A conceptual design and modeling framework for integrated additive manufacturing

A conceptual design and modeling framework for integrated additive

manufacturing

Hossein Mokhtarian^1,2 Hossein.mokhtarian@tut.fi

ASME Membership Number: 100691564 Eric Coatanéa¹

Eric.coatanea@tut.fi Henri Paris²

henri.paris@g-scop.inpg.fr Mouhamadou Mansour Mbow²

mouhamadou-mansour.mbow@grenoble-inp.fr Franck Pourroy²

franck.pourroy@g-scop.inpg.fr Philippe René Marin²

Philippe.marin@g-scop.inpg.fr Jorma Vihinen¹

jorma.vihinen@tut.fi Asko Ellman¹

asko.ellman@tut.fi

1 Tampere University of Technology, Mechanical Engineering and Industrial Systems.

Tampere University of Technology, MEI Laboratory, P.O. Box: 589, 33101 Tampere, Finland.

2 Univ. Grenoble Alpes, CNRS, G-SCOP laboratory

46 Avenue Félix Viallet 38031 GRENOBLE Cedex 1, France

ABSTRACT

Modeling and simulation for additive manufacturing (AM) is commonly used in industry. Nevertheless, a central issue remaining is the integration of different models focusing on different objectives and targeting different levels of details. The objective of this work is to increase the prediction capability of characteristics and performances of additively manufactured parts and to co-design parts and processes. The article contributes to this field of research by integrating part’s performance model and additive technology process model into a single early integrated model. The paper uses the Dimensional Analysis Conceptual Modeling (DACM) Framework in an AM perspective to generate causal graphs integrating the AM equipment and the part to be printed. DACM offers the possibility of integrating existing knowledge in the model. The Framework supported by a computer tool produces a set of governing equations representing the relationships among the influencing variables of the integrated model. The systematic identification of the weaknesses and contradictions in the system and qualitative simulation of the system are some of the potential uses of the model. Ultimately, it is a way to create better designs of machines and parts, to control and qualify the manufacturing process, and to control 3D printing processes. The DACM Framework is tested on two cases of a 3D printer using the Fused Filament Fabrication Powder Bed Fusion. The analysis, applied to the global system formed of the 3D printer and the part, illustrates the existence of contradictions. The analysis supports the early redesign of both parts and AM process (equipment) and later optimization of the control parameters.

Keywords: DACM Framework, Fused Filament Fabrication, Liquefier, 3D part, Design for Additive Manufacturing, colored causal graph, integrated model

INTRODUCTION

The manufacturing and production of parts with repeatable desired quality is a crucial step toward the integration of Additive Manufacturing (AM) technologies in manufacturing processes. AM processes involve complex multi-physics phenomena [1]

[2]. According to Witherell et al., modeling and simulation have a significant role in

extracting knowledge about different sophisticated AM technologies [3]. This paper aims to present a design and modeling Framework for AM. It enables the combined modeling and qualitative simulation of parts and AM processes. The Framework named

‘Dimensional Analysis Conceptual Design (DACM) Framework’ has been tested and validated on different case studies and for different usages [4], [5] [6]. This article proposes some extensions to the original Framework, which focuses on systems behavior [5][7], to make it more suitable for Design for Additive Manufacturing. The article is organized as follows. First, background information is provided to analyze the research problem. In the methodology section, the theories and methods integrated into the DACM Framework are briefly explained. The case study section applies DACM approach on two case studies. The conclusion discusses future research work.

BACKGROUND

Design for Manufacturability (DFM) is the process of tailoring the product design by anticipating manufacturing issues during the early stages of the design process. DFM approaches aim at improving the quality, performance, reliability, and profitability of a product by reducing the development time and cost [8]. Design For Manufacturing and Assembly (DFMA) methods systematically involve identifying available manufacturing and assembly techniques and understanding their associated capabilities and limitations [9].

AM technologies are reducing the manufacturing constraints and increasing the freedom of design while providing unique capabilities for shape, material, and functional complexity [10]. On the other hand, AM technologies are relatively poor for criteria such as surface roughness, dimensional accuracy, or thermal dissipation. However, AM

capabilities offer a unique opportunity to reconsider DFM methods. An analysis of AM impact on the Design Theory and Methodology (DTM) indicates that conventional DTMs are not designed to fully benefit from the unique capabilities of AM [11]. In their critical review, Yang et al. categorize the design methods related to AM into three categories [11]. In the first category, researchers provide guidelines to improve designs by taking advantage of AM capabilities. ASTM released design guidelines [12] such as the ASTM WK38342 with a broad scope. Other publications are more case-specific [13], [14]. For instance, Klahn et al. present guidelines for additively manufactured snap-fit joints [8].

The second category modifies conventional DTM for AM to improve the design process in an AM context [15]. The third category contains specific approaches for Design for Additive Manufacturing (DFAM) [11]. This category identifies the commonality in manufacturing capabilities, the constraint and limitations of the available AM, and proposes design methodologies suitable for AM [16] [17] [18]. Dinar et al. proposed an ontology-based DFAM to guide designers in understanding the limitations and capabilities of various AM technologies [19]. Other guidelines are more specific to topology optimization [20] and lattice structures [21].

Other researchers have proposed different classifications for AM technologies. Kurth proposes a classification based on working material and shape build techniques [22], while others used the physical state of raw material and equipment as the classification criteria [23]. In a related research study, Williams et al. utilize the concept of a function decomposition criterion to find similarities between AM technologies and to categorize them [24]. They decomposed the primary function of converting raw material into connected solid primitives in an additive manner into several limited lists of sub-functions

and considered different possible solutions for fulfilling the sub-function using a Zwicky morphological matrix [25]. This approach is a functional Framework for conceptual DFAM [24]. William et al. stated that future development in AM is becoming application-driven and it either involves new AM processes that specialize in the specific type of parts or tailors AM processes to adapt the manufacturing of the desired parts [24]. In the review of AM design theory and methodology by Yang et al. [11], the authors outline that most of DFAM methodologies are in the phase of semantic representation and provide limited support to use the full potential of AM technologies. Existing DFAM methods have some benefits, but higher details and precision are required to fully support the AM process. It is necessary to move in the direction of computable conceptual models for AM allowing a level of insight not attainable with current approaches. The American Society of Mechanical Engineers (ASME) [26] states that a conceptual model enables: 1) assumptions and determination of the components that will be included in a computable model; 2) the approach to the behavior of the modeling system or phenomenon; 3) the elimination of unimportant features, and 4) the selection of interface and boundary types.

A computable Framework for DFAM complies with those characteristics. The analysis of the literature above motivates the authors to propose and apply DACM Framework in AM field. DACM contributes to DFAM by integrating the models describing AM machine design, process parameters settings and the part design, and simultaneously analyzing the impacts of those models on the desired system performance and quality of the final part. Existing machine design, process parameter settings and part design are highly interconnected and should be studied simultaneously. Due to the limitation of the machine, tailoring process parameter settings (optimization) cannot always enhance part

quality. Identification of those limitations and constraints leads to establishing the design rules and guidelines to tailor the part design for existing AM system. DACM goes further and detects the root causes and system weaknesses causing those design limitations. The systems weaknesses and contradictions guide designers to enhance system behavior and part quality by tailoring the machine design. In summary, DACM investigates three aspects of DFAM: tailoring part design by establishing optimized design rules per process, tailoring process parameters, and tailoring AM machine design. The Framework is presented in the methodology section of the article.

METHODOLOGY

The Dimensional Analysis Conceptual Modeling (DACM) Framework was initially developed as a specification and verification approach for complex systems [27] [28].

DACM provides an approach integrating several theories, concepts, and methodologies related to engineering design, modeling, and simulation. The approach provides modeling and simulation capabilities for functional representations. DACM can be an efficient approach to the creation of surrogate models. The modeling starts with a designation of the system border and definition of model objectives. Function modeling is used to represent the sequence of functions taking place in the system of interest. Those sequences of functions generate a system’s behavior. In DACM, the variables describing the functions are then defined to establish causal relationships among them. DACM transforms the initial functional model into a Generic functional model formed around a limited set of fundamental functions [29] and uses the causal rules extracted from different modelling theories such as Bond Graph [30], [31]. Dimensional analysis is applied

nodes of the graph to form behavioral equations. A color pattern is applied to the different variables to highlight their different design nature. The primary result of this modeling is a colored hypergraph and a list of governing equations of the system for the system of interest. The model can be used for qualitative or quantitative simulations and to search for contradictions. Those uses are explored in the article. Figure 1 visualizes the different steps in a modified DACM Framework proposed in this article. DACM is used in an incremental innovation context in this article. Using the separation between knowledge and concepts of the CK theory [32], the DACM process navigates between those spaces, iteratively accumulating knowledge about the model. The DACM modeling process ends when a computable model of the system of interest is available at the required level of detail.

_Figure 1 _

Interested readers can find the theoretical details of the DACM Framework in the authors’ previously published paper [7]. The current research paper is the extension of authors’ previous efforts, applied to additive manufacturing.

In this work, Generic functions represented by Bond Graph organs are used as an intermediate level between the classical functional models and the final causal graphs. To facilitate the systematic assignment of variables to the Generic functional representation, regardless of the energy domain, the variables are classified into five generalized categories; Flow, Effort, Momentum, Displacement, and Connecting [30] [33]. The mathematical relationship between generic variables describes how those variables relate to each other. In each energy domain, ‘Displacement’ is the result of the integration of the ‘Flow’ over time. Equation (1) indicates that the integration of the electrical current

(I) over time is equal to the charge (q). The charge is equivalent to the ‘Displacement’ (see Table 1). The generalized ‘Momentum’ is the result of the integration of ‘Effort’ over time.

Flux linkage (known as ‘Momentum’) is defined as (2), where U (known as ‘Effort´) is the potential difference between two terminals of an electrical element.

∫ I. 𝑑𝑡 = 𝑞 (1)

∫ U. 𝑑𝑡 = 𝜆 (2)

The ‘Connecting’ variables proposed by Coatanéa [33] cover the other variables that are not in the four above-mentioned categories and are used to describe the material properties, geometry dimensions, etc. For instance, consider Ohm’s law in Equation (3), which indicates the relation between the voltage and current in a conductor. The connecting variable (R), known as the resistance, creates the relation between ‘Effort’ and

‘Flow’.

𝑈 = 𝐼𝑅 (3)

Figure 2 visualizes these relations, where the state variables (Momentum, Connecting, and Displacement) are located inside the elements, and the power variables are located outside.

_Figure 2_

Table 1 illustrates the mapping between the different types of energy and the names of specific generic variables. Complementary information on the dimensions of the variables is also available in the table.

_Table 1_

The sequence of functions in the functional model provides an initial insight into global causality. Mapping the functions to the Generic functional elements enables the extraction of the causality among the variables characterizing those functions [7]. Table 2 summarizes the causal rules in the DACM approach.

_Table 2_

Figure 3 represents a causal extraction algorithm. First, the algorithm checks if a Generic functional organ is defined for each functional box. Then the algorithm explores each functional box from its source to the end of the model to verify that there is no conflict in the coherence of the Generic functional representation in terms of causality.

Finally, according to the categories of assigned variables (Table 1) and using the causal rules (Table 2), the cause-effect relationships between variables are established.

_Figure 3_

The DACM Framework colors the causal graph generated in the previous step. The variables are classified into four main classes (i.e. colors):

- Exogenous variables (shown in black) are outside the system border and part of the environment of the system. They cannot be modified by the designer but are imposed on the system.

- Independent design variables (shown in green) are not influenced by any other variables in the system. Their values can be modified by designers;

- Dependent design variables (shown in blue) are influenced by other variables such as exogenous and independent variables. It is more difficult to modify and control the dependent variables;

- Performance variables (shown in red) are the objective variables. They usually belong to the category of dependent variables as well. They are selected by the designers to evaluate the performance of a system.

To exemplify the methodology explained above, the modeling procedure is applied to model the RLC circuit and extract the associated colored causal graph. Figure 4 illustrates the modeling procedure and extracted colored causal graph of the RLC case study. ‘e’ and

‘f’ stand for effort and flow respectively. Table 3 summarizes the variables assigned to the Generic functional representation of the RLC circuit. We assume to model an existing RLC circuit with a variable resistor that enables the tuning functionality. Thus, variables ‘L’ and

‘C’ are shown as exogenous variables because they have been fixed and cannot be modified and ‘R’ is considered as a dependent design variable. Those are initial design choices and the nature of the color pattern. Ultimately, it facilitates the comparison of models since the designers’ viewpoints are embedded in the colors of a causal graph.

_Figure 4_

_Table 3_

The equations in the junctions are in the form of algebraic summations and equality between variables [34]. The template for this kind of equation for junctions is shown in Table 2. The other equations are calculated using Dimensional Analysis (DA) [35]. DA proposes an approach for reducing the complexity of modeling physical problems and supports knowledge deduction by discovering possible mathematical relationships among variables. The variables are described in fundamental dimensions (i.e., Mass, Length, time, Temperature, etc.). Dimensional homogeneity is the most familiar principle of dimensional analysis theory and can be verified by checking the dimensions on both

sides of an equation. For example, in Newton’s law (F=m.a), the fundamental dimension of force [MLT^-2] is identical to the multiplication of the dimensions of mass [M] and acceleration [LT^-2] on the other side of the equation. The other widely-used theory in DA is the Vashy-Buckingham ∏ theorem, stated and proved by Vashy and Buckingham [36].

The theorem identifies the number of independent dimensionless numbers (Pi-number) characterizing a given physical problem. The method offers a way to simplify the complex problems by grouping the variables into dimensionless primitives. Every law which takes the form yo=f(x1, x2, x3, …, xn) can take the alternative form shown in Equation (4), where

∏i is the dimensionless product for the variable xi.

П₀= 𝑓(П₁, П₂, … , П_𝑛) (4)

Equation (4) is the final form of the dimensional analysis and is the consequence of the Vashy-Buckingham theorem for the variable xi which takes the form shown in (5).

𝜋_𝑦= 𝑦_𝑖. 𝑥_𝑗^𝛼𝑖𝑗. 𝑥_𝑙^𝛼𝑖𝑙. 𝑥_𝑚^𝛼𝑖𝑚 (5)

where ‘x’ are called the repeating variables, yi the performance variable, and αij are the exponents. Equation (5) presents the dimensionless form of Reusable Modeling Primitives (RMPs) used to develop the Framework in the current research. Reynolds number and the Froude number are the examples of RMPs. A backward propagation algorithm is used in this work for detecting contradictions in the system of interest, taking benefit of the mathematical machinery developed by Bhaskar and Nigam [37]. A dimensionless group can be expressed in the manner shown in Equation (6).

𝑦_𝑖 = 𝜋_𝑦. 𝑥_𝑗^{−𝛼𝑖𝑗}. 𝑥_𝑙^{−𝛼𝑖𝑙}. 𝑥_𝑚^{−𝛼𝑖𝑚} (6)

The partial derivative between the variables yi and the variable xj can be written as follows

𝜕𝑦_𝑖

𝜕𝑥_𝑗= −𝛼𝑖𝑗.𝑦_𝑖 𝑥_𝑗

(7)

From Equation (7), the sign of the derivative (^𝜕𝑦_𝜕𝑥^𝑖

𝑗) can be determined by simply verifying the sign of the term (−𝛼_𝑖𝑗.^𝑦𝑖

𝑥𝑗 ). The machinery provides a way for propagating qualitative optimization objectives (i.e. Maximizing or Minimizing) in a causal network. A dimensionless number, derived from Ohm’s law, is presented in Equation (8) with the objective to maximize Current (I). Equations (9) and (10) are the partial derivatives of this equation. From the sign of the derivatives, we can conclude that to increase the current (I) we need to increase the voltage (U) and reduce the resistance (R).

П_𝐼 = 𝐼. 𝑈⁻¹. 𝑅¹ (8)

𝜕𝐼

𝜕𝑈= +1.𝐼 𝑈

(9)

𝜕𝐼

𝜕𝑅= −1.𝐼 𝑅

(10)

This approach is used to propagate backward the qualitative objective(s) in a causal graph. Contradictions appear if variables have to be minimized and maximized simultaneously. Contradictions can be detected automatically using a prototype software tool. The principle has been published in other papers by the authors [4] [5] [6] [7]. The DACM systematic modeling procedure is demonstrated on an RLC circuit. The Figure below represents the analogy between three different fields, such as the electrical, hydraulic and thermal domains. The resulting causal graphs share similarities because of the functional model similarities. The principles apply to AM since each AM process includes the combination of different interacting energy domains. In the FFF liquefier case, the model considered three energy domains – thermal, mechanical and hydraulic.

_Figure 5_

DFAM aims at maximizing the manufacturability of parts dedicated to AM.

Traditionally, the DFAM method includes tools such as topology optimization, design for lattices or cellular structures, multi-material design, mass customization and part support design. In a design process, those tools are usually employed during the detailed design phase [38], which is quite late in the development process. It means that manufacturing considerations are also taken into account also quite late in the process. The manufacturing concerns should be considered earlier, ideally at the conceptual design stage or embodiment design stage. DACM can play this role to enrich the application domain of the DFAM methods. The immediate positive impacts will be a reduction of the redesign tasks resulting from late considerations of additive manufacturing aspects.

Additionally, a better specification of the part can be expected because of the earlier integration of manufacturing concerns. The aim is to expend the toolbox of DFAM using the DACM. The DACM Framework cannot provide a similar level of detail than FEM.

Nevertheless, it can provide an early guidance to compare and select the most appropriate design concepts guiding the later topology optimization effort. For example, Figure 6 represents the formalism of the DACM approach, using the Concept-Knowledge separation [32]. This example demonstrates how the knowledge of a certain domain can be encoded using the DACM Framework. The DACM Framework generates a causal graph, representing the model of a beam subjected to an effort F1. The variables selected to model the beam are presented using nodes and colors. The choice reflects the nature of the requirements faced by the system to be designed. The machinery developed by Bhaskar and Nigam [37] and presented above in the article is applied here for backward

propagation. Each node with incoming edges is associated with a behavioral law. In the example shown in Figure 6, the functional model, the initial design concepts and the set of variables together allow the creation of a causal graph. In the beam subjected to a deflection (d) under a bending effort F1, a behavioral law is created using the matrix in Figure 6. The final law obtained is similar to a generic bending law. The model can be used to analyze a design concept and gives a direction for the optimization of the parameters and redesign. The cantilever example presented in case study 2 provides details of the use of DACM in a DFAM context. This case study section uses DACM as a tool for Design For Additive Manufacturing (DFAM).

_Figure 6_

CASE STUDY

This section includes two separate case studies. The first case study focuses on the concurrent modeling of a 3D printer using the Fused Filament Fabrication (FFF) technology and the part to be printed. In the first case study, a simple geometry is considered with few design requirements. The model is describing the behavior of the machine extruder and then enriching the model by integrating the model of the part. The second case study focuses on characterizing the part’s geometry and support structure to reduce the curling defect in the Powder Bed Fusion technology. The case studies demonstrate the capabilities of the DACM Framework as a support tool for the DFAM method.

Case study 1: FFF Process + Part

Fused Filament Fabrication (FFF) is an AM process that builds up parts from thermoplastic polymer (i.e., ABS or PLA). The filament is fed into a liquefier which is heated by an electrical heater. The molten polymer is extruded from a nozzle located on the head of the liquefier. At the same time, the extruder is moved to deposit the melted material at the appropriate coordinates, according to a predefined pattern. Once the deposition of the material on a layer is completed, either the extruder is incremented up, or the platform is lowered down to a controlled height so that the next layer can be deposited. The process is continued layer by layer until the desired complete geometry is reached. The dimensional accuracy, bonding quality, and final mechanical properties of the part are dependent on the melted polymer flow rate and its temperature [39], [40].

They are directly affected by the filament feed rate and the temperature of the heat source. The temperature is usually measured via a non-contact temperature sensor. A temperature regulator adjusts the heat command according to the expected target temperature. Studies show that the liquefier has complex behavior because of (but not limited to): 1. the uneven distribution of the input heat flow; 2. the gradual change in the physical state of the filament inside the liquefier, and 3. the complexity of modeling the heat transfer in the liquefier [41] [42]. Most studies focus on the effect of the process parameters on parts’ quality but few on the behavior of the liquefier itself. Bellini et al.

[42] investigate the behavior of the liquefier by developing a mathematical model of the physical phenomenon taking place in the liquefier and comparing the results with experimental data. Other authors applied the Finite Element Method (FEM) to predict the melt front in the liquefier [43], the pressure drop of the melt flow and the temperature

gradient of the melt [44]. Jerez-Mesa et al. [45] analyzed the performance and thermal behavior of a RepRap liquefier. They investigated the influence of the airflow generated by the cooling fan on the heat transfer condition using FEM. The thermal behavior of the liquefier is thus dependent not only on the geometry of the liquefier and the heat transfer conditions, but also on the input process parameters (e.g., material feed rate). Literature indicates that models have been produced but with a limited scope. The article aims at developing a model with broader scope for the usage at conceptual design stage.

Figure 7 shows the geometry of the liquefier. The thermal energy, provided from the source, flows to the neighboring materials and finally melts the filament. A portion of the initial thermal energy is used to melt the polymer (i.e., PLA, with a melting temperature of 200-220^oC), and the other portion is dissipated by flowing to the aluminum part on the top of the extruder or by convection to the ambient air. A portion of the thermal energy is stored in the material, and the other portion is transferred (via conduction or convection) to the next interface material. The functions, ‘To store thermal energy’ and

‘To resist the heat transfer’, appear in each material. In order to analyze the sequence of the heat transfer in the liquefier, the detection of the different materials and interfaces is needed. Figure 7 depicts the thermal interfaces between the different blocks of materials in the liquefier. The direction of the arrows in Figure 7 indicates the direction of the heat flow in the liquefier.

_Figure 7_

The following modelling phase is attributing functions to each interface and material and consequently transforming Figure 7 into a functional representation. The function ‘To connect’ is attributed to each interface. The function ‘To store thermal energy’ is used to

characterize the thermal energy capacity and the function ‘To resist the heat transfer’ is used to give resistance to the heat transfer of each material. Figure 8 illustrates this transformation (T1) for materials, interfaces, and sources.

_Figure 8_

The functional mapping presented in the previous publication [7],transforms the functional model into the Generic functional representation (T2) in Figure 8. The transformation T2 is followed by a transformation T3 associating generic functions with Bond graph elements. In thermal energy problems, the effort junction (0) is attributed to each function ‘To connect’ to describe the temperature at each interface between materials. At the same time, variables are associated with the generic functions and the Bond graph organs. The resistive elements (R) characterize the conduction and the convection heat transfer. The coefficient of conduction (k) and the average distance (d) between two interfaces define the conduction. The coefficient of convection (h) and exchange surface (S) define the convection in Resistive elements. The capacitive elements are defined by the mass and specific heat capacity (Cp). The machine feeds the filament inside the liquefier. The solid part of the filament acts as a plunger to push the melted polymer to the nozzle tip. This part is modeled with analogy to hydraulic energy domain, using the following procedure. First, an effort junction (0) is attributed to each pressure change and a flow junction (1) is used to connect the effort junctions. Inertial and resistive organs are attached to the flow junctions (1), which cause the pressure drop in the liquefier. The key variables associated with those organs are the viscosity and the volume of the melted polymer to characterize the resistive element. Fluid inertia is the main variable defining the inertial element. The fluid inertia is related with the density and

length of the hydraulic tube, and to the cross-section of the tube. The set of causality rules is applied based on the type of assigned variables (see Table 3 and Figure 2).

Figure 8 represents the procedure for generating a causal graph for the heat exchanger of the FFF liquefier. The next step is to produce a combined global causal graph of the liquefier and the part to be manufactured (in Figure 9). The aim is to analyze the impact of the process parameters on the part quality and consequently contribute to design for additively manufactured parts. This is done by collecting existing knowledge about the FFF process and integrating into a generic causal graph.

The temperature of the cylinder wall melts the polymer filament inside the liquefier.

The melting of the moving filament happens gradually, and there is no real point at which the state of the polymer changes from solid to liquid. Nevertheless, an approximation of the location of the melt front is helpful in connecting the model of ‘flow of thermal energy’ and the model of ‘flow of material’ together. According to Yardimci et al., the location of the melting point is influenced by the filament feed rate (u), thermal diffusivity (α), and a dimensionless temperature (θ) [43]. Dimensionless temperature, shown in (12), is the ratio between melt temperature (Tm), wall temperature (Tw), and the initial temperature (Ti) of the filament. Thermal conductivity (k), specific heat capacity (Cp), and density (ρ) characterize the thermal diffusivity (α). Equations (11) to (14) summarize the relations between the influencing variables for calculating the melt front location. Based on the relations (11) to (14), a partial causal graph encoding this knowledge is constructed.

𝑌_{𝑚𝑒𝑙𝑡} = 𝑓(𝑄̇, 𝛼, 𝜃 ) (11)

𝜃 =𝑇_𝑚− 𝑇_𝑤 𝑇_𝑖− 𝑇_𝑤

(12)

𝛼 = 𝑓(𝑘, 𝐶_𝑝, 𝜌) (13)

𝑄̇ = 𝑢. 𝑑² (14)

After extracting the influencing variables and the causal graph from the literature review [43], the equations of the problem are constructed using the dimensional analysis if an equation is not existing or by reusing existing equations. For example, equation (15) presents the dimensionless product for the melt front location.

𝜋_{𝑌𝑚𝑒𝑙𝑡}= 𝑌_{𝑚𝑒𝑙𝑡}. 𝑄̇⁻¹. 𝑘. 𝐶_𝑝⁻¹. 𝜌⁻¹. 𝜃 (15)

Up to this stage, the modelling process has focused on the behavior of the liquefier, but it is necessary to integrate the model of the part in order to analyze the relationship between the geometrical features of the part and the variables of the process. The key characteristics of the desired part presented in Figure 7 are the specified flatness tolerance (tv), relatively small radius in the corners (R) and uniform bonding quality between layers. The printing process deposits the material on a predefined path, layer upon layer. Any curve-like path is created by incrementally changing the coordinate or changing the direction of movement of the nozzle. The movement of the nozzle in the X- and Y-directions and the time of the movement in each direction forms the causality for nozzle travel speed. The geometrical flatness tolerance is affected by the variation in material deposited per length (Δ(M/L)). The travel velocity of the nozzle and melted material flow rate are the causes of the amount of material deposited per length. The bonding quality, also known as coalescence, plays an essential role in the part’s final mechanical properties. One of the key variables in determining the bonding quality is the

temperature of the fused filament [40]; minimizing the variation in the temperature of the fused filament supports uniform bonding quality on the part. Figure 9 shows the generated partial causal graph integrating the liquefier model and the part model. It is composed of four zones. The two zones called ‘Thermal Energy Flow’ and ‘Flow of Material’ are created using the Generic functional representation of the liquefier (Figure 8) and the associated causality algorithm and elementary causal rules from Figure 3 and Table 2. Some cause-effect relationships such as ‘melt front’ location are directly drawn from equation (15). Finally, the ‘Part’ zone represents the characteristics of the part in the form of measurable variables. The selected performance criteria are the variation in the mass deposited per length (ΔM/L) and corner radius (R). The other performance variable is the variation of the temperature of the deposited material already present in the ‘Thermal Energy Flow’ zone of the graph. The selected performance criteria are:

- Minimizing the flatness defects by minimizing the variation in material deposited per length: (Min Δ(M/L))

- Reducing the fillet radius: (Min R)

- Minimizing the variation in the temperature of the melted material to enhance the uniformity of the bonding quality: (Min ΔT)

- Increasing the printing speed by maximizing the nozzle velocity: (Max V)

The backward propagation algorithm described in the methodology section is used to propagate those performance criteria in the graph. This is leading to the identification of the existence of possible contradictions and weaknesses in the system. The results of the backward propagations are shown in Figure 9. The two contradictions discovered are highlighted in yellow circles.

_Figure 9 _

As an example, the equations below represent how the causal graph and associated Pi-numbers are used to conduct the backward propagation for the objective of minimizing the variation in the mass per length (Min Δ(M/L)). A ‘+’ sign in the partial derivative indicates that the variable considered in the partial derivative varies in the same direction as the variable considered in the objective. Otherwise, in the case of a negative sign, the variable varies in the opposite direction. The analysis of the equations below indicate that a minimization of the variation in the mass per length (Min Δ(M/L)) is obtained by minimizing the variation in the volume flow rate (Min 𝑄₆̇ ). On the other side, to increase the printing speed (Maximizing V) and be able to print smaller radius in the corners there is a need to maximize the variation in the volume flow rate (Max 𝑄₆̇ ). A contradiction is discovered: the variation of the volume flow rate needs to be maximized and minimized simultaneously.

𝜋𝑀

⁄𝐿 = (𝑀

𝐿) . 𝑉. 𝑀̇⁻¹ (16)

𝜋_𝑀̇ = 𝑀̇. 𝑄̇⁻¹. 𝜌⁻¹ (17)

Objective: Min (Δ𝑴 𝑳⁄ )

𝜕𝑀

⁄𝐿

𝜕_𝑀̇ = +1.

𝑀⁄𝐿 𝑀̇

 Min (Δ𝑀̇)

𝜕𝑀

⁄𝐿

𝜕_𝑉 = −1.

𝑀⁄𝐿 𝑉

 Max (ΔV)

𝜕_𝑀̇

𝜕_𝜌 = +1.𝑀̇

𝜌

 Min (ρ): Only possible by changing material

𝜕_𝑀̇

𝜕_𝑄6̇ = +1.𝑀̇

𝑄₆̇

 Min (Δ𝑄₆̇ )

Identifying the contradictions at the early stages of design guides the designer toward the most valuable and required part design and process improvements. Table 4 summarizes the contradictions found in the case study and presents possible ways to overcome them by modifying the part design, dimension and tolerances (tailoring part design), optimizing process parameter settings (tailoring process), or by adjusting the process technology (tailoring AM machine). The objective (Min Δ(M/L)) can be attained by minimizing the variation in the polymer flow rate and consequently by being able to maintain a constant temperature in the nozzle. Technically, maintaining a constant temperature in the nozzle can be obtained by efficiently controling of the temperature produced by the heating resistor. The existing fan on the equipment and flowing air on the nozzle and part can also play the role of maintaining a constant temperature by providing forced convection when needed. The objective (Min R), on the contrary, implies being able to vary the polymer flow rate leaving the nozzle extremely quickly. Removing these contradictions implies two possible directions: Redesigning the nozzle or tailoring the part’s geometry. The first is to redesign the nozzle to precisely control the temperature. One solution is to move the temperature sensor of the FFF machine inside the nozzle. The second is to use a valve at the outlet of the nozzle that enables quick opening and closing. Indeed, the thermal inertia of the nozzle does not permit modifying the polymer flow rate quickly enough, even with the support of forced convection provided by the fan. Those elements can support a redesign of the nozzle in the FFF machine. If this redesign option is not selected, then modification of the radius and acceptance of variation in the mass flow rate and consequently an increase in the flatness

tolerance on the part is an alternative option, adapting the part design to the capabilities of the equipment.

_Table 4_

To evaluate this qualitative analysis, test parts were printed. Excluding the starting point, the test part has two round corners (R1=1mm and R2=2mm) and a sharp corner.

The initial printed parts demonstrate the predicted defects around corners and poor bonding quality near the starting point (see Figure 10). The defects appeared around all the corners except the corner with the radius of 2 millimetres. The contradictions found in the causal graph (see Figure 9) demonstrate that acting on the variation of polymer volumetric flow rate in nozzle outlet (Δ𝑄₆̇ ) can remove or reduce the defect. Variable (Δ𝑄₆̇ ) is the difference of volumetric flow rate before and after the radius. Nevertheless, volumetric flow rate (𝑄̇) is a dependent variable. The slicer software (here Repetier) calculates the filament feed rate (u) and consequently volumetric flow rate (𝑄̇) according to the input value of nozzle travel speed (V). On the other hand, the inspection of the intial printed part illustartes that the excess of deposited material after the radius causes the defect (Figure 10). Therefore, we have reduced the filament flow rate the in the G- code generated by slicer. Furthermore by adjusting temperature, we managed to remove the defect around the corner with R=1mm and improve the bonding quality near the starting point (tailoring process settings). In the sharp corner’s zone, the defect was reduced but never removed. Moreover, the minimum achievable radius was 0.6 millimetres. This limitation led to a DFAM rule for exisiting machine setup. Therefore, the part design should be modified and consider a corner radius superior to 0.6 mm (tailoring

part design). This is the limitation induced by the available FFF machine design and can be improved by the redesign of the machine (tailoring AM machine). The concurrent consideration of the part and process models in DACM anticipated the system’s weaknesses for fulfilling design requirements at the early design stages and proposed several feasible solutions. The experimental tests verify those weaknesses and propose redesigning the part (considering R>0.6 mm).

_Figure 10_

Case study 2: Curling defect in powder bed fusion process

The Curling defect is one of the recurring defects in metal Powder Bed Fusion (PBF) technologies. It occurs in the areas of the parts that are not supported by material:

‘overhang surfaces’. The excessive heat energy input (overheating) leads to a cumulative thermal constraint on the part being processed. The cumulative thermal constraint finally results in the deflection of the overhang surfaces upward. According to Béraud et al., reducing the curling defect is an important issue, since it is the result of internal stresses and possibly results in residual stress on the final part [46]. Several studies investigate different ways to reduce the curling defect. Béraud et al. investigated the effect of trajectories in Electron Beam Melting (EBM) and proposed new trajectories to reduce the curling effect [46]. Considering an efficient support structure is another way to reduce this defect. Tounsi and Vignat experimentally investigated the effect of different support structure shapes on reducing the defect [47]. The support structure is used for two main reasons: to dissipate excessive heat and to resist distortion by increasing the inertia of the part. Design and manufacturing strategies generate contradictory effects; for

instance, applying a more dense support structure to minimize the curling effect increases the manufacturing time, material cost and difficulties involved in removing the supports.

This case study aims to demonstrate that the DACM Framework can be seen as a new tool for DFAM in the conceptual and embodiment design stage of parts and supports. The design space in the functional model shown in Figure 11 is divided into three domains:

cyclic functions of the AM process, useful functions of the support structure and non- desired functions. The behavioral laws and the key factors of the model are collected from the literature. They are not created in this case study by using the DACM algorithm generator of behavior laws. By having the functional model and the variables of the problem, it is possible to generate the causal graph.

The cyclic functional model of the AM process simply describes the sequence of functions required to build one layer of the part upon the previous layer. The induced heat energy melts the powder, and the excess energy is transferred to the supports to be dissipated. The functional model of the support structure includes two main functionalities of the supports, the function ‘to dissipate’ heat energy, which is defined by the convection variables, and the function ‘to increase inertia’ which contains the variables characterizing the geometry of the supports and material density. The non- desired functional model characterizes the generation of a thermal constraint that results in the creation of the bending moment and the function ‘to resist’ against the deflection.

Figure 11 illustrates the connection between these three functional models with their assigned variables. Table 5 represents the variables with their associated dimensions.

_Table 5_

The associated causal graph and governing equations are extracted with the same procedures explained in the section above. The design variables, material properties variables and performance variables are shown in green, black and red, respectively. This choice is often imposed by the requirements, but often the design freedom allows the status of variables to be modified. The governing equations are the following:

𝜋_𝛥𝑇 = 𝛥𝑇. ℎ. 2𝑛. 𝑆. 𝑞⁻¹ (18)

𝜋𝑀_𝑠 = 𝑀𝑠. 𝑛⁻¹. 𝑡⁻¹. 𝑆⁻¹. 𝜌⁻¹ (19)

𝜋𝜎= 𝜎. 𝐸⁻¹. 𝛼⁻¹. 𝛥𝑇⁻¹ (20)

𝜋_𝑀 = 2. 𝑀. 𝜎⁻¹. 𝑤⁻¹. 𝑏⁻¹ (21)

𝜋_𝛿 = 𝛿. 𝐸. 𝐼_𝐺𝑧. 𝑀⁻¹. 𝐿⁻² (22)

𝐼_𝐺𝑧 =(𝐻 + 𝑏)³. 𝑤 12

(23)

The objectives we seek in this case study are minimizing the curling defect (δ), minimizing the total mass of the support structure (Ms) and possibly modifying the design of the part to support those objectives. The backward propagations of both objectives are shown in different colors in Figure 11. The two backward propagations generated several design contradictions related to the dimensions of the beam and supports (i.e., w, H, t), the total surface of the support structure (S) and the number of elements (n) in the support structure. The DACM Framework applied to DFAM consists of removing or reducing the contradictions by providing innovative design solutions [48] [7]. The

‘segmentation’ of the support structure suggests increasing the number of elements (Maximizing n) while keeping the thickness of elements (t) as small as possible. This reduces the contradictions for ‘S’ and ‘t’ and maintains the capability of the support structure to reduce ‘ΔT’. In the same way, the desire to minimize the mass of the supports

(Ms), ‘S’ or ‘H’ needs to be minimized. This minimization can be achieved by varying the value of ‘t’ in the support structure. There is no need to keep a constant value for thickness; ‘t’ can be smaller at the bottom and bigger in the zone of the attachment to the cantilever. Ideally, minimizing the need for the support structure means removing it totally. In the context of the cantilever beam, this can possibly be achieved by rotating the cantilever structure to print it on the side or to print the long beam part with the dimension ‘L’ first, followed by the prismatic part of height ‘H’ and width ‘w’. The root cause of the problem is the temperature difference between layers (ΔT). An option is to reduce the energy input (q). The study of Béraud et al. confirms the effect of the heat energy input on curling defects [46]. They investigated the beam trajectory as an important parameter influencing the energy input [46]. The ‘porous material’ principle suggests using a porous structure, lattice structure, or topology-optimized structure for the part. Another possibility will be to modify the beam shape to integrate the deflection and the distortions of the shape (‘Preliminary anti-action’).

_Figure 11_

CONCLUSION

This paper attempts to tackle the issue of integrated modeling in Additive Manufacturing. The key objective was to present a conceptual design and modeling Framework enabling this integration and consequently co-design of parts and processes.

For this purpose, the authors proposed the DACM Framework. The Framework can integrate different models in the form of a colored causal graph and generate governing equations among the influencing variables of the system. The integrated model can be

simulated using qualitative or quantitative simulation. The Framework provides an approach to identifying the contradictions and weaknesses in the multi-objective system of interest (i.e. the combination of the manufacturing process and the part to be printed).

This paper presents and applies DACM as a DFAM tool. It is used to integrate models describing AM machine design, process parameters settings and part design into a single model. DACM qualitatively analyzed the desired system performance and the final part’s quality. The analysis of the system highlighted the existence of contradictions and system weaknesses causing defects and design limitations. The contradictions detected in the case study were used as reasoning and improvement support for designers. DACM investigated three aspects of DFAM: tailoring part design by establishing optimized design rules per process, tailoring process parameters, and tailoring AM machine design. The quantitative simulation is the purpose of a future article where the causal graph model and the equations are transformed into a neural network, a Bayesian network or a system dynamic model.

ACKNOWLEDGMENTS

The main author of the article, Hossein Mokhtarian, has been sponsored by the HYBRAM project from the Finnish Funding Agency for Innovation, TEKES (project number: 230553).

NOMENCLATURE

T Temperature

𝑞̇ Heat flow rate

q Heat Energy

k Coefficient of Conduction h Coefficient of Convection

Cp Specific heat capacity

M Mass

d Conduction Length S Heat Exchange Surface

P Pressure

PM Pressure Momentum

If Fluid Inertia FC Fluid Capacitance

µ Viscosity

τ Torque

ω Angular Velocity A Filament Cross-section

F Force

V Translational nozzle velocity

L Length

ρ Density

𝑄̇ Volume flow rate

Q Volume

𝑀̇ Mass flow rate

Ymelt Location of melt front θ Dimensionless temperature α Thermal expansion

E Young's modulus

Ms Mass of support structure t Thickness of support n Number of supports

δ Deflection height (curling defect)

M Moment

I Moment of inertia

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Figures and Figure Captions List

Figure 1: Visual representation of DACM Framework approach

Figure 2: Representation of the generic variables and their interconnections in the Bond Graph theory

Figure 3: An algorithm for extracting causality between assigned variables in DACM

Figure 4: A representation of DACM Framework on RLC circuit case study with the different transformations (blue arrows). Left: Concept space, Generic functional model. Right: Knowledge

space, extracted colored causal graph for the circuit

Figure 5: Functional model and generated causal graph by analogy between three energy domains (Electrical, hydraulic, thermal)

Figure 6: A presentation of the DACM Framework using the CK theory organization of the design process with the concept and knowledge spaces. The dimensionless numbers are the result of the application of the DACM Framework to the beam topology optimization problem.

Figure 7: A: Typical machine liquefier, B: Thermal interfaces between block materials in RepRap liquefier, C: Geometry of the part to be manufactured

Figure 8: Systematic transformation between Interface analysis (A), Functional model (B), and Generic functional representation (C) and extracting causal graph for thermal heat exchange in

FFF liquefier

Figure 9: Partial causal graph of FFF liquefier and the part to be manufactured. (Qualitative objectives are underlined. The backward propagations on the graph are shown with the same

colors)

Figure 10: Printing result before and after process parameter modification

Figure 11: A presentation of DACM Framework for Curling Defect modeling (DACM for DFAM support)

Tables and Table Caption List

Table 1: Mapping between the type of energy and specific variable categories with the associated dimensions

Energy Domain Effort (e) Flow (f) Generalized Momentum

Generalized Displacement

Mechanical (Translation)

Force (F) [MLT^-2]

Velocity (u) [LT^-1]

Momentum (P) [MLT^-1]

Displacement (x) [L]

Mechanical (Rotation)

Torque (τ) [ML²T^-2]

Angular velocity (ω) [T^-1]

Angular

momentum (Pω) [ML²T^-1]

Angle (θ) [---]

Electrical Voltage (U) [ML²T^-3A^-1]

Current (I) [A]

Flux Linkage (λ) [ML²T^-2A^-1]

Charge (q) [AT]

Thermal (Pseudo Bond Graph)

Temperature (T)

[t]

Heat flow rate (𝑞̇)

[ML²T^-3]

--- Heat energy (q)

[ML²T^-2]

Hydraulic Or Pneumatic

Pressure (P) [ML^-1T^-2]

Volume flow rate (𝑄̇) [L³T^-1]

Pressure

Momentum (PM) [ML^-1T^-1]

Volume (Q) [L³]