20th Aerospace and Mechanical Engineering Graduate Student Conference

An Investigation of Reliability-based Topology

Optimization

Chandan MozumderAdvisor: Dr. John E. Renaud

20th Aerospace and Mechanical Engineering Graduate Student Conference

University of Notre Dame

19th October, 2006

20th Aerospace and Mechanical Engineering Graduate Student Conference

Synopsis

• Introduction• Reliability-based Design Optimization

(RBDO) Formulation• Reliability-based Topology Optimization

(RBTO)FormulationDifferent approaches

• RBTO with Hybrid Cellular Automata (HCA) method

• Numerical experiments and results• Conclusion

20th Aerospace and Mechanical Engineering Graduate Student Conference

Why Reliability-based Design?

• Mathematical modeling and simulation for design of systems

• Optimization strategies to avoid burden of manual iterations, manipulating inputs and reviewing outputs

• Models are only abstraction of realities• Deterministic optimization techniques do

not consider impact of uncertaintieserror in design decisions

20th Aerospace and Mechanical Engineering Graduate Student Conference

Reliability-based Approach

•Deterministic Design: may lead to unsafe design

•Factor of Safety Approach: lead to conservative design

•Reliability-based Approach: design is insensitive to input and model uncertainties

20th Aerospace and Mechanical Engineering Graduate Student Conference

Reliability-based Design Optimization

min f(x, p, y(x, p))

subject to gR(V, η) ≥ 0

gjD(x, p, y(x, p)) ≥ 0 j = 1,…,Ndet

xl ≤ x ≤ xu

x = design variablep = fixed parameterP = failure probability

Reliability constraints

•Reliability constraints can be formulated by Performance Measure Approach (PMA) or Reliability Index Approach (RIA)

PMA: grc are formulated as constraints on performance that satisfies a probability requirement

RIA: grc are formulated as constraints on probability of failure

)*,( uRi

rci Gg

iallow

rci PPg

i

20th Aerospace and Mechanical Engineering Graduate Student Conference

RBDO formulation

•Rosenblatt Transformation: •random vector (V) to standard normal vector (U)•zero mean and unit variance

•Limit state function: GiR(u,η) = 0

0),(0),(

)()()(

u

UV uuvRi

Ri Gxg

i ddxfP

•Probability of failure corresponding to a failure mode:

•Approximation to the multi-dimensional integral using First Order Reliability Method (FORM), which computes the Most Probable Point (MPP) of failure

20th Aerospace and Mechanical Engineering Graduate Student Conference

MPP of failure

• Solve the following optimization problem in U-spacemin ||u||subject to GR(u,η) = 0

• First order approximation to probability of failure

Pf = Φ(-βp)

where βp = ||u*|| safe region

unsafe region

βp

G = 0

G > 0

G < 0

u2

u1

20th Aerospace and Mechanical Engineering Graduate Student Conference

Topology Optimization

• Optimization process systematically and iteratively eliminates and re-distributes material throughout a design domain to obtain an optimal structure

• Homogenization approach by Bendsøe and Kikuchi [Bendsøe and Kikuchi ’88]

• Density approach or SIMP approach by Bendsøe[Bendsøe ’89]

Simpler to implement

Topology Optimization

20th Aerospace and Mechanical Engineering Graduate Student Conference

Reliability-based Topology Optimization

• RBTO extends reliability notion to topology optimization

• Reliability-based constraints with SIMP approach for continuum structure [Kharmanda et al. ’02, ’04]

improved reliability level of structure without increasing weight

• RBTO using HCA for continuum structure [Patel et al. ’05]

increase in weight in resulting structure for increased reliability level

• Reliability-based constraints using discrete frame elements [Mogami et al. ’06]

20th Aerospace and Mechanical Engineering Graduate Student Conference

RBTO approach by Kharmanda et al.

• Initial sensitivity analysis to identify random variables which have significant effect on the objective function

• Limit state function used is a linear combination of the random variables

04321 uuuuG u1 = applied loadu2, u3 = the number of elements used to discretize the design domain in 2Du4 = volume fraction

no physical significance with respect to the failure probability of the structure

[Kharmanda et al. ’02, ’04]

20th Aerospace and Mechanical Engineering Graduate Student Conference

Some observations …

•Physical significance of limit state function?

•Reliability analysis independent of boundary and loading condition?

Driving the random variables to satisfy the following equation irrespective of the problem definition:

2221 ......min ni uuu subject to G ≤ 0

•Dependence on the initial point?

20th Aerospace and Mechanical Engineering Graduate Student Conference

Some observations …

Reliability Intial pointDesign point

Mass Fraction

Topology

Deterministicnelx=60 nely=20F = -1.0

NA 0.5

β = 3.0nelx=60nely=20F = -1.0

nelx=69nely=17F = -1.15

0.425

β = 3.0nelx=69nely=17F = -1.15

nelx=79nely=17F = -1.15

0.319

•Dependence on the initial point?

20th Aerospace and Mechanical Engineering Graduate Student Conference

Hybrid Cellular Automata (HCA)

• Cellular Automata (CA) computing & control theory are used to distribute material within a discretized design domain

• CAs are by definition, dynamical systems that are discrete in space and time and operate on a uniform, regular lattice.

• CAs are characterized by local interactions.

Neighborhood:

Von NeumannN = 4

MooreN = 8

EmptyN = 0

Boundary:

Fixed

0

Periodic

X X

20th Aerospace and Mechanical Engineering Graduate Student Conference

HCA Algorithm

Material distribution rule

FEAS* S

Update

[Tovar et al. ’04]

20th Aerospace and Mechanical Engineering Graduate Student Conference

RBTO using HCA

• Decoupled reliability and structural analysis• Strain energy density as target • PMA to search for MPP• Random variables:

modulus of the material E0

the loads Pi on the structure

• Limit-state function:Failure mode with respect to maximum

allowable displacement

20th Aerospace and Mechanical Engineering Graduate Student Conference

RBTO using HCAStart

Structural optimization

(HCA)

Reliability assessment

End

Convergence test|uTu|<ε3

|*max(t+1)–*max (t)|<ε4

Initial Density

x0(0), P(0), E(0)

x0(t), P(t), E(t)

x(t+1)

P(t+1), E(t+1)

yes

no

20th Aerospace and Mechanical Engineering Graduate Student Conference

Some observations …

• Gradient free methodNo approximation of gradientsLess numerical instabilities

• Limit state function is based on a physical failure mode

20th Aerospace and Mechanical Engineering Graduate Student Conference

Numerical Experiments

PP P1

P2P3

P1

P2P3

Mitchell-type Structure Three-bar truss

•Design domain discretized into 5000 elements•Maximum allowable displacement of 1cm for Mitchell-type and 2cm for three-bar truss•Standard deviation of 5% for the applied load(s)

20th Aerospace and Mechanical Engineering Graduate Student Conference

Results

ReliabilityMichell-type Structure Three-bar Truss

Mass Fraction

TopologyMass

FractionTopology

Deterministic 0.359 0.269

β = 0.5 0.388 0.278

β = 1.0 0.392 0.288

β = 2.0 0.431 0.309

β = 3.0 0.478 0.338

20th Aerospace and Mechanical Engineering Graduate Student Conference

Numerical Verification

Reliability

Michell-type structure Three-bar truss

Expected reliability

Reliability from MC simulation

Expected reliability

Reliability from MC simulation

β = 1 0.8413 0.8294 0.8413 0.8325

β = 2 0.9772 0.9743 0.9772 0.9773

β = 3 0.9987 0.9987 0.9987 0.9984

•Monte-Carlo Simulation with 10,000 sample points

20th Aerospace and Mechanical Engineering Graduate Student Conference

Observations

• Mass increases to obtain a six-sigma design as compared to deterministic design Mitchell-type structure: 33.15% Three-bar truss: 25.65%

• Good correlation between expected and MC predicted reliability levels

• Decoupled approach to reliability-based optimization with the HCA method for structural topology synthesis is an efficient approach to topology optimization of continuum structure with desired reliability level

20th Aerospace and Mechanical Engineering Graduate Student Conference

Future Studies …

• Multiple failure criteria

• Design of compliance mechanism considering geometric and material non-linearity

20th Aerospace and Mechanical Engineering Graduate Student Conference

Thank You!!!