an investigation of reliability-based topology optimization
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An Investigation of Reliability-based Topology Optimization. Chandan Mozumder Advisor: Dr. John E. Renaud 20 th Aerospace and Mechanical Engineering Graduate Student Conference University of Notre Dame 19 th October, 2006. Synopsis. Introduction - PowerPoint PPT PresentationTRANSCRIPT
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