# US 2021/0240880 A1 · Surface developability constraint for density-based topology optimization

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- Number: US 2021/0240880 A1
- Document ID: US20210240880A1
- Status: Published application
- Kind: A1
- Application number: 16/778,082
- Filed: 2020-01-31
- Published: 2021-08-05
- Earliest claimed: 2020-01-31
- Assignees: Toyota Motor Engineering & Manufacturing North America, Inc.; The Regents of the University of Michigan
- Applicants: Toyota Motor Engineering & Manufacturing North America, Inc.; The Regents of the University of Michigan
- Inventors: Yuqing Zhou; Tsuyoshi Nomura; Kazuhiro Saitou
- Web page: https://patentgrep.com/patent/US20210240880A1

## CPC

- G06F 30/17: Mechanical parametric or variational design
- G06F 3/00: Input arrangements for transferring data to be processed into a form capable of being handled by the computer; Output arrangements for transferring data from processing unit to output unit, e.g. interface arrangements
- G06F 17/11: for solving equations {, e.g. nonlinear equations, general mathematical optimization problems (optimization specially adapted for a specific administrative, business or logistic context G06Q10/04)}
- G06F 17/16: Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)}
- 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]
- G06T 17/10: Constructive solid geometry [CSG] using solid primitives, e.g. cylinders, cubes
- G06F 17/13: Differential equations (using digital differential analysers G06F7/64)
- G06F 2111/06: Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- G06T: IMAGE DATA PROCESSING OR GENERATION, IN GENERAL

## Abstract

Methods are provided for designing a structure with developable surfaces using a surface developability constraint. The surface developability constraint is developed based on the discovery of a sufficient condition for surface piecewise developability, namely surface normal directions lie on a small, finite number of planes. Automated methods and algorithms may include providing a design domain and a characteristic function of a material in the design domain to be optimized. The methods include defining a nodal density of the material, and determining surface normal directions of a plurality of planes. A density gradient that describes the surface normal directions is then determined. The methods include performing a topology optimization process on the design domain using a surface developability constraint that is based, at least in part, on the characteristic function. A geometric domain is then created for the structure using results from the topology optimization.

## Claims

### Claim 1

An automated method for designing a structure with developable surfaces using a surface developability constraint, the automated method comprising:
providing a design domain and a characteristic function of a material in the design domain to be optimized;
defining a nodal density of the material;
determining surface normal directions of a plurality of planes;
determining a density gradient that describes the surface normal directions;
performing a topology optimization process on the design domain using a surface developability constraint that is based, at least in part, on the characteristic function; and
creating a geometric domain for the structure using results from the topology optimization.

### Claim 2 (depends on 1)

The automated method according to claim 1, further comprising:
using a filter to regularize a scalar design variable, ϕ, as provided in the following equation:
−R2∇2{tilde over (ϕ)}+{tilde over (ϕ)}=ϕ,
where R is a filter radius; and
defining the nodal density, ρ, by a smoothed Heaviside function {tilde over (H)} as provided in the following equation:
ρ={tilde over (H)}({tilde over (ϕ)}).

### Claim 3 (depends on 2)

The automated method according to claim 2, wherein the nodal density is bounded between zero (0) and 1.

### Claim 4 (depends on 3)

The automated method according to claim 3, wherein the plurality of planes, K, are described by directional vectors v(1), v(2), . . . v(K), and the surface normal directions of density variables are computed as provided in the following equation:
∇
ϕ
~
=
(
∂
ϕ
~
∂
x
,
∂
ϕ
~
∂
y
,
∂
ϕ
~
∂
z
)
.

### Claim 5 (depends on 4)

The automated method according to claim 4, wherein the surface developability constraint is summarized with in the following equation:
∫
D
⁢
∏
k
=
1
K
⁢
⁢
(
v
(
k
)
·
∇
ϕ
~
)
2
⁢
d
⁢
⁢
Ω
=
0.
further wherein the surface developability constraint is satisfied only if a surface normal direction is perpendicular to a directional vector.

### Claim 6 (depends on 5)

The automated method according to claim 5, wherein the directional vectors are design variables.

### Claim 7 (depends on 1)

The automated method according to claim 1, wherein the step of creating the geometric domain for the structure comprises patching together at least two geometric patterns.

### Claim 8 (depends on 1)

The automated method according to claim 1, wherein the step of creating the geometric domain for the structure comprises generating a 2-dimensional design for creating a thin-walled structure.

### Claim 9 (depends on 1)

The automated method according to claim 1, wherein the characteristic function is minimizing compliance.

### Claim 10 (depends on 1)

The automated method according to claim 1, wherein the characteristic function is minimizing structural compliance, and the method provides a structure for aligning composite fibers such that the fibers are orientated in a manner that minimizes structural compliance.

### Claim 11 (depends on 1)

The automated method according to claim 1, wherein the characteristic function is minimizing thermal compliance, and the method provides a structure for optimizing a minimum thermal compliance under a constraint of a solid conductive material volume fraction.

### Claim 12 (depends on 1)

A solid volumetric structure with developable surfaces made according to the method of claim 1.

### Claim 13

A non-transitory computer readable medium comprising instructions that, when executed, perform an automated method of for designing a structure with developable surfaces using a surface developability constraint, the instructions comprising:
receiving a design domain and a characteristic function of a material in the design domain to be optimized;
defining a nodal density of the material;
determining surface normal directions of a plurality of planes;
determining a density gradient that describes the surface normal directions;
performing a topology optimization process on the design domain using a surface developability constraint that is based, at least in part, on the characteristic function; and
creating a geometric domain for the structure using results from the topology optimization.

### Claim 14 (depends on 12)

The non-transitory computer readable medium of claim 12, further comprising instructions of:
using a filter to regularize a scalar design variable, ϕ, as provided in the following equation:
−R2∇2{tilde over (ϕ)}+{tilde over (ϕ)}=ϕ,
where R is a filter radius; and
defining the nodal density, ρ, by a smoothed Heaviside function {tilde over (H)} as provided in the following equation:
ρ={tilde over (H)}({tilde over (ϕ)}).

### Claim 15 (depends on 14)

The non-transitory computer readable medium of claim 14, wherein the nodal density is bounded between zero (0) and 1.

### Claim 16 (depends on 15)

The non-transitory computer readable medium of claim 15, wherein the plurality of planes, K, are described by directional vectors v(1), v(2), . . . v(K), and the surface normal directions of density variables are computed as provided in the following equation:
∇
ϕ
~
=
(
∂
ϕ
~
∂
x
,
∂
ϕ
~
∂
y
,
∂
ϕ
~
∂
z
)
.

### Claim 17 (depends on 16)

The non-transitory computer readable medium of claim 16, wherein the surface developability constraint is summarized with in the following equation:
∫
D
⁢
∏
k
=
1
K
⁢
⁢
(
v
(
k
)
·
∇
ϕ
~
)
2
⁢
d
⁢
⁢
Ω
=
0.
further wherein the surface developability constraint is satisfied only if a surface normal direction is perpendicular to a directional vector.

### Claim 18 (depends on 17)

The non-transitory computer readable medium of claim 17, wherein the directional vectors are design variables.

### Claim 19 (depends on 13)

The non-transitory computer readable medium of claim 13, wherein the instruction of creating the geometric domain for the structure comprises an instruction for patching together at least two geometric patterns.

### Claim 20 (depends on 13)

The non-transitory computer readable medium of claim 13, wherein the instruction of creating the geometric domain for the structure comprises an instruction for generating a 2-dimensional design for creating a thin-walled structure.