2011_Gitam

Altair Confidential

Optimization and Lifing of

Aircraft Structures and Engines

Dr. J. S. Rao

Chief Science Officer

[email protected]

January 8 2011

GITAM University

Visakhapatnam

mailto:[email protected]

Altair Confidential

Design a column such that

Stress induced is < yield and buckling

Mean diameter d to be in the range 2 to 14 cm

Thickness t to be in the range 0.2 to 0.8 cm

Cost to be lowest, given by 5 times weight + 2

times the mean diameter.

E = 0.85×106 kg/cm2

Density = 0.0025 kg/cm2

Length l = 250 cm

P = 2500 kg

Yield stress = 500 kg/cm2

Optimization – Mathematical Programming

Altair Confidential

Design Vector

Formulation

t

d

x

xX

2

1

Objective Function 121 282.92525 xxxddtldWXf

Behavioral Constraints

5002500

21

xxdt

P

stressyieldstressinduced

2

2

2

2

1

62

21 2508

1085.02500 xx

xx

stressbucklingstressinduced

Side Constraints

8.02.0

0.140.2

t

d

Altair Confidential

Given the Design Vector

Statement of Problem

2

1

x

xX

Minimize the Objective Function 121 282.9 xxxXf

Subject to Constraints 05002500

21

1 xx

Xg

0

2508

1085.025002

2

2

2

1

62

21

2

xx

xxXg

08.0

02.0

00.14

00.2

26

25

14

13

xXg

xXg

xXg

xXg

Altair Confidential

Problem with Two Design Variables

Constraint 1

593.1

05002500

21

21

1

xx

xxXg

3.47

02508

1085.02500

2

2

2

121

2

2

2

2

1

62

21

2

xxxx

xx

xxXg

8.0;2.0;0.14;0.2 2211 xxxx

Plot P1 Q1 593.121 xx

Plot P2 Q2

Constraint 2

3.472

2

2

121 xxxx

Plot all side constraints 3 to 6

Choose different values of C

and plot; C = 50, 40, 31.8, 26.53

CxxxXf 121 282.9

Altair Confidential

Graphical Solution – Design Space – Optimum Point

Graphical Optimization

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

d(X1)

f(X

2)

X1*X2 = 1.593X1*X2*(X1**2 + X2**2) = 47.3

C = 50C = 40

C = 31.8C = 26.53

Optimum Point

(5.44, 0.293)

Altair Confidential

Iterations X1 X2

1 2 0.2

2 2.44 0.2

3 2 0.244

4 2.4 0.238

5 2.30879 0.2856

6 2.36005 0.321197

7 2.80846 0.385437

8 3.37015 0.45867

9 4.01048 0.550404

10 4.36014 0.445827

11 4.66099 0.356662

12 5.23217 0.386645

13 5.27673 0.309316

14 5.31056 0.313182

15 5.33557 0.309316

16 5.45705 0.292034

Objective_1 Constraint_1 Constraint_2

7.928 -1.193 -45.684

9.67216 -1.105 -44.3751

8.79216 -1.105 -45.3189

10.4092 -1.0218 -43.9775

11.0928 -0.93361 -43.7313

12.1641 -0.83496 -42.9996

16.2469 -0.51052 -38.6011

21.9199 -0.04721 -29.4179

29.6975 0.614384 -11.1278

27.8091 0.350868 -9.95916

25.6467 0.069398 -10.9731

30.3301 0.429989 8.38288

26.5814 0.039176 -1.69776

26.9535 0.070174 -0.23196

26.8778 0.057375 -0.15874

26.5637 6.44E-04 0.293614

Response Surface Analysis

Altair Confidential

Classification of Optimization Problems

Constraints Design Variables Physical Structure

Constrained

Equality and

Inequality

Unconstrained

No constraints

Parameter or

Static –

Minimum weight of a

Prismatic Beam

Dynamic –

Minimum weight of a

Beam with variable

Cross-section –

Design Variables

continuous function

through space

Optimal Control

Two types of variables

Control (design) and

State variables

Non-optimal

Nature of Equations

Nonlinear

Programming

Geometric

Programming

Quadratic

Programming

Linear

Programming

Stochastic Programming Multi-objective Programming

Altair Confidential

Static Optimization –

Prismatic Beam

Static/Dynamic Optimization Problems

d

b

x

xX

2

1 lbdXf

0

0

max

d

b

Xtip

Dynamic Optimization –

Variable Cross-section Beam

Depends not only on the

variables but also on the

trajectories through some sort

of space.

td

tbtX dttdtbtXf

l

0

lttd

lttb

lttXtip

0,0

0,0

0,max

Altair Confidential

Optimal Control Problem

l

i

iii yxfXf1

,

Control (design) variables x thrust and

state variables y velocity of a rocket;

Control variables govern the evolution of

the system from one stage i to the next i+1

and the state variables describe the system

in any stage; q may represent the thrust

velocity relationship in each stage, g may

be the maximum thrust available in a

given stage ….

Mathematical programming involving a

number of stages, where each stage

evolves from the previous stage.

Determine the control variables x such that

the total objective function over l number

of stages, time, is minimized.

lkyh

ljxg

liyyyxq

kk

jj

iiiii

,...,2,1,0

,...,2,1,0

,...,2,1,, 1

ix

x

x

X 2

1

Altair Confidential

Nonlinear Programming

w

d

d

d

X3

2

1Three stepped cone pulley drive, dias di all

having same width w and shaft center

distance a to be designed for minimum

weight to transmit a minimum power from

a constant speed input N and desired

output speeds and tension ratio of the belt.

3

1

2

2 14 N

NdwXf i

i

Same length c of all belts

Equality constraints 0

0

31

21

cc

cca

a

dN

N

N

Ndc

ii

iii 2

4

1

12

2

2

Constraint on ratio of tensions

- Inequality constraint2

21sin2exp 1

a

d

N

N ii

Constraint on Power -

Inequality constraint min

1

17600

'

21sin2exp1 HP

Nd

a

d

N

NT iiii

3,2,1,0;0 idw i

Altair Confidential

Geometric Programming

Polynomial Function

N

i

a

j

n

ji

a

n

aa

N

a

n

aa

ij

NnNNn

xc

xxxcxxxcXh

11

1211211 ......... 2111211

o

ij

N

i

ji

p

j

n

ji xcxcXf

11

0,0,Objective Function

Constraints mjaxaXgj

ik

N

i

ij

q

k

n

kijj ,...2,1,0,0

11

Cost minimization of a shell and tube type heat

exchanger with a constraint on thermal energy

Altair Confidential

Linear Programming

XCxcxcxcxcXfT

n

j

jjnn 1

2211 ...Objective Function

Constraints

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

...

...

...

...

2211

22222121

11212111

0,..1,0

,..1,

1

Xnjx

BXA

mibxa

j

n

j

ijij

Statement of the Problem

n variables, m constraint equations

Altair Confidential

Linear Programming – Dantzig’s Simplex Method - 1947

54321 32103 xxxxxXf

10040102015080

300060010060030002500

54321

54321

xxxxx

xxxxx0ix

n = 5, m = 2 No. of Basic Solutions

10!!

!

mmn

n

• The optimum feasible basic solution out of the 10 basic solutions is given here.

Choose basic variables x1, x5

1048

30625

51

51

xx

xx

26

5,

13

21 51 xx

26

14f

Let

• To obtain a basic solution set

n-m variables (non-basic) = 0

• Choose m out of n variables

as basic and the rest n-m as non

basic variables.

• Feasible solution is one which

satisfies the inequality constraints

• If the solution obtain is not optimal, find a neighboring basic feasible solution which has

a lower value of f and repeat the process until an optimum is found.

Altair Confidential

Associated with every linear programming problem, called the primal, there

is another linear programming problem called its dual. These two problems

possess very interesting and closely related properties. If the optimal solution

to any one is known, the optimal solution to the other can readily be obtained.

In fact, it is immaterial which problem is designated the primal since the dual of a

dual is the primal. Because of these properties, the solution of a linear

programming problem can be obtained by solving either the primal or the dual,

whichever is easier.

Primal and Dual Problems

Altair Confidential

Primal Problem

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

...

...

...

...

2211

22222121

11212111

nixxcxcxcf inn ,..1,0 ...2211

Altair Confidential

Dual Problem

nmmnnn

mm

mm

cyayaya

cyayaya

cyayaya

...

...

...

...

2211

22222112

11221111

maximized be tois and ,..1,0

...2211

vmiy

ybybybv

i

mm

Dual problem is formulated by transposing the rows and columns of Primal including

the right-hand side and the objective function, reversing the inequalities and

maximizing instead of minimizing. Denoting the dual variables as y1, y2 … ym,

symmetric primal-dual pairs

The dual of the dual is the primal

Altair Confidential

Topology Optimization

• Topology optimization generates optimal shape of a structure.

• The shape is generated within a pre-defined design space.

• In addition, the user provides structural supports and loads.

• Without any further decisions and guidance, the method will form the

structural shape providing a first idea of an efficient geometry.

• Therefore, topology optimization is a much more flexible design tool

than classical structural shape optimization, where only a selected

part of the boundary is varied without any chance to generate a

lightness hole, for example.

Altair Confidential

• Finite element discretization of the design space

• Strain Energy Density in each element can be used as a measure to identify the

material necessary to carry the given load

Strain Energy Density Approach

Key to Topology Optimization

► Find how the load is transmitted in the structure before optimization

► Identify the material taking the load vis a vis idle material

Altair Confidential

Load Path with Strain Energy Density

Manual Topology optimization based on strain energy

Iteration IT0 Iteration IT1



Altair Confidential

Solid Isotropic Material with Penalization - SIMP

Function between “relative density“ and

Elasticity using penalty factor p

E/E0

/0

/0)p

1

1

Step 1: Finite element discretization of the design space - Arbitrarily assign for each element volume

fraction (relative density)

Step 2: Convert for each element /0 into E/E0 and find Young’s modulus E in accordance to the value of

penalty parameter p. Start with using penalty = 1

Step 3: Set up each element stiffness matrix and assemble them to form global stiffness matrix

Step 4: Set up [K]{x}={F} and solve for {x}. This will be the response for the structure to begin with.

This of course is not what we want.

Step 5: For each element set up different values of /0 with chosen step size say 0.5 above and below the

starting values. Set up DOE and find responses.

Step 6: Minimize the response vector {x} with E/E0 (i.e., /0) as design variables and obtain the new set

of /0.

Step 7: Go back to step 1 and repeat the iterations until /0 have converged.

p = 4

13 iterations

p = 1

6 iterations

FxK

Altair Confidential

Simple Topology Example - Model details

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

Element IDs

Element ID

Node ID

50 N each

18 Elements and 56 Nodes

Altair Confidential

Baseline Analysis

E = 210000 MPa

= E0

Node ID v/s Displacement

-2.00E-03

-1.50E-03

-1.00E-03

-5.00E-04

0.00E+00

5.00E-04

0 10 20 30 40 50 60

Node ID

Dis

pla

cem

en

t

E/E0

/0

/0)p

1

1

p = 1

0.001826 Elements 7-8

Altair Confidential

Iteration 1

= 0.5E0

Node ID v/s Displacement

-0.004

-0.0035

-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0 10 20 30 40 50 60

Node ID

Dis

pla

ce

me

nt

Reduce density by half

For p = 1, E is reduced by half

Find displacements

Max displ doubles to 0.003652

Keep same displacement as baseline

and reduce weight or find where

material is necessary, i.e find the load

path by iterations

Altair Confidential

= 0.2E0

= 0.8E0

= 0.5E0

Node ID v/s Dispalcement

-3.00E-03

-2.50E-03

-2.00E-03

-1.50E-03

-1.00E-03

-5.00E-04

0.00E+00

5.00E-04

0 10 20 30 40 50 60

Node ID

Dis

palc

em

en

t

Iteration 2

Reduce density where deflections are

small, i.e., E, for example as above.

Find displacements

Max displ reduced to 0.002524

Find the improved load path

Altair Confidential

Iteration 3

= 0.1E0

= E0

= 0.8E0

Node ID v/s Diplacement

-2.50E-03

-2.00E-03

-1.50E-03

-1.00E-03

-5.00E-04

0.00E+00

5.00E-04

0 10 20 30 40 50 60

Node ID

Dip

lacem

en

t

The above E (or ) distribution

Gives the same displacement as

baseline

We need a truss like structure with

red elements above and the other

elements can be removed.

Altair Confidential

Function between density and

elasticity using penalty factor p

Density = 1

Density = 0

E/E0

/0

/0)p

1

1

Problem solution using finite elements and

iterative optimization procedure


Altair Confidential

…Engineer defines Requirements… …Altair OptiStruct Creates

Optimal Design Concepts

Building a Bridge…

Design Conceptualization

Altair Confidential

Fundamental Procedure of Topology Optimization

“old” design

design proposal

Altair Confidential

Optimization Methods in OptiStruct

• Automatic selection of best optimization algorithm (all local approximation methods)

- Optimality criteria method (a primal method)

- Convex approximation method (a dual method)

- Method of feasible directions (a primal method)

- Sequential quadratic programming (a primal method)

• The design sensitivity analysis of the structural responses (with respect to the design

variables) is one of the most important ingredients to take the step from a simple

design variation to a computational optimization.

• The design update is computed using the solution of an approximate optimization

problem, which is established using the sensitivity information.

• Among the four optimization methods listed above, the last three are based on a

convex linearization of the design space.

• The dual or primal methods are used depending upon the number of constraints and

design variables. The dual method is of advantage if the number of design variables

exceeds the number of constraints (common in topology and topography

optimization). The primal method is used in the opposite case, which is more

common in size and shape optimizations. However, the choice is made automatically

by OptiStruct.

Altair Confidential

Optimization Methods in HyperStudy

• Method of feasible directions – Local Approximation Method

• Sequential quadratic programming – Local Approximation Method

• Adaptive Response Surface Method – Global Approximation Method

• Genetic Algorithm – Exploratory Method

Altair Confidential

Concept Design Synthesis - Topology Optimization


Geometry Extraction


Design Space and Loads

Size and Shape Optimization

Fine-tuning the Design

Topology

Optimization

Stiffness Material

Layout

Altair Confidential

Pre World War II – Truss Type Designs

Dornier 728

Henkel

To obtain the airfoil lifting surface, early aeronautical structural engineers noticed Truss

type of structures and adopted a similar design for wings

Altair Confidential

Post World War II Wing Designs

Spar location chart for Boeing 7X7 Series; they were evolved from practice

and testing and got standardized in each aircraft

Altair Confidential


Spar location chart for Airbus Jets

Altair Confidential


Spar location chart for DC-Series Jets

Altair Confidential

Average rib spacing for Boeing 7X7 series


Altair Confidential

Average rib spacing for Airbus jets


Altair Confidential

Average rib spacing for DC-series jets


Altair Confidential

Airbus A380 Droop Nose Leading Edge

Wing is over 36m long and 3m thick at its deepest section

14 Wing Ribs vary in Size from

• 3m x 2m x 0.12m to

• 1m x 0.89m x 0.12m

Courtesy of Airbus

Altair Confidential


Geometry Extraction

ICAD Solid

Geometry Extraction


Geometry Extraction

Topology Optimization Stiffness

Material Layout


Buckling and Stress

Topology Optimization Design

Space and Load

Mass Savings of 44% (500kg) 13 Ribs developed in 7 weeks

Concept Design Synthesis - Topology Optimization

Altair Confidential

Topology Derives Conventional Design Practice

13500 mm

4740 mm

Aluminum

Young’s Modulus = 70000 MPa

Poisson’s ratio = 0.33

Density = 2.60E-09 tn/mm3

Given the airfoil needed from CFD – develop the structure from here

Altair Confidential

Aerofoil Section

Chord Length

0% 13% 40% 100%

Chord Length

At Root = 2680 mm

At Tip = 1390 mm

Pressure values were given at 0, 13, 40

and 100 percent of chord length

Altair Confidential

Pressure Distribution

0

2000

4000

6000

8000

10000

12000

0 10 20 30 40 50 60 70 80 90 100

Chord Length (%)

Pre

ssu

re (

N/m

^2)

0 500 1000 1500 2000 2500 3000 3500 4000 4500

5000 5500 6000 6500 7000 7500 8000 8500 9000 9500

10000 10500 11000 11500 12000 12500 13000 13500

Wing Station (mm)

Pressure Distribution on Wing Stations

Vortex Lattice Method

Altair Confidential

0% 13% 40% 100%

Loads – Flight Load

Pressure Distribution – Linear Interpolation

Cruise Conditions

Weight of Aircraft = 22000 Kg

Cruise Velocity = 510 Km/hr

Cruise Altitude = 25000 ft

Altair Confidential

Fuel and Engine Loads

Altair Confidential

Fuel Load = 1730 Kg per Wing

Fuel tank starts from 20% of first panel fuel tank ends at 85% of first panel

C.G location:

1.27 m from leading edge

2.5m wing root

Loads – Fuel Load

Altair Confidential

Loads – Engine Load

Engine weight = 800 kg per engine

Located between 85% to 98% span of first panel

First Rib Second Rib

Front Spar 250 Kg 150 Kg

Rear Spar 250 Kg 150 Kg

Altair Confidential

Iteration 1 Material Density Plot

Nature provides Stalk and Veins to support skins

Optimization follows nature – wants Spars and Ribs

Fuel Tank Region

Altair Confidential

Bird Wings

Altair Confidential

Objective: Minimize Wcomp

Constraints: Vol frac ≤ 0.3

Iteration 1 – Iso Density Plot

Altair Confidential



Extrusion Constraint

Non Design Region

Iteration 2

Altair Confidential



3 Extrusion Constraints

Non Design Region

Iteration 3

Altair Confidential

Topology Optimization of Wing – Phase II

OB Flap Hinges

Aileron Hinges

Lugs

IB Flap Hinges

Proposed Geometry derived from initial topology iterations

Altair Confidential

Material Density Plot - Spars

Side View

Box section for front spar and channel section for rear spar suggested

Altair Confidential

Iso Density Plot for Spars > 0.3

Side View

Box section for front spar and channel section for rear spar suggested

Altair Confidential

Aileron Hinges

OB Flap Hinges

IB Flap Hinges

Iso Density Plot for Hinges > 0.3

Predominantly V-shaped members for hinges, as usually practiced

Altair Confidential

Material Density Plot – Mid Ribs

Altair Confidential

Iso Density Plot – Nose Ribs

Altair Confidential

Iso Density Plot – Mid Ribs > 0.3

Altair Confidential

Iso Density Plot – End Ribs > 0.3

Altair Confidential

Iso Density Plot – Front Ribs > 0.3

Truss pattern for the ribs suggested

Altair Confidential

Final CAD Model Derived for the Wing

Altair Confidential

Composite

• Consists of high strength and modulus fibers embedded in a matrix

or bonded to a matrix. Both the fibers and the matrix retain their

distinct properties and together they produce properties which

cannot be achieved individually.

• The fibers are usually the principal load carrying elements in the

composite and the purpose of the matrix is essentially to keep the

fibers in the desired location.

• The matrix also serves the purpose of protecting the fiber from heat,

corrosion and other environmental damages.

• The fibers may be glass, carbon, boron, silicon carbide, aluminum

oxide etc. One of the popular commercial fiber is Kevlar 49. These

fibers may be embedded in the matrix either in a continuous form or

in discontinuous form (chopped pieces of different lengths).

• The matrix usually consists of a polymer, metal or a ceramic.

Altair Confidential

Laminate

• A laminate consists of several laminae (Plies) each with a fiber

oriented at a particular angle.

• This is generally made by stacking several thin layers of fibers

(Plies) at the desired locations and angles in a matrix and

consolidating them to give the required thickness.

• The fiber orientation in each thin layer can be arranged in a specific

manner so as to achieve the required properties of the structural

member.

• Since the fibers are of high strength in their axial direction and at the

same time very light compared to conventional metals, they find

many applications in aerospace, automobile industry etc.

Altair Confidential

fmffmmff

mmff

c

c

mmffcc

mf

cmmmm

cffff

cmf

vvvv

AAA

AAA

PPP

EE

EE

1

1

0o Laminae – Load Parallel to Fibers

v is volume fraction

fmff

c

c vEvEE 111

12 is (major) Poisson’s ratiommffvv

12

E11 is longitudinal modulus

Altair Confidential

0o Laminae – Load Perpendicular to Fibers

E22 is transverse modulus

21 is (major) Poisson’s ratio

fmmf

mf

vEvE

EEE

22

12

11

22

21

E

E

Altair Confidential

0o Laminae – Shear Load

fmmf

mf

vGvG

GGG

12

Altair Confidential

Orthotropic Material – Stiffness Matrix [C]

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

C

C

C

C

CC

CCC

66

55

44

33

2322

131211

00

000

000

000

Altair Confidential

Orthotropic Material - Compliance Matrix [S]

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

S

S

S

S

SS

SSS

66

55

44

33

2322

131211

00

000

000

000

Altair Confidential

Orthotropic Material - Compliance Matrix [S]

xyxzyz

zz

yy

yz

yy

xx

xz

xx

xy

xx

GS

GS

GS

ES

ES

ES

ES

ES

ES

1

1

1

1

1

1

665544

33

2322

131211

Altair Confidential

Specially Orthotropic Lamina

12

665544

33

23

22

22

13

11

12

12

11

11

1 0 0

0

0 1

0 1

GSSS

S

SE

S

SE

SE

S

12

22

11

12

22

11

12

11

12

22

11

0

0

0

1

0

000

0000

00001

00001

0

0

0

G

E

EE

Altair Confidential

Angle Ply Lamina

sin

cos

n

m

Altair Confidential

Angle Ply Lamina

xy

xx

yy

yx

xxxy

xy

yy

xx

E

E

nmGEEE

E

nmGEEGG

E

mnm

EGE

n

E

E

nnm

EGE

m

E

22

122211

12

11

12

22

122211

12

12

22

4

22

11

12

1211

4

22

4

22

11

12

1211

4

1121

11214

11

211

211

Altair Confidential

Angle Ply Lamina – Stress Strain Relations

Altair Confidential

Angle Ply Lamina – Stress Strain Relations

xy

yy

xx

xy

yy

xy

yy

xx

xy

xx

xy

xx

xy

yy

xx

G

EE

EEE

1

1

1

mnnmEGE

nm

E

mnE

mnnmEGE

mn

E

nmE

xx

xy

xyyyxx

yyyx

xx

xy

xyyyxx

xxxy

22

33

22

33

2122

2122

Altair Confidential

Composite Optimization Stages

• Free-sizing optimization

• Sizing optimization

• Ply-stacking optimization

Altair Confidential

Free Sizing

Composite free-sizing optimization is to create design concepts that utilize all the potentials

of a composite structure where both structure and material can be designed simultaneously.

By varying the thickness of each ply with a particular fiber orientation for every element, the

total laminate thickness can change ‘continuously’ throughout the structure, and at the same

time, the optimal composition of the composite laminate at every point (element) is achieved

simultaneously.

Manufacturing constraints like lower and upper bound thickness on the laminate, individual

orientations and thickness balance between two given orientations can be defined.

Altair Confidential

• Free size optimization was performed to

determine the thickness and laminate

family within the chosen regions of the

CAD model.

• The wing is divided in to regions for size

optimization as shown.

• Objective functions in this optimization

are mass and compliance and

minimization of the same.

Free Size Optimization

Altair Confidential

Design Space: Ribs Material

Suggested minimum number of layers with orientation

Altair Confidential

CFRP Properties

Symbol Units Properties

E E1 N/mm2 1.52×105

E E2 N/mm2 9650

G G12 N/mm2 5930

n NU12 0.321

r Rho Tonne/mm3 1.53×10-9

Altair Confidential

Phase I - Concept: Free-Size or Topology optimization

Wing Ribs – Ply1

Wing Ribs – Ply4Wing Ribs – Ply3

Wing Ribs – Ply2

Max Thickness 10.16 mm Max Thickness 10.16 mm

Max Thickness 10.16 mm Max Thickness 10.16 mm

Altair Confidential

Sizing Optimization

Sizing optimization is performed to control the thickness of each ply bundle, while

considering all design responses and optional manufacturing constraints. Ply thicknesses are

directly selected as design variables.

Composite plies are shuffled to determine the optimal stacking sequence for the given design

optimization problem while also satisfying manufacturing constraints like control on number

of successive plies of same orientation, pairing 45° and -45° orientations together etc.

Altair Confidential

Level setting Ply-Bundles (+45°/-45° plies)Level setting Ply-Bundles (+0° plies)

Level setting Ply-Bundles (+90° plies)

0

90

45

-45

Illustration of Super-Ply Ply-Bundle Sizing optimization

Bundle 1 Bundle 2

Bundle 3 Bundle 4

Altair Confidential

Phase 2: Ply-Bundle Sizing with Ply-Based FEA

modeling

• The size optimization of the Plies obtained in Phase 1 with Free Size

or Topology optimization was next performed to determine the

thickness and laminate family within all the Ribs, Skin and Spars.

• Objective functions are again mass and compliance and

minimization of the same.

Altair Confidential

Phase II - System: Ply-Bundle Sizing optimization

Super Ply thickness Angle +45oSuper Ply thickness Angle 90o

Super Ply thickness Angle -45oSuper Ply thickness Angle 0o

Max Thickness 3.81 mm




Altair Confidential

Max Thicknesses mm of Super-Plies in Ribs

Super Ply Angle Max Thickness

00 3.81

450 2.54

-450 2.54

900 5.08

Altair Confidential

So far

• A review of the design process up to now reveals that we

established the optimum ply shape and patch locations in phase 1

(free size optimization) and subsequently optimized the ply bundle

thicknesses in phase 2 (ply bundle sizing optimization), allowing us

to determine the required number of plies.

• These ply bundles represent the Optimal Ply Shapes (Coverage

Zones).

Altair Confidential

Optimal Ply Shapes (Coverage Zones): 0 Deg

Ply thickness Angle 0o

Altair Confidential

Ply thickness Angle -45o

Optimal Ply Shapes (Coverage Zones): -45 Deg

Altair Confidential

Ply thickness Angle +45o

Optimal Ply Shapes (Coverage Zones): +45 Deg

Altair Confidential

Ply thickness Angle 90o

Optimal Ply Shapes (Coverage Zones): 90 Deg

Altair Confidential

Optimal Ply shapes of Super-Plies in Ribs

Super Ply

Angle

Max

Thickness

Optimal Plies

00 3.81 3×1.27+1×0

450 2.54 2×1.27+2×0

-450 2.54 2×1.27+2×0

900 5.08 4×1.27

Altair Confidential

Total Optimal Ply Shapes and thickness: Ribs

Altair Confidential

Optimal Ply shapes of Super-Plies in Ribs

Super Ply

Angle

Max

Thickness

Optimal

Plies

Total Max

Thickness

00 3.81 3×1.27+1×0 3.81

450 2.54 2×1.27+2×0 2.54

-450 2.54 2×1.27+2×0 2.54

900 5.08 4×1.27 5.08

Altair Confidential

Composite free size/sizing Optimization

Smeared Superply LevelSuperply Level

0°

+45°

-45°

90°

0°

+45°

-45°

90°

Optimized Ply-Bundle

* * *

• Baseline model with Super ply

• Free Size Optimization (FSO)

• Ply-Bundle Size Optimization

• Final Laminate stacking

Phases

Nondimensional FS optimization Dimensional

Altair Confidential

Illustration Final Laminate stacking

• Symmetric Stack Required

• Number of plies in any one direction placed sequentially in the stack is limited

• Stack is balanced, i.e. the number of 45º and -45º plies is the same.

• Outer plies for the laminate should contain a particular ply (ie. ±45º )

• Minimize the number of occurrences of the 0° to 90° (or 90° to 0°) change in any two adjacent plies.

• Minimize the number of occurrences of 45° to -45° (or -45° to 45°) change

inside the stack by putting one 0° or 90° ply between them

Typical Ply Level Manufacturing Rules

Sized Ply Bundles

Algorithm based Conversion

Hyper Shuffle

Into local ply configuration

45 -45 0 0 0

45 -45 45 90

90 -45 45 -45 0 0

-45 90 45

(a) Ply Level (b) Superply Level

Final Laminate Stacking

0°

+45°

-45°

90°

Optimized Ply-Bundle

Altair Confidential

45°

-45°

0°

90°

PCOMP

STACK

Free-sizing

optimization

Sizing optimization Ply stacking optimization

OU

TP

UT

,FS

TO

SZ

OU

TP

UT

,SZ

TO

SH

Pre Design Detailed Design Validation

Phase III – HyperShuffle for detailed stacking design

Altair Confidential

Hyper Shuffle for detailed stacking design

45°

-45°

0°

90°

Altair Confidential

Optimized thickness for one of the ribs

Total ply thickness in the ribs

varies from 2.54 to 13.97 mm

Mass saving is around 35.7% for the Wing

For the Full Wing Structure:

Optimized Composite mass~1350kg

Original Al mass ~2100kg

Altair Confidential

Phase I Free Sizing optimization of Spars

Ply thickness Angle 0o Ply thickness Angle -45o

Ply thickness Angle +45o Ply thickness Angle 90o

Altair Confidential

Phase II Size Optimization of Spars



Altair Confidential

Phase II – Total Size Optimization of Spars

Altair Confidential

Phase I - Concept: Free-Size Topology Optimization

Skin with Integrated StringersPly thickness Angle 0o Ply thickness Angle -45o


Altair Confidential

Phase II – Size Optimization - Skin with Integrated Stringers



Altair Confidential

Phase II – Total Size Optimization of Skin with Stringers

Altair Confidential

Blade Root Shape Optimization

X

Y

Z X

Y

Z X

Y

Z

Blade

Wheel

LETE

Cyclic Symmetry

Boundary Conditions

Pressure faces

Pre-twisted 290 mm long 60 blades

Blade root bottom radius 248 mm

Using mapped meshing options a

solid element mesh with 8 nodes is

generated, by capturing all the

critical regions with finer mesh.

Mesh around the singularities, blade

and disc dovetail root fillet regions

at higher radius, where the peak

stresses are expected are captured

with 2 to 3 layers of elements with

element size as low as 0.235 mm.

305524 elements and 344129 nodes.

Altair Confidential

1825 MPa at node

153608 Average

Stress 256 MPa

Elastic Analysis

Stress Concentration Factor 7.1289

sector of the disk with one

blade Common nodes on the

pressure faces at six positions,

where the load transfer between

blade and disc takes place are

joined together. The blade and

disc are assumed to be made up

of same material with Yield

stress of 585 MPa, Young’s

Modulus 210 GPa, Density 7900

Kg/m3 and Poisson’s Ratio 0.3.

Stress contour beyond yield

across 3 elements over a depth of

1.22 mm.

60

1

Altair Confidential

Elasto Plastic Analysis

Peak von Mises stress 768 MPa

at a node 176017 in the same

region which is beyond the yield

value 585 MPa. Plastic region

has not changed from the simple

elastic analysis result.

Peak stress value dropped from

1825 MPa to a value 768 MPa

just above the yield.

The material in the root region

around the stress raiser location

flows thus easing the stress and

raising the strain in accordance to

the hardening law given.

The peak strain observed at the

node 153608 in the same region

closer to peak stress location is

0.0153.

0

100

200

300

400

500

600

700

800

900

1000

0 0.002 0.004 0.006 0.008 0.01 0.012

Strain

Str

ess (

MP

a)

EP

Elastic

768 MPa

Average Stress 216.86 MPa

Altair Confidential

Altair OptiStruct Shape Optimization

Shape variables are generated using mesh morphing

technique in Altair HyperMesh. Using the baseline

finite element model a suitable number of shapes in

the vicinity of baseline consistent with the available

design space is defined by modifying the grid point

locations, which are saved as perturbation vectors.

The shapes generated are combinations of

parameters shown. Shapes are then defined as

variables by assigning the lower and upper bounds

to it. Table gives the minimum and maximum

values adopted for defining the shape variables.

Shape variables can then be assigned as indicated to

perturbation vectors, which control the shape of the

model within a given bound. This is helpful in

generating the required shape bounds without re-

meshing the model. A shape optimization is then

carried out using OptiStruct code.

R2

R1

θ

W1

H

VW2

R2

R1

θ

W1

H

VW2

Minimum Value (mm) Maximum Value (mm)

W1 = 22.17 W1 = 25.76

W2 = 13.65 W2 = 13.86

R1 = 1.70, H = 5.67,

V = 4.13, R2 = 4.0

R1 = 2.14, H = 4.85,

V = 4.06, R2 = 3.37

θ = 29.860

θ = 16.250

Altair Confidential

Mesh Morphing with Design Variables

Altair Confidential

Mesh Morphing with Design Variables

Altair Confidential


OptiStruct uses an iterative procedure known as the local approximation method to

solve the optimization problem. The design update is computed using the solution

of an approximate optimization problem, which is established using the sensitivity

information. OptiStruct has three different methods implemented: the optimality

criteria method, a dual method, and a primal feasible directions method. The primal

method is more common in shape optimizations. This approach is based on the

assumption that only small changes occur in the design with each optimization

step. This method determines the solution of the optimization problem using the

following steps:

1. Analysis of the physical problem using finite elements.

2. Convergence test, whether or not the convergence is achieved.

3. Design sensitivity analysis.

4. Solution of an approximate optimization problem formulated using the

sensitivity information.

5. Back to 1.

Altair Confidential


323

MPa

1466

MPaBaseline Configuration

(mm)

Optimized Shape (mm)

4000RPM

Optimized Shape (mm)

8500RPM

W1 = 22.17 W1 = 25.72 W1 = 25.72

W2=13.65 W2 = 13.86 W2 = 13.86

R1 = 1.70, H = 5.67,

V = 4.13, R2 = 4.0

R1 =1.95, H = 5.05,

V = 4.03, R2 = 3.97

R1 = 2.10, H = 4.91,

V = 4.04, R2 = 3.98

θ = 29.860

θ = 16.250

θ = 16.250

Stress Concentration Factor 5.73 as

against 7.1289 of baseline. This

reduction enhances life considerably

Optimized Shape

Baseline Shape

Region of

interest

Altair Confidential

Optimization through Altair HyperStudy

Altair HyperStudy is a multipurpose

DOE/Optimization/stochastic tool that

can be applied in the multi-disciplinary

optimization of a design combining

different analysis types. Once the finite

element model and shape variables are

developed, an optimization can be

performed by linking HyperStudy to a

particular solver of choice that can

include nonlinear analyses.

HyperStudy uses global optimization

methods which is very general in that

they can be used with any analysis

code, including non-linear analysis

codes.

Global optimization methods use higher order

polynomials to approximate the original

structural optimization problem over a wide

range of design variables. The polynomial

approximation techniques are referred to as

Response Surface methods. A sequential

response surface method approach is used in

which, the objective and constraint functions

are approximated in terms of design variables

using a second order polynomial. One can

create a sequential response surface update by

linear steps or by quadratic response surfaces.

The process can also be used for non-linear

physics and experimental analysis using wrap-

around software, which can link with various

solvers.

Altair Confidential


Shape optimization is carried by using the

baseline model, having the cyclic symmetry

boundary conditions imposed on the disc, with

the objective to minimize the peak stresses.

Shape variables generated in the previous

section are used as design variables.

Optimization is carried out at two speeds,

4000 and 8500 RPM, with elastic properties.

Maximum stress has decreased from 404 to

325 MPa by 79 MPa (19.5% as against 20%

achieved in Optistruct) from baseline for 4000

RPM and from 1825 to 1501 MPa by 325 MPa

(17.7% as against 19.5% achieved in

Optistruct) for 8500 RPM from complete

elastic analysis. Note that the analysis in this

study is more accurate as the cyclic symmetry

conditions are directly applied instead of cut

boundary conditions in Optistruct analysis.

325

MPa

1501

MPa

Altair Confidential


Variation of Objective and Shape variables

during optimization

Optimized Shape

Baseline Shape

Region of

interest

Objective

value

Shape

Variables

Altair Confidential

EP Optimization through Altair HyperStudy 8500 RPM

Maximum stress decreased marginally from 768 to 746 MPa by 22 MPa (2.86%) from

baseline elasto-plastic analysis for 8500 RPM; however the peak plastic strains reduced

from 0.0153 to 0.01126 by 26.4%. This is the major advantage in optimization for a blade

root shape. Many existing machines have roots designed by experience and there can be

considerable margin in lowering peak strains and therefore enhanced life.

• Von Mises Stress 746 MPa

Altair Confidential

Fatigue Stress Concentration Factor

7.5256

1466

Nominal

Max S

SKt

r

KK

N

tf

1

11

N

6.4fK

ω = flank angle = 30 deg (At fillet region)

r = notch radius = 1.75 mm (At fillet region)

= Neuber’s parameter = 0.12 mm

Altair Confidential

Nominal Stress Range

S

Peak steady stress = 0.9547 MPa

Peak dynamic stress = 0.9547 x 221.239

= 211.2168 MPa

Average von Mises stress = 0.13373 x 221.239

= 29.586 MPa

Alternating nominal stress range = 2 x 29.586

= 59.1725 MPa

Altair Confidential

Local Stress and Strain and Life

'/12

'2

12

1n

f

KEE

SK

E

SK f

2

= 270 MPa

= 0.00125

cif

b

imf NNE

221

2

1 ''

Ni = 8.7E+06 cycles

At 344.50707 Hz life of the blade is 420.9 minutes which is 4.14 times the base line

Altair Confidential

Weight Optimization – CAD Model

Altair Confidential

Blade and Disk Separated

46970 Solid 45 elements

41811 nodes.106066 Solid 45 elements

121948 nodes.

Altair Confidential

Surfaces constrained

using constrained

equations so as to

have cyclic symmetry.

This surface is constrained in all dofs.

ωz =

1156.62 rad/sec

Boundary Conditions

Altair Confidential

Max. Von Mises Stress

787 MPaMax. Von Mises Stress

721 MPa

Baseline Results

Altair Confidential

R

Blade Root and Shank

8 holes radius 1.75 mm provided in root and two cutouts in shank are allowed to

reduce the weight.

The blade root without any cutouts was 7890.34 mm3 and chosen as Objective

function and minimized subject to the peak stress limited to yield 820 MPa.

Altair Confidential

W1b

D1W1a

Variable Range mm

D1 1.1 to 2.0

D2 1.1 to 2.0

W1a 4.75 to 6.5

W1b 13.94 to 15.5

W2a 4.44 to 6.0

W2b 13.64 to 16.0

R 1.75 to 2.25

Design Variables

Altair Confidential

Variable Value mm

D1 1.64

D2 1.64

W1a 5.09

W1b 15.35

W2a 5.57

W2b 14.56

R 2.0

Optimization

Altair Confidential


810 MPa


798 MPa

Optimized Results

Volume decreased from 7890.34 to 7098.93 mm3,

i.e., a reduction of 10.03%.

Altair Confidential

Thank You

Any Questions?

Date post:	02-Apr-2015
Category:	Documents
Upload:	premkumar165
View:	43 times
Download:	0 times

2011_Gitam

Documents