The Karush–Kuhn–Tucker optimality conditions in minimum weight design of elastic rotating disks with variable thickness and density Pages 765-774 Download PDF

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International Journal of Industrial Engineering Computations

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The Karush–Kuhn–Tucker optimality conditions in minimum weight design of elastic rotating disks with variable thickness and density

Sanaz Jafari

Department of Mechanical Engineering, Babol University of Technology, P.O.Box 484, Babol, Iran

A R T I C L E I N F O A B S T R A C T

Article history: Received 1 May 2010 Received in revised form June, 20, 2011 Accepted 21 June 2011 Available online 22 June 2011

Rotating discs work mostly at high angular velocity. High speed results in large centrifugal forces in discs and induces large stresses and deformations. Minimizing weight of such disks yields various benefits such as low dead weights and lower costs. In order to attain a certain and reliable analysis, disk with variable thickness and density is considered. Semi-analytical solutions for the elastic stress distribution in rotating annular disks with uniform and variable thicknesses and densities are obtained under plane stress assumption by authors in previous works. The optimum disk profile for minimum weight design is achieved by the Karush–Kuhn– Tucker (KKT) optimality conditions. Inequality constrain equation is used in optimization to make sure that maximum von Mises stress is always less than yielding strength of the material of the disk.

Rotating annular disk Variable thickness Variable density Optimization

Karush-Kuhn-Tucker method

1. Introduction

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76 o (V & ad th T p k p ar 2 2 A (F S Ja d d w In (T σ

σ

w ci d 66 ptimization Vanderplaat & Campbell,

ddition, in a he finite elem The main obj

revious wor kept under th

arameters th re discussed . Theoretica .1Governing Assuming tha Fig.1), varyi Simplifying t afari, 2008;

( r)

d

hr h

dr σ − σθ

where ρ =ρ(r

n the plane Timoshenko ⎢⎣ ⎡ − = 1 E 2 r μ μ σ 2 1 E θ

σ

μ

⎡ = _⎢ − ⎣

where

u

is ircumferenti denotes differ

process. Fo s, 1990) als

1973) used a different ap ment method bjective of th rks to minim he allowabl hat fulfill the d in details in

al Backgrou

g equation o

at stresses d ing with radi

the radial eq Timoshenko

2 2_,

h r

θ = − ρω

)

r isthe distr stress (

σ

z =

o, 1970; Güv

⎥⎦ ⎤ ′ +u r u μ , , u u r

μ

⎡ ₊ _′⎤ ⎥ ⎣ ⎦

the radial d ial compone rentiation w

ox (1970) o o used the g sequential l pproach, Ch d with the fe

his work is mize the wei le value. In

ese constrain n (Rao, 2009

und

of rotating d

do not vary o ius, i.e. h=h

Fig. 1. Disk quilibrium co

o, 1970; Güv

ribution func 0

= ) the stres ven, 1995):

displacemen ents of the no with respect t

optimized a gradient of c linear progra hen Cheu (1

asible direct the utilizati ght of the d fact, the fo nts using KK 9; Peressini e

disk

over the thic )

(r h .

k profile and ondition for ven, 1995):

ction of dens ss-displacem

nt,

E

is the ormal stress

o the radius

rotating di constraints. amming met

990) optimi tion method. on of the so disk under a ocus of this KT condition et al., 1988;

ckness of th

acting force an infinitesi

sity, and ω i

ment relation

e modulus and

μ

is the . The disk is

isk using th Zienkiewicz thods for opt zed a rotatin .

olutions ach certain cond s research is

ns for optim Wide et al.,

he disk, let h

es on an elem imal elemen

is the angula ns are given

of elasticity Poisson’s ra s symmetric

he feasible z and Campb

timization o ng disk prof hieved by the

dition of vo s to identify mization. The

1970).

h be the thic

ment

nt of the disk

ar velocity of by Timoshe

y,

σ

rand σ

atio. In the E with respect

direction m bell (Zienki f rotating di file by comb

e authors in n-Mises stre y those geom e KKT cond

ckness of the

k gives (Hojj

f the disk. enko and Go

θ

σ are radia Eqs. (2-3) a p

(3)

and its profiles for thickness and density are assumed to vary as functions of radius (r) (Hojjati & Jafari, 2007; Hojjati & Hassani, 2008; Hojjati & Jafari, 2008; Hojjati & Jafari, 2009):

n

b r h r

h₍ ₎₌ ₍ ₎−

D ,

(4)

m

b r r) ( ) ( ρ_D

ρ = , (5)

where

n

and

m

are geometric parameters (0≤n≤1,m≥0), b is the outer radius of the disk and hD is the thickness of the disk at r=b. Rotating disks with variable density can be considered as functionally graded materials (FGM) and the literatures devoted to this field is rich (Gamer, 1985; Güven, 1992; Bayat et al., 2008). Although, other material properties such as modulus of elasticity can also be assumed to vary with disk radius, in this research it is arbitrary decided to assume that Young’s modulus is constant as the case reported in the previous studies (Gamer, 1985; Güven, 1992; Hojjati & Jafari, 2008; Hojjati & Jafari, 2009). Substitution of Eqs. (2-5) in Eq. (1) yields the governing equation for the radial displacement:

2

2 2

0

2 2

( 1) ( 1)

( ) ( ) ( ) ( )m (1 ) /

d n d n r

u r u r u r r E

dr r dr r b

μ _ρ _ω _μ

− + +

+ − = − − . (6)

This is the differential equation governing on rotating disks in elastic form. In this research we have to use the results of the pervious works by the authors (Hojjati & Jafari, 2008; Hojjati & Jafari, 2009) for starting the disks optimization. Therefore solution of the elastic non-uniform thickness and density rotating disk (Eq.6) obtained by Homotopy perturbation method is derived as (Hojjati & Jafari, 2008):

) n 3 n 8 m ) 6 n ( m (

) 1 ( ) b r ( r r

c r

c ) r (

u ₂

2 m 3 2 2

n 4 4 n 2 n

2 2

n 4 4 n 2 n

1

2 2

+ + − − + −

− −

+

= ⎟⎟

⎟ ⎠ ⎞ ⎜⎜

⎜ ⎝ ⎛ ₊ ₊

− ⎟⎟

⎟ ⎠ ⎞ ⎜⎜

⎜ ⎝ ⎛ ₊ ₊

+

μ Ε

μ ω

ρ μ

μ

D

,

(7)

where , are integration constants. The solution must be completed by incorporating a certain set of boundary conditions. Here, free boundary conditions are considered for annular disks. If both surfaces at inner and outer radius are free of any traction, then the boundary conditions are

0 ) ( )

(a = _r b =

r

σ

.

2.2KKT Conditions

The necessary conditions for equality constrain that contained the Lagrange Multiplier theorem generally discussed in textbooks on calculus (Rao, 2009; Peressini et al., 1988; Wide et al., 1970). Then, the necessary conditions for the general constrained problem are obtained as an extension of the Lagrange Multiplier theorem. These are known as KKT necessary conditions. All optimum designs must satisfy these conditions. We now examine the problem for this optimization with equality and inequality constraints in the standard form:

)

(

min

f

x

subject to ( ) 1, ,

( ) 1, ,

i i

j j

h x b i m

g x c j r

= =

≤ =

"

(8)

In these relations x=

(

x1, ," xn

)

′ is the vector of design variables,f(x) is an object function,

j j x b

h ( )= is an equality constraint and gj(x)≤cj is inequality constraint, respectively and they are

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768

constraint surface, the associated constraint is called an active constraint. At constrained minimum, only the active constraints are significant. Hence the first- order optimality condition (Eq. (9)) must be applied to inequality constraint (Peressini et al, 1988):

. 0 , 0 , 0 ) ( ) ( + ∇ = ≥ =

∇_f _x∗

_λ

T _g _x∗ T

_λ

T_g

(9) Those constraints satisfy Eq. (9) are known as active constraint and the necessary conditions are known as KKT conditions as follow (Rao, 2009),

. 0 , 0 , 0 , 0 . 2 . 0 ) ( , 0 ) ( . 1 = ≥ ≠ = ∇ + ∇ + ∇ ≤ = ∗ ∗ ∗ ∗ ∗ g u where g h u f x g x h T T T T j i

λ

(10)

where u=

(

u₁,....,u_m

)

_{is the vector of a Lagrange multiplier for equality constraints and}

(

1

,....

r

)

λ

=

λ

′

is the vector of a Lagrange multiplier for inequality constraints. Here

x

∗, the minimizer, is assumed to be a regular point. A point that satisfies the KKT conditions is called a KKT point and may not be a minimum since the conditions are not sufficient. Second-order information is necessary to verify the nature of a KKT point. The second-order sufficiency conditions are as follow: If a KKT point x∗exists, such that the Hessian of the Lagrangian on the subspace tangent to the active constraints is positive definite atx∗, then

x

∗ is a local constraint minimum. We have defined Hessian matrix as the second partial derivatives of

f

(

x

)

as follow (Rao, 2009):

int 0 tra activecons for j j L g g J x x XX T j j x x = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∇ ∇ = ∗ ∗ = = (11) where,

n

j

n

i

x

L

x x j i

xx

1 ....

,

1 ....

2

=

⎥

⎦

⎤

⎢

⎣

⎡

∂

=

∗ = (12)

We summarized these results here for the sake of simplicity, Model: min

f

subject to

0 , 0 ≤ = g

h

Lagrangian function:

L

₌

f

₊

u

T

h

₊

_λ

T

g

(13) KKT conditions: 0 , 0 , 0 0 , 0 . = ≥ ≠ ≤ = = + ∇ + ∇ g u g h o g h u f T T T T

λ

(14)

Then for Eq. (8), the Lagrangian function equals to:

(

)

_∑

(

)

∑

= = − + − + = r j j j j m i i i

i h x b g x c

u x f u x L 1 1 ) ( ) ( ) ( ) , ,

( λ λ (15)

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1 1

*

( , , ) ( ) ( ) ( ) 0

0, ( ) 0

( ) , 1, , .

m r

x i i j j

i j

j j j j

j j

L x u f x u h x g x

g x c

g x c j r

λ

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

= =

∗ ∗ ∗

∇ = ∇ +∑ ∇ +∑ ∇ =

⎡ ⎤

≥ _⎣ − _⎦=

≤ = "

(16)

Simultaneous solution of the above equations provides the calculations ofx∗.

3. Weight optimization of rotating disk

3.1Weight of annular disk

The weight of disk can be written as weight= w= ×γ Vwhere Vis volume of annular disk and

g

ρ

γ

=

is the specific gravity of disk where g is the gravity constant.

V

w

weight

=

γ

*

The volume of the body results from rotating the region given in Fig. 2 about vertical axis is:

(

)

∫

−

=

b

a

r

f

r

g

r

dr

V

2 π

(

)

(

)

(17)

where,

( ) ( ) ,

( ) ( ) , 0.1, ,0.6

n

r f r h r h

b r

g r h r h r

b

−

⎛ ⎞

= _{= ⎜ ⎟}

⎝ ⎠ ⎛ ⎞

=− =− _{⎜ ⎟} =

⎝ ⎠

D

D "

(18)

Fig. 2. Disk profile for n=1, a=0.1m, b=0.6m Using these relations, the volume of an infinitesimal element can be found as:

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

radial coordinate (m)

th

ic

kn

es

s (

m

(6)

770

4 .

n

r

dV

h

b

π

−

⎛ ⎞

=

_{⎜ ⎟}

⎝ ⎠

D

(19)

Specific density of disk ρ(r)is assumed to vary with r and it is defined as:

m

b

r

⎟

⎠

⎞

⎜

⎝

⎛

=

ρ

_D

ρ

(

)

(20)

where

ρ

Dis the density at inner radius i.e. r=b. Thus, the specific gravity of the disk equals to:

( )

.

m

r

g

b

γ

=

ρ

⎛ ⎞

_{⎜ ⎟}

⎝ ⎠

D

(21)

By using Eq. (19) and Eq. (21) and setting 9.81 / , the weight of the disk corresponds to:

(

)

⎟⎟

⎠ ⎞ ⎜

⎜ ⎝ ⎛

× × ×

+ ×

−

× −

+ − −

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟

⎠ ⎞ ⎜ ⎝ ⎛ =

=

+ − − +

− − =

=

−

∫

n m n m m n

m m

n r

r

n m

dr b r h r g b r dV

r w

5 3 66667 . 1 36 10

166667 . 1

66667 . 1 22676 . 976 2 1

4 )

( 0.6

1 .

0 ρD π D

γ (22)

3.2Governing constraint of rotating annular disk

That constrain used in this optimization is that the disk must remain in elastic region based on von-Mises criteria. In a plane stress rotating disk von von-Mises stress corresponds to:

θ θ

σ

_e

=

_r2

+

2

−

_r ₍₂₃₎

Those stresses components exist in Eq. (23) are evaluated by using Eqs. (2-3) after substitution of Eq. (7 ) for radial displacement (u). The final relation for von-Mises stress does include the parameters n

and m used in Eqs. (4-5).

3.3Preparing weight and constraint functions for disk weight minimization

The optimization problem is reduced to minimization of objective function W_disksubject to the stress constraint, whose complete statement is expressed as,

min subject to

disk

e W

σ

≤

σ

_D

The differences in scale of the objective function and the constraint may cause some difficulties. Hence, they should be normalized. The weight is normalized by dividing it over the disk at maximum weight conditions i.e. for m=0 and n=1.

0 , 1

) , (

= =

=

m n

disk W

W m n

W (24)

Equivalent von Mises stress can be dimensionless by dividing it by yield strength of the material:

D

σ

e

(7)

From Fig. 3, it is obvious that for different values of m and n parameters, von Mises stress has its maximum value at inner radius i,e. at r=0.1 m in our case study. Therefore, the Eq. (25) can be reduced to the following equation which has no terms of r:

( )

D

σ

₍ _, ₎₌ e r=0.1

e n m (26)

The angular velocity of rotating disk can also be normalized as (Hojjati & Jafari, 2009):

D

σ

ρω

2_b2

=

Ω (27)

Fig. 3. Equivalent von Mises stresses on annular rotating disk with variable thickness and density for free boundary condition (m=1 and different value of at 1.5346 normalizd angular velocity)

4. Case Study

To show the performance of KKT optimization method in weight minimization of a rotating disk with variable thickness and density, application of the proposed method is presented here for two cases. The first case is when σ_e ≤σD, is assumed as the constraint and the other case is when σe ≤0.8 ,σD is used as failure stress

criterion. The later means that we wish to include a safety factor in the design, for instance, making the maximum von Mises stress as 0.8 times of yield strength of the material. Table (1) shows the material and the geometry properties used in the numerical examples (Jafari & Hojjati, 2011).

Table 1

Geometry and material properties of rotating disks

b (m) a(m) h_D (m) ρD (kg/m 3₎

E(GPa) υ ω (rad/s) σ_D_(MPa)

0.6 0.1 0.1 7850 207 0.3 500 300

The optimization problem for the two constraints in normalized form state as follow:

min

W

disk

subject to σ_e −A≤0,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

radial coordinate

equi

va

lent

s

tr

es

s

(8)

772

where A is a real number that has value of 1 for the first constraint as σe ≤σD and 0.8 for second constraint as σ_e ≤0.8σ_D in normalized form (Jafari & Hojjati, 2011).

The optimization problem with KKT conditions is formulated as,

minimize

W

(

n

,

m

)

(28) subject to σ_e

(

n,m

)

≤ A (29) Lagrangianfunction:

(

A

)

W

m

n

L

(

,

λ

)

=

+

λ

σ

_e

−

(30) KKT conditions:

(

−

)

=

0 ∇

+

∇

W

λ

σ

_e

A

(

)

0

,

0 − =

≥

λ

σ

e A

λ

(31)

These relations can now be summarized as:

(

)

0, 1 0.8 ,

0 , 0 ,

0 ,

0 W A A or

m W n

W

e − = =

≥ = =

= λ λσ

δλ δ δ

δ δ

δ ₍₃₂₎

Table 2 presents the results for the first constraint case while Table 3 shows the same results for the second constraint case, i.e. A=1. All the presented results are for Ω =1.5346.

Table 2

The optimal results for disk weight optimization under σ_e ≤σ_D constraint with A=1

Value of n Value of m Lagrange multiplier Weight reduction

KKT 0.43069 0.88913 0.25942 68%

Table 3

The optimal results for disk weight optimization under σe ≤0.8σ_D constraint with A=0.8

Value of n Value of m Lagrange multiplier Weight reduction

KKT 0.73873 0.85014 0.43020 61%

The values of

λ

for both cases are positive, which confirm that the calculated optimum point gives the minimum weight of disk. Based on Eq. (11), one can realize that the Hessian matrix for the above values is positive definite. Therefore, the sufficient condition is satisfied and these values minimize weight of the disk for given constraint.

5. Discussions

As it was mentioned before, KKT optimization method has been used for weight minimization of a rotating disk with non uniform thickness and material properties. For the first case of constraint defined as σe ≤σD , Fig. 4 shows the disc profile before (n=1) and after optimization. Using the

optimization method reviewed here, after design/optimization process, a weight reduction of 68% and maximum von Mises stress of 300 MPa (equal to the yield stress) was achieved. In Fig. 5 von Mises stress distribution for different radii has been shown for the optimized values of m and n. The obtained disk profile for the second case of constraint defined by σe ≤0.8σDis shown in Fig. 6.

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se ex

6

In m st A th g ra th

een that the xpected.

Fig. 4

n=0.4

Fig. 6

n=0.7

. Concludin

n this paper minimum we tress. The L A plane stres he authors h

eometrical p adii have bee he maximum

von Mises s

4. Thicknes 43069 under

6. Thickness 73873 under

ng remarks

r, rotating a eight subject

agrange mu ss condition

has been us parameters o

en found wh m von Mises

stress throug

s profilefo

D

σ

σe ≤ c

profile for

σ σ_e ≤0.8

annular disk t to fulfillme ltiplier along

has been as sed to evalu of the shape hich reduces stress is sati

ghout the dis

or n=1 and constraint

n=1 and

D

σ constraint

ks of variab ent of certain

g with KKT sumed. Sem uate the mag e of the prof

the weight o isfied.

sk thickness

Fig

m= velo

t

F

m v

ble thicknes n condition T conditions mi exact solu gnitude of t file of a giv of the disk b

0.1 0

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

eq

ui

val

en

t s

tres

s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

eq

ui

va

le

nt

s

tr

es

s

has been ke

g. 5. von Mi =0.88913 at

ocity under

Fig. 7. von M m=0.85014 a velocity unde

s and densi on the value theorem hav ution of the d the stress c ven disk wit by as much a

0.2 0.3 0.4

r

0.2 0.3 0.4

ept under the

ses stress fo 1.5346 no

D

σ

σ_e ≤ co

Mises stress fo at 1.5346 norm er σ_e ≤0.8σ

ity have be e of the max ve been used disk previou components

th predefined as 68% while

0.5 0.6 0.7

adial coordinate n=

4 0.5 0.6 0

radial coordinate n

e desired val

or n=0.43069 ormalizd an

onstraint

or n=0.73873 malizd angul

D

σ constraint

en optimize ximum von M

d in optimiz usly develop of the disk d inner and e the conditi

0.8 0.9

=0.43069,m=0.88913

0.7 0.8 0.9

n=0.73873,m=0.85014

lue as

9 and ngular

and ar t

ed for Mises zation. ped by k. The

outer ion on

1

(10)

774

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