# US 2014/0207428 A1 · Methods for Orienting Material Physical Properties Using Constraint Transformation and Isoparametric Shape Functions

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- Number: US 2014/0207428 A1
- Document ID: US20140207428A1
- Status: Published application
- Kind: A1
- Application number: 13/747,955
- Filed: 2013-01-23
- Published: 2014-07-24
- Earliest claimed: 2013-01-23
- Assignees: Toyota Motor Engineering & Manufacturing North America, Inc.
- Applicants: Toyota Motor Engineering & Manufacturing North America, Inc.
- Inventors: Tsuyoshi Nomura; Ercan Mehmet Dede
- Web page: https://patentgrep.com/patent/US20140207428A1

## CPC

- G06F 30/23: using finite element methods [FEM] or finite difference methods [FDM]
- 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 2111/04: Constraint-based CAD
- G06F 2113/26: Composites
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

Constraint-based methods for determining orientations of material physical properties using an isoparametric shape function are disclosed. In one embodiment, a method of defining an orientation of an material physical property includes defining nonlinear and/or discontinuous design constraints of design values in a geometric domain associated with one or more physical attributes of the material physical property, and translating the nonlinear and/or discontinuous design constraints into continuous, first order design constraints of the design values by applying an isoparametric shape function. The method further includes performing a topology optimization using the continuous, first order design constraints of the design values, and reverse-translating results of the topology optimization back into the geometric domain using the isoparametric shape function. The results of the topology optimization in the geometric domain are indicative of the orientation of the material physical property.

## Claims

### Claim 1

A method of defining an orientation of a material physical property:
defining nonlinear and/or discontinuous design constraints of design values in a geometric domain associated with the material physical property;
translating the nonlinear and/or discontinuous design constraints into continuous, first order design constraints of the design values by applying an isoparametric shape function;
performing a topology optimization using the continuous, first order design constraints of the design values; and
reverse-translating results of the topology optimization back into the geometric domain using the isoparametric shape function, wherein the results of the topology optimization in the geometric domain are indicative of the orientation of the material physical property.

### Claim 2 (depends on 1)

The method of claim 1, wherein the nonlinear design constraints are represented by Cartesian coordinates providing quadratic constraints √(xi2+yi2)<1 for i=1, 2, . . . , n.

### Claim 3 (depends on 1)

The method of claim 1, wherein the isoparametric shape function is defined by a plurality of nodes arranged about a unit circle.

### Claim 4 (depends on 3)

The method of claim 3, wherein a number of nodes of the plurality of nodes is at least two times greater than a number of preferred discrete orientations of the material physical property.

### Claim 5 (depends on 3)

The method of claim 3, wherein the plurality of nodes comprises eight nodes defined by: (cos(nπ/4), sin(nπ/4)), for n=0, 1, . . . , 7.

### Claim 6 (depends on 1)

The method of claim 1, further comprising applying a penalty factor to alter a shape of the isoparametric shape function to influence the orientation of the m.

### Claim 7 (depends on 6)

The method of claim 6, wherein application of the penalty factor alters the shape of the isoparametric shape function such that the isoparametric shape function is star-shaped.

### Claim 8

A method of determining an orientation of a plurality of material physical properties of a device, the method comprising:
providing initial design values ξi and ηi for each material physical property on a normalized coordinate system, wherein −1<ξ<1 and −1<η<1 for i=1, 2, . . . , n;
projecting the initial design values (ξi, ηi) to projected initial design values (xi, yi) on a Cartesian coordinate system using a isoparametric shape function (ξ, η), Ny(ξ, η) having a plurality of nodes such that xi=Nx(ξi, ηi), yi=Ny(ξi, ηi), and √(xi2+yi2)<1;
obtaining convergence in an objective function by execution of a topology optimization of xi, yi; and
determining an orientation of each material physical property based on final design variables xi, yi for each material physical property i of the plurality of material physical properties.

### Claim 9 (depends on 8)

The method of claim 8, wherein obtaining convergence in the objective function further comprises:
executing the topology optimization for the projected initial design values (xi, yi);
determining a physical state of the device based on the topology optimization for the projected design values (xi, yi);
calculating the objective function based on the physical state of the device;
if there is not convergence in the objective function then:
calculating sensitivities of the objective function with respect to xi, yi;
converting the sensitivities of the objective function with respect to xi, yi into sensitivities with respect to (ξi, ηi);
updating the initial design values ξi and ηi using the sensitivities with respect to (ξi, ηi) and a gradient-based analysis; and
projecting updated design values ξi and ηi to projected updated design values (xi, ηi) on the Cartesian coordinate system using the isoparametric shape function Nx(ξ, η), Ny(ξ, η) and repeating the topology optimization for the projected updated design values (xi, yi).

### Claim 10 (depends on 8)

The method of claim 8, wherein the isoparametric shape function is circular.

### Claim 11 (depends on 8)

The method of claim 8, wherein the isoparametric shape function is star-shaped.

### Claim 12 (depends on 8)

The method of claim 8, wherein a number of nodes of the plurality of nodes is at least two times greater than a number of preferred discrete angles for the plurality of material physical properties.

### Claim 13 (depends on 8)

The method of claim 8, wherein the plurality of nodes of the isoparametric shape function comprises alternating discrete angle nodes and intermediate nodes.

### Claim 14 (depends on 13)

The method of claim 13, wherein the discrete angle nodes correspond to angles of which individual material physical properties of the plurality of material physical properties may be oriented.

### Claim 15 (depends on 13)

The method of claim 13, further comprising, after obtaining convergence in the objective function, determining if each material physical property of the plurality of material physical properties is oriented in one of a plurality of preferred discrete angles.

### Claim 16 (depends on 15)

The method of claim 15, further comprising, if at least one material physical property of the plurality of material physical properties is not oriented in one of the plurality of preferred discrete angles, applying a penalty factor to the isoparametric shape function to obtain a modified isoparametric shape function, wherein the penalty factor shifts coordinates of the intermediate nodes.

### Claim 17 (depends on 16)

The method of claim 16, wherein the penalty factor encourages individual material physical properties of the plurality of material physical properties to be oriented in one of the plurality of preferred discrete angles upon subsequent topology optimizations.

### Claim 18 (depends on 16)

The method of claim 16, further comprising:
projecting current design values (ξi, ηi) to projected current design values (xi, yi) on the Cartesian coordinate system using the modified isoparametric shape function; and
obtaining convergence in the objective function by execution of the topology optimization of xi, yi to determine an updated design variable (xi, yi) for each material physical property i of the plurality of material physical properties.

### Claim 19 (depends on 8)

The method of claim 8, wherein the plurality of material physical properties comprises a plurality of magnet material physical properties.

### Claim 20 (depends on 19)

The method of claim 19, wherein the objective function corresponds to a temperature profile of a heat receiving surface heated by a heat generating device and a magnetic fluid influenced by magnetic fields generated by the plurality of magnets.