industrial mathematics institute and industrial mathematics competence center johann radon institute...

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 InstituteJohannes Kepler Universität Linz,

Johann Radon Institute for Computational and Applied

MathematicsAustrian Academy of Sciences

andIndustrial Mathematics Competence Center



“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“



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!



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”



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)



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



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



What are Inverse Problems?

Problems, where

CAUSES (INPUTS or SYSTEM PARAMETERS)

are to be determined from

OBSERVERED EFFECTS or

DESIRED EFFECTS.

Inverse Problems Definition



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



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



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



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 Txx t( , ) ( , ), ( , ) für 0 0

u t u t t Tx x( , ) ( , ) ( , )0 0 0 für

u x T g x( , ) ( ) ist gegeben

To be determined:

f x u x( ) ( , ) 0

Inverse Problems Reverse Heat Conduction

for

given

for



The Direct Problem

Calculate the final temperature for a given initial temperature. Solution:• Expansion into a Cosine-Fourier-Series

f x a nxnn

( ) cos( )

0

The temperature distribution is then given as:

u x t a nx enn t

n( , ) cos( )

2

0

Terms of frequency n are damped with exp(-n2t), forward problem smoothes strongly!




The Inverse Problem

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

g x b nxnn

( ) cos( ),

0

Hence, the initial temperature is obtained as

f x b nx enn T

n( ) cos( )

2

0

Terms with frequency n get amplified by exp(n2T)!!!High-frequency noise in the final temperature has an enormous effect on the result!!




Optimal Design: Free-Form Surfaces for Light Reflector

The problem- Construction of a 3D 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



Optimal Reflector Design



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



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)




Solution: Thickness Optimisation – Rocker Arm

Initial design: Optimised design:




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



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



Blast Furnace Lining: Results

Without regularisation: With regularisation:

Inverse Problems Applications



Transient Calculation of Temperature FieldsHeating 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



Temperature distribution in a disk brake / brake lining /carrier plate after an emergency braking

Source : MathConsult GmbH



Transient Temperature Field CalculationTemperature 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



Calibrator

Profile Mesh

Window Profile + Calibrator

Cooling holes



Temperature distribution in profile and calibrator after10 seconds



Temperature Pattern in the Window Profile

Initial temperature200 °C

Source : MathConsult GmbH



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



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



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



Modular construction



Results

Pressure distribution (coloration)

Gas flow (arrows)



Results: Carbon content in the furnace bottom

top: 2%, bottom: 4.5 %



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



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





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




Velocity distribution in the solid

Fe0-share



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



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 (2D, 3D)

- Use of special solvers for large sparse matrices

- Both 2D and 3D 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.



Hot Rolling of Steel - Results

Velocity + Vertical Displacement Stress Distribution



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?



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:

Package UnRisk® ® (www.unriskderivatives.com) (www.unriskderivatives.com)



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)

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