Issue 
Mechanics & Industry
Volume 25, 2024



Article Number  7  
Number of page(s)  19  
DOI  https://doi.org/10.1051/meca/2024004  
Published online  01 March 2024 
Regular Article
Lightweighting structures using an explicit microarchitectured material framework
^{1}
ICA, Université de Toulouse, ISAESUPAERO, MINES ALBI, UPS, INSA, CNRS, Toulouse, France
^{2}
Università di Pisa, DICI, Pisa, Italy
^{*} email: a.dirienzo3@studenti.unipi.it
Received:
21
May
2023
Accepted:
18
January
2024
In this paper, a new approach to design ultralight structures is developed based on a previous work called Efficient Multiscale Topology Optimization. A parameterized (or explicit) trussbased cell is introduced to generate intrinsically wellconnected microstructures and to get clear interpretable optimal multiscale structures. The method uses a precomputed database of optimal microcells to be computational efficient without losing in structural performances. The parameterization allows to generate a lightweight database just storing the set of parameters, that define the optimal cells, and the cells properties, that are obtained through inverse homogenization. The method has been successfully tested on twodimensional compliance problems. Several examples demonstrate its versatility and give quantitative results. Moreover, it allows to obtain structures compatible with additive manufacturing processes, to naturally solve concurrent multiscale problems, as well as controlled porosity and optimal fiber orientation problems.
Key words: Structural optimization / multiscale design / architectured material design / additive manufacturing / homogenization
© A. Di Rienzo et al., Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
1.1 Context
Topology optimization is a mathematical method to find the optimal distribution of material or structural elements within a design space, for given loads and boundary conditions with the objective to maximize the performances of the structure while respecting constraints. The first general theory of topology optimization was formulated by Prager and Rozvany [1] and lots of different methods have been developed starting from this point: the homogenization method [2], the Solid Isotropic Material with Penalization (SIMP) [3], the inverse homogenization method [4], the evolutionary structural optimization (ESO) [5], level set method (LSM) [6] and others.
The structures resulting from a topology optimization have better performances but higher geometry complexity. Therefore, due to manufacturing difficulties, in the late 1990s, the so called “monoscale” approaches are mainly developed, optimizing the distribution of a homogeneous isotropic material (SIMP) [7,8]. However, the latest advancements in additive manufacturing (AM also known as 3D printing) led to a new interest in the design of multiscale structures, where each macro point can be represented as a local “microscopic” structure. Therefore, the design of architectured materials or metamaterials is having a great development since materials with exceptional properties, that cannot be found in nature, can be obtained, such as maximized bulk or shear modulus [4,9,10], negative Poisson's ratio [11], negative thermal expansion [12]. The previous approaches studied periodically repeated microstructures but in the industrial field the assemblies of optimized microstructure, with spatiallyvarying properties, are of great interest since they allow to achieve structures with optimized characteristics. The issue is how to get optimized microstructures that are compatible with their neighbours.
The design of architectured materials can be achieved by a multiscale topology optimization (MTO), which consists in optimizing both the macroscale and the microone. An exhaustive review of several methods about MTO was done by Wu et al. [13]. The MTO faces two main issues that are:
Connecting neighbouring microstructures without reducing too much the solution space. A bad connection (or compatibility) does not allow the load to be transferred correctly and therefore the properties of the microstructures obtained through homogenization are not representative of how the whole structure will react.
Optimizing simultaneously the two scales has a high computational cost.
Various approaches and methods in literature have been developed by considering these two issues. For instance, Wang et al. [14] presented a framework that uses a multiscale isogeometric topology optimization to optimize the relative density of lattice materials. In that last work, four unitcells with welldefined topologies are predefined and their properties are expressed as functions of the relative density and directly used in the optimization loop, improving the computational efficiency. A similar approach is the one of Watts et al. [15] that considered three open truss unit cells, compatible with many additive manufacturing techniques, for which accurate surrogate models of the properties as function of the relative density are obtained through homogenization. Once selected one of the three possible topologies of the unit cell, it is kept constant over the design domain with the possibility to vary its geometry through the domain within defined bounds.
To improve the connectivity between cells Zhou and Li [16] presented three methods namely connective constraint, pseudo load and unified formulation with nonlinear diffusion. In the first two methods, the unit cells are optimized individually imposing constraints to connect them with predefined common regions. In the third method, the unit cells are optimized all together and a nonlinear diffusion term is introduced in the objective function suppressing checkerboard patterns and blurred boundaries, thereby ensuring proper topological interconnections. Also Garner et al. [17] presented a method to assure connectivity between adjacent cells, optimizing simultaneously the physical properties of the individual cells as well as those of neighbouring pairs. Therefore, they ensure material connectivity and the smooth variation of the physical properties.
Xia and Breitkopf [18] proposed a finite element square nonlinear multiscale analysis framework for concurrent design of the materials and structures. However, the computational cost is quite massive due to a large number of instant local material optimizations. Therefore, Xia and Breitkopf [19] introduced a reduced database model to approximate the material behaviour. The approximate constitutive model for locally optimized material is built offline and the structural optimization using the precomputed constitutive model is performed online, improving the computational efficiency. Precomputed databases have also been proposed for parametrized lattice cells, obtaining a polynomial model to access cells in between database points [20–23].
It is also worth mentioning lately some works such like Pantz and Trabelsi [24], Allaire et al. [25], GoeffroyDonders et al. [26], or Groen and Sigmund [27], which also combine topology optimization and periodic homogenization theory in a different way. Indeed, they rather considered some postprocessing dehomogenization (or “inverse homogenization”) treatments that consist in replacing an optimized homogenized design by a periodic lattice. The process involves notably as well an orientation regularization and a projection algorithm onto a specific family of parametrized microcells (with cubic or square symmetry). Their inverse homogenization strategies differ from the one presented hereafter and that intends to use a precomputed database of microcells (defined as clusters of beams) in order to be computational efficient without losing in structural performances.
The remaining part of this paper is organized as follows. Section 2 of this article briefly describes the main contributions of the Efficient Multiscale Topology Optimization (EMTO) method and provides the objective of this work. In Section 3 a new explicit approach (ExEMTO) is developed by defining a parametrized trussbased unit cell, describing the microscale and macroscale optimization problems and explaining the database construction with a first comparison with the original EMTO. Section 4 provides some comparison with the original EMTO and with other strategies on classical problems and shows clear interpretable structures obtained with the new approach. In Section 5, the versatility of the method is demonstrated by solving different types of problems. Finally, Section 6 concludes the paper with final remarks and with suggested future work.
2 Efficient multiscale topology optimization
We extend here previous works of the authors of EMTO for the 2D problems [28]. The EMTO consists in optimizing the micro and the macro scales separately. The microscale topology optimization is used to create offline a database of microstructures to use in the macroscale topology optimization to obtain competitive structure reducing the computational cost. The EMTO method tries to answer to the previous issues of multiscale approach (connectivity and computational cost) while assuring microstructures to follow local stress or strain. The main contributions are:
The definition of the socalled “adaptive transmission zones” in the 2D microstructure. In order to ensure well connected microstructure, a number of connection points is set along the border of the unit cell. Depending on these connection points, the transmission zones are defined as portions of the borders of the unit cell where the material eventually is enforced to be. In this way stresses can only be transmitted through those zones from a unit cell to the neighbour cells. The transmission zones are adaptive because depending on the conditions they can change their sizes, resulting in very low design constraint.
The introduction of wellchosen variables to bridge the macroscale and the microscale. In the simplest forms of MTO, only the density variable is used. In the EMTO, the orientation of the cell (that corresponds to a rotation of the cell with respect to the outofplane direction) and a variable called cubicity (that quantifies the relative importance of the two principal directions in 2D case) are added as variables too. This allows to follow better the principal stresses and to consider their relative intensity difference. Therefore, density, orientation and cubicity are called macrovariables. So, at the end of the macroscale optimization a set of these three variables is given for each macro element. Examples of how the microstructure changes depending on the macrovariables are shown in Figure 1.
A database containing both the structural architecture and the properties of the unit cells, to have a faster computation in the macroscale optimization. The database is built on a regularly spaced 3dimensional grid of the three macro variables in order to have microstructures close to any point of the design space. This means that, for any given set of macrovariables assumed by a macroelement, during and at the end of the macrooptimization, is possible to find inside the database a microstructure with relative properties that satisfy quite well the conditions given by the set of macrovariables.
The objective of this paper is a further validation of the original EMTO method and also a demonstration of its flexibility. Original EMTO shows that the fullscale structure tends to be trusslike as shown in Figure 2 with lot of internal forces redistribution. Therefore, a new explicit approach, called Ex EMTO, to build the database of the unit cells (parametrized with a manufacturable assembly of beams) is developed and the existing EMTO macrocodes are adapted to the new obtained database, with the aim at obtaining clear interpretable microstructures too. The framework is opensource and available on GitHub for reproducible research purpose (https://github.com/mid2SUPAERO/ExEMTO).
Following the original EMTO, the microscale optimization gives as results optimal microstructures depending on the values assumed by the macrovariables. The stiffness tensors of the microstructures are then used in the macroscale optimization to assemble the global stiffness matrix K of the macrostructure. Subsequently, this matrix is inverted to solve the equilibrium problem u = K^{–1}f and compute the global compliance c. The macrodesign variables are iteratively updated until reaching the optimal values from which obtain the optimal macrostructure.
In the new developed approach, the mechanical properties of the cell can be determined through the classical solution of the strain and equilibrium problems following either EulerBernoulli's or Timoshenko's theory. However, since the relative density of the unit cell can also assume values approaching to one, the beam theory loses accuracy. Consequently, the behaviour of the unit cell is obtained by projecting it on a Finite Elements mesh and solving the equilibrium equations as Xia and Breitkopf [29] where an energybased homogenization approach is adopted. The stiffness tensor is obtained in terms of element mutual energies by imposing three unitstrain tests to the unit cell as shown in Figure 3. They correspond to two normal strain fields and one shear strain field. The nodal displacements for each unitstrain test are computed applying the periodic boundary conditions [30]. To better understand the inverse homogenization method, the periodic boundary condition and the Matlab implementation the related papers are suggested [4,29,30].
Fig. 1 Influence of the 3 macrovariables on a microstructure Boxed in green the unit cell having density 0.5, orientation π/4 and cubicity 0. In each subfigure, two of these variables are fixed, while the third varies. 
Fig. 2 Original EMTO half MBB beam solution. In red some cells are highlighted to show the apparent truss structure. 
Fig. 3 The unit test strain fields imposed on a 2D unit cell. 
3 The new explicit approach
In this section the new explicit approach is explained starting from how the trussbased unit cells are built. The new microscale optimization and the definition of the lighter database are described. The macroscale optimization is briefly reported too. In Figure 4 the flowchart diagram of the proposed method is shown.
Fig. 4 Method flowchart. 
3.1 Trussbased unit cell
The considered unit cell is made by superimposing a square structure and an xshaped structure as illustrated in Figure 5, as Wang et al. [20], Wu et al. [31] and Zhang et al. [32]. This design allows to ensure an intrinsically wellconnection between neighbouring cells, avoiding the definition of the transmission zones. In the meanwhile, 6 beams that connect the 4 vertices of the square design space can be identified: AB, BC, CD, AD, AC and BD.
The unit cell is moreover parametrized by 4 geometric parameters β_{i}(i =1,2,3,4) that allow to cluster the beams in 4 “groups”, each group being related to one of the β_{i}parameters as described in Figure 6. These parameters are defined as the ratio between the actual thickness of the associated beams and the maximum physical thickness. Therefore, each parameter can range from 0 to 1. For example, considering the group (a) in Figure 6, the beams AD and BC can assume a thickness from 0 to half the side of the design space, in particular here β_{1} is equal to 0.2.
The unit cell design is built starting from a fullwhite cell design and giving as inputs the dimensions of the cell and the values of the 4 parameters. As stated before, the thicknesses of the beams are related to the parameters. 4 vertices are defined for each element, as shown in Figure 7, and for each jth vertex the distance d_{j} from the boundary (or the axes for the diagonal beams) of the beam is evaluated. The maximum and minimum distances are compared to the thickness t_{i} (or the halfthickness for the diagonal beams) of the ith beam to understand if the element is completely covered, partially covered or not covered by the ith beam. If the element is covered (as for element A, all the distances are less than t _{1} that is the thickness of beam AD), the value 1 is assigned to its density, otherwise (as for element B, all the distances are greater than t_{1}) the value remains 0. If the element is partially covered (as for element C, d_{1} or equally d_{4} are less than t_{1} and d_{2} or equally d_{3} are greater than t_{1}), a value from 0 to 1 has to be assigned in order to have a smooth curve for the volume fraction of the structure depending on the parameters. The value assigned is the area of the element covered by the beam; for the element C, since the beam occupy half element, the value assigned is 0.5 as shown in Figure 8, where the fictitious density of the element C is plotted with respect to the beam thickness. The designs of each group of beams are evaluated one by one independently, therefore, 4 values of fictitious density are associated to each element. Finally, the structure is obtained by keeping only the maximum fictitious density associated to each element. The design is represented with a grey scale, where white stands for a fictitious density 0, black for 1 and grey for the intermediate values. It is important to note that the grey elements can only appear at the border of the beams. These grey elements are fundamental to have smooth partial derivatives of the objective function and the constraint with respect the parameters in the microoptimization (Eq. 4 and Eq. 5).
The implemented Matlab function to build the unit cell can be called from the command window $x=Cell\_4p(nelx,nely,beta)$ where nelx and nely are the dimensions of the design space respectively along the x and y directions, and beta is the vector of the parameters β = [β_{1},β_{2},β_{3},β_{4}], the output x are the fictitious density of the elements of the design space. The design in Figure 7 is obtained by calling from the command window x = Cell_4p(10, 10, [0.5, 0, 0, 0]).
Fig. 5 Base unit cell with the beams AB, BC, CD, AD, AC and BD. 
Fig. 6 The 4 βdependent groups of beams. (a) AD and BC are defined by the parameter β_{1}; (b) AB and CD are defined by the parameter β_{2}; (c) BD is defined by the parameter β_{3}; (d) AC is defined by the parameter β_{4}. Here, the 4 designs are obtained by assigning the value 0.2 to the related parameter. 
Fig. 7 Detail to understand the implementation for the unit cell: the element A is completely inside the beam so is black; the element B is completely outside the beam so is white; the element C is half covered by the beam, so the value assigned is 0.5 and is grey. 
Fig. 8 Fictitious density plot depending on the thickness t_{1} considering the element C from the example in Figure 7. The distances d_{1} (that is equal to d_{4}) and d_{2} (that is equal to d_{3}) are reported with red dotdashed lines, t_{1} is set along vertical black dashed lines and the fictitious density for t_{1} is reported with a red star in the graph. 
3.2 The microscale optimization
Once implemented the code for the structural architecture of the unit cell, the microoptimization code is developed to compute the database of optimal microstructures on a regularlyspaced 3dimensional grid of the three macrovariables (density x_{dens}, orientation x_{or} and cubicity x_{cub} introduced in Sect. 2).
The design space is a discretized square of N (= nelx × nely) micro elements. The design variables are the parameters β, that define the structure assigning to each element a fictitious density x_{e} (a value from 0 to 1) thanks to the unit cell code explained in Section 3.1. The Young's modulus of each micro element is given by a linear relationship with x_{e}:
$${E}_{e}\left({x}_{e}\right)\text{}=\text{}{E}_{min}\text{}+\text{}{x}_{e}({E}_{0}\text{}\text{}{E}_{min})$$(1)
where E_{e} is the element Young's modulus, E_{0} is the Young's modulus of the material and E_{min} is a very small stiffness assigned to void regions in order to prevent the stiffness matrix from becoming singular. A linear combination is used since the grey elements can only appear at the border of the beams and so it is not necessary to penalize them as the SIMP approach [33].
Once the structural architecture of the unitcell is defined, the structural properties can be evaluated by means of the inverse homogenization and the periodic boundary conditions as in Xia and Breitkopf [29]. In this work, an energybased homogenization approach allows to obtain the 4th order homogenized stiffness tensor ${\left({E}_{klqp}\right)}_{k,l,q,p\in \left\{\mathrm{1,2}\right\}}$ of the unitcell by the equation (2).
$${E}_{klqp}\text{}=\frac{1}{N}{\displaystyle \sum}_{i+1}^{N}{({u}_{i}^{A\left(kl\right)})}^{T}{k}_{i}{u}_{i}^{A\left(pq\right)}\text{},$$(2)
where N is the number of micro elements, k_{i} is the stiffness matrix of the ith microelement, and ${u}_{i}^{A\left(pq\right)}$ are the displacement solutions for the for ith microelement corresponding to the unittest strain fields in Figure 3 applied to the considered unitcell. In Figure 3 the unittest strain corresponds to (p, q) = (1, 1), (2, 2) and (1, 2). Therefore, the stiffness tensor is equal to
$$E\text{}=\text{}\left[\begin{array}{ccc}\hfill {E}_{1111}\hfill & \hfill {E}_{1122}\hfill & \hfill {E}_{1112}\hfill \\ \hfill {E}_{2211}\hfill & \hfill {E}_{2222}\hfill & \hfill {E}_{2212}\hfill \\ \hfill {E}_{1211}\hfill & \hfill {E}_{1222}\hfill & \hfill {E}_{1212}\hfill \end{array}\right],$$
Following the original EMTO, the objective function e(β) is defined as the weighted sum of the two principal components of the homogenized stiffness tensor rotated of an angle α as follows
$$e\left(\beta \right)\text{}=\text{}\text{}\left[\left(1\text{}\frac{\surd {x}_{cub}}{2}\right){E}_{1111}^{\alpha \left({x}_{or}\right)}\text{}\left(\beta \right)+\frac{\surd {x}_{cub}}{2}\text{}{E}_{2222}^{\alpha \left({x}_{or}\right)}\text{}\left(\beta \right)\right],$$(3)
where α (= x_{or}π/4) is related to the orientation macrovariable x_{or} belonging to the interval [0,1], the weights depend on the cubicity macrovariable x_{cub} belonging to the interval [0,1] and the rotated stiffness tensor Eα is obtained using the following equation
$${E}^{\alpha}\text{}=\text{}{M}_{\alpha}^{T}\text{}\times \text{}E\text{}\times {M}_{\alpha}\text{}\equiv \text{}{\left({E}_{klpq}^{\alpha}\right)}_{k,l,p,q\in \left[\mathrm{1,2}\right]},$$
where M_{α} is a rotation tensor defined for a rotation angle α measured with respect to a chosen global basis vector. For example, assuming x_{or} = 0 (that stands for an angle α = 0) and x_{cub} = 0, the objective function becomes e(β) = –E_{1111} and the optimization procedure is going to search for an optimal microstructure with the highest possible first principal component respecting the volume fraction constraint given by x_{dens}.
Therefore, the mathematical formulation of the microoptimization problem for the unitcell is
minimize
$${\beta}_{j},j=\mathrm{1...4}e=\left[\left(1\frac{\sqrt{{x}_{cub}}}{2}\right){E}_{1111}^{\alpha \left({x}_{or}\right)}+\frac{\sqrt{{x}_{cub}}}{2}{E}_{2222}^{\alpha \left({x}_{or}\right)}\right]$$
subject to
$$F\left(p,q\right)=K{u}^{A\left(p,q\right)}$$
$$\frac{V\left(\beta \right)}{{V}_{0}}=v\ge {x}_{dens}$$
$$0\le \epsilon \le {\beta}_{j}\le 1j=\mathrm{1...4}.$$
Here K is the unitcell assembled stiffness matrix, u^{A} and F are the global displacement vector and the external force vector of the unitcell, V(β) is the volume occupied by the material, V_{0} is the total volume of the design space, ν is the material volume fraction, β_{j} are the parameters whose lower bound is a very small value ∊ and the upper bound is 1. The microoptimization of the new approach is not a topology optimization as for the original EMTO, but is a sizing optimization.
The sensitivity analysis is carried out by finite differences for both the objective function e and the constraint v for simplicity, since the analytic derivates are not easy to compute. For each jth parameter, the following derivatives are evaluated.
$$\frac{\partial e}{\partial {\beta}_{j}}\cong \frac{e\left({\beta}_{j}+h\right)e\left({\beta}_{j}\right)}{h}$$(4)
$$\frac{\partial v}{\partial {\beta}_{j}}\cong \frac{v\left({\beta}_{j}+h\right)v\left({\beta}_{j}\right)}{h}.$$(5)
Once the derivatives are obtained, the optimality criteria method (OCM) is used to solve the optimization problem with the heuristic updating scheme following Sigmund [34].
The iterative process terminates when the difference for each jth parameter ${\beta}_{j}^{new}{\beta}_{j}$ is less than the convergence tolerance 10^{−5}, when 150 iterations are reached, or when 50 iterations without finding a new minimum are reached.
The Matlab function can be called by the command window as follows
$$\left[tens,\text{\hspace{0.17em}}obj,\text{\hspace{0.17em}}beta\right]=unitCell\_4p\left(nelx,\text{\hspace{0.17em}}nely,\text{\hspace{0.17em}}x\_dens,\text{\hspace{0.17em}}penal,\text{\hspace{0.17em}}x\_or,\text{\hspace{0.17em}}x\_cub,\text{\hspace{0.17em}}beta\_0\right)$$
here nexl and nely are the dimensions of the design space respectively along the x and y directions, x_dens, x_or and x_cub are the macrovariables, penal is the penalization coefficient (set to 1) and beta_0 is the initial guess vector of the parameters β. The outputs are the homogenized stiffness tensor, the objective function and the vector of the parameters named respectively tens, obj and beta.
For illustration, let us consider a random example with a 100 × 100 initial full white design space (beta_0 = [0,0, 0, 0]) and for x_{dens} = 0.55, x_{or} = 0.63 (that corresponds to an orientation angle α of about π/6) and x_{cub} = 0.09. The obtained structure with the related stiffness tensor E and rotated stiffness tensor E^{α} are shown in Figure 9. The optimal unit cell is found at the 61th iteration and both the objective function e and the volume fraction v rapidly reach the optimal point after 15 iterations, as shown in Figure 10, with the vector of the parameters equal to β = [β_{1}, β_{2}, β_{3}, β_{4}] = [0,0.11, 0.49, 0] and the objective function $e\text{}=\text{}\left[\left(1\text{}\frac{\surd {x}_{cub}}{2}\right){E}_{1111}^{\alpha \left({x}_{or}\right)}\text{}+\frac{\surd {x}_{cub}}{2}\text{}{E}_{2222}^{\alpha \left({x}_{or}\right)}\right]\cong \text{}0.4005.$
Fig. 9 Example of optimal unit cell for x_{dens} = 0.55, x_{or} = 0.63 (α of about π/6) and x_{cub} = 0.31. The stiffness tensor and the rotated stiffness tensor are shown too. 
3.3 The macroscale optimization
Before presenting the new database and the results obtained, a brief explanation of the macroscale optimization is given without going too much in the implementation details. We refer to Duriez et al. [28] for a more detailed description.
For a design space of M macroelements, the objective function is the compliance of the structure, and the design variables are the density, orientation and cubicity explained in Section 2, named in the code respectively x_{dens}, x_{or} and x_{cub}. Therefore, each macroelement has three variables assigned during the optimization process. The problem formulation is
minimize
$$x={[}_{x},{x}_{or,}{x}_{cub}]c\left(x\right)={U}^{T}KU$$
subject to
$$\underset{j=1}{\overset{M}{}}{x}_{dens}^{j}\le M{f}_{v}$$
$$0\le {x}^{j}\le 1,j=\mathrm{1...}M$$
where U is the global displacements matrix, K is the stiffness matrix, F is the vector of the external forces and f_{v} is the volume fraction constraint.
Actually, due to the periodicity of the orientation variable, the applied filtering method [33] could lead to optimization issues. To mitigate these issues, the orientation variable x_{or} is replaced by two other variables, from which the orientation is derived:
$${x}_{or}\text{}=\text{}arctan\left(\frac{{x}_{sin}}{{x}_{cos}}\right),$$
where x_{sin} and x_{cos} are not the real sine and cosine but have the same ratio. Defining these variables allows to make the design space redundant, since any set of x_{sin} and x_{cos} with the same ratio corresponds to the same orientation.
During the optimization the stiffness tensor of the jth macroelement is obtained from the database for the set $\left[{x}_{dens}^{j},\text{}{x}_{or}^{j},\text{}{x}_{cub}^{j}\right]$ by using the NadarayaWatson's kernelweighted average [35] obtaining ${E}_{pred}\left({x}^{j}\right)$ and its derivatives with respect to the design variables. The surrogate prediction of ${E}_{pred}\left({x}^{j}\right)$ is reported in Appendix A. E_{pred} enables to find the stiffness matrix K and to solve the equilibrium equation by a finite element analysis. After the sensitivity analysis, the variables are updated with the method of moving asymptotes (MMA, [36]) and the convergence is checked. The process goes iteratively ahead until one of the termination criteria is satisfied: 100 maximum number of iterations, 15 maximum number of iterations without a new minimum or 10^{−3} convergence tolerance. Finally, the macro variables and the compliance of the final design, computed using the homogenized stiffness tensors, are obtained as outputs.
Once obtained the final optimal set of macro variables for each macro element, the full scale structure can be obtained by replacing each macroelement with the closest cell in terms of Euclidean distance in the microstructure database. Due to the limits of using homogenization to obtain the overall compliance of the final structure, the fullscale structure can be evaluated by computing the compliance with a finite element analysis. The EMTO also includes the possibility to use a postprocessing code to improve the structure performances by assuring connection between microstructures and getting rid of unstressed elements. The postprocessing consists in three steps. In the first step, a full finite element analysis is carried out on the design to compute the stress in all the microelements. These stresses are compared to the mean stress value through the structure times a given constant C_{post} to assess if the elements are or not useful. If they are lower than C_{post} times the mean stress value, the related element is “deleted” setting density to 0. The process is repeated three times considering three different constants. In the second step, a classical density filter is applied having a filter radius defined with respect to the microelement size, improving manufacturability and making the thin structural members thicker. Then, all the elements are set to a density value 0 or 1 depending on a threshold that is adjusted through an iterative process to ensure the desired volume fraction. The final step gives the fullscale final design. The original EMTO work [28] is suggested to have further details about the postprocessing.
3.4 Database construction
The database is constructed offline in order to have a faster macrooptimization. The inputs of the database are the three macroscale design variables and the outputs are the four parameters to build the cell and the six independent terms of the homogenized stiffness tensor of the corresponding cell. Actually, the database is divided into two databases, one containing the tensors, and the second one the parameters. Only the tensors are loaded during the macrooptimization. Once the macrooptimization is finished, the database of parameters is used to get the fullscale design by calling the unitcell code in Section 3.1.
The database is computed considering cells of 100×100 elements for the 3dimensional grid as in Figure 11 whose coordinates are given by:
32 values from 0 to 1 (0 and 1 excluded) for the volume fraction (density variable);
32 values from 0 to 1 for the orientation variable (or equally angle α), where 0 stands for a rotation angle of 0 rad and 1 for a rotation angle of π/4 rad;
32 values from 0 to 1 for the cubicity variable, where 0 stands for stiffness only along the first principal direction and 1 for equal stiffness along both directions.
Considering a multistart strategy with 4 possible initial guesses, the total number of designs evaluated is equal to 32^{3} × 4 = 131072. The best solution for each input combinations is kept and added to the database. The unit cells for volume fractions 0 and 1, computed only once because they correspond to void and full material unit cells, are added too. So, a first database containing 34816 unitcells and their properties is obtained.
Since the original EMTO follows SIMP with penalization p = 3 to obtain the optimal unitcells, to compare the parametrized cells of the ExEMTO with the free ones of the original EMTO, the properties and the objective functions for the parametrized cells in Figure 12 have been recomputed by considering a penalization p = 3. The worsening of the performances of the parametrized cells due to the introduction of the penalization is very low since the grey elements can only appear at the border of the beams as explained in Section 3.1. In fact, considering a sample group of 3584 very different cells, the mean deviation of the objective function computed with p = 3 with respect to the ones with a linear relationship for the Young's modulus (that can be seen as having p = 1) is about 2.5%.
The unit cells obtained with the parametrized approach have more regular shapes than the free cells by the original EMTO as shown in Figure 12. Due to the parameterization, the design space is restricted and the performances are worse: the objective function value increases by about 210%. However, it is an acceptable trade off to diminish the database size of about the 97% since the structure database needs to store just the 4 parameters β = [β_{1}, β_{2}, β_{3}, β_{4}] and not the N microelements of the structure.
Besides, the design space has to be artificially extended to avoid local minima at the border. Therefore, the database is extended by applying symmetries and rotation to the computed unit cell without the need of solving further microoptimization problems. The final database contains unit cells for an orientation variable ranging from 0 to π rad (redefining x_{or} from 0 to 1) and cubicity variable from stiffness only along the first principal direction considered (value 0) to stiffness only along the second principal direction (value 1), whereas a value of 0.5 means that stiffness is along both the principal directions. Finally, the final database is obtained with a total number of unit cells equal to 267750. In this way, the gradientbased macrooptimizer has multiple paths from a cell to another as shown in Figure 13.
Due to the restricted unitcell design space, the transition from one cell to another by changing only one macro variable is not always smooth, as shown in Figure 13b. In that figure, the three intermediate unit cells are almost equal and before and after them there is a quite sharp change to different cells. Despite this, the cells result to be well connected without the need to define the transmission zones.
Fig. 11 The 3dimensional grid over which the database is computed. Some cells are shown to show the effects of the three macrovariables. 
Fig. 12 Random examples of cells with the related homogenized stiffness tensor and objective functions from the new database (left) compared to the ones from the original EMTO database (right). The orientation variable from 0 to 1 means an orientation angle from 0 rad to π/4 rad; the cubicity variable ranges from 0 to 1, where 0 stands for stiffness only along the first principal direction and 1 for equal stiffness along both directions. 
Fig. 13 Unit cells examples to show the redundant design space. There are two paths from the cell on the left (density = 0.5, orientation = 0 rad, cubicity = 0.2) to the cell on the right ((a) density = 0.5, orientation = 0 rad, cubicity = 0.8 or (b) density = 0.5, orientation = π/2 rad, cubicity = 0.2). 
4 Results and discussion
Classical test cases from the literature as shown in Figure 14 are solved with the ExEMTO approach and the results are analysed and discussed hereafter compared to the one of the original EMTO. The computational times are not reported since they are almost the same of the original EMTO ones [28]. This is related to the fact that the two approaches have in common the macrooptimization code and the two databases are structured in the same way. A test case for a comparison with another multiscale approach is analysed too. For all the cases, the Young's modulus for the solid material is E_{0} = 1, the Poisson's ratio 0.3 and the force F = 1.
Fig. 14 Beam problems. 
4.1 MBB beam
In Figure 14a just half of the MBB problem is represented and considered for the computation thanks to the problem symmetry. A 30 × 10 macroscale grid is considered. The objective is to minimize the compliance of the macro design with a constraint for the global volume fraction of 0.5. For both the EMTO and ExEMTO, the fullscale designs over a microscale grid 3000 × 1000 are obtained for a penalization coefficient p = 3 without and with the postprocessing (PP) and are shown in Figure 15.
The designs are similar and, despite the worse performances of the unit cells of the new ExEMTO database, the compliances are almost equal too, as can be seen in Table 1. In that table, the objective functions and the %variation with respect to the “homogenized” compliance (the one obtained by the computation by using the stiffness tensors from the database) of EMTO are reported for each step of the optimization. By comparing Figures 15a and 15c, the limitation of the new approach is evident for the cells where there is a transition from normal stress to shear stress (for example the cells highlighted in red). In fact, as anticipated in Section 3 with Figure 13, the restricted design space of the parametrized cells does not allow to obtain optimal cells for transition stress cases as good as the free cells. Moreover, after the postprocessing in Figure 15d (ExEMTO) those cells are slightly distorted, whereas in Figure 15b (EMTO) they are almost the same as before the postprocessing. However, the performances of ExEMTO are only diminished by 1% with respect to the ones of original EMTO, meaning that the previous limitation is not detrimental. In addition, the fact that the performances are similar between original EMTO and ExEMTO, demonstrates that the parametrized unit cells are intrinsically wellconnected as expected. Moreover, the convergence performances are similar too as shown for the homogenized step in Figure 16. Both the codes have a rapid drop for the compliance before the 15th iteration and then converges to a final value approximately equal, instead the volume fraction is quite regular. EMTO converges after 67 iterations, ExEMTO after 90 iterations.
Fig. 15 Comparison of MBB designs with 100 × 100 cells for the compliance minimization: (a) and (b) original EMTO respectively without postprocessing and with postprocessing, (c) and (d) ExEMTO respectively without postprocessing and with postprocessing. 
Fig. 16 Homogenized step iteration histories for the MBB beam. 
Comparison of compliances from EMTO vs ExEMTO for the MBB beam problem for the different steps in the method.
4.2 Cantilever beam
The cantilever beam is shown in Figure 14b and a 20 × 10 macroscale grid is considered. The objective is to minimize the compliance with a constraint for the global volume fraction of 0.5. Both the EMTO and the ExEMTO fullscale designs over a 2000 × 1000 microscale grid without and with postprocessing are shown in Figure 17. As for the MBB beam problem, the cantilever beam designs are similar but the performances with the new database are worse of only the 3% with respect to the one of the original EMTO, as reported in Table 2. The difference in the performances between EMTO and ExEMTO are higher for the cantilever beam than for the MBB beam, but they are still small. This is probably due to the limitation of the parametrized unit cell, as in the MBB beam design, for the “transition” cells. Therefore, the observations are similar to the MBB beam ones. However, for the cantilever beam, the designs with the post processing (Figs. 17b and 17d) results to be almost equal to the “homogenized” step ones. In Figure 18, the iteration histories for the homogenized step are shown for both the code. The compliance rapidly drops before the 10th iteration, instead the volume fraction shows a less regular behaviour until the 25th iteration. The final convergence is reached for both the code after the 95th iteration.
Fig. 17 Comparison of Cantilever beam designs which minimize the compliance: (a) original EMTO without postprocessing, (b) original EMTO with postprocessing, (c) ExEMTO without postprocessing and (d) ExEMTO with postprocessing. 
Fig. 18 Homogenized step iteration histories for the Cantilever beam. 
Comparison of compliances from EMTO vs ExEMTO for the Cantilever beam problem for the different steps in the method.
4.3 Lshaped beam
The Lshaped beam problem in Figure 14c with a 14 × 14 macroscale grid is considered. The objective is to minimize the compliance with a volume fraction constraint of 0.5. The fullscale designs over a 1400 × 1400 microscale grid are shown in Figure 19. The designs obtained by ExEMTO show more regular shapes and have performances comparable with the ones of EMTO as reported in Table 3 and shown in Figure 20. However, both the approaches for the fullscale designs have performances worse by more than 20% with respect to the “homogenized” step. This discrepancy is due to the low number of macroelements used to be able to compute the fullscale model. Consequently, the loads are not perfectly transmitted from one cell to another. Using more macroelements would give lower discrepancies.
Fig. 19 Comparison of Lshaped beam designs which minimize the compliance: (a) original EMTO without postprocessing, (b) original EMTO with postprocessing, (c) ExEMTO without postprocessing and (d) ExEMTO with postprocessing. 
Comparison of compliances from EMTO vs ExEMTO for the Lshaped problem for the different steps in the method.
4.4 Comparison with works from literature
ExEMTO has been tested also for a half MBB beam problem with a macroscale grid 32 × 24 and 0.3 global volume fraction constraint to compare the results with the ones obtained for the same problem by Garner et al. [17]. The design obtained by Garner in Figure 21a has been projected over a microscale grid of 800 × 600 as in Figure 21b and the compliance evaluated with the same code used to evaluate the fullscale designs of the EMTO. For the comparison, the original EMTO has been considered too. To reduce the computational and memory efforts of EMTOs, the 100 × 100 cells of the databases have been reshaped to 25 × 25 cells, allowing also to have the same microscale grid of 800 × 600 elements. It is immediate to reshape the parametrized cells since the code to build the cells allows to do that by inserting as input the desired dimensions and the vector of parameters (stored in the database). In Figure 22, some reshaped cells both for ExEMTO and original EMTO, with their stiffness tensors and objective functions, are shown. The parametrized cells are less affected by the reshaping, as can be seen by comparing the objective function values of the cells in Figure 12 with the reshaped cells in Figure 22. This is probably due to the much more regular shapes of the parametrized cells as can be seen in Figures 22b and 22c. Considering 3584 cells, the reshaping leads to a mean worsening of the objective function of about 5.8% for the parametrized cells and 6.9% for the free cells. Despite the loss in performance of the unit cells, substituting in the full scale design the reshaped cells (for example the 25 × 25 cells instead of the 100 × 100 cells) does not affect the results in terms of compliance as shown for the half MBB problem in Table 1 in Section 4.1. However, since the database of the stiffness tensors is related to the 100 × 100 cells, full scale results with reshaped cells in line with the “homogenized” step ones are not always sure.
In Table 4 the compliances of the various designs are reported and compared to the ones of top88 (monoscale approach) with the 800 × 600 grid. The designs of top88 (on a 32 × 24 grid and 800 × 600 grid) are shown in Appendix B in Figure B1. As expected from the previous test problems ExEMTO and EMTO have similar results for each step of the methods. Both the EMTOs show an “homogenized” compliance lower of about the 12% with respect to the one of top88 on an 800 × 600 grid. However, after the introduction of the cells, obtaining the fullscale designs with the postprocessing, the compliance for the EMTOs is slightly worse than the one of top88, about 5% worse, but with a lower computational effort and time as discussed in the previous work [28]. The total computational time to obtain and evaluate the EMTO fullscale design with the postprocessing (800 × 600 grid) is about 6 times lower than the top88 time on the same 800 × 600 grid.
The postprocessed designs deviate from the theoretical results of about the 16%, that is considered acceptable. This deviation with respect to the theoretical results could be also due to the reshaping of the cells. Therefore, designs with higher resolutions for the unit cells should be tested to find a lower limit for the dimensions of the cells reshaping. Despite the slightly worse performances of the EMTOs with respect to the top88 method, the results are instead much better than the ones obtained by Garner's method.
The previous comparisons and the new approach for the EMTO demonstrate the validity and versatility of the method. The introduction of the macro variables and the database allow to have a lot of freedom in the design of the unit cells. It is possible to reduce the solution space over which the database is built or to reduce the resolution of the unit cells without losing too much in the performances of the method.
Fig. 20 Homogenized step iteration histories for the Lshaped beam. 
Fig. 21 Comparison of half MBB designs. (a) and (b) design from Garner et al. [17] and the same design reprojected on a 800 × 600 grid, (c) and (d) ExEMTO on a 32 × 24 grid of 25 × 25 cells (800 × 600 microscale grid) without and with postprocessing respectively, (e) and (f) EMTO on a 32 × 24 grid of 25 × 25 cells (800 × 600 microscale grid) respectively without and with postprocessing. 
Fig. 22 Examples of 25 × 25 cells with the related homogenized stiffness tensor and objective functions obtained by reshaping the 100 × 100 cells in Figure 12 from the new database (left) and the original EMTO database (right). 
Comparison of compliances from different methods for the MBB beam problem.
4.5 Ultralight design
Ultralight structures are of great interest in several industries like automobile, aerospace and aircraft, and to get such structures, ultralight materials are widely studied, including trusslike materials [37,38]. The ExEMTO allows to get trusslike ultralight designs with very simple shapes. It has been tested for a 20 × 10 half MBB beam problem for compliance minimization with volume fractions 0.126, 0.127 and 0.128. The post processed designs, shown in Figure 23, are obtained with 50 × 50 cells, so over a 1000 × 500 microscale grid, and compared with the one of the original EMTO. The compliances are reported below each structure.
Thanks to the more regular shapes of the parametrized cells, the designs by ExEMTO results to be very simple and well connected. Moreover, the designs of the ExEMTO have better performances with respect to the one of the original EMTO since the compliances are much lower as can be seen from Figure 23. In particular, in Figure 23a the compliance of the new approach is about the half of the original one. The Pareto curve of the half MBB beam problem is shown in Figure 24.
Fig. 23 Examples obtained for a 20 × 10 macroscale grid half MBB beam for different volume fractions, both for the new database (top figures) and the original EMTO one (bottom figures). The compliance is reported below each structure. 
Fig. 24 “Theoretical compliance vs volume fraction” Pareto curve for the 20 × 10 half MBB beam problem. 
5 Demonstration of the versatility of ExEMTO
The new ExEMTO approach to build the database demonstrates the versatility of the EMTO to be used with different types of unit cells. The versatility of the method is also related to the possibility to accommodate different types of problems and constraints. Therefore, it is interesting to see how these two strengths of the methods can coexist, by solving with the new database the problems of fiber orientation, controlled porosity [39] and fixed topology test cases.
5.1 The fiber orientation test case
Since one of the macrovariables is the orientation of the unit cell, EMTO can be easily adapted to get optimal fiber orientation in a topology optimization framework. In Figure 25, the orientations solutions for 80 × 40 half MBB beam problems are shown.
In Figure 25a, the problem has a constraint related to the usage of macro elements in percentage, that is set to 1, and a constraint for the unit cell volume fraction set to 0.5. Therefore, the global volume fraction constraint can be obtained by multiplication of the two constraints giving as a result 0.5. The images are the results of the ExEMTO (top) and the original EMTO (bottom) and show the orientation variable database indexes of each cell. The indexes go from 1 to 32, where 1 stands for 0 rad and 32 for π/4 rad. ExEMTO shows a smoother and more regular transition of the orientation variable.
In Figure 25b the problem the usage of macro elements in percentage is equal to 0.5 and the unit cell volume fraction 0.5, therefore a global volume fraction constraint of 0.25. In this case, the ExEMTO shows a better topology and also, as a consequence, a more regular orientation variation. These results are just preliminary results and further improvements are needed to correctly solve the fiber orientation test case.
Fig. 25 Examples obtained for a 80 × 40 half MBB beam for the new database (top figures) and the original one (bottom figures). 
5.2 The controlled porosity test case
In Figure 26a the designs obtained by ExEMTO and the original EMTO are shown in the case of porosity fixed to 0.5 for a cantilever beam with global volume fraction constraint of 0.5. In Figure 26b the designs in the case of porosity constrained to a minimum value of 0.4 are shown. These two cases are easier to implement imposing limits to the density variable x_{dens}: a same porosity (p_{fixed}) as in equation (6) or a minimum porosity (p_{min}) as in equation (7).
$${x}_{dens}\le 1{p}_{min},$$(7)
In Figure 26 the theoretical compliances or “homogenized” ones (i.e. the compliances computed by using the stiffness tensors form the database) are reported below the designs. The compliances of the ExEMTO are a bit higher than the original EMTO ones. This is due to the restrained design space of the unit cells as explained in Section 3.4.
Fig. 26 Examples obtained for a 50 × 20 cantilever beam with restrained porosity both for the new database (top figures) and the original EMTO one (bottom figures). 
5.3 The fixed topology test case
In Figure 27, the design of ExEMTO and original EMTO are shown for a 120 × 40 cantilever beam in the case of fixed unit cell topology with a global volume fraction constraint of 0.5. The cubicity and the orientation variables are fixed respectively to 0.5 (the principal directions have the same importance) and 0.25 (that stands for an orientation angle of π/4 rad). The designs are really similar and also the values of the theoretical compliances are almost the same. ExEMTO seems to give a totally symmetric result. For the optimum, it seems to be slightly better, but it should be remembered that the new database is obtained with a penalization coefficient equals to 1 so a better result for the compliance was expected.
Fig. 27 Example obtained for a 120 × 40 cantilever beam by limiting the design space. 
6 Conclusions
In this paper a different approach to build the unit cell for the EMTO is developed. Here we restricted ourselves to the twodimensional case, postponing its application to the threedimensional cases for other communications. In the 2D case, the connectivity is assured by considering intrinsically wellconnected unit cells instead of using the transmission zones. Despite the reduction of the design space of the microstructures, the efficiency and versatility of the EMTO method is demonstrated thanks to the results of the parametrized cells similar to the ones of the free cells. Moreover, the new developed approach to construct the unit cells leads to some advantages:
A lighter database, since it is not necessary to store all the 100 × 100 elements of the optimal microstructure, but just the 4 parameters;
The possibility to introduce more easily additive manufacturing constraints, since minimum and maximum thicknesses for the beams can be defined both in the unit cell code and in the microoptimization;
The resulting macrostructures are surely made of beams, without the risk to have very strange shapes;
The possibility to reshape more easily the microstructures reducing the computational cost of the fullscale evaluation.
Future works include speeding up the codes, adding additive manufacturing constraints and validate the results by testing printed structures obtained by the method. Lately, it is also worth mentioning that the foregoing twodimensional analysis is currently being extended to threedimensional case. The 3D case has to deal with a higher number of parameters to build the 3D unit cell shown in Figure 28, as well as higher computational effort and complexity. It will be presented in a forthcoming communication.
Fig. 28 3D unit cell obtained by superimposing beams. 
Acknowledgements
The authors are grateful to the anonymous reviewers for their expertise, their very helpful and referenced comments, proofreading and checking.
Appendix A Surrogate prediction
The Nadaraya–Watson's kernelweighted average with a Gaussian Kernel G
$${E}_{pred}\left({x}^{i}\right)\text{}=\frac{{{\displaystyle \sum}}_{\mathrm{l=1}}^{k1}G\left({x}^{i},\text{}{x}_{l}\right){E}_{db}\left({x}_{l}\right)}{{{\displaystyle \sum}}_{l}^{k=1}G\left({x}^{i},{x}_{l}\right)}$$
$$G\left({x}^{i},\text{}{x}_{l}\right)\text{}=\text{}exp\left(\text{}\frac{{d}_{eucl}{\left({x}^{i},\text{}{x}_{l}\right)}^{2}}{2{b}^{2}}\right)$$
Here E_{pred} (x^{i}) is the predicted stiffness tensor of the cell corresponding to the set of macrodesign variable ${x}^{i}\text{}=\text{}\left[{x}_{dens}^{i},\text{}{x}_{or}^{i},\text{}{x}_{cub}^{i}\right]$x_{l} are the points in the database and E_{db} (x_{l}) are the database stiffness tensors of the cells corresponding to those points; b is the kernel radius; d_{eucl} (x^{i}, x_{l}) measures the Euclidean distance between x^{i} and x_{l}.
Appendix B top88 results
Fig. B1 Comparison of half MBB designs. a top88 design on a 32 × 24 grid and b top88 design on a 800 × 600 grid. 
Funding
The authors would like to thank Erasmus + Programme for funding this research.
Conflict of interest
The authors declare that they have no conflict of interest.
Data availability
The codes and the unitcell databases are available on GitHub for reproducible research purpose (https://github.com/mid2SUPAERO/ExEMTO) [40].
Author contribution statement
Each named author has substantially contributed to conducting the underlying research and drafting this manuscript.
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Cite this article as: A. Di Rienzo, E. Duriez, M. Charlotte, J. Morlier, Lightweighting structures using an explicit microarchitectured material framework, Mechanics & Industry 25, 7 (2024)
All Tables
Comparison of compliances from EMTO vs ExEMTO for the MBB beam problem for the different steps in the method.
Comparison of compliances from EMTO vs ExEMTO for the Cantilever beam problem for the different steps in the method.
Comparison of compliances from EMTO vs ExEMTO for the Lshaped problem for the different steps in the method.
All Figures
Fig. 1 Influence of the 3 macrovariables on a microstructure Boxed in green the unit cell having density 0.5, orientation π/4 and cubicity 0. In each subfigure, two of these variables are fixed, while the third varies. 

In the text 
Fig. 2 Original EMTO half MBB beam solution. In red some cells are highlighted to show the apparent truss structure. 

In the text 
Fig. 3 The unit test strain fields imposed on a 2D unit cell. 

In the text 
Fig. 4 Method flowchart. 

In the text 
Fig. 5 Base unit cell with the beams AB, BC, CD, AD, AC and BD. 

In the text 
Fig. 6 The 4 βdependent groups of beams. (a) AD and BC are defined by the parameter β_{1}; (b) AB and CD are defined by the parameter β_{2}; (c) BD is defined by the parameter β_{3}; (d) AC is defined by the parameter β_{4}. Here, the 4 designs are obtained by assigning the value 0.2 to the related parameter. 

In the text 
Fig. 7 Detail to understand the implementation for the unit cell: the element A is completely inside the beam so is black; the element B is completely outside the beam so is white; the element C is half covered by the beam, so the value assigned is 0.5 and is grey. 

In the text 
Fig. 8 Fictitious density plot depending on the thickness t_{1} considering the element C from the example in Figure 7. The distances d_{1} (that is equal to d_{4}) and d_{2} (that is equal to d_{3}) are reported with red dotdashed lines, t_{1} is set along vertical black dashed lines and the fictitious density for t_{1} is reported with a red star in the graph. 

In the text 
Fig. 9 Example of optimal unit cell for x_{dens} = 0.55, x_{or} = 0.63 (α of about π/6) and x_{cub} = 0.31. The stiffness tensor and the rotated stiffness tensor are shown too. 

In the text 
Fig. 10 Iteration history for the example in Figure 9. 

In the text 
Fig. 11 The 3dimensional grid over which the database is computed. Some cells are shown to show the effects of the three macrovariables. 

In the text 
Fig. 12 Random examples of cells with the related homogenized stiffness tensor and objective functions from the new database (left) compared to the ones from the original EMTO database (right). The orientation variable from 0 to 1 means an orientation angle from 0 rad to π/4 rad; the cubicity variable ranges from 0 to 1, where 0 stands for stiffness only along the first principal direction and 1 for equal stiffness along both directions. 

In the text 
Fig. 13 Unit cells examples to show the redundant design space. There are two paths from the cell on the left (density = 0.5, orientation = 0 rad, cubicity = 0.2) to the cell on the right ((a) density = 0.5, orientation = 0 rad, cubicity = 0.8 or (b) density = 0.5, orientation = π/2 rad, cubicity = 0.2). 

In the text 
Fig. 14 Beam problems. 

In the text 
Fig. 15 Comparison of MBB designs with 100 × 100 cells for the compliance minimization: (a) and (b) original EMTO respectively without postprocessing and with postprocessing, (c) and (d) ExEMTO respectively without postprocessing and with postprocessing. 

In the text 
Fig. 16 Homogenized step iteration histories for the MBB beam. 

In the text 
Fig. 17 Comparison of Cantilever beam designs which minimize the compliance: (a) original EMTO without postprocessing, (b) original EMTO with postprocessing, (c) ExEMTO without postprocessing and (d) ExEMTO with postprocessing. 

In the text 
Fig. 18 Homogenized step iteration histories for the Cantilever beam. 

In the text 
Fig. 19 Comparison of Lshaped beam designs which minimize the compliance: (a) original EMTO without postprocessing, (b) original EMTO with postprocessing, (c) ExEMTO without postprocessing and (d) ExEMTO with postprocessing. 

In the text 
Fig. 20 Homogenized step iteration histories for the Lshaped beam. 

In the text 
Fig. 21 Comparison of half MBB designs. (a) and (b) design from Garner et al. [17] and the same design reprojected on a 800 × 600 grid, (c) and (d) ExEMTO on a 32 × 24 grid of 25 × 25 cells (800 × 600 microscale grid) without and with postprocessing respectively, (e) and (f) EMTO on a 32 × 24 grid of 25 × 25 cells (800 × 600 microscale grid) respectively without and with postprocessing. 

In the text 
Fig. 22 Examples of 25 × 25 cells with the related homogenized stiffness tensor and objective functions obtained by reshaping the 100 × 100 cells in Figure 12 from the new database (left) and the original EMTO database (right). 

In the text 
Fig. 23 Examples obtained for a 20 × 10 macroscale grid half MBB beam for different volume fractions, both for the new database (top figures) and the original EMTO one (bottom figures). The compliance is reported below each structure. 

In the text 
Fig. 24 “Theoretical compliance vs volume fraction” Pareto curve for the 20 × 10 half MBB beam problem. 

In the text 
Fig. 25 Examples obtained for a 80 × 40 half MBB beam for the new database (top figures) and the original one (bottom figures). 

In the text 
Fig. 26 Examples obtained for a 50 × 20 cantilever beam with restrained porosity both for the new database (top figures) and the original EMTO one (bottom figures). 

In the text 
Fig. 27 Example obtained for a 120 × 40 cantilever beam by limiting the design space. 

In the text 
Fig. 28 3D unit cell obtained by superimposing beams. 

In the text 
Fig. B1 Comparison of half MBB designs. a top88 design on a 32 × 24 grid and b top88 design on a 800 × 600 grid. 

In the text 
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