Development of a Novel Beam-Based Finite-Element Approach for the Computationally Efficient Prediction of Residual Stresses and Displacements in Large 3D-Printed Polymer Parts

Abstract

Recently, 3D printing of large, structural polymer parts has received increasing interest, especially for the creation of recyclable structural parts and tooling. However, the complexity of large-scale 3D polymeric printing often dictates resource-intensive trial and error processes to achieve acceptable parts. Existing computational models used to assess the impact of fabrication conditions typically treat the 3D-printed part as a continuum, incorporate oversimplified boundary conditions and take hours to days to run, making design space exploration infeasible. The purpose of this study is to create a structural model that is computationally efficient compared with traditional continuum models yet retains sufficient accuracy to enable exploration of the design space and prediction of part residual stresses and deformations. To this end, a beam-based finite element methodology was created where beads are represented as beams, vertical springs represent inter-bead transverse force transfer and multi-point, linear constraints enforce strain compatibility between adjacent beads. To test this framework, the fabrication of a large Polyethylene terephthalate glycol (PETG) wall was simulated. The PETG was modeled as linearly elastic with an experimentally derived temperature-dependent coefficient of thermal expansion and elastic modulus using temperature history imported from an ABAQUS thermal model. The results of the simulation were compared to those from a continuum model with an identical material definition, showing reasonable agreement of stresses and displacements. Further, the beam-based model required an order of magnitude less run time. Subsequently, the beam-based model was extended to allow separation of the part from the printing bed and the inclusion of part self-weight during fabrication to assess the significance of these effects that pose challenges for existing continuum models.

1. Introduction

With recent advances in technology, 3D printing for structural applications has become increasingly popular. One type of 3D printing is material extrusion, or fused deposition modelling (FDM). A computer model of the desired part is created, and a program called a slicer converts the geometry into instructions for the printer, called G-code. The printer uses the G-code to construct the part layer by layer, where molten polymer is pushed out of a moving nozzle and deposited in a bead or raster. Depending on the part geometry, a single layer can consist of several beads. After the layer cools for a predetermined amount of time, the next layer is printed on top in the same fashion. FDM allows for creation of complex parts with relative ease and low labor requirements [1]. Originally, FDM printers were quite small, with build volumes allowing part sizes on the scale of inches [2]. Within the last decade, researchers have begun working with large-scale printers with print volumes on the order of meters. Furthermore, these printers can use short-fiber reinforcing material which can increase strength and stiffness [3], enabling the applicability of FDM in construction. Large-scale 3D printers have been used to produce concrete form work [4,5], a culvert diffuser [6], a mold for a vacuum-infused prototype FRP bridge girder [7] and even a small house [8]. These examples show the potential of large-scale 3D printing.

However, because of the manufacturing process which induces repeated heating and cooling of the polymer, there are some unique pitfalls to navigate when 3D printing. If the part cools too much between the placement of layers, this can cause warping and poor inter-layer adhesion. However, printing another layer on an excessively hot part can cause collapse [9]. Furthermore, due to the directional nature of the printing process, the strength of these parts is highly anisotropic. Both the strength and unwanted distortions are dependent on the printing parameters, e.g., print speed, layer height, raster direction, bed temperature and chamber temperature to name a few [10]. It is not usually clear a priori which combination of settings produces a successful print, and this can cause an inefficient trial and error manufacturing process that wastes time and materials [11]. In order to remedy this, some researchers have turned to computational models to help predict the behavior of 3D-printed parts.

Many different computational models of the FDM printing process have been created. Laminate analysis, typically used for fiber reinforced composites, has been applied to 3D-printed specimens. Each layer of the part is assumed to be a linear elastic orthotropic lamina with a direction of reinforcement, in this case the printing direction. Layers are typically assumed to be perfectly bonded together. Researchers have looked at Acrylonitrile Butadiene Styrene (ABS) [12,13,14,15,16,17] or Polylactic Acid (PLA) [14,15,18,19], either neat [12,14,15,16] or reinforced with short carbon fiber [13,17,18], long carbon fiber [20] or wood flour [19], obtaining their material properties from either their own uniaxial coupon testing [12,13,14,16,18,19,20] or from existing published literature [15]. Many have focused on using classical lamination theory (CLT) [12,13,14,17,18,19,20], first-order shear deformation theory [15] and/or homogenization [15,16,17] to predict the stiffness of uniaxial coupons or multi-layered plates and some have predicted failure using the Azzi–Tsai [16], Hill [13], Tsai–Hill [12,19] or Tsai–Wu [20] failure criterion. Comparisons of predicted to experimental stiffness and yield strength are typically within 10%. Though laminate analysis shows promise for simple parts made of unidirectional layers, many FDM parts have complex geometry and infill, which cannot be modeled using this method. As a result, some researchers have used beam [21,22,23] or shell [24] elements to model the infill of small-scale FDM parts. Other researchers have used various continuum models employing 2D or 3D solid elements to predict 3D-printed specimen behavior. They have focused on creating material models of various reinforced and unreinforced plastics, including isotropic [25,26,27,28], orthotropic [29,30], viscoelastic [31,32,33,34,35], elastic–plastic [31,33,35,36,37,38], damage [35,37,38] and hyperelastic [31,39]. Some have used modelling strategies including multiscale or representative volume element (RVE) simulation [28,29,30,32,33,37], cohesive elements [37], XFEM [33], embedded elements [26], or the explicit inclusion of voids [27,31]. Though continuum models have been successfully applied to the behavior of FDM parts, such models do not accurately capture the parts’ structure, which is not homogenous but consists of voids, beads, and inter-bead bonds. While multiscale modeling can incorporate this heterogeneity, none of the papers above took into account the thermal loading on their modeled parts during printing, which can induce significant residual stresses. Also, while one of the strengths of FDM is the ability to fabricated complex structures, the vast majority of the papers above looked at simple geometries, either coupons or plates.

Sequentially coupled thermal mechanical analysis (SC-TMA) is one way to simulate the effects of the printing process and printing-induced stresses. This method was first applied to the FDM of polymers by Wang [40] and Zhang and Chou [41]. As interest in 3D modelling increased, commercial finite element programs incorporated built-in tools for using this method to simulate FDM, which explicitly model the deposition and thermal behavior of material during printing. Some typical characteristics of these commercial codes are small strains, no self-weight, rectangular bead cross-sections, perfect bond between beads, part fixity to the bed, and treatment of the part as a continuum. Some authors used these tools with material properties provided by the software to model tensile coupons or cubes [42,43,44,45,46]. While SC-TMA can more closely simulate the stress state of a part, it has been shown that 3D printing also affects the strength of parts compared to neat polymer or traditional manufacturing methods and so using general material properties is most likely inaccurate. Cattenone et al. [47] manufactured and modeled the printing of two complex geometries of ABS (termed planar spring and bridge) employing SC-TMA in ABAQUS v.2017 using 3D solid brick elements. An elastic–plastic material model was used, with temperature-dependent material properties taken from the literature [47]. During printing, the part was fixed to the bed, but once cooled, a portion of the part was released based upon experimental observations [47]. Vertical displacements from the model were compared to those measured after actual printing and agreed relatively well [47]. Similar to Cattenone et al., Trofimov et al. [48], Syrlybayev et al. [49] and Corvi et al. [50] all modeled various geometries (plate and bridge, tensile coupon, and plate, respectively) with an elastic–plastic model with thermally dependent properties using commercial software. All three groups compared measured versus simulated distortions at the edge of the part and errors ranged from 10% to well over 100% [48,49,50]. Other groups have also applied SC-TMA to the printing of FDM parts. Akbar et al. [51] used SC-TMA in ABAQUS to model the printing and behavior of rectangular prisms of polyurethane-based amorphous thermoplastic using experimentally derived material properties in a linearized thermo-elastic model. Their comparisons between experimental and simulated warpage had a maximum difference of 12% [51]. Jiang et al. [52] modeled short carbon fiber PLA square and circular pipes using Digimat AM. Their material model was thermoelastic with parameters determined through experimental testing and included the effects of crystallization [52]. The model predicted warpage better for the square pipes than the circular with a maximum percent difference overall of 25% [52]. Samy et al. [53,54,55,56,57] examined the effects of various printing parameters using a SC-TMA model built in COMSOL that included gravity effects. Short rectangular prisms made of polypropelene were represented using a thermo-viscoelastic material model that included crystallization [53,54,55,56,57]. While Samy et al. [53,54,55,56,57] state that contact between the bed and part was modeled using a spring foundation, no details of the implementation are provided and their cited source has a fixed then released boundary condition [58]. The warpage at one or two points per specimen is compared to experiments and agrees very well with measured values [53,54,55,56,57].

FEA-based SC-TMA is not the only method being used to model FDM. Wang and Papadopoulos [59] presented a fully coupled thermo-mechanical model of a two-dimensional wall. Unlike in a sequentially coupled analysis where the thermal problem is independent and can be solved before the structural problem, in a fully coupled analysis, both responses are dependent on the other and the two problems must be solved simultaneously [59]. They compared their results to those from a SC-TMA and found very small differences in final displacement [59]. Sreejith et al. [60] devised a framework to model deposition from the nozzle and liquid, transition and solid phases of the polymer during FDM. They used the Arbitrary Lagrangian–Eulerian method to solve their coupled governing equations [60]. They applied the method to a hypothetical polystyrene wall one bead thick and four beads tall in plane strain conditions [60]. Xia et al. [61] applied the one fluid formulation to fully modelling FDM which is a finite volume, front tracking method with two meshes, one stationary and one moving. They used the method to examine the effects of overhang using an inverted cone, spacing between two adjacent filaments using a two-bead-wide by two-bead-high wall and bridging by printing a single filament connecting two spaced out single bead walls four beads high [61]. Though these methods have the potential to more accurately capture the physics of FDM, their complexity and computational demand are quite high. None of these methods were compared against experimental data and the one that was compared to a SC-TMA found only small differences in displacements. Furthermore, these models consist of novel code that would be difficult to implement for wider classes of simulations and for the reduction in trial and error during large-format FDM.

While the discussion above shows there has been much work on simulating small-scale 3D printing, the simulation of large-scale 3D printing poses other challenges. Radiative heat transfer is a significant mode of heat transfer, unlike in small-scale printing [62]. Also, because of the reduced surface area to volume ratio, large-scale parts cool much more slowly and can therefore be more prone to sagging [62]. While most findings from small-scale SC-TMA are most likely applicable to large-scale SC-TMA in a general sense, it is still very important to have models specifically designed for use at large scale. Talagani et al. [63] created a structural simulation including element activation of the printing of a car made of short carbon ABS. Material properties were determined using a multi-scale homogenization approach that included damage, and stresses in each layer were calculated using CLT [63]. Measured temperatures were input into the structural model [63]. Locations of high stress in the model qualitatively matched where the printed structure cracked [63]. Kim et al. [64] performed a SC-TMA on the printing of a 122 cm long by 38.1 cm tall single bead wall made of carbon fiber ABS. The wall was printed on an aluminum bar that allowed for deflection measurements and the material properties of the wall were temperature dependent and determined through experiments and multiscale modeling [64]. Measured and simulated deflections of the end of the bar throughout printing were qualitatively similar [64]. Friedrich and Choo [65] modeled a pyramid of ABS using a SC-TMA. The ABS was modeled as thermoviscoelastic with properties from the literature [65]. The size of the model was reduced by using quarter symmetry [65]. Simulated versus measured final displacements along one line of the structure were within roughly 10% [65]. Brenken et al. [11] created a simulation tool in ABAQUS AM that includes the effects of thermoviscoelasticity and crystallization. The tool was tested by simulating the printing of plates and rings made of short carbon-reinforced polyphenylene sulfide [11]. Material properties were determined using experimentation and micromechanical modeling. Comparisons between experiment and simulation of the curvature of the plate and deformation of the ring were within 7% [11]. Though this model is accurate, it is also very computationally expensive, with a simulation of an autoclave mold taking around 11 h with two computers, each with 10 cores and 128 GB of RAM [11].

To accurately simulate the printing of a large 3D-printed part, the height of an element must be at most the bead height, which is typically on the order of millimeters or perhaps centimeters. This limits the size of traditional solid elements resulting in meshes with many elements. Further, because of complex geometries and/or printing paths, models can rarely take advantage of symmetry. Model runs on the order of hours or more makes a comprehensive exploration of a part design space impractical using available FEA-based SC-TMA strategies and tools, motivating the development of more computationally efficient models. There has been some work on speeding up thermal simulations of small [66,67] or large [67,68] parts, using, for example, mesh merging [66], modified beam elements [67] or a one-dimensional model [68]. Bhandari and Lopez Anido [69] developed a discrete event simulation thermal model and modeled the printing of a “miniature… ashtray model” made of PLA or ABS. The authors then used the model as part of a SC-TMA of a large-scale culvert outlet diffuser printed with PLA reinforced with wood flour, simulated using a visco-elastic–plastic material model [9]. Viscous plastic deformations due to self-weight (no thermal shrinkage) were calculated using a finite element model, which were then used to predict whether the print will fail or not [9]. Bhandari and Lopez Anido also developed a different model to simulate the printing of a 0.5 m × 0.5 m × 0.5 m open top box made of carbon fiber-reinforced ABS [70]. The material is assumed to be orthotropic and its properties are not dependent on temperature [70]. A SC-TMA is written in MATLAB, using a fourth-order Runge–Kutta finite difference approximation with mesh merging for the thermal simulation and finite element analysis using CLT with solid hexahedral elements for the structural simulation [70]. Thermal stresses begin to develop once an element cools to below its glass transition temperature and self-weight is ignored [70].

Though there has been some work performed on computationally efficient SC-TMA of large scale FDM, the vast majority of prior work has focused on advancing thermal simulations. Available continuum structural models rely on brick elements that do not accurately capture bead geometry and make it challenging, if not impossible, to simulate imperfect bond at interfaces between beads. Such models also require a large number of elements to discretize each bead, resulting in models that quickly become computationally intractable. Further, the simulation of part-bed separation and other complex boundary conditions is challenging with solid models. In the big picture, the ongoing aim of this work is to develop a beam-element-based, computationally efficient structural model of large 3D-printed parts to determine residual stresses and deformations developed during printing. The use of beam elements allows for the more realistic simulation of bead cross-sections, decreases the difficulty of simulating imperfect inter-bead bonds and simplifies the implementation of boundary conditions such as part-bed separation. Because maximum element lengths are no longer constrained by bead height, models can employ far fewer degrees of freedom than brick element-based continuum models, greatly increasing simulation speed. The model provides a convenient framework for addressing the shortcomings of existing solid models while allowing designers to determine optimal printing parameters before a print and reduce costly trial and error iterations. Toward those ends, the purpose of this paper is to establish the model methodology and evaluate its predictions and performance relative to an existing continuum model to assess its potential moving forward. More details of the model, its application to a wall print and comparison to an ABAQUS model can be found in Section 2. Section 3 discusses the results, Section 4 their implications and Section 5 provides ideas on future work and concludes the paper.

2. Development of Beam Element-Based Simulation

2.1. Model Description

A two-dimensional, beam-based, small-strain and deformation finite element model is presented here. As discussed in the introduction, an FDM part consists of layers made up of one or more beads, which all bond to their neighbors. As shown in Figure 1, the beads are represented in the model with beam elements and the inter-bead bond is simulated using multipoint constraints and vertical axial-only elements (termed vertical springs for the rest of this paper). The model is structural only and requires temperatures versus location and time as input.

Examining the FDM process, the print head continuously deposits molten material (which forms a bead) at a given temperature, moving at a certain speed and extrusion rate, pauses to allow the bead to cool if directed by the G-code and then repeats the process for the next bead. This continuous process is approximated using discrete time steps in the model and in each time step only introducing elements whose centroids have already been printed. The settings defining this process determine the cross-section of the bead in addition to the area of the inter-bead bond, and the aspects relevant to the model are captured via the elemental cross-sectional areas ($A_{bead}$ or $A_{vert}$) and moment of inertia ($I_{bead}$ or $I_{vert}$). As the bead’s temperature fluctuates due to cooling and rewarming due to the placement of adjacent beads, the stiffness of the bead increases and decreases, respectively. This phenomenon is incorporated into the model by defining the elastic modulus ($E_{bead}$ or $E_{vert}$) as a function of temperature. Changes in bead temperature also induce thermal strains, which are included in the model as element pre-strains. It is assumed that thermal strains accumulate only once the bead temperature falls below the glass transition temperature of the polymer ($T_g$). The stiffness of the inter-bead bond is dependent on printing parameters and is accounted for in the vertical direction in the model by allowing $E_{vert}\ne E_{bead}$, where both moduli of elasticity can be calibrated to the specific printer and parameters through material testing. Linear, multipoint constraints are employed to enforce inter-bead shear transfer and assume full composite action (i.e., no interlayer slip). Flowcharts for both the process and the model’s representation of that process are shown in Figure 2. The grey dotted lines denote correspondence between process and model steps. All aspects of the model are elaborated on in the following sections.

Employing the principle of virtual work for the current model produces Equation (1),

$$\delta W_{in{t}_b}+\delta W_{in{t}_{ax}}+\delta W_{in{t}_f}-\delta W_{ext}=0$$

(1)

where sub-subscript $b$ denotes bending, $ax$ denotes axial and $f$ denotes a spring foundation supporting the base of the printed part, which is discussed later. Substituting the well-established expressions for each virtual work term produces the weak form of the governing differential equation for a beam element in the model, shown in Equation (2),

$${{\displaystyle \int }}_0^{l}EI{v}^{″}\delta v″dx+{{\displaystyle \int }}_0^{l}A\sigma \delta ϵdx+{{\displaystyle \int }}_0^{l}kv\delta vdx-{{\displaystyle \int }}_0^{l}q\delta vdx=0$$

(2)

where $E$ is modulus of elasticity; $I$ is the second area moment of inertia; a prime denotes differentiation with respect to $x$, the position along the element; $v$ is the transverse displacement along the element; $\delta$ denotes the variation of a displacement, strain or curvature; $A$ is the cross-sectional area of the element; $\sigma$ is the axial stress; $ϵ$ is the axial strain; $k$ is the stiffness of the foundation; $q$ represents the applied vertical loads due to self-weight; $u$ is the axial displacement and $U^{el}$ is a vector of element nodal displacements. Thermal pre-strains are incorporated in the axial stress–strain relationship. To obtain Equation (3), the standard shape functions for an Euler–Bernoulli beam denoted by $N_{ax}$ and $N_b$ for axial and bending, respectively, are substituted into Equation (2) for the corresponding displacements, virtual displacements and their derivatives. Further, applying $\sigma =E\left(ϵ-{ϵ}_T\right)$ where ${ϵ}_T$ is a thermally induced strain and letting $ϵ=u\prime$ from standard small strain kinematics produces

$${{\displaystyle \int }}_0^{l}EI{N}_b^{″}{U}^{el}{N}_b″\delta U^{el}dx+{{\displaystyle \int }}_0^{l}AE{N}_{ax}^{\prime }{U}^{el}{N}_{ax}\prime \delta U^{el}dx+{{\displaystyle \int }}_0^{l}k{N}_b{U}^{el}{N}_b\delta U^{el}dx-{{\displaystyle \int }}_0^{l}q{N}_b\delta U^{el}dx-{{\displaystyle \int }}_0^{l}AE{ϵ}_T{N}_{ax}\prime \delta U^{el}dx=0$$

(3)

Simplifying and combining like terms produces Equation (4),

$$\delta {U^{el}}^T\left(\left({{\displaystyle \int }}_0^{l}EI{N_b^{″}}^T{N}_b^{″}dx+{{\displaystyle \int }}_0^{l}AE{N_{ax}^{\prime }}^T{N}_{ax}\prime dx+{{\displaystyle \int }}_0^{l}k{N}_b^T{N}_{b}dx\right)U^{el}-{{\displaystyle \int }}_0^{l}q{N_b}^{T}dx-{{\displaystyle \int }}_0^{l}AE{ϵ}_T{N}_{ax}{\prime }^{T}dx\right)=0$$

(4)

Because the virtual displacements are arbitrary, Equation (5) must hold.

$$\left({{\displaystyle \int }}_0^{l}EI{N_b^{″}}^T{N}_b^{″}dx+{{\displaystyle \int }}_0^{l}AE{N_{ax}^{\prime }}^T{N}_{ax}\prime dx+{{\displaystyle \int }}_0^{l}k{N}_b^T{N}_{b}dx\right)U^{el}={{\displaystyle \int }}_0^{l}q{N}_{b}dx+{{\displaystyle \int }}_0^{l}AE{ϵ}_T{N}_{ax}{\prime }^{T}dx$$

(5)

Equation (5) is the discretized version of the weak form for a single element which, in conjunction with the multi-point constraints detailed later, drives the model. The terms multiplying the nodal displacements are the bending, axial and foundation contributions to the element stiffness matrix. The axial and bending stiffness matrices are the standard ones for an Euler–Bernoulli beam. The foundation contribution is non-zero only for the bottom layer of elements, and the treatment of this is detailed in Section 2.1.3.

2.1.1. Element Activation

Elements are activated as they are printed through a two-step process. It is assumed that just-printed elements contribute self-weight but not stiffness to the structure as shown in Figure 3. Element activation times, corresponding to when the centroid of an element is printed and easily derived from G-code, are input to the model. During the first time step after activation, the self-weight of the new elements is applied. In the next time step, new nodes corresponding to the new elements are activated with an initial vertical position based on the deformed position of the nodes below. Then the new elements are added to the model and their self-weight is applied to the centerline of these new elements. New elements are assigned an initial displacement based on the current position of their nodes, which does not cause stress or strain in the element.

2.1.2. Multipoint Constraints

Because the beads are modeled using beams, which are one-dimensional elements, additional strategies must be employed to transfer forces between the beads. Vertical forces are transferred via vertical springs connecting a beam’s end nodes to those of the beams immediately above and below it. Interface shear force transfer between beads is tackled with multipoint constraints (MPCs). For example, MPCs have been used for shear transfer in a composite beam [71], shear transfer between a bridge deck and girder [72] and during the simulation of warping of a composite beam’s cross-section [73]. In the current model, multipoint constraints are employed to ensure strain compatibility at every bead–bead interface. A pair of constrained elements, with notation used throughout this paper, is shown in Figure 4.

Full composite action is assumed between layers, meaning beads do not slip relative to each other. This implies that from the time of activation of the upper element, the change in longitudinal strain at the top of the layer below (the lower element in Figure 4) is equal to the change in longitudinal strain at the bottom of the layer above (the upper element in Figure 4). The total longitudinal strain at any point within a beam element $ϵ\left(x,y\right)$ is given in Equation (6),

$$ϵ\left(x,y\right)={ϵ}_a-v″\left(x\right)y,$$

(6)

where ${ϵ}_a$ is the axial strain, $v″$ is the element bending curvature and $y$ is the signed distance from the neutral axis of the beam (centerline of the bead). Using the conventional Euler–Bernoulli beam element, axial strain is constant along the length of the beam and is given by Equation (7),

$${ϵ}_a={\displaystyle \frac{u_j-u_i}{L}},$$

(7)

where $u_i$ and $u_j$ are the axial displacements of element starting and ending nodes $i$ and $j$, respectively, and $L$ is the element length. The curvature of a beam element is given in Equation (8),

$$v″\left(x\right)=N_b^{″}\left(x\right)U^{el},$$

(8)

where $N_b^{″}$ as defined previously is the second derivative of the element bending shape functions with respect to $x$ and $U^{el}={\left[ u_i v_i {\theta }_i u_j v_j {\theta }_j\right]}^T$ is the element displacement vector of $x$-direction nodal displacements $u$, $y$-direction nodal displacements $v$ and rotations $\theta$ at nodes $i$ and $j$ of an Euler beam element. Since axial strain is a function of $x$-direction nodal displacements, the corresponding coefficients can be added to $y{N}_b″$ in the correct locations to obtain $B$, the matrix that relates strains to $U^{el}$. Total longitudinal strain at any point in a beam element can then be expressed as shown in Equation (9),

$$ϵ\left(x,y\right)=B\left(x,y\right)U^{el}.$$

(9)

To enforce strain compatibility between two beads in the model at a location $x$ at any time $t$ after the activation of the upper element at $t_u^{act}$, the strain at the bottom of the upper element ${ϵ}_u$ must equal the strain accumulated in the top of the lower element ${ϵ}_l$ following activation of the upper element. Equation (10) offers the full constraint expression at any point $x$ within a pair of upper and lower elements that must be enforced to ensure inter-bead shear transfer and ${ϵ}_u={ϵ}_l$.

$${ϵ}_u\left(x,-{\displaystyle \frac{h}{2}},t\right)-\left({ϵ}_l\left(x,{\displaystyle \frac{h}{2}},t\right)-{ϵ}_l\left(x,{\displaystyle \frac{h}{2}},t_u^{act}\right)\right)=0$$

(10)

Substituting Equation (9) into Equation (10) offers the required strain compatibility Equation (11) at the bead interface in terms of element displacements in a form suitable for inclusion in a displacement-based finite element analysis,

$$B_u\left(x,-{\displaystyle \frac{h}{2}}\right)U_u^{el}\left(t\right)-B_l\left(x,{\displaystyle \frac{h}{2}}\right)U_l^{el}\left(t\right)+B_l\left(x,{\displaystyle \frac{h}{2}}\right)U_l^{el}\left(t_u^{act}\right)=0.$$

(11)

For the structure as a whole, the multipoint constraints required to ensure strain compatibility at the interfaces of all beads are expressed in the form given in Equation (12),

$$G^{T}U=C,$$

(12)

where $G^T$ is a matrix of constraint coefficients, $U$ is a vector of global nodal displacements and rotations, and $C$ is a vector of constraint constants. Equation (11), which applies within a single element, is a function of local element displacements that are easily transformed to global nodal displacements using standard techniques. As discussed in Section 2.1.1., elements have an initial displacement $U^0$, which does not cause strain, that is defined when the element is activated and must be subtracted from the total nodal displacements as shown in Equation (13) to obtain the strain producing displacement $U^{el},$

$$U^{el}=U-U^0.$$

(13)

Because the total longitudinal strain due to both bending and axial force within a single element vary linearly along the length of the element at any depth, the strain compatibility must be enforced at two locations within each pair of bonded elements to ensure strain compatibility along the full length in contact. For simplicity, the end nodes of the elements are chosen. Once activated, a constraint is present in every subsequent time step. Substituting Equation (13) into Equation (11), recognizing that the constants in $G^T$ are terms in $B_u$ and $B_l$ at the element ends, and isolating the unknowns offers the final constraint equations for a single element pair, shown in Equations (14) and (15),

$$G_u\left(0\right)U_u\left(t\right)-G_l\left(0\right)U_l\left(t\right)=G_u\left(0\right)U_u\left(t_u^{act}\right)-G_l\left(0\right)U_l\left(t_u^{act}\right)$$

(14)

$$G_u\left(L\right)U_u\left(t\right)-G_l\left(L\right)U_l\left(t\right)=G_u\left(L\right)U_u\left(t_u^{act}\right)-G_l\left(L\right)U_l\left(t_u^{act}\right),$$

(15)

where vectors $G_l$ and $G_u$ are given below in Equations (16)–(19).

$$G_l\left(0\right)=\left[-{\displaystyle \frac{1}{L}}-{\displaystyle \frac{3h}{L^2}}-{\displaystyle \frac{2h}{L}} {\displaystyle \frac{1}{L}} {\displaystyle \frac{3h}{L^2}}-{\displaystyle \frac{h}{L}}\right],$$

(16)

$$G_u\left(0\right)=\left[-{\displaystyle \frac{1}{L}} {\displaystyle \frac{3h}{L^2}} {\displaystyle \frac{2h}{L}} {\displaystyle \frac{1}{L}}-{\displaystyle \frac{3h}{L^2}} {\displaystyle \frac{h}{L}}\right],$$

(17)

$$G_l\left(L\right)=\left[-{\displaystyle \frac{1}{L}} {\displaystyle \frac{3h}{L^2}} {\displaystyle \frac{h}{L}} {\displaystyle \frac{1}{L}}-{\displaystyle \frac{3h}{L^2}} {\displaystyle \frac{2h}{L}}\right],$$

(18)

$$G_u\left(L\right)=\left[-{\displaystyle \frac{1}{L}}-{\displaystyle \frac{3h}{L^2}}-{\displaystyle \frac{h}{L}} {\displaystyle \frac{1}{L}} {\displaystyle \frac{3h}{L^2}}-{\displaystyle \frac{2h}{L}}\right].$$

(19)

2.1.3. Boundary Conditions

In the model, the contact between the bed and print can be modeled in two ways. The bottom layer of the print can be fixed to the bed, simulating a perfect bond between the first layer of the part and the bed. To fix vertical displacement, vertical springs are added extending from the centroid of the lowest bead to the height of the bed. The bottom ends of these elements are fixed. To lock horizontal displacements, constraints of the form of Equations (14) and (15) are added, where the first bead is the upper layer and the constraint constants and values for the lower bead are all zero.

The second option for boundary conditions in the model simulates no adhesion of the part to the bed, which means the part should be able to slide horizontally and freely lift off from the print bed but not penetrate into it. The simulation of this condition has significant practical importance, as it is not unusual to sandwich a bond breaker between a part and the print bed. To ensure numerical stability, very soft horizontal springs are attached to the lowest bead. To allow only positive vertical displacement, bilinear beam on elastic foundation (BBOEF) elements were used for the lowest bead that have distributed vertical springs with a very large compressive stiffness and very small tensile stiffness relative to other elements in the model. The derivation of the contribution of the springs to the element stiffness matrix is as follows. From the discretized weak form given in Equation (5), the contribution of the springs to element stiffness is given in Equation (20),

$$K_{se}={{\displaystyle \int }}_0^{l}k{N}_b^T{N}_{b}dx.$$

(20)

Because $k$ is not constant over the length of the element, the integral must be split as shown in Equation (21),

$$K_{se}={{\displaystyle \int }}_{L_c}{k}_c{N}^{T}Ndx+{{\displaystyle \int }}_{L_t}{k}_t{N}^{T}Ndx,$$

(21)

where $L_c$ and $L_t$ are the portions of the element where the springs are in compression and tension, respectively. The compressive stiffness $k_c$ is large to prevent inter-penetration of the bed, and $k_t$ is set to a very small value to allow free separation of the part from the bed if it is unbonded. Because $L_c$ and $L_t$ are not known a priori, the use of these elements introduces nonlinearity, requiring an iterative solution strategy as discussed in Section 2.1.4.

The formulation of the BBOEF element was verified through comparison of the model prediction using 100 elements with the analytical solution of Zhang and Murphy [74] for the displacement of a free–free beam on a tensionless foundation with a point load at the center. The authors normalize all values by $\beta$, given in Equation (22),

$$\beta ={\left({\displaystyle \frac{k_c}{4EI}}\right)}^{{\displaystyle \frac{1}{4}}},$$

(22)

where $E$ is the Young’s modulus of the beam material and $I$ is the second moment of area of the beam cross-section. Important values are normalized as shown in Equation (23),

$$\begin{gathered} l=\beta L \\ F={\displaystyle \frac{P}{4{\beta }^{2}EI}} \\ \zeta =\beta x \\ w=\beta v \end{gathered}$$

(23)

where $L$ is physical beam length, $P$ is the magnitude of the applied point load and $v$ is transverse displacement. Zhang and Murphy [74] chose to show the results of a beam with normalized length of four and two cases of normalized applied load of 0.1 and 0.2. For the BBOEF beam, $EI$ was chosen to be 4300 kN-m² and $k_c$ was 1.76 × 10⁶ kN/m². From this, $\beta , L$ and $P$ were calculated and the simulation was run. The two solutions are compared in Figure 5, which indicates excellent agreement.

2.1.4. Solution Strategy

The general problem to be solved is given in Equation (24),

$$F_{int}=F_{ext},$$

(24)

subject to some boundary conditions discussed later, where $F_{int}$ are internal forces and $F_{ext}$ are external forces, both in the global coordinate system. There are two types of external forces in the model: the self-weight of each element and thermal forces. Thermal forces develop in an element as it cools, as given by the inputted thermal profile. While element temperatures generally change between time steps, within one time step it is assumed that the temperature is constant for the whole element. The thermal force in one element, $f_{th}$, is given in Equation (25),

$$f_{th}=EA\alpha \mathsf{\Delta }T,$$

(25)

where $E$ is the modulus of elasticity for the material at the current temperature, $A$ is the cross-sectional area of the element, $\alpha$ is the coefficient of thermal expansion at the current temperature and $\mathsf{\Delta }T$ is the change in temperature from deposition to the current time. Thermal forces in the longitudinal and height directions are carried by the beams and vertical springs, respectively, and there can be a different coefficient of thermal expansion for each direction. Since the thickness of the wall is much smaller than the height or length, thermal expansion through the thickness of the wall is ignored.

There are two kinds of internal forces in the model, constraint and member forces. From Section 2.1.2., the system of equations of all constraints is given by Equation (12). The constraints are enforced using the penalty method, where violations are penalized by correcting forces applied by very stiff springs. These forces, $F_c$, given in Equation (26),

$$F_c=G\kappa \left(G^{T}U-G_0\right),$$

(26)

are calculated by pre-multiplying the error in the constraints by $G\kappa$, where $\kappa$ is a diagonal matrix of spring stiffnesses corresponding to each constraint equation. The other type of internal forces in the model are member forces, shown in Equation (27) for a single element,

$$f_{el}=k_{el}{u}_{el},$$

(27)

where $k_{el}$ and $u_{el}$ are the element stiffness matrix and element displacement vector, both in local coordinates. Since strain-producing element displacements are equal to nodal displacements minus initial element displacements, distributing the element stiffness matrix and assembling the two parts separately (into $KU$ and the correction for non-strain causing displacements, $F_{e{l}_0}$, respectively) produces the global internal force vector, $F_{el}$, shown in Equation (28),

$$F_{el}=KU-F_{e{l}_0},$$

(28)

where $K$ and $U$ are the global stiffness and nodal displacement vectors. Assembling the local element thermal and self-weight forces into the global external force vector, $F_{ext}$, and isolating terms with the unknowns, which are the global vector of nodal displacements, produces Equation (29),

$$\left(K_{bc}+G\kappa G^T\right)U=F_{ext}+G\kappa C+F_{e{l}_0},$$

(29)

where $K_{bc}$ is the global system stiffness matrix after applying relevant boundary conditions.

Because the bilinear boundary springs make the problem nonlinear, Equation (29) is solved iteratively using Newton’s method. The contribution of the BBOEF elements to system stiffness is recalculated at every time increment, and initial member forces due to the BBOEF elements must also be recalculated. To increase computational efficiency, $F_{e{l}_0}$ is split into two parts: $F_{m_0}$, the initial member force correction calculated once per increment, and $F_{s_0}$, the initial force correction due to the BBOEF elements, calculated every iteration. The entire solution algorithm is below.

For each time step $t_i$,

• Set $U_i=U_{i-1}$;
• Determine $T_i$, the current temperature, and compute $\mathsf{\Delta }{T}_i=T_i-T_0$ for all elements;
• Update $K$ by updating elastic moduli and adding new elements, if any;
• Compute and assemble $F_{th}=EA\alpha \mathsf{\Delta }{T}_i$ for each element and update $F_w$ if there are new elements;
• Update $G$ if there are new elements;
• Compute $K\prime =\left(K_{bc}+\kappa G{G}^T\right)$ and $F^{\prime }=F_{th}+F_w+\kappa GC+F_{e{l}_0}$;
• Calculate $R=F^{\prime }-K^{\prime }{U}_i$;
• While $R>tol$,
  • Solve $K^{\prime }\mathsf{\Delta }U=R$;
  • $U_i=U_i+\mathsf{\Delta }U$;
  • $K_{bc}=K$;
  • For each BBOEF element,
    • Calculate $v\left(x\right)=N{U}_e$;
    • Find $v\left(x\right)=0$, in the interval $\left(0,L\right)$;
    • Break into subintervals using the roots determined in ii and find state of springs in each;
    • Calculate $K_{se}$;
    • Add $K_{se}$ to $K_{bc}$;
  • $K\prime =\left(K_{bc}+\kappa G{G}^T\right)$;
  • Recalculate $F_{s_0}$;
  • $F^{\prime }=F_{th}+F_w+\kappa GC+F_{e{l}_0}$;
  • $R=F^{\prime }-K^{\prime }{U}_i$.

3. Wall Print

To test the model, it was applied and compared to the print and continuum model created by Robles-Poblete et al. [75]. The authors describe the printing of a 1 m tall by ¾ m long single bead wide wall made of polyethylene terephthalate glycol (PETG) reinforced with short carbon fiber. The first layer of the wall was a brim slightly larger than the wall with a length of 0.87 m and a width of 0.135 m. Thermocouples were placed in the wall throughout printing. Each layer was printed left to right, making the problem unsymmetric. A SC-TMA was created in ABAQUS, which utilized a combination of experimentally derived properties and properties chosen such that error between measured and simulated temperatures was minimized. Temperature-dependent specific heat, coefficient of thermal expansion and elastic modulus in two directions were all determined experimentally, as was thermal conductivity, but only at room temperature. Room temperature shear moduli and Poisson’s ratios were taken from the literature and the multifactor approach was used to estimate the temperature dependence of the stiffnesses. It was assumed that the material was transversely isotropic and that above the glass transition temperature, elastic properties were constant, and no thermal strains were induced. To minimize error between simulated and measured temperatures at relevant locations, a bed-part conductance value of 10 W/m²K and a convection coefficient varying from 3 to 15 W/m²K with increasing wall height were chosen. Perfect bonding between the bottom of the part and the bed was assumed and self-weight was ignored. The mesh consisted of cube-shaped hexahedral elements where side length was equal to the height of a bead which resulted in 148 elements along the wall width and 3 elements through the thickness. Full details are given in [75].

The thermal results of Robles-Poblete’s model, and the event series controlling element activation were all input into the beam-based model and a copy of the ABAQUS 3D structural continuum model discussed in [75] for comparison. In the two-dimensional beam-based model, applied temperatures were averages though the thickness of the wall. The material properties for both models were from Robles-Poblete with slight modifications. In particular, elastic properties (shown in Figure 6) were at a finer resolution with the room temperature definition updated and the coefficient of thermal expansion values (depicted in Figure 7) were shifted such that the location of zero strain corresponded to the material deposition temperature of 200 degrees Celsius.

To match the continuum model, the part was fixed to the bed and self-weight was ignored in the beam-based model. Though the actual brim was several beads wide, in the beam-based model it was simulated as one large bead with the same area and moment of inertia because the model is two-dimensional. To choose mesh density of the beam-based model, a convergence study was run. The beads were meshed with increasing numbers of equal length elements varying from 15 to 240 elements per layer and constant time steps of 100 s to 2 s were also examined. The beam-based model was then extended to include self-weight and the BBOEF elements.

4. Results and Discussion

4.1. Convergence of Beam-Based Model

To check convergence of the model, the effect of time step and mesh density on both stresses and displacements must be examined. For vertical stresses, one area of interest was the corner where the lowest layer of the wall intersects the brim, because this is where the maximum stresses are located. These results and their location are depicted in Figure 8.

As Figure 8 shows, the vertical corner stress is converging to a value between 5.2 and 5.3 MPa as the number of elements per layer is increased, while time step has virtually no effect on results. Based on this, 60 elements per layer, which exhibited a 4% difference in peak stress compared to the finest mesh, is a reasonable level of mesh refinement.

In the case of displacements, no obvious point of interest was visible and maximum magnitude only varied 3% from the coarsest time step and mesh density to the finest. Therefore, convergence of change in average nodal displacement for the entire wall was examined and is shown in Figure 9, which indicates that average vertical displacements are converging with decreasing time step and increasing mesh density. Mesh density has a modest effect on convergence, mainly at larger element sizes. Time step has a much larger effect on the convergence of displacements. A time step of 20 s was chosen for subsequent simulations.

Figure 10 shows the impact of time step and mesh density on model run time. As expected, simulation times grow significantly with decreasing time step and mesh density, with run times in Figure 10 varying by four orders of magnitude. For reference, the run times for the ABAQUS structural model are also given in Figure 10. The ABAQUS run times are significantly slower than all beam element model runs for the same time step, highlighting the gains in computational efficiency achievable with the beam element model. For example, the chosen configuration for the beam-based model of 60 elements per layer with a 20 s time step, shown with a red circle in Figure 10, ran in roughly six minutes, compared to 34 min and 3 h for the ABAQUS model runs with time steps of 100 and 10 s, respectively. This is due to the much larger mesh resulting from the 3D formulation with brick elements. It should be noted, however, that the ABAQUS mesh was chosen for simplicity and it is quite possible that fewer elements could be used while retaining acceptable accuracy, which could significantly reduce ABAQUS computation time. However, this is countered by the fact that the beam element model is implemented entirely in MATLAB, which is inherently slow due to the interpreted nature of the code; significant speedup would be achieved were the beam element model written in a compiled language such as FORTRAN or C. Further, while not explored here, we note that meshes with varying beam element lengths could be used to ensure convergence of peak stresses (e.g., vertical corner stress discussed above) while reducing computation costs. Overall, the time comparison indicates the promise of the beam element model for reducing run time for simulating residual stresses in large, 3D-printed polymeric parts.

4.2. Comparison between Beam-Based Model and ABAQUS Structural Model

To verify the model formulation and predictions, the results of the beam-based model were compared to those of the ABAQUS continuum model. It is important to note here that at present, neither model has been validated with experimental data and so this comparison is not a validation of model accuracy. However, the previous review of the literature shows that many researchers have used continuum models incorporating ABAQUS AM Modeler, which makes it a reasonable benchmark against which to assess the beam-based model predictions. Because the two modeling approaches differ significantly (element type, dimensionality, etc.), some variation between the two sets of results is expected. The continuum results displayed are the average through the wall thickness.

Table 1. Summary of comparison of modeling approaches.

Type	Max-BB	Max-Cont	Min-BB	Min-Cont	Max Diff	Min Diff
X-displacement (mm)	0.174	0.173	−0.171	−0.172	0.803%	−0.583%
Y-displacement (mm)	1.03 × 10−5	0	−0.355	−0.361	100%	−1.89%
Axial Stress (Mpa)	2.60	1.99	−1.12	−1.21	23.4%	−7.57%
Vertical Stress (Mpa)	5.09	3.97	−1.80	−1.25	22.0%	30.6%

offers a summary of the results of the comparison of the final state of the part between the two models, where “Beam” denotes the beam-based model and “Cont” the continuum model. A more detailed full-field comparison of quantities follows.

Shown in Figure 11 are the full field displacements in the horizontal and vertical directions for both models at the end of simulation, after the part has cooled to ambient room temperature.

The spatial variation of the displacements in both directions is very similar between the two models, with the beam-based model having less displacement overall. In the case of the $x$-displacements (Figure 11a), the maximum difference was roughly 0.03 mm, about 20% of the maximum displacement. The largest differences are in the first third of the layers, which suggests the simplified, single-element width brim at the base of the beam-based model may be playing a significant role in the discrepancies. Examining the $y$-displacements (Figure 11b), there is a maximum difference of 11% of the maximum displacement or 0.04 mm. The small differences between the models in the lowest part of the wall suggests that the simplified brim discretization used in the beam-based model might not impact vertical $y$-displacements as significantly. Finally, the pattern of results is nearly symmetric about the wall centerline in both models, showing that the effect of the un-symmetric printing pattern is small.

Figure 12 shows $x$-displacements of the right edge of the wall at six points in time: four during printing, at the end of printing and at the end of cool down. At a given time step, Figure 12 shows that the shape of the curves of the two models are very similar. Though the displacements are different in the bottom of the wall between the two models, they converge at a height of roughly 300 mm and stay close after this for all but the first time step where the wall height is less than 300 mm. Figure 13 shows the vertical displacement throughout time at one point, located where the magnitude of displacement after cooling is maximum, indicating that both the trend and magnitude of $y$-displacements at the location shown match very well for both models.

Overall, displacements in both directions for both models match well, but stresses still must be examined. Figure 14 shows axial and vertical stress over the entire wall after cooldown and indicates that the largest discrepancies in stress between the two models are concentrated in the lowest third of the wall. Though the spatial variations are very similar, the magnitudes of the beam-based model are significantly higher, with the largest difference in the axial direction of 0.9 MPa being around 32% of maximum stress. In the case of vertical stresses, the largest difference of roughly 1.7 MPa coincides with the maximum stress in the beam-based model of 5.1 MPa, corresponding to a 33% difference. These discrepancies could be caused by the use of the simplified brim in the beam-based model. Larger magnitudes of stresses in the beam based versus continuum models are consistent with the smaller magnitudes of displacements observed previously.

Figure 15 shows axial ($x$-direction) stress at the vertical centerline of the wall as a function of height for multiple discrete times. Comparison of axial stresses shows a similar trend as $x$-displacements where the shapes of the curves are very similar between the models and the magnitudes match very well for portions of the wall higher than 300 mm for all time shown. The worse agreement for heights below 300 mm could be due to the beam element model’s simplified discretization of the brim.

The post-cooldown maximum vertical stress in the beam-based model occurs at the interface between the brim and the second layer. Stress versus time at this location is shown in Figure 16 for both the beam-based and continuum model. Examination of Figure 16 shows that though the shapes of the two curves are very similar, at almost all times the stress in the beam-based model is larger than that of the continuum model, with a final difference of roughly 33%. It is important to note that the locations of maximum final stress of the two models do not coincide: the maximum vertical stress in the continuum model is 4 MPa and is located one bead higher, between the second and third layers. Again, these discrepancies between the magnitudes and locations of maximum vertical stress may be due to the beam-based model’s simplified representation of the brim. It is important to stress that because the two modeling strategies are significantly different, discrepancies between the models are not in themselves cause for concern and without experimental data the relative accuracy of the two models cannot be compared.

Overall, temporal and spatial trends for both displacement and stresses matched well between the two models. The magnitudes of peak displacements and stresses from the two models did not match as well, especially in the lower part of the wall. However, this could be due in part to the simplified discretization of brim in the beam-based model. It is also important to note that the results of the continuum thermal model showed a significant variation in temperature through the thickness of the brim, which could not be captured in the two-dimensional beam-based model.

Comparisons were further complicated by the fact that certain algorithms in ABAQUS are not explained in documentation. For example, in the beam-based model, elements are only included in the stiffness matrix after they are activated, but inactivated elements in ABAQUS have very small non-zero stresses, suggesting they are included in the stiffness matrix before activation to some extent. This makes it impossible to determine how differences in element activation schemes impact the comparison between model results. Further, a part made using FDM is not a continuum but rather a pattern of discrete beads, which implies that a continuum FE model itself is not an ideal representation. Given these caveats, the result of the comparison between the beam-based model and ABAQUS indicate that beam-based models show significant promise for the prediction of deformations and residual stresses in large, 3D-printed polymeric parts.

4.3. Effect of Boundary Conditions on Beam-Based Model Results

An important practical detail present in many prints is de-bonding the part from the bed during the print process. To explore the significance of this, a second beam-based model was created. This model included self-weight and the elliptical cross-section of the beads with otherwise identical properties to the first beam-based model. There were two cases for boundary conditions, one which implemented a fixed boundary condition like in the first beam-based model and one which used the BBOEF elements derived previously to simulate de-bonding and part lift-off.

Table 2. Summary of comparison of boundary conditions of the beam-based model.

Type	Max-Spring	Max-Fixed	Min-Spring	Min-Fixed	Max Diff	Min Diff
X-displacement (mm)	0.174	0.1744	−0.171	−0.171	−0.229%	−0.467%
Y-displacement (mm)	0.232	1.03 × 10−5	−0.330	−0.355	2260000%	−6.91%
Axial Stress (Mpa)	1.50	2.60	−1.52	−1.12	−42.5%	35.4%
Vertical Stress (Mpa)	1.25	5.09	−1.87	−1.80	−75.4%	4.29%

presents a summary of the results of the comparison with BBOEF results termed “Spring” and the fixed results denoted “Fixed.” Detailed results and a discussion are presented in the remainder of this section.

A comparison between final displacements predicted by the two beam-based models is shown below in Figure 17.

For both $x$- and $y$-direction displacements, the two models have similar patterns of variation with the BBOEF model having larger magnitudes in general. This is consistent with the boundary conditions, where the BBOEF model is freer to move than the fixed one. It also makes sense that the largest differences are near the bottom of the wall, close to the boundary. Interestingly, though the bottom corners of the BBOEF part lift off ~0.2 mm, this deformation does not significantly affect vertical displacements in the upper half of the wall.

Figure 18 shows significant differences in magnitude and pattern of stresses between the two models. Both axial and vertical stresses are much lower for the BBOEF model, and in particular the high 5.3 MPa vertical stress near the bottom corners of the model with the fixed base is completely relieved in the BBOEF model, which predicts no stress at those locations. Such vertical stresses are a major contributor to inter-layer de-bonding during or following printing, which highlights the importance of accurately simulating boundary conditions. We also emphasize the relative ease of incorporating such boundary conditions with the beam-based modeling framework developed here.

5. Conclusions

A beam-based finite element framework for the simulation of the fabrication of large 3D-printed polymer parts was developed. Because the model is beam-based, it has the potential to use many fewer DOFs than existing continuum models which typically use brick elements. Not only do brick elements have more DOFs per element, but the size of the bricks is limited by layer height whereas the beam elements are not. Fewer DOFs suggests the model is faster than a comparable continuum model. Additionally, using beam elements allows for the simulation of part lift-off with relative ease. Finally, the use of beams allows for more accurate modeling of the ellipse-shaped bead cross-section, which is normally modeled as rectangular in a continuum model.

The framework is tested by comparing resulting stresses and displacements to a continuum model made with ABAQUS AM modeler. Reasonable agreement is observed, with the beam-based model also extended to allow separation of the part from the bed during printing. Specific conclusions are highlighted below.

• Spatial and temporal convergence studies showed generally smooth and adequate convergence of the beam-based model. Run times for the beam-based model were much less than for the ABAQUS AM model. Although the ABAQUS AM model was not optimized for solution efficiency, the beam-based model was developed using the MATLAB programming language, which is interpreted and relatively inefficient. Overall, convergence and run-time studies indicate that beam-based models have potential to be substantially more computationally efficient than widely available continuum modeling tools.
• The beam-based model and ABAQUS AM predicted very similar patterns of displacements and stresses. While the beam-based model tends to predict smaller displacements and somewhat larger stresses than the ABAQUS AM continuum model, these discrepancies are likely due in part to the simplified model of the brim necessitated by the two-dimensional nature of the current beam-based model.
• Simulations clearly show that simulating the practical boundary condition where part-bed separation is allowed during the print results in significantly reduced residual part stresses, indicating that accurately modeling boundary conditions is critical to accurate predictions. Part-bed separation is challenging to simulate with available continuum models, highlighting the value of this particular feature of the beam-based model developed here.

Despite its advantages, the beam-based model requires significant further development to make it a broadly useful tool for simulating the development of residual stresses and deformations in large, 3D-printed polymeric parts. Future efforts should focus on extending the beam-based model to 3D to allow the simulation of complex 3D-printed parts, obtaining experimental temperature and displacement information during the printing of large 3D polymeric parts to validate the model, and incorporating more complex phenomena like inter-layer slip, viscoelasticity and progressive fracture between printed layers.