theory and computations M. V. Sivaselvanbechtel.colorado.edu/~sivaselv/SivaCEAETalk.pdf · 2010....

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

[email protected]



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




Acknowledgements

Kajima Corporation, Japan (CUREE-Kajima Joint Research Project)

National Science Foundation (CAREER award)




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





Outline





Curiosities





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





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





Outline





Curiosities





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





Complex behavior (2)

Reinforced concrete bridge pier after an earthquake

Plot of Force vs.

Displacement

Laboratory model of a reinforced concrete pier





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 ?





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





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





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)




Complementarity problemsOptimization problemsGeneralized Standard MaterialMixed Lagrangian Formalism (MLF)

Outline





Curiosities





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





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?





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





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





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





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





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





Theoretical building blocks

OptimizationKKT conditions

Standard MaterialEnergy and dissipation functions

Set-valued derivative

Extending Fenchel duality

MLFEuler-Lagrange equations

Variational integrators

Mathematical programming

ComplementarityTensegrity


A Beautiful Mind

KKT conditions

Emphasis on duality





Outline





Curiosities





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















A Beautiful Mind

KKT conditions

Emphasis on duality





Outline





Curiosities





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





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





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





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−





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.





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





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















A Beautiful Mind

KKT conditions

Emphasis on duality





Outline





Curiosities





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





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





MLF (continued)

Look like Euler-Lagrange equationsTime discretization results in mathematical programs





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

∙∙















A Beautiful Mind

KKT conditions

Emphasis on duality




CapabilitiesCuriosities

Outline





Curiosities





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





Earthquake collapse (continued)

0 1 2 3 4 5 6 7−10

0

10

time (s)

x−di

sp (

m)

15th story


0 1 2 3 4 5 6 7−10

0

10

time (s)

x−di

sp (

m)

10th story


0 1 2 3 4 5 6 7−10

0

10

time (s)

x−di

sp (

m)

5th story


0 1 2 3 4 5 6 7−10

0

10

time (s)

z−di

sp (

m)

15th story


0 1 2 3 4 5 6 7−10

0

10

time (s)

z−di

sp (

m)

10th story


0 1 2 3 4 5 6 7−10

0

10

time (s)

z−di

sp (

m)

5th story


Horizontal displacement Vertical displacement





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


Snapshot at 10s Number of iterations





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)


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)


Horizontal displacement Vertical displacement





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


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


−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





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





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





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





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





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)





Outline





Curiosities





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





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





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




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


theory and computations M. V. Sivaselvanbechtel.colorado.edu/~sivaselv/SivaCEAETalk.pdf · 2010....

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