theory and computations m. v. sivaselvanbechtel.colorado.edu/~sivaselv/sivaceaetalk.pdf · 2010....
TRANSCRIPT
BackgroundTheoretical building blocks
Computational examplesSummary
Exploring structural failures— theory and computations
M. V. SivaselvanDept. of Civil, Environmental and Architectural Engineering
University of Colorado at BoulderEmail: [email protected]
Department Faculty MeetingOctober 27, 2010
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Acknowledgements
Kajima Corporation, Japan (CUREE-Kajima Joint Research Project)
National Science Foundation (CAREER award)
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Introduction
Extreme events such has earthquakes, hurricanes, blasts etc. often result in failures ofengineering structures
Loss of lives and severe economic consequences
Failures of critical infrastructure, hospitals, bridges, power structures etc. — affect abilityto respond to disaster
Civil engineers are interested in understanding failure processes, and designing structuresto perform better
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Engineering questions
How do we design new structures to be safe ?
How do we evaluate the safety of existing structures, and decide how much tofix them up ?
It is uneconomical/impractical to design structures to be safe under allcircumstances
Modern codes of practice advocate a performance-based approach
EQ Intensity50% Chance of Exceeding in 50 years
20% Chance of Exceeding in 50 years
10% Chance of Exceeding in 50 years
PerformanceObjective
Operational
Immediate Occupancy
Life Safety
Collapse Prevention
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Engineering questions
Forensic engineering: How do we systematically investigate the causes of afailure ?
How do we assess the reserve strength of a structure after an extreme event ? —Important for decision support
Mathematical modeling and numerical simulation would be helpful
In this talk, we describe a strategy for failure simulation
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Complex behavior (1)
Portion of a steel building
Limited force capacity(plasticity)
Portion of a steel building after an earthquake Plot of Bending
Moment vs. Rotation of the Decrease in
specimen strength and stiffness
(damage)
Laboratory specimen representing a typicalrepresenting a typical
connection
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Complex behavior (2)
Reinforced concrete bridge pier after an earthquake
Plot of Force vs.
Displacement
Laboratory model of a reinforced concrete pier
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Sensitivity ?
Picture from the 1999 Mamara earthquake inTurkey
Two apparently identicalapartment buildings, one ofwhich has collapsed, and theother is relatively intactIs the dynamic response sensitiveto variability in materials,construction, earthquake ?
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Modeling and simulation of collapse processes
Types of behaviorElastoplastic and viscoplastic responseFracture, damage and fatigueGeometric nonlinearity, P-∆ effects and bucklingContact and frictional interface behaviorDynamic response and impactFragmentation and projectile motion
Complex dynamic behavior, possible sensitivity
Existing tools often have difficulties with these computations
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Goals
Ability to simulate failure of structures of practical interest
Develop a theoretical foundation where questions about mathematicalwell-posedness can be asked
ExistenceMultiplicity of solutionsSensitivity to perturbations
Provide analyst with assurancesConvergence of numerical methodsQuality of numerical solution
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Engineering questionsComplex dynamics of failure processes
Approach
Many existing approachesHave theoretical support for small class of problemsBut applied outside this class
Rather than extend existing approaches, rethink modeling andsimulation strategies
Theoretical building blocksMathematical programmingMixed Lagrangian Formalism (MLF)
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
A scene from “A Beautiful Mind”
Adam Smith needs revisionn-player game — each player has a finite number of choices to pickfrom (strategy)Nash equilibrium
DefinitionA strategy set in which no player benefits from unilaterally changing hisstrategy
2-player game example — Prisoner’s dilemma
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Prisoner’s dilemma
Two prisoners interrogated independentlyDeal
Both confess — each gets 5 years in prisonOne of them confesses — he walks, other person gets 10 yearsBoth do not confess – each gets one year
Called a bimatrix game
Prisoner II does not confess Prisoner II confessesPrisoner I does not confess (1,1) (10,0)
Prisoner I confesses (0,10) (5,5)
Is there a Nash Equilibrium ? (5,5)Something to think about — Is the situation described in the movie aNash equilibrium?
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Prisoner’s dilemma
Two prisoners interrogated independentlyDeal
Both confess — each gets 5 years in prisonOne of them confesses — he walks, other person gets 10 yearsBoth do not confess – each gets one year
Called a bimatrix game
Prisoner II does not confess Prisoner II confessesPrisoner I does not confess (1,1) (10,0)
Prisoner I confesses (0,10) (5,5)
Is there a Nash Equilibrium ? (5,5)Something to think about — Is the situation described in the movie aNash equilibrium?
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Prisoner’s dilemma
Two prisoners interrogated independentlyDeal
Both confess — each gets 5 years in prisonOne of them confesses — he walks, other person gets 10 yearsBoth do not confess – each gets one year
Called a bimatrix game
Prisoner II does not confess Prisoner II confessesPrisoner I does not confess (1,1) (10,0)
Prisoner I confesses (0,10) (5,5)
Is there a Nash Equilibrium ? (5,5)Something to think about — Is the situation described in the movie aNash equilibrium?
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Prisoner’s dilemma
Two prisoners interrogated independentlyDeal
Both confess — each gets 5 years in prisonOne of them confesses — he walks, other person gets 10 yearsBoth do not confess – each gets one year
Called a bimatrix game
Prisoner II does not confess Prisoner II confessesPrisoner I does not confess (1,1) (10,0)
Prisoner I confesses (0,10) (5,5)
Is there a Nash Equilibrium ? (5,5)Something to think about — Is the situation described in the movie aNash equilibrium?
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Complementarity
Computing the Nash equilibrium can be formulated as
Complementarity problem
w = Mz + q
z ≥ 0; w ≥ 0; zTw = 0
Called Linear Complementarity Problem (LCP)
Turns out to be an appropriate setting to formulate collapse models instructural mechanics1964 — Lemke and Howson invented an algorithm to compute Nashequilibrium for bimatrix games (more general Lemke, 1965)This and other algorithms developed in economics are the basis for acollapse simulation algorithmNote: we checked the uniqueness of the Nash equilibrium byenumeration; we will come back to this later
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Mechanics example — Tensegrity structures
Structures comprised of bars and prestressed cables — cables need tobe prestressed for stability
Source: http://www1.ttcn.ne.jp/a-nishi/tensegrity
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Tensegrity structures — Complementarity
Taut cable Slack cable
force > 0 slack > 0
Complementarity conditions
force ≥ 0; slack ≥ 0; force× slack = 0
z ≥ 0 w ≥ 0 zTw = 0
Free vibration of a tensegrity grid
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Long-term energy balance
0 2 4 6 8 10 120
1000
2000
3000
4000
5000
6000
7000
Time [s]
Ene
rgy
[Nm
]
Initial strain energyStrain energyKinetic energyTotal energy (Strain+kinetic)
1 2 3 4 5 6 7 8 9 10 11 12−0.004
−0.003
−0.002
−0.001
0
0.001
0.002
0.003
0.004
Time [s]
Err
orEnergy Energy error
Variational integrator for non-smooth systemsNewmark is an instance
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Tensegrity examples
Marine energy production (WPSB) Human foot Cytoskeleton
Source: Skelton and de Oliveira (2009) Source: Skelton and de Oliveira (2009) Source: Ingber (1998)
Art — A tensegrity structure in Denver
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Theoretical building blocks
OptimizationKKT conditions
Standard MaterialEnergy and dissipation functions
Set-valued derivative
Extending Fenchel duality
MLFEuler-Lagrange equations
Variational integrators
Mathematical programming
ComplementarityTensegrity
Prisoner’s dilemma
A Beautiful Mind
KKT conditions
Emphasis on duality
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Optimization problem
Minimize a function subject to equality and inequality constraintsKarush Kuhn Tucker (KKT) necessary conditionsFor convex problems
KKT conditions are also sufficientThere is a dual problem with the same solution
Example — Castigliano’s first and second theorems of elementarystructural mechanics
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Theoretical building blocks
OptimizationKKT conditions
Standard MaterialEnergy and dissipation functions
Set-valued derivative
Extending Fenchel duality
MLFEuler-Lagrange equations
Variational integrators
Mathematical programming
ComplementarityTensegrity
Prisoner’s dilemma
A Beautiful Mind
KKT conditions
Emphasis on duality
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Linear spring model
Model
x
F = k x
k
F(x) = kx
Energy convex (quadratic) andsmooth
F(x) = ψ′(x)
Energy function
x
ψ(x)
ψ(x) = 12 kx2
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Soft contact model
Model
x0 x0
x ,F
k k
F(x) = k max(|x| − x0, 0) sgn(x)
Energy function convex andcontinuously differentiable
F(x) = ψ′(x)
Energy function
x
ψ(x)
-x0 x0-x0 x0
ψ(x) =
{0 if − x0 < x < x012 k (|x| − x0)
2 otherwise
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Hard contact model k→∞
Model
x0 x0
x ,F
Energy function convex andnonsmooth
F(x) = ?
Energy function
x
ψ(x)
-x0 x0-x0 x0
ψ(x) =
{0 if − x0 < x < x0
∞ otherwise
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Hard contact model — generalized derivative
Model
x0 x0
x ,F
F(x) ∈ ∂ψ(x)
Energy function
x
ψ(x)
-x0 x0-x0 x0
∂ψ(x0) = R+
∂ψ(−x0) = R−
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Nonsmooth models
Linear spring Nonsmooth version−−−−−−−−−−→ Hard contact
Linear dashpot Nonsmooth version−−−−−−−−−−→ Ideal plasticityNote also the corresponding complementarity conditions
force ≥ 0; slack ≥ 0; force× slack = 0yield function ≤ 0; plastic flow ≥ 0; yield function× plastic flow = 0
Generalized Standard MaterialTwo scalar functions — Stored energy ψ and Dissipation φMaterial behavior is obtained as (generalized) derivatives of these functions.
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
A stiffness degrading model
ψc(F, ζ) =
{−ζ if F2
2 + ζ ≤ 0∞ otherwise
ϕ(F, ζ) = −ζ − Fy
2
Planar in the region
F2/2 + ζ ≤ 0
∞ outside
−5 0 5 10−1.5
−1
−0.5
0
0.5
1
1.5
Displacement
For
ce
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Strength degradation model
By an extension of Fenchel duality to saddle functions (a key concept)
k1
k0
-k2
-k3
Fy
r1Fy
r2Fy
F
r3Fy
e
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Theoretical building blocks
OptimizationKKT conditions
Standard MaterialEnergy and dissipation functions
Set-valued derivative
Extending Fenchel duality
MLFEuler-Lagrange equations
Variational integrators
Mathematical programming
ComplementarityTensegrity
Prisoner’s dilemma
A Beautiful Mind
KKT conditions
Emphasis on duality
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Mixed Lagrangian Formalism (MLF)
“A good notation has a subtlety and suggestiveness which make it seem, attimes, like a live teacher”
— Bertrand Russell, Introduction to Wittgenstein’s Tractatus Logico-Philosophicus,also in J. R. Newman (ed.) The World of Mathematics, New York: Simon andSchuster, 1956
Extension of Lagrangian formalism of classical mechanics to failuremodelsAllows thinking about complicated problems as
complicated problem = simple problem + technicalitiesTime discretization results in mathematical programs
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Governing dynamics, gentlemen!
Governing equations in the MLF
ddt
(∂Fψ
c(F, ζh, ζs)
)︸ ︷︷ ︸
elastic strain rate
+ ∂Fφc(F, ζh, ζs)︸ ︷︷ ︸
plastic strain rate
− Bv︸︷︷︸total strain rate
3 0
deformationcompatibility
ddt
(∂v(kinetic energy))︸ ︷︷ ︸inertia force
+ ∂v(Rayleigh dissipation)︸ ︷︷ ︸damping force
+ BT F︸︷︷︸internal forces
= P
momentumconservation
ddt
(∂ζhψ
c(F, ζh, ζs)
)︸ ︷︷ ︸
reversible internal variable rate
+ ∂ζhφc(F, ζh, ζs)︸ ︷︷ ︸
irreversible internal variable rate
3 0
ddt
(−∂ζs
[−ψc
(F, ζh, ζs)])
︸ ︷︷ ︸reversible internal variable rate
+ ∂ζsφc(F, ζh, ζs)︸ ︷︷ ︸
irreversible internal variable rate
3 0
internal variableevolution
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
MLF (continued)
Look like Euler-Lagrange equationsTime discretization results in mathematical programs
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Mathematical programs from the MLFMLF representation
Mixed Nonlinear
Complementarity Problem
(MNCP)
Variational integrator (or other)
time discretization
No deterioration
(no softening,
convex energy),
Associated Flow
Quadratic (non-convex) energy,
Piecewise planar yield surfaces
(and possibly non-associated flow)
System of nonlinear equations
and generalized inequalities
Some restrictions
Second Order
Cone Program
(SOCP)
…
Other restrictions
Other restrictions
∙∙∙Convex
Minimization
Convex Quadratic
Program (QP)
Quadratic energy,
Piecewise planar
yield surfaces
Mixed Linear
Complementarity
Problem (MLCP)
Optimization Complementarity
∙∙
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
Theoretical building blocks
OptimizationKKT conditions
Standard MaterialEnergy and dissipation functions
Set-valued derivative
Extending Fenchel duality
MLFEuler-Lagrange equations
Variational integrators
Mathematical programming
ComplementarityTensegrity
Prisoner’s dilemma
A Beautiful Mind
KKT conditions
Emphasis on duality
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Earthquake collapse simulation — Optimization approachGraphics View 1
LARSA 4D Sivaselvan
University of Colorado at BoulderC:\Siva\Publications\EESD_SpecialIssue\AnalysisData\ModelADynamicGeomOn\Model_A_Dynamic.lar
Last Analysis Run : 1/1/2001 1:01:00 AM
Page 1
0 1 2 3 4 5 60
1
2
3
4
5
6
7
8
9
10
time (s)
Num
ber
of It
erat
ions
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
Snapshot at time 7s Number of iterations
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Earthquake collapse (continued)
0 1 2 3 4 5 6 7−10
0
10
time (s)
x−di
sp (
m)
15th story
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
0 1 2 3 4 5 6 7−10
0
10
time (s)
x−di
sp (
m)
10th story
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
0 1 2 3 4 5 6 7−10
0
10
time (s)
x−di
sp (
m)
5th story
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
0 1 2 3 4 5 6 7−10
0
10
time (s)
z−di
sp (
m)
15th story
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
0 1 2 3 4 5 6 7−10
0
10
time (s)
z−di
sp (
m)
10th story
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
0 1 2 3 4 5 6 7−10
0
10
time (s)
z−di
sp (
m)
5th story
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
Horizontal displacement Vertical displacement
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Collapse due to erosion — Optimization approachGraphics View 1
LARSA 4D Sivaselvan
University of Colorado at BoulderC:\Siva\Publications\EESD_SpecialIssue\AnalysisData\ModelARemoval\Model_A_Removal.lar
Last Analysis Run : 1/1/2001 1:01:00 AM
Page 1
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
time (s)
Num
ber
of It
erat
ions
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
Snapshot at 10s Number of iterations
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Erosion collapse (continued)
0 2 4 6 8 10 12−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time (s)
y−di
sp (
m)
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
0 2 4 6 8 10 12−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
time (s)
z−di
sp (
m)
Increment = 0.2sIncrement = 0.02sIncrement = 0.002s
Horizontal displacement Vertical displacement
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Simulation with strength degradation model —Complementarity approach
0 1 2 3 4 5 6 7 8 9 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (s)
x di
spla
cem
ent (
m)
15th story
Increment = 0.02s; Start = LemkeIncrement = 0.02s; Start = ResidualIncrement = 0.002s; Start = LemkeIncrement = 0.002s; Start = ResidualIncrement = 0.001s; Start = LemkeIncrement = 0.001s; Start = Residual
0 1 2 3 4 5 6 7 8 9 10−0.6
−0.4
−0.2
0
0.2
0.4
0.6
time (s)
x di
spla
cem
ent (
m)
10th story
Increment = 0.02s; Start = LemkeIncrement = 0.02s; Start = ResidualIncrement = 0.002s; Start = LemkeIncrement = 0.002s; Start = ResidualIncrement = 0.001s; Start = LemkeIncrement = 0.001s; Start = Residual
0 1 2 3 4 5 6 7 8 9 10−0.4
−0.2
0
0.2
0.4
time (s)
x di
spla
cem
ent (
m)
5th story
Increment = 0.02s; Start = LemkeIncrement = 0.02s; Start = ResidualIncrement = 0.002s; Start = LemkeIncrement = 0.002s; Start = ResidualIncrement = 0.001s; Start = LemkeIncrement = 0.001s; Start = Residual
−0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01−4
−3
−2
−1
0
1
2
3
4x 10
6
Rotation (radians)
Mom
ent (
N−
m)
increment = 0.001s
Horizontal displacements Typical moment-rotation behavior
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Implementation matters — a comparison
OpenSEES — popular research platform in Earthquake Engineering
Compare in problems where algorithms in both codes can compute
500 increments of earthquake input
Model A (15 stories) Model B (50 stories)2,500 DOF; 1,200 elements 30,000 DOF; 16800 elements
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Implementation matters — a comparison (continued)
Code based on MLF Algorithm in OpenSEES
Model A 30 sec 118 sec
Model B 15 min 6 12 hours
Ability to compute relatively large problems
For practical use
Some theoretical difficulties only show up in large problems
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
CPU Time for 15 story model — Optimization approach
Increment 0.2 s 0.02 s 0.002 s
5x El Centro Geom. nonlin. On 17.60 s 121.29 s 1038.39 s
Removal of bottom rows of columns 25.21 s 129.83 s 1163.65 s
CPU times for different simulations of the 15-story structure using a system with a 1.8 GHzIntel Pentium M processor and 1 GB RAM
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
CPU time for 15 story model— Complementarity approach
Time increment Covering vector Number of increments CPU time
(a) Softening only0.02 s Lemke 500 130 s0.02 s Residual 500 88 s0.002 s Lemke 5000 1282 s0.002 s Residual 5000 378 s0.001 s Lemke 10000 2495 s0.001 s Residual 10000 691 s
(b) Ideal plasticity0.02 s Lemke 500 130 s0.02 s Residual 500 85 s0.002 s Lemke 5000 942 s0.002 s Residual 5000 355 s0.001 s Lemke 10000 1752 s0.001 s Residual 10000 631 s
(c) Combined hardening and softening0.02 s Lemke 500 269 s0.02 s Residual N/A Cycling at increment 1490.002 s Lemke 5000 1573 s0.002 s Residual N/A Cycling at increment 11600.001 s Lemke 10000 3151 s0.001 s Residual N/A Cycling at increment 2320
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Linear algebra speedup in complementarity approach
100
101
102
103
250
310
390
480
600
Max number of updates before refactorization + 1
CP
U ti
me
(sec
onds
, log
arith
mic
sca
le)
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Outline
1 BackgroundEngineering questionsComplex dynamics of failure processes
2 Theoretical building blocksComplementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)
3 Computational examplesCapabilities
15 story building modelImplementation mattersCPU times
Curiosities
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Portal frame model
7200
3600
Properties I-sectionColumnsd = 203.2 mmtw = 7.2 mmbf = 203.1 mmtf = 11.1 mm
Beamd = 303.3 mmtw = 7.5 mmbf = 203.3 mmtf = 13.1 mm
W8 ×
31 W12×40
W8 ×31
k1
k0
-k2
-k3
Fy
r1Fy
r2Fy
F
r3Fy
e
Portal frame model Moment-rotation model
0 2 4 6 8 10 12 14 16 18 20−5
−4
−3
−2
−1
0
1
2
3
4
5
time (s)
Gro
und
acce
lera
tion
inpu
t (m
/s2 )
Ground acceleration input
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Computed displacement
Compute using four solution algorithms (two developed by us, and two fromthe PATH solver)
0 2 4 6 8 10 12 14 16 18 20−150
−100
−50
0
50
100
150
time (s)
disp
lace
men
t (m
m)
Approach 1Approach 2Approach 3Approach 4
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
CapabilitiesCuriosities
Recall enumeration from Prisoners’ Dilemma
Solutions by enumeration Approaches 1–3 Approach 4
velocities
206.4301 212.9097 206.4247 206.4182 206.4233
0.1713 -1.1647 0.1724 0.1724 0.17130.2396 -1.6288 0.2412 0.2411 0.2396
206.4301 212.9097 206.4247 206.4182 206.4233-0.1180 0.8021 -0.1188 -0.1187 -0.1180
γ for bottom hinge 0 0 0 0 0γ for top hinge 0 0 0 0 0
λs for bottom hinge
1.1090 0 1.2378 1.2377 1.1089
0 0 0 0 00 0 0 0 00 0 0 0 0
λs for middle hinge{ 0 0 0 0 0
0 0 0 0 0
λs for top hinge
0.1330 6.2970 0 0 0.1330
0 0 0 0 00 0 0 0 00 0 0 0 0
M. V. Sivaselvan Failure simulation
BackgroundTheoretical building blocks
Computational examplesSummary
Summary
Building blocksComplementarity (Prisoner’s dilemma, tensegrity etc.)Optimization (duality)Generalized Standard Material (generalized derivative for nonsmoothenergy functions)Mixed Lagrangian Formalism (MLF)
What is made possibleRobust and efficient simulation of large-scale problemsExploration of multiple solutions etc. (sensitivity)A framework to ask questions about existence, uniqueness
High performance computational plaftformUsed at Kajima Corporation, JapanEarlier versions used in the US through Larsa, Inc.
Further work (topics of CAREER award)Further exploration of theoretical questionsPhysical experiments
M. V. Sivaselvan Failure simulation