A systematic approach for product modelling and function integration to support adaptive redesign of product variants

When a product variant offers functionality that is high in demand, firms may decide to leverage that design to enhance other variants in their product line. This can be achieved by extracting functions and their realisations from one product variant and integrating them into another variant, resulting in a third product variant that has a new combination of functions and physical features. This article introduces a systematic approach called the ﻿Adaptive Redesign Method (ARM) to support this function integration process. The ARM is based on a new product model called the Detailed Design Model (DDM). In comparison to existing approaches, the DDM allows the architecture of an existing product to be modelled on a sufficiently detailed level to identify geometric features and parts that realise particular operating functions of a product. This detailed information provides a basis for systematic determination of the redesign activities needed to derive a new variant design, down to the detailed level of adding, removing and integrating specific parts and features. The main benefit is to assist with planning the redesign process while ensuring nothing is overlooked, which might be especially useful if the task is to be divided among several designers or if designers are not fully familiar with the designs at hand. A secondary benefit is to show how this type of redesign process can be decomposed into systematic steps, which could potentially reveal opportunities for computer support. The new approach has been developed and tested through reverse engineering studies of consumer products, confirming its applicability.


Introduction
This article introduces a method that provides detailed guidance for function integration in engineering design. Function integration involves extracting particular functions and their physical realisations from existing products and integrating them into other product variants to form a new variant with a unique combination of features and operating functions. One situation in which function integration occurs is when a product variant has functions or technology that are high in demand. In such cases, companies may want to implement that new technology into their other existing product variants (Li et al. 2009;Zhang et al. 2011;Liu et al. 2014).
Function integration is a form of redesign in which an existing product variant is adapted. In general, product redesign can be grouped into two major categories: parametric and adaptive redesign (Otto and Wood 1998). Parametric redesign is concerned with improving a product's architecture or its performance by adjusting parameters of an existing design. Adaptive redesign, which is the focus of this article, is concerned with modifying the functionality of an existing product to derive a new product. In research literature, studies of adaptive redesign often involve the reverse engineering of existing products (Otto and Wood 1998). The purpose of reverse engineering in this context is to support the reuse of existing elements of the design, with a particular Abstract When a product variant offers functionality that is high in demand, firms may decide to leverage that design to enhance other variants in their product line. This can be achieved by extracting functions and their realisations from one product variant and integrating them into another variant, resulting in a third product variant that has a new combination of functions and physical features. This article introduces a systematic approach called the Adaptive Redesign Method (ARM) to support this function integration process. The ARM is based on a new product model called the Detailed Design Model (DDM). In comparison to existing approaches, the DDM allows the architecture of an existing product to be modelled on a sufficiently detailed level to identify geometric features and parts that realise particular operating functions of a product. This detailed information provides a basis for systematic determination of the redesign activities needed to derive a new variant design, down to the detailed level of adding, removing and integrating specific parts and features. The main benefit is to assist with planning the redesign process while ensuring nothing is overlooked, which might be especially useful if the task is to be divided among several designers or if designers are not fully familiar with the designs at hand. A secondary benefit is to show how this type of redesign process can be decomposed into systematic steps, which could potentially reveal opportunities for computer support. The new approach has been developed emphasis on appreciating how those elements influence each other and how they must be considered in combination when introducing changes. The changes may involve a combination of adding, removing, redesigning, and reusing parts to achieve desired functions (Lee and Park 2014).
One of the important issues to consider in redesign is the product architecture. The architecture of a product to be redesigned can be described in terms of functions and physical parts. Three types of interactions are possible on this level of description: interactions between functions, interactions between parts, and interactions between functions and parts. Each function or part can interact with several others, exhibiting complex many-to-many relationships (Ulrich and Seering 1988). On a more granular level, physical part interactions are realised at the interfaces between specific part surfaces. In a CAD model, these surfaces are typically defined by geometric features of the respective parts. Overall, the complex patterns of connections that comprise product architectures need to be considered when integrating existing design elements into an existing product to develop a new product variant.

Research questions
To provide systematic support for function integration between product variants, this article addresses three research questions: -RQ1: How can product variants be systematically modelled at an appropriate level of detail to support function integration at a part and feature level? -RQ2: How can the resulting models of product variants be systematically analysed to identify redesign activities required for function integration? -RQ3: Is the resulting systematic approach useful and what future research opportunities does it reveal?
In this context, a systematic approach is expected to provide several benefits. First, a systematic approach is helpful to model existing designs in a structured way and better appreciate how the design solutions work (Otto and Wood 1998; Tang et al. 2010a). A systematic approach is particularly useful when modelling products having moving parts that are each involved in multiple functions, such that function realisation is complex. Second, a systematic process for modelling design information supports the sharing and reuse of that information (Gietka et al. 2002;Tang et al. 2010b).
Reusing existing design information should support the development of reliable products, since it facilitates adoption of proven principles and solutions (Smith et al. 2012). Third, a systematic approach allows for detailed planning of the (re)design process and may reduce design rework, since designers will be less likely to initially overlook some redesign activities, requiring later correction (Tang et al. 2010b). This may be especially useful if the redesign work needs to be coordinated between several team members, or if a designer is not fully familiar with the designs at hand. Finally, decomposing the redesign process into well-defined smaller steps may suggest opportunities for computer support of that process.

Research method
To address the research questions a representative redesign task based on a simple product (redesigning a ballpoint pen) was generated as a test case. Existing approaches from the literature were assessed in light of this test case. From this, limitations of the existing product models and function integration methods were appreciated. This led to the generation of a new approach for product modelling, and based on it, a new systematic approach for function integration. The new approaches were then applied to a series of more complex consumer products to test and further improve them. Overall, the development was highly iterative. Literature review also proceeded concurrently with the activities described above.

Article outline
The rest of this article proceeds as follows. In Section 2, literature is reviewed and the research gap is pinpointed. Section 3 introduces the basis of our solution, which is to model existing variants using a new product model called the Detailed Design Model (DDM). Section 4 then explains the Adaptive Redesign Method (ARM) itself. Section 5 discusses some application cases and insights drawn from them. Section 6 discusses advantages, limitations and some suggestions for further work. Concluding remarks are offered in Section 7.

Literature review
As established in Sect. 1, function integration requires extracting existing design solutions from a product and integrating it into another product to derive a new variant. It was determined that a method to support this should satisfy four method requirements (MRs), for the reasons outlined below: -MR1: Determine low-level and high-level functions of a design and link them to the parts and physical features. This information is needed to identify the aspects of a design that realise each function that is to be integrated. -MR2: Determine the physical interactions between features of a design. This information is needed to identify supporting features that help primary features realise desired functions that are to be integrated.
-MR3: Ensure that product data obtained for different product variants is consistently represented, using similar function terminology. This is needed so that the variant designs can be directly compared for function integration. -MR4: Systematically process the obtained product data to determine how parts and specific features of parts can be carried over, removed and modified from existing variants to generate a new variant design that provides a unique desired combination of existing functions and features. The benefits of such function integration, as well as the reasons for a systematic approach, were outlined in Sect. 1.
In the next subsections, product models and function integration methods from literature are reviewed against these method requirements. This confirms the gap addressed by this article.

Product models for function integration
Most product modelling approaches in literature are based on hierarchical decomposition of a design (typically involving tree diagramming), on block diagramming, and/or on matrix-based modelling. Some models of these types are discussed in the next subsections.

Product models based on hierarchical decomposition and tree diagramming
This subsection discusses decomposition-based product modelling approaches based on (1) Axiomatic Design, (2) Function-Behaviour-State modelling, (3) The Chromosome model and (4) Function-Means Tree approaches. Firstly, tree-based product models building on axiomatic design (Suh 1995) essentially consist of two tree diagrams, to model the functions and parts domains of a product respectively. In a reverse engineering context, the trees can be formed progressively while disassembling a product by considering how the functions relate to each assembly, subassembly, and so on. While convenient for decomposition, this type of model does not emphasise the relationships between parts, features or between functions at a particular level of the tree. Hence, it is not ideal to identify supporting features (as required by MR2). This limitation was partially addressed by the transdisciplinary product development life-cycle model (TDPL) of Gumus et al (2008). TPDL captures the interaction between parts and according to Gumus et al. (2008), it can help to verify whether the parts can realise the functional requirements of a product. However, it does not capture low-level functions (as required by MR1). Lo and Helander (2007) also adopted axiomatic design concepts, introducing a goal domain and user actions domain to capture the coupling between user requirements/ interface and product architecture elements. The coupling between these domains was used to determine whether the user interface of the product needs to be redesigned when there are changes to user requirements. However, this model uses mathematical equations to link elements between these four design domains, which may be difficult to formulate from a product decomposition and are not necessary to meet MR1-MR4.
The second type of hierarchical product model to be discussed is the Function-Behaviour-State diagram (Umeda et al. 1990). In this approach, the behaviour domain of a design exists between the function and structure domains. Including this domain enables consideration of the performance attributes of a product during redesign. However, this extra domain also increases modelling effort and is not required for MR1-MR4. Furthermore, the extra domain will unnecessarily increase the complexity of tracing between function and structure domains, as required by MR1. Deng et al. (1999) extended this approach to form the FEBS model, additionally capturing the interaction between the product and its operating environment. Tor et al. (2002) in turn extended FEBS to form B-FES, a model to support product variant concept generation. B-FES can be used to extract and reuse certain physical attributes from existing products when given a set of requirements. However, B-FES does not meet MR2 because it does not consider the supporting features required by the primary parts that realise desired functions (requirements).
The third type of hierarchical product model builds on the Chromosome Model (Andreasen 1992), which is based on Domain Theory (Andreasen 1980). This model represents product information in three domains: activities, organs, and parts, in addition to the process of using the product (Andreasen et al. 2014). It captures the connections between elements within and across these domains, and hence could be used to extract features that realise high level functions (MR1) and to determine necessary supporting features (MR2). However, in common with other approaches this does not directly address the need for consistent modelling of different products, required to support their comparison and integration (MR3).
The final type of hierarchical model to be discussed is the function-means tree. This model represents a product by decomposing top-level functions into the subsystems realising them, then decomposing those subsystems into constituent subsystems and so on until individual parts (potentially features) are reached. Malmqvist (1997) extended the function-means tree by including information such as functions relevant to different product life-cycle phases, alternative design solutions, parametric constraints and design objectives. The relationship between these elements are also captured. Borgue et al. (2019) built on this work to model connections between product functions, design solutions and additive manufacturing constraints. Müller et al. (2020) linked the function-means tree to an automation approach, drawing connections between the function domain and the physical domain to enable CAD models of design concepts to be generated from different combinations of design solutions. Advantages of all function-means tree based models are that they are well-suited to systematic product decomposition, and that they directly show the relationship between functions and the parts that realise them. However, they do not capture the relationships between parts involved in function realisation. Such approaches are therefore not ideal to identify supporting features, as required by MR2.
Overall, hierarchical models support product modelling in a reverse engineering context by structuring analysis and description of a product at progressively increasing levels of detail. A general disadvantage is that they are not well suited to represent connections across branches of the tree. Although several tree-based approaches allow such connections to be made, e.g. by adding diagonal lines that cut across the tree diagram, the diagrams are likely to become difficult to visually manipulate and trace as the modelled product increases in complexity. These limitations could perhaps be addressed by appropriate software support. However, such modelling software is not readily available at present.

Models based on block diagramming
Other models used to represent product information are based on block diagrams. For instance, the operation of a product can be modelled using concepts from the Theory of Technical Systems (Hubka and Eder 1988). Here, the main function of a design to realise a transformation process is represented as a black box, with input arrows to represent the human, knowledge and management systems needed to operate the design to perform the function. Arrows are also used to indicate the operands being transformed while using a design. Hubka and Eder (2002) integrated this model with the Chromosome model to provide a more comprehensive block diagram-based product model. A related and very well-established approach is function-flow modelling ( Pahl and Beitz 1996). This depicts how sub-functions operate together to realise the overall function of a product. Function-flow models could be useful for assessing functional changes during redesign, but do not relate these functions to physical parts and features as needed for MR1. Another drawback is that function-flow models are constructed at a high level of abstraction and can be very different when constructed by different people, which means this approach is not ideal to meet MR3. To reduce this variation, some researchers have developed sets of vocabulary, grammar and topology rules to improve the consistency of function structure models (Szykman et al 1999;Stone and Wood 2000;Hirtz et al. 2002;Caldwell and Mocko 2012;Mohammed and Shammari 2021). Gietka et al (2002) also found that model variation can also be reduced if designers describe each function based on its input and output flows rather than identifying them based on a higher-level function.
Other block diagram models of product structure are based on standardised modelling notations. For instance, SysML has been used for this purpose, as has Object-Process Methodology (Dori 2011).

Models based on matrices
The final group of product architecture models to be discussed are based on matrices. In one well-known approach, Pimmler and Eppinger (1994) used the Design Structure Matrix (DSM) to represent the interactions between parts of a product. The DSM consists of a square matrix with identical row and column headings, used to represent the product elements, while the content of the matrix cells indicates connections between pairs of elements. For instance, Pimmler and Eppinger (1994) used letters in the cells to differentiate spatial, signal, material, and energy interactions between parts. Tilstra et al. (2012) developed systematic steps for modelling a product structure using DSMs, adopting a hierarchical approach in which a product is decomposed and subsystems are modelled separately, prior to combination of those models. They demonstrated the importance of predefining the types of interactions to ensure that consistency is maintained when a product is modelled by different people. They also emphasised the importance of producing consistent DSMs to allow for product architecture comparisons, which is required for function integration (this supports MR3). DSMs have also been used to represent and evaluate multiple design alternatives by Wyatt et al. (2008). However, while providing a concise overview, the DSM does not directly capture the detailed logic of interactions, which can be problematic for applications in which logic is important (Karniel and Reich 2009). For further information on DSMs the reader is referred to the reviews by Yassine and Braha (2003) and Browning (2015).
Researchers have also used more elaborate matrices to show connections across product information domains. For example, the Domain Mapping Matrix (DMM) is a nonsquare mapping matrix that can be used to map the function elements of a product to its physical elements (Danilovic and Browning 2007). Lindemann and Maurer (2007) integrated DSMs and DMMs covering multiple domains to form a multi-domain matrix (MDM) to more comprehensively model product architectures and product families, among other applications. Tang et al. (2010b) applied MDMs in a database-based approach to trace how elements of an existing product will be impacted when there are changes to functions. Eisenbart et al (2017) adopted the MDM concept to form the Integrated Function Model (IFM) to show how lowlevel functions and parts relate to a particular high-level user function. This model combines block diagrams and matrices.

Methods to support function integration in product redesign
Having discussed a number of product modelling approaches, this subsection moves on to discuss how such approaches have been used as the basis of methods to support function integration. Our review revealed that there are relatively few methods of this type in literature. In one publication, Kalyanasundaram and Lewis (2014) proposed a product integration method to support the derivation of re-configurable products and multi-functional products from two existing products. Their method combines the functionality of two existing products to form a new design, using a matrix model to compare low-level functions between parts (as required by MR3) and interactions between physical parts (partially addressing MR2). However, this approach does not model products at the feature level and therefore cannot identify physical features of a product variant involved in function integration (MR1). Since it does not capture parts at a features level, the method also does not provide detailed guidelines for part modification (MR4). A similar method was developed by Kang and Tang (2013) for developing multi-functional products. This method also uses matrices to model existing products, resulting in similar advantages and drawbacks. However, it does consider supporting parts for primary parts that realise the functions, which better addresses MR2 in comparison to the aforementioned method. Lu et al. (2017) also combined existing products to produce multi-functional products using matrix models. However, they used a different approach to model the function of existing products. Instead of modelling products using the functional decomposition approach by Pahl and Beitz (1996), they derive their function structures from detailed flows and parts. This increases the consistency of the product model which should make product comparison more effective (MR3). Liu et al. (2014) proposed a method for integrating products with interrelated functions. They compared the functions of existing parts using a table, listing the functions of products as the row headings and the products being compared as the column headings. The corresponding parts of each product that realise the function are recorded in the cell of the table. An advantage of this comparison table format is that it enables more than one variant to be compared. However, it cannot represent multiple functions for a given part. As a result, the approach cannot accurately compare the function realisations across product variants, which is needed to determine the redesign activities required for a new variant design (MR4). The method does, however, consider conflicts between product functions which occur when a newly added function reduces the performance of other functions. In this scenario, the researchers suggested using the Theory of Inventive Problem Solving (TRIZ) to resolve the performance conflicts. Alternatively, the Advanced Systematic Inventive Thinking (ASIT) approach can be used to generate ideas for deriving the new combination design (Moon et al. 2012). Finally, Smith et al. (2012) proposed a method to combine more than two existing products to derive a new product. Instead of selecting a single product as a foundation for the new variant design, they selected parts from different products based on customers' needs and combined them to form a new design. Their method is similar to a morphological analysis. A drawback is that it does not model the architectures of the products. Therefore, it can only determine parts to carry over to the new variant design based on the user's needs. It does not compare product parts functionally to determine the physical modifications required for a part to fulfil the desired functions, as is required by MR4.

Critique and the need for a new approach
To recap, a relatively small number of function integration methods have been proposed in literature, based on analysis of products using hierarchical models, flowchart-based models and matrix models. Hierarchical models offer systematic procedures for identifying detailed functions and parts of products. However, they do not clearly (or sometimes at all) represent the relationship between detailed functions and parts as required by MR1 and MR2. Block diagram models capture flow information and allow function elements to be placed in their sequence of operation to depict how a product operates. However, they typically do not capture the parts and features that realise the functions, and are not graphically well-suited to this due to the dense structures of dependency involved. Hence, block diagramming-based models are not ideal to meet MR1, MR2 and MR3. Overall, neither tree-nor block diagramming-based approaches are ideally suited to reflect the complexity of function-form relationships in a product. In comparison, matrices are very well suited to graphically represent the dense structures of relationships between functions and parts/features, which allows existing design solutions to be comprehensively represented, extracted, compared and traced without specialised software. These characteristics make a matrix-based product model the most suitable basis for a function integration method.
Regarding the methods themselves, the majority of function integration methods in literature simply merge all high-level functions of existing products by comparing the corresponding low-level functions of existing products to determine which parts to carry over into the new design. They do not capture design information at the features level which prevents them from identifying redesign activities at Table 1 Assessment of function integration methods in literature with respect to requirements MR1-MR4 MR1: Determine low-level and high-level functions of a design and link them to the parts and physical features. This is needed to identify the aspects of a design that realise each high-level function. MR2: Determine the physical interactions between features of a design. This information is needed to identify supporting features that help primary features to realise a desired function MR3: Ensure that product data is consistently represented across variants, using similar function terminology. This is needed so that variant designs can be directly compared for function integration.
MR4: Systematically process the obtained product data to determine how parts and specific features of parts can be carried over, removed and modified from existing variants to generate a new variant design that provides a new desired combination of existing functions.
Kang and Tang  Partially met: Systematically determined: Parts to carry over. Briefly mentioned: to modularise, to modify parameters the level of specific modifications to parts. Overall, none of the existing approaches fully address the requirements set out at the start of Sect. 2. This gap, addressed by the current article, is summarised in Table 1.

Detailed Design Model (DDM)
This section introduces a new product model called the Detailed Design Model (DDM) which was developed to capture design information of existing product variants. Section 4 will then detail how the information captured in this new model is used to support function integration.
To demonstrate the DDM and the reverse engineering procedure used to construct such a model, a BiC ballpoint pen is used as an example. This product is complicated enough to illustrate all aspects of the model, while also being simple enough to explain in full detail. It is referred to here as the retracting pen. Variations of it are available from several manufacturers. The design allows the user to extend the ballpoint tip by pressing a button at the opposite end of the barrel. The tip can then be retracted into the barrel by pressing a latch on the side of the barrel. An exploded view depicting parts and features of the retracting pen is provided in Fig. 1.
Before moving on to describe the DDM in detail using the retracting pen as an example, the next sub-section introduces key concepts and terminology used in the model.

Elements of the DDM product model
The DDM is a matrix-based product model that represents a design in both functional and physical domains, in terms of functions, flows, parts, features, states and state transitions, as well the interactions among these elements. It provides enough detail to capture how all these elements and their interactions contribute to the design's functionality in use.
The following product elements are included in the DDM: -Function: In the DDM the concept of function focuses on what a product does as a physical artefact. Under this definition, functions enable the use of a product but do not describe the multitude of ways in which it could be used, which additionally depend on the user, the task to be performed, the use environment, and so on. Two types of functions are distinguished in the DDM: -Operating function: A transformation in the state of a product, that is associated with operation (use) of the product. -Technical function: A transformation process occurring when a flow (defined below) interacts with feature(s) of a design (also defined below).  -State: A set of partially-constrained connections (and related flows) that can change configuration (i.e. relative motion can occur between the features and flow processes can occur) when a product is operated. Two types of states are distinguished in the DDM: -Static state: A stable physical configuration.
-Transition state: A temporary configuration involving flow processes and relative motion of features, while a product variant is in process of changing from one static state to another.
All these elements of the DDM are required for the Adaptive Redesign Method (ARM), that is to be described in Sect. 4.

Procedure for modelling a product with the DDM
A DDM representation of a product variant is developed in six steps, described below.

Step 1. Model the product in CAD (or obtain an existing CAD model)
The first step is to model the existing variant using CAD or to obtain an existing model. Basing the analysis on a CAD model ensures that no physical elements of the product are overlooked. It also provides a basis and nomenclature to reference specific features of each part, which is necessary for the Adaptive Redesign Method (ARM). The parts must be modelled with appropriate detail to capture the geometric features. For instance, Fig. 2 shows how the Upper Barrel part of the retracting pen is decomposed into six features in the CAD model. Only geometric features of a part that physically realise an operating function need to be modelled. This is because the aim of the ARM method (to be discussed in Sect. 4) is to derive product variants by adapting existing features from two product variants to realise a desired new combination of operating functions. The method only currently considers product use functionality. For instance, it does not consider manufacturing and assembly processes required for the new design. Therefore, minor features such as rounds do not need to be modelled, except where important to realising an operating function.
In general, only 3D CAD features that are derived from a 2D sketch need be considered, because these CAD features generate the spatial geometry of a part. CAD features that only modify the spatial geometry of an existing CAD feature need not be modelled. This is to prevent repetitive referencing of the same physical feature in the DDM. For instance, considering Fig. 2, the Upper Barrel.Tip should be recognised as a feature in the DDM because this domeshaped geometric feature is formed by connecting the surfaces between two differently-sized 2D circles. Whereas the Upper Barrel.Tip Hole is not regarded as a feature for this analysis because the hole feature only modifies the existing feature Upper Barrel.Tip.

Step 2. Form the feature interaction matrix
A DDM of a product is represented as a matrix called the DDM matrix. An overview schematic of the DDM matrix is shown in Fig. 3. The CAD features that form the parts are used as the row and column headings for the top-left field of the DDM matrix, as shown for the retracting pen in Fig. 6. Features that physically interact with another feature are indicated by the letter C or P in the corresponding cell:  These features do not move relative to one another once the product is assembled -C indicates feature connections that are fully-constrained.
For example, the connection between the Upper Barrel. Body and Lower Barrel.Slot of the retracting pen is fullyconstrained because, as shown in Fig. 4, these two features do not move relative to each other in the assembled product. Therefore C is placed in cell S9 and, symmetrically, cell I19 of Fig. 6. -P indicates connections between features that are partially constrained. An example of a partially constrained connection occurs between the features Upper Barrel. Tip and Clicker.Shaft. This connection is partially constrained because the Clicker.Shaft can slide through the hole of the Upper Barrel.Tip as shown in Fig. 5. Hence, the letter P is placed in the cells D20 and T4 of Fig. 6.
This differentiation between partially-constrained and fullyconstrained connections is necessary to distinguish features and interactions that are directly involved in realising a function (P) from those that provide support structures (C). The modeller should physically operate each mechanism of the variant to identify all the possible feature-feature interactions. This is recommended to avoid overlooking any feature-feature interactions which are not explicitly captured in the CAD model. For example, when modelling the retracting pen, it should be retracted and extended to observe which features interact with each other in each stable configuration, and in the transition between configurations. All must be included in the DDM matrix.

Step 3. Identify flows and flow-feature interactions
The next step of the DDM modelling procedure is to identify and model flows and their interactions with the features. In the DDM, flows can be in the form of material, energy or signal as set out in Sect. 3.1. Three situations should be considered: -Flows applied to a feature. For example, a flow of human energy and a human finger are applied to the Clicker.Tip of the retracting pen to extend its ball point tip. Note that only flows that are involved in operating the product need be included. -Flows caused by a feature. For instance, a flow of elastic energy released from the Spring.Coil is used to retract the ballpoint tip of the retracting pen. Note that only flows that involve modelled features need be included. -Flows occurring between features. For example, ink flows from the ink chamber, through the plastic nib, ballpoint socket and ball when the retracting pen is being operated for writing.
Flows that only describe physical connections between features (e.g. "flow of force") are not modelled, because they are already captured in the feature-feature interactions section of the DDM matrix. Flows are listed in the DDM matrix headings immediately after the feature headings, as shown in Fig. 3. As indicated in the bullet points above, every flow interacts with at least one feature. In the DDM, each interaction between a flow and a feature is described using a technical function descriptor to support consistent modelling. The technical function descriptors used for the ballpoint pen example are defined in Table 2. This is a subset of the list provided by Hirtz et al. (2002). For brevity we show only the terms necessary for modelling the pens and internet protocol (IP) cameras discussed in this article.
Numbers are placed in the top-rightmost field and the middle-leftmost field of the DDM matrix to indicate the features responsible for each flow, and the type of technical function involved. For example, in Fig. 6, the numbers 11 and 33 both appear in cell AB8 to indicate that the feature Ink Chamber. Chamber Body (in row 8) Guides (technical function type 11) and also Contains (technical function type 33) the Material.Ink flow (in column AB). Note that unlike feature-feature connections, this relationship is directionalthe feature or flow in the matrix row is the producer of the technical function while the feature or flow in the column is the receiver.

Step 4. Identify motions, states and operating functions
Operating functions describe how a design can be mechanically operated. In the DDM, an operating function is defined in terms of an initial (static) state, an intermediate (transition) state, and a subsequent (static) state. (The ARM will subsequently determine whether there are any operating functions that are dependent on other operating functions. This will be explained in Sect. 4.2.1). The total number of operating functions in a variant is determined by the physical configurations it can be used in, which is in turn dependent on the arrangements of connections within the design.
To identify operating functions of a variant, the possible independent motions are identified by, firstly, physically moving each partially-constrained connection one-at-a-time. Once the motions are identified, the state transitions generated by each motion can be established. Two situations are common: (1) motions that occur when the user changes the product configuration from one static state into another, and (2) motions that are essentially continuous while operating the product.
In the running example, an example of the former type of motion occurs when the pen user pushes the clicker to extend the tip of the pen (see Fig. 5). Here, the transition state (the state of the pen during the motion) involves energy input from the user and a certain set of partially-constrained connections whose features are in relative motion. This transition state can be described as Ballpoint tip extending. It can only occur from an initial (static) state of Ballpoint tip retracted. When the transition is completed, the pen is left in the subsequent (static) state of Ballpoint tip extended. In the DDM, each combination of initial-transition-subsequent state is listed to define an operating function. The features and flows involved in each state are also identified. This is necessary because different features may be involved in operating functions that are complementary to each other. For instance, although both the operating functions of the retracting pen To extend ballpoint tip and To retract ballpoint tip contain the same static states: Ballpoint tip retracted and Ballpoint tip extended, it does not imply that their respective transition states Ballpoint tip extending and Ballpoint tip retracting involve identical features. In this case, extending the ballpoint tip of the retracting pen involves applying force to the Tip of the Clicker whereas retracting the ballpoint tip of the pen involves applying force on the Latch which is located on the side of the pen body. This example demonstrates that the features involved in a transition state are not always identical to its reverse state. Each of these operating functions and its three states are shown at the bottom of the DDM matrix in Fig. 6.
An example of the continuous motion type is rolling the ballpoint ball in its socket. In the DDM, this is viewed as a transition state Ball rolling. It is only possible from a certain initial (static) state Ballpoint tip extended and, when the motion finishes, leads to a subsequent (static) state which is the same as the initial state. In this case, the three-state combination defines the operating function To deposit ink.
To summarise, the procedure for identifying operating functions is to first identify motions in the design, then to identify the initial, transition and subsequent states for each motion, and finally to denote this combination of product states as an operating function. The operating functions and their corresponding states are shown in the bottom part of the DDM matrix. Representing operating functions in terms of three states has proven helpful for the reverse engineering of complex motions to identify the parts, features and flows involved when a product is operated. The three-states approach has proved sufficient for all products modelled to date and also, has the benefit of being possible to visualise in a simple table format. However, it is recognised that a more elaborate representation might be necessary to efficiently describe some situations.
Note how the modelling procedure outlined in this step differs from a top-down identification of product functions starting from considerations of how the product would be used. The advantage of the bottom-up approach is that, while the possible functions of a device are subject to interpretation, the motions and states possible within it can be unambiguously identified. Hence, following the procedure outlined in this section leads to a more objective model of product functions than a top-down analysis.

Step 5. Link operating functions to feature connections and flows
The penultimate step of the DDM modelling procedure is to link the states of each operating function to the featurefeature and feature-flow interactions that are active in each state. This is required by the ARM method to be described in the next section, so that the features responsible for specific operating functions can be identified. For each transition state, the modeller must identify the active flow processes and the active connections involved in the motion. An active connection is one in which relative motion is occurring between the two features, or in which force transmission between the two features is necessary to enable the motion. These flows and interactions are represented in the rows of the transition states in the DDM matrix by referencing the coordinates. Only the numerical coordinates (rows) are recorded. The alphabetical coordinates (columns) do not need to be recorded because the numerical coordinate is placed in the corresponding column. For example, in the Ballpoint tip extending state, a "human energy" flow is being applied to the Clicker.Tip of the retracting pen as shown in cell E25 of Fig. 6. Therefore 25 appears in column E of the Ballpoint tip extending row.
Next, it is necessary to identify the unique connections that occur in the initial and subsequent states. Unique connections are those in which the two features are in direct physical contact in that state, but not in the complement state. For example, in the Ballpoint tip extended state of the retracting pen, the Clicker.Latch is engaged with the Upper Barrel.Latch hole. This contact only occurs when the pen is extended and is not present when the pen is retracted or in the transition state. Thus, in Fig. 6 the coordinates G21 and U7 appear respectively in columns G and U in the row defining the state.

Step 6. Verify the product model
The final step of the DDM modelling procedure is to check that the DDM matrix has been correctly completed. The following checks may be helpful to reveal errors: 1. Check that the feature-feature portion of the DDM matrix is diagonally symmetrical. 2. Check that each feature and flow listed in the DDM is shown to interact with at least one other feature or flow. Features with empty rows or columns should be revisited to check that no interactions were missed, and if not, to reconsider whether the feature is necessary to include in the model. 3. Review any CAD features that were not included at the beginning of the modelling process, and confirm they are not involved in any identified operating function.

Adaptive Redesign Method (ARM)
Having completed description of the Detailed Design Model and modelling procedure, we now move on to the Adaptive Redesign Method (ARM) itself. To support description of the ARM, we return to the ballpoint pen example and consider the situation in which the To retract and To extend operating functions of the retracting pen are to be carried across into another pen design, referred to here as the basic pen. As shown in Fig. 8, the basic pen has a hexagonal barrel which mimics the shape of a pencil and a cap which covers the ballpoint tip. More detail is provided in the Supplementary Materials. The objective of the running example is to combine these two variants to produce a unique new design, i.e. a retractable pen with hexagonal barrel. This case is deliberately simple to allow the new method to be explained in full depth within the space constraints of this article. In Sect. 5, application of the method to more complex products is discussed to demonstrate that it is scaleable and to indicate the effort involved. The ARM comprises of three phases, which are depicted in Fig. 7 and detailed in the next subsections.

ARM Phase I: Develop DDM for each variant
As input to the ARM, a DDM matrix must be formed for each of the two product variants to be combined. Although this is quite time-consuming, each DDM matrix is a reuseable resource that could be drawn upon in future each time a new variant is required. As additional variants are modelled, a data library would be built up, broadening the possibilities for forming new variants without additional modelling effort.
The DDM for the first pen variant to be used in the running example was already discussed and is shown in Fig. 6. Fig. 8 shows the DDM matrix for the second variant, namely the basic pen. Note how the basic pen has a different set of operating functions than the retracting design-the only common operating function between the two pens is To deposit ink. It is important that technical function descriptors are used consistently when modelling the different product variants, so that the variants can be compared. For example, noting that the number 6 is used to describe import functions in the retracting pen DDM, the same number is used to describe this type of interaction in the basic pen DDM.

ARM Phase II: Identify adaptive redesign steps
In the method, the variant to be adapted/redesigned is described as the base variant. The variant from which additional operating functions are to be drawn is described as the source variant. Parts and features from the source variant that realise a desirable operating function are to be integrated into the base variant. For the running example, the basic pen is chosen as the base variant while the retracting pen (offering the desired retraction and extension functionality) is chosen as the source variant.
The second phase of the method involves systematic analysis of the DDM matrices for these two variants to identify the features of the base variant that will need to be removed, those that will need to be carried across from the source variant, and which parts need to be redesigned to generate the new variant design. The phase comprises eight steps, described in the next subsections.

Step 1: Categorise operating functions in the base variant and source variant
The set of desirable operating functions for a new product variant depends on market needs, product strategy and other factors. Identifying that set is beyond the scope of this article. For the worked example, the desired operating functions of the new variant are: To extend ballpoint tip; To retract ballpoint tip; and To deposit ink. For each desired operating function, the modeller must identify whether it requires any prerequisite operating function, and if so, classify it as also being desired. For example, the To deposit ink operating function of the retracting pen requires the pen to be extended, which means that the operating function To extend must be executed beforehand. If To deposit ink was a desired operating function of the retracting pen, it would therefore require To deposit ink to be categorised in this way as well. The modeller should also check for redundancy or other conflicts among the functions that are intended for the new variant design. In the running example, considering that the To retract function of the source variant is desired in the new variant, the modeller can observe that two operating functions of the base variant, namely To cover ballpoint tip and To uncover ballpoint tip (that are realised by the cap of the pen), will no longer be necessary and therefore are labelled as Uo (Undesired operating function) in the base variant DDM matrix as shown in Fig. 8.

Step 2 Identify undesired features of the base variant (to be removed)
This step proceeds as follows: 1. Identify features of the base variant that are involved in the previously-identified undesired operating functions. These can be readily identified by tracing the DDM matrix dependencies from all three states involved in that operating function. 2. Examine the DDM matrix columns for each of these undesired features to determine whether they are also involved in any of the previously-identified desired operating functions. If not, the feature should be classified as undesired.
To illustrate, since the To clip operating function is to be removed from the basic pen, the Cap.Lower clip inner face must be removed because it is not involved in any other desired operating functions, as can be identified by reading down column L of the operating functions section of the DDM matrix in Fig. 8. On the other hand, the Barrel.Body feature cannot be removed because column G of Fig. 8 contains entries relating the Barrel.Body to the desired function To deposit ink. The base variant DDM matrix is annotated by adding Uf (Undesired feature) next to the row and column of each undesired feature, to indicate that their interactions are to be ignored later in the analysis. In Fig. 8 these are highlighted in dark pink. To provide another example, these are also identified in Fig. 6 (although identifying undesired features for the source variant is not strictly required for the method).

Step 3. Identify undesired supporting features of the base variant's undesired features (to be removed)
Features of the base variant whose only purpose is to support undesired features are to be classified as undesired supporting features. Such features are identified from their connections to the features to be removed, as shown in the base variant DDM matrix. These supporting features may be from the same part or from another part as the features to be removed. Continuing the previous example, the Cap.Lower clip inner face was identified as an undesired feature of the base variant to be removed. Column L of the DDM matrix in Fig. 8 shows that the feature has two supporting features: the Barrel.Body and Cap.Lower clip outer face. The Barrel.Body cannot be removed, since it is involved in desired functions. However, the Cap.Lower clip outer face can and should be removed, because tracing its dependencies in the DDM indicates that it is not supporting any features required for any desired operating function. For each undesired supporting feature that is identified for removal, the matrix should be checked again to ensure that no additional features have become redundant, in which case those are to be removed as well (this check is repeated until no more features are candidates for removal).
Once identified, all undesired supporting features are annotated with Usf in the base variant DDM matrix. In Fig. 8 these are highlighted in light pink. To provide another example, such features are also identified in Fig. 6 (although identifying undesired supporting features for the source variant is not strictly required for the method)

Step 4. Identify desired features of the two variants
The next step is to identify desired features of the source variant, which are features that contribute to realising any of the desired operating functions. As before, this is achieved by tracing the numbers in the rows of the desired operating functions located in the bottom field of the source variant DDM matrix. Recall from Sect. 3.2.5 that these numbers indicate the features required for the function.
To distinguish desired features in the source variant DDM matrix they are annotated Df. The flow of dependencies from the desired operating functions through to the desired features is highlighted in blue in Fig. 6.

Step 5. Identify desired supporting features of the two variants' desired features
Supporting features that are necessary to the working of carried-across features also need to be carried over to the new variant design. They can be identified by reading down the columns of the desired features in the source variant DDM matrix. For example, reading down column R of Fig. 6 shows that the Lower barrel. Slot (in row 9) of the retracting pen is a supporting feature for the desired feature Spring.Coil. This can be explained by considering Fig. 9, which shows how the Lower barrel.Slot keeps one end of the Spring.Coil in place during the retract/extend state. The source variant DDM matrix also reveals several other features that interact with the Spring.Coil, but these need not be classified as supporting features because they were already classified as desired features (as labelled Df and highlighted in blue). Supporting features for the desired operating functions are marked with Dsf and highlighted in light blue in the DDM matrix of the source variant. These are to be carried over into the base variant.

4.2.6
Step 6. Determine the functional similarity between each desired/desired supporting feature of the source variant and each desired/desired supporting feature of the base variant The next step is to complete a pairwise comparison of the features across the two variants and compute their similarity. This is required so that, in Step 7, it will be possible to determine which features need to be drawn from each of the two existing variants to create the new variant. Each part of the source variant that contains at least one desired feature or desired supporting feature (as identified in the previous step) is compared to each of the desired features and desired supporting features of the base variant. A comparison matrix is formed with these base variant features in the column headings and the source variant features in the row headings, as shown in Fig. 10. Empty columns represent base variant features that were identified as undesired or undesired supporting features (Uf or Usf) in Steps 2 and 3. Similarly, empty rows represent source variant features that are not desired. For each of the non-empty cells in the matrix, the flow interactions and partially-constrained connections of the corresponding pair of features are next compared (by analysing the two DDM matrices) to compute the degree of functional similarity between that pair of features. Fully-constrained connections between features are not included in this comparison since they provide structural support but are not directly involved in realising operating functions. Functional similarity is defined here as the total number of interactions that meet the above criteria and are common to both features, divided by the total number of interactions that meet the criteria. The result is interpreted as follows: -100% indicates that the two features are functionally identical. -0% indicates that the two features are functionally disjoint. -Greater than 0% and less than 100% indicates that the existing feature in the base variant realises some, but not all of the functions of the source variant feature. The two features are functionally similar.
To illustrate, consider the comparison between the Ink chamber.Chamber body features of the two variants. Comparing the DDMs of Figs. 6 and 8 reveals that these features participate in two identical technical functions with respect to the same flows, namely To store (33) Ink and To channel (11) Ink. Figure 8 shows that the feature of the basic pen has no partially-constrained connections to other features of that design, while Fig. 6 shows that the feature of the retracting pen has one partially-constrained connection, namely to the Spring.Coil. In this case the feature of the source variant (retracting pen) has three interactions in total. Two of them also appear in the feature of the base variant being compared, i.e. the two flow interactions, while one does not. This leads to a similarity score of 2∕3 = 67% , so the two features are classified as functionally similar. Note the importance of comparing technical functions in the flows part of the DDM, which allows the more accurate discernment of functional similarity of the two features from different variants. The required accuracy would not be possible if only the physical connections to other parts of the respective variants were considered. The technical functions and flows modelled in the DDM matrices allow features performing similar functions, even across variants that have slightly different architectures to be compared.
For functionally similar feature pairs, if the base variant feature contributes to all the desired technical functions of the source variant feature, a ' − ' sign is placed alongside the similarity score. On the other hand, if the base variant feature does not meet all the desired technical functions of the source variant feature, the matrix cell is denoted with '+' to indicate that additional technical functions would need to be met by the base variant feature if it were to realise the technical functions of that source variant feature.

Step 7. Determine which features to carry across from the source variant into the new variant
Recall that the base variant is the baseline design to be changed when creating the new variant. To determine which parts/features and supporting features of the source variant are to be carried across to the new variant design, the rows of the comparison matrix are next worked through systematically. Each part is considered one-at-a-time as follows: -If all desired/desired supporting features of the source variant part are functionally disjoint from all features of the base variant, the entire part of the source variant is to be carried over into the new variant. For example, the part Clicker of the retracting pen will be used in the new variant design because no parts of the base variant (basic pen) have features with any functional similarity to it. This is identified by noting that in Fig. 10 all the entries are 0% for every row describing features of the Clicker. The same is true for the Plunger and Spring. -If the desired feature of the source variant has 100% in one or more of the cells, when reading across its row, then the base variant feature already offers the desired functionality. This means that those base variant features are to be retained when creating the new variant design. An example of this can be found between the Ballpoint tip ball.Ball and Ballpoint tip.Socket features of the two pens. -If the desired feature of the source variant is functionally disjoint from all features of the base variant, i.e. all entries across the row are 0%, then that source variant feature must be carried over into the base variant. This is done by modifying the part of the base variant that is functionally most similar to the part of the source variant that contains the desired feature being carried across. Functional similarity between two parts is calculated as the average of the functional similarity of all pairwise comparisons of their features. For example, the Upper barrel. Tip and Upper barrel. Latch hole features of the retracting pen will be carried over by adding them to the Barrel of the basic pen because the Upper barrel of the retracting design (from which the features are drawn) is the most functionally similar part to the Barrel of the basic pen design. -If the source variant feature is functionally similar to one or more base variant features, and is not functionally identical to any feature, its row contains at least one entry > 0% and no entries = 100%. In this case, it is necessary to visually/geometrically compare the feature of the source variant against each of the functionally most similar features of the base variant to determine what changes to the base variant feature(s) might be needed to produce the desired functionality of the source variant feature. This comparison requires design judgement. For example, the Ink chamber. Chamber body of the basic pen needs to be geometrically compared to the Ink chamber. Chamber body of the retracting pen (source variant). This is because the '+' sign in front of the 67% (calculated in Step 6) indicates that the basic pen needs fulfil a desired interaction to realise the operating function. This desired interaction happens to be with the Spring.Coil which can be traced by comparing the DDM of the two pens as discussed in Step 6. In this example, the existing geometry of the basic Ink chamber.Chamber body is identical to that of the retracting pen Ink chamber.  -If the desired feature of the source variant has < 0% in one or more cells when reading across its row, then it also indicates that the base variant feature already provides the desired functionality of the source variant feature. In this case, the base variant feature is also retained. Note the percentage shown in the matrix cell may not neces-

Step 8: Compile list of redesign activities to create the new variant design
Finally, the results of the previous steps are compiled to form a list of redesign activities that will be needed to form the new variant design as shown in Fig. 11.
The list shows what redesign activities are needed in terms of operations on features and parts. It does not show the geometric and dimensional details, such as where on a part a feature should be added and what dimensional adjustments will be necessary. These details are usually quite obvious when viewing the two parts for each step, as illustrated in the next subsection.

ARM phase III: execute adaptive redesign steps to generate the new variant design
The final phase of the method is for the human designer/ method user to execute the identified redesign activities to form the new variant design on CAD. Firstly, undesired features of the base variant are removed by removing CAD features from the CAD model. Before a CAD feature is removed, it is important to check that there are no dependent sketches and features built upon it in the model tree that are desired to be retained. Secondly, features from the source variant that realise the desired operating function are added to the base variant. To integrate these features to the base variant, redesign of existing parts of the base variant are required. An example of two steps from the basic pen's barrel being redesigned is provided in Fig. 12. Features of the base variant which do not require feature modification can be reused in the new variant design, although some adjustment to dimensions may be required. Figure 13 shows the completed design of the new pencilshaped pen with the retractable function, that was generated by following the redesign steps laid out in the redesign activities table.

Application cases
As previously mentioned, the ballpoint pens example was chosen to illustrate the DDM and ARM because it is simple enough to present the approaches in full detail. To also illustrate that the new approaches can support modelling and redesign of more complex products, this section discusses their application to a pair of Foscam IP cameras (security cameras).
The cameras to be discussed are from a product range in which different variants offer different functions. Depending on the variant, these functions include motorised panning, motorised tilting, recording, motion detection, voice detection, audio output, and single press calling. In this section, the motorised panning function is integrated into a simple IP camera (here called the fixed camera) that is capable of being manually tilting upwards and downwards. The motorised panning function was sourced from a motorised IP Fig. 11 The generated list of redesign activities needed to derive the new variant design camera that can be panned and tilted using software control (here called the PT camera).
The PT camera (the more complex variant) contains 37 parts, and as can be seen in Fig. 14, its mechanical assembly is representative of many moderately-complex consumer products. More detail for both cameras is provided in the Supplementary Material.

Application of the DDM modelling procedure to the two IP cameras
The detailed design modelling procedure was applied to generate DDM matrices for the two product variants, as shown in Fig. 15. More detail is provided in the Supplementary Materials. In overview, the DDM of the PT camera comprises 37 parts, 157 features with 155 interactions, 59 technical functions, and 7 operating functions. Of the seven operating functions, the automatic panning function will be carried across from the PT camera into the fixed camera. The DDM revealed that this operating function requires 25 features to realise it. The equivalent information for the fixed position IP camera is provided in Fig. 15. In comparison to the ballpoint pens case study, there are energy and signal flows in both camera variants. The IP cameras contain various electronic components that were not modelled in detail. This is because this article focuses on mechanical considerations. Geometric features relating only to part manufacturing processes were also not  Overall, this application confirmed that the DDM modelling procedure can be applied to products that are more complex than the ballpoint pens discussed earlier. The matrices generated for the IP cameras are rather large, but also extremely sparse. The majority of interactions are fully-constrained connections between features of the same part (i.e. within the clusters shown in Fig. 15). In total, generating the DDMs for the two camera variants required about 8 h of effort. Additionally, about 24 h was required for the CAD modelling of the two variants, but this would not be needed for an application in an industrial context where CAD models would already be available.

Application of the Adaptive Redesign Method for function integration between the two IP cameras
Once the DDM matrices were completed, by following the steps of the Adaptive Redesign Method, it was possible to generate the design activities table in about 1.5 h. An additional 5 h were then required to follow the steps by completing the new variant design in CAD-but this would be necessary whether or not the new method was used. The effort would have been substantially less if the straightforward matrix tracing steps and calculations were automated. This indicates that the method, if implemented in special-purpose software, could potentially help to identify redesign activities for even quite complex products fairly rapidly, provided that existing CAD models were available. Reflecting on the ballpoint pens example and the more complex IP cameras analysis, the latter added insight by confirming that the ARM could be used to compare parts having substantially different geometric features across the two designs, and still identify redesign activities at the parts level for the new design. It was also observed that the effort required for the first phase of the approach (DDM generation) and the final phase (executing redesign steps) increased significantly with the complexity of the product. At the same time, the majority of this effort (and about 75% of the total effort according to the estimates above) was devoted to CAD activities that would need to be done to generate the variants, regardless of whether the method was used in support of the redesign process. Put another way, the overhead of using the method (without any specialised software) appears to be approximately 33%. Overall the cameras analysis provided a measure of confidence in the DDM and ARM, although more studies will be required to substantiate the benefits set out in Sect. 1.1.

Initial assessment of the useability of the DDM and ARM
Noting that the DDM and ARM approaches are quite intricate, we sought to assess whether they could be applied by a person other than the authors. In an initial assessment, an undergraduate mechanical engineering student was tasked to apply the emerging method to two cases. In the first case, the student applied the method to implement a sheet-fed scanning function into a budget inkjet printer having a singlesheet scanner. In the second case study, the student applied the method to implement an automatic juicing function into a simple hand-operated juicer. While not a comprehensive evaluation, these cases along with the pen and camera cases provided additional confidence that the method can work with a variety of products and also, that it is useable. The application cases also highlighted that the method relies on the user to detect whether there are conflicts between operating functions across variants being integrated (this was subsequently included in the method description, see Sect. 4.2.1). A conflict occurs when more than one operating function of the new variant design addresses the same task from a user's perspective. For example, the ARM does If a conflict is detected by the method user while performing a function integration, then the features (and potentially parts) related to the redundant operating function should be removed from the base variant. If not identified early on, these conflicts become obvious towards the end of the redesign phase while the physical form of the new design is being manipulated in CAD. In such cases, it is possible to return to steps 2 and 3 of the ARM to identify the features and parts that should be removed. Several iterations among steps of the method may be necessary to finalise the design.

Summary
While a comprehensive evaluation of the method with practitioners has not yet been attempted, applications by different researchers to four different types of product (pens, IP cameras, printers and juicers) build confidence that: 1. The DDM provides a basis for systematically identifying and modelling functions, flows, parts, features, states and interactions in existing product designs.
2. The information captured by the model is at an appropriate level of detail to support the adaptive redesign activities needed for function integration-namely adding, removing, redesigning and carrying over features and parts to achieve a desired new combination of existing operating functions. 3. The method can be applied to different types of products and those with moderate levels of mechanical complexity. 4. The systematic steps of the method are possible to perform by different users (not only the researchers).

Recap of contributions
To summarise, this article offers the following contributions. Firstly, the DDM and ARM provide means to systematically extract selected operating functions and their physical realisations from existing product designs and integrate them into other existing product variants. This is achieved by modelling the parts of the products at the geometric level using CAD features and capturing their interactions with other features, flows, states and state transitions to realise operating functions. Function integration is not comprehensively supported by prior methods, that are discussed in  Table 1 none analyse  a product down to the features, states and state transition  levels. Secondly, the Adaptive Redesign Method (ARM) supports identification of redesign activities required to derive a desired new variant design. In particular, it helps to identify the redesign activities for removing unnecessary operating functions and their physical realisations, as well as the redesign activities required for modifying an existing part. This is achieved by analysing and comparing data at the features level of the DDM for the two product variants under consideration. This level of detail is not offered by previous function integration methods discussed in Sect. 2.2.
Thirdly, the two approaches consider not only the primary features but also the supporting features for each operating function. This is achieved by distinguishing partiallyconstrained interactions (involving primary features) from fully constrained interactions (involving supporting features). Considering the latter allows identification of supporting features and supporting parts that might need to also be modified when primary features are carried over or removed. Existing function integration methods do not comprehensively consider supporting features.
Finally, the DDM provides a more objective approach to modelling the high-level functions of existing products by forming operating function descriptions based on the possible physical configurations of a product. In other words, the modelling procedure is based on describing what a product can do instead of what a product can be used for. Low-level technical functions are then identified based on the interaction between features and flows based on a set of vocabulary by Hirtz et al (2002). As previously stated in literature, identifying function descriptors based on flows is beneficial to reduce subjectivity of functional modelling (Gietka et al 2002).

Limitations
In this section, some limitations will be discussed with respect to functional and physical domains of the DDM and ARM.
Some limitations concern mainly the functional domain. Firstly, the DDM does not consider what the user uses the product for, i.e. use case functions. This was a deliberate choice to reduce subjectivity in the modelling and analysis. However as a result, the ARM is unable to identify repeating use cases between different operating functions for the new variant design. Recall from Sect. 5.2, that for the ballpoint pens redesign, the ARM did not detect that the cap of the basic pen was not needed once that pen was made retractable. Therefore, future work is needed to model the relationship between the use case functions and operating functions of existing products to more systematically avoid redundant functionality in the new variant design. A second limitation of the DDM and ARM in the functional domain is that they do not consider the relationships between the technical functions involved in each operating function. In particular, the sequence of operation for the technical functions is not represented and, if it is important, it will need to be considered separately by a designer using the approach.
Other limitations concern the physical domain. Firstly, the features listed in the DDM matrix depend on how each physical part is modelled with CAD and since parts can be modelled in different ways, this list may vary. However, because the ARM outputs a redesign activities list in terms of the features of the input CAD models, the utility of the method is not greatly dependent on CAD modelling choices. Secondly, the DDM does not consider the spatial relationships and interface constraints between features and between parts. As a result, the geometry of features from the source variant that are being integrated into a base variant part may need to be scaled to ensure physical fit with other parts of the base variant. These parametric adjustments are not accounted for in this article. Another constraint-related limitation is that the ARM assumes that all existing parts of a product are possible to redesign. However, in practice some parts of a product cannot be easily modified, e.g. because they are purchased from a supplier or form part of a product platform. Future work could incorporate such constraints in the method.
Regarding the relationships between functional and physical domains, the case studies reported in this article confirmed that these relationships are captured by the DDM in sufficient detail for the function integration task. However, future applications may reveal opportunities for improvement in this area as well.

Additional areas for future work
By considering the limitations above and also by considering the PSI analysis approach developed by Reich and Subrahmanian (2022), three additional areas of future work have been identified.
Firstly, we hope to undertake empirical studies in companies with product families to explore how function integration is done in practice. Additional application studies are also needed to test the practicality of the method and to test it against the expected benefits set out in Sect 1.1.
Secondly, the method could be extended to account for different contexts of use. For instance, in the context of product families there are multiple variants with different operating functions and architectures available for integration. In a product family context, the scope of the method could be expanded to determine which variant to source each desired operating function from, and to determine the best sequence of implementing multiple functions to avoid redesigning the same parts multiple times. Also noting that a design is influenced by many considerations beyond the use of a product, and hence that multiple reference frames are possible for design analysis, the methods reported in this article could potentially be expanded to support product modelling and integration from the viewpoints of other lifecycle phases apart from product use, such as assembly, inspection and repair, etc. To achieve this would require expanding the DDM to map design elements onto considerations in different lifecycle phases.
Finally, future work could investigate opportunities to reduce the data requirements and effort-intensiveness of the DDM and ARM. Data requirements might be reduced by developing an initial assessment approach to determine the parts of a product that need to be modelled down to the feature level and those that only need to be modelled at the parts level for instance, because they are not involved in any function expected to change, or because they cannot be changed-for instance because they are off-the-shelf parts. Effort-intensiveness could be reduced by automating some of the steps in the DDM modelling method and the ARM to reduce the overall analysis time. Automatable steps include (1) extracting features and their interactions from the CAD model tree to form the DDM, (2) calculating similarity between features and (3) identifying redesign activities based on the similarity value. By reducing effort and data requirements, this would also improve the scalability of the DDM and ARM.

Concluding remarks
Companies refresh and update their product offerings to remain competitive in changing markets. These updates can involve leveraging technology from one variant to improve other existing variants, which involves function integration. To support the function integration process, this article has introduced the Adaptive Redesign Method (ARM). The method provides more detailed redesign guidance than previous methods, which is achieved by modelling and comparing existing variants using a new product model called the Detailed Design Model (DDM). Several case studies indicate that these approaches provide sufficient detail to identify the low-level tasks required to redesign existing parts to produce new product variants, showing that this can be done in reasonable time and does not require prior expertise about the designs being worked with. It is hoped that in future, the approaches can be partially automated and expanded for application to a range of redesign situations.