Simultaneous optimization of build orientation and topology for self-supported enclosed voids in additive manufacturing

Highlights

• Topology optimization of self-supported enclosed voids for additive manufacturing.

• Identification of enclosed voids with small surface slope through pseudo heat transfer problem.

• Density gradient based formulations for surface slope dependent heat flux loading.

• Differentiable to build orientation.

• Applicable to both 2D and 3D problems.

Abstract

The paper proposes a heat-flux based approach to optimize build orientation and topology simultaneously for self-supported enclosed voids in additive manufacturing. The enclosed overhangs that require supports in additive manufacturing are removed from the optimized design by constraining the maximum temperature of a pseudo heat conduction problem. In the pseudo problem, heat flux is applied on the non-self-supported open and enclosed surfaces. Since the density-based topology optimization involves no explicit boundary representation, we impose such surface slope dependent heat flux through a domain integral of a Heaviside projected density gradient. In addition, the solid materials and the void materials in the pseudo problem are assumed to be thermally insulating and conductive, respectively. As such, heat flux on the open surfaces can be successfully conducted to external heat sink through the void (or conductive) materials. However, heat flux on the non-self-supported enclosed surfaces is isolated by the solid (or insulating) materials and thus leads to locally high temperature. Hence, by limiting the maximum temperature of the pseudo problem, self-supported enclosed voids can be achieved, and the slope of the open surfaces would not be affected. Due to the differentiability of the heat flux loading in the pseudo problem, the calculated temperature and constraint on it are differentiable to both the build orientation and the density field. Numerical examples on 2D and 3D linear elasticity problems are presented to demonstrate the validity and effectiveness of the proposed approach in the design of self-supported enclosed voids.

Introduction

Topology optimization allows designers to find parts of complex geometry in a given design domain for desired performance [1]. Since the seminal work of Bendsøe and Kikuchi [2], topology optimization approaches have been widely used in the design of industrial products [1], [3], [4]. It is generally challenging to fabricate the optimized parts of complex geometry through traditional manufacturing techniques. Additive manufacturing makes it possible to produce such parts by depositing successive layers of materials on a build plate [5]. Combining topology optimization and additive manufacturing hence enables us to exploit the design space and produce industrial products of complex geometry. In this paper, we propose a topology optimization approach for designing parts with self-supported enclosed voids that are favorable for additive manufacturing.

Support structures are required during the additive manufacturing for undercuts or overhanging surfaces of small overhang angle. The fabrication of such support structures will waste a huge amount of materials and significantly increase the build time. Furthermore, the removal of support structures after build can be a laborious process for some complex designs. For the enclosed voids, the internal support structures are even inaccessible for machining tools and thus cannot be removed. We can eliminate support structures by constraining the lower bound of the surface slope of the optimized part, i.e. $\alpha \ge \bar{\alpha }$. However, imposing the slope constraint on both the enclosed surfaces and the open surfaces (see Fig. 1) can significantly impair the performance of the part. Since the support structures of the open surfaces can be removed after build, a compromise solution is to control only the surface slope of the enclosed voids. In the paper, we achieve this goal by constraining the maximum temperature of a pseudo heat transfer problem under the surface slope dependent heat flux (see Fig. 1). The proposed constraint is differentiable to the build orientation and the density field.

Additive manufacturing related constraints have been considered in topology optimization. Such constraints include the minimal feature size constraint [6], [7], [8], [9], [10], [11], elimination of enclosed voids [12], [13], surface slope control for self-support [14], [15], [16] and surface roughness [15], and component build volume control [17], [18] etc. Since surface slope depends on the representation of boundary, it is more challenging to impose the surface slope constraint in the density-based topology optimization such as the SIMP (Simplified Isotropic Material with Penalization) method [1], [19]. Density gradient based formulations have been proposed in Qian [14] to control undercut and surface slope of overhangs. Wang et al. [15] extended the density gradient based formulations to control surface slope for both self-support and surface roughness. The differentiability of the density gradient formulations to build orientation has been found by Wang and Qian [16]. In [16], the authors optimized the build orientation and topology simultaneously to control both the internal and external supports of the optimized part. Zhang et al.[20], [21] utilized the density gradient formulation to impose overhang angle constraint, and proposed an extra constraint to remove boundary oscillations. Mezzadri and Qian [22] formulated a second-order measure of the boundary oscillations, and proposed an adaptive anisotropic filter and a cost penalty to suppress such oscillations in topology optimization. Filter schemes have been also proposed in [23], [24], [25], [26] to impose the surface slope constraint. In such approaches, unsupported elements are suppressed from the optimized design through density filtering. In [27], [28], a front propagation based approach is proposed to remove overhanging surfaces of small angle. Wang et al. [29] control overhang angle in B-splines based topology optimization through the density-gradient formulation, and suppress boundary oscillations by limiting the density values at detection points. The surface slope constraint for self-support has been also considered in the topology optimization approaches with explicit boundary representation [30], [31], [32], [33], [34], [35]. Besides reducing the use of support structures through the surface slope constraint, topology of supports has been optimized for saving manufacturing cost [36], enhancing heat dissipation [37], [38], [39], improving stiffness [40], [41] and reducing residual stresses [42]. The readers can refer to [43], [44] for an overview of research on additive manufacturing related constraints.

Imposing surface slope constraint only on enclosed voids is more challenging due to the difficulty of identifying internal surfaces in the density-based topology optimization. Recently, Luo et al. [45] identified the enclosed voids by solving a nonlinear heat conduction problem and controlled the surface slope of the enclosed voids through the density gradient based formulations [14], [15], [16], [20]. Ven et al. [46] combined the filter scheme proposed in [25] with geometric boolean operations to control the surface slope of the enclosed overhangs. In both approaches, the build orientation is fixed during the optimization. However, build orientation plays an important role in the additive manufacturing. It determines the overhang angle of a part’s surfaces with respect to the build plate, and hence affects the amount of support structures. For a different build orientation, the enclosed voids can be self-supported without large change of the structural topology. Due to the lack of explicit boundary representation, it is challenging to optimize the build orientation in the density-based topology optimization. Although the build orientation has been optimized by Wang and Qian [16] for surface slope control, their formulations cannot be used to design self-supported enclosed voids.

In the paper, a heat-flux approach is proposed to design self-supported enclosed voids for additive manufacturing. The proposed approach enables us to optimize build orientation and topology simultaneously. We solve a pseudo heat conduction problem in which surface slope dependent heat flux is imposed on the surfaces of small overhang angle (see Fig. 1). By setting the void and solid materials to be thermally conductive and insulating, respectively, the input heat flux on the non-self-supported enclosed voids would lead to high temperature in the pseudo problem. Hence, such non-self-supported enclosed surfaces can be eliminated from the optimized design through constraining the maximum temperature of the pseudo heat conduction problem. Since the density-based topology optimization approach involves no explicit boundary representation, it is challenging to explicitly identify the overhanging surfaces and impose the surface slope dependent heat flux. In our work, we impose the surface slope dependent heat flux through a volume integral of a Heaviside projected density gradient. For the given build orientation $\mathbf{b}$, the density gradient based Heaviside projection enables us to implicitly identify those solid/void material interfaces that require supports in additive manufacturing [14], [15], [16], [37], and impose boundary loading as body forces [47], [48]. Furthermore, since the density gradient based heat flux term is differentiable to both the build orientation and the density field, we can optimize the build orientation and topology simultaneously for self-supported enclosed voids.

The remainder of the paper is organized as follows. Section 2 proposes the heat flux based approach for self-supported enclosed voids. Then the optimization formulation and sensitivity analysis are presented in Sections 3 Topology optimization formulation, 4 Sensitivity analysis . We present both 2D and 3D numerical examples in Section 5 to demonstrate the effectiveness of the proposed approach. Finally, conclusions are drawn in Section 6.