# US 2022/0035967 A1 · Efficient method for representation and optimization of porous structures

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- Number: US 2022/0035967 A1
- Document ID: US20220035967A1
- Status: Published application
- Kind: A1
- Application number: 17/349,270
- Filed: 2021-06-16
- Published: 2022-02-03
- Earliest claimed: 2020-07-28
- Assignees: Dalian University of Technology
- Applicants: Dalian University of Technology
- Inventors: Shengfa WANG; Jiangbei HU; Xiaopeng ZHENG; Na LEI; Zhongxuan LUO
- Web page: https://patentgrep.com/patent/US20220035967A1

## CPC

- G06F 30/10: Geometric CAD
- G06F 30/20: Design optimisation, verification or simulation (optimisation, verification or simulation of circuit designs G06F30/30)
- G06F 30/23: using finite element methods [FEM] or finite difference methods [FDM]
- G06F 2111/04: Constraint-based CAD
- G06F 2111/06: Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]
- G06F 2111/10: Numerical modelling
- G06F 2113/10: Additive manufacturing, e.g. three-dimensional [3D] printing
- G06F 2119/18: Manufacturability analysis or optimisation for manufacturability
- Y02P 90/02: Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- Y02P: CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS

## Abstract

An efficient method for representation and optimization of porous structures belongs to the field of computer aided design. Firstly, a representation method of a multi-scale porous structure described by a function is provided. Based on the function representation, an optimization frame is designed. Then, an optimization problem model is established by taking structural energy minimization as a goal and taking a volume and a gradient as constraints. Finally, topological optimization is conducted firstly, and then geometric optimization is conducted. The topology and the thickness of the porous structure are optimized to obtain an optimization model filled with the porous structure. The present invention completely represents, analyzes, optimizes and stores the porous structure by functions, which greatly reduces the calculation complexity and greatly shortens the design and optimization period. Moreover, the present invention can provide an optimization model with strong structural hardness and stiffness under the volume constraint. The structure is suitable for frequently-used 3D printing manufacturing technologies. The internal structure does not need additional support in the printing process, which can save printing time and printing material.

## Claims

### Claim 1

An efficient method for representation and optimization of porous structures, comprising the following steps:
(I) multi-scale porous shell structure
firstly, establishing a multi-scale porous surface:
{tilde over (φ)}P(r)=cos(2π·t(r)·x)+cos(2π·t(r)·y)+cos(2π·t(r)·z) (1)
wherein r=(x, y, z)∈R3 is a point in a space, and t(r)>0 is a continuous periodic distribution function, which controls a continuous change of a hole size;
then, constructing a multi-scale porous shell structure with thickness: using a closed region determined by two different isosurfaces with a same periodic distribution as an inner space of a porous shell, defining:
ϕ
s
⁡
(
r
)
=
min
⁡
(
ϕ
1
⁡
(
r
)
,
ϕ
2
⁡
(
r
)
)
⁢
⁢
ϕ
1
⁡
(
r
)
=
φ
~
⁡
(
r
)
+
c
~
⁡
(
r
)
⁢
⁢
ϕ
2
⁡
(
r
)
=
c
~
⁡
(
r
)
-
φ
~
⁡
(
r
)
⁢
⁢
c
~
⁡
(
r
)
=
c
0
⁡
(
r
)
⁢
t
⁡
(
r
)
t
0
⁡
(
r
)
(
2
)
wherein ϕs(r)>0 represents a region defined as Ωs, Ωs is an interior of the porous shell structure based on a triply periodic minimal surface; {tilde over (φ)}(r) is a multi-scale triply periodic minimal surface; and c0(r) and t0(r) are initial thickness and period value;
for a given model M, representing a model filled with the porous structure by a function:
ϕM=ϕs∩ϕVDF=min(ϕs,ϕVDF) (3)
wherein ϕVDF is a distance field of the model M; and then ϕM≥0 represents the interior of the model filled with the porous structure;
(II) establishing an optimization model and discretization thereof
constructing a problem model, filling an inner space of the problem model using the constructed multi-scale porous shell structure by taking structural energy minimization as a goal and taking a model volume and a hole size distribution gradient as constraints, so that strength and stiffness of the problem model are still strong under a condition of given material volume constraints; calculating the problem model by a finite element analysis method and discretizing a problem; increasing calculation efficiency while ensuring calculation accuracy by a element method; uniformly dividing a design domain into elements firstly and then uniformly subdividing the elements into sub-divide elements; interpolating a displacement vector by a element mesh system; and describing the model by a sub-divide element system;
discretizing a problem in the problem model as follows:
min
c
⁡
(
r
)
,
t
⁡
(
r
)
⁢
I
=
F
T
⁢
U
(
6
)
so that
KU
=
F
⁢
⁢
V
=
1
8
⁢
∑
j
=
1
N
b
⁢
∑
l
=
1
8
⁢
H
η
⁡
(
ϕ
l
j
)
⁢
v
b
≤
V
¯
⁢
⁢
G
=
1

Ω
M

⁢
∑
j
=
1
N
b
⁢
∑
l
=
1
8
⁢
ℏ
⁡
(

∇
t
l

g
¯
)
⁢
v
b
≤
1
(
7
)
wherein I is structural energy of the model; U is a displacement vector; F is a force vector of a node point; K is a total stiffness matrix; V is a volume fraction; V is a designated volume constraint; ϕlj is a value of ϕs at a l node point of a j sub-divide element; G is a gradient of the periodic distribution; g is a designated gradient constraint value; vb is the volume of the sub-divide elements; Nb is the number of the total sub-divide elements in a solution domain; ∥∇tl∥ is a value of the periodic distribution gradient ∥∇t(r)∥ at the l node point of the j sub-divide element; ∥ΩM∥ is the volume of the design domain; and Hη(X) is a regularization Heaviside function, and h(x) is defined as follows:
ℏ
⁡
(
x
)
=
{
(
x
-
1
)
2
+
1
,
if
⁢
⁢
x
≥
1
1
,
if
⁢
⁢
x
<
1
(
8
)
(III) optimization solution
based on the problem model, only two unknown parameters t(r) and c(r) need to be calculated and optimized; therefore, at fixed thickness, t(r) is optimized firstly, then a topological parameter is fixed, and c(r) is optimized; a specific optimization process is as follows:
step 1. topological optimization: interpolating the topological parameter t(r) by a radial basis function (RBF) interpolation method and converting function optimization into the optimization of the parameter at an interpolation node point; randomly selecting an interpolation point {pi}i=1nr∈ΩM in the design domain ΩM, so that the form of interpolation t(r) is:
t
⁡
(
r
)
=
∑
i
=
1
n
,
⁢
R
⁡
(
r
)
⁢
a
i
t
+
∑
j
=
1
m
⁢
q
j
⁡
(
r
)
⁢
b
j
t
(
9
)
wherein Ri(r)=R(∥r−pi∥); logarithmic RBF R(x)=x2 log(|x|) is selected; {qj(r)} is a polynomial for coordinates; ait and bjt are undetermined coefficients; through derivation, (9) is simplified as:
t
⁡
(
r
)
=
∑
i
=
1
n
r
⁢
S
i
⁡
(
r
)
⁢
t
i
(
10
)
wherein ti=t(pi) is a periodic value at a control point pi; Si(r) is a polynomial form derived from matritization; thus, the optimization of a topological structure is converted into the optimization of a parameter the {ti}i=1nr, the parameter {ti}i=1nr is the set of ti=t(pi);
finally, calculating the derivatives of an objective function and a constraint function for optimization variables as follows:
∂
I
∂
t
i
=
-
∑
k
=
1
N
s
⁢
U
k
T
⁡
(
1
8
⁢
∑
j
=
1
N
b
⁢
(
∑
l
=
1
8
⁢
∂
H
η
⁡
(
ϕ
l
kj
)
∂
t
i
)
⁢
K
0
)
⁢
U
k
⁢
⁢
∂
V
∂
t
i
=
1
8
⁢
∑
j
=
1
N
b
⁢
∑
l
=
1
8
⁢
∂
H
η
⁡
(
ϕ
l
j
)
∂
t
i
⁢
v
b
⁢
⁢
∂
G
∂
t
l
=
1

Ω
M

⁢
∑
j
=
1
N
b
⁢
∑
l
=
1
8
⁢
ℏ
′
⁡
(

∇
t
l

g
¯
)
⁢
∂

∇
t

∂
t
⁢
v
b
(
11
)
wherein Ns is the number of the elements; Nb is the number of the sub-divide elements in the design domain; Uk is a node point displacement vector corresponding to a k element; K0=E0BTD0Bvb; E0 is Young's modulus; B is a strain matrix; D0 is a constitutive matrix of solid material filled by the elements; ϕlij is a value of ϕs at the/node point of the j sub-divide element in the i element; then, sensitivity information is substituted into a solution optimization algorithm MMA (Method of Moving Asymptotes) which is commonly used in the field of mechanics to obtain a solution of a problem under topological optimization, wherein, the value of an optimal parameter {ti}i=1nr; the topological form of the porous structure is determined;
step 2. geometric optimization: topological optimization is regarded as rough optimization: after topology of whole porous structure is determined, more detailed optimization, wherein, geometric optimization is conducted;
converting the function optimization into the optimization of a parameter {ci}i=1nr at the control point using the same method as topological optimization by the RBF, {ci}i=1nr is the set of cl=c(pl); randomly selecting an interpolation point {pi}i=1nr∈QM in the design domain λM, so that the form of interpolation c(r) is:
c(r)=Σi=1mrRi(r)aic+Σj=1mqj(r)bjc (12)
wherein aic and bjc are undetermined coefficients, which are also simplified as:
c
⁡
(
r
)
=
∑
i
=
1
n
r
⁢
S
i
⁡
(
r
)
⁢
c
i
(
13
)
herein cl=c(pi) is a thickness value at the control point pi; Si(r) is a polynomial form derived from matritization; then, the sensitivity information about the optimization variables is calculated and substituted into the MMA to obtain the solution of a problem under geometric optimization;
∂
l
∂
c
t
=
-
∑
k
=
1
N
b
⁢
U
k
T
⁡
(
1
8
⁢
∑
j
=
1
N
b
⁢
(
∑
l
=
1
8
⁢
∂
H
η
⁡
(
ϕ
l
kj
)
∂
c
t
)
⁢
K
0
)
⁢
U
k
⁢
⁢
∂
V
∂
c
t
=
1
8
⁢
∑
j
=
1
N
b
⁢
∑
l
=
1
8
⁢
∂
H
η
⁡
(
ϕ
l
j
)
∂
c
i
⁢
v
b
(
14
)