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  • 1

    Recent Advances in Nonlinear Response Structural Optimization Using the Equivalent Static Loads

    Method

    Oct. 28, 2014 VR&D 2014 Users Conference

    Monterey, CA

    Gyung-Jin Park Professor, Hanyang University

    Ansan City, Korea

  • Optimization

    2

    Problem formulation is important.

    Do we have to understand the details of the optimization theory?

    Various software systems with various algorithms are available.

    UL

    ,...,10)(,...,10)( subject to

    )( minimize toFind

    bbb

    bbb

    b

    ===

    mjglih

    fR

    j

    i

    n

  • Linear Static Response Structural Optimization

    3

    Popular

    Easy

    Well developed software systems are available.

    UL

    ,...,10),()(: subject to

    ),( minimize to,Find

    bbb

    zbfzbKh

    zbzb

    ==

    mjg

    fRR

    j

    ln

  • Linear Static Response Structural Optimization

    4

    Size Optimization: The FEM data are fixed.

    Shape Optimization: Node and element data of FEM analysis are changed

    during optimization.

    Topology Optimization: Material distribution is optimized.

  • Not Linear Static Response Structural Optimization

    5

    (1) Dynamic Response Optimization

    (2) Structural Optimization for Multibody Dynamic Systems

    (3) Structural Optimization for Flexible Multibody Dynamic Systems

    (4) Nonlinear Static Response Structural Optimization

    (5) Nonlinear Transient Response Structural Optimization

    fzbKzbMh =+ )()(:

    fzzbKh =),(:

    system dynamicmultibody ofequation Governing:h

    fzzbKzzbMh =+ ),(),(:

    systems dynamicmultibody flexible ofequation Governing:h

  • The Equivalent Static Loads Method

    6

    Not a Linear Response Analysis

    Linear Static Optimization

    Disp. Field

    Multiple Loading Conditions

    Equivalent Static Loads

    Updated design

    Analysis Domain Design Domain

    This method has been developed for not-linear static response structural optimization.

    Analysis is performed in the analysis domain.

    Equivalent loads are calculated.

    Linear response optimization is performed using the equivalent static loads in the design domain.

    The process proceeds in a cyclic manner.

  • Contents

    7

    (1) Linear Dynamic Response Optimization

    (2) Structural Optimization for Multibody Dynamic Systems

    (3) Structural Optimization for Flexible Multibody Dynamic Systems

    (4) Nonlinear Static Response Structural Optimization

    (5) Nonlinear Transient Response Structural Optimization

  • 8

    Nonlinear Static Response

    Optimization Using Equivalent Loads

    (NROEL) Installed in NASTRAN

  • Nonlinear Response Optimization

    9

    General Formulation

    ,m,ig

    f

    i 1;0)()(subject to)(minimize to

    Find

    ==

    zb,fzzb,K

    zb,b

  • NROEL

    10

    Definition: An Equivalent Load is a load in a linear static system that makes an identical response to that in a nonlinear system.

    Design-Oriented Loads

    displacement stress etc.

    Responses in optimization formulation

  • Flow of NROEL

    11

    START

    ? No

    Nonlinear Analysis fzzb,K =NN )(

    Calculate Equivalent Loads

    NLeq zKf =

    Update Design

    Linear Response Optimization

    ,m,ig

    f

    Li

    eqLL

    L

    1;0)()(subject to

    )(minimize toFind

    =

    =

    zb,fzbK

    zb,b

    END

    Optimization process using equivalent loads

    )1()( keqk

    eq ff

    k=0 k=k+1

    The one for stress constraints is separately defined.

  • Shape Optimization 1

    12

    11.22E+6 N

    b1

    Geometric and Material Nonlinearity -Linear hardening

    E = 200.0 GPa y = 300.0 Mpa Eh = 50.0 GPa

    Shape change using domain element

    2. A plate

    Nonlinear Response Optimization

    2001;00.1.350/)(subject to

    Massminimize tochange) shape(Find 321

    ,,j

    , b, bb

    j

    NN

    ==

    fzzb,K

    b2 b3

    b1 b2

    b3

  • Shape Optimization 1

    13

    Results of Optimization Initial design

    Conventional method

    NROESL Initial design

    Con. Meth. (215)

    NROEL (7)

  • Large Scale Size Optimization 1

    14

    Only Geometric Nonlinearity E = 68.9 GPa

    Loading and Boundary Conditions

    4. 280 shell structure

    Nonlinear Response Optimization

    ( )( )315,,1005.0

    280,,100.1.60/)(subject to

    Massminimize to)280,,1(Find

    =

    ==

    =

    j

    i

    it

    j

    i

    NN

    i

    fzzb,K

    NRO using EL Abaqus 6.4 Optistruct 7.0

  • Large Scale Size Optimization 1

    15

    Results of Optimization

    < Design history graph > < Optimum thickness contour >

    Optimization using the conventional method is fairly expensive.

  • 0.5kN 0.5kN

    0.5kN 0.5kN

    16

    Shape Optimization with Linear Contact

    6. A Ring: Problem Definition

    Problem model

    - Loading condition: The forces are applied at the elements of the upper parts.

    - The element property: PSOLID

    - The total number of elements: 672 ( 64 CPENTA + 608 CHEXA)

    - Only the boundary nonlinearity is considered.

    - NASTRAN is used for contact analysis and linear response optimization.

    - NASTRAN DMAP is utilized for calculating the equivalent loads.

    Solver: SOL 101

    - Linear contact: Linear analysis + Nonlinear contact parameters

    ccontact boundaries ( )

  • 17

    Design condition

    Design perturbation vectors

    - Perturbation vectors are utilized for shape change in shape optimization.

    - Each arrow is a perturbation vector.

    - Seven design variables are selected based on the perturbation vectors.

    Shape Optimization with Linear Contact

    b1

    b2 b2

    b1 b3 b3

    b4 b4

    b5 b5

    b6 b6

    b7

    )672,,1(0KPa0.2subject tomassmin. to

    ring) theof(shape,,,Find 721

    = i

    bbb

    i

    Formulation

  • 18

    Optimization result shape change

    Shape Optimization with Linear Contact

    Initial shape Optimum shape

  • 19

    Nonlinear Dynamic Response Optimization Using Equivalent Static

    Loads Installed in GENESIS and OptiStruct

  • NDROESL

    20

    Equivalent Static Loads for Displacement

    App

    lied

    forc

    e

    Time Time step: Load set:

    0t 1t 2t nt0s 1s 2s ns

    )t()t()t()t( NNN f)zzK(b,zM =+

    )t()t( NLz zKf =eq

    )s()s( zLL eqf(b)zK =

    Nonlinear transient analysis

    Transformation to equivalent loads

    Linear analysis with ESLs

    Linear response optimization

    )t(Nz

    st

    Find

    to minimize

    subject to

    0zz allowableL )s(

    b)z(b L,F

    )s()s(LL eqf(b)zK =

    newb

  • Flow of NDROESL

    21

    START

    ? No

    Nonlinear Dynamic Analysis

    )()()( tt(t) NNN fzzb,KzM(b) =+

    Calculate Equivalent Static Loads

    )()( tt NLeq zKf =

    Update Design

    Linear Response Optimization

    ,m,igqttt

    f

    Li

    eqLL

    L

    1;0)(

    ,,1);()()(subject to)(minimize to

    Find

    =

    ==

    zb,fzbK

    zb,b

    END

    Optimization process using equivalent static loads

    )1()( keqk

    eq ff

    k=0 k=k+1

  • Nonlinear Transient Size Optimization

    22

    Loading and Boundary Conditions

    1. 160 shell structure

    Nonlinear Response Optimization

    ( )205,,1005.0)(subject to

    Massminimize to)160,,1(Find

    =

    =+

    =

    j

    it

    j

    NNN

    i

    fzzb,KzM(b) NRO using ESL

    Abaqus 6.4 Optistruct 7.0

    Time Loading time: 0.01 sec Total analysis time: 2.0 sec

    Only Geometric Nonlinearity

  • Nonlinear Transient Size Optimization

    23

    Results of Optimization

    < Design history graph >

    < Optimum thickness contour >

    Initial

    Optimum

    < Tip displacement >

  • Roof Crush Optimization

    24

    3. Roof crush problem - Ford Explorer Model: developed by the GWU

    Finite Element Model

    Number of Parts : 394

    Number of Elements : 432,596

    Number of Nodes : 431,629

    Number of total DOFs : 2,589,774

    FMVSS 216 Standard (Roof crush resistance)

    The current FMVSS 216 standard requires that a passenger car roof withstand a load of 1.5 times the vehicles unloaded weight in kilograms multiplied by 9.8 or 22,240 Newtons, whichever is less, to either side of the forward edge of the vehicles roof with no more than 127 mm of crush.

  • 25

    Roof Crush Optimization

    Definition of design variables

    DV 1: Thickness of A-Pillar (t1)

    DV 2: Thickness of B-Pillar (t2)

    DV 3: Thickness of Roof-Rail (t3)

    < Roof Structure >

    DV 1

    DV 3

    DV 2

  • 26

    Roof Crush Optimization

    0.23dv6.0.022dv.60.021dv.60

    67.5ms)(t 0.0 crush roof theof distance127mm)(subject to

    massminimize to )3,2,1(Find

    step

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