous work (Yuan and Liu Citation2014; Yuan, Zhang, and Liu Citation2015).
为了连接功能解决方案和原理解决方案之间的差距，提出了功能效应（Yuan 等，2016）的概念，以反映功能的本质，它也可以用来将产品的功能映射到实现它所需的物理效应。目前，没有办法分解材料流形态变化的功能。为了帮助设计者在概念设计中确定流的形态状态，并找到塑造或变形材料的过程，我们之前的工作（Yuan 和 Liu，2014；Yuan、张和 Liu，2015）关注了材料流的结构和几何形状表示以及形态分解。

3. Method overview  3. 方法概述

To conduct automated functional decomposition for mechanical products based on the morphological changes of material flows, a formal representation is proposed to model the morphological information. Thus, the overall function of a system for processing material flows can be represented by its input and output material flows, with their detailed morphological semantics. Then, the functional decomposition process is executed on the overall function to obtain the structure changes between flows, the geometric form change of a solid material flow and other property changes of flows. Finally, the sub-function planning algorithm is presented to determine the logical implementation order for the sub-functions of the local shape form changes. To achieve the above morphological information-based functional decomposition process, three issues are considered in this work as follows.
基于物料流的形态变化，对机械产品进行自动化功能分解，提出了一种形式化表示方法来建模形态信息。因此，处理物料流的系统的整体功能可以通过其输入和输出物料流及其详细的形态语义来表示。然后，在整体功能上执行功能分解过程，以获得流之间的结构变化、固体物料流的几何形态变化以及其他流属性变化。最后，提出了子功能规划算法，以确定局部形态形态变化子功能的逻辑实现顺序。为实现上述基于形态信息的功能分解过程，本研究考虑了以下三个问题。

• The formal representation of a material's morphological information. A material flow's morphological information contains its structure and shape semantics. Firstly, the structure of the material is modelled hierarchically according to its components and the relation constraints between them. Next, a graph-based model is used to represent the shape of the material flow. In this model, the global shape of the material is divided into several local shape elements, and each of them is described by an independent model. Then, the local abstract models are connected with certain constraints to generate a morphological representation graph.
  材料形态信息的正式表示。材料流的形态信息包含其结构和形状语义。首先，根据材料的组成部分及其相互间的关系约束，分层建模材料的结构。接下来，使用基于图的模型来表示材料流的形状。在该模型中，材料的整体形状被划分为多个局部形状元素，每个元素由一个独立的模型描述。然后，将局部抽象模型通过特定约束连接起来，生成形态表示图。

• The morphological decomposition. The idea of divide and conquer is employed here. The input and output morphological representation models of the material flow are parsed and compared to determine the changes between the two models. Then, the overall morphological change of a material flow is broken into several local structure changes and local shape element deformations, each of which is a logical sub-function for the morphological change.
  形态分解。这里采用了分而治之的思想。解析和比较物料流的输入和输出形态表示模型，以确定两个模型之间的变化。然后，将物料流的整体形态变化分解为几个局部结构变化和局部形状元素变形，每个都是形态变化的一个逻辑子功能。

• The sub-function planning algorithm. This algorithm is used to determine the implementation order of the local shape form change sub-functions. A morphological changes graph is created first, each node of which refers to a shape element of the material flow and each edge of which refers to the local relation change between two shape elements. Then, the graph is traversed with changeability checking and the priority setting for each local morphological change. Finally, a sub-function sequence for these local morphological changes is planned, which will inspire designers to consider the sequential relation between sub-functions to construct the functional structure of this product.
  子功能规划算法。该算法用于确定局部形状形式变化子功能的实现顺序。首先创建一个形态变化图，其中每个节点指代物料流中的一个形状元素，每条边指代两个形状元素之间的局部关系变化。然后对图进行遍历，进行变化性检查并对每个局部形态变化设置优先级。最后，规划出这些局部形态变化的子功能序列，这将启发设计者考虑子功能之间的顺序关系，以构建该产品的功能结构。

4. Morphological representation for material flow
4. 物质流的形态表示

A formal representation of function is the basis of automating the functional decomposition process. In our previous work (Yuan et al. Citation2016), the flow type and flow property were used to describe the input/output flow's semantic information, based on the flow classification in FB (Hirtz et al. Citation2002). To enable the functional decomposition for changes of material flow, the morphological information of material flow is added to the previous function representation model, which consists of two parts: the hierarchical structure, to show how a material flow is constituted by its components; and the graph representation, to describe the shape of an object at different levels of abstraction (or resolution).
功能的形式化表示是自动化功能分解过程的基础。在我们的前期工作（Yuan 等人，2016 年）中，基于 FB（Hirtz 等人，2002 年）中的流动分类，使用流动类型和流动属性来描述输入/输出流动的语义信息。为了实现针对物质流动变化的功能分解，将物质流动的形态信息添加到先前功能表示模型中，该模型由两部分组成：层次结构，用于展示物质流如何由其组成部分构成；以及图表示，用于描述对象在不同抽象层次（或分辨率）下的形状。

4.1. Hierarchical representation of the structure of a material flow
4.1. 物质流结构的层次表示

The nouns ‘mixture’ and ‘composite’ defined in FB (Hirtz et al. Citation2002) can be used to describe the material flow, which is ‘a substance mixed by multiple components and can be separated by physical means’, or ‘the composite material with its components in fixed proportions respectively’. However, they are not suitable for assemblies, packaged objects or connected mechanical parts. Moreover, these two words can only identify a flow's type at an abstract level without information about how the components are associated with each other. However, various spatial and geometric relations can exist between material flows, which are important for the functional decomposition of morphological changes.
FB（Hirtz 等人，2002 年）中定义的名词“混合物”和“复合材料”可用于描述物质流，即“由多个成分混合而成的物质，可以通过物理方法分离”，或“各成分按固定比例组成的复合材料”。然而，它们不适用于组装件、包装物体或连接的机械部件。此外，这两个词只能抽象地识别流的类型，而无法提供关于成分如何相互关联的信息。然而，物质流之间可能存在各种空间和几何关系，这些关系对于形态变化的功能分解非常重要。

Therefore, the attributes ownedComponent and ownedRelation have been defined (Yuan and Liu Citation2014) to represent a material flow that has a complex structure. The former is used to describe the components that compose an input or output flow, whereas the latter refers to the relation between the components with the help of the class relationConstraint, the structure of which is shown in Figure 1. Here, the attributes ‘source’ and ‘target’ have the type of flow object. They indicate that the relationship is between the source object and the target object. The attribute relationType refers to the type of relation as a geometric constraint type (Chen et al. Citation2009) or spatial constraint type (such as in/on). The ‘value’ is a qualitative or quantitative value for a certain relation type, such as the value of the degree for relationType ‘angle’ and the length for relationType ‘distance’. Because the ownedComponents of a flow are also flow objects, which can have their owned components. Thus the structure of a complex material flow can be represented hierarchically.
因此，为了表示具有复杂结构的物质流，已定义了 ownedComponent 和 ownedRelation 属性（Yuan and Liu 2014）。前者用于描述构成输入或输出流的组件，而后者则借助类 relationConstraint 来指代组件之间的关系，其结构如图 1 所示。在此，属性“source”和“target”的类型为 flow object。它们表明关系存在于源对象和目标对象之间。属性 relationType 指的是关系类型，可以是几何约束类型（Chen et al. 2009）或空间约束类型（如 in/on）。‘value’是某种关系类型的定性或定量值，例如 relationType 为‘angle’时的角度值和 relationType 为‘distance’时的长度值。由于流的 ownedComponents 也是 flow object，它们可以拥有自己的组件。因此，复杂物质流的结构可以分层表示。

Figure 1. The meta-model for flow semantics.
图 1. 流语义的元模型。

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An overall function and the definitions for the flow objects of its input and output flows are shown in Figure 2. A function called ‘milk filling’ is defined in Figure 2(a). The output flow of this function is a flow object called ‘bottled milk’, shown in Figure 2(b), and it has three components: ‘milk’, ‘bottle’ and ‘cap’. Two ownedRelations are used to describe how bottled milk is composed from the components: ‘milk in bottle’ and ‘cap on the bottle’. Thus, the flow object ‘bottled milk’ has a formal and semantic representation about its structure.
整体功能及其输入和输出流的流对象定义如图 2 所示。图 2(a)中定义了一个名为“牛奶灌装”的功能。该功能的输出流是一个名为“瓶装牛奶”的流对象，如图 2(b)所示，它包含三个组成部分：“牛奶”、“瓶子”和“盖子”。使用两个 ownedRelations 来描述瓶装牛奶如何由这些组成部分构成：“瓶子中的牛奶”和“瓶子上的盖子”。因此，流对象“瓶装牛奶”具有关于其结构的正式和语义表示。

Figure 2. The definition of the milk filling function and its output flow ‘bottled milk’.
图 2. 牛奶灌装功能的定义及其输出流“瓶装牛奶”。

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4.2. Representation of the shape property of a material flow
4.2. 材料流形状特性的表示

The morphological property is proposed to model the geometric form of a solid material flow. The type of attribute refers to the geometric shape to which the flow can be abstracted. The regular geometric shapes can be described by the known geometric types. In this study, some basic shape types are defined, as shown in Figure 3. They are reusable when modelling the shape of a specific material flow. If the shape of a solid material can be abstracted as a basic and regular shape, it can be represented by a basic element with the shape type in Figure 3. Note that more basic shape types can be defined in the future if necessary. Moreover, the value property is employed to describe the geometric parameters. For example, the shape of a box can be abstract as a cuboid, and its value properties are the length, width and height.
提出使用形态特性来模拟固体材料流的几何形态。属性类型指的是流可以被抽象出的几何形状。规则的几何形状可以通过已知的几何类型来描述。在本研究中，定义了一些基本形状类型，如图 3 所示。在建模特定材料流的形状时，这些形状类型是可重复使用的。如果固体材料的形状可以被抽象为基本且规则的形状，它就可以用图 3 中所示的形状类型的基本元素来表示。请注意，如果需要，未来可以定义更多基本形状类型。此外，使用值特性来描述几何参数。例如，盒子的形状可以被抽象为长方体，其值特性是长度、宽度和高度。

Figure 3. The rough shape types.
图 3. 粗糙形状类型。

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However, the solid material flow can have various geometric shapes. Some of them might be complex and irregular and thus cannot be easily abstracted as a regular geometry shape. In this case, the composite element is used, which is composed of basic elements in a certain manner. This mechanism is similar to the creation of a complex 3-D model in CAD tools, which is achieved by the Boolean operations between basic models. Li, Zhou, and Liu (Citation2015) provided a hierarchical partition graph for B-rep model retrieval, which is adapted in this study to model the composite element's structure in a hierarchical fashion, called hierarchical shape graph (HSG), which includes the following two parts.
然而，固体物料流可以有多种几何形状。其中一些可能比较复杂和不规则，因此难以抽象为规则的几何形状。在这种情况下，使用复合元素，它由以某种方式组合的基本元素构成。这种机制类似于在 CAD 工具中创建复杂 3D 模型，通过基本模型之间的布尔运算实现。Li、Zhou 和 Liu（2015）为 B-rep 模型检索提供了一个层次划分图，本研究将其应用于以层次方式建模复合元素的结构，称为层次形状图（HSG），它包括以下两个部分。

1. Hierarchical structure tree (HST): It is an abstract model for the structure of the shape of a material flow with complex form. The root of an HST is a composite element that indicates the abstract shape of a material flow. Each child node is a local shape descriptor corresponding to the sub-element. The child node can be a leaf node (a basic element) or a non-leaf node (a composite element).
   层级结构树（HST）：它是一个用于复杂形态材料流结构的抽象模型。HST 的根是一个复合元素，表示材料流的抽象形态。每个子节点是对应子元素的局部形态描述符。子节点可以是叶节点（基本元素）或非叶节点（复合元素）。

2. Local adjacent graph (LAG): It describes the connection relationship between the local shape elements of a composite element. It is modelled by an adjacent graph with the vertex being the shape element and the edge being the connection relationship between a pair of shape elements. If the child nodes of a composite element are all basic elements, the LAG is exactly the descriptor of the entire shape structure. Otherwise, the LAG only shows the connections of the composite element's sub-structures at an abstract level.
   局部邻接图（LAG）：它描述了复合元素局部形状元素之间的连接关系。它通过一个邻接图来建模，其中顶点是形状元素，边是形状元素之间的一对连接关系。如果复合元素的子节点都是基本元素，则 LAG 正好是整个形状结构的描述符。否则，LAG 仅以抽象级别显示复合元素子结构的连接。

Figure   图 4(a) depicts a real pyramidal box. The cardboard shown as Figure 
(a) 展示了一个真实的金字塔形盒子。图示的纸板如图 4(b) is folded into a pyramid, as in Figure 
(b) 折叠成一个金字塔，如图所示4(c). The input and output HSG of the cardboard and pyramid are shown in Figure 
(c). 纸板和金字塔的输入和输出 HSG 如图所示5. The shape structure of the cardboard is very simple, as presented in Figure 
. 纸板的形状结构非常简单，如图5(a). Five shape elements, labelled part0 to part4, are integrated to generate the entire shape of this cardboard. Each of these shape elements is a basic element. Because they are linked on the same plane, the angles between each of the two of them are straight angles. The LAG of the root node is shown as Figure 
(a)所示。五个形状元素，标记为 part0 到 part4，被集成起来生成这个纸板的整个形状。这些形状元素中的每一个都是一个基本元素。因为它们在同一平面上连接，所以它们之间的角度都是直角。根节点的 LAG 如图5(b). Figure   (b)所示。图5(c) shows the HSG of the shape structure of the output pyramidal box. The HST shows that the overall shape descriptor (the root) of this box has two shape elements at the first level of detail: the basic element part5 as the base and the pyramidal side. The pyramidal side is composed of part1, part0 and part3 at the second level with more detail. Then, the LAG of the root node and side shows how the shape elements connect to each other. The relationConstraint semantics of each edge in the two LAGs are shown in Table 
(c)显示了输出金字塔盒子的形状结构的 HSG。HST 显示，该盒子的整体形状描述符（根）在第一级细节中具有两个形状元素：基本元素部分 5 作为基座和金字塔侧面。金字塔侧面由第二级更详细的 part1、part0 和 part3 组成。然后，根节点和侧面的 LAG 显示了形状元素如何相互连接。两个 LAG 中每条边的 relationConstraint 语义在表中显示。1. This hierarchical shape modelling method reflects the designer's cognitive processes of recognising the morphology of a material flow. The advantage of the method is that the complexity caused by describing all of the details of the overall structure of the material flow is avoided.
这种方法学反映了设计师识别材料流形态的认知过程。该方法的优势在于避免了描述材料流整体结构所有细节所带来的复杂性。

Figure 4. The folding of a pyramidal box.
图 4. 金字塔形盒子的折叠。

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Figure 5. The HST of the input/output shape of the pyramidal box.
图 5. 金字塔形盒子的输入/输出形状的 HST

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Table 1. The semantics contained in the input/output LAG.
表 1. 输入/输出 LAG 中包含的语义

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4.3. Representation of the morphological changes of a material flow
4.3. 物质流形态变化的表现

The plain text of verbs and nouns without detailed semantic information are used widely in the existing method to represent functions. To enforce the semantics of a function and to reveal the essential effect that a function acts on the flows, the concept ‘functional effect’ is proposed in the author's previous work (Yuan et al. Citation2016) as typeChange and valueChange to show the type and property changes of a flow. For the material's morphological changes, two types of functional effect, shapeChange and relationChange, are extended as shown in Figure 6. Note that these effects are not mutually exclusive. Therefore, one or more of them can be captured at the same time, according to the changes that occur in the flow.
现有方法广泛使用动词和名词的纯文本形式来表示功能，而缺乏详细的语义信息。为了强化功能的语义并揭示功能对流程产生的本质效果，作者在先前的工作中（Yuan 等人，2016 年）提出了“功能效果”的概念，将其分为 typeChange 和 valueChange 两种类型，以展示流程的类型和属性变化。对于物质形态变化，扩展了两种功能效果：shapeChange 和 relationChange，如图 6 所示。请注意，这些效果并非相互排斥。因此，根据流程中发生的变化，可以同时捕捉其中一种或多种效果。

Figure 6. The definitions of functional effects.
图 6. 功能效应的定义。

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Here, the ShapeChange is used to describe the changes of the shape of a material flow. For example, the bending machine changes a straight pipe into a bent one with certain bending angle and bending radius. The RelationChange is used to show the change of a relation constraint between two objects (two flows or two shape elements). To describe the change trend of a relation, the definition of values of the attribute changeTrend can be ‘generate’, ‘remove’, ‘increase’, ‘decrease’, ‘change’ and ‘keep’. They represent generating a constraint, removing a constraint (to decouple two objects), the increasing or decreasing of the value of a specific type of constraint, the change of a constraint's type or the maintaining of a relation constraint, respectively. The remainder of the attributes of RelationChange are defined the same as relationConstraint's. For example, to describe that a mechanical arm inserts a screw into a cap, the relationChange of the screw through the cap will be <screw, cap, joint, generate>.
这里使用 ShapeChange 来描述物料流形状的变化。例如，弯曲机将直管弯曲成具有一定弯曲角度和弯曲半径的弯管。RelationChange 用于表示两个对象（两个流或两个形状元素）之间关系约束的变化。为了描述关系的变化趋势，属性 changeTrend 的值定义可以是“generate”、“remove”、“increase”、“decrease”、“change”和“keep”。它们分别代表生成约束、移除约束（使两个对象解耦）、特定类型约束值的增加或减少、约束类型的改变或保持关系约束。RelationChange 的其余属性与 relationConstraint 的定义相同。例如，为了描述机械臂将螺丝插入盖子中，螺丝通过盖子的 relationChange 将是<screw, cap, joint, generate>。

5. Decomposition principles for morphological changes
5. 形态变化分解原则

The aforementioned hierarchical structure and HSG-based shape representation provide the morphological description for the material flow. To support automated functional decomposition, based on the morphological semantics of input and output material flows, the global morphological change of a material flow should be broken up into several local shape elements' changes or the relation changes between them. In this study, several types of principles – the material conservation principle, shape element conservation principle and energy minimisation principle – are proposed to facilitate the functional decomposition automatically.
上述的层次结构和基于 HSG 的形状表示为物质流提供了形态描述。为了支持自动功能分解，基于输入和输出物质流的形态语义，物质流的全局形态变化应被分解为多个局部形状元素的变化或它们之间关系的变化。在本研究中，提出了几种类型的原理——物质守恒原理、形状元素守恒原理和能量最小化原理——以促进自动功能分解。

6. Morphological change-based automated functional decomposition
6. 基于形态变化的自动功能分解

Usually, there are several input and output flows for a mechanical system, and complex changes are made in them. It is not a trivial task to find principle solutions for a mechanical system directly. In this study, the automated functional decomposition method based on morphological change is divided into two steps: (1) Sub-task decomposition: the overall function is decomposed into multiple independent and explicit sub-tasks, according to the semantics of the input and output material flows; and (2) Shape change decomposition: the shape change sub-function is further decomposed with the help of decomposition principles and the semantics contained in the shape model of the material flow.
通常，一个机械系统有多个输入和输出流，这些流的变化很复杂。直接为机械系统寻找原理解决方案并非易事。在本研究中，基于形态变化的自动化功能分解方法分为两个步骤：（1）子任务分解：根据输入和输出物料流的语义，将整体功能分解为多个独立和明确的子任务；以及（2）形态变化分解：借助分解原理和物料流形态模型中包含的语义，进一步分解形态变化子功能。

6.1. Sub-task decomposition
6.1. 子任务分解

In this work, the composite relationships between a function and its sub-functions can seem to be parthood relations. The sub-task decomposition is used to divide the multiple input and output flows with their various attributes in the overall function. Thus, the sub-functions are relatively independent, each of which is a mapping of a particular attribute of a flow in the input set to a particular attribute of a flow in the output set. Moreover, the hierarchical structure of a material flow and HSG-based shape representation allow the morphological model of a flow to have structural and hierarchical characteristics. Such characteristics are considered in the decomposition principles. Thus, with the help of the aforementioned decomposition principle, the sub-tasks' decomposition process is shown in Figure 7 and is described as follows.
在这项工作中，功能与其子功能之间的复合关系似乎可以被视为部分与整体的关系。子任务分解用于将整体功能中的多个输入和输出流及其各种属性进行划分。因此，子功能相对独立，每个子功能都是将输入集中流体的特定属性映射到输出集中流体的特定属性。此外，物质流的层次结构和基于 HSG 的形状表示使得流体的形态模型具有结构和层次特征。这些特征被考虑在分解原则中。因此，借助上述分解原则，子任务的分解过程如图 7 所示，并描述如下。

1. Map each input flow to the corresponding output flow(s) according to the material conservation principle. If a flow is branched to its ownedComponents or composed to other flows, a sub-function with its function type of ‘branch’ or ‘connect’ will be determined.
   根据物质守恒原理，将每个输入流映射到相应的输出流。如果一个流分支到其所属组件或与其他流组合，将确定一个类型为“分支”或“连接”的子功能。

2. Generate the sub-function of relation change. For the ‘branch’ and ‘connect’ sub-functions, the ownedRelation of its input flow or output flow refers to the change of relation constraints between their ownedComponents.
   生成关系变化子功能。对于“分支”和“连接”子功能，其输入流或输出流的 ownedRelation 指的是它们 ownedComponents 之间关系约束的变化。

3. Check whether the type and properties of the input/output flows are changed. Then, the sub-functions with typeChange, valueChange (Yuan et al. Citation2016) or shapeChange effects can be determined.
   检查输入/输出流的类型和属性是否发生变化。然后，可以确定具有 typeChange、valueChange（Yuan 等人，2016 年）或 shapeChange 效果的部分功能。

4. Decompose the general shape change of a material flow. If the inShape and outShape in the shape change of a material flow have the same shape type, with different values of their geometric parameters, the shapeChange can be further decomposed to valueChanges of these geometric parameters. If the inShape and outShape refer to complex shape forms that are represented by HSG descriptors, then the shape change can be decomposed into multiple local relation changes between shape elements by the shape change decomposition process.
   分解材料流的一般形状变化。如果材料流形状变化中的 inShape 和 outShape 具有相同的形状类型，但它们的几何参数值不同，则该形状变化可以进一步分解为这些几何参数的值变化。如果 inShape 和 outShape 指的是由 HSG 描述符表示的复杂形状形式，那么形状变化可以通过形状变化分解过程分解为形状元素之间的多个局部关系变化。

5. Determine the implementation order of relationChange effects. The local relationChanges should be executed to determine the overall shape change. Thus, a planning process for obtaining the order of sub-functions is performed as the end step.
   确定 relationChange 效应的实施顺序。局部 relationChanges 应被执行以确定整体形状变化。因此，作为最终步骤，执行一个获取子功能顺序的规划过程。

Figure 7. The flowchart of sub-task decomposition.
图 7. 子任务分解流程图

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6.2. Shape change decomposition
6.2. 形状变化分解

As mentioned above, the HSG structure modelled by designers is a formal abstraction of the shape of a material flow. It provides semantic information for the recognition and comparison of the input and output shapes and automates the decomposition process of shape changes to reveal the local relation changes between shape elements or the value changes of a basic element.
如前所述，设计师建模的 HSG 结构是物质流形状的形式抽象。它为输入和输出形状的识别与比较提供语义信息，并自动化形状变化的分解过程，以揭示形状元素之间的局部关系变化或基本元素值的改变。

6.2.1. Generation of global adjacent graph (GAG)
6.2.1. 全局邻接图（GAG）的生成

Before the shape change decomposition, the GAG of the input/output shape descriptor should be obtained by connecting the LAGs of composite elements in input/output HSG. The construction process of a GAG is described as follows.
在进行形状变化分解之前，应通过连接输入/输出 HSG 中复合元素的 LAGs 来获得输入/输出形状描述符的 GAG。GAG 的构建过程描述如下。

• An adjacent graph object is created, which refers to a GAG. The vertex set and edge set of this graph are initialised to empty sets.
  创建了一个相邻图对象，该对象引用了一个 GAG。该图的顶点集和边集被初始化为空集。

• The depth-first traversal is employed on the HST of the HSG. If the currently visited vertex is a basic element, it will be placed in the vertex set. Otherwise, obtain its LAG. At the same time, each edge of the LAG is placed in the edge set with its semantics of relationConstraint. The GAG for that HSG is constructed when the traversal is finished.
  对 HSG 的高阶结构树（HST）进行深度优先遍历。如果当前访问的顶点是一个基本元素，则将其放入顶点集。否则，获取其 LAG。同时，将 LAG 的每条边以其 relationConstraint 语义放入边集。当遍历完成后，为该 HSG 构建 GAG。

6.2.2. The detailed shape change decomposition
6.2.2. 详细形状变化分解

Based on the obtained GAG, the shape change decomposition is shown in Figure 8 and is detailed as follows.
基于获得的 GAG，形状变化分解如图 8 所示，具体如下。

(1)	Compare the vertex sets and edge sets of the input and output GAG, respectively, and obtain their existing states. The existing state refers to the knowledge that a vertex, which indicates a basic element, or an edge, which indicates a relation constraint, is generated, is removed or continues to exist. 分别比较输入和输出 GAG 的顶点集和边集，并获取它们当前的状态。当前状态指的是一个表示基本元素的顶点或一个表示关系约束的边被生成、被移除或继续存在的情况。
(2)	Construct the original shape change graph for the changes of basic elements and relation constraints. The vertex set and the edge set are the union of vertex sets and edge sets of input/output GAGs, and the semantics of each vertex/edge records its change type, which can be ‘generated’, ‘removed’ or ‘keep’. 构建基本元素和关系约束变化的原形状变化图。顶点集和边集是输入/输出 GAGs 的顶点集和边集的并集，每个顶点/边的语义记录其变化类型，可以是“生成”、“移除”或“保持”。
(3)	Match the newly generated vertices to the removed ones in the original shape change graph, based on the shape element conservation principle and the minimum energy principle. 根据形状元素守恒原理和最小能量原理，将新生成的顶点与原始形状变化图中的移除顶点进行匹配。
(4)	Replace the branched and composed basic elements with the corresponding basic elements. For a basic element e, which is branched, use the corresponding basic elements E′ to replace it in the input GAG. Thus, the relationConstraint between e and the other shape element is transformed into the relationConstraint between one element in E′ and other basic elements. It indicates that, if a relationConstraint r exists between e and a basic element a in the input, and there exists a relationConstraint r′ with the same relationType of r between an e′ that belongs to E′ and a in the output, then the source or target of r will be replaced by e′, and the other end of this edge will remain unchanged. Additionally, like the branch effect on these elements in E′, it indicates that the associations between them are broken up, indicating that the relationConstraint with relationType ‘joint’ will be removed. 将分支和组合的基本元素替换为对应的基本元素。对于分支的基本元素 e，在输入 GAG 中使用对应的基本元素 E′来替换它。因此，e 与其他形状元素之间的关系 Constraint 被转换为 E′中的一个元素与其他基本元素之间的关系 Constraint。这表明，如果 e 与输入中的一个基本元素 a 之间存在关系 Constraint r，并且 E′中的一个属于 e′的元素与 a 之间存在与 r 相同关系类型的 r′，那么 r 的源或目标将被替换为 e′，而这条边的另一端将保持不变。此外，就像 E′中这些元素上的分支效果一样，它表明它们之间的关联被打破，这意味着关系类型为“joint”的关系 Constraint 将被移除。

Figure 8. The flowchart of shape change decomposition.
图 8. 形状变化分解的流程图。

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Similarly, the basic element that is newly generated in the output flow is replaced by its corresponding basic element, which is composed of the new basic element in the output GAG with corresponding relationConstraints. At the same time, because the elements are connected together, a new relationConstraint is generated with relationType ‘joint’.
类似地，输出流中新生成的基元被由输出 GAG 中新基元和相应 relationConstraints 组成的对应基元所替换。同时，由于元素相互连接，会生成一个新的 relationConstraint，其 relationType 为‘joint’。

(5)

Generate the relation change graph by comparing the refined input and output GAG as follows. 通过比较精炼的输入和输出 GAG 来生成关系变化图。 Construct the relation change graph. After replacement among the basic elements, the vertex sets of input and output GAG are exactly the same; they are copied to the vertex set of the relationChange graph, and the edge set of the relationChange graph is the union of edge sets of input/output GAG. 构建关系变化图。在基本元素替换后，输入和输出 GAG 的顶点集完全相同；它们被复制到关系变化图的顶点集中，关系变化图的边集是输入/输出 GAG 边集的并集。 Set the semantic information for each edge in the relation change graph. Each of the edges refers to a relationChange effect, whose attributes source, target and relationType are exactly like the corresponding relationConstraint in the edges of input/output GAG, and its change trend can be obtained by the semantic information about the corresponding edge in the original shape change graph. 为关系变化图中的每条边设置语义信息。每条边都指代一个 relationChange 效果，其属性 source、target 和 relationType 与输入/输出 GAG 边中的相应 relationConstraint 完全相同，而其变化趋势可以通过原始形状变化图中相应边的语义信息获得。 Create a relationChange effect with the change trend being ‘generate’ for each new generated edge. The other attribute of relationChange is the same as the relationConstraint of the corresponding edge in the output GAG. 为每个新生成的边创建一个 relationChange 效果，其变化趋势为“生成”。relationChange 的其他属性与输出 GAG 中相应边的 relationConstraint 相同。 Create a relationChange effect with change trend being ‘remove’ for each new removed edge. The other attribute of relationChange is the same as the relationConstraint of the corresponding edge in the input GAG. 为每个新移除的边创建一个 relationChange 效果，其变化趋势为“remove”。relationChange 的其他属性与输入 GAG 中对应边的 relationConstraint 相同。 Create a relationChange effect for each edge, which is labelled ‘keep’. The change trend of it can be obtained by qualitative reasoning, based on the different value of relationConstraint in the corresponding input GAG and output GAG. The other attribute of relationChange is the same as the relationConstraint. 为每条边创建一个 relationChange 效果，并标记为“keep”。通过基于相应输入 GAG 和输出 GAG 中 relationConstraint 的不同值进行定性推理，可以获得其变化趋势。relationChange 的其他属性与 relationConstraint 相同。

For example, according to the HSGs of the input cardboard and the output pyramid box, the GAG can be obtained, as shown in Figure 5(b) and Figure 5(d), respectively, and the related information is listed in Table 1. The original graph of shape change is constructed as Figure 9. Part2 and part4 are removed together with the straight angle relations that are between part1 and part2 and between part3 and part4, respectively. Part5 is a newly generated shape element, together with the generation of relations of acute angles from it to part0, part1 and part2. Moreover, an acute angle relation is generated between part1 and part3; the straight angles between part0 and part1 and between part0 and part3 are changed to acute angles. Thus, the total energy consumption is E₁ = 3*3 + 2*6 + 1*2 = 23. However, the shape element conservation principle and the minimum energy principle can be applied to optimise the relationChanges, which are needed to achieve this shapeChange. Because the newly generated part5 has angle relations to part1 and part3, and also the removed part2 has an angle relation to part1, as well as the angle relation between the removed part4 and part3, part5 can be considered the combination of part2 and part4. Based on this assumption, the semantics in Figure 9 are analysed and summarised in Table 2. Here, a pair <N1, N2> is used to refer the edge between part N1 and part N2. Moreover, if part2 and part4 are used to replace part5 together with the replacements of related relation constraints, the relation change graph that reveals the actual shape model's structural change is constructed as shown in Figure 10. The red edges refer to the newly generated relation constraints, and the black edges indicate the relation constraints that must be changed. The total energy consumption of this solution is E₂ = 1*4 + 2*4 = 12, which is smaller than E₁ and thus is a better solution.
例如，根据输入纸板和输出金字塔纸箱的 HSG，可以得到 GAG，分别如图 5(b)和图 5(d)所示，相关信息列于表 1。形状变化的原始图构建如图 9 所示。将 Part2 和 Part4 一起移除，同时移除位于 Part1 和 Part2 之间以及 Part3 和 Part4 之间的直角关系。Part5 是一个新产生的形状元素，并从它到 Part0、Part1 和 Part2 之间生成了锐角关系。此外，在 Part1 和 Part3 之间生成了锐角关系；Part0 和 Part1 之间以及 Part0 和 Part3 之间的直角关系被改变为锐角。因此，总能耗为 E ₁ = 3*3 + 2*6 + 1*2 = 23。然而，形状元素守恒原理和最小能耗原理可以应用于优化实现该形状变化所需的 relationChanges。 由于新生成的部件 5 与部件 1 和部件 3 存在角度关系，而移除的部件 2 与部件 1 存在角度关系，以及移除的部件 4 与部件 3 之间的角度关系，因此部件 5 可以被视为部件 2 和部件 4 的组合。基于这一假设，对图 9 中的语义进行分析并总结在表 2 中。这里，用一对来指代部件 N1 和部件 N2 之间的边。此外，如果用部件 2 和部件 4 共同替换部件 5，并替换相关的关系约束，则构建出揭示实际形状模型结构变化的关联变化图，如图 10 所示。红色边表示新生成的关联约束，黑色边表示必须更改的关联约束。该解决方案的总能耗为 E ₂ = 1*4 + 2*4 = 12，这比 E ₁ 小，因此是一个更好的解决方案。

Figure 9. The original shape change graph.
图 9. 原始形状变化图

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Figure 10. The relation change graph.
图 10. 关系变化图

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Table 2. The relation changes determined by semantic information.
表 2. 基于语义信息确定的关系变化

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7. Sub-function planning for local geometric form changes
7. 局部几何形状变化的功能子规划

After the functional decomposition is conducted, the hierarchy of functional effects with sub-functions and the overall function is clear. The sub-functions reflect the local structure changes or shape deformations needed to cause the required morphological change of a material flow. However, the sequential order of sub-functions, which indicates the order of implementation of the local geometric form changes, remains unknown. Thus, a planning algorithm is proposed to find a feasible order for the local relation changes. When these functional effects are implemented with this order in practice, the global geometric changes of a material flow will be achieved.
功能分解完成后，包含子功能的功能效应层次结构以及整体功能变得清晰。子功能反映了为使材料流产生所需形态变化所需要进行的局部结构变化或形状变形。然而，子功能的顺序，即局部几何形式变化实施的顺序仍然未知。因此，提出了一种规划算法来寻找局部关系变化的可行顺序。当按照这种顺序在实际中实现这些功能效应时，材料流的全球几何变化将被实现。

7.2. Planning algorithm for the sub-functions
7.2. 子功能规划算法

Note that path planning is an NP problem. A variety of sequential orders can be obtained for different start paths. With the consideration of efficiency, a vertex that refers to a basic element is selected by the designer, and the relation changes corresponding to the edges between the selected vertex and other vertices are considered first. Figure 11 illustrates the algorithm for the traversal of the relation change graph and the priority value setting for each edge in this graph. Let the parameter S represent a set of edge-vertex pair. Each element in this list is a candidate ‘path’ waiting to be visited. All of the paths in set S are checked to determine whether it is an ‘open’ path, indicating that the edge and vertex on this path are not in conflict with the sequential planning principles (other constraints might exist). Based on the above analysis, the algorithm for sub-function sequence planning is detailed as follows.
请注意路径规划是一个 NP 问题。对于不同的起始路径可以获得多种顺序。考虑到效率，设计者选择一个指向基本元素的顶点，并首先考虑所选顶点与其他顶点之间的边对应的关系变化。图 11 说明了遍历关系变化图以及为该图中的每条边设置优先值的算法。令参数 S 表示一个边-顶点对集合。该列表中的每个元素都是一个待访问的候选“路径”。检查集合 S 中的所有路径，以确定它是否是一条“开放”路径，这表示该路径上的边和顶点不与顺序规划原则冲突（可能存在其他约束）。基于上述分析，子功能顺序规划的算法详细如下。

1. Select a vertex in the graph as the start. Link all the <edge, vertex> pairs to the start vertex and place them into set S.
   在图中选择一个顶点作为起点。将所有<边, 顶点>对连接到起点，并将它们放入集合 S 中。

2. Call the algorithm for graph traversal and order the relation changes.
   调用图遍历算法，并排序关系变化。

3. Obtain the execution sequence of sub-functions after the priority of each of the edges in the graph is set. Copy the edge set of the relation change graph into a list, and reorder them by their priorities in descending order.
   在为图中每条边的优先级设置后，获取子函数的执行顺序。将关系变化图的边集复制到列表中，并按优先级降序重新排序。

Figure 11. The algorithm for graph traversal and plan of the local relation changes.
图 11. 图遍历算法及局部关系变化计划

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For example, to perform the sub-function planning process for the graph in Figure 10, the vertex 0 is chosen as the start, and all of the edges in List are selected to be reordered at each turn. Because the relation change semantics corresponding to <0, 2> and <0, 4> have the change type ‘generate’, they are not in the open path before their coupling edges have been ordered. However, <0, 1> and <0, 3> can be considered simultaneously with the same priority of ‘1’. Then, edges <1, 2>, <3, 4> will have the same priority of ‘2’. Because the coupling edges of edge <1, 3> have been already ordered, the relation change of edge <1, 3> is also set with a priority of ‘2’. Finally, the edges <0, 2>, <0, 4> and <2, 4> can be ordered with a priority of ‘3’ because all of their coupling edges have been ordered. Therefore, the execution sequence planning of sub-functions is obtained to fold the pyramidal box.
例如，为了对图 10 中的图执行子功能规划过程，选择顶点 0 作为起点，并在每次迭代中选择列表中的所有边进行重新排序。由于与<0, 2>和<0, 4>对应的关系变化语义具有变化类型“生成”，在它们的耦合边被排序之前，它们不在开放路径中。然而，<0, 1>和<0, 3>可以同时考虑，具有相同的优先级“1”。然后，边<1, 2>和<3, 4>将具有相同的优先级“2”。由于边<1, 3>的耦合边已经被排序，边<1, 3>的关系变化也被设置为优先级“2”。最后，由于它们所有的耦合边都已经被排序，边<0, 2>、<0, 4>和<2, 4>可以以优先级“3”进行排序。因此，获得了子功能执行顺序规划，用于折叠金字塔盒子。

8. Demonstration  8. 示例

The proposed formal representation and decomposition method for the morphological changes of material flows can be used for the conceptual design of various systems. However, because box folding is a common functional requirement for packing machines with complicated morphological changes, the functional design of a sugar packing machine is used for the demonstration.
所提出的针对物料流形态变化的正式表示和分解方法可用于各种系统的概念设计。然而，由于箱式折叠是具有复杂形态变化的包装机的一种常见功能需求，因此以糖包装机的功能设计为例进行演示。

The proposed method is implemented with the IBM Rhapsody platform as a plug-in. Firstly, to model the shape of input and output material flows that participate in an overall function, the meta-classes of SysML/UML are extended. A set of stereotypes are established in a profile as the meta-model elements. The block definition diagram (BDD) of SysML is used to define the overall function with its input and output flows. The HSTs of the input and output flows' shapes are also defined in BDD to model the structure of the shape elements, which compose the material's shape. Then, the internal block diagram is used to describe the LAG, which abstracts the local connection relations between the shape elements at a certain level of detail. Therefore, designers can use the meta-model elements in SysML to describe the abstract model of the material's shape in the early stage of conceptual design. Moreover, the automated functional decomposition is implemented, based on the APIs of IBM Rhapsody.
所提出的方法以插件形式在 IBM Rhapsody 平台上实现。首先，为了对参与整体功能的输入和输出物料流进行建模，扩展了 SysML/UML 的元类。在构架中建立了一套构架元素作为元模型元素。使用 SysML 的块定义图（BDD）定义整体功能及其输入和输出流。输入和输出流的形状的 HSTs 也在 BDD 中定义，用于对组成物料形状的形状元素结构进行建模。然后，使用内部块图描述 LAG，该 LAG 抽象了在某一详细层次上形状元素之间的局部连接关系。因此，设计人员可以在概念设计早期使用 SysML 中的元模型元素来描述物料的形状抽象模型。此外，基于 IBM Rhapsody 的 API 实现了自动功能分解。

With the predefined function modelling profile, the function of a sugar packaging machine is modelled in Figure 12 with detailed semantics of input and output flows. The input flows are the sugar flow and the cardboard flow. The flow ‘sugar’ has no change during the packaging function. The information of its attributes does not need to be modelled. The shape of cardboard is a composite element called cardBoard_Shape to describe its shape in detail. The output flow is a solid object that packages the sugar. Therefore, the ownedComponents are ‘box’ and ‘sugar’, and its ownedRelation defines the relationship between the ownedComponents is box ‘carry’ sugar. The flow ‘box’ also has its own shape property, which is a composite element named box_Shape.
通过预定义的功能建模配置文件，图 12 中详细展示了糖包装机功能，包括输入和输出流的详细语义。输入流包括糖流和纸板流。在包装功能过程中，糖流没有变化，其属性信息无需建模。纸板的形状通过一个名为 cardBoard_Shape 的复合元素详细描述。输出流是一个包装糖的固体对象。因此，其拥有的组件是“盒子”和“糖”，而其拥有的关系定义了组件之间的关系，即盒子“承载”糖。流“盒子”也有自己的形状属性，该属性是一个名为 box_Shape 的复合元素。

Figure 12. The overall functional model of sugar packaging.
图 12. 糖包装的整体功能模型。

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The cardboard_shape is a composite element called ‘flat’, which is composed of 5 parts, including a base and four sides that are modelled by 5 shape elements respectively. As shown in Figure 13, the HST of the cardboard is modelled as a BBD. The shape of the cardboard is composed of five shape elements, called ‘base’, ‘part3’ (the left part), ‘right’ part, ‘up’ part and ‘down’ part at the first level of abstraction. They are composite elements except for the base and part3. Therefore, the composite elements at level 1 are defined by the LAGs in Figure 14. Note that each basic element in LAGs has its own shape type information, and each edge in Figure 14 also has its own semantic information. However, this information is omitted due to limited space. The output shape models of the cardboard (a box), including the corresponding HST and the LAG for each composite element in the HST, are shown in Figures 15 and 16, respectively.
cardboard_shape 是一个由“平面”组成的复合元素，它由 5 个部分组成，包括一个基座和四个分别由 5 个形状元素建模的侧面。如图 13 所示，纸板的 HST 被建模为 BBD。纸板的形状由五个形状元素组成，在抽象的第一层中称为“基座”、“part3”（左侧部分）、“右侧部分”、“上侧部分”和“下侧部分”。除了基座和 part3 之外，它们都是复合元素。因此，第一层的复合元素由图 14 中的 LAG 定义。请注意，LAG 中的每个基本元素都有其自己的形状类型信息，图 14 中的每条边也有其自己的语义信息。但由于空间有限，这些信息被省略了。纸板（一个盒子）的输出形状模型，包括相应的 HST 和 HST 中每个复合元素的 LAG，分别如图 15 和图 16 所示。

Figure 13. The HST of the shape of the cardboard.
图 13. 纸板形状的 HST

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Figure 14. The LAGs of the shape of the cardboard.
图 14. 纸板形状的 LAGs

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Figure 15. The HST of the box's shape.
图 15. 箱子形状的 HST

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Figure 16. The LAGs of the box's shape.
图 16. 箱子形状的 LAGs

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At the beginning of the sub-function decomposition, as shown in Figure 17, the material conservation principle and hierarchical principle are used to obtain the coarse-grained functional effects; they are used to shape the cardboard to box shape and to connect the box and sugar together, respectively. Then, shape change reasoning and sub-function sequential planning are performed on the sub-function called ‘BoxShapeChange’, and the result is shown in Figure 18.
在子功能分解的初始阶段，如图 17 所示，利用物质守恒原理和层级原理获得粗粒度的功能效应；它们分别用于将纸板塑形为箱形，以及将箱子和糖连接在一起。然后，对名为“BoxShapeChange”的子功能进行形状变化推理和子功能顺序规划，结果如图 18 所示。

Figure 17. The first level of sub-functions.
图 17. 第一级子功能。

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Figure 18. The results of shape change decomposition.
图 18. 形状变化分解结果

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Nineteen sub-functions with their functional effects are obtained by functional decomposition, including 7 relation change effects with changeTrend ‘change’ (referring to the changes of angles between cardboard parts) and 12 new relation constraints (referring to the combination of parts). In this case, there are additional relation constraints for the order of sub-functions. For part 3 linked to parts 7 and 8 in the output state, it constrains the relation change related to part 3 which cannot occur before part 7 and part 8 change to their desired states. Therefore, these constraints are also used to determine whether a path is ‘open’ in the planning algorithm. Based on the sub-function sequence list in Figure 18, sub-functions with the same priority can be considered to be implemented simultaneously or sequentially. Thus, the designer can obtain the sub-functions with the constraints of their priorities to build the function structure for the morphological change of the cardboard. One of the feasible function structure models is shown in Figure 19. The sub-functions for shaping the box are re-allocated to 9 functional modules because some of them can be implemented simultaneously. These functional modules are linked to material flows, which refer to the intermediate morphological states during the process that the cardboard is folded into a box.
通过功能分解，获得了十九个具有功能效应的子功能，包括七个关系变化效应（指纸板部件之间角度的变化）和十二个新的关系约束（指部件的组合）。在这种情况下，子功能顺序还存在额外的关系约束。对于输出状态中与部件 7 和 8 相连的部件 3，它约束了与部件 3 相关的变化关系，该关系不能在部件 7 和 8 达到期望状态之前发生。因此，这些约束也用于规划算法中确定路径是否为“开放”状态。根据图 18 中的子功能序列列表，具有相同优先级的子功能可以认为同时或顺序执行。因此，设计师可以获得具有优先级约束的子功能，以构建纸板形态变化的函数结构。图 19 展示了一个可行的函数结构模型。 用于成型盒子的子功能被重新分配到 9 个功能模块，因为其中一些可以同时实现。这些功能模块与物料流相连，物料流指的是卡纸折叠成盒子过程中的中间形态状态。

Figure 19. The function structure for the box folding processes.
图 19. 箱子折叠过程的函数结构。

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9. Discussion  9. 讨论

In this section, a comparative analysis is performed to compare the proposed method with other function representation and functional decomposition methods. The validity of the proposed method is also discussed.
在本节中，通过比较分析，将所提出的方法与其他功能表示及功能分解方法进行对比，并讨论了所提出方法的有效性。

1. The semantics and modelling efforts. Compared to other FB-based function representations, the proposed method supports building a function model with rich semantics, which is readable for both a designer and a computer. Moreover, the level of the detailed morphological information modelling process conforms to the human designer's cognition process. The hierarchy of the component parts of material flow will be recognised and modelled by designers first; then, the overall outline or shape of each component and finally the details of morphological features of a component are described. Compared to other functional modelling methods, additional efforts to model the morphological information of material flows are needed for the designers in the definition of the overall function. However, because the automated decomposition process can be performed based on these semantics, it saves much time by avoiding duplication of the designer's work in comparing the input and output states of a material flow and digging into every detail of the morphological changes manually.
   语义和建模工作。与其他基于功能表示的方法相比，所提出的方法支持构建具有丰富语义的功能模型，该模型对设计师和计算机都具有良好的可读性。此外，详细的形态信息建模过程符合人类设计师的认知过程。设计师首先识别和建模物料流组件部分的层次结构；然后，描述每个组件的整体轮廓或形状；最后，描述组件形态特征的细节。与其他功能建模方法相比，在定义整体功能时，设计师需要额外的工作来建模物料流的形态信息。然而，由于自动化分解过程可以基于这些语义进行，它通过避免在比较物料流的输入和输出状态以及手动挖掘每个形态变化细节时重复设计师的工作，从而节省了大量时间。

2. Validity of the results. The proposed functional decomposition relied on the morphological semantics of material flows modelled by designers. Thus, the validity of decomposition heavily depends on the quality of the morphologic information model. Moreover, the limitations of material conservation principles make it difficult to decompose the complicated structure change between the material flows with ‘branch’ and ‘connect’ that occur in multiple flows. Therefore, based on the provided information, it can still provide accurate decomposition results by detecting all of the local morphological changes. However, the planning algorithm is not a general method now. It is only validated for box packing machines because we did not find other examples in engineering that have morphological changes with the same degree of complexity. Based on the logical order obtained by this algorithm, the original cardboard can be ultimately transformed into its desired form. However, the designer should reconsider the order of the sub-functions with the same priority, or it might make mistakes regarding the relation between the panel parts. For example, there might be a mistake in which part11 might be ‘on’ part10, rather than ‘under’ it.
   结果的有效性。所提出的函数分解依赖于设计师对物料流形态语义的建模。因此，分解的有效性很大程度上取决于形态信息模型的质量。此外，物料守恒原理的局限性使得难以分解物料流中出现的复杂结构变化，这些变化在多个流中通过“分支”和“连接”发生。因此，基于所提供的信息，通过检测所有局部形态变化，它仍然可以提供准确的分解结果。然而，规划算法目前并非通用方法。它仅经过装箱机验证，因为我们未在工程中找到其他具有同等复杂程度形态变化的实例。基于该算法获得的逻辑顺序，原始纸板最终可以转化为其期望形态。然而，设计师应重新考虑具有相同优先级的子函数顺序，否则可能会在面板部件之间的关系上出错。 例如，可能存在一个错误，其中部件 11 可能在部件 10 的“上面”，而不是“下面”。

3. Rationality. There is no consensus among researchers for the concept ‘function’ or the composite relations between function and sub-functions. Functional decomposition is a difficult problem due to the absence of a formal definition of their relations. It was proved by Vermaas (Citation2013) that, when functions are represented as the transformations of flows, the composite relation between a function and its sub-function is not the parthood relation. The above conclusion is valid for FB-based function representation methods. However, in the proposed method, flows are represented with properties described by their own descriptors. The different semantics of flow in the input and output states reflect the changes of flows in a function. An overall function contains all of the information about the changes that occur in a single flow or between flows, with each sub-function referring to a subset of the changes of the overall function and reflecting the transformation from parts of the input flows to their corresponding output flows; thus, the composite relation between an overall function and its sub-function is the parthood relation. The rationality of this method is proved based on the set theory. The decomposition process is exactly like the subset partition process in generating sub-functions. Thus, a function of the morphological change of material flows can be partitioned into sub-functions, each of which refers to a local geometric change.
   合理性。研究人员对于“功能”这一概念或功能与子功能之间的复合关系尚未达成共识。由于它们之间关系缺乏正式定义，功能分解是一个难题。Vermaas（2013）证明，当功能被表示为流的转换时，功能与其子功能之间的复合关系并非部分与整体的关系。上述结论适用于基于功能表示方法。然而，在所提出的方法中，流被用其自身描述符所描述的属性来表示。输入状态和输出状态中流的语义差异反映了功能中流的变化。一个整体功能包含了单个流或流之间发生变化的全部信息，每个子功能引用整体功能变化的一个子集，并反映输入流的部分到其对应输出流的转换；因此，整体功能与其子功能之间的复合关系是部分与整体的关系。 基于集合论，该方法的有效性得到了证明。分解过程与生成子函数中的子集划分过程完全相同。因此，材料流形态变化的功能可以被划分为子功能，每个子功能都对应一个局部几何变化。

4. Automaticity. The functional decomposition method for the morphological changes of material flows has been implemented in the IBM Rhapsody platform, and it is an automated decomposition process. In the existing methods, the task-decomposition process is usually achieved manually by designers (Umeda et al. Citation1996). Due to the lack of sematic information in the FB-based function model, functional decomposition is unable to be undertaken without additional design experience knowledge or rules. In the proposed method, the functional decomposition process performs automatically, based on the semantics contained in the functional representation model. The coupling relationship among shape elements are also considered in a sub-function planning process. The hierarchical relationship between the functions and their sub-functions, together with the order of the sub-function sequence, can be used to inspire designers to build the function structure.
   自动化性。基于物料流形态变化的功能分解方法已在 IBM Rhapsody 平台上实现，这是一个自动化的分解过程。在现有方法中，任务分解过程通常由设计人员手动完成（Umeda 等人，1996 年）。由于基于功能模型（FB）的功能模型缺乏语义信息，如果没有额外的设计经验知识或规则，功能分解就无法进行。在提出的方法中，功能分解过程基于功能表示模型中包含的语义自动执行。在子功能规划过程中，也考虑了形状元素之间的耦合关系。功能及其子功能之间的层次关系，以及子功能序列的顺序，可用于启发设计人员构建功能结构。

10. Conclusions  10. 结论

It is difficult to describe material flow's morphological information in the conceptual stage; thus, the morphological changes have been neglected in existing functional modelling and decomposition methods. The proposed functional representation and decomposition method can be widely used in engineering design for the analysis of morphological change functions, such as for pipe benders, packing machines and assembly robots.
在概念阶段难以描述物料流的形态信息；因此，现有的功能建模和分解方法中已忽略形态变化。所提出的功能表示和分解方法可广泛应用于工程设计，用于分析形态变化功能，例如用于弯管机、包装机和装配机器人。

With the proposed representation, the intentional requirement descriptions of morphological changes can be translated into formal functional models. The input and output states of the morphological property of a material flow can be obtained directly from the requirement description or from the analysis of a use case. Thus, the morphological information can be included in the functional modelling context, which indicates the manner of the material flow's morphological changes. Therefore, the automated decomposition process could be performed to break the overall changes into several local shape and structure changes, freeing designers from having to determine these trivial details manually. As a result, the sub-functions of local morphological changes and the logical implementation order of them obtained by the planning algorithm could inspire designers to determine the technical process and the physical structure of the product. Thus, the proposed approach could bridge between the requirement descriptions and formal functional representations of morphological changes. Additionally, it can close the gaps between the pure functional modelling and the physical structure solution of products designed for processing material flows. The main contributions of this study are as follows:
通过所提出的表示方法，形态变化的功能需求描述可以转化为形式化的功能模型。材料流的形态属性输入和输出状态可以直接从需求描述中获得，或者通过用例分析得到。因此，形态信息可以包含在功能建模环境中，这表明了材料流形态变化的方式。因此，自动化分解过程可以将整体变化分解为多个局部形状和结构变化，从而让设计者无需手动确定这些琐碎细节。结果，局部形态变化的子功能以及规划算法获得的它们逻辑实现顺序，可以启发设计者确定产品的技术工艺和物理结构。因此，所提出的方法可以在需求描述和形态变化的形式化功能表示之间建立桥梁。 此外，它还能缩小纯功能建模与为处理物料流而设计的产品物理结构解决方案之间的差距。本研究的主要贡献如下：

1. A function representation for modelling the morphological semantics of material flows is proposed. In addition, the type and physical properties of material flows, the components that compose the flow and the relation constraints are used to represent the structure of material flows hierarchically. Moreover, the shape of a material flow is modelled by the HSG descriptor. This hierarchical representation is consistent with the intuitiveness of designers, enabling the modelling of the structure and geometric form of material flows in the conceptual design phase.
   提出了一种用于建模物料流形态语义的功能表示方法。此外，利用物料流的类型和物理属性、构成流体的组分以及关系约束，分层表示物料流的结构。同时，通过 HSG 描述符对物料流的形状进行建模。这种分层表示与设计者的直观性相符，能够在概念设计阶段对物料流的结构和几何形态进行建模。

2. An automated functional decomposition strategy is proposed based on the parthood relation between a function and its sub-functions. With the help of decomposition principles, an overall function of the morphological change of material flows is broken into a set of sub-functions. Each of them reflects the independent local morphological change of a material flow. Therefore, it renders the function semantics more clear and can facilitate the following design tasks, such as working principle determination.
   提出了一种基于功能与其子功能之间部分关系（parthood relation）的自动化功能分解策略。借助分解原则，将材料流形态变化的整体功能分解为一组子功能。每个子功能都反映了材料流独立的局部形态变化。因此，这使功能语义更加清晰，并有助于后续设计任务，例如工作原理的确定。

3. A sub-function planning process is provided based on the coupling relationship between shape elements and the structural and hierarchical characteristics of a material's shape model. By traversing the relation change graph, the logical implementation order of the local relation change functions is determined. Therefore, designers can be inspired to design the manufacturing process of the material flow and the physical structure of the product.
   基于形状元素之间的耦合关系以及材料形状模型的结构和层次特征，提供了一种子功能规划过程。通过遍历关系变化图，确定了局部关系变化函数的逻辑实现顺序。因此，设计师可以从中获得启发，设计材料流的制造过程和产品的物理结构。

There are still some shortcomings of this work. It makes high demand of users to ensure the correctness of the function model constructed by the proposed representation method. Therefore, more constraints and rules should be defined as model elements to ensure the model's correctness and to facilitate user operation. Currently, because of the limitation of material conservation principles, the decomposition can only address functions with explicit changes from its input state to the output state. When the complicated structure change between material flows with ‘branch’ and ‘connect’ occurs in multiple flows, user interaction is needed to match the input flows and the corresponding output flows. Moreover, the feasibility of the sub-function planning remains limited. The logical functions will be considered to improve the generality of this method in future work.
这项工作仍存在一些不足。它对用户提出了很高的要求，以确保所提出的表示方法构建的功能模型的正确性。因此，应将更多的约束和规则定义为模型元素，以确保模型的正确性并方便用户操作。目前，由于物质守恒原理的限制，分解只能处理从输入状态到输出状态有明确变化的功能。当在多个物料流中发生具有“分支”和“连接”的复杂结构变化时，需要用户交互来匹配输入流和相应的输出流。此外，子功能规划的可实现性仍然有限。未来工作中将考虑逻辑功能来提高该方法的一般性。