## Abstract

Traditional design-for-manufacturability (DFM) strategies focus on efficiency and design simplification and tend to be too restrictive for optimization-based design methods; recent advances in manufacturing technologies have opened up many new and exciting design options, but it is necessary to have a wide design space in order to take advantage of these benefits. A simple but effective approach for restricting the design space to designs that are guaranteed to be manufacturable is needed. However, this should leave intact as much of the design space as possible. Work has been done in this area for some specific domains, but a general method for accomplishing this has not yet been refined. This article presents an exploration of this problem and a developed framework for mapping practical manufacturing knowledge into mathematical manufacturability constraints in mechanical design problem formulations. The steps for completing this mapping and the enforcing of the constraints are discussed and demonstrated. Three case studies (a milled heat exchanger fin, a 3-D printed topologically optimized beam, and a pulley requiring a hybrid additive–subtractive process for production) were completed to demonstrate the concepts; these included problem formulation, generation and enforcement of the manufacturability constraints, and fabrication of the resulting designs with and without explicit manufacturability constraints.

## 1 Introduction

Recent years have seen much advancement in the sophistication of mechanical design methods, both for the design of individual parts and integrated assemblies/systems. Many of these new techniques for design genesis focus on design automation, in which large areas of a given design space can be explored in a quick and efficient way and a large number of candidate designs can be compared quickly; some good examples are the development of generative design [1,2], topology optimization [3,4], candidate architecture analysis for mechanical systems [5,6], and machine learning for analyzing and selecting potential designs [7,8]. Such methods are steadily increasing in their level of maturity, but problems remain which restrict their usefulness in final product design. A particular concern that has not yet been fully addressed is the manufacturability of the final designs. While these and other advanced design methods can produce very sophisticated and nearly ideal parts in terms of performance and other metrics, the designs are often extremely complex and not easily manufacturable using conventional fabrication methods, including with additive manufacturing techniques [9–12]. This is the case for both macro-level user products and design problems at smaller scales (e.g., structured material design, micro-scale design features, and similar areas of interest). In this article, the term “mechanical design” refers to the design of mechanical devices, assemblies, and systems (including electrical devices with mechanical components such as motors and switches and the design of structured materials).

Realization of the final design is one of the most important considerations of a product lifecycle but it is often overlooked or deprioritized by designers, especially at the earlier stages and in requirements definition [13,14]. When the manufacturing process can be selected after the completion of the design, this can speed up the design process and reduce the number of design requirements. However, this presents the risk of mismatch between the final product and any available fabrication processes [9,13,15,16]. When this mismatch is encountered, the final design may need to be sent back for additional iterations (i.e., “repeated design”). This generally increases the cost and schedule risk significantly and may, in extreme cases, require revising the requirements from scratch after lessons learned from the manufacturing domain [17,18].

This risk may be low if the product is simple or is a member of an established product family that was shown to work well in the past. However, when the design is relatively complex (as is often the case in generated or geometrically optimized designs), the risk can be high that the mismatch will occur and that the product is completely nonmanufacturable with any available process [9,10,16]. Traditionally, this problem was addressed using design-for-manufacturability (DFM) principles (see Section 1 in the Supplemental Materials on the ASME Digital Collection); these principles were developed as a set of guidelines or rules to simplify the manufacturing requirements to the point where several processes could be feasible and the risk of mismatch is low [16,19–21]. According to Bralla [16] and Boothroyd [18], traditional DFM prioritizes simplicity in design and material selection and integration of simple expert intuition into decision-making processes. When applied to assemblies and systems, as many existing or commercially available components as possible should be used. In all cases, the tolerances should be as loose as possible.

In the past couple of decades, partially aided by emerging advanced design and production methods and a renewed focus on user-centered design, the traditional mass-production focus has been shifting to a mass-customization environment [22–24]. In such an environment with small-batch, high-value part production, it is vital for designers to have access to as much of the product design space as possible in order to produce useful designs for complex problems such as those encountered in the medical and aerospace industries [25–31]. Therefore, it is necessary for a DFM technique to be developed and used which guarantees (or at least better ensures) manufacturability while restricting the design space as little as possible; this will allow more rigorous problem formulations and will prevent missing potentially feasible regions of the design space. In this context, the *design space* is the set of feasible solutions for a given design problem which satisfy the constraints and objectives of the problem. The different points or regions of the design space may represent the outcomes of different design decisions or tradeoffs.

This new paradigm in DFM will only be possible if the general design restrictions imposed by traditional DFM are replaced with well-defined, manufacturing-process-driven design constraints which can be customized for each design problem or even each design feature. This would involve explicitly imposing the manufacturability constraints in the problem formulation or requirements definition, instead of simply checking manufacturability post-design or simplifying the design at the point of low-risk manufacturing. In this article, this concept is referred to as *minimally restrictive design-for-manufacturability* or MR-DFM. Significant work has been invested into developing concepts related to optimizing, automating, and improving DFM for some specific problems and domains (such as in Refs. [9,10,32–51]), but any kind of general process- and solver-independent method for capturing, defining, mapping, and enforcing useful manufacturing constraints does not yet exist in the literature. The “minimally restrictive” nature of the constraints generated refers only to their restrictiveness on the domains of the design variables, not to minimization (optimality) in the mathematical sense, and does not establish a “tolerance” on the design variables (the concept of tolerance allocation has been extensively explored [52–55] and is not the focus of this work).

The novelty and value of the work presented in this paper and its Supplemental Materials on the ASME Digital Collection is the dynamic nature of MR-DFM; in contrast to the more static or linear nature of traditional DFM, MR-DFM enables better capture and control of design information during the design phase of product or mechanical system development. This allows better integration of the manufacturing (and often material-based) information into design decisions and, therefore, allows the constraints to be as liberal as possible, providing the largest possible design space. While most implementations of MR-DFM will require a significant amount of problem-specific information, many general principles are defined and discussed in this work. Beginning with the baseline given here, designers, manufacturing engineers, DFM practitioners, and other design stakeholders can more easily apply the principles to the problem at hand in a useful and dynamic way. In addition, MR-DFM has a better potential to be automated based on historical or collected data than most traditional DFM approaches, which often rely mostly on expert intuition or simple general guidelines.

This article was organized into several major sections, beginning with conceptualization, development, and definitions of manufacturability constraints in Sec. 2. The final part of Sec. 2 is dedicated to demonstrating the relationship between the proposed method and classic DFM, as well as the novelty of this approach in the engineering literature and practice. From here, Sec. 3 examines the important properties of the constraints, particularly related to restrictiveness and dominance. A general framework for applying MR-DFM to realistic design problems is given in Sec. 4, followed by three detailed case studies in Sec. 5. After the case studies, some remarks on the practical implementation and automation of the proposed process are given in Sec. 6. The final part of the paper (Sec. 7) presents some conclusions and future work directions. This article is accompanied by a significant amount of the Supplemental Materials on the ASME Digital Collection.

## 2 Manufacturability Constraints: Definition, Generation, and Enforcement

A general framework for applying rigorous (i.e., repeatable, as complex or simple as needed, and with low uncertainty) manufacturability constraints in design practice requires development in three domains, namely (1) process and material behavior modeling, (2) mapping and problem formulation, and (3) practical implementation, including verification and validation strategies. Figure 1 shows some of the major technical knowledge areas within each domain. The first, the process and material modeling domain, is mostly concerned with mechanics and materials science and includes rigorous process and material modeling and definition. The second domain is concerned with collecting and mapping design knowledge and formulating useful and rigorous problem formulations. The third is the practical implementation, consisting mainly of design method (to solve the problems formulated in Domain 2), automation, and verification, validation, certification, and standards development for the design problem or problem family under consideration. Much previous work has been completed in the first and third domains, but very little in the second domain. Hence, there is a clear and specific need for an MR-DFM concept which is general and can serve as the connection from the process/material modeling and practical design application.

To address this need, this article focuses on the second domain to develop a framework for practical MR-DFM with a focus on mechanical design problems. As with classic DFM, this framework is agnostic to the design solution, manufacturing process or processes, and materials selected. This method is based on a simple, direct mapping of the “practical” knowledge from the selected manufacturing processes (and by extension the materials) into a set of mathematical manufacturability constraints which can be imposed in the design problem formulation or requirements, or only selected features. Whether these constraints actually restrict the design space (i.e., the set of all possible design solutions or options) more than the other design constraints in the problem may be established during the mapping process or after a preliminary solution is found (depending mainly on the complexity and form of the constraints) [56–58]. In an ideal situation, all of the nonconstant (“boundary”) constraints are linear or at least monotonic [56,59–61] and there is only one for each design variable over the entire space. Realistic problems are unlikely to be so simple, but effort should be made to impose the smallest possible number of these constraints and ensure that they are as minimally restrictive as possible while ensuring a significant increase of manufacturability under the conditions established by the stakeholders/designers. A fundamental tradeoff exists between manufacturing constraint fidelity (how accurately constraints quantify the boundary between manufacturable and nonmanufacturable designs) and complexity (number of constraints, constraint nonlinearity, computational evaluation expense [62–64]). This framework aids engineers in determining tradeoff decisions that are appropriate for a given design effort.

### 2.1 Mapping the Manufacturability Constraints.

Three related levels of analysis (separately from what is shown in Fig. 1) can be defined to map the practical knowledge from process mechanics into enforceable manufacturability constraints. These are the *manufacturing considerations* (basic mechanical knowledge about the processes and materials), *manufacturing constraints* (constraints on the process), and *manufacturability constraints* (design constraints imposed by the choice of the manufacturing process). These are further described and developed in the following sections and shown in relation to each other in Fig. 2. Some mathematical discussion of design problem formulation and descriptions of major manufacturing process families are covered in the Supplemental Materials on the ASME Digital Collection in Sections 2 and 3, respectively.

#### 2.1.1 Manufacturing Considerations.

Three things can be gained at this level: (1) process advantages (which expand the design space) and (2) process limitations (which restrict the design complexity), and best-practices or guidelines for proper use of the process. This is the broadest level of analysis and the applicability may be to an entire industry or family of manufacturing processes. Technical ownership at this level belongs to technicians and process engineers who have the most practical knowledge about manufacturing processes. The use of DFM generally implies that a specific process has been selected early in the design lifecycle; if the process is not yet specified, the manufacturing considerations level would be the most appropriate place to compare processes to aid in the selection process. An example of manufacturing considerations (for a machining process) is the requirement that all features in the design be (1) reachable by the cutting tools and (2) able to dissipate the friction heat and stress from the cutting without damaging the product.

#### 2.1.2 Manufacturing Constraints.

Mapped from the design considerations, these are natural constraints on the use of the process in question and are typically not changeable within a particular process. In most cases, incompatible manufacturing constraints necessitate the selection of a different manufacturing process to fabricate the design in question. The level of analysis is moderate in scope, being restricted to a single manufacturing process or several very similar processes within the same family. It should be noted that manufacturing constraints, by their nature, are more likely to be equality constraints and may take the form of discrete or combinatorial functions (such as a list of available machining tools for milling). The constraints generated here can be redundant, duplicate, or inactive, so it is important to consider some kind of refinement at this level in order to facilitate the formal constraint definition once mapped to the design itself. Following the example manufacturing considerations from the previous step, the equivalent manufacturing constraints would be (1) the quantification of the cutting tool range and (2) the limitation of machining to features strong and thick enough to withstand the associated heat and stress.

#### 2.1.3 Manufacturability Constraints.

These constraints are mapped from the manufacturing constraints and are constraints on the design, not on the process. There are different methods of enforcing these, depending on the nature of the problem, but in most cases they can be described mathematically and imposed onto the problem via mathematical constraints for typical mechanical design problems. Carrying on the example from the previous two steps, the manufacturability constraints on a design to be made using a machining process would be the (1) maximum complexity allowed considering the type of tool used and (2) the minimum feature thickness required for the machining loads. Note that the design space could be expanded through use of higher fidelity constraints, such as feature thickness constraints that depend on neighboring geometry instead of a uniform limit, but this trades off with the effort required to create and use constraints in the design problem.

#### 2.1.4 High-Level Mapping Scheme.

Figure 2 shows the conceptual mapping process, with each of the levels shown relative to each other.

First (Fig. 2(a)), the process advantages, disadvantages, and best-practices are analyzed and then mapped to manufacturing constraints (Fig. 2(b)). The needed domain-specific knowledge here is a fundamental understanding of the manufacturing process or processes that may be used.

These constraints are then subject to a refinement process (Fig. 2(b)), where they are identified, specified carefully, ranked in terms of importance, and combined when possible to reduce the number of them that need to be mapped to the design domain (Fig. 2(c)). Knowledge about the mechanics of manufacturing processes is needed for this step, but most of the technical data will carry over from the first step.

Finally, the manufacturing constraints are mapped onto the design domain, where the focus shifts from the process mechanics to the details of the design (Fig. 2(c)).

Once these constraints and the other problem constraints imposed by the stakeholders (material constraints, cost constraints, technology limitations, performance and reliability, etc.) are applied and enforced, the remaining design space is available for the designers to explore. This mapping method ensures that all of the important manufacturing process information is used in the design process, while restricting the design space as little as possible.

### 2.2 Constraint Representation and Uncertainty.

One of the major concerns remaining with the derived manufacturability constraints is the quality of representation and level of uncertainty for them. Classic DFM tends to provide a very flexible design representation, proxy comparison metrics, and a highly simplified system representation. Therefore, the classic DFM constraints tend to be very general and simple. On the other hand, MR-DFM gives a much more structured design representation, a higher-fidelity system representation, and more realistic metrics. Figure 3 shows the design formulation space for classic DFM and MR-DFM [56,65]. This is heavily dependent on the inputs from Domain 1 (Fig. 1) and uncertainties in the location and effect of the constraints certainly may come from uncertainties in process and problems modeling. This is especially true in the cases where expert intuition is used to determine the constraints.

To illustrate the concept, the constraints can be visualized like mathematical constraints within a level set, as shown in Fig. 4. In these examples, the more static and linear traditional DFM constraints would typically be simple bounds, while the more dynamic MR-DFM would allow the constraints to be more completely defined and possibly allow more liberal constraints for the same design problem (and hence a larger design space) [56–58,66–68,]. When defining classic DFM constraints on design problems, it is common to give very general guidelines such as “simplify as much as possible.” This simply limits general design complexity, often resulting in a feasible region similar to the one shown in Fig. 4(a) [66,67]. In the case where MR-DFM is used, more complex but tight constraints can be used, as shown in Fig. 4(b). These cases are far more simplified than most design problems and only represent level sets for two variables; however, they illustrate the benefit of using MR-DFM. Most product and system design problems have many local optima, so these represent what could happen around only one of the available solutions. More extensive discussion of this topic and a realistic example of how the constraints compare is given in Section 4 of the Supplemental Materials on the ASME Digital Collection.

Note that the MR-DFM constraints in Fig. 4(b) are “fuzzy” and less well-defined than those for classic DFM. This represents the uncertainties that come from simplifications in process and material modeling and possible errors when using expert intuition to derive the constraints. However, it should be noted that often the MR-DFM constraints still provide a significant benefit even if there is some uncertainty about their exact location and path. One of the expected outcomes from this kind of problem formulation is that a large set of constraints will be derived, most of which will likely not be active. However, at the formulation phase of the problem it is difficult to determine activity [56], especially in a way that is solution method agnostic. For problems which are convex, have a small number of variables, or have a clear and obvious solution within the original feasible domain, this is not difficult using monotonicity analysis. For very simple problems with 2–3 variables, visual analysis (such as plots or level sets) may be effective. However, these kinds of design problems most likely have large design domains and a number of local optima that need to be examined. In the case of multi-objective problems, this could become even more complicated and difficult to address.

### 2.3 Relationship to Classic Design-for-Manufacturability.

This proposed method of identifying and using manufacturing knowledge in design is distinct from (and potentially complementary to) classic DFM in several major ways. Specifically,

MR-DFM uses the basic DFM principles and modifies them using specific manufacturing process knowledge and problem formulation techniques from classical and modern optimal design methods.

In contrast to classic DFM (which generally relies on generic design rules which the designer or other stakeholders then apply to a problem), MR-DFM generates a clear, clean, screened/sorted set of constraints which can be directly integrated into a design problem; this point will be elaborated in later sections of this paper.

Both classical DFM and MR-DFM focus on constraint generation. However, classical DFM can also be used to drive objective function and solution method selection.

Using the classical definitions [9,16,17,19], DFM forces a general design simplification (typically as a heuristic requirement or tool for decision making after the initial design is completed). This tends to be unnecessarily restrictive for many modern design methods, especially those which can be formulated as a mathematical program, where the imposition of the constraints is a simple task once they are known. As shown in the previous section and in Figs. 3 and 4, MR-DFM works to create the smallest possible restriction on the design space for each individual design problem by identifying realistic constraints.

Due to the restrictiveness and focus on simplicity, classical DFM does not work well in conjunction with topology optimization and similar design automation strategies. Successful constraining of these designs to the manufacturable domain requires process-specific and carefully formulated constraints [9,34,35,38–42].

Due to the potential time and resource cost of applying MR-DFM fully, it is better suited for complex and computationally expensive design problems (e.g., many problems in the aerospace, automotive, and medical devices industries) where higher-value designs and products can better justify the extra cost. On the other hand, classical DFM is the best for simple problems and products which are well-established (e.g., consumer goods and part/product families with relatively simple designs).

Classical DFM relies primarily upon expert intuition and decision-making processes derived from it [13,16,18]. MR-DFM may be based on expert intuition or may be based on explicit or implicit mathematics, depending on the form and needs of the problems. This makes MR-DFM solution method independent (since it focused on solid formulations) and able to handle many different types (and mixes) of input data. It also helps remove a lot of dependency on the experience of the decision-maker; since MR-DFM is far more data- and model- driven than classical DFM is, it is far easier to apply effectively for regular engineers without decades of practical experience.

As shown in Fig. 3, the problem formulation decision space (i.e., a framework approach for conceptualizing design problem formulation decision options) for MR-DFM is very different from classical DFM since it is a distinct method with different goals. Specifically, classic DFM relies on a very simple predictive model (i.e., as much as possible, everything is primitive shapes and as simple as can be made), proxy comparison methods (necessary since DFM is usually very generic unless applied carefully to a specific problem), and a very flexible application (also since the principles are typically generic). On the other hand, MR-DFM relies on a higher-fidelity predictive model (i.e., the manufacturing knowledge), a more structured design representation, and more realistic comparison methods (since the constraints are directly mapped).

Once a set of data about a specific process is collected and mapped, this mapping can generally be used again for other designs with small or no modifications needed. This helps open the door to easier automation of the process.

See the Supplemental Materials on the ASME Digital Collection for additional information about these concepts and distinctions. This method is novel and relevant to the modern design world due to these major contributions. In addition, it is easy for practicing engineers and students to apply, as it is far more specific and knowledge-driven than classic DFM. A final major contribution that can be attributed to this method is that applying it does not require many years of developed expert intuition to apply, whereas most DFM methods do. It does require knowledge about a specific process (or family of processes) and their effects on materials during processing; this is typically a much smaller domain than the DFM requirement of intuitively understanding the complexities between design and manufacturing on a general level. Therefore, the designer does not need as much practical experience or developed judgement to successfully apply MR-DFM. In addition, an important future work direction in this area will be the use of digital twins to drive the constraints, removing most or all required experience on the part of the user and helping to better automate the process.

## 3 Constraint Restrictiveness and Dominance

Let *G* be the set of all possible manufacturability constraints (active and inactive) on the set of all possible design variables *X* (including those that may be defined as constants). Assuming that the potentially useful set of constraints $g\xaf\u2208G$ on the set of selected design variables $x\xaf\u2208X$ is defined and ordered after mapping (Fig. 2) and that the nonmanufacturability constraints are known, screening can begin. The exact screening process will depend on the nature of the problem, but the general goal is to evaluate each of the manufacturablity constraints for each design variable and determine if this constraint *g*_{i} ∈ *G* restricts the design space in any way for that variable *x*_{i} ∈ *X*. If so, it should be classified as a “restrictive” constraint and kept in the initial set. After a set of potentially useful manufacturability constraints is defined, they need to be screened (in a sequence as done in previous steps) for duplication, redundancy, and dominance. After all the screening steps, the manufacturability constraints could be classified into five categories:

*Restrictive*: The manufacturability constraint restricts the design space to feasible designs for a specific process [34]. These constraints are potentially active but their status will need to be established during the problem solution [56,69–71]. Inactive constraints can be removed from the model once activity can be tested mathematically. However, this does not affect the initial formulation and restrictive constraints are useful in estimating the feasible design space without unnecessarily restricting it. An example of a useful and restrictive constraint (which is very likely but not guaranteed to be mathematically active) could be the minimum feature thickness on a manufactured part to be designed for minimum mass.*Not useful/inactive*: The constraint is either nonrestrictive in the defined design space or it is obviously inactive for the problem. An example of a nonuseful constraint would be a maximum feature size constraint when the objective function seeks to minimize mass or size; in this case, an upper bound on the size is obviously not an active constraint and can be safely removed. Depending on the needs of the designer, this determination may be made based on expert intuition or may be easily automated. In cases of doubt, the designer may decline to reject the constraint and retain it in the restrictive category.*Duplicate*: The constraint applied is mathematically or effectively identical to one that was already imposed and is therefore not needed at all. This is relatively common for manufacturability problems, as lower or upper bound values for the design may be identical for several constraint sources (e.g., heat dissipation and minimum thickness-to-height ratio to withstand force of machining may produce identical lower bounds on wall thickness for a machined part).*Internally dominated*: The constraint was restrictive when added to the model but later found to be less restrictive to another manufacturability constraint (it is assumed here that the list of constraints will be examined in a sequence) and therefore is a dominated constraint and no longer necessary.*Externally dominated*: Identical to internally dominated but dominated by a nonmanufacturability constraint.

This process can be easily automated in many cases, with the possible exception of determining some of the rejected constraints for the “not useful” category since these may need to be determined by expert opinion. However, if the resources are available to retain questionable constraints until their activity can be established (to avoid mistakenly rejecting active constraints and opening up the design to manufacturing process mismatch), automation of the process can still be done, even if not as efficiently as with a smaller number of constraints. When possible, defining constraints as bounds or low-order functions will prevent this problem. In the case where a hybrid manufacturing process is selected or it is necessary to consider more than one process, it should be noted that constraints may be different in different stages of the process. Depending on the problem, this could be a significantly more difficult problem or may be leveraged to improve the design (e.g., if a sequential hybrid process [11] is used the order of the processes may affect the constraints significantly). Additional discussion, remarks, and considerations about the form and use of these constraints can be found in Section 5 of the Supplemental Materials on the ASME Digital Collection.

## 4 General MR-DFM Framework

### 4.1 High-Level Framework.

Combining the discussion from the previous two sections, a general framework for generating, mapping, and screening the set of MR-DFM constraints can be formulated for use within a general mechanical design process; this framework is shown in Fig. 5. The inputs consist of stakeholder preferences, the selection of a manufacturing process to use, and any needed nonmanufacturability constraints imposed or potentially imposed on the system. The inputs and outputs of the activity blocks are shown in Table 1.

The first step (Block 1) is to collect the manufacturing considerations (including both advantages and limitations, as well as any relevant best practices), which can then be directly translated to manufacturing constraints. The best method for converting these will often be problem-specific, but some general principles can be developed, as will be shown in the following sections. It is assumed that the stakeholders specify a manufacturing process or an acceptable set of processes in the design requirements, but if this is not the case, several processes can be compared at this step to see which are the least restrictive within the desired domain. It is necessary, however, to select a process or small set of processes before going any further.

Activity block | Input | Output |
---|---|---|

1 | Given from stakeholders | Raw set of manufacturing considerations |

2 | Raw set of manufacturing considerations | Ranked, ordered, and specified manufacturing constraints |

3 | Full set of manufacturing constraints | Raw set of manufacturability constraints |

4 | Raw set of manufacturability constraints | Set of restrictive or possibly restrictive manufacturability constraints |

5 | Set of restrictive or possibly restrictive manufacturability constraints | Screened set of restrictive or possibly restrictive constraints with duplicate, redundant, and dominated constraints removed |

Activity block | Input | Output |
---|---|---|

1 | Given from stakeholders | Raw set of manufacturing considerations |

2 | Raw set of manufacturing considerations | Ranked, ordered, and specified manufacturing constraints |

3 | Full set of manufacturing constraints | Raw set of manufacturability constraints |

4 | Raw set of manufacturability constraints | Set of restrictive or possibly restrictive manufacturability constraints |

5 | Set of restrictive or possibly restrictive manufacturability constraints | Screened set of restrictive or possibly restrictive constraints with duplicate, redundant, and dominated constraints removed |

### 4.2 Preliminary Constraint Identification and Screening.

Once a preliminary set of manufacturing constraints are defined, they should be subjected to an identify, specify, rank, and combine (I–S–R–C) process (Block 2). There are a variety of methods for accomplishing this from optimization, decision analysis, and systems engineering, depending on the specifics of the problem. However, a good general (widely applicable) method is discussed extensively in the NASA Systems Engineering Handbook [15] for collecting sets of system requirements and other constraints. In this approach, four steps are taken to ensure that the list of requirements or constraints are both as complete as possible and can be feasibly implemented. These steps are:

Identify all of the relevant requirements and constraints that should be considered. This includes both the manufacturing-related constraints and the ones from other sources. For example, a set of dimensional constraints on a part feature may come from both performance requirements and the minimal thickness needed to successfully machine the part.

As much as possible, specify all the requirements in the same terms and language to make them easier to compare against each other. For example, for all the identified dimensional constraints, they should be specified in the same units at the same temperature and use conditions to make them directly comparable.

Since in a large system or complex product, it is impossible to account for every single possible constraint or requirement [15,56,72,73] (hence the common use of factors of safety and design assumptions). Therefore, the set of desired or needed constraints or requirements should be ranked in terms of influence and impact. It may only be possible to account for some of them in the final design (this would of course be problem-specific) and having a ranked list would help identify the urgency of each one to the stakeholders. If only part of the set could realistically be accomplished, then a ranked list will make the decision of which to keep and which to reject more straightforward. The ranking may be done manually by the stakeholders or using some common decision analysis techniques (such as rank scoring or analytic hierarchy process (AHP) or other appropriate methods).

Combine as many of the constraints together as possible. It is very common that some of the requirements for a system or product will be redundant and can be combined to reduce the size of the constraint set. For example, if the minimum thickness of a part feature is specified as 3 mm by heat dissipation requirements during machining as well as by performance or interface requirements, only one of these constraints needs to be kept in the final set.

The output of this process is a set of manufacturing (and other) constraints which are well-defined, clearly specified, ranked in order of importance, and combined into the smallest practical number of constraints. This is then mapped onto a set of manufacturability constraints (Block 3).

### 4.3 Manufacturing Constraints and Manufacturability Constraints Mapping.

Revisiting earlier definitions, *manufacturability constraints* are limitations on the capability of the manufacturing process itself, while *manufacturability constraints* are on the design that will be manufactured. The conversion of these will take place within Block 3 inFig. 5. The exact steps needed to complete this conversion will be problem-specific in most cases and will require either some measure of expert intuition or historical design data. However, some general principles can be identified for most mechanical design problems.

Most general design constraints on final parts can be divided into three categories: (1) dimension (e.g., length or height), (2) form (e.g., straightness or roundness), and (3) surface finish [74–76].

Therefore, most manufacturability constraints will be related to a dimension, form, or required surface finish.

Generally, the manufacturing constraints will be the process constraints that drive the dimension, form, or surface finish of the part.

For example, heat dissipation needed, vibration and compliance, and applied force during processing (whether from direct force of a tool or force from shrinkage and residual stresses) are all process aspects that affect the final part in terms of dimension, form, and surface finish. This applies regardless of the manufacturing process used, as these general principles apply to machining, casting, additive manufacturing processes, and most others. Therefore, a general approach for converting manufacturing constraints into manufacturability constraints will be to:

Consider the identified manufacturing constraints and determine the influence on dimension, form, or surface finish each would have on a manufactured part.

Each effect on dimension, form, or surface finish from each manufacturing constraint will have an equivalent manufacturability constraint driven by it.

The potential set of manufacturability constraints will be this set of equivalent constraints, condensed for clarity and to remove obviously redundant or useless constraints.

### 4.4 Detailed Constraint Screening.

After the potential set of manufacturability constraints is defined (set *C*1), the constraints are then individually screened to determine if they are restrictive or obviously not restrictive. Uncertain constraints at this point should be retained in the set. The collection of restrictive or potentially restrictive constraints then make up set *C*2 ⊆ *C*1 (Block 5). This set is then screened as a set for duplicate and dominated constraints, which are rejected from the set. Note that this includes comparison with known nonmanufacturability constraints as well.

### 4.5 Final Constraint Set.

The set *C*_{final} should consist of the smallest possible number of manufacturability constraints, which can then be effectively imposed into the design problem, restricting the design space only just enough to ensure manufacturability (i.e., “minimally restrictive”) and leaving as much of it as possible for the designer to explore. Mathematical activity is not clearly established in this set, as the problem has only been formulated and not yet solved. After building the set of useful and possibly useful constraints, they can be fed into a solution method (classic optimization techniques, topology optimization, procedure/rule-based design, etc.) and the initial solution should give needed information about the activity of the constraints. At this point, the formulation can be finalized and a set of feasible designs generated for the stakeholders to examine.

### 4.6 Data Types and Automation.

The data that are mapped from the manufacturing knowledge to the design problem could take a variety of forms and the mapping may be explicit, implicit, or manual depending on the needs and formulation of the problem. This may depend on the processes involved in the form of the design problems. In the most common case, it is anticipated that the design problem will involve geometric constraints related to manufacturability; in this case, the mapping could be based on primitive shapes (squares, circles, triangles, etc.), nodes in a mesh, points or lines on a shell model, a toolpath (i.e., g-code), or similar. The form of this will be one of the decisions made by the stakeholders when using MR-DFM very early in the design lifecycle.

## 5 Case Studies

Due to space limitations, this section provides brief summaries of three extensive case studies meant to explore and demonstrate the concepts developed in this article. Section 6 in the Supplemental Materials on the ASME Digital Collection provides the full details for the case studies.

### 5.1 Case 1: Milled Aluminum Heat Exchanger Fin.

This case study explores a design problem using a single well-defined manufacturing process as the basis for the manufacturability constraints. A heat exchanger fin in a natural-convection environment must be machined from 6061 aluminum under a given set of performance and temperature parameters. The design objective was to minimize total fin volume. A figure describing the problem, the list of assumptions, and modeling parameters, and the full description is provided in Section 6.6.1 of the Supplemental Materials on the ASME Digital Collection. A summary of the mapping process from manufacturing considerations to useful manufacturability constraints is shown in Table 2.

Mfg considerations | Resulting Mfg constraints | Manufacturability constraints |
---|---|---|

Heat of machining | Heat of machining Q = 40 W | t ≥ 1.17 mm for cutting force |

Machining force | Cutting force F = 200 N | t ≥ 1.27 mm for heat during machining |

Compliance/vibration | (also accounted for in cutting force constraint) | |

Nonmanufacturability constraints | t ≥ 0.85 mm for thermal performance | |

$t\u2265negligible(O\u2264\mu m)$ for buckling |

Mfg considerations | Resulting Mfg constraints | Manufacturability constraints |
---|---|---|

Heat of machining | Heat of machining Q = 40 W | t ≥ 1.17 mm for cutting force |

Machining force | Cutting force F = 200 N | t ≥ 1.27 mm for heat during machining |

Compliance/vibration | (also accounted for in cutting force constraint) | |

Nonmanufacturability constraints | t ≥ 0.85 mm for thermal performance | |

$t\u2265negligible(O\u2264\mu m)$ for buckling |

Figures 6(a) and 6(b) show the final designs (with and without manufacturability constraints, respectively), while Figs. 6(c) and 6(d) present the final manufactured designs. It can be clearly observed that the thin (0.85 mm) fin has numerous manufacturing defects (Fig. 6(e)), while the one with the manufacturability constraint was successfully fabricated. In this unconstrained (thin) fin, three major defects were observed: (1) (Note (A)) the fin thickness was inconsistent, with the bottom of the fin being the nominal thickness and the top being 20% thinner, (2) (Note (B)) the top corner was chipped by the end mill during a cutting pass due to its flexibility under cutting force, and (3) (Note (C)) the top of the fin displayed a jagged, almost “scalloped” surface finish. It was observed that all of these defects were cause by the fin deflecting under the load from the end mill, with the top thinning implying that the deflection was at least $10%$ of the fin thickness, 100 times the allowable deflection.

None of the fin surface defects or thinning or extreme heating were observed in the machining of the 1.27 mm fin under identical manufacturing conditions, showing that the imposed constraints were restrictive and effective in ensuring manufacturability. Note that the cost of manufacturability and accuracy was a 33% increase in the mass of the fin. However, this study showed that this was the least-restrictive constraint under which the fin can be effectively fabricated using the specified conditions and process assumptions.

It should be noted that this particular case study could have been completed using a variety of design methods, including classic DFM and MR-DFM (as shown). If classic DFM had been used, it is likely that the original thickness based on performance would have been calculated and a factor of safety applied. Using a typical factor of safety of 2.0 for a problem like this, the fin thickness would have been about 1.7 mm thick—in this case, it would both perform correctly and be manufacturable, but the design would be inferior (i.e., be heavier) than the one produced using MR-DFM. Since MR-DFM provided more detailed constraints directly derived from the manufacturing process, a thinner fin that was both manufacturable and functional was able to be produced.

### 5.2 Case 2: FDM/SLA TO Cantilever Beam.

This case study used a design problem that must consider the constraints from two manufacturing processes from the same family, unlike Case Study 1 which used only a single process. In this problem, a simple symmetric cantilever beam was to be designed via topology optimization (TO) for minimum mass and minimum compliance (i.e., maximum stiffness). The final design was required to be symmetric along the length and thickness directions. The full set of a schematic of the problem, parameters, assumptions, and methods is provided in Section 6.2.1. in the Supplemental Materials on the ASME Digital Collection. Similar to Case Study 1, the mapping for the problem is shown in Table 3.

Mfg considerations | Resulting Mfg constraints | Manufacturability constraints |
---|---|---|

Shell/infill print pattern | Two shells + infill for FDM | Minimum length scale for FDM is 2 mm |

FDM = 0.4 mm/line | One shells + infill for SLA | Minimum length scale for SLA is 1 mm |

SLA = 0.2 mm/line | ||

Nonmanufacturability constraints | Mass fraction $\u224850%$ | |

Factor of safety ≥1 for TO | ||

Beam design symmetry |

Mfg considerations | Resulting Mfg constraints | Manufacturability constraints |
---|---|---|

Shell/infill print pattern | Two shells + infill for FDM | Minimum length scale for FDM is 2 mm |

FDM = 0.4 mm/line | One shells + infill for SLA | Minimum length scale for SLA is 1 mm |

SLA = 0.2 mm/line | ||

Nonmanufacturability constraints | Mass fraction $\u224850%$ | |

Factor of safety ≥1 for TO | ||

Beam design symmetry |

The Pareto method was used to generate the topology under the given conditions and constraints, using a voxel count of 500,000. Printing orientation was not expected to have a large impact on the final product, so the parts were printed wall-out (from the base up) to replicate cantilever beams; the orientation of the layers can be seen on inspection of Fig. 7.

Figure 7(c) shows an attempt to fabricate the 1 mm length scale solution using FDM, resulting in numerous major manufacturing defects and missing features (highlighted yellow regions) since the features were too small for the process to accurately create with the 0.4 mm bead size. In contrast, the FDM fabrication was successful for the 2 mm length scale solution (Fig. 7(d)). Both geometries were successfully produced by the SLA process, as shown in Figs. 7(e) and 7(f). The Pareto software did not allow the use of a zero-thickness minimum feature size constraint, so it was not possible to generate a topology that was not manufacturable using SLA with this method at the small part size involved. However, the manufacturability constraints imposed clearly had a major impact on the part geometry and were clearly restrictive at least for the FDM fabrication.

It is clear from the results of this case study that the constraints were necessary for the designs to be manufacturable. Traditional DFM methods would have provided some guidelines for setting up the general design problem (cantilever beam under load), but would have been very difficult to integrate directly with the topology optimization problem. Since the formulation of the problem requires the manufacturability constraints to be known before a solution is attempted, it is was necessary to use MR-DFM to find them. The TO problem could have been set up using some general guidelines (e.g., a general minimum feature thickness), but it would not have been easily customized for each of the processes or have accounted for specific constraints for each case.

### 5.3 Case 3: Hybrid AM/SM PLA Pulley.

The final case study presented is an updated solution approach to the generator pulley problem presented by Patterson and Allison [11]. In this problem, a belt-drive pulley for a generator was designed and required to be manufactured from PLA plastic using a combined additive-subtractive hybrid manufacturing process. Therefore, this is also a two-process problem but with the processes in different families. A detailed problem diagram, along with the assumptions, modeling parameters, and sample manufacturing details are given in Section 6.3.1 in the Supplemental Materials on the ASME Digital Collection. The constraint mapping for Case Study 3 is shown in Table 4.

Mfg considerations | Resulting Mfg constraints | Manufacturability constraints |
---|---|---|

Hybrid process | Two shells + infill for FDM | Minimum length scale for FDM is 2 mm |

AM + SM in sequence | Print orientation | Minimum shell count = 5 |

FDM + lathe | Shell thickness | Minimum roof layer count = 5 |

Anisotropic material | Part base/roof thickness | Minimum base layer count = 5 |

Work holding | Maximum size ≤ lathe chuck jaw diameter | |

Nonmanufacturability constraints | Factor of safety ≥ 1for TO | |

Pulley design symmetry |

Mfg considerations | Resulting Mfg constraints | Manufacturability constraints |
---|---|---|

Hybrid process | Two shells + infill for FDM | Minimum length scale for FDM is 2 mm |

AM + SM in sequence | Print orientation | Minimum shell count = 5 |

FDM + lathe | Shell thickness | Minimum roof layer count = 5 |

Anisotropic material | Part base/roof thickness | Minimum base layer count = 5 |

Work holding | Maximum size ≤ lathe chuck jaw diameter | |

Nonmanufacturability constraints | Factor of safety ≥ 1for TO | |

Pulley design symmetry |

Based on the requirements, a series of TO solutions were found (Pareto TO, three million voxels), with the solution selected being the minimum-mass solution found before no more feasible designs were found (Fig. 8(a)); the final design (Fig. 8(b)) had a mass fraction of 0.66 and a compliance of about 5.00 mm/kN, well within the maximum allowed. Figure 8(c) shows the printed pulley before lathe operations, while Fig. 8(d) is the successful finished pulley. To show that the manufacturability constraints are restrictive, a second pulley was produced using the same TO solution, but which used only two shells; the activity of the minimum feature size constraint was established in Case Study 2, so only the shell constraint was tested here. Figure 8(e) illustrates the results for the two-shell pulley, which was clearly a failure. Several areas of surface tearing, layer delamination, and plastic melting were observed, as highlighted in the figure.

As can be reasonably expected when using a hybrid process, the effects from both the additive and subtractive processes influenced the final outcome of the design. In this case, the design problem (re-designing the pulley web under two dissimilar processes) likely would have been too complex to use traditional DFM principles to accomplish without a large amount of expert intuition and even guessing. It is clear that it was necessary to find the true manufacturability constraints at the beginning of the problem formulation for the design to be successful. As seen in Fig. 8(a), a number of feasible designs could have been produced, all of them manufacturable. If a traditional DFM process had been followed to complete this task, it is most likely that only a single design would have been accomplished and the wed re-design would have been based on a primitive (triangle, square, or similar) or minimal surface (which can be easily calculated in general cases) instead of a topologically optimal, manufacturable, 3-D design which is the best within the goals and constraints of the problem.

## 6 Remarks on Inputs and Practical Implementation

### 6.1 Process and Material Modeling.

One of the most important mantras, reinforced throughout this article, MR-DFM is that the quality of the constraints depends heavily on the quality of the inputs. In design, it is very common to use simplified representations of process and material models in order to reduce computational cost. While this may be useful in many cases, this could be the source of significant error in the formulation of MR-DFM constraints and should be avoided when possible. It is best to use direct experimental results to generate the constraints whenever possible via a fitted model or random variable generation. In both cases, the estimated uncertainty and potential errors may be calculated and documented. With these, the amount of “fuzziness” (Fig. 4(b)) can be estimated to ensure that the calculated constraints are actually useful for the design at hand. An important area of future research should be to find how much uncertainty is allowable in these constraints before the method fails to provide a design superior to one generated using classic DFM.

### 6.2 Constraint Automation and Design Tradeoffs.

An obvious consideration with the MR-DFM method is that it can rapidly become unmanageable for a single designer or decision-maker as the problem size increases. The case studies presented are reasonably complex for academic studies but when dozens of features are to be considered or system constraints must be considered (i.e., interfaces, tolerances, and reliability) it can become impossible to implement manually. Therefore, the mapping and enforcement should be automated as far as possible. This should be relatively easy to do for most manufacturability problems, as many DFM and MR-DFM constraints can be reasonably defined as boundaries and simple polynomial constraints, with some binary and discrete functions on occasion. However, as the method and computational methods improve in capacity, more accurate models of interactions between geometry and considerations such as temperature, stress, and deformation during manufacturing can be used. These higher-fidelity constraints can expand the design space even more (in many cases) but only are worth pursuing where the increase in design quality is worth the investment of time and resources.

Some practical considerations for automating the constraint generation are as follows:

For explorations and for mapping new processes and process–material combinations, the mapping will likely involve a significant amount of manual or mathematical work. However, the mapping should be stable and consistent for each process so the mappings can be cataloged.

These catalogs of mappings and constraints can be used as the starting place for automating constraints.

Once a set of mappings are completed (e.g., for the ten most common manufacturing processes), design automation studies can begin. Similarities and differences can be identified and some mappings combined or simplified in the ideal case.

It is very important at the beginning of the design process to identify or specify the data type (mathematical functions, points in a mesh, etc.) to be used in the mapping. Consistent application of this will aid in automating the constraints.

A future work direction may be to see if a new data type is needed to map these constraints, but this may not be necessary for general mechanical design problems. In formulating most mechanical design problems, most of the relevant constraints will be geometric and will be in the form of distances, thicknesses, norms between points, and similar.

A very helpful technology that can support MR-DFM is digital twins of manufacturing processes. When properly developed and verified, this may be able to replace actual manufacturing processes and make the identification of manufacturing considerations much easier.

From the other perspective, MR-DFM and other constraint/formulation methods may help drive the development and refinement of these digital twins. It would be especially useful if digital twin models could be developed and formulated so that they provide the manufacturing considerations and constraints as default outputs.

Clearly, there will be design tradeoffs when using this method as there are with classic FDM and other methods. The most important considerations for most problems would be as follows:

Cost, both in terms of labor during initial mappings or explorations and in terms of computational expense once automated.

Different classes of manufacturing processes will have drastically different costs associated with using them in design. For example, simple geometric constraints based on a die casting tool will be far less expensive (both in terms of design cost and in terms of production cost, assuming enough are to be made to justify tooling costs) than designing a free-form lattice structure with different location densities.

As previously discussed, MR-DFM is far more focused and based on technical knowledge instead of expert intuition. For simple problems and those with few or no active manufacturablity constraints, the flexibility offered by using expert-intuition-driven DFM could provide better design outcomes.

Continuing from the last point, not every mechanical design problem will have active manufacturability constraints and discovering this after using MR-DFM could waste time and resources. Design problems which are convex in nature are less likely to require many (or any) manufacturability constraints due to the fact that only a single global solution is possible.

### 6.3 Expert Intuition.

For any DFM or MR-DFM method, some level of expert intuition will be necessary. For classic DFM, several excellent guides are available but still require interpretation and application to the problem at hand. With MR-DFM, the ability to capture process knowledge (manufacturing considerations) and experience is one of the strengths of the presented method; this allows more realistic constraints and a wider design space. However, there will always be some uncertainty in the collected information. To minimize this, the design team should be careful to collect only the most reliable information and make an effort to communicate well with the expert, who will often be a technician and not a researcher or engineer. Respect for the experience of the expert (who will not necessarily have advanced degrees), the inclusion of the expert in design decisions, and a real effort at partnership with the design team will be vital for collecting the best quality design knowledge. When practical, it would also be very useful for at least some members of the design team to have some degree of practical knowledge and experience with the selected manufacturing process or processes. Whether this involved hiring designers with hands-on experience (of which there are very few in most industries), providing additional coursework, or some kind of hands-on training will depend on the type of problem being solved and the desired outcomes.

### 6.4 Design Method Selection.

The presented framework is design-method agnostic and can provide useful constraints for a wide variety of problems. However, the design method may affect the formulation of the problem and this should be expected for any design method outside of classic optimization methods. Therefore, care should be taken that the final model (objective function and constraints) is as mathematically rigorous and clear as possible to avoid any mistakes or misunderstandings during any needed reformulation. All variables should be distinct and clearly and consistently defined in a way that is easily understandable by someone not in the original design team.

### 6.5 Verification, Validation, and Certification.

As with all design methods (all three domains in Fig. 1), the final step and ultimate success metric is the completion of the verification, validation, and certification (VV&C) process. While this is largely independent of the specific constraint formulation, it may be necessary to have the constraints very clearly specified for VV&C. Therefore, it is vital that everything be well-documented during the entire process and, when possible, a calculation of the expected uncertainty and error in the placement of the constraints.

## 7 Conclusions and Future Work

In this work, a conceptual framework and approach was developed for generating and imposing minimally restrictive manufacturability constraints (MR-DFM) for mechanical design problems. The technique is based on mapping of practical manufacturing knowledge into enforceable manufacturability constraints; these can be screened and eliminated as needed to ensure that the imposed constraints restrict the design space only enough to guarantee (or at least greatly improve) manufacturability while leaving as much as possible intact for design exploration.

This new approach to DFM is useful and provides value beyond the state-of-the-art due to its dynamic nature. Unlike more static and linear traditional DFM, it better allows the more effective capture and use of practical manufacturing information which can be problem- or process-specific. This enables better consideration of manufacturing information into product and system design and helps provide a wide design space. MR-DFM is also much less dependent on expert judgement and, as discussed in the previous section, can be automated (using models, digital twins, historical data, and similar) for many important applications. The method was shown to be straightforward and useful in three case studies, all of which were not manufacturable using the specified manufacturing conditions without the additional constraints.

Note that the case studies made some simplifying assumptions, clearly stated within each problem, which may impact the validity of the case studies if the conditions were changed. The reader should keep in mind these assumptions when following the case studies and applying the approach to other (similar) problems. The impact of using simplifying assumptions for generating manufacturability constraints is the risk of false positives (the generated constraints are not sufficient to guarantee manufacturability) and false negatives (the constraints exclude more of the design space than actually required); care should be taken to avoid this when applying the method.

The case studies presented covered three different types of mechanical design problems under manufacturability: (1) design under a single process, (2) design under two different but related processes, and (3) design under a hybrid of two dissimilar processes. In terms of design problems,

The design focus of Case Study 1 was to find the least restrictive constraint on the fin thickness such that it could both meet its design requirements and be manufacturable. This involved modeling and testing several aspects of the design (heat transfer, beam bending, and buckling) using a common design variable; the constraint found was the least-restrictive one which was applicable to all of the problems encountered containing the design variable.

Case Study 2 explored the formulation of an algorithmic design problem (topology optimization) under two possible manufacturing processes, requiring constraints which were valid for the entire problem. This also captured the potential scenario in which uncertainty exists in the selection of the process and so the design must allow some flexibility.

Finally, Case Study 3 explored the complexity of a hybrid manufacturing problem, where a larger number of manufacturability constraints needed to be considered even for a just two design variables. This problem also clearly demonstrated the way that the constraints can be quickly tested and eliminated when not useful.

This method is applicable for any mechanical design problems in which the physical design constraints (part architecture and performance) can be defined and understood in terms of its manufacturing processes in some way. Future work will focus on refinement of the method and extension of it to other design domains, such as design of tailored materials, on the automation of the process for larger problems, tolerance allocation along with the constraints, method sensitivity, the use of simulations and experiments to establish the limits of problem complexity, and examining the impact of the various assumptions on the proposed framework.

## Acknowledgment

The authors thank William Bernstein (National Institute of Standards and Technology), Sherri Messimer (University of Alabama in Huntsville), Katherine Matlack (University of Illinois), Dan Herber (Colorado State University), Yong Hoon Lee (University of Illinois), Danny Lohan (Toyota Research Institute of North America), Krishnan Suresh (University of Wisconsin-Madison), and Niao He (ETH Zurich) for their discussion and comments on the presented work at various stages of development. All opinions and conclusions offered in this article are solely those of the authors.

## Funding Data

No external funding was used to complete this project or fund its publication.

## Conflict of Interest

This article does not include research in which human participants were involved. Informed consent not applicable.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.