Contents CourseOutline CourseOutline Topicstobediscussed: ReducedBasisMethods – Structuralmechanics – ParametrizedPDEs ProperOrthogonalDecomposition

Contents1 Course Syllabus 1

1.1 General Information . . . . . . . . . . . . . . . . . . . . . . . 11.2 Course Material . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Course Outline . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Model Order Reduction 32.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Reduced Basis Method 83.1 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 A Posteriori Error Estimation . . . . . . . . . . . . . . . . . 123.3 Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Reduced Basis Applications 154.1 Concrete Delamination . . . . . . . . . . . . . . . . . . . . . 154.2 Problems in Elasticity . . . . . . . . . . . . . . . . . . . . . . 174.3 Contaminant Transport . . . . . . . . . . . . . . . . . . . . . 184.4 GMA Welding Process . . . . . . . . . . . . . . . . . . . . . 22

1 Course Syllabus

1.1 General InformationDate & Time

• Lecture

– Monday & Wednesday, 10.00-11.30am, Room 224.3

– Start: 04.04.2012 (total 26 lectures)

– Any conflicts?

• Recitation

– Place and time to be determined

– Homework requires programming in Matlab (or C/C++)

• Website

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

1

• Assessment (9 ECTS Credits)

– Final exam (oral)

– Date to be determined

Instructors

• Martin Grepl

– Room 126, Templergraben 55

– Email: [email protected]

– Phone: 0241/80-96470

– Office hours: Monday, 14:00pm-15:00pm (or by appointment)

• Karen Veroy-Grepl

– Room 421/a, Schinkelstrasse 2

– Email: [email protected]

– Phone: 0241/80-99146

– Office hours: by appointment

1.2 Course MaterialCourse Material

Primary Source

• Lecture notes are available before class on the website

Reference Texts

• A.T. Patera and G. Rozza, Reduced Basis Approximation and A Poste-riori Error Estimation for Parametrized Partial Differential Equations,Version 1.0, Copyright MIT 2006-2007, to appear in (tentative rubric)MIT Pappalardo Graduate Monographs in Mechanical Engineering.

• A.C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM,2005.

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

2

1.3 Course OutlineCourse Outline

Topics to be discussed:

• Reduced Basis Methods

– Structural mechanics

– Parametrized PDEs

• Proper Orthogonal Decomposition

– Empirical data to generate eigenfunctions

• Balanced Truncation

– Control theory

– Balancing transformation (observability vs. controllability)

Lecture Outline

• Course Syllabus

• Model Order Reduction

– Definition

– Motivation

– Methodologies

• Reduced Basis Method

– Short Introduction

– Applications

2 Model Order Reduction

2.1 Definition

GoalReplicate input-output behavior of large-scale system Σ over a certain (re-stricted) range of

• forcing inputs and• parameter inputs

3

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

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

(wide range of validity) (restricted range of validity)7?

dim(ΣN ) = N N = dim(ΣN )

Problem StatementGiven large-scale system ΣN of dimension N , find a reduced order model

ΣN of dimension N N such that:

• The approximation error is small, i.e., there exists a global error boundsuch that

‖u(µ)− uN(µ)‖ ≤ εdes, and

|s(µ)− sN(µ)| ≤ εsdes, ∀µ ∈ D.

• Stability and passivity are preserved.

• The procedure is computationally stable and efficient.

2.2 MotivationExample: Contaminant Transport

x2

x1

!M8 !M

2(xs

1, xs2)

!M1!

Source: Univ. of Houston Civil EngineeringGround Water Contaminant Transport

Course 7332 website

• Governing PDE: Convection-diffusion or Darcy flow

• Parameters: fluid flow, source location, release time, diffusivity

• Inverse Problem:Given measurement data for pollutant concentration,compute possible source location and release time.

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

4

Example: WeldingPe = vLc/!

!D

!

dW

x2

1

Measurement 1 Measurement 2

3.5 5x1

!N

• Governing PDE: Convection-diffusion

• Parameters: Heat source profile, material properties, Peclet number

• Outputs: Average temperature at measurement regions

• Real-Time Estimation & Control:

Given measurement data for temperature, compute heat source

profile and strength to achieve desired welding depth

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)

• Governing PDE:

– Convection-diffusion

– Elasticity

• Parameters:

– Geometry (thicknesses, angle), material properties

– Fluid properties, flow (Reynolds number), heat transfer coefficients

• Design & Optimization:

Given constraints on structural and heat transfer performance,

find design which minimizes cost

5

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

• Governing PDE: Heat diffusion

• Parameters: Delamination width, relative conductivity

• Outputs: Average temperature at measurement regions

• Parameter Estimation:

– Determine delamination width, given uncertainty in conductivity– Assess structural reliability

Forward Problem: Parametrized PDE

• System governed by a PDE(µ) for µ in design space D

• 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))

• Parameters µmay include geometry, material properties, loading or bound-ary conditions

• “Outputs of interest” or performance metrics are often linear functionalsof field variables

– Output = ¯(u(µ)) (or LT u(µ))– For example: average stress, deflection, temperature,

flowrate, etc.

6

Generalized Inverse Problem

• Given PDE(µ) constraints, find value(s) of parameter µ which:

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

• Full solution computationally very expensive due to repeated evaluationfor 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

• Goal: Low average cost or real-time online response

• Parameter EstimationGiven: Measurements → feasible parameter µfeas

• Optimal ControlGiven: Objective Function → control input u(t)

• Parameter OptimizationGiven: Objective Function → opt. parameter µ∗

2.3 MethodologiesModel Order Reduction Techniques

• Reduced Basis Methods Reduced Basis Methods

– Structural mechanics– Parametrized PDEs

• Proper Orthogonal Decomposition

– Empirical data to generate eigenfunctions

• Balanced Truncation

– Control theory– Balancing transformation (observability vs. controllability)

• Krylov Subspace Methods

– Arnoldi, Lanczos methods (factorization)

7

3 Reduced Basis Method

3.1 ApproximationThe Main Idea – Key Obervation

u(µ)

VFE SPACE

FE SPACE

V

8

FE SPACE

u(µi)SNAPSHOTS

V

FE SPACE

V

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

FE SPACE

V

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

uN(µ)APPROXIMATION

9

FE SPACE

APPROXIMATION

V

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

uN(µ)

ERROR BOUND

uN(µ)APPROXIMATION

VFE SPACE

∆N(µ)

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

Assumption: Affine Decomposition

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

a(u(µ), v;µ) = f(v) for all v ∈ X

or A(µ)u(µ) = F

where a is continuous, coercive, and permits an affine decomposition

a(w, v;µ) =

Q∑

q=1

Θq(µ) aq(w, v)︸︷︷︸µ−independent

or A(µ) =

Q∑

q=1

Θq(µ)︷︸︸︷Aq

10

• Example: the Laplacian (∇2) on a rectangle

Reduced Basis Space and Galerkin Projection

• Take “snapshots” at different parameter values µi, i = 1, . . . , N , and setXN = spanu(µi), i = 1, . . . , N.

• Parameter samples µi are “optimally” chosen

• Given a new µ, calculate approximation uN(µ) to u(µ) by a linearcombination of the snapshots

• 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)

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

Offline-Online Computational Decomposition

• Expanding a(w, v;µ) and uN(µ), and choosing v = ζi

Q∑

q=1

N∑

i=1

Θq(µ) aq(ζj, ζi)uN i(µ) = f(ζi)

Q∑

q=1

Θq(µ)ZTAqZ uN(µ) = ZTF

• 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 N

11

Real-Time Approximation

• Method converges very fast: the error

‖u(µ)− uN(µ)‖Xor [(u− uN)TX(u− uN)]

12

decreases rapidly with N

• N can be taken to be very small compared to NFEM

• Online cost to compute uN(µ) is very small compared to a full solve foru(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?

3.2 A Posteriori Error EstimationAn Upper Bound to the Error

We introduce a rapidly computable error bound

∆N(µ) =εN(µ)

αLB(µ)≥ ‖u(µ)− uN(µ)‖X

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.

Dual Norm of the Residual

• The dual norm of the residual is given by

εN(µ) = supv∈X

a(uN(µ), v;µ)− f(v)

‖v‖X

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

] 12

• By expanding a(·, ·;µ) and uN(µ), the quantity εN(µ) can be computedusing an offline-online decomposition [1ex]

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 N

12

Lower Bound to the Coercivity Constant

• We also require αLB(µ)

0 < αLB(µ) ≤ α(µ) = infw∈X

a(w,w;µ)

‖w‖2X(1)

or = infw∈RN

wTA(µ)w

wTXw

which we find

– "by inspection" for easy problems, or

– using the Successive-Constraint-Method [Huynh, et al (2007)] [1ex]OFFLINE: Solve standard eigenproblems

ONLINE: Solve a linear program with Q variables

3.3 Greedy AlgorithmGreedy Algorithm for Optimal Samples

Given XN , we choose the next sample as follows:

µ1

∆N

µ2 µ3

∆N

µ2µ1 µ3

∆N

µ2µ1

µN+1 = arg maxµ∈DJ

∆N(µ)

‖uN(µ)‖X

XN+1 = XN ⊕ spanu(µN+1)

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

• Error bound ∆N ⇒ "optimal" samples⇒ good approximation uN

In Summary . . .

• Reduced basis approximation provides certifiably accurate inexpensive ap-proximations to solutions of parametrized PDEs

13

3.4 SummaryRB Opportunities

Computational Opportunities

I. We restrict our attention to the typically smooth and low-dimensionalmanifold induced by the parametric dependence.

⇒ Dimension reduction

II. We accept greatly increased offline cost in exchange for greatly decreasedonline cost.

⇒ Real-time and/or many-query context

RB RelevanceReal-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.

RB Key Challenges

• A Posteriori error estimation

– Rigorous error bounds for outputs of interest

– Lower bounds to the stability “constants”

• Offline-online computational procedures

– Full decoupling of finite element and reduced basis spaces

– A posteriori error estimation

– Nonaffine and nonlinear problems

• Effective sampling strategies

– High parameter dimensions

14

RB Outline

1. Affine Elliptic Problems

• (non)symmetric, (non)compliant, (non)coercive

• (Convection)-diffusion, linear elasticity, Helmholtz

2. Affine Parabolic Problems

• (Convection)-diffusion equation

3. Nonaffine and Nonlinear Problems

• Nonaffine parameter dependence, nonpolynomial nonlinearities

4. Reduced Basis (RB) Method for Fluid Flow

• Saddle-Point Problems (Stokes)

• Navier-Stokes Equations

5. Applications

• Parameter Optimization and Estimation (Inverse Problems)

• Optimal Control

4 Reduced Basis Applications

4.1 Concrete DelaminationConcrete 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

15

Concrete Delamination – Problem StatementGiven (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.

Concrete Delamination – Problem Statement1

∆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).

Concrete Delamination – ResultsTemperature distribution: wdel/2 = 5, κ = 1

k = 10 k = 20

k = 40 k = 60

16

Concrete Delamination – ResultsThermal 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

Concrete Delamination – ResultsMATLAB DEMO

4.2 Problems in ElasticityProblems in Elasticity [V]

• 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)

Problems in Elasticity [V]

• 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

• Outputs: Average stress, average deflection

17

• 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

Problems in Elasticity [V]

• Goal: Design & Optimization

• Problem Statement:minµ∈D

Material Cost (Area)

subject to constraints:stress < σmax

deflection < δmax

• 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

4.3 Contaminant TransportContaminant Transport

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

• Application: Identification of sources

18

Airborne contaminants Airborne contaminantsin urban canyon. in LA basis.

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

Contaminant Transport

• 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−xs

1)2+(x2−xs2)2)

(say, µ ≡ (κ, xs1, xs2))

Contaminant Transport – Problem StatementScalar 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.

Contaminant Transport – Sample SolutionsField variable: µ = (0.05, 2.9, 0.3) (N = 3720)

19

Contaminant Transport – Sample SolutionsField variable: µ = (0.05, 3.1, 0.5) (N = 3720)

Inverse ProblemDetermine: µ∗ ∈ D (actual value)Given experimental datameasurements : 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

Inverse Problem – (Regularized) SolutionGiven noisy measurements, z(tk), k ∈ IKexp, solve

• Output least squares problem

20

µ = arg minµ∈D

12

IKexp∑k=1

‖sN (tk;µ)− z(tk)‖2W

s.t. PDEN (µ) being satisfied; or

• Regularized problem

µ = arg minµ∈D

12

IKexp∑k=1

‖sN (tk;µ)− z(tk)‖2W + 12δRR(µ)

s.t. PDEN (µ) being satisfied.



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



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

21

4.4 GMA Welding ProcessGMA Welding Process [SH93],[SH94]

• 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)

GMA Welding Process – Problem StatementScalar 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.

GMA Welding Process – Sample SolutionField variable: µ = (0.3, 0.4) (N = 3720)

22

GMA Welding Process – ResultsApproach 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 problem

4. 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)

GMA Welding Process – ResultsParameter 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 )|

ThermalBlockGoverning PDE: Parameters: Outputs:

Heat Diffusion Conductivities µi Average Temp.

Source: A.T. Patera

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

23

SummaryMany 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.

24

Documents

Contents CourseOutline CourseOutline Topicstobediscussed: ReducedBasisMethods – Structuralmechanics – ParametrizedPDEs ProperOrthogonalDecomposition