US 2022/0019704 A1 · 3D OBJECT INTERNAL HOLLOWING FORM LIGHTWEIGHT METHOD BASED ON FUNCTION REPRESENTATION

Abstract

The present invention discloses a 3D shape internal hollowing form lightweight method based on function representation, and belongs to the field of computer-aided design. First, function representation is used and effective analytical calculation of shape optimization is explored; then, under the constraint of given external conditions, the stress structure design of a 3D object as well as the problems of center of mass, stand stability, tumbler design and buoyancy of an object are modeled by building an energy function model, and a corresponding discrete computation is given; finally, the above modeling problems are geometrically optimized to obtain an optimized internal shape of the object under given constraint conditions. The present invention greatly shortens the design and optimization cycles of this kind of cavity structures and can give theoretically optimal results.

A 3D object internal hollowing form lightweight method based on function representation, comprising the following specific steps:

(I) shape function representation of 3D object with cavities
a 3D object with cavities is expressed as ϕ°(r)≥0, wherein ϕ°(r) is a representation function of a model:
ϕ°(r)=min(ϕ(r),−ϕ(r)) (1)
wherein r=(x,y,z) is the coordinate of a point on the model, ϕ(r) is an external surface function of the object, and ϕ(r)=ϕ(r)−t(r) is an inner surface function of the object;
and t(r)≥0 is a continuous function of thickness field, which is expressed as follows:
t(r)=Ei=1ncaiRi(r)+Q(r) (2)
wherein Rij=R(|Pi−Pj|) is a radial basis function which represents the distance between points Pi and Pj, {Pi}Ei=1nc are uniformly sampled on the external surface of the model, nc is the number of control points, Q(r)=b1x+b2y+b3z+b4 is an offset term, {ai} is the weight of Ri(r), and {bi} is the weight of the offset term Q(r);
(II) 3D object internal hollowing form lightweight modeling and optimization based on function representation model stress and boundary conditions are given, a given problem is modeled by the function representation of the 3D object, so as to reduce material consumption as much as possible in the given material volume and boundary constraint conditions, and the specific steps are as follows:
1. problem modeling
1.1 modeling of the problem of structural strength
for the given model stress and boundary conditions, the problem of structural strength is modeled as follows:
⁢
min
t
⁡
(
r
)
⁢
I
=
∫
Ω
M
⁢
H
⁡
(
ϕ
o
⁡
(
r
)
)
⁢
f
·
udV
+
∫
τ
s
⁢
s
·
udS
⁢
⁢
s
.
t
.
⁢
∫
Ω
M
⁢
H
⁡
(
ϕ
o
⁡
(
r
)
)
⁢
𝔼
:
ɛ
⁡
(
u
)
:
ɛ
⁡
(
v
)
⁢
dV
=
∫
Ω
M
⁢
H
⁡
(
ϕ
o
⁡
(
r
)
)
⁢
f
·
vdV
+
∫
τ
s
⁢
s
·
vdS
,
∀
v
∈
U
ad
⁢
⁢
⁢
u
=
u
_
,
on
⁢
⁢
τ
u
⁢
⁢
⁢
∫
Ω
M
⁢
H
⁡
(
ϕ
o
⁡
(
r
)
)
⁢
dV
≤
V
_
(
3
)
wherein ΩM is the whole region occupied by a given model M, ϕ° (*) is a representation function of the model, f is a body force, s is a surface force defined on a Riemann boundary τs, S is the area of the Riemann boundary τs, u is a displacement field, v is a test function defined on the region ΩM, Uad={v|v∈Sob1(ΩM), v=0 on τu}, Sob1 is the first order soblev space, ε is the second order linear strain tensor, and is the fourth order isotropic elasticity identity tensor which is determined by elastic modulus and Poisson ratio;
ū is a prescribed displacement defined on a Dirichlet boundary τu, V is the volume of the model M, V is a volume constraint value, and H(x) is a regularized Heaviside function which is expressed as:
H
⁡
(
x
)
=
{
1
,
if
⁢
⁢
x
>
β
,
3
⁢
(
1
-
α
)
4
⁢
(
x
β
-
x
2
3
⁢
β
2
)
+
(
1
+
α
)
2
,
if
-
β
≤
x
≤
β
,
α
,
if
⁢
⁢
x
<
β
,
(
4
)
wherein α and β are threshold parameters;
1.2 modeling of the problems of mass and center
for the problems of mass and center of the 3D object, the mass m and center of mass c of a model are respectively expressed as follows:
m
=
M
1
⁢
⁢
c
=
[
c
x
,
c
y
,
c
z
]
T
=
1
m
⁡
[
M
x
,
M
y
,
M
z
]
T
(
5
)
wherein,
Mμ=∫ΩMH(ϕ°(r))μdV,μ=1,x,y,z (6)
ϕ°(r) is a representation function of the model, ΩM is the whole region occupied by a given model M, V is the volume of the model M, and H(x) is a regularized Heaviside function;
1.2.1 model of stand stability of object
for the stand stability of an object, modeling is carried out as follows:
min
t
⁡
(
r
)
⁢
S
⁡
(
t
)
=
c
x
2
+
c
y
2
+
c
z
2
⁢
⁢
s
.
t
.
⁢
(
c
x
+
c
y
)
2
-
(
r
-
ɛ
)
2
≤
0
(
7
)
wherein t(r)≥0 represents a function of thickness field to be solved, S(t) is an objective function, S(t) is minimized to make the center of mass of the object as low as possible, cx, cy and cz are the centers of mass of the object respectively in x, y and z directions, r is the radius of the maximum inscribed circle of a contact point convex hull, and ε is a safety factor;
1.2.2 model of tumbler
for the problem of a 3D tumbler, the problem is modeled as follows:
min
t
⁡
(
r
)
⁢
R
⁡
(
t
)
=
c
z
⁢
⁢
s
.
t
.
⁢
c
x
=
0
⁢
⁢
c
y
=
0
⁢
⁢
c
z
-
r
+
ɛ
≤
0
(
8
)
wherein t(r)≥0 represents a function of thickness field to be solved, R(t) is an objective function, R(t) is minimized to make the center of mass of the object in z-axis direction as low as possible, cx, cy and cz are the centers of mass of the object respectively in x, y and z directions, r is the radius of the maximum inscribed circle of a contact point convex hull, and ε is a safety factor;
1.2.3 model of buoyancy
for the problem of buoyancy of a 3D object, modeling is carried out as follows:
min
t
⁡
(
r
)
⁢
B
⁡
(
t
)
=
(
ρ
l
⁢
V
l
-
ρ
m
⁢
V
m
)
2
⁢
⁢
s
.
t
.
⁢
c
x
-
c
buoy
,
x
=
0
⁢
⁢
c
y
-
c
buoy
,
y
=
0
⁢
⁢
c
z
-
c
buoy
,
z
≤
0
(
9
)
wherein t(r)≥0 represents a function of thickness field to be solved, B(t) is an objective function, B(t)=0 represents that an object floats in water, cx, cy and cz are the centers of mass of the object respectively in x, y and z directions, ρl is the density of a liquid, Vl is the volume of the object submerged in a given liquid, ρm is the density of the object, Vm is the volume of the object, and cbouy,x, cbouy,y and cbouy,z are centers of mass of the corresponding space of the liquid occupied by immersion respectively in x, y and z directions;
2. problem optimization
a coarse and fine element strategy is used to solve a problem model, i.e., each coarse element is further divided into more fine elements inside;
the sensitivity analysis of variables is obtained by discrete computation of a problem, and is finally substituted into an optimizer to obtain the optimization results;
the details are as follows:
for problem modeling, a corresponding parameter value {ti}i=1nc at a control point of a function of thickness field t(r) need to be calculated, and the function of thickness field t(r) is expressed as:
t(r)=Σi=1ncNi(r)ti, (10)
wherein Ni(r)=[RQ]U−1, Ri,j=R(|Pi−Pj|) is a radial basis function which represents the distance between points Pi and Pj, Q is an offset matrix of a corresponding offset term,
U
-
1
=
[
R
Q
Q
T
0
]
-
1
,
and nc is the number of control points;
then the problem of model optimization is transformed into the problem of optimization of the parameter {ti}i=1nc, and the derivation of the objective function and the constraint function with respect to the optimized variables is carried out as follows:
∂
V
∂
t
i
=
1
8
⁢
∑
j
=
1
N
b
⁢
∑
k
=
1
8
⁢
∂
H
⁡
(
ϕ
jk
o
)
∂
t
i
⁢
⁢
∂
M
μ
∂
t
i
=
1
8
⁢
∑
j
=
1
N
b
⁢
∑
k
=
1
8
⁢
∂
H
⁡
(
ϕ
jk
o
)
∂
t
i
,
⁢
∂
c
x
∂
t
i
=
1
m
2
⁢
(
∂
M
x
∂
t
i
⁢
m
-
∂
m
∂
t
i
⁢
M
x
)
,
⁢
∂
c
y
∂
t
i
=
1
m
2
⁢
(
∂
M
y
∂
t
i
⁢
m
-
∂
m
∂
t
i
⁢
M
y
)
,
⁢
∂
c
z
∂
t
i
=
1
m
2
⁢
(
∂
M
z
∂
t
i
⁢
m
-
∂
m
∂
t
i
⁢
M
z
)
,
(
11
)
wherein Nb is the number of fine integration elements;
and the calculation results of formula (11) are substituted into an optimizer to obtain an optimized {ti}i=1nc, and thus to obtain a final optimization model, i.e., the internal shape of the object optimized in the given constraint conditions.

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3d object internal hollowing form lightweight method based on function representation

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3d object internal hollowing form lightweight method based on function representation

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Abstract

Claims1