A new overhang constraint for topology optimization of self-supporting structures in additive manufacturing

A topology optimization problem consists in the formulation of a lay-out problem, where the goal is to find the optimal distribution of material in a specific region, according to the applied loads, the possible support conditions, the volume of the structure to be constructed and possibly some additional design restrictions. The first topology optimization method (the so called homogenization method) was proposed by Bendsøe and Kikuchi (1988). In this method porous microstructures are assumed to relax the optimization problem since the binary discrete setting for structural compliance designs is known to be ill-posed. However, the determination and evaluation of optimal microstructures and their orientations was cumbersome if not irresolvable, and the resulting structures could not be built as there was not associated length scale to these microscopic materials. Alternatively, density based methods were proposed, where the design variables are the elemental densities representing solid material or void. In order to approximate the solution to a discrete solution, material properties are scaled according to the density value of each element. One can find several approaches as the Solid Isotropic Material with Penalization approach (Bendsoe 1989), the Rational Approximation of material property approach (Stolpe and Svanberg 2001) and the SINH method (Bruns 2005) in the literature.

At present the SIMP method can be stated as the most popular topology optimization method, but apart from this approach there is an extensive amount of other alternative methods. Some are based on more heuristic concepts, such as the evolutionary methods, which have undergone great development over the past decades. Among all its variants, the Bi-directional Evolutionary Structural Optimization (BESO) method and the Sequential Element Rejection and Admission (SERA) approaches have been extensively investigated (Huang and Xie 2010; Querin et al. 2017). Some other methods explore shape and boundary variation and can be considered relatively new, like Topological Derivative methods (Sokolowski and Zochowski 1999) and Level Set methods (Allaire et al. 2004; Wang et al. 2003). Other recent approaches that recall shape and size optimization procedures for topology optimum design are the Moving Morphable Conponent (MMC) and the Moving Morphable Void (MMV) methods, introduced in Guo et al. (2014) and Norato et al. (2015), where the topology optimization problem can be solved based on explicit geometry description. These methods were further improved in (Zhang et al. 2016, 2017) and (Gou et al. 2016). A comprehensive review of these methods can be found in the monograph by Bendsoe and Sigmund (2003), the general survey by Deaton and Grandhi (2014) and the comparative review by Sigmund and Maute (2013).

Even if topology optimization has shown to be a powerful technique that allows for increasingly efficient designs, because of the complexity and intricacy of the solutions obtained, it was often constrained to research and theoretical studies (Zegard and Paulino 2016). Recently, Additive Manufacturing is filling the gap between topology optimization theory and its application. AM is a rapidly evolving field that first emerged in the 1980s, and by now, because of the important technological developments, one can directly manufacture end-used parts, opening the design process to new challenges (materials, shape and internal structures) and enhancing the design freedom (Ponche et al. 2014). Hitherto, along with technological development, a large variety of AM processes have been proposed, for both metallic and plastic materials. However, and although AM overcomes the limits currently imposed by the conventional manufacturing technologies, the processes based on powder beds and polymer materials still present construction limitations and there are some technical constraints that must be satisfied in order to generate consistent geometries (Doubrovski et al. 2011). These constraints include the minimum member size, the overhanging distance and the member inclination (Chang and Tang 2001; Leary et al. 2013). Extremely thin members cannot be properly printed if that thickness is below the minimum allowable thickness of the tool. The fabrication overhang angle is another example of a rule which is of great importance to ensure that the part will not collapse during the layer adding process when additive manufacturing is used. Generally the inclination of a member is described as the angle between the base plate and the down facing contour. This is similar to considering the angle between the material growing direction and the vector normal to the contour. In this work, the member inclination referred to the material growing direction will define the overhang and whether it is properly supported or not. The theoretical threshold for the overhanging angle from which the member is considered non self supported is commonly considered to be 45 degrees (Daniel 2009), though there are some authors that propose that not only the overhanging angle but the combination of overhanging angle and overhanging length should be considered (Vayre et al. 2012). Other authors propose different values for that angle, ranging from 20 to 45 degrees (Leary et al. 2014). A structure satisfying such an overhang angle constraint is called self-supporting. For a non self-supporting structure to be manufactured, its geometry has to be modified or additional support structures need to be generated. Such support structures are formed during the fabricating process so that the primary object can be manufactured layer by layer without collapsing. This is very time-consuming and a waste of material, rising the issue of further post-processing activities to remove the unwanted supports.

Modifying the geometry manually in order to fulfill the overhang constraint may ultimately reduce the structure’s physical performance and lead to structures with a compliance that is far from the optimum support-needed structure. Since the idea of coupling topology optimization and AM was formulated, many different considerations have been proposed but generally we can group them in three main groups, depending on the level of engagement between the two technologies. In the first group we find the standard topology optimization process, where once the structure is optimized, the supporting material is added where it is required during the manufacturing process. A second level engagement (Leary et al. 2014; Hu et al. 2015) gathers strategies that consist in adapting the optimized design to AM processes. Once the topology optimization is over, the generated geometry may or not be suitable for AM. In case the use of scaffold structures is not an option, the designer proceeds to introduce some manual changes on the geometry adapting it to the fabrication process. These engagement strategies include a critical step where designer’s experience plays an important role. This step is where the designer introduces manually the changes on the post topology optimization geometry and creates some variations of that geometry in order to avoid bad supported overhanging members. Hence, there is a level of interference and uncertainty with the new level of optimality and deterioration of the structures’ physical performance. Finally, the third level engagement considers the strategies involving a total integration of topology optimization and AM processes by introducing the overhang constraint into the design process, which enables a final result that can be directly manufactured and shows and acceptable compliance ratio with respect to the non-supported reference optimum structure. The main advantage of this approach is a total integration of topology optimization and design process with the AM procedure.

This paper progresses in this line of work and presents a new topology optimization approach that helps to design optimal self-supporting structures, which are ready to be fabricated via additive manufacturing without the usage of manual modifications or additional support structures, so that their associated optimal compliance is close to that of the reference structure. A strategy to solve the issue of designing a completely self-supporting structure via topology optimization was presented by Brackett et al. (2011), where the overhang angle was included in the optimization problem but did not produce a complete self-supporting structure. Gaynor and Guest (2016) introduced a wedge-shaped filter during the optimization process to obtain self-supporting structures. Recently, Langelaar presented a novel self-supporting filter into the topology optimization process, which is achieved via building smooth approximations to the minimum and maximum functions (Langelaar 2016a, b). Also very recently, Qian (2017) has proposed a density gradient based approach for undercut and overhang angle control in topology optimization. Recently, Guo et al. (2017) introduced a novel formulation for AM–oriented topology optimization where the corresponding problem can be solved in a geometrically explicit way and simultaneous optimization of the self-supporting structural topology and the angle of working plane is achieved for the first time. Theoretical issues associated with AM–oriented topology optimization are also discussed in this work, which are important to examine the effectiveness of different approaches.

The approach proposed in this paper follows a different way and develops a novel global inequality constraint for the overhanging angle which is included explicitly in the topology optimization formulation problem. This constraint can be easily added to the problem formulation as an additional inequality constraint along with the regular volume constraint for compliance minimization problems, and coupled with traditional mathematical programming optimization algorithms, like the Method of Moving Asymptotes (Svanberg 1987). The constraint that is proposed in this paper refers to the ratio between acceptable and not acceptable material members, and is based on the evaluation of self-supported “contours”, instead of the traditional formulation of supported and unsupported “elements”. The methodology used to identify and control the inclination of members is based on an edge detection algorithm developed by Smith and Brady (1997) in the field of image processing, known as the Smallest Univalue Segment Assimilating Nucleus (SUSAN). The implemented procedure allows the designer to specify not only any critical overhang angle and printing direction, but also the tightness of the constraint through a harshness parameter that controls the ratio of self-supported contours. The capacities of this approach are demonstrated with extensive numerical examples and obtained results are compared with previous studies.

Finally, making a comparison of the proposed method with other methods existing in the literature, the following aspects could be highlighted. Most of previous works need the incorporation of filters that must be included in the traditional density-based topology optimization procedures (Serphos 2014), or they introduce some sort of additional constraints to control several problems like lateral overhangs and the grayness level of the density field (Qian 2017). On the contrary, the method proposed here needs only a single overhang constraint, defined as a relative value instead of an absolute value, which allows controlling the impact of the restriction easily and makes it straightforward to choose an adequate value for it. Concerning the SUSAN algorithm adopted for edge detection, it must be said that the choice of this algorithm was not casual, since it has been demonstrated that it is extremely efficient for contour detection. The inclination of the members is detected precisely and locally along the full contour of the structure, while other methods may require introducing a series of auxiliary constraints to efficiently detect not self-supported edges and singular parts (Guo et al. 2014). Its extension to 3D problems is also straightforward, as it can be checked in the work by Walter et al. (2009a, 2009b).

In this section the contour detection algorithm and the self-supporting constraint are formulated, which will be integrated within the modified SIMP framework in terms of the element density, also known as RAMP parameterization. First we shall develop a tool based on edge detection capable of finding forms and shapes in a grey-scale topology optimization model that will be used to evaluate the amount of feasible and non-feasible contours during the topology optimization iterations. Following on from this, we will present the proposed explicit overhang constraint and use it for analysis and sensitivity analysis of the manufacturability of the structure. A brief description of the well known Heaviside projection technique for density filtering is also included at the end of the section, since it will be used later for the sensitivity analysis.

This paper considers the maximum stiffness topology design where a new global overhang constraint is included in order to control and minimize the need of scaffold structures in additive manufacturing of optimized designs. This problem will be formulated in the following way:where u is the nodal vector of displacements, K is the global stiffness matrix that depends on element’s E_e Young modulus, F represents the nodal force vector, and k₀.

is the element stiffness matrix for a solid element. V(ρ) and V₀ correspond to the volume fraction of the actual iteration and the maximum allowable volume fraction, respectively. Finally, \( \tilde{\varPhi}\left(\rho \right) \) represents the overhang constraint and corresponds to the ratio between the values of allowable and total number of contours, which should be larger than a specified value Φ₀.

The RAMP method that converts intermediate densities less efficient in the optimization will be used. Each element is assigned a density and its Young’s modulus is still given by the well known density-stiffness interpolation scheme, but replacing the regular density variable ρ_e with the physical density \( {\overline{\rho}}_e \).where p is the penalization parameter, E₀ is the Young’s modulus of the solid isotropic phase and E_min represents the modulus of the void material, which is taken to be 10⁻⁹ in this work.

In the following, the approach for the sensitivity analysis of the functions depicted in the topology optimization problem formulation is described. In order to compute the derivatives of any function f(ρ) involved in the problem with respect to the density of each element, we will apply the chain rule, so that the filtering and projection operations are considered. The sensitivities of the compliance and the volume and overhang constraints will therefore be calculated using the following rule:

The second and third terms in the right side of the equation are obtained for any f(ρ) function as

The derivative of the first term on the right hand side of (24) will vary depending on the nature of function f(ρ) considered in each case. It is well known that the derivative of the compliance with respect to variable \( {\overline{\rho}}_e \) can be easily obtained in the following way:

The derivatives of the volume constraint with respect to the projected densities for substitution in the chain rule of (24) can be obtained in the following way:where v_e represents the volume of finite element e. Finally, let us consider the sensitivity computation of the overhang constraint function with respect to \( {\overline{\rho}}_e \). First we should normalize the inequality constraint shown in (14) in the following way:where a different name has been adopted to avoid confusion. Taking derivatives in (29) and omitting density dependence for simplicity we have:

The sensitivities of the allowable and non-allowable contours, φ⁻ and φ⁺, can be calculated taking derivatives of expressions (11) and (12):

Finally, it is necessary to compute the derivatives of the contour values of each mask φ_m for substitution in (31) and (32). Recalling the function built in (10) for contour evaluation, the derivatives can be obtained with the following expression:where the derivatives of the coordinates of the gravity center can be easily obtained with the relations given in (1) and (2) and remembering that we are using the physical density variables for contour evaluation once the smoothing and projection filters were applied:

Once the sensitivities of objective and constraint functions are obtained, the topology optimization problem formulated in (18)–(22) is solved using the Method of Moving Asymptotes (MMA) developed by Svanberg (1987). MMA minimizes a set of sequential convex approximations of the original functions enabling the solution of problems with more than one constraint and is well known to be very efficient in the field of structural topology optimization. Figure 4 shows the flowchart of the proposed procedure that summarizes the different loops considered and the sequence of their application. Once a suitable reference domain is chosen, the basic data for structural analysis should be defined, including loads, boundary conditions, fixed void and solid regions, finite element mesh, etc. Then the topology optimization problem is defined for computation of the optimal material distribution over the reference domain, starting with an initial homogeneous distribution of material. Volume and overhang constraint bounds should be specified, as well as the normalized growing direction and the critical allowable inclination of members for additive manufacturing. After variables are smoothed and projected in order to obtain the physical density field, the contour evaluation loop starts, where the gradients for all the masks in the design domain are calculated and the overhang constraint value is obtained. For this distribution, the volume and compliance are computed by the finite element method. Finally, sensitivity analysis is performed and densities are updated with the help of the method of moving asymptotes. This loop is repeated until only a marginal improvement of the compliance is achieved over the last design. Once the optimization problem is over and in order to send the component to the 3D printing machines, the optimal material distribution should be interpreted so that the designer can build a CAD representation of the shape. Depending on the harshness parameter selected for the overhang constraint, there may be a little amount of elements that are badly supported and violate the overhang constraint, which could be fixed in this stage. However, this is not always a required step.

To illustrate the functionality of the developed optimization procedure and the characteristics of the overhang constraint parameters, several examples are discussed in this chapter. In all cases a Young’s modulus E₀ = 1 and a lower bound E_min = 10⁻⁹ are used in the RAMP law. As it was mentioned in the paper, the common practice of a continuation method will be applied on the RAMP exponent p and the Heaviside parameter β. The penalization parameter is initially set to 10 and increased by 2 every continuation step until reaching a maximum penalization of 18 while the Heaviside parameter is initially set to 5 and increased by 5 until a maximum magnitude of 25 (Gaynor and Guest 2016; Guest et al. 2011). Finally, the filter radius in the density filter is chosen equal to 4 times the size of the element and the threshold parameter T of the Heaviside function is set to 0.5 unless expressly indicated. The values adopted for these parameters have shown to be effective and lead to a very accurate geometry definition. It should be noted that the value of the filter radius does not produce particular effects on the proposed strategy and its influence is similar to traditional topology optimization. Although it is actually very important in other strategies in the literature where large filter values are required to regularize oscillations, no remarkable effect is present in this case. The printing direction is always considered vertical and upwards. It must be noted that although scaffold structures can be avoided in any 2D example if the structure layout is placed in the horizontal plane of the printing machine, they are still valid to demonstrate the proposed approach’s ability in handling overhang constraints if required.

The method presented in this paper offers direct control over the inclination of structural members in topology optimization problems, so that self supporting, print-ready designs can be generated for additive manufacturing. The proposed approach presents some new features compared with previous works in this area. The inclination of members is calculated by an efficient edge detection algorithm developed in the field of image processing and applied here to evaluate the overhang angle of contours. This paper proposes also a new constraint function to control the amount of self-supporting contours, defined as the ratio between the admissible and the total amount of contours. That constraint can be easily added to the conventional volume requirement of density-based topology optimization procedures and combined with any general purpose optimizer like the popular MMA method. This technique has been combined with the commonly applied Heaviside projection to enhance black and white designs. Results obtained from the numerical examples suggest that the methodology developed allows the designer to efficiently control the amount of self-supporting members so that unprintable designs with infeasible overhanging sections can be banned from the design space, producing self-supporting topologies that are ready for additive printing. Moreover, the examples included in the paper demonstrate that the proposed strategy is effective at generating designs that can be printed without additional supports for any critical overhang angle defined by the user. Even if important aspects such as deformation of members due to overheating and residual stresses have not been included, authors expect that the approach suggested in this work could be of considerable practical value as a first approximation in early design stages. Future works include an implementation of the method for 3D structures optimization, where it would be especially valuable.