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
1.20e03
1.08e03
9.60e04
8.40e04
7.20e04
6.00e04
4.80e04
3.60e04
2.40e04
1.20e04
1.73e12
Z
Y
X
Contours of Velocity Magnitude (m/s)
FLUENT 5.0 (3d, segregated, lam)
Apr 09, 1999
1.20e03
1.08e03
9.60e04
8.40e04
7.20e04
6.00e04
4.80e04
3.60e04
2.40e04
1.20e04
1.73e12
Z
Y
X
Contours of Velocity Magnitude (m/s)
FLUENT 5.0 (3d, segregated, lam)
Apr 09, 1999
1.20e03
1.08e03
9.60e04
8.40e04
7.20e04
6.00e04
4.80e04
3.60e04
2.40e04
1.20e04
1.73e12
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)
Comments 0
Log in to post a comment