model order reduction techniques - introduction · coursesyllabus modelorderreduction...

Course SyllabusModel Order ReductionReduced Basis Method

Reduced Basis Applications

Model Order Reduction TechniquesIntroduction

M. Grepla & K. Veroy-Greplb

aInstitut für Geometrie und Praktische MathematikbAachen Institute for Advanced Study in

Computational Engineering Science (AICES)

RWTH Aachen

Sommersemester 2012

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General InformationCourse MaterialCourse Outline

Date & Time

I LectureI Monday & Wednesday, 10.00-11.30am, Room 224.3I Start: 04.04.2012 (total 26 lectures)I Any conflicts?

I RecitationI Place and time to be determinedI Homework requires programming in Matlab (or C/C++)

I WebsiteI http://www.igpm.rwth-aachen.de/MOR1_RB

I Assessment (9 ECTS Credits)I Final exam (oral)I Date to be determined

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http://www.igpm.rwth-aachen.de/MOR1_RB




Instructors

I Martin GreplI Room 126, Templergraben 55I Email: [email protected] Phone: 0241/80-96470I Office hours: Monday, 14:00pm-15:00pm (or by appointment)

I Karen Veroy-GreplI Room 421/a, Schinkelstrasse 2I Email: [email protected] Phone: 0241/80-99146I Office hours: by appointment

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Course Material

Primary SourceI Lecture notes are available before class on the website

Reference TextsI A.T. Patera and G. Rozza, Reduced Basis Approximation and

A Posteriori Error Estimation for Parametrized PartialDifferential Equations, Version 1.0, Copyright MIT 2006-2007,to appear in (tentative rubric) MIT Pappalardo GraduateMonographs in Mechanical Engineering.

I A.C. Antoulas, Approximation of Large-Scale DynamicalSystems, SIAM, 2005.

I Website on reduced basis methods:http://augustine.mit.edu/index.htm

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http://augustine.mit.edu/index.htm




Course Outline

Topics to be discussed:

I Reduced Basis MethodsI Structural mechanicsI Parametrized PDEs

I Proper Orthogonal DecompositionI Empirical data to generate eigenfunctions

I Balanced TruncationI Control theoryI Balancing transformation (observability vs. controllability)

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Lecture Outline

I Course Syllabus

I Model Order ReductionI DefinitionI MotivationI Methodologies

I Reduced Basis MethodI Short IntroductionI Applications

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DefinitionMotivationMethodologies

GoalReplicate input-output behavior of large-scale system Σ over acertain (restricted) range of• forcing inputs and• parameter inputs

Forcing

Inputs

Parameter

Inputs µ

Outputs of

Interest s(µ)A(µ)u(µ) = F (µ)

s(µ) = L(µ)Tu(µ)

ΣN :

Large-Scale Model

(wide range of validity)

Outputs of

Interest sN(µ)

Forcing

Inputs

Parameter

Inputs µ

AN(µ)uN(µ) = FN(µ)

sN(µ) = LN(µ)TuN(µ)

ΣN :

Reduced-Order Model

(restricted range of validity)7?

dim(ΣN ) = N N = dim(ΣN)

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Problem Statement

Given large-scale system ΣN of dimension N , find a reduced ordermodel ΣN of dimension N N such that:

I The approximation error is small, i.e., there exists a globalerror bound such that

‖u(µ)− uN(µ)‖ ≤ εdes, and

|s(µ)− sN(µ)| ≤ εsdes, ∀µ ∈ D.

I Stability and passivity are preserved.

I The procedure is computationally stable and efficient.

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Example: Contaminant Transport

x2

x1

!M8 !M

2(xs

1, xs2)

!M1!

Source: Univ. of Houston Civil EngineeringGround Water Contaminant TransportCourse 7332 website

I Governing PDE: Convection-diffusion or Darcy flowI Parameters: fluid flow, source location, release time, diffusivityI Inverse Problem:

Given measurement data for pollutant concentration,compute possible source location and release time.

I Optimization ProblemDesign exit strategy to minimize casualties,or pumping strategy to minimize water contamination

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Example: Welding

Pe = vLc/!

!D

!

dW

x2

1

Measurement 1 Measurement 2

3.5 5x1

!N

I Governing PDE: Convection-diffusionI Parameters:

Heat source profile, material properties, Peclet numberI Outputs:

Average temperature at measurement regionsI Real-Time Estimation & Control:

Given measurement data for temperature, compute heat sourceprofile and strength to achieve desired welding depth

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Example: Multifunctional Structures [V]

(a) (b)

Figure 1-1: Examples of (a) periodic (honeycomb) and (b) stochastic (foam) cellular structures(Photographs taken from [16]).

load

heat, q!!

coolant Phigh, T0

Plow

ft

!S

!tN

!flux

Figure 1-2: A multifunctional (thermo-structural) microtruss structure.

2

(a) (b)


load

heat, q!!

coolant Phigh, T0

Plow

ft

!S

!tN

!flux


2

Source: Gibson & Ashby (1997)

(a) (b)


load

heat, q!!

coolant Phigh, T0

Plow

ft

!S

!tN

!flux


2

Source: Veroy (2003)I Governing PDE:

I Convection-diffusionI Elasticity

I Parameters:I Geometry (thicknesses, angle), material propertiesI Fluid properties, flow (Reynolds number), heat transfer

coefficientsI Design & Optimization:

Given constraints on structural and heat transfer performance,find design which minimizes cost

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Example: Concrete Delamination

Delamination

Heat Flux

FRP laminate

q(t)

x1

Concrete slab

x2

!

!F

wdel

"2 , Measurement 2

!del

, Measurement 1"1

"0,FRP"FRP, cP,FRP, kFRP

"0,C"C, cP,C, kC 1 [kC]

y0(x, t = 0; µ) = 0

I Governing PDE: Heat diffusionI Parameters: Delamination width, relative conductivityI Outputs: Average temperature at measurement regionsI Parameter Estimation:

I Determine delamination width, given uncertainty inconductivity

I Assess structural reliability12 / 72




Forward Problem: Parametrized PDE

I System governed by a PDE(µ) for µ in design space DI PDE(µ) ≈ Finite element approximation ≡ “Truth”with A a linear (distributional) operator, and F a linear functional. The precise definition

of Y , A and F are alluded to the following chapters.

Figure 1-3: Finite element mesh.

For the solution of (1.3), a triangulation Th of the computational domain is introduced,

as in Figure 1-3. We assume that the triangles, also referred to as elements, cover the

computational domain ! ,¯! = !Th!Th

Th (Th is the closure Th) and that each of the elements

do not overlap, T ih " T j

h = 0, #T ih, T j

h $ Th. The subscript h denotes the diameter of the

triangulation defined as:

h = supTh!Th

supx,y!Th

|x % y|; (1.4)

here | · | is the Euclidean norm.

Discrete Problem

Using then the triangulation Th, we define the space Yh as the space of continuous functions

which are piecewise linear over each of the elements Th $ Th:

Yh = v $ C0(!)|v|Th$ P1(Th), #Th $ Th. (1.5)

If N is the number of nodes in the triangulation, we introduce the functions !i $ Yh , such

that !i(xj) = "i j, i = 1, . . . ,N , where xj are the coordinates of node j, and "i i = 1 if i = j,

or "i j = 0 if i &= j . Each function !i has compact support over the region defined by the

elements surrounding node i (shaded area on Figure 1-3). Then, it is not hard to see, that

21

a(u(µ), v;µ) = f(v) for all v ∈ X(Ω(µ))

or A(µ)u(µ) = F

(Mesh - Source: Rovas (2003))

I Parameters µ may include geometry, material properties,loading or boundary conditions

I “Outputs of interest” or performance metrics are oftenlinear functionals of field variables

I Output = ¯(u(µ)) (or LTu(µ))

I For example: average stress, deflection, temperature,flowrate, etc.

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Generalized Inverse Problem

I Given PDE(µ) constraints, find value(s) of parameter µwhich:

I (OPT) minimizes (or maximizes) some functional;I (EST) agrees with measurements;I (CON) makes the system behave in a desired manner;I or some combination of the above

I Full solution computationally very expensive due torepeated evaluation for many different values of µ

Our optimization problem can then be stated as: find µ! = t!t , t!b , t

!, !!, which satisfies

find µ! = arg minµ

J (µ) (7.42)

subject to

!""""""""""""""""""""""""#""""""""""""""""""""""""$

f0(µ) = "(µ) ! "0 = 0,

f1(µ) = ttop ! 0.022 " 0,

f2(µ) = 0.22 ! ttop " 0,

f3(µ) = tbot ! 0.022 " 0,

f4(µ) = 0.22 ! tbot " 0,

f5(µ) = t ! 0.022 " 0,

f6(µ) = 0.22 ! t " 0,

f7(µ) = ! " 0,

f8(µ) = 45 ! ! " 0,

g1(µ) = !1#max ! #ave(µ) " 0,

g2(µ) = !2$Y ! $ave(µ) " 0,

h1(µ) = "(µ) ! "0 = 0,

7.4.2 Solution Methods

We now consider methods for solving general optimization problems of the form (7.32). In par-ticular, we focus on interior point methods, computational methods for the solution of constrainedoptimization problems which essentially generate iteratetes which are strictly feasible (i.e., in theinterior of the feasible region) and converge to the true solution. The constrained problem is re-placed by a sequence of unconstrained problems which involve a barrier function which enforcesstrict feasibility and e!ectively prevents the approach to the boundary of the feasible region [9].The solutions to these unconstrained problems then approximately follow a “central path” to thesolution of the original constrained problem; this is depicted in Figure 7-11. We present here aparticular variant of IPMs known as primal-dual algorithms.

Figure 7-11: Central path.

To begin, we introduce the modified optimization problem

find µ!! = arg min

µJ!(µ) , (7.43)

152

I Goal: Low average cost or real-time online response14 / 72




Model Order Reduction Techniques




I Krylov Subspace MethodsI Arnoldi, Lanczos methods (factorization)

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Model Order Reduction Techniques




I Krylov Subspace MethodsI Arnoldi, Lanczos methods (factorization)

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ApproximationA Posteriori Error EstimationGreedy AlgorithmSummary

The Main Idea – Key Obervation

u(µ)

VFE SPACE

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FE SPACE

V

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FE SPACE

u(µi)SNAPSHOTS

V

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FE SPACE

V

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

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FE SPACE

V

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

uN(µ)APPROXIMATION

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FE SPACE

APPROXIMATION

V

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

uN(µ)

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ERROR BOUND

uN(µ)APPROXIMATION

VFE SPACE

∆N(µ)

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

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Assumption: Affine Decomposition

I Given µ ∈ D, the “truth” solution u(µ) ∈ X satisfies

a(u(µ), v;µ) = f(v) for all v ∈ X (1)

or A(µ)u(µ) = F

where a is continuous, coercive, and permits an affinedecomposition

a(w, v;µ) =

Q∑

q=1

Θq(µ) aq(w, v)︸︷︷︸µ−independent

(2)

or A(µ) =

Q∑

q=1

Θq(µ)︷︸︸︷Aq (3)

I Example: the Laplacian (∇2) on a rectangle24 / 72




Assumption: Affine Decomposition

Original Domain Ω(µ)

µ = h

a(w, v;µ) =

∫Ω(µ)

∂w

∂x

∂v

∂xdΩ

+

∫Ω(µ)

∂w

∂y

∂v

∂ydΩ

Reference Domain Ω

1

a(w, v;µ) =1

h

∫Ω

∂w

∂x

∂v

∂xdΩ

+ h

∫Ω

∂w

∂y

∂v

∂ydΩ

since∂

∂x=

1

h

∂

∂x∂

∂y=

∂

∂y

dΩ = h dΩ

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Reduced Basis Space and Galerkin Projection

I Take “snapshots” at different parameter values µi,i = 1, . . . , N , and set XN = spanu(µi), i = 1, . . . , N.

I Parameter samples µi are “optimally” chosenI Given a new µ, calculate approximation uN(µ) to u(µ) by a

linear combination of the snapshotsI Compute uN(µ) in XN using Galerkin projection

a(uN(µ), v;µ) = f(v), ∀v ∈ XN ,

or ZTA(µ)Z︸︷︷︸ uN(µ) = ZTF︸︷︷︸AN(µ) uN(µ) = FN

Columns of Z are orthonormalized basis functions ζi “=” u(µi)

I Reduced-basis dimension dim(XN) = N NFEM

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Offline-Online Computational Decomposition

I Expanding a(w, v;µ) and uN(µ), and choosing v = ζiQ∑

q=1

N∑

i=1

Θq(µ) aq(ζj, ζi)uN i(µ) = f(ζi)

Q∑

q=1

Θq(µ)ZTAqZ uN(µ) = ZTF

I OFFLINE: Calculate solutions u(µi) (and compute the ζi)Form and store the ZTAqZ ∈ RN×N , ZTF ∈ RN

ONLINE: Given a new µ,Compute the sum at cost O(QN2)Solve for uN(µ) at cost O(N3)

⇒ Online cost independent of N27 / 72




Real-Time Approximation

I Method converges very fast: the error

‖u(µ)− uN(µ)‖Xor [(u− uN)TX(u− uN)]

12

decreases rapidly with NI N can be taken to be very small compared to NFEM

I Online cost to compute uN(µ) is very small compared to afull solve for u(µ)

BUTHow do we know the error is small?How do we know what value of N to take?How do we choose the sample points µi optimally?

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An Upper Bound to the Error

We introduce a rapidly computable error bound

∆N(µ) =εN(µ)

αLB(µ)≥ ‖u(µ)− uN(µ)‖X (4)

for all µ ∈ D, where

εN(µ), the dual norm of the residual a(uN(µ), v;µ)− f(v),

αLB(µ), a lower bound to the coercivity constant of a(·, ·;µ),

also permit an offline-online decomposition.

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Dual Norm of the Residual

I The dual norm of the residual is given by

εN(µ) = supv∈X

a(uN(µ), v;µ)− f(v)

‖v‖X(5)

or =[(A(µ)Z uN − F )TX−1(A(µ)Z uN − F )

]12

I By expanding a(·, ·;µ) and uN(µ), the quantity εN(µ) canbe computed using an offline-online decomposition

OFFLINE: O(QNN ∗FEM) to do the "X-solves"

O(Q2N2NFEM) to do the µ-independent products

ONLINE: O(Q2N2) to evaluate the sum

⇒ Online cost independent of N30 / 72




Lower Bound to the Coercivity Constant

I We also require αLB(µ)

0 < αLB(µ) ≤ α(µ) = infw∈X

a(w,w;µ)

‖w‖2X(6)

or = infw∈RN

wTA(µ)w

wTXw

which we findI "by inspection" for easy problems, orI using the Successive-Constraint-Method [Huynh, et al (2007)]

OFFLINE: Solve standard eigenproblems

ONLINE: Solve a linear program with Q variables

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Greedy Algorithm for Optimal Samples

Given XN , we choose the next sample as follows:

µ1

∆N

µ2

µN+1 = arg maxµ∈DJ

∆N(µ)

‖uN(µ)‖X

XN+1 = XN ⊕ spanu(µN+1)

I Key point: ∆N(µ) is sharp and inexpensive to compute(online)

I Error bound ∆N ⇒ "optimal" samples⇒ goodapproximation uN

In Summary . . .I Reduced basis approximation provides certifiably accurate

inexpensive approximations to solutions of parametrized PDEs

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µ3

∆N

µ2µ1


∆N(µ)

‖uN(µ)‖X






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µ3

∆N

µ2µ1


∆N(µ)

‖uN(µ)‖X






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µ3

∆N

µ2µ1


∆N(µ)

‖uN(µ)‖X






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µ3

∆N

µ2µ1


∆N(µ)

‖uN(µ)‖X





inexpensive approximations to solutions of parametrized PDEs36 / 72




RB Opportunities

Computational Opportunities

I. We restrict our attention to the typically smooth andlow-dimensional manifold induced by the parametricdependence.⇒ Dimension reduction

II. We accept greatly increased offline cost in exchange for greatlydecreased online cost.⇒ Real-time and/or many-query context

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RB Relevance

Real-Time Context (control, . . . ):

µ → sN(µ), ∆sN(µ).

t0 (“need”) t0 + ∂tcomp (“response”)

Many-Query Context (design, . . . ):

µj → (sN(µj), ∆sN(µj)), j = 1, . . . , J .

t0 t0 + ∂tcomp J as J →∞

⇒ Low marginal (real-time) and/or low average (many-query) cost.

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RB Key Challenges

I A Posteriori error estimationI Rigorous error bounds for outputs of interestI Lower bounds to the stability “constants”

I Offline-online computational proceduresI Full decoupling of finite element and reduced basis spacesI A posteriori error estimationI Nonaffine and nonlinear problems

I Effective sampling strategiesI High parameter dimensions

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RB Outline

1. Affine Elliptic ProblemsI (non)symmetric, (non)compliant, (non)coerciveI (Convection)-diffusion, linear elasticity, Helmholtz

2. Affine Parabolic ProblemsI (Convection)-diffusion equation

3. Nonaffine and Nonlinear ProblemsI Nonaffine parameter dependence, nonpolynomial nonlinearities

4. Reduced Basis (RB) Method for Fluid FlowI Saddle-Point Problems (Stokes)I Navier-Stokes Equations

5. ApplicationsI Parameter Optimization and Estimation (Inverse Problems)I Optimal Control

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Concrete DelaminationProblems in ElasticityContaminant TransportGMA Welding Process

Concrete Delamination [HJN], [S]

Delamination

Heat Flux

FRP laminate

q(t)

x1

Concrete slab

x2

κ

ΓF

wdel

Ω2 , Measurement 2

Γdel

, Measurement 1Ω1

Ω0,FRP%FRP, cP,FRP, kFRP

Ω0,C%C, cP,C, kC 1 [kC]

y0(x, t = 0;µ) = 0

Input (parameter): µ ≡ (wdel/2, κ ≡ kFRP/kC)

Output of interest: si(t;µ) =∫Ωiy0(x, t;µ), i = 1, 2

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Concrete Delamination – Problem Statement

Given (µ1, µ2) ∈ D ≡ [1, 10]× [0.4, 1.8], evaluate the outputs,

for k = 1, . . . , 200, (∆t = 0.05, tk ∈ (0, 10]),

Si(tk;µ) =

1

|Ωi|

∫

Ωi

y0(tk;µ), i = 1, 2

TS(tk;µ) = S1(tk;µ)− S2(tk;µ) ,

where y0(tk;µ) ∈ Y0(Ω0(µ1)) satisfies†

† Here, Y0 ≡ v ∈ H1(Ω0(µ1))| v|Γbottom = 0; y0(t0;µ) = 0.

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Concrete Delamination – Problem Statement

1

∆t

∫

Ω0(µ1)(y0(tk;µ)− y0(tk−1;µ)) v0

+ µ2

∫

Ω0,FRP(µ1)∇y0(tk;µ) · ∇v0

+

∫

Ω0,C(µ1)∇y0(tk;µ) · ∇v0 = u(tk)

∫

ΓF

v0 ,

∀v0 ∈ Y0,

where u(tk) is specified “in the field.” †

† Reduced Basis is trained on impulse (LTI).

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Concrete Delamination – Results

Temperature distribution: wdel/2 = 5, κ = 1

k = 10 k = 20

k = 40 k = 60

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Thermal signal TSe(tk;µ)

κ = 1

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time t

Ther

mal

Sig

nal

µ1 = 1

µ1 = 2

µ1 = 3

µ1 = 5

µ1 = 10

wdel/2 = 3

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time t

Ther

mal

Sig

nal

µ2 = 0.4

µ2 = 0.6

µ2 = 1

µ2 = 1.8

45 / 72





MATLAB DEMO

46 / 72




Problems in Elasticity [V]

I Application: Lightweight Multifunctional Materials

(a) (b)


load

heat, q!!

coolant Phigh, T0

Plow

ft

!S

!tN

!flux


2

(a) (b)


load

heat, q!!

coolant Phigh, T0

Plow

ft

!S

!tN

!flux


2

Source: Gibson & Ashby, 1997

(a) (b)


load

heat, q!!

coolant Phigh, T0

Plow

ft

!S

!tN

!flux


2

open cells; the coolant enters the inlet at a temperature T0, and is forced through the cells by apressure drop !P = Phigh! Plow from the inlet to the outlet. In addition, the microtruss transmits

a force per unit depth ft uniformly distributed over the tip "N through the truss system to thefixed left wall "D. We assume that the associated Reynolds number is below the transition valueand that the structure is su#ciently deep such that a physical model of fully-developed, laminarfluid flow and plane strain (two-dimensional) linear elasticity su#ces.

1.1.1 Inputs

The structure, shown in Figure 1-3, is characterized by a seven-component nondimensional param-eter vector or “input,” µ = (µ1, µ2, . . . , µ7), reflecting variations in geometry, material property,and loading or boundary conditions. Here,

µ1 = t = thickness of the core trusses,µ2 = tt = thickness of the upper frame,µ3 = tb = thickness of the lower frame,µ4 = H = separation between the upper and lower frames,µ5 = ! = angle (in degrees) between the trusses and the frames,µ6 = k = thermal conductivity of the solid relative to the fluid, andµ7 = p = nondimensional pressure gradient;

furthermore, µ may take on any value in a specified design space, Dµ " IR7, defined as

Dµ = [0.1, 2.0]3 # [6.0, 12.0] # [35.0, 70.0] # [5.0 # 102, 1.0 # 104] # [1.0 # 10!2, 1.1 # 102],

that is, 0.1 $ t, tt, tb $ 2.0, 6.0 $ H $ 12.0, 35.0 $ ! $ 70.0, 5.0 # 102 $ k $ 1.0 # 104, and1.0 # 10!2 $ p $ 1.1 # 102. The thickness of the sides, ts, is assumed to be equal to tb.

tb

ttt

!Hts(= tb)

Figure 1-3: Geometric parameters for the microtruss structure.

1.1.2 Governing Partial Di!erential Equations

In this section, (and in much of this thesis) we shall omit the spatial dependence of the fieldvariables. Furthermore, we shall use a bar to denote a general dependence on the parameter; forexample, since the domain itself depends on the (geometric) parameters, we write $ % $(µ) todenote the domain, and x to denote any point in $. Also, we shall use repeated indices to signifysummation.

3

Source: Veroy (2003)

47 / 72





I Governing Equations: PDE(µ) of Linear Elasticity

a(u(µ), v;µ) =

∫

Ω

∂vi

∂xjCijkl(µ)

∂uk

∂xl= f(v), for all v in X

I Outputs: Average stress, average deflectionI Sample results:

0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

400

ttop

Ave

rag

e D

efle

ctio

n

Finite Element Solution (n=13,000)Reduced Basis Solution (N=30)

0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

400

tbot

Ave

rag

e D

efle

ctio

n


Figure 7-5: Plots of the average deflection as a function of ttop and tbot.

143

0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

400

ttop

Ave

rag

e D

efle

ctio

n


0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

400

tbot

Ave

rag

e D

efle

ctio

n


Figure 7-5: Plots of the average deflection as a function of ttop and tbot.

143

48 / 72





I Goal: Design & OptimizationI Problem Statement:

minµ∈D

Material Cost (Area)

subject to constraints:

stress < σmax

deflection < δmax

I Sample results:

necessarily optimal. In Scenario 2, we minimize the area of the structure while allowing ttop tovary. We then find that the optimal value is ttop = 0.507, resulting in a 30% reduction in the costfunction (area) compared to the results of Scenario 1. We also note that in this case, the yieldingconstraint on the stress is active. In Scenario 3, we allow both ttop and tbot to vary. We then findthat the cost can be reduced further by 30% compared to the results of Scenario 2. Finally, weallow ttop, tbot, and t to vary, and find that the cost can still be reduced by another 10%; note thatin this case, both the deflection and stress constraints are active.

Scenario ttop(mm) tbot(mm) t(mm) !(!) V(mm2) "+N (mm) #+

N (MPa) time (s)

1 1.500 0.500 0.500 54.638 50.04 0.0146 09.227 0.680

2 0.507 0.500 0.500 54.638 35.14 0.0200 30.000" 1.020

3 0.523 0.200" 0.500 53.427 25.65 0.0277 30.000" 1.050

4 0.521 0.224 0.345 52.755 23.02 0.0300" 30.000" 1.330

Table 7.14: Optimization of the microtruss structure (for H = 9mm) using reduced-basis outputbounds. (These results were obtained in collaboration with Dr. Ivan Oliveira of MIT, and are usedhere with permission.)

The solution of the optimization problem for each scenario requires O(10) deflection and stresscalculations. As shown in Table 7.14, our reduced-basis solution method therefore e!ectively solves— on-line — O(10) partial di!erential equations within a single second. In contrast, matrix assem-bly and solution (using non-commercial code) of the finite element equations for a single value ofµ takes approximately 9 seconds. The online computational savings e!ected by the reduced-basismethod is clearly no small economy.

7.5 Prognosis: An Assess-(Predict)-Optimize Approach

The design of an engineering system, as illustrated in Section 1.1.4, involves the determination ofthe system configuration based on system requirements and environment considerations. Duringoperation, however, the state of the system may be unknown or evolving, and the system may besubjected to dynamic system requirements, as well as changing environmental conditions. The sys-tem must therefore be adaptively designed and optimized, taking into consideration the uncertaintyand variability of system state, requirements, and environmental conditions.

For example, we assume that extended deployment of our microtruss structure (for instance,as a component in an airplane wing) has led to the developement of defects (e.g., cracks) shown inFigure 7-12. The characteristics of the defects (e.g., crack lengths) are unknown, but we assumethat we are privy to a set of experimental measurements which serve to assess the state of thestructure. Clearly, the defects may cause the deflection to reach unacceptably high values; a shimis therefore introduced so as to sti!en the structure and maintain the deflection at the desiredlevels. However, this intervention leads to an increase in both material and operational costs. Ourgoal is to find, given the uncertainties in the crack lengths, the shim dimensions which minimizethe weight while honoring our deflection constraint.

155

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Contaminant Transport

I Application: control of emission (Source: Dede (2008))

I Application: Identification of sources

Airborne contaminants Airborne contaminantsin urban canyon. in LA basis.

Source: Bashir et. al. 2008 Source: Akcelik et. al. 2006

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Contaminant Transport

I Application: Identification of Sources

Dispersion of a pollutant Ω = [0, 4]× [0, 1]

x2

x1

ΩM8 ΩM

2(xs1, x

s2)

ΩM1κ

Source: gPS(x;µ) = 50πe−50((x1−xs1)2+(x2−xs2)2)

(say, µ ≡ (κ, xs1, xs2))

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Contaminant Transport – Problem Statement

Scalar Convection-Diffusion y(x, t = 0;µ) = 0

∂∂ty(t;µ) + U · ∇y(t;µ) = κ∇2y(t;µ) + gPS(x;µ)u(t),

INPUTS : µ ≡ (κ, xs1, xs2) ∈ D ⊂ IRP=3, where

D = [0.05, 0.5]× [2.9, 3.1]× [0.3, 0.5];

U(Gr = 105) from Pr = 0

Natural Convection (Navier-Stokes);

u(t) “control” input (source strength).

OUTPUTS : Measurements sq(t;µ), 1 ≤ q ≤ 8.

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Contaminant Transport – Sample Solutions

Field variable: µ = (0.05, 2.9, 0.3) (N = 3720)

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Contaminant Transport – Sample Solutions

Field variable: µ = (0.05, 3.1, 0.5) (N = 3720)

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Inverse Problem

Determine: µ∗ ∈ D (actual value)

Given experimental data

measurements : z(tk) ∈ Zkexp, ∀k ∈ IKexp, where

Zkexp ≡ [sN (tk;µ∗)− εexp, sN (tk;µ∗) + εexp]

observations : IKexp ⊂ IK

error : εexp ∈ IR (bounded, “white”)

input : u(tk) = δ1k, ∀k ∈ IK

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Inverse Problem – (Regularized) Solution

Given noisy measurements, z(tk), k ∈ IKexp, solve

I Output least squares problem

µ = arg minµ∈D

12

IKexp∑k=1

‖sN (tk;µ)− z(tk)‖2W

s.t. PDEN (µ) being satisfied; or

I Regularized problem

µ = arg minµ∈D

12

IKexp∑k=1

‖sN (tk;µ)− z(tk)‖2W + 12δRR(µ)

s.t. PDEN (µ) being satisfied.

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µ = arg minµ∈D

12

IKexp∑k=1

‖sN (tk;µ)− z(tk)‖2W

s.t. PDEN (µ) being satisfied; or


µ = arg minµ∈D

12

IKexp∑k=1

‖sN (tk;µ)− z(tk)‖2W + 12δRR(µ)

s.t. PDEN (µ) being satisfied.

⇒ Solution very expensive: N -dependent cost

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µ = arg minµ∈D

12

IKexp∑k=1

‖sN(tk;µ)− z(tk)‖2W

s.t. PDEN(µ) being satisfied; or


µ = arg minµ∈D

12

IKexp∑k=1

‖sN(tk;µ)− z(tk)‖2W + 12δRR(µ)

s.t. PDEN(µ) being satisfied.

⇒ Surrogate model approach: N -dependent cost

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Robust Parameter Estimation

⇒ “Classic” solution neglects measurement errors εexp

(Truth) “Uncertainty Region” Approach

UN ≡ µ ∈ D | sN (tk;µ) ∈ Zkexp, ∀k ∈ IKexp(OR BN ≡ [µmin = min

µ∈UNµ, µmax = max

µ∈UNµ])

. . . all parameter values in D consistent with experimental data→ µ∗ ∈ UN ⊂ BN

but expensive to construct.Goal: Approximation UN to UN , such that

– UN ⊂ UN , and hence µ∗ ∈ UN RELIABILITY

– UN is inexpensive to construct EFFICIENCY

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Robust Parameter Estimation

Define

IkN(µ) ≡ [sN(tk;µ)−∆sN(tk;µ), sN(tk;µ) + ∆s

N(tk;µ)],

∀k ∈ IKexpand

UN ≡µ ∈ D | IkN(µ) ∩ Zkexp 6= ∅, ∀k ∈ IKexp

.

We then obtain: UN ⊂ UN → µ∗ ∈ UNUN reflects uncertainty in

– experimental data through Zkexp

– RB approximation through IkN(µ)

UN → UN as ∆sN(tk;µ)→ 0 ACCURACY

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Contaminant Transport – Sample Solution

Solve for µIC & UN : TCPU = 1.35 + 70.0 sec †

(892 forward solutions, J = 72)

2.95 2.955 2.96 2.965 2.970.45

0.455

0.46

0.465

0.47

µ1

µ 2

µ*

µIC

Here, εexp = 1.0%, N = 120, M = 40,IKexp = 10, 20, . . . , 200, Sexp = 1, 2, 3, 4

†MATLAB 7.5 on Intel DualCore 1.8GHz61 / 72




Contaminant Transport – Sensitivity: ∆sN,M and εexp

2.95 2.96 2.97 2.980.445

0.45

0.455

0.46

0.465

0.47

µ1

µ 2

µ*

N = 100, M = 35N = 120, M = 40N = 140, M = 45N = 160, M = 50

2.92 2.94 2.96 2.98 3 3.020.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

µ1

µ 2

µ*

εexp

= 5 %

εexp

= 2.5 %

εexp

= 1 %

εexp

= 0.5 %

Here, IKexp = 10, 20, . . . , 100, Sexp = 1, 2, 3, 4

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Contaminant Transport – Sensitivity: uncertainty in κ

Increased uncertainty in (µ2, µ3) due to unknown κ

2.93 2.94 2.95 2.96 2.97 2.98 2.990.43

0.44

0.45

0.46

0.47

0.48

0.49

µ2

µ 3

Here, µ∗ = (0.06, 2.96, 0.46), εexp = 1.0%, N = 120,M = 40, IKexp = 10, 20, . . . , 200, Sexp = 1, 2, 3, 4

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Contaminant Transport – Sensitivity: release time

2.92 2.94 2.96 2.98 30.34

0.36

0.38

0.4

0.42

µ2

µ 3

µ*

k = 50k = 51k = 52k = 49k = 48

Here, εexp = 1.0%, µ∗ = (0.06, 2.96, 0.38), tkrel = 50,N = 120, M = 40, IKexp = 10, 20, . . . , 200,Sexp = 1, 2, 3, 4

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GMA Welding Process [SH93],[SH94]

I Application: Real-time parameter estimation and control

Welding Process Ω = [0, 5]× [0, 1]Pe = vLc/κ

ΓD

κ

dW

x2

1

Measurement 1 Measurement 2

3.5 5x1

ΓN

Torch: qw(x;µ) = ηw

2πσ2we−((x1−3.5)2+(x2−1)2)/(2σ2

w),

µ ≡ (ηw, σw)

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GMA Welding Process [SH93],[SH94]

I Application: Real-time parameter estimation and control

Introduce “fictitious” output s3 Ω = [0, 5]× [0, 1]Pe = vLc/κ

ΓD

κ

x2

3.5 5x1

0.5

1

1.5

ΓN

s1 s2

s3

Torch: qw(x;µ) = ηw

2πσ2we−((x1−3.5)2+(x2−1)2)/(2σ2

w),

µ ≡ (ηw, σw)

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GMA Welding Process – Problem Statement

Scalar Convection-Diffusion y(x, t = 0;µ) = 0

∂∂ty(t;µ) + Pe · ∂

∂xy(t;µ) = κ∇2y(t;µ) + qw(x;µ)u(t),

INPUTS : µ ≡ (ηw, σw) ∈ D ⊂ IRP=2, where

D = [0.1, 0.4]× [0.15, 0.65];

Torch velocity Pe ;

u(t) “control” input (source strength).

OUTPUTS : Measurements 1 & 2.

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GMA Welding Process – Sample Solution

Field variable: µ = (0.3, 0.4) (N = 3720)

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GMA Welding Process – Results

Approach to real-time parameter estimation and control:

1. Start welding with nominal control un(t)

2. Take temperature measurements z1,2(t) of outputs s1,2(t;µ)

3. Solve parameter estimation problem for µ∗

⇒ PDE(N)(µ)-constrained optimization problem4. Given µ∗, solve optimal control problem for u∗(t)⇒ PDE(N)(µ)-constrained optimization problem

5. Apply optimal control law u∗(t)(6. Go to 2. - Model Predictive Control)

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GMA Welding Process – Results

Parameter estimation & control: µ∗ = (0.34, 0.46), sd,3(t) = 1

µIC = (0.339, 0.463)

εexp = 1%, fs = 5 Hz

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

u* (tk )

0 1 2 3 4 5 6 7 8 9 100

0.250.5

0.751

1.25

s 3(µ* ,tk )

0 1 2 3 4 5 6 7 8 9 1010

−410

−310

−210

−110

0

time t

|s3* −

s3(µ

* ,tk )|

µIC = (0.334, 0.473)

εexp = 5%, fs = 5 Hz

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

u* (tk )

0 1 2 3 4 5 6 7 8 9 100

0.250.5

0.751

1.25

s 3(µ* ,tk )

0 1 2 3 4 5 6 7 8 9 1010

−410

−310

−210

−110

0

time t

|s3* −

s3(µ

* ,tk )|

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ThermalBlock

Governing PDE: Parameters: Outputs:

Heat Diffusion Conductivities µi Average Temp.

Source: A.T. Patera

Reduced Basis Methods — Supercomputing on a phoneCourtesy of: D.B.P. Huynh, D.J. Knezevic, and A.T. Patera

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Summary

Many problems in computational engineering require

many or real-time evaluations ofPDE(µ)-induced

input-output relationships.

Reduced-basis methods enable

certified, real-time calculationof outputs of PDE(µ)

for parameter estimation, optimization, and control.

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