Simultaneous Topology and Path Optimization for 3D Concrete Printing Based on Discrete Frame Structures

Abstract

This article presents a novel methodology for the simultaneous optimization of both structural topology and printing path in 3D concrete printing (3DCP), addressing a critical gap between digital design and physical manufacturability. Unlike conventional sequential approaches, our framework is grounded in discrete frame structures, which inherently reflect the filament-based nature of 3DCP, thereby enhancing geometric and mechanical fidelity. The proposed formulation strategically leverages the inherent anisotropy of printed concrete by aligning the printing direction along the longitudinal axis of each frame member to maximize strength and material efficiency. Key manufacturing constraints are integrated directly into the optimization process: member widths are restricted to integer multiples of the nozzle size, and the printing path is enforced as a globally continuous, non-intersecting, and non-overlapping Eulerian circuit through a mixed-integer linear programming model. The efficacy of this simultaneous optimization approach is demonstrated through a series of benchmark problems, which confirm that the resulting designs not only satisfy strict structural displacement and stress constraints with minimal material usage but are also readily manufacturable via direct “one-stroke” printing. This work establishes a foundational integration of structural performance and manufacturability, paving the way for more efficient and reliable 3DCP applications.

Keywords

3D concrete printing topology optimization path optimization frame structure mixed-integer linear program

Introduction

3D concrete printing (3DCP), a burgeoning additive manufacturing technology, has revolutionized the construction of civil structures.¹ It extrudes cementitious materials² layer by layer, with each layer deposited through a nozzle filament by filament, offering exceptional flexibility in terms of free-form structures.^3,4 This flexibility has been validated by the remarkable success of numerous large-scale building and bridge projects.⁵ Furthermore, this flexibility offers a unique opportunity to enhance structural performance through optimization.

For 3DCP, two major optimization problems need to be studied: structural topology optimization and printing path optimization. The topology optimization is concerned with the optimal structural layout under the prescribed design conditions. The layout freedom can be well fulfilled by the constructional flexibility of 3DCP. Therefore, incorporating topology optimization technology into 3DCP structural design has attracted extensive attention. Vantyghem et al.⁶ discussed three potential topology optimization techniques, that is, compliance, stress, and multi-physics, that are applicable to 3DCP. Later, they and coworkers made demonstrations of a post-tensioned 3DCP girder⁷ and a 3DCP bridge⁸ designed through topology optimization. Kinomura et al.⁹ manufactured a topology optimized pedestrian bridge consisting of 44 prestressed 3DCP segments. Alabbasi et al.¹⁰ fabricated optimized 3DCP columns for housing projects in Saudi Arabia. Li et al.¹¹ constructed a cable-supported composite 3DCP structure based on a multi-material topology optimization method.

In contrast, the path optimization is concerned with converting the virtually designed structural model into reality through feasible and efficient path planning of the nozzle. Several manufacturing constraints, such as path continuity, intersection, and overlapping, should be considered during this optimization to guarantee the printing quality. Currently, three strategies are widely used for the path planning¹²: direction-parallel paths (e.g., raster or zigzag path),^13–15 contour-parallel paths (successive offsets of boundary contour),^16,17 and space-filling curves (e.g., spiral,¹⁸ Fermat spiral,¹⁹ Hilbert path,^20,21 or Hamiltonian path²²). Since almost all the strategies are derived from “hot-joint” additive manufacturing technologies for other materials such as thermoplastics and metals, their direct applicability to “cold-joint” 3DCP remains a challenge. Issues like path discontinuity, underfilling and overfilling caused by sharp turns, or self-intersection due to dimensional non-integer multiple of the nozzle size are inevitable when using a single strategy. Bi et al.²³ and Wan et al.²⁴ have elaborately discussed the advantages and disadvantages of each strategy for 3DCP. To optimize the printing path, one common solution is the combination of two or more strategies.²⁵ For instance, Bi et al.²³ first generated primary contours by double offset and outward contour schemes, then filled local underfilling areas by zigzag paths, and finally connected all paths into a globally continuous path. A post-processing optimization algorithm was further implemented to improve the overall smoothness of the path through local adjustment. This kind of hybrid methods is superior to the single strategy. However, the disadvantages inherited from the single strategy cannot be fully eliminated. Another solution is the adaptive path,^24,26,27 which controls the extrusion parameters, such as the printing speed and the material extrusion rate, in process to vary filament size satisfying the planned non-uniform path. Due to the rheological complexity of printable concrete (being flowable within the printing system and solidifying quickly after deposition), this solution poses a high demand for the material mixture and the printing system.

The aforementioned studies explored the topology and path optimization separately. However, it is aware that there exists a gap between the topology optimized model and its manufacturing. On the one hand, the topology optimized design may have issues, such as non-smooth boundaries, dimensional non-integer multiple of the nozzle size, and non-uniform member sizes, that are infeasible for direct 3D printing and require further post-processing adjustment, which deviates from the original optimal design. On the other hand, the printing path produces anisotropic material properties, which are stronger in the printing direction compared with the transverse and stacking directions.^28,29 The anisotropy affects the structural performance and has not been considered during the topology optimization, rendering the optimized design nonoptimal. Therefore, the topology optimization and the path optimization need to be integrated. Martens et al.³⁰ demonstrated that the material anisotropy and manufacturing constraints such as overhangs of 3DCP should be taken into account in the design optimization. Carstensen³¹ and Fernández et al.³² embedded the nozzle size constraint into the topology optimization. Pastore et al.³³ proposed a Bézier curve-based genetic algorithm to address the path continuity and nozzle size constraints. Bi et al.³⁴ addressed the material anisotropy and various manufacturing constraints, including self-support, printing path continuity, and domain segmentation, of 3DCP within the bidirectional evolutionary structural optimization framework. Jewett and Carstensen³⁵ considered both the nozzle size constraint and the weak inter-bead bonding. Mogra et al.³⁶ integrated buildability requirements during the printing process into the optimization problem. Yang et al.³⁷ and Yang et al.³⁸ proposed a topology optimization method integrated with the anisotropy, nozzle size, path continuity, and overhang angle constraints.

Nevertheless, the optimization studies above were based on continuum structures. As stated earlier, 3DCP extrudes materials filament by filament, and the adjacent filaments are cold jointed, forming clear interfaces. The geometric feature of the printed member is closer to discrete structures such as frames. More relevant discussions between the discrete and continuum topology optimization for additive manufacturing have been recently given by Carstensen et al.³⁹ Therefore, the optimization of 3DCP is to be tackled on the basis of discrete frame structures in this study. In doing so, naturally, the boundary of each member is smooth and the size of each member is uniform,^39,40 which is potential for direct printing. In addition, a 3DCP methodology optimizing the structural topology and the printing path simultaneously, instead of optimizing the topology first and then generating the tool path,⁴⁰ is proposed. Although the concept of simultaneous optimization has been introduced for other additive manufacturing technologies,^41–43 the difference between the cold-joint and hot-joint printing renders 3DCP unique. The proposed methodology utilizes the anisotropy of the printed material and the graph theory,⁴⁴ guaranteeing the mechanical efficiency of the optimized structure and the dimensional integer multiple of the nozzle size for the “one-stroke” path. The optimization problem is mathematically formulated as a mixed-integer linear program (MILP), which is readily solvable. The obtained structure is printable directly and efficiently.

The remainder of this article is organized as follows. First, the optimization problem for simultaneous topology and path optimization is mathematically formulated. Then, the optimization formulation is tested by a series of problems. The effectiveness of the simultaneous optimization needs verification. Finally, conclusion is drawn.

Problem Formulation

To optimize the structural topology and printing path of a frame simultaneously, this study follows the popular ground structure approach.^39,45 The given design domain provided with appropriate boundary conditions and external loads is decomposed by often evenly distributed nodes, as illustrated in Figure 1. The link between two arbitrary nodes represents a potential frame member, and the ground structure is composed of all potential members within the design domain.

FIG. 1.

Illustration of a frame ground structure with member intersection and overlapping.

Topology optimization

The topology optimization seeks the minimum weight of the structure with constraints on the nodal displacements and on the local member stresses by determining the existence of each potential member and finding the cross-sectional areas of the existing members. Considering the 3DCP’s material anisotropy, which exhibits the strongest performance in the printing direction, each existing member is preferred to be printed along its own length direction to maximize the member’s load-carrying capacity. This idea utilizes the anisotropy aptly and is similar to the additive manufacturing along the principal stress lines in the continuum setting.^46–49 Consequently, the manufacturing constraint of the nozzle size requires the width of the member to be an integer multiple of the nozzle size, that is, the $i$ -th member can take an area $A_{j}$ from a given set $A_{j} \in {A_{f}, 2 A_{f}, \dots, n_{f} A_{f}}$ where $A_{f}$ is the cross-sectional area of one single filament and $n_{f}$ is the number of allowable filaments, or else the member is not present in the design. A binary design variable, $x_{i j}$ , can be used to denote the presence of the $i$ -th member with area $A_{j}$ . $x_{i j} = 1$ denotes the presence and $x_{i j} = 0$ denotes the absence.

In addition, the frame topology optimization problem can be formulated in a nested analysis and design (NAND) form or a simultaneous analysis and design (SAND) form.⁵⁰ The former uses $x_{i j}$ as the only discrete design variable, and the nodal displacements and member forces are solved from finite element equations, resulting in non-linear constraints on the nodal displacements and member stresses, rendering the problem an integer non-linear program, which is difficult to solve mathematically. In contrast, the latter utilizes the nodal displacements and member forces as additional continuous design variables, imposing the equilibrium and stiffness conditions explicitly as linear constraints, rendering the problem an MILP which is easier to solve. Therefore, the SAND approach is adopted in this study. Following the formulation proposed by Van Mellaert et al.⁵¹ with a minor modification that permits the absence of the member, the topology optimization problem is defined as follows:

\begin{matrix} minimize \\ x_{i j}, u, q_{i j} \end{matrix} \sum_{i = 1}^{n_{e}} L_{i} \sum_{j = 1}^{n_{f}} A_{j} x_{i j} (volume)

subject to \sum_{j = 1}^{n_{f}} x_{i j} \leq 1 (single area or absence of the member)

\sum_{i = 1}^{n_{e}} \sum_{j = 1}^{n_{f}} B_{i}^{T} T_{i}^{T} R_{i} q_{i j} = f (force equilibrium)

(1 - x_{i j}) {\underline{q}}_{i j}^{'} \leq K_{i j} T_{i} B_{i} u - q_{i j} \leq (1 - x_{i j}) {\bar{q}}_{i j}^{'} (member stiffness condition)

x_{i j} {\underline{q}}_{i j}^{'} \leq q_{i j} \leq x_{i j} {\bar{q}}_{i j}^{'} (member forces)

{\underline{d}}_{i j}^{'} + x_{i j} ({\underline{d}}_{i j} - {\underline{d}}_{i j}^{'}) \leq D_{i} T_{i} B_{i} u + {\tilde{d}}_{i j} \leq {\bar{d}}_{i j}^{'} + x_{i j} ({\bar{d}}_{i j} - {\bar{d}}_{i j}^{'}) (member displacements)

{\underline{s}}_{i j}^{'} + x_{i j} ({\underline{s}}_{i j} - {\underline{s}}_{i j}^{'}) \leq S_{i j} T_{i} B_{i} u + {\tilde{s}}_{i j} \leq {\bar{s}}_{i j}^{'} + x_{i j} ({\bar{s}}_{i j} - {\bar{s}}_{i j}^{'}) (member stresses)

x_{i j} \in {0, 1}, \forall i \in {1, \dots, n_{e}}, \forall j \in {1, \dots, n_{f}},

(1)

where

n_{e}

is the total number of potential frame members,

L_{i}

is the length of the

i

-th member,

q_{i j} \in R^{3}

is the independent member force vector usually choosing the left axial force and the bending moments at the two ends,

R_{i} \in R^{6 \times 3}

is a matrix relating the three independent member force components to the full six components,

T_{i} \in R^{6 \times 6}

is the transformation matrix rotating the member from the global to the local coordinate system,

B_{i}

is a

6 \times n_{dof}

binary location matrix mapping the global degrees of freedom to the member degrees of freedom with

n_{dof}

being the number of global degrees of freedom,

f \in R^{n_{dof}}

is the global external load vector,

u \in R^{n_{dof}}

is the global nodal displacement vector,

K_{i j} \in R^{3 \times 6}

is the frame member stiffness matrix containing the three rows corresponding to the independent member force components,

D_{i}

is the member displacement matrix at the location of interest,

S_{i j}

is the member stress matrix at the location of interest,

{\tilde{d}}_{i j}

and

{\tilde{s}}_{i j}

are respectively the displacements and stresses at the location of interest due to the member loads by clamping the member ends,

{\underline{d}}_{i j}

and

{\bar{d}}_{i j}

are respectively the lower and upper bounds of displacements,

{\underline{s}}_{i j}

and

{\bar{s}}_{i j}

are respectively the lower and upper bounds of stresses, and

{\underline{q}}_{i j}^{'}

{\bar{q}}_{i j}^{'}

{\underline{d}}_{i j}^{'}

{\bar{d}}_{i j}^{'}

{\underline{s}}_{i j}^{'}

, and

{\bar{s}}_{i j}^{'}

are, respectively, the artificial bounds of member forces, displacements, and stresses.

For $n_{p}$ distributed nodes in the ground structure, the total number of potential frame members is $n_{e} = n_{p} (n_{p} - 1) / 2$ , as illustrated in Figure 1. The reader is referred to the Appendix A of Van Mellaert et al.⁵¹ for the complete expressions of $R_{i}$ , $T_{i}$ , $K_{i j}$ , $D_{i}$ , and $S_{i j}$ . The $r$ -th component of the artificial bounds ${\underline{q}}_{i j}^{'}$ and ${\bar{q}}_{i j}^{'}$ is obtained easily by solving the following linear program (LP)⁵²:

{\underline{q}}_{r, i j}^{'} = \begin{matrix} minimize \\ u \end{matrix} K_{r, i j} T_{i} B_{i} u subject to \underline{u} \leq u \leq \bar{u}

{\bar{q}}_{r, i j}^{'} = \begin{matrix} maximize \\ u \end{matrix} K_{r, i j} T_{i} B_{i} u subject to \underline{u} \leq u \leq \bar{u}

where

\underline{u}

and

\bar{u}

are the lower and upper bounds of nodal displacements, respectively. The artificial bounds

{\underline{d}}_{i j}^{'}

{\bar{d}}_{i j}^{'}

{\underline{s}}_{i j}^{'}

, and

{\bar{s}}_{i j}^{'}

are computed similarly by minimizing and maximizing

D_{r, i} T_{i} B_{i} u + {\tilde{d}}_{r, i j}

and

S_{r, i j} T_{i} B_{i} u + {\tilde{s}}_{r, i j}

, respectively.

Simultaneous path optimization

To optimize the printing path simultaneously, the graph theory⁴⁴ is employed. First, an efficient and high-quality printing requires the path to be globally continuous and to be covered only once. A Eulerian circuit fulfills this requirement as a “one-stroke” printing in one layer, and starting and ending at any identical position. For the subsequent layers, the printer simply necessitates a precise elevation of the nozzle, followed by a repetition of the previous path. In accordance with the graph theory, the existence of a Eulerian circuit is guaranteed whenever there is an even number (inclusive of zero) of filaments interconnecting all the distributed nodes. Therefore, this optimization constraint is expressed as

\sum_{i} \sum_{j = 1}^{n_{f}} j x_{i j} \equiv 0 (mod 2), i \in {member numbers connecting at node k}, \forall k \in {1, \dots, n_{p}}

In terms of integer LP,⁵³ such a constraint can be modeled as a linear constraint with respect to $x_{i j}$ by introducing an additional integer variable, $y_{k}$ , written as

\sum_{i} \sum_{j = 1}^{n_{f}} j x_{i j} - 2 y_{k} = 0, i \in {member numbers connecting at node k}

y_{k} \geq 0, y_{k} \in N, \forall k \in {1, \dots, n_{p}}

(5)

In this study, to narrow the optimization design space, the constraint is differently reformulated by introducing a binary variable, $y_{k l}$ . For each distributed node, the possibly maximum number of filaments is $(n_{p} - 1) n_{f}$ . Therefore, the number of positive even numbers, $n_{e v}$ , that are less than or equal to $(n_{p} - 1) n_{f}$ can be determined. $y_{k l} = 1$ denotes that the total number of filaments at node $k$ is the $l$ -th positive even number, that is, $2 l$ . The corresponding constraint is expressed by

\sum_{i} \sum_{j = 1}^{n_{f}} j x_{i j} = \sum_{l = 1}^{n_{e v}} 2 l y_{k l}, i \in {member numbers connecting at node k}

\sum_{l = 1}^{n_{e v}} y_{k l} \leq 1 (single choice of even - number filaments)

y_{k l} \in {0, 1}, \forall k \in {1, \dots, n_{p}}, \forall l \in {1, \dots, n_{e v}}

(6)

In practical computational scenarios, to further narrow the design space and enhance efficiency, it is advisable to truncate the right-hand side of the first equation above after the first term, which means that at most two filaments connect at the node. Subsequently, this truncation can be iteratively augmented by successively adding one term at a time (the maximum number of filaments at the node is increased by two), facilitating the exploration of a converged feasible solution. This process continues, if no feasible solution is found, until the condition $l = n_{e v}$ is met.

Second, there may exist necessary member intersection and overlapping in the initial ground structure, as illustrated in Figure 1, to not restrict the optimization design space. Consequently, the intersection and overlapping of members may be inherited in the Eulerian circuit-based frame. In practical printings, however, the intersection and overlapping phenomena should be avoided to ensure continuity and high quality. All the possible intersection and overlapping scenarios⁵⁴ are plotted in Figure 2. It can be observed that if two members are non-parallel and the intersection point is located inside at least one member (excluding the two endpoints), the two members intersect; if two members are a pair of colinear lines and share a common segment, the two members overlap. By associating each pair of potential members in the ground structure, the geometric relation between the two members can be determined mathematically through the member orientations and point coordinates. The algorithmic pseudocode is given in Appendix. If the $s$ -th member intersects or overlaps with the $t$ -th member, the following constraint is to be satisfied:

\sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j} \leq 1

(7)

FIG. 2.

All possible scenarios for intersection and overlapping of two members.

This constraint assures that at least one of the two members does not exist and hence the intersection or overlapping is eliminated, and this constraint is a linear constraint with respect to $x_{i j}$ .

Moreover, in 3DCP, sharp turns in the printing path should be avoided to prevent defects such as underfilling and overfilling. A straightforward way to achieve this is by imposing an angular constraint between two potential members meeting at a common node. Such a constraint can also be conveniently enforced by Equation(7). Specifically, if the angle between the two members falls below a given threshold, the formulation ensures that at least one of them is excluded from the final layout, thereby precluding the occurrence of a sharp corner.

Last, since the external load is treated as equivalent nodal loads in the topology optimization, the members subjected to interior loads may be absent after the optimization, which will be shown later in a test problem. In this case, the members carrying the interior loads are mandatory, and hence the following path constraint can be provided:

\sum_{j = 1}^{n_{f}} x_{i j} = 1, i \in {member numbers carrying interior loads}

(8)

This constraint is also a linear constraint with respect to $x_{i j}$ .

In summary, Equations (1) and (6 )–(8) define the simultaneous topology and path optimization problem for 3DCP. Owing to the requirement of structural topology optimization that the force transmission path must remain continuous and unbroken from the loading point to the supports, multiple disjoint paths are inherently excluded under the volume minimization objective. As will be demonstrated in the subsequent benchmark tests, this formulation consistently yields a single Eulerian circuit as the optimal solution. The overall optimization problem is formulated as an MILP, which is readily solvable. The “intlinprog” MILP solver, which predominantly relies on a branch-and-cut algorithm, in the general computing tool MATLAB⁵⁵ is employed subsequently to solve the test problems. And the solver is run on a laptop with a 1.6 GHz dual-core processor and 16 GB of memory.

Test Problems and Optimization Results

In this section, the simultaneous topology and path optimization formulation presented above is applied to five popular problems. The first problem is a simply supported beam subjected to a simple mid-span load. It is used to verify the optimization formulation because it can also be solved by complete enumeration. The second problem changes the single mid-span load into two concentrated loads, resulting in a four-point bending beam. The third problem changes the simply supported condition into a cantilever constraint. The optimal design of the cantilever location is to be investigated. The fourth problem is a simply supported beam subjected to a uniformly distributed load. In this case, due to the distributed load, not only the nodal displacements and stresses but also the displacements and stresses at the interior locations of the loaded members can be constrained. The first four problems analyze standard rectangular design domains, whereas the last one investigates a more complex L-shaped domain.

Simply supported beam subjected to a mid-span load

Figure 3a shows the design domain for a simply supported beam structure subjected to a mid-span load on the top surface. The span is $W = 1.2$ m and the height is $H = 0.3$ m. The cross sections along the printed layers (perpendicular to the plane) remain constant and thus only one layer in the plane needs to be optimized. The printed filament is approximated as a rectangle with a nozzle width $b = 0.02$ m and a layer height $h = 0.01$ m, resulting in a filament area $A_{f} = 2 \times 10^{- 4}$ m². According to the experimental data,⁵⁶ the Young’s modulus of the hardened material along the printing direction is taken as $E = 20$ GPa, and the compressive and tensile strengths are set as $σ_{c} = 50$ MPa and $σ_{t} = 5$ MPa, respectively. Since the external load is applied at one node, only the nodal displacements and the normal stresses at the two ends of each member are bounded. The allowable nodal horizontal and vertical displacements are both $u_{allow} = 0.01$ m, about $1 / 100$ of the span, and the allowable nodal rotation is $θ_{allow} = 0.01$ rad (about 0.57°), a strict bound to fully harness the compressive capacity of the concrete material and prevent tensile fracture.

FIG. 3.

Design domains for a simply supported beam subjected to a mid-span load on: (a) top surface and (b) bottom surface.

Since members straightly connecting the loading point and the two supports are mechanically the most efficient ones to transfer the external load, it is sufficient to decompose the design domain by evenly distributed $n_{p} = 3 \times 2$ (horizontal by vertical) nodes ( $n_{e} = 15$ , $n_{dof} = 15$ , and 19 intersection or overlapping cases) for a small load, say $F = 10$ N. If only the topology optimization problem in Equation (1) is considered and the number of allowable filaments is $n_{f} = 1$ ( $n_{e v} = 2$ ), the computational time was 0.6 s and the optimal design is plotted in Figure 4a, that is, two single-filament members connecting the loading point and the two supports, respectively. This is understandable because of the aforementioned reason. Densifying the node distribution (the loading and support nodes are fixed) and/or increasing $n_{f}$ will not change the design since the present volume is the minimum. With this optimal design, the maximum displacement is the vertical displacement at the loading point, $u_{max} = 0.003$ m, and the maximum rotations are the two support rotations, $θ_{max} = 0.0075$ rad. The maximum compressive stress, $σ_{c max} = 4.51$ MPa, occurs at the top fiber of the loading point, and the maximum tensile stress $σ_{t max} = 4.49$ MPa, which is close to the tensile strength $σ_{t} = 5$ MPa, occurs at the bottom fiber of the loading point. The optimal volume is $V = 2.68 \times 10^{- 4}$ m³.

FIG. 4.

Optimal designs of the simply supported beam subjected to a top mid-span load $F = 10$ N: (a) without path optimization and (b) with path optimization.

However, the path of the optimal design above is not an Eulerian circuit. Therefore, the path optimization constraints in Equations (6) and (7) are further supplied. The computational time was 1.5 s, and the optimal design obtained is shown in Figure 4b. A bottom single-filament member connecting the two supports is added to form an Eulerian circuit path, which is convenient for direct printing. Due to the stiffness increase of the whole structure, the vertical displacement at the loading point is reduced to $u_{max} = 7.17 \times 10^{- 6}$ m and the support rotation is reduced to $θ_{max} = 1.24 \times 10^{- 5}$ rad. Meanwhile, the maximum compressive stress at the top fiber of the loading point is reduced to $σ_{c max} = 0.067$ MPa, and the location of the maximum tensile stress, $σ_{t max} = 0.054$ MPa, switches to the bottom fiber of the bottom member. The current optimal volume is $V = 5.08 \times 10^{- 4}$ m³.

Since the maximum tensile stress above is about $1 / 100$ of the tensile strength, the load is amplified 100 times, that is, $F = 1000$ N. If the number of allowable filaments is still $n_{f} = 1$ and only the topology optimization in Equation (1) is considered, the computational time was 1.4 s, and the optimal design is given in Figure 5a with the optimal volume $V = 7.48 \times 10^{- 4}$ m³. Obviously, the printing path is not an Eulerian circuit, and member overlapping at the bottom of the design is observed, which is impossible for printing. This emphasizes the necessity of path optimization in addition to the topology optimization. By adding the path optimization constraints in Equations (6) and (7), it is found that there is no feasible design under the requirement of $n_{f} = 1$ for $n_{p} = 3 \times 2$ nodes because the member tensile stress constraint cannot be satisfied. More nodes and thus more potential members can be utilized to stiffen the structure. For instance, using $n_{p} = 3 \times 3$ nodes ( $n_{e} = 36$ , $n_{dof} = 24$ , $n_{e v} = 4$ , and 134 intersection or overlapping cases), the computational time was 44.8 s and the optimal design is shown in Figure 5b with the optimal volume $V = 10.63 \times 10^{- 4}$ m³; while using $n_{p} = 3 \times 4$ nodes ( $n_{e} = 66$ , $n_{dof} = 33$ , $n_{e v} = 5$ , and 510 intersection or overlapping cases), the computational time was 1539.4 s and the optimal design is shown in Figure 5c with a smaller volume $V = 10.45 \times 10^{- 4}$ m³. As the number of nodes aligned vertically increases (note that increasing the number of nodes aligned horizontally is not the most helpful for the load transfer), the two top side nodes move toward the supports, and the central node approaches the bottom member, rendering the widths of the inclined and bottom members doubled. This design trend can be verified by setting $n_{f} = 2$ for $n_{p} = 3 \times 2$ nodes ( $n_{e v} = 5$ ). The computational time was 16.9 s, and the optimal design, as shown in Figure 5d, is achieved by doubling the filament numbers of each member in Figure 4b, resulting in the optimal volume of $V = 10.17 \times 10^{- 4}$ m³. In addition, if $n_{f} = 3$ ( $n_{e v} = 7$ ), the computational time was 44.3 s and the optimal design further produces a smaller volume, $V = 9.88 \times 10^{- 4}$ m³, as shown in Figure 5e. The two inclined members are single-filament members, and the bottom member is a triple-filament one. For even larger $n_{f}$ , the optimal design remains.

FIG. 5.

Optimal designs of the simply supported beam subjected to a top mid-span load $F = 1000$ N: (a) $n_{f} = 1$ and $n_{p} = 3 \times 2$ nodes without path optimization; (b) $n_{f} = 1$ and $n_{p} = 3 \times 3$ nodes with path optimization; (c) $n_{f} = 1$ and $n_{p} = 3 \times 4$ nodes with path optimization; (d) $n_{f} = 2$ and $n_{p} = 3 \times 2$ nodes with path optimization; and (e) $n_{f} = 3$ and $n_{p} = 3 \times 2$ nodes with path optimization.

Similarly, the mid-span load may be applied on the bottom surface of the design domain, as shown in Figure 3b. When $F = 10$ N, the optimal design for $n_{f} = 1$ and $n_{p} = 3 \times 2$ nodes is the same as that shown in Figure 4b with 1.1 s of computational time, but the different loading manner produces much higher maximum vertical displacement, $u_{max} = 1.12 \times 10^{- 3}$ m, at the loading point and much higher support rotation, $θ_{max} = 1.48 \times 10^{- 3}$ rad, which leads to larger maximum compressive stress, $σ_{c max} = 2.71$ MPa, at the top fiber of the loading point and larger maximum tensile stress, $σ_{t max} = 2.77$ MPa, at the bottom fiber of the loading point.

For $n_{p} = 3 \times 3$ , $n_{p} = 3 \times 4$ , and $n_{p} = 3 \times 5$ ( $n_{e} = 105$ , $n_{dof} = 42$ , $n_{e v} = 7$ , and 1323 intersection or overlapping cases) nodes, the computational times were 15.3, 224.5, and 1170.7 s, respectively. And the optimal designs are plotted in Figure 6a–c, respectively. The tendency of lowering the top point of the structure to decrease the volume is observed. It is reasonable to deduce that the limit situation will be reached when the top point falls onto the bottom member. This deduction is verified by allowing $n_{f} = 2$ for $n_{p} = 3 \times 2$ nodes. The computational time was 2.0 s, and the optimal design, as shown in Figure 6d, reduces to a simple double-filament beam possessing a volume $V = 4.8 \times 10^{- 4}$ m³. This is possibly the lightest design which connects the loading point and the two supports directly and forms an Eulerian circuit. Densifying the node distribution and/or further increasing $n_{f}$ will not change the design.

FIG. 6.

Optimal designs of the simply supported beam subjected to a bottom mid-span load $F = 10$ N: (a) $n_{p} = 3 \times 3$ nodes with $n_{f} = 1$ ; (b) $n_{p} = 3 \times 4$ nodes with $n_{f} = 1$ ; (c) $n_{p} = 3 \times 5$ nodes with $n_{f} = 1$ ; (d) $n_{p} = 3 \times 2$ nodes with $n_{f} = 2$ ; and (e) $n_{p} = 3 \times 4$ nodes, $n_{f} = 1$ with angular constraints.

As the top point moves downward, the bottom angles become progressively sharper. For instance, in the optimal design shown in Figure 6b, the angle is reduced to 9.5°. Assuming a minimum threshold of 10° is required to prevent underfilling or overfilling, this design violates the constraint. To address this, the angular constraint Equation 7 between the top inclined members and the bottom member was imposed. The resulting optimal design, shown in Figure 6e, successfully increases the angle to 18.4°, which is well above the required threshold.

Four-point bending beam

This problem mimics the four-point bending on a beam, as shown in Figure 7. Similar to the first problem, $n_{p} = 4 \times 2$ nodes ( $n_{e} = 28$ , $n_{dof} = 21$ , and 90 intersection or overlapping cases) are a good start for the node distribution. When the loads are small, say $F = 10$ N, Figure 8a shows the optimal design for $n_{f} = 1$ ( $n_{e v} = 3$ ), with 12.5 s of computational time and $V = 5.20 \times 10^{- 4}$ m³ of volume. This design is understandable because the members connect the loading points and the supports directly and form an Eulerian circuit.

FIG. 7.

Design domain for a four-point bending beam.

FIG. 8.

Optimal designs of the four-point bending beam: (a) $F = 10$ N, $n_{p} = 4 \times 2$ nodes with $n_{f} = 1$ ; (b) $F = 710$ N, $n_{p} = 4 \times 3$ nodes with $n_{f} = 1$ ; (c) $F = 710$ N, $n_{p} = 4 \times 2$ nodes with $n_{f} = 2$ ; and (d) $F = 710$ N, $n_{p} = 4 \times 2$ nodes with $n_{f} = 3$ .

It is further found that when the loads are slightly beyond 700 N, there is no feasible solution for $n_{f} = 1$ and $n_{p} = 4 \times 2$ nodes. Densifying the node distribution or increasing $n_{f}$ is necessary. Consequently, when $F = 710$ N, the optimal designs for $n_{p} = 4 \times 3$ nodes ( $n_{e} = 66$ , $n_{dof} = 33$ , and 497 intersection or overlapping cases) with $n_{f} = 1$ ( $n_{e v} = 5$ ), $n_{p} = 4 \times 2$ nodes with $n_{f} = 2$ ( $n_{e v} = 7$ ), and $n_{p} = 4 \times 2$ nodes with $n_{f} = 3$ ( $n_{e v} = 10$ ) are shown in Figure 8b–d, respectively. The computational times were 1819.5, 496.0, and 1041.4 s, respectively. The trend of stiffening the structure by more or stronger members is well observed.

Cantilever beam

This problem is a cantilever beam subjected to an end load, as shown in Figure 9. When $F = 10$ N, the optimal design is shown in Figure 10a for $n_{p} = 2 \times 3$ nodes and $n_{f} = 1$ ( $n_{e} = 15$ , $n_{dof} = 9$ , $n_{e v} = 2$ , and 19 intersection or overlapping cases) with 0.7 s of computational time. By increasing the number of nodes aligned vertically ( $n_{p} = 2 \times 5$ [ $n_{e} = 45$ , $n_{dof} = 15$ , $n_{e v} = 4$ , and 265 intersection or overlapping cases] and $n_{p} = 2 \times 7$ nodes [ $n_{e} = 91$ , $n_{dof} = 21$ , $n_{e v} = 6$ , and 1227 intersection or overlapping cases]), designs with smaller volume are obtained, as shown in Figure 10b and c. The computational times were 4.3 and 60.7 s, respectively. It can be observed that the smaller volume is achieved by adjusting the location of the fixed support, and the two cantilever members tend to merge together. This tendency is verified by allowing $n_{f} = 2$ for $n_{p} = 2 \times 3$ nodes ( $n_{e v} = 5$ ), as shown in Figure 10d. The computational time was 0.9 s. The optimal design is reduced to a double-filament cantilever beam, and it remains while densifying the node distribution and/or increasing $n_{f}$ .

FIG. 9.

Design domain for a cantilever beam.

FIG. 10.

Optimal designs of the cantilever beam: (a) $n_{p} = 2 \times 3$ nodes with $n_{f} = 1$ ; (b) $n_{p} = 2 \times 5$ nodes with $n_{f} = 1$ ; (c) $n_{p} = 2 \times 7$ nodes with $n_{f} = 1$ ; and (d) $n_{p} = 2 \times 3$ nodes with $n_{f} = 2$ .

Simply supported beam subjected to a uniformly distributed load

The fourth problem designs a simply supported beam structure subjected to a uniformly distributed load, as shown in Figure 11. Since the top-chord member is bent by the distributed load, besides the regular nodal displacements and stresses, specific constraints are imposed on the displacements and stresses at five equidistant locations, marked by “x” in Figure 11. To accommodate these five critical locations, two distinct strategies for node distribution are adopted. The first strategy involves positioning five uniformly spaced nodes horizontally, each directly corresponding to one of the marked locations. Alternatively, the second strategy utilizes three horizontally uniform nodes, necessitating additional constraints at the midpoints of the two halves of the top-chord member to ensure coverage of the designated locations.

FIG. 11.

Design domain for a simply supported beam subjected to a uniformly distributed load.

When $q$ is small, say $q = 10$ N/m, the two strategies ( $n_{p} = 5 \times 2$ and $n_{f} = 1$ [ $n_{e} = 45$ , $n_{dof} = 27$ , $n_{e v} = 4$ , and 268 intersection or overlapping cases] vs. $n_{p} = 3 \times 2$ and $n_{f} = 1$ [ $n_{e} = 15$ , $n_{dof} = 15$ , $n_{e v} = 2$ , and 19 intersection or overlapping cases]) yield identical optimal design, as evident from Figure 12a and b, thereby mutually validating their efficacy. The computational times were 65.3 and 2.0 s, respectively. However, as $q$ approaches 33 N/m, there is no feasible solution for $n_{p} = 3 \times 2$ nodes and $n_{f} = 1$ . Densifying the node distribution or increasing $n_{f}$ is required. Figure 13 shows the optimal designs obtained for $q = 35$ N/m, using $n_{p} = 3 \times 3$ nodes with $n_{f} = 1$ , $n_{p} = 3 \times 4$ nodes with $n_{f} = 1$ , and $n_{p} = 3 \times 2$ nodes with $n_{f} = 2$ . The computational times were 52.3, 425.9, and 9.2 s, respectively.

FIG. 12.

Optimal designs of the simply supported beam subjected to a uniformly distributed load $q = 10$ N/m: (a) $n_{p} = 5 \times 2$ nodes with $n_{f} = 1$ and (b) $n_{p} = 3 \times 2$ nodes with $n_{f} = 1$ .

FIG. 13.

Optimal designs of the simply supported beam subjected to a uniformly distributed load $q = 35$ N/m using three uniform nodes horizontally: (a) $n_{p} = 3 \times 3$ nodes with $n_{f} = 1$ ; (b) $n_{p} = 3 \times 4$ nodes with $n_{f} = 1$ ; and (c) $n_{p} = 3 \times 2$ nodes with $n_{f} = 2$ .

Figure 14a shows the optimal design for $q = 35$ N/m using $n_{p} = 5 \times 2$ nodes with $n_{f} = 1$ (80.9 s of computational time), but without incorporating the constraint defined in Equation (8). As previously discussed, due to the treatment of the distributed load as equivalent nodal loads, a crucial segment of the top-chord member is conspicuously absent from the design, rendering it incapable of withstanding the actual load. By incorporating the constraint from Equation (8) into the problem formulation, the optimal design, as shown in Figure 14b, now includes the essential top-chord member in its entirety, emphasizing the pivotal role of this constraint in optimizing structures subjected to interior loads. The computational time was 45.6 s. In addition, Figure 14c and d shows the optimal designs obtained using $n_{p} = 5 \times 3$ nodes with $n_{f} = 1$ ( $n_{e} = 105$ , $n_{dof} = 42$ , $n_{e v} = 7$ , and 1330 intersection or overlapping cases) and $n_{p} = 5 \times 2$ nodes with $n_{f} = 2$ ( $n_{e v} = 9$ ), respectively, demonstrating that lighter structures can be achieved through the employment of more nodes or larger $n_{f}$ . The computational times were 1443.9 and 393.7 s, respectively.

FIG. 14.

Optimal designs of the simply supported beam subjected to a uniformly distributed load $q = 35$ N/m using five uniform nodes horizontally: (a) $n_{p} = 5 \times 2$ nodes, $n_{f} = 1$ without constraint Equation (8); (b) $n_{p} = 5 \times 2$ nodes, $n_{f} = 1$ with constraint Equation (8); (c) $n_{p} = 5 \times 3$ nodes with $n_{f} = 1$ ; and (d) $n_{p} = 5 \times 2$ nodes with $n_{f} = 2$ .

L-shaped bracket

The last problem evaluates a more complex design domain, specifically an L-shaped domain, as shown in Figure 15. Therefore, any member connecting two arbitrary nodes that lie partially or entirely outside this domain cannot be incorporated into the ground structure. Figure 16a and b plots the optimal designs for $F = 5$ N and $F = 7$ N, respectively, using $n_{f} = 1$ and $n_{p} = 8$ nodes as shown in the figures ( $n_{e} = 25$ , $n_{dof} = 21$ , $n_{e v} = 3$ , and 62 intersection or overlapping cases). Their computational times were 1.1 and 0.8 s, respectively. When the load is increased close to $F = 7.7$ N, there is no feasible design. Densifying the node distribution or increasing $n_{f}$ is required.

FIG. 15.

Design domain for an L-shaped bracket subjected to a point load.

FIG. 16.

Optimal designs of the L-shaped bracket using $n_{f} = 1$ and $n_{p} = 8$ nodes for: (a) $F = 5$ N and (b) $F = 7$ N.

Figure 17a and b shows the optimal designs for $F = 8$ N using more nodes ( $n_{p} = 21$ , as shown in the figure) and larger number of allowable filaments ( $n_{f} = 2$ ), respectively. For the former case, $n_{e} = 185$ , $n_{dof} = 60$ , $n_{e v} = 10$ , and there are 4302 intersection or overlapping scenarios. The computational time was 27.2 s. And for the latter case, the computational time was 1.2 s. It is observed that both cases yield lighter structures.

FIG. 17.

Optimal designs of the L-shaped bracket for $F = 8$ N: (a) $n_{f} = 1$ and $n_{p} = 21$ nodes and (b) $n_{f} = 2$ and $n_{p} = 8$ nodes.

Conclusion

In this article, an innovative methodology for simultaneous topology and path optimization of 3DCP is introduced. This formulation, grounded in discrete frame structures, meticulously approximates the geometric intricacies of 3DCP, thereby enhancing the fidelity of the optimization process. Furthermore, the methodology astutely harnesses the mechanical anisotropy inherent in 3DCP, further optimizing the designs’ performance. The efficacy of this simultaneous optimization approach is rigorously demonstrated through a series of test cases. In these tests, the topology optimization component ensures that the resulting designs not only adhere to stringent structural displacement and stress constraints but also achieve minimal weight, optimizing material usage. Simultaneously, the path optimization guarantees global continuity, ensuring the design is traversed seamlessly, once and only once, without intersections or overlaps. In addition, it guarantees that the member widths are integer multiples of the nozzle size, facilitating seamless integration with the printing process. Consequently, the optimal designs generated by this methodology are poised for direct “one-stroke” printing, seamlessly bridging the gap between the optimized digital model and its physical realization. This integration not only streamlines the manufacturing process but also underscores the potential of advanced optimization techniques in revolutionizing the field of 3DCP. While this study optimizes single-layer structures with uniform cross-sections and in-plane loading, future work will extend this framework to three-dimensional applications. A key objective will be the incorporation of additional manufacturing constraints, such as maximum self-supporting overhang angles.

Authors’ Contributions

M.L. designed the article concept and methodology from the original idea and wrote the main article text. X.S. implemented the investigation and prepared the visualization. Z.C. acquired the funding and supervised the implementation. All authors reviewed the article.

Footnotes

Author Disclosure Statement

The authors declare that there are no conflicts of interest.

Funding Information

This study was funded by the National Key Research and Development Program of China (No. 2021YFF0500804).

Appendix

Algorithmic pseudocode for identifying intersection or overlapping of two potential members

% In: ne – total number of potential members

% (x1, y1), (x2, y2) – coordinates of the two endpoints of the first member

% (x3, y3), (x4, y4) – coordinates of the two endpoints of the second member

% Out: CS – constraints

for s = 1:ne-1

for t = s + 1:ne

A = [y1-y2, x1-x2;

y3-y4, x3-x4]; % member slope matrix

B = [y1-y2, x2-x1, x2*y1-x1*y2;

y3-y4, x4-x3, x4*y3-x3*y4]; % coefficient matrix of two member lines

C = GE(B); % Gaussian elimination. C(1,3), C(2,3) – x and y values of intersection point

if det(A)∼ = 0% non-parallel members, intersecting potentially

If x1 = x2 and y3 = y4% first member vertically and second member horizontally

if ${min (x 3, x 4) < C (1, 3) < max (x 3, x 4) and min (y 1, y 2) < C (2, 3) < max (y 1, y 2)}$

or ${min (x 3, x 4) < C (1, 3) < max (x 3, x 4) and [C (2, 3) = y 1 or C (2, 3) = y 2)]}$

or ${min (y 1, y 2) < C (2, 3) < max (y 1, y 2) and [C (1, 3) = x 3 or C (1, 3) = x 4)]}$

CS= $\sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j}$ ;

end if

else if y1 = y2 and x3 = x4% first member horizontally and second member vertically

if ${min (x 1, x 2) < C (1, 3) < max (x 1, x 2) and min (y 3, y 4) < C (2, 3) < max (y 3, y 4)}$

or ${min (x 1, x 2) < C (1, 3) < max (x 1, x 2) and [C (2, 3) = y 3 or C (2, 3) = y 4)]}$

$or {min (y 3, y 4) < C (2, 3) < max (y 3, y 4) and [C (1, 3) = x 1 or C (1, 3) = x 2)]}$

$CS = \sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j};$

end if

else if (y1 = y2 and x3∼ = x4) or (y3 = y4 and x1∼ = x2) % one member horizontally and the other not vertically

if ${min (x 1, x 2) < C (1, 3) < max (x 1, x 2) and min (x 3, x 4) < C (1, 3) < max (x 3, x 4)}$

or ${min (x 1, x 2) < C (1, 3) < max (x 1, x 2) and [C (1, 3) = x 3 or C (1, 3) = x 4)]}$

or ${min (x 3, x 4) < C (1, 3) < max (x 3, x 4) and [C (1, 3) = x 1 or C (1, 3) = x 2)]}$

$CS = \sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j};$

end if

else % both members not horizontally

if ${min (y 1, y 2) < C (2, 3) < max (y 1, y 2) and min (y 3, y 4) < C (2, 3) < max (y 3, y 4)}$

or ${min (y 1, y 2) < C (2, 3) < max (y 1, y 2) and [C (2, 3) = y 3 or C (2, 3) = y 4]]}$

or ${min (y 3, y 4) < C (2, 3) < max (y 3, y 4) and [C (2, 3) = y 1 or C (2, 3) = y 2)]}$

CS= $\sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j}$ ;

end if

else % parallel members

if C(2,3) = 0% colinear members, overlapping potentially

if y1 = y2% horizontal members

if min(x3,x4)<x1<max(x3,x4) or min(x3,x4)<x2<max(x3,x4) or

min(x1,x2)<x3<max(x1,x2) or min(x1,x2)<x4<max(x1,x2)

CS = $\sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j}$ ;

end if

else % not horizontal members

if min(y3,y4)<y1<max(y3,y4) or min(y3,y4)<y2<max(y3,y4) or

min(y1,y2)<y3<max(y1,y2) or min(y1,y2)<y4<max(y1,y2)

CS = $\sum_{j = 1}^{n_{f}} x_{s j} + \sum_{j = 1}^{n_{f}} x_{t j}$ ;

end if

end for

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