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Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Mathematics and Industry
-

a Relationship for

Mutual Benefit


Heinz W. Engl



Industrial Mathematics Institute

Johannes Kepler Universität Linz,

Johann Radon Institute for Computational and Applied Mathematics

Austrian Academy of Sciences

and

Industrial Mathematics Competence Center





Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

“Classification of Mathematics“


Pure Mathematics


Applied Mathematics

*

Applicable

*

Applied


Industrial

mathematics
:

Mathematics,

which

is

motivated

by

problems

from

industry
.



Difference

only

in

the

motivation,

not

in

the

method

(mathematical

rigor
;

proof!)
:

Ideal

case
.

Application

problems

are

very

often

too

complex

to

enable

us

to

meet

this

demand
:

Compromise
:

mathematical

rigor

(e
.
g
.

proof

of

convergence)

at

least

for

model

problems
.




Scientific

Computing“




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Treating for Problems from Applications



Translation into a

mathematical model



(many mathematical questions, such as,

Which terms are
important?“


asymptotic analysis; compromise between
simplicity and exactness)




Development of efficient solution methods


(analytical / numerical / symbolic / ...)




Efficient Implementation




Re
-
interpretation of the results


Several iterations are often necessary!




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics


Historical Development of Mathematics


Wave

motion“

between

emphasis

a

theory/basics

and

applications
.

~

1960
:

„Bourbakism“


Felix

Klein
:


Göttingen

Association

for

Applied

Physics

and

Mathematics”
:



Support

of

mathematics

in

scientific,

technical

and

economic

applications




Interaction

between

science

and

technology




Motivation
:

input

for

scientific

work,

“Unification

of

spirit

and

industry”




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Before

World

War

I
:

more

than

50

industrial

members

(e
.
g
.

CEO‘s

of

Krupp,

Siemens,

AEG)


Prandtl
:


Klein

tried

to

bridge

the

gap

between

pure

science

and

the

“working

world”
.



Techno
-
mathematics/Industrial

mathematics
:

A

more

recent

attempt

of

closing

this

gap

in

teaching

and

research

in

the

spirit

of

Felix

Klein

(„Felix
-
Klein
-
Award“

of

the

EMS)




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Mathematics

as

a


Cross
-
Sectional

-
Science



Different

real

world

problems

can

lead

to

very

closely

related

mathematical

models

and

therefore

can

be

treated

with

similar

methods
.


Examples
:


American

options



melting

of

steel


Heat

conduction



diffusion

in

porous

media


Gas

dynamics



semi
-
conductor

models



models

for

road

traffic


Reaction
-
diffusion
-
equations

in

chemistry



spread

of

epidemics




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

What

do

we

do?


Basic

research

in

the

field

of

inverse

problems


Applications
-
oriented

research
:

Use

of

modern

mathematical

methods

for

problems

in

industry

and

finance
;

modelling

and

numerical

simulation
.


Development

of

individual

software


Consulting






Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

What are Inverse Problems?


Problems, where


CAUSES

(INPUTS or SYSTEM PARAMETERS)


are to be determined from


OBSERVERED EFFECTS

or


DESIRED EFFECTS.


Inverse Problems Definition




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Where do we find Inverse Problems?



Differentiation!



Computerized

tomography
:

Which

density

distribution

in

the

patient’s

body

causes

the

measured

distribution

of

absorption

of

X
-
rays?

Similar
:

Non
-
destructive

material

testing,

impedance

tomography

(Johann

Radon)
.




Inverse

heat

conduction

problems
:

How

does

one

have

to

adjust

the

secondary

cooling

of

a

continuous

casting

machine

to

obtain

an

intended

solidification

front

in

the

strand?




Inverse

scattering

problems
:

Where

are

reinforcement

bars

located

inside

concrete

walls

which

cause

a

measured

scattering

of

a

magnetic

field?



Inverse Problems Example 1




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Further Examples



Parameter

identification
:

Calculate

temperature
-
dependent

thermal

conductivity

of

sand

for

casting

forms

from

measurements

of

the

temperature

evolution

in

thermocouples
.



Non
-
destructive

material

testing
:

Determine

the

thickness

of

a

blast

furnace

wall

from

temperature

measurements

using

thermo
-
elements

in

the

wall
.



Inverse

problems

in

optical

systems
:

Which

specific

shape

of

a

reflector

provides

a

prescribed

illumination

distribution?



Inverse Problems Example 2




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Hadamard’s Questions (1923)



Does a solution
exist
for all data?



If there is a solution, is it
unique
?



Does the solution
depend continuously on the data
?


If

3

x

YES
:

the

problem

is

called


well

posed

:

„correct

modelling

of

a

relevant

physical

problem“


Inverse

problems

are

typically

“ill
-
posed”
;

first

appearance
:

Geophysics

(search

for

ore

or

oil

deposits),

A
.

Tikhonov



Inverse Problems Ill

posed Problems




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

An Inverse Heat Conduction Problem

Determine the initial temperature in a laterally isolated rod from a
given final temperature:

u
x
t
u
(x,t)
x
t
T
xx
t
(
,
)
(
,
),
(
,
)



für
0
0

u
t
u
t
t
T
x
x
(
,
)
(
,
)
(
,
)
0
0
0




für
u
x
T
g
x
(
,
)
(
)

ist gegebe
n
To be determined:

f
x
u
x
(
)
(
,
)

0
Inverse Problems Reverse Heat Conduction

for

given

for




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

The Direct Problem

Calculate the final temperature for a given initial temperature.
Solution:



Expansion into a Cosine
-
Fourier
-
Series

f
x
a
nx
n
n
(
)
cos(
)




0
The temperature distribution is then given as:

u
x
t
a
nx
e
n
n
t
n
(
,
)
cos(
)





2
0
Terms of frequency
n
are damped with exp(
-
n
2
t),
forward
problem
smoothes

strongly!

Inverse Problems





Reverse Heat Conduction




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

The Inverse Problem

Now the final temperature is given (and expanded into a Cosine
-
Fourier
-
Series):

g
x
b
nx
n
n
(
)
cos(
)
,




0
Hence, the initial temperature is obtained as

f
x
b
nx
e
n
n
T
n
(
)
cos(
)




2
0
Terms with frequency
n
get amplified by
exp(
n
2
T)
!!!

High
-
frequency noise in the final temperature has an enormous
effect on the result!!

Inverse Problems





Reverse Heat Conduction




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Optimal Design: Free
-
Form Surfaces for
Light Reflector

The

problem

-
Construction

of

a

3
D

free
-
form

surface

reflector

with

a

prescribed

illumination


-
Integration

into

an

existing

CAD
-
system

-
Mathematics
:

Monge
-
Ampere

type

equation,

approach

via

stable

minimization

of

a

complicated

nonlinear

functional

Application

example

-
Uniform

illumination

of

a

long

escape

route

with

a

single

reflector,

which

should

be

nearly

circular




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Optimal Reflector Design




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Shape Optimization of Mechanical Parts

Aims:


Weight reduction of structural parts in compliance with
limits for the maximum stress

OR


Reduction of stress peaks subject to equal weight

OR


Obtaining a sufficiently uniform stress distribution to
enhance the life
-
span of a part

Optimisation Structure optimisation




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Methodological Similarities to Inverse
Problems


Efficient combination of optimization methods with “direct
solvers” (e.g. FEM) is necessary



Numerous projects in this area, from engine components
to flour hoppers (as used in bakeries)

Optimisation Structure optimisation




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Solution: Thickness Optimisation


Rocker Arm

Initial design:


Optimised design:

Optimisation Structure optimisation




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Mechanical Design of the Sustainers of Flour Hoppers

Partner company:

hb technik

The

mechanical

stresses


in

the

base

of

flour

hoppers


(up

to

30

t

capacity)

are

to


be

calculated/optimized
.

Von Mises stress distribution in the hopper basis




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Determination of the Thickness of a Blast Furnace
Wall from External Temperature Measurements

The problem:


Through

chemical

and

physical

wear,

the

lining

of

the

blast

furnace

decreases

with

time



Determine

the

thickness

of

the

lining

of

the

blast

furnace

based

on

temperature

measurements

on

the

exterior

of

the

wall


Solution
:


Parameter

identification

problem



sideways

heat

equation

(nonlinear),

highly

ill
-
posed!


Stable

solution

can

be

obtained

only

with

regularisation

methods

Inverse Problems Application




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Blast Furnace Lining: Results

Without regularisation:

With regularisation:

Inverse Problems Applications




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Transient Calculation of Temperature Fields

Heating of disk brakes for a wind power station

The problem:

-
In

case

of

technical

failure

in

wind

power

stations,

masses

with

huge

moments

of

inertia

have

to

be

stopped

by

disk

brakes

within

seconds
.

-
One

is

interested

in

the

transient

temperature

distribution

in

the

brake

to

predict

the

possible

occurrence

of

cracks

due

to

thermal

stresses
.


Modelling

and

solution

method

-
Equation

for

heat

conduction

for

disk

brake,

brake

lining

and

carrier

plate

-
Stable

and

fast

numerical

solution

obtained

with

a

fully

implicit

discretization

method




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Temperature distribution

in a disk brake / brake lining /

carrier plate after an

emergency braking

Source : MathConsult GmbH




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Transient Temperature Field Calculation

Temperature Distribution in Window Profiles

The

problem


-
Window

frames

are

fabricated

by

extrusion

techniques
.

After

leaving

the

mould,

the

frames

pass

through

4

further

calibrators
.

One

is

interested

in

the

temperature

distribution

of

the

frame

after

leaving

the

last

calibrator
.

Optimization/inverse

problem

in

the

background!


Modelling

and

solution

method

-
Taking

into

account

heat

conduction

und

heat

radiation

-
Finite

Element

Method,

fully

implicit

in

time




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Calibrator

Profile

Mesh


Window Profile + Calibrator

Cooling holes




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Temperature distribution


in profile and calibrator after

10 seconds




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Temperature Pattern in the Window Profile

Initial temperature

200
°
C

Source : MathConsult GmbH




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Aim:


Development of a kinetic blast furnace
-
simulation model


Quantities of interest:


Flow of the solid materials (iron ore, coke) and of the wind;
pressure distribution


Temperature distribution


Chemical composition as a function of the position, taking
into account the reaction kinetics

Numerical Simulation of the

Blast Furnace Process




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Potential

flow

for

the

solid

(now

more

complicated

models!)


Wind
:

Flow

through

a

layered

porous

medium


Energy

balance
:

diffusion,

convection,

heat

sources

and


sinks

caused

by

chemical

reactions


Reaction

kinetics

for

at

least

30

compounds



The

outcome

of

this

is

a

system

of

about

40

coupled

non
-
linear

partial

differential

equations
.

Mathematical Model




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Modular, object orientated construction


Problem adapted finite elements in each module


Iterative coupling of single modules


Integration into a larger automation package


Lead to a patent


Numerical Realisation




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Modular
construction




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Results


Pressure distribution
(coloration)


Gas flow (arrows)





Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Results: Carbon content in the furnace bottom

top: 2%, bottom: 4.5 %




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Numerical Simulation of the

COREX
®
-
Process


COREX
®

=

new

technology

for

the

production

of

pig
-
iron




Instead

of

coke

one

uses

coal

(no

coking

plant

necessary,

thus

more

cost
-
effective)
;

cheaper

ore

usable
;

less

harmful

emissions



Process

divided

into

two

reactors
:


Reduction

shaft
:

Reduction

of

the

iron

ore




Melter

gasifier
:

Melting

off

of

the

produced

iron

sponge

in

the

shaft
;

production

of

the

reduction

gas

to

be

used

in

the

shaft
.

Industrial Mathematics Competence Center




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

COREX
®
-
Process
-

Complexity


Modelling

and

calculating



the

flow

of

the

ore

(special

material

law)

and

of

the

reduction

gas

in

the

shaft,



the

chemical

reactions,


the

temperature

distribution

in

ore

and

gas,



the

deposition

of

the

dust

in

the

gas



3

dimensional

model


Coupled

system

of

about

35

nonlinear

partial

differential

equations


Industrial Mathematics Competence Center




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Industrial Mathematics Competence Center

Z
Y
X
Contours of Velocity Magnitude (m/s)
FLUENT 5.0 (3d, segregated, lam)
Apr 09, 1999
1.20e-03
1.08e-03
9.60e-04
8.40e-04
7.20e-04
6.00e-04
4.80e-04
3.60e-04
2.40e-04
1.20e-04
1.73e-12
Z
Y
X
Contours of Velocity Magnitude (m/s)
FLUENT 5.0 (3d, segregated, lam)
Apr 09, 1999
1.20e-03
1.08e-03
9.60e-04
8.40e-04
7.20e-04
6.00e-04
4.80e-04
3.60e-04
2.40e-04
1.20e-04
1.73e-12
Z
Y
X
Contours of Velocity Magnitude (m/s)
FLUENT 5.0 (3d, segregated, lam)
Apr 09, 1999
1.20e-03
1.08e-03
9.60e-04
8.40e-04
7.20e-04
6.00e-04
4.80e-04
3.60e-04
2.40e-04
1.20e-04
1.73e-12
Velocity distribution in the solid




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Industrial Mathematics Competence Center

Velocity distribution
in the solid

Fe0
-
share




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Hot Rolling of Steel

Description

of

the

problem

-
Hot

rolling

of

steel

leads

to

large

plastic

deformations

and

stress

differences

in

the

material

-
Experimental

investigations

are

highly

cost
-
intensive

and

are,

in

general,

restricted

to

surface

deformations


Mathematical

model,

numerical

simulation


Complexity

-
Large

plastic

deformations

within

the

rigid

zones

-
Appearance

of

a

neutral

point

or

zone

in

the

roll

gap

area

-
Contact

problem

with

friction

-
Vertical

displacements

in

the

contact

area,

between

the

working

roll

and

the

slab,

are

vital

for

hot

rolling

processes




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Numerical

Realisation


-
Mixed

Eulerian
-
Lagrangian

description

of

the

velocity

and

the

vertical

displacement

with

pressure
-
coupling


-
Development

of

a

Finite
-
Element

based

software

package

for

the

efficient

solution

of

this

complex

non
-
linear

problem

(
2
D,

3
D)

-
Use

of

special

solvers

for

large

sparse

matrices

-
Both

2
D

and

3
D

simulation

packages

(broadening

is

an

effect

of

major

interest)


-

Application

of

special

techniques

for



Solution

of

contact

problems,


consideration

of

the

neutral

point

and



handling

of

the

stiff

zone
.




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Hot Rolling of Steel
-

Results

Velocity + Vertical Displacement

瑲tss䑩D瑲楢畴楯t




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Financial Mathematics: Pricing and Risk
Analysis of Financial Instruments

Examples

of

Financial

Derivatives
:


Call
-
Option
:

An

option

is

a

contract

that

gives

the

holder

the

right

but

not

the

obligation

to

buy

a

certain

quantity

of

an

underlying

security

at

a

specified

price

(the

strike

price)

up

to

a

specified

date

(the

expiration

date)


Callable

Bond
:

A

bond

which

the

issuer

has

the

right

to

redeem

prior

to

its

maturity

date



Austrian

home

mortgage
:

fixed

interest

rate

for

the

starting

period,

then

a

floating

rate

based

on

an

average

yield

of

government

bonds
.

This

rate

is

equipped

with

caps

and

floors
.

Typically,

not

the

lifetime

of

the

mortgage

is

fixed,

but

the

repayment
.

Possibility

of

early

redemption
.


What is a fair price for such a complicated instrument?




Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Derivative Financial Instruments

Theory
:

Black
-
Scholes
-
Merton

1973

(Nobel

prize

1997
)


For

simple

instruments,

analytic

solutions

can

be

obtained
.

Complex

contracts

are

to

be

evaluated

numerically
.



Development

of

a

new

numerical

method

which

is

very

fast

and

robust

especially

for

complex

derivatives

(e
.
g
.
,

Japanese

convertible

bonds

with

strong

path

dependence),

inverse

problems

like

volatility

identification
:




偡捫慧a

啮剩獫
®


(www
.
unriskderivatives
.
com)





Industrial Mathematics Institute and Industrial Mathematics Competence Center

Johann Radon Institute for Computational and Applied Mathematics

Value

of

an

Up
-
and
-
Out

Call

Option

for

a

share

discrete

divi
-
dends

paying

with

rising

interest

rates
.

Value of the option

Source: MathConsult GmbH

Call
-
Option

for

a

bond

with

discrete

coupons

as

function

of

the

interest

rate

and

the

remaining

term

of

the

option

with

rising

volatility

Price of shares

Time (days)