On equal-width length-scale control in topology optimization

In mathematical morphology, the aim is to gain information about M by probing it by B. The basic morphological operators are erosion, dilation, closing, and opening. Following Heijmans (1995), we define the dilation and erosion of M by B as and respectively. That is, \(\mathcal {D}_{B}(M)\) equals the Minkowski sum M ⊕ B and \(\mathcal {E}_{B}(M)\) equals the Minkowski difference M ⊖ B. The closing of M by B and the opening of M by B are defined as respectively. Figure 1 illustrates these four basic morphological operators defined by the Euclidean unit ball B acting on a set \(M\subset \mathbb {R}^{2}\). Here, M is the union of a rectangular region (upper right part of M) and a polygon with a rectangular hole (lower left part of M). In colloquial terms, given a brush with shape B, \(\mathcal {D}_{B}(M)\) is the area that is filled by placing this brush’s center at all points inside M and \(\mathcal {O}_{B}(M)\) is the largest area that can be filled by using this brush without touching any point outside M. \(\mathcal {E}_{B}(M)\) and \(\mathcal {C}_{B}(M)\) are the dual operators of \(\mathcal {D}_{B}(M)\) and \(\mathcal {O}_{B}(M)\), respectively. Colloquially, given a fully filled space and an eraser with shape B, \(\mathcal {E}_{B}(M)\) is the area that remains filled after having erased as much as possible keeping the eraser’s center outside M and \(\mathcal {C}_{B}(M)\) is the area that remains filled after having erased as much as possible without erasing any point in M.

The minimum and maximum length-scales R_B and L_B can also be defined by using the morphological operators and the convention sup∅ = 0 as and respectively; recall that rB is the unit ball scaled by r. A proof of the equivalence between minimum length-scale definitions (4) and (10) is given by Hägg and Wadbro (2018, Theorem 3). The equivalence of definitions (5) and (11) can be shown similarly.

The equivalence between definitions (4) and (10) as well as definitions (5) and (11) provides us with alternative interpretations of length-scales R_B and L_B. Minimum length-scale R_B is the largest radius r so that M can be completely pained with a brush with shape rB without touching any point outside M. Maximum length-scale L_B is the smallest radius r so that M is completely removed by an eraser with shape rB keeping the eraser’s center outside M.

The medial axis transform, which is extensively used to describe shapes in the field of computer graphics and pattern recognition (Loncaric 1998). Blum (1967) introduced the concept of describing the shape of an object occupying a domain M by its medial axis or skeleton S and a distance map \(d:S\mapsto \mathbb {R}^{+}\). The skeleton consists of all points in M for which the closest point in \(\mathbb {R}^{N} \setminus M\) is not unique, that is where \({\Pi }_{\mathbb {R}^{N} \setminus M}(\boldsymbol {x})\) is the set of projections of x onto \(\mathbb {R}^{N} \setminus M\) or, in other words, \({\Pi }_{\mathbb {R}^{N} \setminus M}(\boldsymbol {x})\) consists of all points in \(\mathbb {R}^{N} \setminus M\) with infimum distance to x. The distance map \(d:S\mapsto \mathbb {R}^{+}\), is defined so that Metrics defined via distance functions have been used extensively as a practical tool in the field of computer visualization, while these metrics primarily have been used as a theoretical tool for shape calculus, see for example Delfour & Zolesio’s book (Delfour and Zolésio 2011).

Note that given M and its skeleton S, there may be multiple distance maps d that satisfy condition (13). To strictly define minimum and maximum length-scales based on the medial axis transform, we need to add a requirement on the distance map d. More precisely, we will, in addition to require that d satisfies condition (13), also stipulate that d(x) is maximal for each x ∈ S. Provided that d is selected as stipulated above, minimum and maximum length-scales can also be defined by using the distance map d associated with the medial axis transform, as suggested in the context of topology optimization by Zhang et al. (2014). The minimum length-scale R_B(M) can also be defined as and the maximum length-scale L_B(M) can also be defined as That is, the minimum and maximum length scales of domain M may be defined as the smallest and largest value of the distance map for all points on S, respectively.

Allaire et al. (2016) proposed an alternative minimum length-scale definition based on the thickness of the design. Consider a point x on the boundary ∂M and assume that its outward directed unit normal n(x) at x is well-defined. Then, the local thickness \(\hat T(M;\boldsymbol {x})\) of M at x is defined as In other words, \(\hat T(M;\boldsymbol {x})\) is the largest t such that for any s ∈ (0, t), it holds that x − sn(x) ∈ M. The minimum thickness T(M) of M is then defined as Later in this manuscript we will also refer to T(M) as the minimum width of the domain M.

We remark that, as also Allaire et al. (2016) acknowledged, the behavior of the minimum thickness T(M) and the minimum length-scale R_B(M) can be rather different. The minimum thickness of a square with side length 1 and rounded corners is 1, while R_B for this domain is the rounding radius of the corners provided that B is the unit sphere. To impose minimum thickness via constraints or penalties in the problem formulation, Allaire et al. (2016) reformulated the thickness definition by using the signed distance function d_M. Note that: for a point in the interior of M, the value of the signed distance function d_M is the negative distance from the point to the boundary of M; for a point on the boundary of M, d_M is 0; and for a point in the exterior of M, d_M is the distance from the point to the boundary.

The end-goal of this work is to find designs with equal member width, or in other words, we want to find designs whose minimum and maximum length-scale coincide. However, this requirement together with length-scale definitions (4) and (5) implies that we cannot have designs that contain equal-width bars that cross. Figure 2 illustrates two bars of width 2 and infinite length that cross. At the crossing point, the local length-scale is \(\sqrt {2}\); away from the crossing region, the local length-scale is one. More precisely, in this case, \(R_{B}= 1<\sqrt {2}=L_{B}\). To avoid this paradoxical consequence of definitions (4) and (5), we choose to use an alternative definition of the maximum length-scale.

Instead of defining the maximum length-scale of M by using the local length-scales or directly by using the distance map and skeleton associated with the medial axis transform, one could try to mix concepts from the medial axis transform and mathematical morphology. First, note that the structural skeleton S may include more points than C, the set of all center points of the balls defining the local length-scale. Formally C may be defined as For the domain in Fig. 2 where two crossing beams comprise M, the left and right images in Fig. 3 illustrate the sets C and S, respectively. The skeleton S includes all points in M that do not have a unique closest point on the boundary of M, while the set of all center points C holds only the center points of the circles that define the local length-scale \(\hat {R}_{B}(M;\boldsymbol {x})\) for any point x ∈ M.

We define the maximum width D_B(M) as twice the smallest scaling factor r such that M is a subset of its skeleton dilated by rB. That is, The factor 2 is included to account for that the thickness of a long bar is twice the radius of the largest ball that fits inside the bar. We remark that by using the maximum length-scale constraint proposed by Zhang et al. (2014) (More precisely, when we let parameter 𝜖₂ → 0 in problem (3.1) in their article.), we get length-scale control according to maximum length-scale definition (19), but not necessarily according to maximum length-scale defintion (5). (Recall that distance map d is not uniquely defined by expression (13).)

In the context when two bars cross at a right angle, recall Fig. 2, definition (19) results in a maximum width of 2, which is the same as the minimal thickness according to definition (17). However, in other cases, such as the one detailed below, maximum width definition (19) produces unintuitive results. Consider a sequence of domains M_n, where each domain is the union of the unit square and 4 ⋅ 2ⁿ balls of radius r_n = 2^−n− 1 centered at (0, a_k), (1, a_k), (a_k,0), and (a_k, 1) for k = 1, 2, … , 2ⁿ and where a_k = (2k − 1)r_n. The domains M₂, M₃, and M₄ are illustrated in Fig. 4. For domain M_n, the minimum length-scale following definition (4) is R_B(M_n) = 2^−n− 1 and the maximum length-scale following definition (5) is L_B(M_n) = 0.5. (Recall that the side length of the unit square is 1, while the radius of the unit ball is 1.) However, the maximum length-scale according to definition (19) is D(M_n) = 2^−n− 1. Note that, the skeleton (illustrated by the black lines in Fig. 4) becomes increasingly dense in each step. In essence, the closest point from a point inside the unit square to the boundary of M_n is one of the points where the boundary of the unit square and the boundary of M_n intersect. In the limit n → ∞, we have that R_B(M_n) = D(M_n) → 0, while L_B(M_n) = 0.5 for all n. Moreover, the local thickness \(\hat T\) at \((r_{n}+r_{n}/\sqrt {2},-r_{n}/\sqrt {2})\) is \((2+\sqrt {2})r_{n}\) and thus \(T(M_{n}) \le (2+\sqrt {2})r_{n} \to 0\) as n → ∞; so T(M_n) → 0 as n → ∞.

Above, we have discussed various possibilities of quantifying the minimum and maximum length-scales of a domain M. To proceed towards our end-goal of imposing equal-width length-scale, we need to select which definitions to use. In this study, we use definition (17) for the minimum length-scale and definition (19) for the maximum length-scale. By using these definitions, we avoid the equal-width paradox discussed in Section 2.4. Therefore, we can use these definitions for the equal-width length-scale control without inconsistency. However, we may still arrive at the case illustrated in Fig. 4, where the maximum length behaves unintuitively. The problem is that, due to the wiggly boundary of the domain, the skeleton includes many lines that are located close together.

We let θ_i = arcsinη_i and θ_j = arcsinη_j; recall that η_i and η_j are the variables that determine the direction of the i th and j th component of Ω, respectively. First, let us start by considering the penalty with respect to angle difference. First, we want this penalty to be symmetric in that the numbering of the two beams should not matter, that is, only the absolute angle difference should matter. To avoid the non-differentiability of the absolute values, we use a penalty function that is based on the squared angle difference. More precisely, we let the penalty function depend on the approximate angle difference where ε is a small positive regularization parameter. The function α is differentiable in θ_i and θ_j and approximates |θ_i − θ_j| with a maximum error dictated by the regularization parameter ε. Moreover, we want the penalty function to have its maximum value when the bars are parallel, that is, when α(θ_i, θ_j) = 0 and its minimum for α(θ_i, θ_j) = π/2 and we require that this function be symmetric around π/2. One example of such a function, which we use in our numerical experiments, is There are many choices of possible penalty function that satisfy the above conditions as well as smooth approximations of the absolute value function. The functions used here are merely used to serve as a proof of concept for the proposed approach to obtain equal-width length-scale control. Figure 7 graphs the angle penalty p_a and its exponentiation (p_a)^q with power q = 6 as functions of the absolute angle difference |θ_i − θ_j|. Here, as well as in the numerical experiments, we use 𝜖 = 10^− 4 to compute α(θ_i, θ_j) following definition (25). Figure 7 shows the p_a and \({p_{a}^{6}}\) as functions of the absolute angle difference |θ_i − θ_j| for two crossing bars. The angle penalty is scaled so that it is 1 when the bars are parallel, or fully overlapping, while the angle penalty is negligible when the crossing angle is close to 90 degrees. For the basic angle penalty, p_a the amount of penalization is small only for a small range of angles around 90 degrees, and as a consequence, many of the bars that cross in the optimized designs will do so with a crossing angle around 90 degrees. By using the exponentiation (p_a)^q, for example, with power q = 6, as shown in Fig. 7, the angle penalty will have relatively small value in a wider range around 90 degrees. Thus, by using \({p_{a}^{6}}\) instead of p_a, bars may cross in a large range of angles with only a minor influence on the value of the angle penalty. In a practical situation, it is likely that one wishes to have more control, for example, be able to select at which angle the penalization “starts”. To obtain such control, one can use some projection-based technique, such techniques have been successfully used in density-based topology optimization (Xu et al. 2010; Wang et al. 2011).

We start by noticing that each point on the centerline of beam i can be written on the form \(p_{i}(s_{i}):=(x_{i}^{c},y_{i}^{c}) + s_{i} (\cos \theta _{i},\eta _{i})\), where s_i is a parameter satisfying |s_i|≤ l_i/2. Here, we are interested in computing the distance between the two beams. To do so, we study the minimum distance between a point on the centerline of beam i and a point on beam j’s centerline. In principle, if this distance is larger than t_eq, the beams do not overlap. However, the beams need not overlap if the distance is smaller, for example, if the end-points of the two-beams are the points of minimum distance. However, if we only consider points \(p_{i}(s_{i}):=(x_{i}^{c},y_{i}^{c}) + s_{i} (\cos \theta _{i},\eta _{i})\) with |s_i|≤ q_i = (l_i − w_sp)/2, then the two beams will (at least partially) overlap if the minimum distance between their centerlines’ mid-sections is smaller than t_eq. The symbol w_sp is defined as the predetermined space along the centerlines at the two ends of the beam. In cases where the length of the component is very short, that is if l_i is smaller than w_sp, then we will consider all points along the centerline by setting q_i = l_i/2. Thus, we compute Next, we expand the objective function above to find that So, evaluating the minimum distance between the two segments boils down to solving a quadratic programming problem. If the lines are not parallel, this problem has a unique solution that can be computed very quickly, for example, by using an active set algorithm. When evaluating derivatives, we can capitalize on that the same constraints will be active in the perturbed case as in the original when considering sufficiently small perturbations. (In some cases, we may need to settle for a one-sided difference.) Note that, if no constraints are active, then the segments cross and the distance between them is zero—after any small perturbation, the segments will still cross and thus the gradient will be zero.

On the other hand, if the lines are parallel, the quadratic problem may no longer have a unique solution. The case when the problem does not have a unique solution is the worst possible case; in this case, the two centerline segments are parallel and there are multiple pairs of closest points between these segments. In this case, one could argue that the gradient with respect to the length as well as rotation of the segments can be considered to be 0.

We note that if the distance between centerlines is greater than the imposed component thickness, then the segments do not overlap. Here, to also penalize configurations with nearly overlapping components, we use the following tanh inspired penalty function: where k is a slope parameter. Figure 8 shows penalty function p_d as a function of inter-component centerline distance β for three values of parameter k. In the numerical experiments, we use k = 5.

As mentioned already at the beginning of this section, we would like to penalize structures that include beams that are nearly parallel and partially overlap. To do so, we form a total penalty function P by summing the product of the angle penalty and distance penalty for all pairs of beams in the structure; that is, We add a scaled version of this total penalty function to the objective function and obtain our penalized optimization problem: where γ is a penalty parameter.

Here, the design domain Ω_D, illustrated in Fig. 5, has length L = 1.5, height H = 1, and is meshed into 300 × 200 elements. A concentrated traction force is applied at the center of Γ_L and at most 40% of the design domain is allowed to be filled with material.

We first solve the problem without considering our end-goal of obtaining a structure with equal-width control. More precisely, we solve problem (31) with parameter γ = 0 to find a design with minimum compliance by the basic MMC method with the lower and upper thickness bounds set to 0.02 and 0.1, respectively. For all runs with central loading, a symmetry condition on the design along the mid-line in the horizontal direction is enforced throughout the optimization.

The initial layout, illustrated in Fig. 9, consists of 24 components: 4 × 3 crosses uniformly spaced in the design domain. Figure 10 shows the optimized design from this experiment. The compliance of the optimized beam is 40.00. As a second reference study, we set the lower and upper bounds for the thickness to t_eq = 0.08, but we do not include any penalty to promote design with components of equal-width, so γ = 0 also in this experiment. The compliance of the optimized beam shown in Fig. 11 is 40.90. These reference designs in Figs. 10 and 11 do not have the desired equal-width length-scale control. Each component in the design in Fig. 11 is restricted to have a specified thickness. However, the complete designs include multiple components that are nearly parallel and partially overlap; so, the full structure has a maximum width that is much larger than t_eq. For parameter q = 1, the values of total penalty function (30) for the designs in Fig. 10 and Fig. 11 are 15.67 and 22.61, respectively; the corresponding values for parameter q = 6 are 12.78 and 14.08, respectively.

Having established our reference solutions, we turn to the problem of obtaining a design with equal-width length-scale control. To that aim, we solve optimization problem (31) by MMC, impose lower and upper bounds for the thickness by setting t_eq = 0.08, and use a so-called continuation strategy to increase penalty parameter γ and gradually increase γ from 0 to 60 during the optimization. The space w_sp is set to 0.01; and the distance w used to define distance penalty p_d is set to 0.2. Figures 12 and 13 show the optimized designs obtained when using total penalty (30) with parameter q = 1 and q = 6, respectively. The compliance corresponding to the optimized designs in Figs. 12 and 13 is 81.16 and 55.22, respectively.

From Figs. 12 and 13, it can be seen that all bars have the uniform width and no obvious partially overlapped or paralleled bars exist, which is also reflected by the relatively small value of the total penalty. More precisely, the total penalty P is 0.22 and 0.02 for the results in Figs. 12 and 13, respectively. Note that the total penalty values above are computed with q = 1 and q = 6, respectively, so they are not directly comparable to each other. However, these numbers can be directly compared to and are magnitudes smaller than the corresponding value for the optimized designs (Figs. 10 and 11) for the unpenalized problem.

Next, we investigate the influence of the initial guess for the MMC on the final result. More precisely, we study the same problem as above, with the load at the right-center point; but instead of starting from the design in Fig. 9, we start from the design in Fig. 14. This design consists of 20 components: 2 × 5 crosses uniformly spaced in the design domain. Also, in this case, we first solve the problem by MMC with the lower and upper thickness bounds of the individual components set to 0.02 and 0.1, respectively. The optimized design without equal-width control is shown in Fig. 15 and its compliance is 43.51. By just changing the lower and upper thickness bounds of the individual components to t_eq = 0.08, we obtain an optimized design that is similar in structure to the design presented in Fig. 15 and has compliance 44.81.

Figures 16 and 17 show the optimized designs obtained when using the layout in Fig. 14 as an initial guess; total penalty (30) with parameter q = 1 and q= 6, respectively; w_sp = 0.01; w= 0.2; and a continuation approach gradually increasing penalty parameter γ from 0 to 60. The compliances corresponding to the optimized designs in Figs. 16 and 17 are 84.95 and 52.62, respectively. For the optimized beams in Figs. 16 and 17, all bars have the same width and there are no obvious partially overlapping or parallel bars.

As our second experiment, we aim for equal-width length-scale control while minimizing the compliance of a cantilever beam loaded by a concentrated traction force applied at the lower right corner of the computational domain. Here, the design domain Ω_D, illustrated in Fig. 5, has length L = 1.0 and height H = 0.5, and is meshed into 200 × 100 elements. The maximum allowable volume fraction is set to 40%, that is, \(\overline {V}= 0.4|{\Omega }_{D}|\). The initial design is shown in Fig. 18 and consists of 12 components: 2 × 3 partially overlapping crosses.

Also, in this case, we present reference results obtained without equal-width control. Just as in the previous experiments, we first solve the problem of finding a design with minimum compliance with the lower and upper thickness bounds set to 0.04 and with penalty parameter γ = 0 throughout the optimization. Figure 19 shows final design, whose compliance is 89.02. Similarly as in the experiments for the center loading, simply setting the lower and upper thickness bounds to t_eq only yields minor changes in the overall layout for the optimized beam.

Next, we consider the penalized problem. Figures 20 and 21 show the optimized designs obtained when using the layout in Fig. 18 as initial guess; total penalty (30) with parameter q = 1 and q = 6, respectively; w_sp = 0.01; w = 0.1; and gradually increasing penalty parameter γ from 0 to 20. The compliance corresponding to these optimized designs is 112.76 and 106.74, respectively.

Next, we try to optimize the beam with corner loading and setting the upper and lower thickness bounds to 0.02. That is, the bars have half the width compared to the earlier experiments. We mesh the design domain Ω_D of length L = 1.0 height H = 0.5 by 300 × 150 elements and set the maximum volume \(\overline {V}= 0.3|{\Omega }_{D}|\). The initial design for this test case is shown in Fig. 22 and consists of 16 components: 2 × 4 partially overlapping crosses. We solve problem (31) using total penalty (30) with parameters q = 1, w_sp = 0.01, w = 0.1, and gradually increasing penalty parameter γ from 0 to 20. Figure 23 shows the final design, which has compliance 154.91 and total penalty 2.69.

In our final experiment, we use another set of boundary conditions. Here, we consider the setup illustrated in Fig. 24. The design domain is rectangular and has length L = 1.0 and height H = 0.5 and is just as in the initial experiments for the corner loading meshed into 200 × 100 elements. The structure is fixed at the lower left and right corners of the design domain and is subject to a uniform distributed load over the middle 50% of the design domain.

We use the layout in Fig. 25 as initial guess. Figures 26 and 27 show optimized beams obtained when using total penalty (30) with parameter q = 1 and q = 6, respectively; w_sp = 0.01; w = 0.1; and gradually increasing penalty parameter γ from 0 to 3. Here, the initial design consists of 15 bars, one horizontal bar placed so that the distributed force is applied over its top side; the remaining bars are placed in a pattern with four plus three crosses on two rows. The compliance corresponding to these optimized designs is 9.996 and 11.36, respectively.

Figure 28 shows the iteration history for the optimization process that resulted in the design in Fig. 27. The scaling factor for the penalty is chosen so that both functions are of the same order. Missing data for the compliance graph corresponds to iterations where the compliance is greater than 25.