Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions

We consider the shape and topology optimization problem to design a structure that minimizes a weighted sum of material consumption and (linearly) elastic compliance under a fixed given boundary load. As is well-known, this problem is in general not well-posed since its solution typically requires the use of infinitesimally fine microstructure. Therefore we examine the effect of singularly perturbing the problem by adding the structure perimeter to the cost. For a uniaxial and a shear load in two space dimensions, corresponding energy scaling laws were already derived in the literature. This work now derives the scaling law for the case of a uniaxial load in three space dimensions, which can be considered the simplest three-dimensional setting. In essence, it is expected (and confirmed in this article) that for a uniaxial load the compliance behaves almost like the dissipation in a scalar flux problem so that lower bounds from pattern analysis in superconductors can directly be applied. The upper bounds though require nontrivial modifications of the constructions known from superconductors. Those become necessary since in elasticity one has the additional constraint of torque balance.


Introduction
Using the concept of energy scaling laws, this article predicts the optimal (or rather an almost optimal) shape of a three-dimensional structure under a uniaxial tension or compression load. Optimality here is with respect to a cost that consists of the compliance (a measure of structural weakness), the material consumption, and the perimeter or surface area (a measure of complexity, for instance during production of the structure). Our work is essentially a continuation and extension of [14] as well as [7,4]: The former solves the same problem in two space dimensions, and the latter solves a closely related pattern analysis problem for intermediate states in type-I superconductors, whose lower bounds and basic patterns can (up to minor modifications) directly be applied here. The major difficulty in and contribution of the current work is to turn those basic patterns into feasible constructions for the compliance minimization setting.

Motivation
Elastic shape optimization or compliance minimization is a well-studied field with lots of results since the 1960s on the minimum possible compliance for a given material volume and on the corresponding optimal microstructures, see [11,13,1,15] and the references therein. It is of interest not only for material design, but also for getting some understanding of structures appearing in nature, adopting the hypothesis that biological evolution actually solves an optimization problem. A common example is the microstructure of bones ('spongiosa'). Since compliance minimization in general leads to non-natural, infinitely fine microstructure, there must be some additional complexitylimiting mechanism involved. Various aspects might potentially contribute to this complexity limitation such as growth processes, energy consumption for remodelling and maintenance, etc. In absence of any corresponding biological model we will consider surface area or perimeter as a measure of structural complexity, since remodelling of biological structures typically happens at their surface. Given the fine length scales of many biological structures (pore sizes in human trabecular bone range from tens to hundreds of micrometres) the complexity limitation cannot be very strong. Therefore it is natural to examine the regime of small perimeter penalization. Unfortunately, the optimal structures in this regime are far too complex for a numerical resolution, but they are amenable to asymptotic analysis, for instance in the form of scaling laws as will be proved here.
From a more mathematical viewpoint, elastic shape optimization or compliance minimization serves as a formidable model for the study of energy-driven pattern formation. Similarly to many other systems studied in the literature (such as thin sheets, micromagnetics, martensite models, or nonconvex versions of optimal transport) it seems to exhibit several regimes of very distinct behaviour and a rich class of possible patterns. The task then is to quantify all parameter regimes and study the full phase diagrams of these systems, and our work is in line with this general objective. In the specific setting of compliance minimization the diverse pattern types and their levels of complexity can be controlled by the applied boundary load. This diversity is inherited from the problem without perimeter regularization, as becomes apparent already in two space dimensions: If the macroscopic principal stresses have opposite sign, then the situation is known to be very rigid in that any optimal microgeometry must resemble a rank-2 laminate [2]. In contrast, if the macroscopic principal stresses have equal sign, then there is a high degree of freedom with a multitude of optimal microgeometries besides sequential laminates (such as the "confocal ellipse" construction based on elliptic inclusions at all length scales [9] or the "Vigdergauz construction" based on more complicated inclusions at a single length scale [10]). Sometimes, microstructure might not even be necessary: The optimal geometry for a unit disc domain under hydrostatic pressure is an annulus (which is a special case of the confocal ellipse construction). In three space dimensions it is expected that combinations of these properties may occur. The setting of compliance minimization shares some features with patterns formed in thin sheets or martensites. In the former, there exist regimes of simple hanging drapes with dyadic coarsening structures along essentially just one dimension [19], but also regimes with complicated multidirectional patterns as in crumpling paper [8]. Likewise, in martensites plain dyadic branching patterns occur in the simplest situation [12], while in some shape memory alloys there seems to be enormous flexibility in the possible patterns [16].
In this article we study the borderline case between principal stresses of equal and different sign, the situation of just a single nonzero principal stress. This represents the simplest possible situation and therefore should be studied first. The reason is its resemblance to pattern formation problems of scalar conservation laws or equivalently of vector fields: While in elasticity one in principal has conservation of the vectorvalued momentum and thus has to work with tensor fields (the stress), many other pattern formation problems just deal with vector fields. Examples include micromagnetic structures in ferromagnets [5,6] or the intermediate state of type-I superconductors [7,4] (in both cases the field is the magnetic field). Applying a uniaxial load now implies that mainly the momentum along only one direction (say the vertical, z-direction) is transmitted so that the corresponding row of the stress tensor behaves just like the magnetic field in the above problems.
We will analyse the pattern formation problem by an energy scaling law (a common tool in the theory of pattern formation), that is, we will quantify how the total cost scales in both the strength of the load and of the perimeter regularization. To this end we have to prove upper bounds (via explicit geometric construction) and matching lower bounds. For the latter it turns out we can almost completely adapt the lower bounds from [6,7] for patterns in type-I superconductors. The upper bounds are more challenging: The basic geometric shapes manifesting in type-I superconductors can indeed be adapted as well, but modifications have to be applied that account for the balance of torque in addition to linear momenta. While this is relatively straightforward in two space dimensions [14] it becomes considerably more difficult in three space dimensions.

Problem formulation and main result
We fix a design domain Ω = (0, ) 2 × (0, L) ⊂ R 3 with square base and consider the optimization of the geometry and topology of a structure O ⊂ Ω. The set O here represents the region occupied by an elastic material, while Ω \ O is void. On the boundary ∂Ω of the design domain we apply a uniaxial, vertical boundary load of strength F ∈ R to be supported by O, see fig. 1.1 left. In more detail, abbreviatinĝ σ = F e 3 ⊗ e 3 for the Euclidean basis vector e 3 = (0 0 1) T , we apply the stress (force per area)σn on all of ∂Ω, where n denotes the unit outward normal to ∂Ω (note that this stress is nonzero only at the top and bottom face Γ t = (0, ) 2 × {L} and Γ b = (0, ) 2 × {0} of Ω). Throughout we will employ linearized elasticity to describe equilibrium displacements and stresses, which is appropriate for small displacements and strains as they occur for instance in trabecular bone. Furthermore, for simplicity we will assume O to consist of a homogeneous, isotropic material with zero Poisson ratio (a nonzero Poisson ratio is not expected to qualitatively change the result, see remark 3, but would introduce substantial additional notation). This implies that the material response to stress is governed by a single material parameter, the shear modulus (or second Lamé parameter) µ > 0.
The cost function with respect to which we optimize the structure O is a weighted sum of its so-called compliance comp µ,F, ,L (O) under the applied load, its volume vol(O), and its surface area or perimeter per Ω (O), where α, β, ε > 0 are positive weights and the compliance will depend on the material and configuration parameters µ, F, , L. The corresponding optimization problem is commonly known as compliance minimization with perimeter regularization (in the literature the volume is sometimes constrained instead of being part of the cost function). The compliance is the work performed by the applied load and can in linearized elasticity also be interpreted as the stored elastic energy. If ∂O does not contain the top and bottom face of Ω and thus cannot support the load, one sets comp µ,F, ,L (O) = ∞. Since in linearized elasticity the compliance is known to be invariant under a sign change of the load we will without loss of generality assume a tensile load F > 0 throughout. The volume and perimeter serve as measures for material consumption or weight and structural complexity, respectively. While for vanishing perimeter regularization ε = 0 the cost J α,β,ε,µ,F, ,L admits no minimizer (instead, an infinitely fine microstructure, a so-called rank-1-laminate is optimal [1]) optimal structures O are known to exist for ε > 0 [3]. Note that by per Ω (O) we mean the perimeter relative to the open domain Ω so that components of ∂O inside ∂Ω do not contribute to the cost. The reason is that unlike the surface area of ∂O ∩ Ω in the domain interior, the surface area of ∂O ∩ ∂Ω within the domain boundary is not really connected to structural complexity. Nevertheless one can also consider the alternative cost with per R 3 (O) the full perimeter of O, viewed as a subset of R 3 . In that case the perimeter contribution from ∂O ∩ ∂Ω will sometimes dominate the cost. For the sake of completeness we will treat this setting alongside the one of actual interest. As mentioned previously, we are interested in the case of small perimeter regularization ε 1. Furthermore we will assume F ≤ 4µβ/α since for larger F the optimal structure is known to be O = Ω (with or without perimeter regularization), see remark 6 later. We additionally require the domain to be sufficiently wide. In that regime we prove the following energy scaling law. Right: Sketch of basic construction scheme consisting of layers (here three layers above and below the midplane) of elementary cells, whose width halves from layer to layer.
Theorem 1 (Energy scaling law for compliance minimization under a uniaxial load).
There exist constants c, C > 0 independent of α, β, ε, µ, F, , L such that whose optimal scaling we characterize in theorem 1. This scaling depends on the relation between ε and F or equivalentlyF -the theorem shows four different regimes. Almost the same regimes occur in the pattern analysis for type-I superconductors [7,4]. In each regime a different structure O leads to the optimal scaling, and the main contribution of this work is the construction of these structures, thereby proving the upper bound in theorem 1 (the lower bound will essentially follow from [4] except for the first regime and for energyJ α,β,ε,µ,F, ,L ). Below we briefly summarize these regimes in order of increasingF . As explained before, their optimal patterns will resemble the constructions from [7]. The main part of these patterns will always be composed of elementary cells that all look the same and only differ in size and height-width ratio. These elementary cells are organized side by side in single horizontal layers of identical cells, where the layers at the centre contain the largest elementary cells and where from each layer to the next the cell width is halved so that towards the top and bottom boundary Γ t and Γ b the cells become finer and finer (see fig. 1.1 right). The reason for this ansatz is that near the boundaries the structures have to be very evenly distributed and thus very fine in order to withstand the boundary load, while away from the load it pays off to coarsen since this way one can reduce surface area. The elementary cells will be designed in such a way that each cell connects seamlessly to four cells of half the width in order to be able to stack the different cell layers on top of each other.
Throughout we will use the notation f g for two expressions f and g to indicate that there is a universal constant C > 0 such that f ≤ Cg. Likewise, f g means g f , and f ∼ g is short for f g and f g.
Extremely small force. In this regime the nondimensional forceF is small compared to √ ε,F (Note that in the definition of f in theorem 1 the regime of low force was actually defined with ≤ in place of , but of course there is a transition region between any two neighbouring regimes in which they have the same scaling and thus the constructions of both are valid. Therefore the regime boundaries are in fact only specified up to a constant factor.) In this regime the dominant energy contribution comes from the perimeter regularization near the top and bottom boundary Γ t and Γ b : Since Γ t ∪ Γ b must be contained in ∂O, but O may only have a tiny material consumption on any cross-section between the top and bottom boundary, the shape O must exhibit a surface area of roughly H 2 (Γ t ∪ Γ b ) (where H 2 denotes the two-dimensional Hausdorff measure) for the transition from Γ t and Γ b to an almost empty cross-section nearby. This causes the excess cost of order 2 ε. The actual geometry of the force-transmitting structure in between Γ t and Γ b has substantially smaller cost and thus is not so important; one can for instance construct it similarly to the regime of small forces, shown in fig. 1.2 left. Furthermore, in this regime it does not matter whether per Ω or per R 3 is used as perimeter regularization as the scaling is the same. Here the major cost contribution stems from the coarsest elementary cells near the midplane, for which it is important to find the right balance between compliance and perimeter. The structure differs from the optimal pattern in type-I superconductors by additional cross trusses (corresponding to the base edges of the pyramid). Without these all other struts would be bent towards each other, resulting in a huge compliance. Such cross trusses were already required in the two-dimensional setting in [14]. The structure may be seen as approximating a laminate; just like for laminates, all struts have unit stress inside. Here, too, there is no difference in the energy scaling whether per Ω or per R 3 is used as perimeter regularization, since the boundary contribution to the perimeter is negligible compared to the total excess cost.
Intermediate force. By intermediate forces we want to refer to the regime c ≤F ≤ C for two (arbitrary) positive constants 0 < c < C < 1. This regime did not occur explicitly in theorem 1 but instead forms part of the regimes of small and of large forces, respectively. Indeed, ifF is bounded away from 0 and from 1, the scaling in both these regimes coincides. Nevertheless we here mention the range of intermediate forces separately since in this range the optimal scaling is in fact also obtained by a third construction different from the one for small or large forces: One can consider the optimal pattern for compliance minimization in two space dimensions and constantly extend it along the third dimension, see fig. 1.3. This construction exhibits the same energy scaling independent of whether per Ω or per R 3 is used.
Large force. This regime contains all forces with for some (arbitrary) positive constant C < 1. Those forces are relatively close to the maximumF = 1 from which on O = Ω is known to be optimal, but they still stay some distance (expressed in terms of ε) away from it. The corresponding construction is again composed of elementary cells, where each cell is a solid cuboid perforated by five thin cone-like voids, see fig. 1.2 right. In this regime, for the first time, it can make a difference whether the perimeter is regularized with per Ω or per R 3 since the maximum in the definition off in theorem 1 may arise from any of the terms, depending on the size ofF and . Indeed, ifF gets too close to 1, the perimeter contribution from ∂Ω dominates and masks the cost associated with the fine structure in the domain interior.
In the domain interior, however, again the coarsest elementary cells near the midplane contribute the major part of the cost.
Extremely large force. The regime of extremely large forces is characterized bȳ In this regime the force is large enough such that the optimal geometry O is just a solid block of material O = Ω. Any reduction in volume via material removal would be outweighed by the associated increase in compliance and perimeter. If per R 3 is used instead of per Ω , the boundary cost dominates.
Remark 2 (Two-and three-dimensional constructions). Note that in the regime of intermediate forces both two-dimensional constructions (constantly extended along the third dimension) and three-dimensional constructions (the constructions from the regimes of small and of large forces) achieve the optimal energy scaling. The same holds true in the regime of extremely small forces: The constructions from fig. 1.2 left and fig. 1.3 right both achieve the scaling ε (though the three-dimensional construction is expected to have a better constant): The perimeter contribution from the top and bottom boundary simply dominates the excess cost so much that the construction in between becomes unimportant. In the regime of extremely large forces even a zero-dimensional construction, constantly extended in all three dimensions, achieves the optimal energy scaling: the full material block. However, in the regimes of small and large forces, respectively, only three-dimensional constructions can achieve the optimal energy scaling: As shown in [14] for small forces, the excess cost of any two-dimensional construction can at most scale like the third root inF , which is worse than the power 2 3 obtained here via a threedimensional construction. Likewise, via the same proof technique as in [14] or via an optimal transport-based argument as in section 3.3 one can show that the excess cost of any two-dimensional construction can at most scale like the power 2 3 in 1 −F (we do not know whether this lower bound is sharp, though, cf. section 4.2), which is worse than the scaling (1 −F )| log(1 −F )| 1/3 obtained by the three-dimensional construction.
Remark 3 (Nonzero Poisson ratio). Above we assumed a material with zero Poisson ratio. While switching from zero to nonzero Poisson ratio changes the compliance and thus J α,β,ε,µ,F, ,L (O) by at most a constant factor, this need not be true for the excess cost J α,β,ε,µ,F, ,L (O) − J α,β, * ,µ,F, ,L 0 , which is the quantity we estimate. However, all constructions are based on almost vertical strut-like structures bearing just uniaxial loads (except maybe the construction from the large force regime), whose excess cost is (in addition to the perimeter cost) mainly caused by their slight deviation from vertical alignment. This contribution is actually independent of the Poisson ratio so that a nonzero Poisson ratio is not expected to change the energy scaling.

Model description and relation to superconductivity
In this section we provide a detailed definition of the considered cost functional, introduce related costs in models of type-I superconductors and show that the latter provide lower bounds for our problem.

Cost components
To define compliance we first briefly recapitulate the basic framework of linearized elasticity. Assume an elastic body O ⊂ R 3 with Lipschitz boundary is subjected to a boundary load f : ∂O → R 3 (say in L 2 (∂O; R 3 )). The load represents a surface stress acting on the boundary. As a result the body O deforms, and the equilibrium displacement u : O → R 3 will be a minimizer of free energy where C is the fourth order elasticity tensor of the elastic material, A : B = tr(A T B) denotes the Frobenius inner product between two matrices A and B, and is the symmetrized gradient of the displacement (the skew-symmetric part corresponds to linearized rotations and thus does not contribute to the energy). The first term of the free energy is the internally stored elastic energy, the second is the negative work performed by the load during the deformation. For the minimizer u of the free energy, this work is also called compliance of O under the load f , and it represents a quantitative measure of structural weakness. In linearized elasticity there are multiple equivalent formulations of the compliance. In particular, it is well-known (and essentially follows from convex duality, see for instance [18]) that the compliance can also be expressed as for R 3×3 sym the real symmetric 3 × 3 matrices and n the unit outward normal to ∂O. The minimizing σ is the so-called stress tensor, describing the internal forces in equlibrium, and it is related to the equilibrium displacement u via σ = C (u). As explained in section 1.2, we consider the simple elastic constitutive law of a homogeneous isotropic material with vanishing Poisson ratio and shear modulus µ, which means that the elasticity tensor is given by C = 2µ . Furthermore, in our case O ⊂ Ω as well as f =σn = (0 0 F ) T on ∂Ω ⊃ ∂O and f = 0 else so that we can extend any stress field σ by zero to Ω \ O and thus obtain a slightly simpler formulation of the compliance via for the admissible stress fields This definition of the compliance can even be extended to arbitrary Borel sets O ⊂ Ω, so the condition of Lipschitz boundary can be dropped. The second cost contribution, the material consumption or volume vol(O) is nothing else than the Lebesgue measure of O.
The third cost contribution, the perimeter, is defined as the total variation of the characteristic function Sets with per R 3 (O) < ∞ are commonly called sets of finite perimeter, and there is a measure-theoretic notion of boundary for them, the essential boundary

Nondimensionalization
In this paragraph we briefly nondimensionalize our cost J α,β,ε,µ,F, ,L in order to reduce the number of parameters. As for the compliance, it is straightforward to check that Thus we have where we abbreviated Ω ,L = (0, ) 2 × (0, L). The combination of the above formulas yields the desired result.

A model for the intermediate state in type-I superconductors
Here we briefly recapitulate the model from [7,4] whose lower bounds will be applicable to our setting. So-called type-I superconductors reveal an interesting intermediate state if exposed to a magnetic field. They partition into two regions, the superconducting one, in which the magnetic field is suppressed, and the normal one, where the magnetic field passes through the material without loss. The resulting patterns were analysed via energy scaling laws in [7] (upper bounds) and [4] (matching lower bounds). The authors consider an infinite material plate of thickness L (we adapt the notation slightly to ours), and assuming periodicity they set to represent the material and its complement with periodic boundary conditions along the first two, inplane dimensions (in fact they rescale lengths such that = 1). The plate is exposed to a transverse magnetic field b a e 3 (the 'applied field') of strength b a > 0.
The magnetic field has been rescaled such that the critical field strength above which the material immediately loses its superconducting properties is 1. Consequently, the range of interest for the applied field strength is 0 < b a < 1.
Denoting the induced magnetic field by B : Ω ∪ Ω c → R 3 and the characteristic function of the superconducting region by χ : Ω → {0, 1}, both periodically extended along the first two dimensions, the free energy is given by with the additional constraints divB = 0 in R 3 and Bχ = 0 in Ω due to Maxwell's equations and the magnetic field being suppressed in the superconducting region, respectively. The first and last term express the magnetic energy due to the discrepancy between induced and applied magnetic field within and outside the material, respectively (the occurrence of χ in the first term is motivated via a relaxation argument in [7]). The middle term measures the interfacial area between superconducting and normal regions via the total variation of χ, weighted by ε, representing a (small) magnetic energy loss from the transition between normal and superconducting regions. Actually, in [7,4] the total variation is taken with respect to periodic boundary conditions along the first two dimensions, while we defined |χ| TV(Ω) as the total variation relative to the open set Ω and thus ignore contributions from the periodic boundary. Note that this slight change does not affect the energy scaling law proved in [7,4] and cited below, since none of their constructions shows any contact area between normal and superconducting regions within the periodic boundary and since their lower bound proofs also apply in this setting.
Theorem 7 (Energy scaling in intermediate state of type-I superconductors, [7,4]). Let In fact, the theorem is proved for = 1 in [7,4], but the above version immediately follows from the straightforward relation for any r > 0. We will actually employ the result for L = 1 since in our setting we think it more natural to nondimensionalize the domain height rather than its base area to 1.
The base area then simply enters the energy scaling linearly, as one would expect.

The relation between both models
There are essentially three differences between our compliance minimization problem and the superconductor setting, 1. the former has a matrix-valued state variable σ, the latter only a vector-valued variable B, 2. σ · n is prescribed on the top and bottom domain boundary, while for B · n only the deviation from a preferred value is penalized via Ω c |B − b a e 3 | 2 dx, 3. σ · n is prescribed as zero on the sides of Ω, while B · n is only required to be compatible with periodic boundary conditions.
As a consequence, as already noted in [14], the superconductor energy minorizes the compliance minimization cost.
Now set χ to be the characteristic function of Ω \ O and B to be the last row of σ, extended to Ω c by b a e 3 . Then by construction, B is divergence-free with χB = 0 in Ω and periodic boundary conditions along the first two dimensions. Furthermore, We now use that the total magnetic flux through any cross-section is conserved since B is divergence-free. Indeed, let

This implies
a 2 so that the above estimate can be continued with The result now follows from the arbitrariness of δ > 0.

Lower bounds
In this section we prove the lower bounds of theorem 5. Due to lemma 8 and theorem 7 only three lower bounds remain to be shown, for the case when the perimeter regularization is performed with per R 3 rather than per Ω , (while for larger F this bound is already implied by lemma 8 and theorem 7).
The following sections provide the corresponding estimates. Before, let us briefly introduce some notation. For t ∈ [0, 1] we will abbreviate We will use that any stress σ ∈ Σ F, ,1 ad (O) has the same average vertical tension in all cross-sections. Indeed, for almost all t ∈ (0, 1) we have

which implies
Ot Using Jensen's inequality this implies a bound on the compliance in almost all crosssections, (3.1)

Lower bound on exterior perimeter contribution
Here we estimate the cost contribution from the surface area of the optimal geometry O within ∂Ω. Since necessarily ∂O is a subset of the top and bottom face Γ t and Γ b of Ω (as otherwise the compliance is infinite), we automatically havẽ It thus remains to estimate the cost contribution related to the perimeter at the sides of the domain. To this end we use the following elementary result.
Lemma 9 (Polynomial estimate). For any y, a > 0 we have Proof. Abbreviate f (y) = (1/y − y) 2 + ay, then for any y ∈ (0, 1 2 ) we have where the minimizer is given by z = a −1/3 . Furthermore, for any y ≥ 1 we have Finally, for y ∈ [ 1 2 , 1) we have This expression is minimized by y = 1 − a so that The desired estimate now follows essentially from the isoperimetric inequality.

Proposition 10 (Exterior perimeter estimate). For any
where per R 2 indicates the perimeter of the two-dimensional set O t (projected into R 2 ). Now by Fubini's theorem, (3.1) and lemma 9 we havẽ where we abbreviated y(t) = A t /(F 2 ).

Lower bound for extremely small force
Here we aim to exploit that for small forces almost no material may be used so that most cross-sections are almost empty. The transition to the boundary then produces substantial perimeter cost.
Proposition 11 (Interior perimeter estimate). Let ε < 1 8 and F < 1 Proof. There are two cases, depending on whether A t ≥ l 2 /2 for almost every crosssection t ∈ (0, 1) or not. In the former case, by Fubini's theorem and (3.1) we have as desired. In the latter case let t ∈ (0, 1) be a cross-section with A t ≤ 2 /2. Then for χ O the characteristic function of O and | · | TV((0,1)) the total variation seminorm of a function on (0, 1). Due to as desired, where χ O (·, ·, s) with s = 0, 1 is the trace of the function of bounded variation χ O and equals 1 whenever comp F, (O) < ∞.

Lower bound for small force
Here we show that the lower bound in the regime of small forces actually extends to ε 1/2 F (lemma 8 and theorem 7 only imply that it holds for ε 2/7 F ). In fact, we can simply repeat the proof of [4,Thm. 4.3] (the lower bound for the superconductor energy under a small applied field). We only have to modify the last inequalities of that proof, which actually simplify in our setting. For the sake of completeness we recapitulate the proof below. We slightly changed its structure to provide the reader with a better intuition of why the proof is actually quite direct. Proof. One can argue by contradiction, so assume for some O and some constant c * ∈ (0, 1 2 ) to be specified later. Furthermore, let σ ∈ Σ F, ,L ad (O) be such that comp F, (O) = Ω |σ| 2 dx − δ with δ small enough so that also For t ∈ (0, 1) we will abbreviate A t := H 2 (O t ) and P t := per (0, ) 2 (O t ). The argument is performed in three steps.
Step 1. For a generic cross-section we bound the compliance, volume and perimeter from above in order to obtain estimates for the material volume and the value of the stress in that cross-section. For a contradiction assume that there exists I ⊂ (0, 1) with Lebesgue measure at least 1 2 such that almost all cross-sections t ∈ I satisfy Ot |σ| 2 dH 2 + A t − 2F 2 > 2c * F 2/3 ε 2/3 2 or P t > F 2/3 ε −1/3 2 . By Fubini's theorem we would then have a contradiction. Therefore we may assume that for at least half the cross-sections t ∈ (0, 1) we have From now on let t denote such a cross-section. The above bound directly implies an estimate on the cross-sectional volume. Indeed, if A t > 2F 2 or A t < F 2 2 , then another contradiction. Therefore we have in addition Finally, the above bound also implies that on the cross-section the material stress deviates only little from a vertical tensile unit stress, but we will postpone the quantification of this fact to step 3.
Step 2. Using the previous perimeter and volume bounds we now characterize O t as being (mainly) composed of connected components with typical area ( At Pt ) 2 ∼ (F ε) 2/3 and perimeter (or equivalently diameter) At Pt ∼ (F ε) 1/3 . This could be done directly by elementary methods, but for later purposes it is a little more convenient to instead first replace O t by a slightly nicer setÕ t and then to express the above characterization in terms of a scaling behaviour for the volume of dilations ofÕ t . To this end abbreviate and, denoting the dilation ofÕ t by r > 0 asÕ r t = {x ∈ Ω t | dist(x,Õ t ) < r}, This latter condition encodes that the connected components ofÕ t exhibit the behaviour mentioned above.
Step 3. Finally, one estimates the excess compliance necessary to distribute the approximate vertical unit stress in each of the connected components of O t evenly on the upper and lower boundary of Ω. This redistribution of the stress can actually be viewed as a transport problem of the vertical momentum: We interpret the vertical momentum ρ s (x 1 , x 2 ) = e T 3 σ(x 1 , x 2 , s)e 3 as a temporally changing material distribution, where time runs from s = 0 to s = t. The corresponding temporally changing material flux is then given by the time-dependent vector field ω s (x 1 , x 2 ) = (σ 13 (x 1 , x 2 , s) σ 23 (x 1 , x 2 , s)) T on (0, ) 2 . Indeed, ρ and ω satisfy the transport equation ∂ ∂s ρ s + divω s = 0 in the distributional sense since for any smooth φ : where we used Stokes' theorem and that σ is divergence-free. Now part of the compliance can be viewed as a cost associated with this transport: Using (3.1) and Jensen's inequality we have where π(O s ) stands for the projection of O s into the x 1 -x 2 -plane. Now the Wasserstein-1 optimal transport distance between ρ 0 and ρ t in the Benamou-Brenier formulation is [17, § 6.1] where the last equality is known as Kantorovich-Rubinstein duality (see for instance [17, § 4.2.1] for the last formula). Thus we have ∆J ≥ 1 2F 2 (W 1 (ρ 0 , ρ t )) 2 , and to bound the Wasserstein-1 distance from below we can simply construct a dual variable ψ for the Kantorovich-Rubinstein formula, which we do in the following. Since ρ t is approximately 1 in the connected components of O t (which we still have to show) and those roughly have diameter l and area l 2 , the mass of ρ t in each of these components needs to be spread out to an area l 2 /F in order to reach the density ρ 0 = F . Thus we expect the typical distance that a particle is transported to be l 2 /F . We therefore abbreviate to be the typical transport distance. A good ψ then is given by which is r onÕ t and decreases to 0 linearly with the distance toÕ t . We thus obtain and it remains to estimate (0, ) 2 ψ(ρ t − ρ 0 ) dx = Ωt σ 33 ψ dH 2 − F (0, ) 2 ψ dx. It is the estimate of the first integral that shows that σ approximately has a unit vertical component or at least that σ 33 is more or less greater than or equal to 1. Indeed, we calculate where the summands can be estimated as We bound the two factors via for some fixed constant C so that in summary we obtain By choosing c * small enough (depending on C) and using √ ε < F ≤ 1 2 we get (0, ) 2 ψ(ρ 1 − ρ 0 ) dx ≥ rF 2 16 and thus which is the desired contradiction to our assumption if we choose c * < 2 −15 .

Upper bounds
As already explained in the introduction, the upper bounds are obtained by constructions that are composed of layers of elementary cells. We briefly introduce our corresponding notation. We will only specify the constructions and compute their energy for the upper half (0, ) 2 × ( 1 2 , 1) of Ω since the construction for the lower half is always mirror-symmetric. The different layers of elementary cells in that upper half are numbered in ascending order, beginning with 1 for the layer sitting on the midplane Ω1 2 . The index of the last layer will be denoted n ∈ N ∪ {∞}. Sometimes this last layer will not be composed of elementary cells but will be constructed as a special boundary layer. The height of the ith layer (or equivalently of the elementary cells in that layer) is denoted l i , and L i = l 1 + ... + l i denotes the accumulated height of the first i layers so that the bottom of layer i + 1 is at height 1 2 + L i and we must have 1 2 + L n = 1. The width of the elementary cells in the ith layer is denoted w i and halves from layer to layer,
All our constructions will ensure σ ∈ Σ F, ,L ad (O). Finally, the excess cost of our construction will be abbreviated as We will only explicitly calculate the perimeter relative to Ω; changing from per Ω to per R 3 is trivial for all our constructions.

Small and extremely small force
Here we detail the construction from fig. 1.2 left, whose elementary cells consist of struts along the edges of a pyramid. We employ the same construction for the regimes of small and of extremely small forces. The different energy scaling for extremely small forces then only comes from the boundary layer dominating the total cost. We first describe the construction within a single elementary cell and estimate its excess cost contribution. We then describe the construction of a boundary cell in the boundary layer and estimate its excess cost contribution. Finally, we describe the assembly of all cells into the full construction and estimate its excess cost.
Elementary cell construction. The material distribution within an elementary cell is illustrated in fig. 4.1. We will just detail the front right quarter of the construction (shown in black), the other quarters being mirror-symmetric.
Let coA denote the convex hull of a set A ⊂ R 3 . We specify the material distribution O within the front right quarter of the unit cell as the union of (not necessarily disjoint) simple geometric shapes elementary cell as indicated in fig. 4.1 so that the elementary cell of width w and height l (for simplicity we drop the index i indicating the layer) occupies the volume (− w 2 , w 2 ) 2 × (0, l). We will assume w ≤ l which will be ensured throughout the construction. We further fix the lengths a and s, using their relation shown in fig. 4.2

left, via
2s tan α (they are chosen such that all struts will have unit stress), where α denotes the angle of the z-axis with the upwards pointing trusses. We do not provide an explicit formula for α and just note that it is implicitly and uniquely determined by the construction as a function of the lengths w, l, and s. However, using F ≤ 1 2 it is straightforward to see that as long as w ≤ l. With this the point coordinates are given as P 1 = (s, s, 0), P 2 = (0, s, 0), P 3 = (0, 0, 0), P 4 = (s, 0, 0), P 5 = (s, s, a 2 ), P 6 = (0, s, a 2 ), P 7 = (0, 0, a 2 ), P 8 = (s, 0, a 2 ), P 9 = (0, 0, a), In addition to the material distribution we have to specify an admissible stress field σ in the front right quarter of the elementary cell (the stress field σ in the front left quarter is then obtained as σ (x 1 , x 2 , x 3 ) = diag(1, −1, 1)σ(x 1 , −x 2 , x 3 )diag(1, −1, 1) for diag(r, s, t) denoting a diagonal matrix with entries r, s, t, and the stress field σ in the back half is obtained as σ (x 1 , x 2 , x 3 ) = diag(−1, 1, 1)[σ + σ ](−x 1 , x 2 , x 3 )diag(−1, 1, 1)). Denoting the characteristic function of shape O j by χ O j , we set σ = 11 j=1 χ O j σ j for constant stresses σ j specified below. We will abbreviate the identity matrix by 1, the Euclidean standard basis by {e 1 , e 2 , e 3 } and the tangent vector to the upwards pointing truss by s a a/2 α · · · · · · · · · P 9 P 1 P 5 Elementary cell excess cost. The volumes of the simple geometries can readily be calculated as so that, using |F − P 1 | ≤ l/ cos α, the material volume of the elementary cell can be estimated from above as  In the following we will frequently use the estimates a ≤ sw l , 1 cos 2 α = 1 + tan 2 α ≤ 1 + w 2 2l 2 , and 1 cos α With those, the volume becomes vol cell ≤ 4s 2 l + 3s 2 w 2 l + 32 3 s 3 w l ≤ 4s 2 l + 41 3 s 2 w 2 l .
We here evenly distributed J * ,F, 0 = 2F 2 over the total volume Ω so that the amount corresponding to the elementary cell is 2F w 2 l. We now pick the minimizing elementary cell height which is still no smaller than w, (In fact, had we not simplified the excess cost using the assumption w ≤ l, we would have arrived at the same choice of l as the minimizer of the non-simplified elementary cell excess cost.) With this choice we obtain Boundary cell construction. The last layer has to evenly distribute the stress from the previous layer of elementary cells over the top boundary of Ω. It will again be partitioned into identical cells of width w and height l which we term 'boundary cells' (again we drop the index n indicating the layer). We again place a coordinate system at the bottom centre of the boundary cell so that it occupies the volume We choose l = √ 3w/2 and fill the boundary cell completely with material. The stress field in the boundary cell is again specified as σ = We abbreviate by B Z (r) the ball with centre Z and radius r and by C Z,v (r) the infinite cylinder with centre Z, axis v ∈ R 3 , and radius r. With this preparation the domains are given as and the stresses as Again by checking that the normal stresses add up to zero at all domain interfaces (except the bottom face of O 12 and the top face of O 16 ) we obtain that σ is divergencefree with normal tensile stress of magnitude F at the top boundary of the boundary cell and normal tensile stress of magnitude 1 on (−s, s) 2 × {0}, where it is attached to the elementary cell underneath, whose stress it exactly balances.
w l x 1 x 2 Boundary cell excess cost. Since the full boundary cell volume is occupied by material, the volume and perimeter contribution of the boundary cell can be estimated by (note that only the bottom face counts to the perimeter; the top face lies in ∂Ω and is thus not counted, while the sides are adjacent to the neighbouring cells and thus do not form an interface with the void). Using that the stress nowhere exceeds Frobenius norm √ 3, the compliance can be estimated from above as Thus, the excess cost contribution of a boundary cell becomes Full construction. The layers of elementary cells are stacked as previously described, where the last, nth layer is composed of boundary cells. We let index N refer to the last layer whose elementary cells satisfy w < l (thus w j = l j for j = N + 1, . . . , n − 1). For the time being let us assume N ≥ 1. In principle one could stop layering at index N and directly introduce the boundary layer, however, it turns out that in that case the optimal energy scaling in the small force regime is only reached for F ε 2/7 , which is why we will continue adding layers.
Let us now identify the width w 1 of the coarsest elementary cells. Since all layers have to sum up to total height 1, we have as long as 1 < N < n, where we exploited the geometric series. Due to F 1 . Hence we pick In more detail, w 1 shall be the largest width smaller than F − 1 6 ε 1 3 /8 such that is an integer multiple of w 1 . For this to satisfy w 1 ∼ F − 1 6 ε 1 3 we require ≥ F − 1 6 ε 1 3 . Note that with this choice of w 1 the heights of all layers actually add up to less than one, which is remedied by increasing l 1 until the total height exactly equals 1. This little change of l 1 increases the excess cost ∆J cell of the coarsest elementary cells at most by a bounded factor.
We next specify the number n − N of layers with w = l. Since the excess cost of the boundary layer at least scales like ε, which is achieved by the choice w n ∼ ε, we set n to be the first index for which w n ≤ ε (thus n = 1 + log 2 2w 1 ε for the floor function · ). Furthermore, N is the last index for which w N < l N or equivalently We are finally in the position to estimate the total excess cost. Note that in the ith layer there are ( /w i ) 2 many elementary or boundary cells. Denoting the excess cost of cells in layer i by ∆J i cell , the total excess cost thus is In the regime of small forces the first summand dominates, in the regime of extremely small forces the second one. So far we had assumed N ≥ 1 or equivalently w 1 < l 1 . For our above choice of w 1 this only holds for ε √ F . If this is violated, we instead have N = 0, and the calculation above yields ∆J 2 [(n − N ) √ F ε + ε]. Since again n − N ≤ | log 2 √ F |, the scaling is not impaired.

Intermediate force
In the regime of intermediate forces the optimal energy scaling is also obtained by a two-dimensional construction, which is constantly extended along the third dimension. Indeed, letΩ = (0, ) × (0, 1) and letŌ ⊂Ω andσ :Ω → R 2×2 sym be such thatσ = 0 on Ω \Ō, divσ = 0 inΩ andσn = F (e 2 · n)e 2 on the boundary ∂Ω. Then Hence we have reduced the three-dimensional construction to the problem of findingŌ andσ with ∆J : SuchŌ andσ were constructed in [14], where ∆J F 1/3 ε 2/3 is proved for any F ≤ 1 − 1 16 . The three-dimensional extension of this construction is illustrated in fig. 1.3 right. In this section we want to complement this construction by yet an alternative, valid for F > 1 16 and satisfying ∆J (1 − F ) 1/3 ε 2/3 . Its three-dimensional extension is illustrated in fig. 1.3 left. As before, we first specify the two-dimensional elementary cells and estimate their excess cost, then we specify the boundary cells and estimate their excess cost, and finally we describe the full construction.
The angle α is again implicitly and uniquely determined as a function of w, l, and s. The union of these simple shapes forms the material region. The stress is defined as with τ = (− sin α, cos α) T . It is straightforward to check thatσ is divergence-free and has unit normal tensile stress on ] × {l}, balanced exactly by the elementary cells on the next and previous layer.
Elementary cell excess cost. The volumes of the simple geometries can readily be calculated as so that the material volume of the elementary cell can be estimated from above as Throughout the construction we will ensure (1 − F )w ≤ l, from which we obtain Consequently the volume estimate becomes Likewise, the surface area shared between void and each simple geometry can readily be calculated as so that the perimeter contribution from the elementary cell can be estimated from above as per cell ≤ 2l + 4 l cos α + 4(1 − F )w ≤ 12l. Finally, noting that the stress nowhere exceeds Frobenius norm √ 2, the squared L 2 -norm of the stress on the elementary cell can be estimated via Summarizing, the excess cost contribution of an elementary cell can be estimated via We now specify the elementary cell height as the minimizer so that the excess cost contribution of the elementary cell becomes x 2 and B Z (r) to be the ball of radius r centred at Z. The stress field in the boundary cell is then specified asσ fig. 4.6). Again it is straightforward to check thatσ is divergence-free and has boundary stresses compatible with the below layer of elementary cells and the boundary load applied on ∂Ω.
Boundary cell excess cost. Since the boundary cell is filled completely with material and since the stress nowhere exceeds Frobenius norm √ 2, the volume, free perimeter and compliance of the boundary cell can be estimated as Its contribution to the excess cost thus becomes Full construction. Let us first identify the width w 1 of the coarsest elementary cells. Since all layers have to sum up to height 1 we have so that we choose w 1 to be the largest width smaller than As before, the layer heights now sum up to something less than 1, which is remedied by sufficiently increasing l 1 . We will stop layering as soon as (1 − F )w n ≥ l n , thus w n ∼ (1 − F )ε. Denoting again the excess cost contribution of the cells in layer i by ∆J i cell , the total excess cost of all cells can then be bounded via Note that for the superconductor problem the scaling actually would be (1−F ) 2 3 ε 2 3 . Our different power of (1−F ) stems from the excess compliance in regions O 4 and O 2 ∩O 3 . A more careful construction might be able to recover the better power, however, we do not attempt this here since F is bounded away from 1 anyway in the regime of intermediate forces.

Large force
Here we describe the construction from fig. 1.2 right in which the elementary cells consist of full material blocks with small roughly conical holes drilled inside. The construction of an admissible stress field with optimal scaling requires substantially more effort than in the superconductor setting. Due to the estimate of the previous section it suffices to provide a construction of the correct energy scaling for 1 − F < 1 64 , which we will assume in the following.
Elementary cell material distribution. As before, we use coordinates such that the elementary cell occupies the volume The elementary cell is a full block of material with five cone-like holes arranged as in fig. 4.7 left: a big central one pointing upwards and four identical downward pointing ones of half the diameter, centred within each quarter of the elementary cell. Their bases are such that four elementary cells of half the width can be attached on top of ω such that the cones seamlessly fit together. In more detail, abbreviate the solid of revolution around the x 3 -axis with radius function f by If we denote the cone tips by P 0 = (0 0 l) T and P 1 , ..., P 4 = (± w 4 ± w 4 0) T and the x 3 -dependent radius of the four smaller cones by R(x 3 ), then the material distribution O within the elementary cell is given by for the cones Note that for symmetry reasons we chose the central cone to be identical to the four smaller ones, only flipped upside down and dilated by the factor 2 along the first two dimensions. We next specify the radius function R(x 3 ). For ε = 0 it is known that the optimal microstructures everywhere exhibit a material volume fraction of F . We aim to match this exactly on each horizontal slice of the elementary cell and thus require Furthermore, the elementary cells will be imposed with a normal stress at their top and bottom boundary, and since this normal stress cannot immediately be redirected to a nonvertical direction without producing infinite compliance at the cone tips, we require This is one of the central differences to the corresponding construction in the superconductor setting. Together with the previous condition this also implies R (l) = 0. One possible solution is to take the cross-sectional area of the cones to be the quintic polynomial fulfilling the above conditions, Note that the material volume of the elementary cell is F w 2 l by construction. Elementary cell stress field construction. The construction of the stress field within the elementary cell is substantially more complicated than in the superconductor setting. We are looking for a divergence-free σ : ω → R 3×3 sym with σ = 0 on C 0 ∪ . . . ∪ C 4 and unit tensile stress σn = n on the top and bottom material boundary . We will first construct a radially symmetric stress field within cylinders around the cones and in a second step construct the stress field in the complement of these cylinders. In more detail, let C[r] denote the infinite cylinder around the x 3 -axis of radius r, then we define . . , 4 to be disjoint cylinders around the cones K 0 , . . . , K 4 . Since we assumed 1 − F < 1 64 , these cylinders contain the cones completely. We then compose the stress field via where σ cyl , σ rad , and σ shear will be specified below.
The stress field in C 0 , . . . , C 4 is reduced to a stress fieldσ : ω ∩ C[ w 16 ] → R 3×3 sym with σ = 0 on K[R] and unit tensile stress at the top and bottom boundary via By construction, σ cyl is divergence-free ifσ is. We takeσ radially symmetric, where the simple intuition behind our ansatz is that the material (C[ w 16 ] ∩ ω) \ K[R] is decomposed into infinitely many infinitesimally thin chalice-like shells that are each in equilibrium on their own and do not interact. We parameterize the family of chalices by their radius s at x 3 = 0 and denote their radius at height x 3 by R s (x 3 ) (thus R 0 = R). The vertical stress component in each of these chalices shall be 1 (which is known to be preferred from the case ε = 0). The balance of forces acting in vertical direction on the union of chalices for s ∈ [0, S] with S ∈ [0, w 16 ] arbitrary then requires Using cylindrical coordinates (r, ϕ, z) and denoting the radial, circumferential and vertical unit vectors by e r , e ϕ and e z , respectively, the tangent plane to a chalice is spanned by t(r, ϕ, z) = R s(r,z) (z) e r + e z and e ϕ , where s(r, z) = r 2 − R(z) 2 is the parameter of the chalice containing the circle of radius r at height z. Since the chalices do not interact, they can only have in-plane stress, and due to radial symmetry all shear stresses are zero so thatσ must be of the form σ(r, ϕ, z) = α(r, z) t(r, ϕ, z) ⊗ t(r, ϕ, z) + β(r, z) e ϕ ⊗ e ϕ .
By constructionσ is divergence-free, zero inside the cones, and it satisfies the correct boundary condition at the top and bottom.
Remark 13 (Improved stress field). By letting the chalices interact via a normal stress component one could reduce the excess compliance further, but it turns out that this only leads to a reduction by a constant factor, leaving the scaling the same.
In the complement of the five cylinders we employ a vertical unit tensile stress plus two stress fields σ rad , σ shear : ω \ (C 0 ∪ . . . ∪ C 4 ) → R 3×3 sym that compensate the boundary stress induced by σ cyl on ∂C 0 ∪ . . . ∪ ∂C 4 . In more detail, σ rad will be a pure inplane stress-field in each horizontal cross-section, accommodating the normal stress on the cylinder walls imposed by σ cyl , while σ shear will be a stress field accommodating the shear stress on the cylinder walls. The stress σ rad will be radially symmetric around each cylinder and will itself be contained in a slightly larger cylinder of radius T = 3w 32 at the four outer cones and radius 2T at the inner one (so that those larger cylinders do not yet overlap). It is reduced to a stress fieldσ : As already suggested,σ shall be of the form σ(r, ϕ, z) = at the outer and inner cylinder boundary (recall that it compensates the normal stress of σ cyl at the inner boundary, which is given byσ rr ). A possible choice ofσ is (this choice is actually the equilibrium stress under the above conditions, which is a straightforward exercise to derive). The stress σ shear finally is chosen as follows: For fixed height z it has to accomodate the (vertical) shear stress on the cylinder walls ∂C i induced by σ cyl , which equals σ rz ( w 16 , ϕ, z) = 16 w R(z)R (z) on ∂C 1 , . . . , ∂C 4 , while it is given by −2σ rz ( w 16 , ϕ, l − z) = −2 16 w R(l − z)R (l − z) = − 32 w R(z)R (z) on ∂C 0 . Thus, for g 2 (z) = R(z)R (z) w/16 we require σ shear n = 2g 2 (x 3 ) e 3 on ∂C 0 , σ shear n = −g 2 (x 3 ) e 3 on ∂C i , i = 1, ..., 4 with n the unit outward normal to ω \ (C 0 ∪ . . . ∪ C 4 ).
Remark 14 (Inplane equilibrium of shear stresses). Since ∂C 0 has exactly twice the circumference of ∂C 1 ∪ . . . ∪ ∂C 4 , the shear forces at cross-section z are in equilibrium (which essentially is a consequence of our choice of R). Therefore, σ shear 33 will be zero. This not only simplifies the construction of σ shear but is also important for the energy scaling: a nonzero σ shear 33 would yield a large compliance in combination with the overall vertical unit tensile stress e z ⊗ e z . Such complications only arise due to the symmetry condition on the stress field and are absent in the superconductor setting.  where ∂ n u denotes the normal derivative of u (a solution exists due to ∂ω ∂ n u dx = 0; this is related to the previous remark). Furthermore, let the two-dimensional stress fielď σ :ω → R 2×2 sym satisfy divσ = ∇u inω, σ · n = 0 on ∂ω (such aσ exists due to ω divσ dx = ω ∇u dx = 0 = ∂ωσ · n dx, where the second equality holds by symmetry of u) and defineσ shear :ω × (0, 1) → R 3×3 sym as Elementary cell excess cost. As usual we denote the elementary cell contribution to compliance, volume, and perimeter by comp cell , vol cell , and per cell , respectively. Furthermore we again evenly distribute J * ,F, 0 = 2F 2 over all of Ω so that its proportion within an elementary cell is 2F w 2 l. Recalling now vol cell = F w 2 l, the excess cost contribution of the elementary cell is  where in the last step we used the boundedness of all integrals as well as a < w 2 . The analogous computation for i = 0 yields the same scaling. Similarly,