COMPUTER IMPLEMENTED LIGHTWEIGHT DESIGN METHOD

Abstract

A computer implemented lightweight design method including: a preliminary homogenization step for defining a material model related to an adopted manufacturing material; a subsequent optimization step for finding an optimal distribution of material density within the design domain; and a final post-processing step to find the geometry for manufacturing; wherein the preliminary homogenization step is performed for deriving the material model for a 2D/3D version of a porous material provided with circular/spherical holes in a Hexagonal Close-Packed (HCP) arrangement, and the post-processing step comprises computing position and size of the circular/spherical holes.

Claims

1. A computer implemented lightweight design method, comprising: a preliminary homogenization step for defining a material model related to an adopted manufacturing material; a subsequent optimization step for finding an optimal distribution of material density within a design domain; and a final post-processing step to find the geometry for manufacturing, wherein said preliminary homogenization step is performed for deriving the material model for a 2D/3D version of a porous material provided with circular/spherical holes in a Hexagonal Close-Packed (HCP) arrangement, and said final post-processing step comprises computing position and size of said circular/spherical holes.

2. The computer implemented method according to claim 1, wherein said preliminary homogenization step comprises a geometrical parameters definition step of said 2D/3D porous microstructure of the porous material.

3. The computer implemented method according to claim 2, wherein said geometrical parameters definition step comprises calculating the density of said 2D porous microstructure as a function of the radius (r) of said circular holes of the 2D porous microstructure, according to the following equation: $\rho =1-\frac{2\pi r^2}{\sqrt{3}{d}^2}\mathrm{for}0\le r\le r_{\max },\mathrm{with}{r}_{\max }=\frac{d-t}{2},$ wherein: r is the radius of said circular holes; d is a reference dimension of said porous microstructure; t is the minimum thickness of the material between two adjacent holes.

4. The computer implemented method according to claim 2, wherein said geometrical parameters definition step comprises calculating the density of said 3D porous microstructure as a function of the radius (r) of said spherical holes of the 3D porous microstructure, according to the following equation: $\rho =1-\frac{8\pi r^3}{3\sqrt{2}{a}^3}0\le r\le r_{\max },\mathrm{with}{r}_{\max }=\frac{d-t}{2},$ wherein: r is the radius of said spherical holes; d is a reference dimension of said porous microstructure; and t is the minimum thickness of the material between two adjacent holes.

5. The computer implemented method according to claim 1, wherein said preliminary homogenization step comprises computing material laws for said 2D/3D porous microstructure.

6. The computer implemented method according to claim 5, wherein said step of computing material laws for said 2D/3D porous microstructure comprise calculating a constitutive tensor of the homogenized material (C.sub.ijhk) according to the following equation: $C_{\mathrm{ijhk}}=\frac{1}{.Math.Y.Math.}{\int }_Y{E}_{\mathrm{pgrs}}{\epsilon }_{\mathrm{pq}}^{A\left(\mathrm{ij}\right)}{\epsilon }_{\mathrm{rs}}^{A\left(hk\right)}\mathrm{dY},$ wherein: C.sub.ijhk is said constitutive tensor of the homogenized material; |Y| is the volume of said base cell; E.sub.pqrs is the constitutive tensor of the material used to manufacture the object; and ε.sub.pq.sup.A(ij) is the strain field induced within the cell when enforcing unit test strains at its boundaries.

7. The computer implemented method according to claim 1, wherein said optimization step comprises defining design domain, boundary conditions and relevant finite element discretization.

8. The computer implemented method according to claim 7, wherein said optimization step comprises preparing finite element analysis for the solution of the state equation using reference element stiffness matrices for full material.

9. The computer implemented method according to claim 8, wherein, with reference to said 2D porous microstructure, said step of preparing finite element analysis comprises calculating said element stiffness matrices, for any prescribed value of density (x.sub.e), according to the following equation: $K_e\left(x_e\right)=\frac{C_{11}\left(x_e\right)}{C_{11}^0}{K}_{11,e}^0+\frac{C_{33}\left(x_e\right)}{C_{33}^0}{K}_{33,e}^0.$

10. The computer implemented method according to claim 8, wherein, with reference to said 3D porous microstructure, said step of preparing finite element analysis comprises calculating said element stiffness matrices, for any prescribed value of density (x.sub.e), according to the following equation: $K_e\left(x_e\right)=\frac{C_{11}\left(x_e\right)}{C_{11}^0}{K}_{11,e}^0+\frac{C_{66}\left(x_e\right)}{C_{66}^0}{K}_{66,e}^0.$

11. The computer implemented method according to claim 8, wherein said optimization step comprises the formulation of a topology optimization problem.

12. The computer implemented method according to claim 11, wherein said topology optimization problem is formulated according to the following equation: $\begin{array}{cc}\{\begin{array}{c}\underset{x_{\min }\le x_e\le x_{\max }}{\min }C={.Math.}_{e=1}^N{U}_e^T{K}_e\left(x_e\right)U_e\\ s.t.K\left(x\right)U=F,\\ \frac{1}{V}\underset{N}{.Math.}{x}_e{V}_e\le V_f,\end{array}& \begin{array}{c}\begin{array}{c}\left(\mathrm{Eqn}.14.1\right)\\ \left(\mathrm{Eqn}.14.2\right)\end{array}\\ \left(\mathrm{Eqn}.14.3\right)\end{array}\end{array}$ wherein C is the structural compliance; F is the load vector; U.sub.e are the element-wise displacement vectors; K(x) is the global stiffness matrix; K.sub.e(x.sub.e) are the element-wise contributions; V is the volume of the whole design domain; V.sub.e, is the volume of the e-th element; and V.sub.f is a volume fraction.

13. The computer implemented method according to claim 1, wherein said final post-processing step comprises processing a map of the optimal distribution of the element-wise material densities (x.sub.e) for the detection of external boundaries of the object.

14. The computer implemented method according to claim 1, wherein said step of computing position and size of the holes comprises determining the coordinates (y.sub.1, y.sub.2, y.sub.3) of the centers of the circular/spherical holes in the adopted HCP arrangement as follows: $\left(i+\frac{1}{2}\mathrm{mod}\left(j+k,2\right)\right)d,\left(\frac{\sqrt{3}}{2}j+\frac{1}{2\sqrt{3}}\mathrm{mod}\left(k,2\right)\right)d,\left(\frac{\sqrt{6}}{3}k\right)d$ wherein i, j and k are integer indices starting at the origin of a prescribed reference system, and wherein the operator mod returns the remainder after division of two terms.

15. The computer implemented method according to claim 1, wherein said step of computing position and size of the holes comprises, with reference to 2D, calculating the radius of a h-th circular hole according to the following equations: if ρ.sub.h>ρ.sub.min, then $r_h={\left(\frac{\left(1-{\rho }_h\right)\sqrt{3}}{2\pi }\right)}^{\frac{1}{2}}d;$ otherwise r=r.sub.max with r.sub.max=(d−t)/2.

16. The computer implemented method according to claim 1, wherein said step of computing position and size of the holes comprises, with reference to 3D, calculating the radius of a h-th spherical hole according to the following equations: if ρ.sub.h>ρ.sub.min, then $r_h={\left(\frac{\left(1-{\rho }_h\right)3\sqrt{2}}{8\pi }\right)}^{\frac{1}{3}}d;$ otherwise r=r.sub.max with r.sub.max=(d−t)/2.

17. The computer implemented method according to claim 1, wherein said final post-processing step comprises processing the geometry of the optimal layout for file exchange.

18. A data processing system comprising means for carrying out the steps of the method of claim 1 for generating 2D or 3D object and a 3D-printer for the fabrication of said object.

19. A 3D-printer comprising means for carrying out the steps of the method of claim 1.

20. A computer implemented lightweight design method comprising: defining a material model related to an adopted manufacturing material; finding an optimal distribution of material density within a design domain; and finding the geometry for manufacturing, wherein said defining is performed for deriving the material model for a 2D/3D version of a porous material provided with circular/spherical holes in a Hexagonal Close-Packed (HCP) arrangement, and said finding of the geometry for manufacturing comprises computing position and size of said circular/spherical holes.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Other characteristics and advantages of the present invention will become better evident from the description of a preferred, but not exclusive embodiment of a computer implemented lightweight design method, illustrated by way of an indicative but non-limiting example in the accompanying Figures, in which:

[0012] FIG. 1 shows a general flowchart of the computer implemented lightweight design method according to the invention;

[0013] FIGS. 2 and 3 show a 2D version of a porous microstructure;

[0014] FIG. 4 shows a three-dimensional view of a single base cell in a 3D version of a porous microstructure;

[0015] FIG. 5 shows an example of interpolation laws fitting results from numerical homogenization, as compared to a conventional SIMP, relating to the 2D version of a porous microstructure;

[0016] FIG. 6 shows an example of interpolation laws fitting results from numerical homogenization, as compared to a conventional SIMP, relating to the 3D version of a porous microstructure;

[0017] FIGS. 7 and 8 are related to an example of possible application of the computer implemented method according to the invention.

EMBODIMENTS THE INVENTION

[0018] According to the invention, the computer implemented lightweight design method of an object to be fabricated, comprises at least the following steps: [0019] obtain a material model related to an adopted manufacturing material; [0020] a subsequent optimization step 2 for finding an optimal distribution of material density within the design domain; and [0021] a final post-processing step 3 to find the geometry for manufacturing, particularly for 3D-printing.

[0022] A flowchart to exemplify the computer implemented method is given in FIG. 1.

[0023] Preferably, the computer implemented lightweight design method according to the invention is applied for fabrication of objects/components with Additive Manufacturing (AM) techniques.

[0024] Different applications with different fabrication methods are not excluded. Furthermore, as shown in the example of FIG. 1, the computer implemented method comprises a preliminary homogenization step 1 to define said material model.

[0025] Particularly, the preliminary homogenization step 1 is performed to derive the material model for the two-/three-dimensional (2D/3D) version of the porous material with circular/spherical holes in an HCP arrangement.

[0026] This preliminary homogenization step 1 is not needed, if material laws for the constituent material are already available from previous simulations.

[0027] The preliminary homogenization step 1 comprises a geometrical parameters definition sub-step 11 of a 2D or 3D porous microstructure of said porous material (step 11).

[0028] Particularly, the 2D/3D porous microstructure according to the invention comprises at least a single base cell comprising circular/spherical holes in a Hexagonal Close-Packed (HCP) arrangement.

[0029] FIGS. 2, 3 and FIG. 4 show the 2D/3D version respectively.

[0030] Particularly, FIG. 2 shows a hexagonal arrangement of circular holes, while FIG. 3 shows a single base cell.

[0031] FIG. 4 shows a three-dimensional view of a single base cell in a 3D version of the porous microstructure.

[0032] The geometrical parameters definition step 11 further comprises calculating the density of the 2D porous microstructure as a function of the radius r of said circular holes of the 2D porous microstructure, according to the following equation (Eqn.1):

[00001]$\rho =1-\frac{2\pi r^2}{\sqrt{3}{d}^2}\mathrm{for}0\le r\le r_{\max },\mathrm{with}{r}_{\max }=\frac{d-t}{2}$

[0033] wherein:

[0034] r is the radius of said circular holes;

[0035] d is a reference dimension of said porous microstructure;

[0036] t is the minimum thickness of the material between two adjacent holes.

[0037] With reference to 3D, the geometrical parameters definition step 11 comprises calculating the density of the 3D porous microstructure as a function of the radius r of said spherical holes of the 3D porous microstructure, according to the following equation (Eqn.2):

[00002]$\rho =1-\frac{2\pi r^2}{\sqrt{3}{d}^2}\mathrm{for}0\le r\le r_{\max },\mathrm{with}{r}_{\max }=\frac{d-t}{2}$

[0038] wherein:

[0039] r is the radius of said spherical holes;

[0040] d is a reference dimension of said porous microstructure;

[0041] t is the minimum thickness of the material between two adjacent holes.

[0042] Hence, the minimum density of the porous material that is feasible with the prescribed set of geometrical parameters reads:

[00003]$\begin{array}{cc}{\rho }_{\min }=1-\frac{{\pi \left(d-t\right)}^2}{2\sqrt{3}{d}^2}\mathrm{related}\mathrm{to}2D\mathrm{and}& \left(\mathrm{Eqn}.3\right)\end{array}$$\begin{array}{cc}{\rho }_{\min }=1-\frac{{\pi \left(d-t\right)}^2}{3\sqrt{2}{d}^2}\mathrm{related}\mathrm{to}3D.& \left(\mathrm{Eqn}.4\right)\end{array}$

[0043] The preliminary homogenization step 1 further comprises computing material laws for said 2D/3D porous microstructure (if not already available) (step 12). Particularly, numerical homogenization is repeatedly performed on a discretized version of the relevant unit cell to compute the material law, i.e. the entries of the constitutive tensor of the homogenized material C.sub.ijhk written as a function of its density ρ.

[0044] To this end, a set of sampling points is considered in the range 0≤r≤r.sub.max. Particularly, the step 12 of computing material laws for the 2D/3D porous microstructure comprises calculating a constitutive tensor of the homogenized material (C.sub.ijhk) according to the following equation (Eqn.5):

[00004]$C_{\mathrm{ijhk}}=\frac{1}{.Math.Y.Math.}{\int }_Y{E}_{\mathrm{pqrs}}{\epsilon }_{\mathrm{pq}}^{A\left(\mathrm{ij}\right)}{\epsilon }_{\mathrm{rs}}^{A\left(\mathrm{hk}\right)}\mathrm{dY}$

[0045] wherein:

[0046] C.sub.ijhk is said constitutive tensor of the homogenized material;

[0047] |Y| is the volume of said base cell;

[0048] E.sub.pqrs(E.sub.0; v.sub.0) is the constitutive tensor of the material used to manufacture the object;

[0049] ε.sub.pq.sup.A(ij) is the strain field induced within the cell when enforcing unit test strains at its boundaries.

[0050] Particularly, the achieved results may be fitted using a fifth-degree polynomial, for which zero stiffness is additionally enforced at ρ=0.

[0051] Furthermore, according to a specific but not limiting example, the equation Eqn.5 is defined considering specific Young modulus E.sub.0 and Poisson's ratio v.sub.0 of the constitutive material used to manufacture the object.

[0052] As a result, with reference to the 2D version of the porous microstructure, which is isotropic, the homogenized constitutive tensor may be written in Voigt notation as follows (Eqn.6):

[00005]$C=C_{11}\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 0\end{array}\right]+C_{33}\left[\begin{array}{ccc}0& -2& 0\\ -2& 0& 0\\ 0& 0& 1\end{array}\right]$

[0053] where C.sub.11(ρ) and C.sub.33(ρ) read (Eqn.7):

C.sub.11=(1.4506ρ.sup.5−2.2357ρ.sup.4+1.1573ρ.sup.3+0.5214ρ.sup.2+0.1065ρ)C.sup.0.sub.11

C.sub.33=(3.5322ρ.sup.5−7.5798ρ.sup.4+5.6944ρ.sup.3+0.6874ρ.sup.2+0.0405ρ)C.sup.0.sub.33;

[0054] for 0≤ρ≤1 and

[00006]$\begin{array}{cc}{C}_{11}^0=\frac{E_0}{1-v_0^2},C_{33}^0=\frac{E_0}{2\left(1+v_0\right)}& \left(\mathrm{Eqn}.8\right)\end{array}$

[0055] FIG. 5 plots results from the numerical homogenization performed on the 2D version of the porous microstructure for different material densities. The fitting interpolation laws are compared to the conventional SIMP to show that the porous microstructure is much stiffer at intermediate densities than the conventional penalization.

[0056] Furthermore, due to the low degree of anisotropy that is peculiar to the 3D version of the porous microstructure, its homogenized constitutive tensor can be reasonably written in Voigt notation as (Eqn.9):

[00007]$C\approx C_{11}\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 1& 1& 1& 0& 0& 0\\ 1& 1& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]+C_{66}\left[\begin{array}{cccccc}0& -2& -2& 0& 0& 0\\ -2& 0& -2& 0& 0& 0\\ -2& -2& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

[0057] where C.sub.11(ϕ) and C.sub.66(ρ) read (Eqn.10):

C.sub.11=(1.7075ρ.sup.5−2.9452ρ.sup.4+1.2826ρ.sup.3+0.9011ρ.sup.2+0.0540ρ)C.sup.0.sub.11;

C.sub.66=(1.5915ρ.sup.5−3.0529ρ.sup.4+1.3610ρ.sup.3+1.0854ρ.sup.2+0.0151ρ)C.sup.0.sub.66;

for 0≤ρ≤1 and

[00008]$\begin{array}{cc}{C}_{11}^0=\frac{E_0\left(1-v_0\right)}{\left(1+v_0\right)\left(1-2v_0\right)},C_{66}^0=\frac{E_0}{2\left(1+v_0\right)}& \left(\mathrm{Eqn}.11\right)\end{array}$

[0058] As an example, FIG. 6 reports results from the numerical homogenization performed on the 3D version of the porous microstructure for different material densities. The fitting interpolation laws for C.sub.11 and C.sub.66 are represented along with the conventional SIMP. Again, the porous microstructure is much stiffer at intermediate densities than the conventional penalization.

[0059] With reference to the optimization step 2 of the computer-implemented method according to the invention, it comprises a first sub-step 21 of defining design domain, boundary conditions and relevant finite element discretization (for example, pixel-/voxel-based discretization).

[0060] Particularly, the 2D/3D geometry of the design domain is provided by the user.

[0061] In case of a hollow object with fixed external boundaries, the design domain consists of the volume inscribed.

[0062] Boundary conditions are prescribed, i.e. loads and supports.

[0063] In view of the adoption of a regular finite element mesh with N square/cubic elements, a pixel-/voxel-based representation of the two-/three-dimensional design domain is adopted whenever possible.

[0064] In case of a design domain with complex geometry, an unstructured discretization is generated.

[0065] The optimization step 2 further comprises preparing finite element analysis for the solution of the state equation using reference element stiffness matrices for full material (step 22).

[0066] Particularly, a density unknown x.sub.e, with x.sub.min≤x.sub.e≤x.sub.max, x.sub.min≥10.sup.−3 and x.sub.max≤1, is assigned to each element in the mesh.

[0067] Two reference element stiffness matrices are computed at the beginning of the procedure.

[0068] In 2D, K.sup.0.sub.11,e and K.sup.0.sub.33,e address the first and second term of the constitutive law in above equation Eqn.6 respectively, when using full material, i.e. ρ=1. According to a possible solution, with reference to said 2D porous microstructure, the step 22 of preparing finite element analysis comprises calculating the element stiffness matrices, for any prescribed value of density (x.sub.e), according to the following equation (Eqn.12):

[00009]$K_e\left(x_e\right)=\frac{C_{11}\left(x_e\right)}{C_{11}^0}{K}_{11,e}^0+\frac{C_{33}\left(x_e\right)}{C_{33}^0}{K}_{33,e}^0.$

[0069] Particularly, Eqn.12 is determined entering Eqn.7 with ρ=x.sub.e.

[0070] In 3D, K.sub.11,e.sup.0 and K.sub.66,e.sup.0 address the first and second term of the constitutive law in Eqn.9 respectively, when using full material, i.e. ρ=1.

[0071] According to a possible solution, with reference to said 3D porous microstructure, said step 22 of preparing finite element analysis comprises calculating said element stiffness matrices, for any prescribed value of density (x.sub.e), according to the following equation (Eqn.13):

[00010]$K_e\left(x_e\right)=\frac{C_{11}\left(x_e\right)}{C_{11}^0}{K}_{11,e}^0+\frac{C_{66}\left(x_e\right)}{C_{66}^0}{K}_{66,e}^0.$

[0072] Particularly, Eqn.13 is determined entering Eqn.10 with ρ=x.sub.e.

[0073] In case of an unstructured discretization, two reference element stiffness matrices are defined for each element in the mesh.

[0074] Furthermore, the optimization step 2 comprises the formulation of a topology optimization problem (step 23).

[0075] Particularly, the step 23 of topology optimization problem can be formulated according to the following equation (Eqn.14):

[00011]$\begin{array}{cc}\{\begin{array}{c}\underset{x_{\min }\le x_e\le x_{\max }}{\min }C={.Math.}_{e=1}^N{U}_e^T{K}_e\left(x_e\right)U_e\\ s.t.K\left(x\right)U=F,\\ \frac{1}{V}\underset{N}{.Math.}{x}_e{V}_e\le V_f,\end{array}& \begin{array}{c}\begin{array}{c}\left(\mathrm{Eqn}.14.1\right)\\ \left(\mathrm{Eqn}.14.2\right)\end{array}\\ \left(\mathrm{Eqn}.14.3\right)\end{array}\end{array}$

[0076] wherein

[0077] C is the structural compliance;

[0078] F is the load vector;

[0079] U.sub.e are the element-wise displacement vectors;

[0080] K(x) is the global stiffness matrix;

[0081] K.sub.e(x.sub.e) are the element-wise contributions;

[0082] V is the volume of the whole design domain;

[0083] V.sub.e, is the volume of the e-th element;

[0084] V.sub.f is a volume fraction.

[0085] Particularly, the objective function in Eqn.14.1 is the structural compliance C, that is a measure of the overall deformability of the object.

[0086] Eqn.14.2 enforces the elastic equilibrium for the load vector F and allows computing the element-wise displacement vectors U.sub.e (a subset of the nodal displacement vector U).

[0087] The global stiffness matrix K(x) gathers the element-wise contributions K.sub.e(x.sub.e) of Eqn.12 and Eqn.13, in 2D and 3D respectively.

[0088] Eqn.14.3 requires that no more than a user-defined volume fraction 0<V.sub.f<1 is used in the design.

[0089] The volume of the e-th element is denoted by V.sub.e, whereas V stands for the volume of the whole design domain.

[0090] With reference to Eqn.14, it is also pointed out that other objective functions can be considered (e.g. thermal or dynamical compliance), as well as constraints (e.g. stress-based and eigenvalue-based enforcements), adopting alternative formulations that exploit the basic concept of material distribution, including the distribution of multiple phases of material(s).

[0091] Eqn.14 can be solved using iterative update schemes of a known type.

[0092] At each iteration, the solution of the linear system in Eqn.14.2 allows computing the objective function and its sensitivities. For instance, the latter quantities are filtered using a convolution for a user-defined filter radius r.sub.f.

[0093] The minimization unknowns are bounded from below by x.sub.min=10.sup.−3, if both the external boundaries and the inner microstructure are sought. In case of a hollow object with prescribed external boundaries, the optimal infill is sought for x.sub.min=ρ.sub.min, where μ.sub.min is that of Eqn.3/Eqn.4 in 2D/3D.

[0094] In both cases, the upper bound of the element-wise densities can be user-defined. For x.sub.max=1, regions of full materials are allowed in the optimal design. For x.sub.max<1, a porous microstructure is enforced throughout the specimen.

[0095] As a result of the optimization step, a map of the distribution of the element-wise material densities x.sub.e is found throughout the design domain.

[0096] The post-processing step (3) comprises processing the map of the optimal distribution of the element-wise material densities x.sub.e for the detection of external boundaries of the object (if not given) (step 31).

[0097] Particularly, a cutoff value is defined, e.g. ρ.sub.cut=0.2. The relevant iso-line/iso-surface is computed processing the map of the pixel-/voxel-based densities.

[0098] The region inscribed in the detected/assigned boundaries defines a surface or a solid, in 2D and 3D respectively.

[0099] Furthermore, the post-processing step 3 comprises computing position and size of said circular/spherical holes in a Hexagonal Close-Packed (HCP) arrangement within the body of the object while accounting for the printing technology (step 32).

[0100] Particularly, holes are allowed only within an inner offset of the edges of the final object, when the distance of the center from the external boundaries is not less than a user-defined value, e.g. t.sub.off=d.

[0101] According to a possible solution, the step 32 of computing position and size of the holes comprises determining the coordinates y.sub.1, y.sub.2 and y.sub.3 of the centers of the circular/spherical holes in the adopted HCP arrangement as follows

[0102] (Eqn.15):

[00012]$\left(i+\frac{1}{2}\mathrm{mod}\left(j+k,2\right)\right)d,\left(\frac{\sqrt{3}}{2}j+\frac{1}{2\sqrt{3}}\mathrm{mod}\left(k,2\right)\right)d,\left(\frac{\sqrt{6}}{3}k\right)d$

[0103] wherein i, j and k are integer indices starting at the origin of a prescribed reference system, and

[0104] wherein the operator mod returns the remainder after division of two terms.

[0105] For the h-th hole, the average density of the elements falling within a neighborhood of its center with diameter d is computed and denoted by ρ.sub.h. Particularly, with reference to 2D, the step 32 of computing position and size of the holes comprises calculating the radius of a h-th circular hole according to the following equations:

[0106] if ρ.sub.h>ρ.sub.min (where ρ.sub.min is that of Eqn.3), then

[00013]$\begin{array}{cc}{r}_h={\left(\frac{\left(1-{\rho }_h\right)\sqrt{3}}{2\pi }\right)}^{\frac{1}{2}}d;& \left(\mathrm{Eqn}.16\right)\end{array}$

otherwise r=r.sub.max with r.sub.max=(d−t)/2.

[0107] With reference to 3D, the step 32 of computing position and size of the holes comprises calculating the radius of a h-th spherical hole according to the following equations:

[0108] if ρ.sub.h>ρ.sub.min (where ρ.sub.min is that of Eqn.4), then

[00014]$\begin{array}{cc}{r}_h={\left(\frac{\left(1-{\rho }_h\right)3\sqrt{2}}{8\pi }\right)}^{\frac{1}{3}}d;& \left(\mathrm{Eqn}.17\right)\end{array}$

otherwise r=r.sub.max with r.sub.max=(d−t)/2.

[0109] Finally, the post-processing step (3) comprises processing the geometry of the optimal layout for file exchange (step 33).

[0110] Particularly, Boolean subtraction of the geometrical entities representing the holes (circles or spheres, in 2D and 3D respectively, from step 32) from the geometrical entity representing the region within the optimal/given external boundaries of the layout (a surface or a solid, in 2D and 3D respectively, from step 31) is performed.

[0111] As an example, the graphical information can be exported using an IGES format.

[0112] Alternatively, a STL file format can be used.

[0113] A preliminary out-of-plane extrusion is needed in the 2D case.

[0114] Hence, a triangular representation of the involved three-dimensional surfaces (external boundaries of the object along with cylindrical/spherical holes in 2D/3D) is performed.

[0115] A possible numerical example is disclosed here below concerning to a cantilever, schematically represented in FIG. 7.

[0116] Geometrical and optimization parameters are assumed as it follows: l.sub.z1=320 mm, l.sub.z2=200 mm, l.sub.z3=10 mm.

[0117] The cantilever is fully clamped along the left edge and is subject to a vertical distributed load q=1 N/mm.sup.2 acting along the lower edge.

[0118] A mesh of N=64000 elements is used.

[0119] The optimization is performed using x.sub.min=10.sup.−3 and x.sub.max=1, whereas V.sub.f=0.4. FIG. 8 depicts results achieved for d=l.sub.z1/16 and t=d/10.

[0120] Particularly, the choice of d and t mainly depends on the adopted manufacturing technology.

[0121] Another aspect of the invention relates to a data processing system comprising means for carrying out the steps of the method disclosed above for generating a 2D or 3D object and comprising also a 3D-printer for the fabrication of said object.

[0122] Therefore, the system according to the invention comprises all the hardware/software for generation and printing of lightweight objects/components. The hardware can be constituted by at least one elaboration unit provided with software implementing the above defined method, wherein the elaboration unit is connected to at least a 3D-printer.

[0123] Furthermore, according to another possible implementation of the invention, a 3D-printer directly comprises means for carrying out the steps of the method disclosed above.

[0124] Therefore, in this case the 3D-printer according to the invention incorporates all the hardware/software for generation and printing of lightweight objects/components.

[0125] In practice it has been found that the described invention achieves the proposed aims.

[0126] Particularly, the distinctive feature of the multiscale method according to the invention is the adoption of a graded porous microstructure made of circular/spherical holes, packed in an HCP (Hexagonal Close-Packed) arrangement.

[0127] This gives the following advantage over existing multi-scale approaches.

[0128] The microstructure is ready for 3D-printing: circular/spherical holes are not affected by overhang issues (they can be printed at a reasonable quality without calling for additional printing supports or tricks such as reducing printing speed/temperature).

[0129] Furthermore, the porous microstructure has a smooth geometry: circular/spherical holes prevent geometrical singularities, such as sharp corners and intersections, to minimize structural weaknesses. No loss of continuity is allowed among adjacent cells.

[0130] The geometry of the graded microstructure is easy to compute, digitally handle and pass to the 3D-printer through an efficient procedure: once the reference dimension of the unit cell is defined, along with the minimum thickness of material between adjacent holes, centers and radii of circles/spheres are the only parameters involved in the description of the geometry.

[0131] The microstructure is isotropic (2D) or transversely isotropic with low degree of anisotropy (3D). From a computational point of view this means that: a few independent parameters are needed to describe the homogenized constitutive tensor; expressions giving tensor components as functions of the material density can be easily derived through numerical homogenization or data available from the literature; modification of SIMP-based codes to account for the proposed material models is straightforward. Moreover, from a mechanical point of view, this implies that a lightening of the full material is performed introducing a minimal anisotropy; robustness towards variation in the loads is expected.

[0132] The microstructure may be easily scaled depending on the printing technology by acting on the reference dimension of the base cell (that can be related to the thickness of the deposed layer of material).

[0133] The approach works both in 2D and 3D.

[0134] Furthermore, while the computer implemented method according to the invention mainly addresses FDM printers, other AM techniques that discard the need for support material, such as Selective Laser Sintering (SLS), can take full advantage of the method itself.

[0135] The following potential applications are considered.

[0136] Generation and fabrication of mechanical components: printing of original parts/lightened spares with optimized performance and robustness towards load variations (automotive and aerospace industries).

[0137] Generation and fabrication of massive structural components or massive technological components with structural function that need for lightening (building industry).

[0138] Generation and fabrication of design pieces through the definition of the optimal shape of the boundaries within a given design domain, along with a print-ready internal microstructure.

[0139] In general, solution to fabrication issues/printing issues encountered in the additive manufacturing of complex hollow objects, through the definition of an optimal perforated microstructure (infill) against critical overhangs and deformability/sagging issues.