a space-time finite element model1

8/3/2019 A Space-time Finite Element Model1

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A SPACE-TIME FINITE

ELEMENT MODELFOR DESIGN AND

CONTROL OPTIMIZATIONOF NONLINEAR DYNAMICRESPONSE

J. E. Barradas Cardoso 1, P. Pires Moita 2, and Aníbal J. J. Valido2

1 Instituto Superior Técnico, Departamento de Engenharia MecânicaAv. Rovisco Pais, 1049-001 Lisboa , Portugal

2 Escola Superior de Tecnologia, Instituto Politécnico de SetúbalCampus do IPS, Estefanilha, 2914-508, Setúbal, Portugal



Introduction

• Integrated methodology

– structures and flexible mechanical systems

– optimal design and optimal control

– Bound formulation is used to solve both themulticriterium and unicriterium problems



Response Analysis

• Space discretization – Virtual work dynamic equilibrium equation

– After Finite element discretization

0 0( )t t t t t t t t W dV d

u u S f u Q u

t t t t t t

S S M U

+ C U + K U = P



Response Analysis

• Time discretization

e e e

S D z P

11 12 11 12

21 22 21 22

,

t t t

S S nxn nxn nxn

e t t t t t t e

S S S nxn S S nxn 1

t t t t t t

S S nxn S S nxn 2

0 0 0

0 0

0 0

K C M P

D D D D D P P

D D D D P

( , ), ( , )e t t t t t t t z z z z U, U

U



After the boundary conditions are applied it becames

Response Analysis

• Solved at-onceTime assembly

S D z P

ˆ ̂ ˆ ˆ

,S c c

K U P P P D U



•Performance functional

•Total variation

Design Sensitivity Analysis Problem

( , , )t t

G t dt z b




• Adjoint system method – adjoint fields replace the arbitrary variation of state fields by into the

virtual work equation as

– define an extended 'action' function

– replace the implicit design variations of the state fields by explicit designvariations and adjoint fields, then determining these adjoint fields by

vanishing the implicit design variation of the 'action' function as

– total design variation of

ˆ ˆ ˆ ˆ( ) 0a a

SW = K U - P U

a A W

0 A

δΨ δA




The sensitivities are firstly performed at the elementlevel and then the sensitivity equations are

assembled and time boundary equations imposed



Optimum Design Problem

• A bound formulation is used to solve themulticriterium and unicriterium problems

•

The diferent objectives are normalizedaccordingly to

0

min , s.t.

, ( 1,2,..., );

0, ( 1,2,..., ); 0

0k

T

j

k m

j m

( ) ( ) , 1min max

0i 0i 0i 0i0i i iΨ Ψ Ψ w Ψ Ψ w



Examples

• Nonlinear ImpactAbsorberv(t=0)=1;

M=1;n=h2

Decision variables:

M

K=K0| u | n

C=C0| u .

| h

u

P(t)

0 0; ;[ / ), 0 12]K C P t t



Case I

Starting design:

Optimal design:

Examples

0 00,5; 0,5;[ / ) 0, 0 12]K C P t t

maxmin . .

1

a s t

u

0 00,00667; 6,96K C

-0,2

0, 0

0, 2

0, 4

0, 6

0, 8

1, 0

0 2 4 6 8 10 1 2t[s ]

P

[ N

]



Examples

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0 2 4 6 8 10 1

t[s ] a [ m / s 2 ]

starting

Ko, CoKo, Co+control

Acceleration response



Case II

•Starting design:

•Normalized weighted bound formulation

•Optimal design

Examples

0 00,597; 0,597;[ / ) 0, 0 12]K C P t t

2 2 20min{ , (100 100 ) } . .

1

C E u u P dt s t

u

wE /wCo K O optimu

m

C O optimu

m

0.5/0.5 0.458 0.015

0.6/0.4 0.453 0.022

0.7/0.3 0.450 0.032

0.8/0.2 0.446 0.050

0.9/0.1 0.442 0.091-0,60

-0,40

-0,20

0,00

0,20

0,40

0 2 4 6 8 10 12

t[s ]

P

[ N

P0,5/0,5

P0,6/0,4P0,7/0,3

P0,8/0,2P0,9/0,1



Examples

0.00

0.02

0.04

0.06

0.08

0.10

53 55 57 59 61 E

Co

Criterium Space



Examples

• Vehicle Suspension – Sinusoidal obstacle of height

0,1016m and wavelenghtof 24,384m

– minimize the maximumacceleration magnitudesuch that the relativevertical displacementbetween masses is not

larger than 0.05 m

c3

c5

1

m4 m3

m1

k1 c1

k2c2k3

k4c4k5

2 6

4

9

11

3 7

5

8

10

Beam element 1 Beam element 2

800kg400kg



Examples

Starting designA = 0.1 m2, k1 = 1.75E4, k2 = k3 = 5.26E4 N/m,

c1 = 1.75E3, c2 = c3 = 4.38E3 Ns/m

Optimal design

A = 0.35579 m2, K1=6.94E4 N/m, K2=6.31E4 N/m, K3=5.02E4

N/m, C1=1.E6 Ns/m, C2=1.34E4 Ns/m, C3=1.E2 Ns/m

-1.4E-03

-1.2E-03

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

0.0 0.2 0.4 0.6 0.8 1.0

t[s]

P [ N ]

( ) 0.,0 1P t t s



Examples

• Driver seat max. acceleration [m/s2]

-6.0E-05

-5.0E-05 -4.0E-05

-3.0E-05

-2.0E-05

-1.0E-05

0.0E+00

1.0E-05

2.0E-05

0.0 0.2 0.4 0.6 0.8 1.0

t[s]

a[m/s ]

Initial Design Optimal design wrt area Optimal design wrt area, K’s and C’s Optimal design wrt area, K’s and C’s + control



CONCLUDING REMARKS

• Current formulation unifies in one single formulation the problems ofoptimum design and optimal control, treating the design variables ascontrol variables that do not change with time.

• Using at-once integration of the equations of motion and itssensitivities, the adjoint method of design sensitivity analysis can beused with all of its advantages and without its main disadvantage indynamics: the necessity of backwards integration of the adjointsystem. So, the needing of backwards integration with its majordrawback - the memorization of the response history - is notassociated with path-dependent problems, as it is frequentlyassumed, but it is associated with the fact of using step-by-step

integration.

• Temporal boundary conditions can be applied at any point in time,not necessarily at the start of the integration, as it is usual for thestep-by-step integration method.

a space-time finite element model1

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