Continuum structural topological optimization with dynamic stress response constraints

Bin Xu1*, Lei Zhao1 and Wenyu Li1

1)

School of Mechanics, Civil Engineering & Architecture, Northwestern Polytechnical University, Xi’an 710072, PR China

Abstract

Stress related problem is of great importance to structure optimization, however ,most works were paid attention to static stress constrained topology optimization,only a limited number of works have been devoted to the topology optimization of the structures with dynamic stress response requirements under random vibration.This paper deals with topology optimization of continuum structures with dynamic stress response constraints.

Based on the ICM method and the bi-directional evolutionary structural optimization method,a method for the topology optimization of continuum structures with dynamic stress response constraints limits is proposed, The topology optimization model with the objective function being the structural weight and the constraint functions being structural dynamic stress responses under random vibration is presented in this paper,In order to greatly reduce the sensitivity cost of dynamic stress reponses, an optimization model with reduced dynamic stress response constraints is built,being incorporated with the rational approximation for material properties(RAMP), an varying dynamic stress response limits schemes,a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon.Therefore, a set of quadratic approximate functions for structural dynamic stress response constraints are formed,Finally,The proposed algorithms, integrated with the dual theory, is to solve the optimization problem.A series of numerical examples, which are under different kinds of excitations, different boundary conditions and different models, are presented to demonstrate the feasibility and effectiveness of the proposed approach.

Keywords:topology optimization,dynamic stress response, stress constraint,

RAMP,MMA approximations,dual theory

1.Introduction

In recent years,structural optimization subjected to stress constraints is of great importance in advance engineering application,most attention has been paid to stastic stress constraints problems, Matteo Bruggi et.al [1]dealt with the imposition of local stress constraints in topology optimization by an approach called as the qp approach, Bruggi and Duysinx et.al [2] proposed a selection method leading to further reduction in the number of active constraints, París et.al [3] investigated a technique of ‘block aggregation’ by using more global constraints to reduce the number of constraints involved in a single global function, The adaptive normalization scheme proposed by Le et al.[4] works well in controlling the maximum stress, Recently, a series of varied constraint limits were introduced and a kind of new structural topological optimization method with different characteristics requirements,was proposed by Rong et al. [5], based on the rationalapproximation for material properties (RAMP) and the dual theory.

However, few attempts have been made to incorporate random dynamic response constraints, Shu et.al [6] proposed level set based topology optimization method for minimizing frequency response. Rong et.al [7-8] used ESO method and SQP method to obtain the optimal topology of the continuum structures under random excitations. Yang et al.[9] studied the topology optimization under static loads and narrow-band random excitations, where the static and dynamic response analyses were processed independently without any superposition. Zhang et al. [10] investigated the optimal placements of the components and the configuration of the supporting structure within a predefined design domain to improve the structural static and random dynamic responses simultaneously. While these optimization works are not devoted to solve dynamic stress response constraints limits problems.

To improve the dynamic characteristics of the structure, it is much important and necessary to consider the dynamic stress response constraints limits in topology optimization.Here, a method for the topology optimization of continuum structures with dynamic stress response constraints limits is proposed in this paper, The topology optimization model with the objective function being the structural weight and the constraint functions being structural dynamic stress responses under random vibration is presented.

The following sections of this paper are organized as follows: Section 2 formulate the dynamic stress reponse under random excitation. Section 3 addresses element topological variables and interpolation functions used in this paper. The topological optimization with dynamic stress response constraints limits for minimizing the structural weight, is introduced in Section 4. Section 5 presents the sensitivity analysis on the dynamic stress response with respect to the design variable, The Section 6 introduces several numerical examples to discuss the effectiveness of the proposed method. Concluding remarks are finally drawn in Section 7.

2.The dynamic stress reponse

The Von Mises stress in the three-dismensional space can be defined as follow:

2 2 2 2 2 2 2 2 2 2 2 2 23( )k k k k k k k

vm x y z x y x z y z xy xz yz (1)

Here, { } { , , , , , }k k k k k k k T

x y z xy xz yz represent the stress vector of the k-th element of the

structure,Equation(1) can be expressed as:

2 { } { } { { }{ } }Tk T

vm A Trace A (2)

where A is a symmetric matrix,and

1 0.5 0.5

0.5 1 0.5

0.5 0.5 1

3

3

3

A

.

Therefore, the mean square response of the Von Mises stress can be written as:

2( ) ({ } { }) ( { { } { } }) { { } { } }Tk k kT k kT

vmE E A E Trace A Trace AE (3)

Where { } { }k kTE is the covariance matrix of the stress vector { }k .In a finite

element analysis,the stress vector { }k of the k-th element can be expressed as:

1

1{ } [ ] [ ] { } { }

hNk k k k k k

h

hh

D B u S uN

(4)

where [ ]kD is the elastic matrix of the k-th element, [ ]k

hB is the strain matrix at the h-th

Guass integration point of the k-th element, hN is the total number for the Guass

integration point,and { }ku is nodal displacement vector of the k-th element.Then the

covariance matrix { } { }k kTE can be expressed as:

{ } { } ({ } { } )({ } { } )

({ } { } { } { } )

{ } [{ } { } ]{ }

k kT k k k k T

k k kT kT

k k kT kT

E E S u S u

E S u u S

S E u u S

(5)

where [{ } { } ]k kTE u u is the the covariance matrix of the nodal displacement vector { }ku .

The finite element equation for a structural dynamic problem can be expressed as:

[ ]{ } [ ]{ } [ ]{ } { ( )}M Y C Y K Y f t

(6)

Where[ ]M , [ ]C and [ ]K are the N N mass. Damping, and stiffness matrices of the

structure repectively. Is the number of degree of freedom. { }Y , { }Y

and { }Y

are the

displacement,velocity and acceleration vector, { ( )}f t is the N 1 white noise random

external excitation vector,which is loaded in the form of spectrum.It can be assumed that power spectral density matrix of the external excitation vector { ( )}f t can be expressed

as:

( )fS S (7)

where S is N N semi-positive real symmetric matrix, S would be a diagonal matrix

if all degree of freedom are unrelated, i.e., , 0( )I II ijS S S i j .

The corresponding undamped free vibration equation of (6) can be written as:

[ ]{ } [ ]{ } 0M Y K Y

(8)

Assume that the solution for equation (8) is sin t ,then the characteristic vector

equation for the equation (8) can be expressed as:

2([ ] [ ]) 0I iK M (9)

In equation (9), i and i are the i-th circle natural frequency and mode

shape.We consider that the mode shapes can be normalized with respect to the mass matrix and satisfy equation (10) under the proportional damping.

2

{ } [ ]{ }

{ } [ ]{ } ( )

{ } [ ]{ } (2 )

T

T

i

T

i i

M

K diag

C diag

I

(10)

Where 1 2{ } { , , , , }i N and i is the damping ratio of the i-th mode,assume

that { } { }Y y (11)

Equation(6) can be written as:

2(2 ) ( ) { } ( ) ( )i i iy diag y diag y f t F t

(12)

Then the correlation matrix of mode response ry and sy can be expressed as:

rs rs rsrs

r s r s

d a rC

(13)

where

21r r , 21s s , { } { }T

rsd S (14a)

r s r r s sa , rs r r s sb , r s r r s se (14b)

1

2 2

rs rs rsg a b

, 1

2 2

rs rs rsQ a e

, r s r s r sg Q (14c)

3.Element topological variables and interpolation functions

A topological optimization for minimizing the structural weight while satisfying dynamic stress response constraints limits is built in this paper,In the topological optimization with dynamic response constraints,numerical examples show that the solution for dynamic response value and its sensitivity would be wrong because of interpolation schemes without enough accuracy,These interpolation schemes,such as Solid Isotropic Material with Penalty(SIMP),which cannot achieve the matching punishment between the stiffness and mass matrix,and lead to localized modes, reduce the accuracy of modal analysis.

Referring to the Rational Approximation for Material Properties(RAMP) and the ICM (Independent, Continuous and Mapping) method,the weigth.stiffness matrix ,stress and mass matrix of an element are recognized by following functions,respectively

0( )i w i iw f w , 0( )K i if KK , 0[ ] ( )[ ]k

s i iS f S 0( )i

M i iM f M (15a)

where

( ) W

w i If , ( )

1 (1 )

iK i

i

f

, ( ) s

s i If , ( ) M

M i If (15b)

where are the weight,stiffness matrix,stress and mass matrix of the i-th

element,respectively,and 0

iw , 0

iK , 0[ ]iS and 0

iM are orginal weight,orginal stiffness and

mass matrix of the i-th element,respectively.In this work, 1.25w M , 5s and

6 are used.Due to the rational function in the RAMP interpolation scheme, the

derivative of the function isn’t zero even when i tends to zero,which can prove the

effect of weak material on stiffness,and prevent the localized modes.

4.Optimization problem statement

The topological optimization model for minimizing the structural weight with dynamic stress response constraints limits can be formulated as follows:

2 2

, ,

min : W

. .: ( )U

vm i vm is t

(16)

For the purpose of obtaining predominantly black-and-white design in this paper,

the vary dynamic response constraints method is adopted,Then,equation(16) can be transferred into equation (17):

0 02 2

1 1

2 2 ( 1),

, ,1,2,

2 2 ( 1),

, ,

1 1

min : (1 )

max ( ) ( )

. . : 1 1( ) ( )

Q Q

i i i i i

i i

k U

vm i vm ii Q

Q Qk U

vm i vm i

i i

f w w

s t

Q Q

(17)

Where is a weighting experience parameter; is a varying constraint limit,and

2 ( 1),

,( ) k U

vm i is expressed as:

2 ( ) 2 ( ) 2 2 ( ) 2 ( ) 2

, , , , , ,2 ( 1),

, 2 ( ) 2 ( ) 2 2 ( ) 2 ( ) 2

, , , , , ,

( ) min( ( ) , ( ( ) ( ) ).( ) ( )( )

( ) min( ( ) , ( ( ) ( ) ) .( ) ( )

k k U k k U

vm i vm i L vm i vm i vm i vm ik U

vm i k k U k k U

vm i vm i L vm i vm i vm i vm i

(18)

Where is a dynamic response limit change factor, L is a relaxation coefficient for

the dynamic response constraints, 2 ( 1),

,( ) k U

vm i are varied according to equation (18)at

each outer loop iteriation step, 2 ( )

,( ) k

vm i is the dynamic stress response of the i-th

element of the structure at the k-th loop iteration step. 5.Sensitivity analysis on the dynamic stress response

Assume that 1

i

i

x

,the sensitivity of the constraint function with respect to ix can

be expressed as: 2

,( ) [{ } { } ]( { } { } ) ( { } { } )

k kTvm k k kT k kTrs

i i i

CE u utrace A S S trace A S S

x x x

(19)

22

2 2

( ) ( )( )

2 2

( ) ( ) ( )

rs rs rs rs s sr r

i r s r s r i s i

rs rs rsrs rs rs rs rs rs

r s r s i i i

C d a r

x x x

d a ra r d r d a

x x x

(20)

1

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2 2

nrs ir is

is ir i

ii j j

r srs sr

i r i s i

r srs sr

i r i s i

r srs sr

i r i s i

rs rs rs rs

rs rs rs rs rs

i i i i

dS

x

a

x x x

b

x x x

e

x x x

r a b ag a b Q a

x x x x

2

rs

rs

i

ee

x

(21)

the sensitivities of the circle natural frequency and the mode shape with the respect

to ix can be respectively written as:

21 [ ] [ ]

2

Tkk k k

i k i i

K M

x x x

(22)

2

2 2

1

[ ] [ ]

1 [ ]

2

T

k k p

i iNij ijkkp p kp k p

pi

T

k k

i

K M

x xk p

b bx

Mk p

x

(23)

0

2

[ ] (1 )[ ]

((1 ) )i

i i

KK

x x

, 0

1

[ ][ ]i

i i

MM

x x

(24)

In order to deal with checkboards problem of topological optimization,the approach

proposed by Sigmund and Petersson is adopted to redistribute the sensitivity of the dynamic stress response. Then the dynamic stress constraint expressions in equation

(17) can be approximated by their one-order approximation functions， and the objective

function in equation (17) can be approximated by its second-order approximation function, and solving theapproximating quadratic programming problem of equation.(17) can be transferred into solving a dualprogramming problem by using the dual theory.

6.Numerical examples

A beam like 2D structure is shown in Fig.1. The beam with dimensions 0.3 m×0.1m is simply supported at both ends. The design domain 0.2 m×0.1 m is equally divided into 80×40 four-node plane stress elements. The material is assumed with Young’s modulus E = 210GPa, Poisson’s ratio v = 0.3, A concentric force with auto-power spectral is applied at the middle point A along vertical directions at the bottom of the plate as shown in Fig. 1 and all mode damping ratios n = 0.02 are

specified, the prescribed dynamic stress response constraint limit is 172.05 10 .

Here, the parameter 0.046 and 1.02L is selected,Fig. 2 is the optimal

topology obtained by the proposed method. The optimization histories of the weight and the maximum dynamic stress response for the truncated mode number 16 are illustrated in Figs. 2 and 3, respectively. The optimaltopology obtained by the proposed method is shown in Fig. 4.

Fig. 1. The initial structure model

Fig. 2. the optimization history of the topology weight

Fig. 3. the optimization history of the maximum dynamic stress response

Fig. 4. The optimal topology of the plane stress plate obtained by the proposed

method

7.Conclusions

(1)A methodology for the topology optimization of continuum structures with with dynamic stress response constraints is proposed for minimizing the structural weight. Ramp interpolation schemes is used to prevent localized modes,and a vary dynamic response constraints method is adopted to make sure that the approximate expression is effective and avoid objective oscillation phenomenon in optimization iterations.The sensitivity of the dynamic stress response with respect to the design variable is derived.

(2)The example is used to prove the validity and efficiency of the proposed method for the dynamic stress response constraints optimization problem,the optimized topology can stably converge to optimal solutions. The obtained optimal design with prescribed dynamic stress response by the proposed method, will have a lower weight than the initial, guess design. The proposed continuum structural topological optimization with dynamic stress response constraints are effective, and undesired localized modes can be prevented.

REFERENCES [1] B.Matteo, On an alternative approach to stress constraints relaxation in topology

optimization,Structural and Multidisciplinary Optimization，Vol.36, pp.125-142, 2009.

[2] Bruggi M,Duysinx P. Topology optimization for minimum weight with compliance and stress constraints. Structural Multidisciplinary Optimization 2012; 46:369–384. [3] París J, Navarrina F, Colominas I, Casteleiro M. Block aggregation of stress constraints in topology optimization of structures. Advances in Engineering Software 2010; 41:433–441. [4] Le C, Norato J, Bruns T, Ha C, Tortorelli D. Stress-based topology optimization for continua. Structural Multidisciplinary Optimization 2010; 41:605–620. [5] Rong JH, Zhao ZJ, Xie YM, Yi JJ. Topology optimization of finite similar periodic continuum structures based on a density exponent interpolation model. Computer Modeling in Engineering and Sciences 2013; 90(3): 211-231. [6] Lei Shu, Michael Yu Wang, Zongde Fang, Zhengdong Ma, Peng Wei, Level set based structural topology optimization for minimizing frequency response, Journal of Sound and Vibration, 2011, 330(24):5820-5834. [7] J.H.Rong, Y.M.Xie, X.Y.Yang, Q.Q.Liang, Topology optimization of structures under dynamic response constraints, Journal of Sound and Vibration, 2000, 234(2):177-189. [8] J.H.Rong, Z.L.Tang, Y.M.Xie, F.Y.Li, Topological optimization design of structures under random excitations using SQP method, Engineering Structures, 2013, 56:2098-2106. [9] Yang ZX, Rong JH, Fu JL. Dynamic topology optimization of structures under narrow-band random excitations. J Vib Shock 2005;24:85–97. [10] Zhang Q, Zhang WH, Zhu JH, Gao T. Layout optimization of multi-component structures under static loads and random excitations. Eng Struct 2012;43:120–8.