Mapping the landscape of product models in embodiment design

The purpose of this contribution is to develop a structured overview of linkable and non-linkable product models to enable researchers to access the knowledge of linking possibilities between product models and simplify the integration of new models into the state of research. A particular relevance of the linking possibilities is expected during the development of the shape of the technical system starting from concepts, since this represents a major change in the product description and the fulfillment of requirements must be ensured by testing activities with models. This development from concept to shape aligns with the embodiment design phase as defined by Pahl and Beitz (1996: 199): “Embodiment design is that part of the design process in which, starting from the working structure or concept of a technical product, the design is developed, in accordance with technical and economic criteria and in the light of further information, to the point where subsequent detail design can lead directly to production […].” According to Pahl and Beitz (1996), this phase is characterized by repeated corrective steps, the constant alternation of analysis and synthesis, and the intersection of function, durability, production, assembly, and operation. Since this multitude of elements is addressed by different models, the repeated steps result in repeated switching between product models and thus also in a greater relevance of links in embodiment design. For this reason, the structured overview of linkable and non-linkable product models is narrowed to the embodiment design phase hereafter referred to as the landscape of product models in embodiment design.

To be able to identify the linkable product models, the components of the links, the inputs and outputs, have to be investigated first. Concerning the objective of identifying links between product models, similarities between inputs and outputs are particularly relevant. Therefore, the following research question is formulated:

RQ1: Which categories of similar inputs and outputs of product models in embodiment design can be identified in the literature?

RQ2: Which links between product models in embodiment design can be formed based on the inputs and outputs from the literature?

The research questions are answered through a literature review, as the literature contains descriptions of the currently used and recently developed product models as well as their inputs and outputs. The three steps of the literature review, consisting of the creation of a literature collection, grounded theory-based literature analysis, and subsequent visualization, are explained in detail in Sect. 2. Section 3 contains the results, with the landscape of product models as the main result, and the discussion in a structure analogous to that of the methods section. Section 4 presents the main conclusions regarding the links between product models, as well as their implications for the development of new product models.

As described in the introduction, Matthiesen et al. (2019b) conducted a literature review regarding product models in embodiment design containing publications between 2009 and 2018. The literature identified by Matthiesen et al. was, therefore, used as a basis and updated to cover the period from 2009 to August 2022. The procedure for the systematic literature search used by Matthiesen et al., which was based on Dresch et al. (2015), was replicated for the period since 2018. The sources, search terms, and inclusion criteria defined by Matthiesen et al. (2019b) were as follows:

The search terms embodiment design, embodiment design AND model, embodiment design AND analysis, embodiment design AND synthesis, embodiment function relation, function model*, product model, and system* modeling were applied to the databases¹ASME digital collection, Google Scholar, IEEE, ResearchGate, Scopus, and Web of Science. The matches in the databases were filtered using three sets of inclusion criteria increasing in requirements. The first set of inclusion criteria applied simultaneously in the search in the databases were:

After applying the first set of inclusion criteria, the remaining matches were transferred to the reference management program Citavi 6.³ Before applying the second set of inclusion criteria, duplicates were filtered using the tool included in Citavi 6 and manually. The second set of inclusion criteria were:

The remaining publications were filtered using the third set of inclusion criteria considering the full-text sources:

The publications following all inclusion criteria were merged with the literature collection by Matthiesen et al. (2019b). Due to the overlap of search periods in 2018, the merged literature collection was again checked for duplicates to achieve a consistent collection of literature for the period from 2009 until August 2022.

The aim of the literature analysis was to answer Research Question 1 by creating a table of model inputs and outputs, and subsequently building categories from them. To begin with, the product models described in the publications had to be identified, as many models and their inputs and outputs were described in multiple publications. The definition for product models, human-made, pragmatic, reductive representations of a technical product carrying attributes similar to the modeled original, specified in the introduction was applied. For the following extraction of the inputs and outputs from the publications and to build categories, the step Analyze of Wolfswinkel et al.’s (2013) grounded theory literature review method was chosen. This method was selected as the inputs and outputs of the product models were expected to be described differently in each publication. The method enables the merging of differently described constructs into overarching categories, as will be shown in Sect. 3.2.

The method started with the highlighting of excerpts containing inputs and outputs, which can be words, sentences, or paragraphs. Due to the specific context of this analysis, figures were considered as well. The extracts were then merged for each product model. The product models, corresponding publications, and merged excerpts of their inputs and outputs were documented in an overview table.

Subsequently, three coding processes were applied to the excerpts to build categories based on similarities of the excerpts. The three coding processes, open coding, axial coding, and selective coding, were defined by Wolfswinkel et al. (2013: 51) as follows: “In sum, open coding is the analytical process of generating higher abstraction level type categories from sets of concepts/variables. Axial coding is the further development of categories and relating them to their possible sub-categories. With selective coding, the categories are integrated and refined.” The coding processes were first applied separately to the inputs and outputs and finally joined to create overarching categories. This resulted in a hierarchical structure of categories of inputs and outputs.

In the following, we describe how we consolidated and visualized the results from the literature analysis to answer Research Question 2. From the overview table, model pairs were formed based on matches between the outputs of one model and the inputs of another model. Gathering every possible model pair and category each model belonged to resulted in the characteristics of each model.

We did not visualize model inputs without corresponding model outputs and vice versa. Due to the missing links, these inputs and outputs do not contribute to answering the research question and reduce the clarity of the network. But if there was a subset of inputs without corresponding outputs, we still displayed the model with all inputs and the corresponding outputs. These could be, for example, mental images as input for sketches as described by Andreasen et al. (2015) or the material behavior in the case of FEM models as described by Rajaguru et al. (2010). While the mental images are supplied by the designers themselves, the material behavior originates from material science and thus does not fall within the product models considered in this publication. Still, other inputs of the FEM model have corresponding outputs of other models, which are, therefore, visualized.

The characteristics of all models were subsequently sorted and transformed into a set of tables as inputs for the network visualization software Gephi.⁵ The categories were represented as their own type of nodes and linked by undirected graphs. The product models were modeled as different types of nodes and linked to each other and the categories by directed graphs. If a model belonged to a category with sub-categories, the model was only linked to the lowest sub-category. The link of such a model to the higher category was established by the link between the sub-category and the higher category. The layout was calculated using the force-directed Yifan Hu layout algorithm.⁶ “Force-directed algorithms model the graph layout problem by assigning attractive and repulsive forces between vertices [i.e. nodes], and finding the optimal layout by minimizing the energy of the system” (Hu 2005: 40). This resulted in the product model landscape as a mixed network.

The literature search was conducted in August 2022. The result was 2853 publications after applying the first set of inclusion criteria. After applying all inclusion criteria, 26 publications remained. Table 1 shows an overview of the number of publications found in each database per set of inclusion criteria. Combining the remaining 26 publications with the 48 publications identified by Matthiesen et al. (2019b) resulted in a literature collection containing 74 publications.

Considering the number of publications in the individual steps across all databases, several aspects are notable. The number of publications after the first step is very high (2853). We can, therefore, assume that due to the openness of the search terms, the majority of the publications relevant to this review have been included into consideration in the following steps. From this initial amount, only 0.9% of the publications remained in the end. Thereby, the second set of inclusion criteria excluded the most publications, evaluating the mentioning of a product model or modeling process and the context of embodiment design. Based on the search terms used, this indicates the presence of many other types of models, e.g., process models, or models outside of embodiment design. This is consistent with the expectations based on the breadth of model use in product design and models identified in literature reviews alongside product models (e.g., Wynn and Clarkson (2018) for process models). A comparison of the number of remaining publications of this search with the number of publications identified by Matthiesen et al. (2019b) shows that the number of identified publications has increased from 4.8 publications per year in the search by Matthiesen et al. (2019b) to 6.5 publications per year in the search of this publication. This is in line with the expectation given the trend of increasing scientific publications observed by Bornmann and Mutz (2015). Considering the institutions involved, in Matthiesen et al. (2019b), the corresponding authors of the 48 publications came from 36 institutions, and the 26 newly identified publications came from 13 institutions. Accordingly, the institutions involved in the publications have become more concentrated. Nevertheless, this is not a situation where a single institution determines the number of publications or research landscape. Hence, the literature collection appears to be an appropriate sample based on the number of considered publications.

The examination of the literature collection containing 74 publications revealed 52 different product models. The publications by Zhan and Huang (2018) and Zheng et al. (2017), which were adopted from Matthiesen et al. (2019b), were excluded as they did not include a product model according to the definition used in this review. An overview of the product models and associated publications is presented in Table 2 in the Appendix. Compared to Matthiesen et al.’s (2019b) list including 34 product models, this table contains 18 more product models. This expansion is partly due to new models identified in the 26 additional publications, and partly due to the breakdown of frameworks that Matthiesen et al. (2019b) counted as one model. For example, the framework of models by He et al. (2013) was subdivided into the individual product models contained therein to achieve a more thorough examination of the individual models.

The literature analysis resulted in the extension of Table 2 by inputs and outputs. The subsequent coding steps resulted in 11 categories of inputs and 14 categories of outputs. Figure 5 and Fig. 6 in the Appendix present the categories developed separately based on the inputs and outputs, as well as descriptions merged under the categories for traceability. These figures thus illustrate the merging of differently described constructs into overarching categories, as in the case of the physical system in Fig. 5 or the relations between components in Fig. 6. Combining these categories of similar descriptions of inputs and outputs resulted in 17 categories across the inputs and outputs which are presented in Fig. 2. The hierarchical relations between categories resulted from the axial coding step. In addition, Fig. 2 presents the number of models belonging to each category. Five models did not belong to any of the developed categories since none of their inputs or outputs could be categorized. These were four feature-based models by He et al. (2013), where the description was limited to the inputs and outputs among each other due to the intended use within the same framework. Thereby, the description was too specific to be comparable to descriptions of other models outside of the framework. In addition, there was the SAPPhIRE Model, for which no classification was possible due to insufficiently described inputs and outputs due to the specific context of the use case.

Concerning the identified product models, an anticipated model, i.e., the technical drawing, is missing. Technical drawings were only mentioned but not sufficiently described for inclusion in the table. It can be assumed that due to the general and thus presupposed understanding of technical drawings in the publications of recent years, a new description has been omitted, as well as the interest in further research has been limited. Nevertheless, comparing the product models included in this sample to the product models included in the literature reviews by Matthiesen et al. (2019b) and Weidmann et al. (2017) shows a substantial overlap of product models in embodiment design, confirming the appropriateness of this sample.

Considering the diversity of inputs and outputs displayed in Table 2, the types of categories derived are particularly relevant. The categories correspond to types of representations, specific models that serve as inputs themselves, as well as exchanged information. The representations, e.g., graph representation, overlap with the “type of depiction” from the state of research used by Matthiesen et al. (2019b) and Weidmann et al. (2017) for classification. Thus, the top-down approach of classification converges with the bottom-up approach of the grounded theory-based analysis. However, since the coding steps and thus also the classification are based on the similarities of the descriptions of the inputs and outputs, there is the possibility of an influence by classifications from the state of research. Besides, if the authors of the product models know the classifications and use them for their model descriptions, these classifications also flow into the bottom-up classification. In terms of categories corresponding to product models, the technical drawing stands out in particular, as it did not emerge as a product model. Although it was not sufficiently described in any of the publications, it can still be used as an input for several product models and consequently will find its way into the landscape of product models as a category. The categories of exchanged information, again, show similarities to the state of research in which the “type of information” was used in the classification. However, with the description of the information instead of the type, these are more specific than the classes in the state of research. Hence, the derived categories cover a wider range of abstraction levels in comparison to the state of research, from exchanged information to types of representation comparable to classes in the state of research.

Due to the varying abstraction levels, these categories may be used for different purposes in model research. The types of representation structure the state of research, enabling researchers to position newly developed models in comparison to existing models and possibly point out blank spaces. Specific product models demonstrate frequent inputs and outputs in the state of research, such as the Function Structure or the Technical Drawing. When developing a new product model, these categories can be used as a reference to see which models can already be created from frequently used inputs and outputs. As references, the models offer the possibility to adapt or learn from components of the model or the way they work with frequently used inputs and outputs.

Summarizing, Table 2 as the result of the literature analysis answers Research Question 1, “Which categories of similar inputs and outputs of product models in embodiment design can be identified in the literature?” The 17 categories identified across the inputs and outputs range from exchanged information to types of representation. As the types of representation converge with the classes known in the state of research, especially the categories on lower abstraction levels, exchanged information, and specific product models, supplement the state of research and provide assistance to model developers.

Consolidating the product models and categories resulted in the landscape of product models in embodiment design presented in Fig. 3. For readability reasons, only three sections are represented with labeled product models and categories. Product models are represented as empty black circles and categories as filled green squares. As hierarchies of categories do not represent links, they are represented as undirected dashed green edges. Links starting at product models are represented as solid gray edges, while links starting at categories are represented as dashed green edges. These links between two product models represent the possibility of using an output of the source model as input for the target model of the directed edge. The size of the nodes (circles and squares) represents their degree, the number of edges connected to a particular node. The network contains 69 nodes and 183 edges. The average degree of the nodes is 5.3. The maximum degree is 23 and is found for the 2D/3D CAD Model. The Gephi file underlying the network is available as supplementary material.

Examining the landscape from the perspective of the force-directed algorithm underlying the structuring of the network, three exemplary areas can be analyzed: (1) the arrangement consisting of three nodes at the bottom left, (2) the single node with edge to the center at the left corner, and (3) the model without an edge. The comparison of (1) and (2) shows that they are repelled from the center as expected. The greater distance of (1) from the center reflects the fact that the arrangement (1) is pushed further away from the center by the lack of attraction of the outer nodes. If we add (3) to the comparison, the smaller distance of (3) seems unjustified due to the lack of an attractive force. Accordingly, a weakness of the structure becomes apparent here, since the distance cannot be considered meaningful in this individual case and possibly resulted from the algorithm. Consequently, the structure and exemplary comparisons within the landscape correspond to the expectation as long as a determining edge is present. The position of the individual model cannot be considered meaningful.

The sections in Fig. 3 illustrate product models with varying amounts of links. For discussion, the network can be divided into three sectors based on the number of links of the product models: surrounding the Graph Representation category and the 2D/3D CAD Model, there is a center of densely linked models (e.g., Design Structure Matrix); the center flows into a margin of models (e.g., Function Means Tree) that are repeatedly but not as densely linked; located away from the center are the satellite sectors, with only weak (e.g., Pseudo Rigid Body Model) or the unlinked SAPPhIRE/-lite Model. The sectors and mentioned examples are pictured in Fig. 4.

The positioning of the 2D/3D CAD Model in or as the center based on the highest degree elevates it as a binding point even over the categories that emerged from the literature analysis (see Fig. 2). Thus, the use of a 2D/3D CAD Model provides the most possible links for the use with other models in embodiment design. As a result, it can serve as a linking point, as also described by Danjou et al. (2008), to switch between existing models with minimum effort, or as an access point to integrate newly developed models into the state of research. Although a 2D/3D CAD Model may contain additional information in addition to the geometry information that is always included, the required information for linked models may exceed this. If this information cannot be integrated into or transferred by the 2D/3D CAD Model, it must be transferred via another interface, possibly the designers themselves. For a more formal transfer, approaches such as category theory can be used as they offer more advanced expressiveness but require a translation into a mediation language (see Breiner et al. 2018; Wisnesky et al. 2017). As a result, the effort of switching via the 2D/3D CAD Model as a linking point can grow due to additional required information until the use of another interface is more efficient.

In the case of the models at the margin in particular, it is noticeable that in some cases, for example, the NVH Model or Pseudo Rigid Body Model, no link leads back to the center. Accordingly, the landscape of product models does not include a solution as to how the outputs and findings of these models are incorporated back into other models or, for that matter, the product design process, as the return is not described in the publications. A possible answer is provided in the publications by Jones et al. (2020) and Maier et al. (2014), who distinguish three types of models: physical, virtual, and cognitive/mental models. As cognitive/mental models are only conceived by the designer and do not exist in virtual or physical space, no cognitive/mental model is included in the landscape. This leads to the assumption that for the models without a link to the center, the output of the model flows back into the design process through a mental model or mental interpretation. In these cases, the contribution or support offered by the product models is potentially dependent on the individual abilities of the designer using the model. In this process, errors such as misinterpretations could occur and diminish the value of the product model. For clarification of how the outputs and insights of these models are integrated back into the product design process and how the potential for error can be reduced, additional research is necessary as this has been out of the scope of the literature review.

A noticeable feature of the satellite sector is the cluster of linked models in the upper right corner of the map. These models were all identified in the publication by He et al. (2013) and described including their links to each other. The positioning apart from the center creates the impression that in the introductory description of these models, more consideration was given to describing the links to one another in their own framework than to describing links to other product models at the state of research. This results in the circumstance that after entering this cluster, there is no known way back to the center available as described for product models in the margin in the previous paragraph. Within the satellite sector, this problem also exists for the completely unlinked SAPPhIRE/-lite Model. In the use case of this model described by Sarkar et al. (2017), it can be seen that this model was used as a data repository within an approach for synthesizing designs. Therefore, although the model does not link to any other product model in the map, it can be integrated into a design process through a suitable technical implementation. As a result, due to very few explicitly described links to models at the state of research, the models in the satellite sector are far from the center, and linking possibilities remain partly unknown.

Summarizing, the links identified in the landscape of product models in embodiment design (see Fig. 3) answer Research Question 2, “Which links between product models in embodiment design can be formed based on the inputs and outputs from the literature?” Based on the number of links, the landscape can be divided into the densely linked center, the more loosely linked margin, and the distant satellite sector. In the center, the 2D/3D CAD Model serves as a linking point between different models. In the margin and satellite sector, insufficiencies of the descriptions in the literature become apparent: In the margin, possibilities to return to the center for a continuous design process are not traceable, and in the satellite sector, very few described links result in unknown or missing linking possibilities.