US 2023/0409767 A1 · METHOD OF OPTIMIZING A DESIGN OF A STRUCTURE BY CONSIDERING CENTRIFUGAL LOADS AND STRESS CONSTRAINTS

Abstract

Disclosed is design optimization method of the structure considering centrifugal loads and stress constraints. Compared with the prior art, this application improves the method for obtaining the relaxation coefficient c, including the calculation of the second predicted maximum stress based on the predicted stress method from steps S9.1 to S9.8. The influence of the existence of jagged boundaries and gray densities on the calculation of the structural stress field is reduced. The most important thing is to decide whether to use linear penalty or nonlinear penalty for the elastic modulus of each element according to the ratio of the number of elements with design variables less than 0.1 and greater than 0.9 to the total number of elements. Introducing the predicted maximum stress can make full use of the allowable stress of materials, improve the quality of optimization design, and obtain a design scheme with a lighter mass.

A method of optimizing a design of a structure by considering centrifugal loads and stress constraints, with an optimization objective of minimizing structure mass under the stress constraints, the method comprising:

a step of discretizing a structure in a design domain,
a step of initializing design variables xe of discrete elements to form an initial design variable set,
a step of using a density method to carry out an iterative design until an optimal design variable set is obtained, and
a step of smoothing the structure according to the optimal design variable set,
wherein the design variables xe of all elements in all design variable sets satisfy 0≤xe≤1,
wherein each iteration of the iterative design executes the following steps:
S1:
obtaining a filtered density {tilde over (p)}e of an element, based on the initial design variable set or the design variable set formed in a last iteration, by filtering the design variables of each element with a density filtering method;
S2:
obtaining a first density ρe1 by employing Heaviside projection function to map the filtered density {tilde over (p)}e of each element;
ρ
e
⁢
1
=
tanh
⁡
(
0
.
5
⁢
β
1
k
)
+
tanh
⁡
(
β
1
k
(
ρ
~
e
-
0.5
)
)
2
⁢
tanh
⁡
(
0
.
5
⁢
β
1
k
)
;
wherein β1k is used to control smoothness of mapping curves, k is a sequence number in this iteration;
when k=1, β1k=4;
iteration updates are performed every 20 times from the first iteration and the value of β1k is 1.19 times the previous value of β1k;
a maximum value of β1k is set to 8;
S3:
calculating a first elastic modulus Ee1 of each element by a rational approximation of material properties (RAMP) method;
E
e
⁢
1
=
E
min
+
ρ
e
⁢
1
1
+
q
1
(
1
-
ρ
e
⁢
1
)
⁢
(
E
max
-
E
min
)
;
wherein Emax is an elastic modulus of a material, and Emin is a value added to avoid matrix singularity, q1 is a first stiffness penalty factor, and q1=4;
S4:
assembling a global stiffness matrix, and calculating a structural displacement according to a static equilibrium equation;
S5:
calculating a first stress σe1 of each element;
S6:
calculating a first von Mises stress σVM1 of each element;
S7:
obtaining a first penalty stress σVM1 of each element by punishing the first von Mises stress σVM1 of each element by using the RAMP method;
σ
¯
VM
⁢
1
=
(
E
min
+
ρ
e
⁢
1
1
+
q
2
(
1
-
ρ
e
⁢
1
)
⁢
(
E
max
-
E
min
)
)
⁢
σ
VM
⁢
1
;
wherein q2 is a second stiffness penalty factor, and q2=−0.95;
S8:
obtaining an aggregate stress {tilde over (σ)} by aggregating the first penalty stress σVM1 of each element;
S9:
relaxing the stress constraints, wherein a formulation of stress relaxation is c·{tilde over (σ)}=σl, σl is a yield stress of the material;
c is a relaxation coefficient in this iteration and is updated every five iterations, a formulation of c is
c
=
σ
¯
p
k
σ
~
,
σpk is a first maximum predicted stress in this iteration;
σ
¯
p
k
=
{
σ
p
k
,
k
=
1
0.4
σ
p
k
+
0.6
σ
p
k
-
1
,
k
>
1
;
wherein σpk is a second maximum predicted stress, acquired by following steps:
S9.1:
obtaining a second density ρe2 by employing a Heaviside projection function to map the filtered density {tilde over (ρ)}e of each element;
ρ
e
⁢
2
=
tanh
⁡
(
0
.
5
⁢
β
2
k
)
+
tanh
⁡
(
β
2
k
(
ρ
~
e
-
0
.
5
)
)
2
⁢
tanh
⁡
(
0
.
5
⁢
β
2
k
)
;
wherein β2k is used to control the smoothness of the mapping curves;
when k=1, β2k=4;
iteration updates are performed every 20 times from the first iteration and the value of β2k is the last value of β2k plus 4;
a maximum value of β2k is 20;
S9.2:
calculating a transition factor φ:
φ
=
N
b
+
N
w
N
;
wherein N is a total number of elements, Nb is a number of elements with the second density ρe2 greater than a first threshold, and Nw is a number of elements with the second density ρe2 less than a second threshold;
the first threshold is greater than 0.75, and the second threshold is less than 0.25;
S9.3:
calculating the second elastic modulus Ee2 of each element;
E
e
⁢
2
=
{
E
min
+
ρ
e
⁢
2
1
+
q
1
(
1
-
ρ
e
⁢
2
)
⁢
(
E
max
-
E
min
)
,
φ
<
0.9
E
min
+
ρ
e
⁢
2
(
E
max
-
E
min
)
,
φ
≥
0.9
;
S9.4:
assembling the global stiffness matrix, and calculating the structural displacement according to the static equilibrium equation;
S9.5:
calculating a second stress σe2 of each element;
S9.6:
calculating a second von Mises stress σVM2 of each element;
S9.7:
obtaining a second penalty stress σVM2 of each element by punishing a second von Mises stress σVM2 of each element by using a linear method;
σVM2=(Emin+ρe2(Emax−Emin))σVM2;
S9.8:
calculating the second maximum predicted stress σpk in this iteration:
σpk=max(σVM2);
S10:
conducting a sensitivity analysis of an objective function Vf:
V
f
=
∑
e
=
1
N
ρ
e
⁢
1
⁢
v
e
∑
e
=
1
N
v
e
;
where ve is a volume of the e-th element;
S11:
obtaining and storing the design variable set formed in this iteration by moving asymptote algorithm to solve optimization problem and updating the design variables of each element;
S12:
judging whether this iteration meets exit iteration conditions, if this iteration meets the exit iteration conditions, exit the iteration, and record the design variable set formed in this iteration as the optimal design variable set;
if the exit iteration conditions are not met, the next iteration will be carried out;
above procedures from S9.1 to S9.8 allow parallel computation with those from S2 to S8.

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Method of optimizing a design of a structure by considering centrifugal loads and stress constraints

Life of this patent

Method of optimizing a design of a structure by considering centrifugal loads and stress constraints

Life of this patent

Abstract

Claims7