Mathematics for innovative technology development

M. KleiberPresident of the Polish Academy of SciencesMember of the European Research Council

Warsaw, 21.02.2008

1. Math as backbone of applied science and technology

2. Applied math in ERC programme

3. Examples of advanced modelling and simulations in developing new technologies (J. Rojek + International Center for Numerical Methods in Engineering – CIMNE, Barcelona)

Mathematics as a key to new technologies

Applied mathematics is a part of mathematics used to model and solve real world problems

Applied mathematics is used everywherehistorically: applied analysis (differential equations, approximation theory, applied probability, …) all largely tied to Newtonian physicstoday: truly ubiquitous, used in a very broad context

Mathematics as a key to new technologies

Real Problem

Mathematical Model

Computer Simulation

modellingvalidation of model

verification of results

algorithm design and implementation

Mathematics as a key to new technologies

Applied math for innovative technologies: used at every level –

product analysis and designprocess planningquality assessmentlife cycle analysis including environmental issuesdistribution and promotional techniques …

Mathematics as a key to new technologies

Members of the ERC Scientific Council

Dr. Claudio BORDIGNON (IT) – medicine (hematology, gene therapy)Prof. Manuel CASTELLS (ES) – information society, urban sociologyProf. Paul J. CRUTZEN (NL) – atmospheric chemistry, climatologyProf. Mathias DEWATRIPONT (BE) – economics, science policyDr. Daniel ESTEVE (FR) – physics (quantum electronics, nanoscience)Prof. Pavel EXNER (CZ) – mathematical physicsProf. Hans-Joachim FREUND (DE) – physical chemistry, surface physicsProf. Wendy HALL (UK) – electronics, computer science Prof. Carl-Henrik HELDIN (SE) – medicine (cancer research, biochemistry)Prof. Michal KLEIBER (PL) – computational science and engineering, solid and fluid mechanics, applied mathematics Prof. Maria Teresa V.T. LAGO (PT) – astrophysicsProf. Fotis C. KAFATOS (GR) – molecular biology, biotechnologyProf. Norbert KROO (HU) – solid-state physics, opticsDr. Oscar MARIN PARRA (ES) – biology, biomedicine Lord MAY (UK) – zoology, ecology Prof. Helga NOWOTNY (AT) – sociology, science policyProf. Christiane NÜSSLEIN-VOLHARD (DE) – biochemistry, geneticsProf. Leena PELTONEN-PALOTIE (FI) – medicine (molecular biology)Prof. Alain PEYRAUBE (FR) – linguistics, asian studies Dr. Jens R. ROSTRUP-NIELSEN (DK) – chemical and process engineering, materials researchProf. Salvatore SETTIS (IT) – history of art, archeologyProf. Rolf M. ZINKERNAGEL (CH) – medicine (immunology)

Mathematics as a key to new technologies

SH1 INDIVIDUALS, INSTITUTIONS AND MARKETS: economics, finance and management.

SH2 INSTITUTIONS, VALUES AND BELIEFS AND BEHAVIOUR: sociology, social anthropology, political science, law, communication, social studies of science and technology.

SH3 ENVIRONMENT AND SOCIETY: environmental studies, demography, social geography, urban and regional studies.

SH4 THE HUMAN MIND AND ITS COMPLEXITY: cognition, psychology, linguistics, philosophy and education.

SH5 CULTURES AND CULTURAL PRODUCTION: literature, visual and performing arts, music, cultural and comparative studies.

SH6 THE STUDY OF THE HUMAN PAST: archaeology, history and memory.

ERC panel structure:Social Sciences and Humanities

Mathematics as a key to new technologies

LS1 MOLECULAR AND STRUCTURAL BIOLOGY AND BIOCHEMISTRY: molecular biology, biochemistry, biophysics, structural biology, biochemistry of signal transduction.

LS2 GENETICS, GENOMICS, BIOINFORMATICS AND SYSTEMS BIOLOGY: genetics, population genetics, molecular genetics, genomics, transcriptomics, proteomics, metabolomics, bioinformatics, computational biology, biostatistics, biological modelling and simulation, systems biology, genetic epidemiology.

LS3 CELLULAR AND DEVELOPMENTAL BIOLOGY: cell biology, cell physiology, signal transduction, organogenesis, evolution and development, developmental genetics, pattern formation in plants and animals.

LS4 PHYSIOLOGY, PATHOPHYSIOLOGY, ENDOCRINOLOGY: organ physiology, pathophysiology, endocrinology, metabolism, ageing, regeneration, tumorygenesis, cardiovascular disease, metabolic syndrome.

LS5 NEUROSCIENCES AND NEURAL DISORDERS: neurobiology, neuroanatomy, neurophysiology, neurochemistry, neuropharmacology, neuroimaging, systems neuroscience, neurological disorders, psychiatry.

ERC panel structure:Life Sciences

Mathematics as a key to new technologies

LS6 IMMUNITY AND INFECTION: immunobiology, aetiology of immune disorders, microbiology, virology, parasitology, global and other infectious diseases, population dynamics of infectious diseases, veterinary medicine.

LS7 DIAGNOSTIC TOOLS, THERAPIES AND PUBLIC HEALTH: aetiology, diagnosis and treatment of disease, public health, epidemiology, pharmacology, clinical medicine, regenerative medicine, medical ethics.

LS8 EVOLUTIONARY POPULATION AND ENVIRONMENTAL BIOLOGY: evolution, ecology, animal behaviour, population biology, biodiversity, biogeography, marine biology, ecotoxycology, prokaryotic biology.

LS 9 APPLIED LIFE SCIENCES AND BIOTECHNOLOGY: agricultural, animal, fishery, forestry and food sciences, biotechnology, chemical biology, genetic engineering, synthetic biology, industrial biosciences, environmental

biotechnology and remediation.

ERC panel structure:Life Sciences

Mathematics as a key to new technologies

PE1 MATHEMATICAL FOUNDATIONS : all areas of mathematics, pure and applied, plus mathematical foundations of computer science, mathematical physics and statistics.

PE2 FUNDAMENTAL CONSTITUENTS OF MATTER : particle, nuclear, plasma, atomic, molecular, gas and optical physics.

PE3 CONDENSED MATTER PHYSICS: structure, electronic properties, fluids, nanosciences.

PE4 PHYSICAL AND ANALYTICAL CHEMICAL SCIENCES : analytical chemistry, chemical theory, physical chemistry/chemical physics.

PE5 MATERIALS AND SYNTHESIS: materials synthesis, structure – properties relations, functional and advanced materials, molecular architecture, organic chemistry.

PE6 COMPUTER SCIENCE AND INFORMATICS : informatics and information systems, computer science, scientific computing, intelligent systems.

ERC panel structure:Physical Sciences and Engineering

Mathematics as a key to new technologies

PE7 SYSTEMS AND COMMUNICATION ENGINEERING: electronic, communication, optical and systems engineering.

PE8 PRODUCTS AND PROCESSES ENGINEERING: product design, process design and control, construction methods, civil engineering, energy systems, material engineering.

PE9 UNIVERSE SCIENCES: astro-physics/chemistry/biology; solar system; stellar, galactic and extragalactic astronomy, planetary systems, cosmology, space science, instrumentation.

PE10 EARTH SYSTEM SCIENCE: physical geography, geology, geophysics, meteorology, oceanography, climatology, ecology, global

environmental change, biogeochemical cycles, natural resources management.

ERC panel structure:Physical Sciences and Engineering

Mathematics as a key to new technologies

PE1 MATHEMATICAL FOUNDATIONS : all areas of mathematics, pure and applied, plus mathematical foundations of computer science, mathematical physics and statistics.

Logic and foundations Algebra Number theoryAlgebraic and complex geometryGeometryTopologyLie groups, Lie algebrasAnalysisOperator algebras and functional analysisODE and dynamical systems Partial differential equationsMathematical physics Probability and statisticsCombinatoricsMathematical aspects of computer scienceNumerical analysis and scientific computingControl theory and optimization Application of mathematics in sciences

Mathematics as a key to new technologies

PE4 PHYSICAL AND ANALYTICAL CHEMICAL SCIENCES: analytical chemistry, chemical theory, physical chemistry/chemical physics

Physical chemistryNanochemistrySpectroscopic and spectrometric techniquesMolecular architecture and StructureSurface scienceAnalytical chemistryChemical physicsChemical instrumentationElectrochemistry, electrodialysis, microfluidicsCombinatorial chemistryMethod development in chemistryCatalysisPhysical chemistry of biological systemsChemical reactions: mechanisms, dynamics, kinetics and catalytic reactionsTheoretical and computational chemistryRadiation chemistryNuclear chemistryPhotochemistry

Mathematics as a key to new technologies

PE6 COMPUTER SCIENCE AND INFORMATICS: informatics and information systems, computer science, scientific computing, intelligent systems

Computer architectureDatabase managementFormal methodsGraphics and image processingHuman computer interaction and interfaceInformatics and information systemsTheoretical computer science including quantum informationIntelligent systemsScientific computingModelling toolsMultimediaParallel and Distributed ComputingSpeech recognitionSystems and software

Mathematics as a key to new technologies

PE7 SYSTEMS AND COMMUNICATION ENGINEERING: electronic, communication, optical and systems engineering

Control engineeringElectrical and electronic engineering: semiconductors, components, systemsSimulation engineering and modellingSystems engineering, sensorics, actorics, automationMicro- and nanoelectronics, optoelectronicsCommunication technology, high-frequency technologySignal processingNetworksMan-machine-interfacesRobotics

Mathematics as a key to new technologies

PE8 PRODUCTS AND PROCESS ENGINEERING: product design, process design and control, construction methods, civil engineering, energy systems, material engineering

Aerospace engineeringChemical engineering, technical chemistryCivil engineering, maritime/hydraulic engineering, geotechnics, waste treatmentComputational engineeringFluid mechanics, hydraulic-, turbo-, and piston enginesEnergy systems (production, distribution, application)Micro(system) engineering,Mechanical and manufacturing engineering (shaping, mounting, joining, separation)Materials engineering (biomaterials, metals, ceramics, polymers, composites, …)Production technology, process engineeringProduct design, ergonomics, man-machine interfacesLightweight construction, textile technologyIndustrial bioengineeringIndustrial biofuel production

Mathematics as a key to new technologies

PE9 UNIVERSE SCIENCES: astro-physics/chemistry/biology; solar system; stellar, galactic and extragalactic astronomy, planetary systems, cosmology; space science, instrumentation

Solar and interplanetary physicsPlanetary systems sciencesInterstellar mediumFormation of stars and planetsAstrobiologyStars and stellar systemsThe GalaxyFormation and evolution of galaxiesClusters of galaxies and large scale structuresHigh energy and particles astronomy – X-rays, cosmic rays, gamma rays, neutrinosRelativistic astrophysicsDark matter, dark energyGravitational astronomyCosmologySpace SciencesVery large data bases: archiving, handling and analysisInstrumentation - telescopes, detectors and techniquesSolar planetology

Mathematics as a key to new technologies

Further Information

Website of the ERC Scientific Council at http://erc.europa.eu

Mathematics as a key to new technologies

Discrete element method – main assumptions

Material represented by a collectionof particles of different shapes,in the presented formulationspheres (3D) or discs (2D) are used(similar to P. Cundall´s formulation)Rigid discrete elements, deformablecontact (deformation is localized in discontinuities)Adequate contact laws yield desiredmacroscopic material behaviourContact interaction takes intoaccount friction and cohesion,including the possibility of breakage of cohesive bonds

Mathematics as a key to new technologies

Micro-macro relationships

Parameters of micromechanical model: kn , kT , Rn , RT

Macroscopic material properties:

Determination of the relationship between micro- and macroscopic parameters

Homogenization, averaging procedures

Simulation of standard laboratory tests (unconfined compression, Brazilian test)

un

Fn

Rn

tensioncompression

kn

un

Fn

Rn

tensioncompression

kn

un < 0

uT

fT

RT

kTnf

nf

un < 0

uT

fT

RT

kT

un < 0

uT

fT

RT

kTnf

nf

Micromechanical constitutive laws Macroscopic stress-strain relationships

tcE ,,,

micro-macro relationships

inverse analysis

Mathematics as a key to new technologies

Simulation of the unconfined compression test

Distribution of axial stresses Force−strain curve

Mathematics as a key to new technologies

Numerical simulation of the Brazilian test

Distribution of stresses Syy Force−displacement curve (perpendicular to the direction of loading)

Mathematics as a key to new technologies

Numerical simulation of the rock cutting test

Failure mode Force vs. time

Average cutting force:

experiment: 7500 N

2D simulation: 5500 N (force/20mm, 20 mm – spacing between passes of cutting tools)

Analysis details: 35 000 discrete elements, 20 hours CPU (Xeon 3.4 GHz)

Mathematics as a key to new technologies

Rock cutting in dredging

Mathematics as a key to new technologies

Model details:92 000 discrete elementsswing velocity 0.2 m/s, angular velocity 1.62 rad/s

Analysis details: 550 000 steps30 hrs. CPU (Xeon 3.4 GHz)

Mathematics as a key to new technologies

DEM simulation of dredging

Model details:48 000 discrete elements340 finite elements

Analysis details: 550 000 steps16 hrs. CPU (Xeon 3.4 GHz)

Mathematics as a key to new technologies

DEM/FEM simulation of dredging – example of multiscale modelling

DEM/FEM simulation of dredging – example of multiscale modelling

Map of equivalent stresses

Mathematics as a key to new technologies

Methods of reliability computation

Monte Carlo Adaptive Monte Carlo Importance Sampling

u 2 G ( u ) = 0

s

f

0 u 1

n ( u,0,I ) = const

u 2

G ( u ) = 0

s

f

0 u 1

n ( u,0,I ) = const

u*

FORM SORM Response Surface Method u 2

G ( u ) = 0

s

f

l ( u ) = 0

*

u*

0 u 1

region of mostcontribution toprobability integral

n ( u,0,I ) = const

u 2

Gv(v) = f v ( v ) – v n = 0

s

f

0 u 1

v n v n

v ~

~

v n = sv ( v )

v*

~

Simulation methods

Approximation methods

Mathematics as a key to new technologies

Real part (kitchen sink) with breakage

Results of simulation

Deformed shape with thickness distribution

Forming Limit Diagram

Failure in metal sheet forming processes

Mathematics as a key to new technologies

Forming Limit Diagram (FLD)

Major principal strains

Blank holding force: 19.6 kN, friction coefficient: 0.162, punch stroke: 20 mm

Experiment - breakage at 19 mm punch stroke

Minor principal strains

Deep drawing of a square cup (Numisheet’93)

Mathematics as a key to new technologies

Limit state surface – Forming Limit Curve (FLC)

Limit state function – minimum distance from FLC = safety margin (positive in safe domain, negative in failure domain)

Metal sheet forming processes – reliability analysis

Mathematics as a key to new technologies

Results of reliability analysis

Results of reliability analysis

Probability of failure in function of the safety margin for two different hardening coefficients

0 , 3 3 1

0 , 1 3 6

0 , 0 1 6 90 , 0 0 0 3 6 9 7 , 2 6 E -0 7

0 , 0 0 4 4 2 0 , 0 0 0 0 9 5 4

0 , 3 8

0 , 1 9 3

0 , 0 4 8 3

0 , 0 0 E + 0 0

5 , 0 0 E -0 2

1 , 0 0 E -0 1

1 , 5 0 E -0 1

2 , 0 0 E -0 1

2 , 5 0 E -0 1

3 , 0 0 E -0 1

3 , 5 0 E -0 1

4 , 0 0 E -0 1

0 , 0 0 1 , 0 0 2 , 0 0 3 , 0 0 4 , 0 0 5 , 0 0 6 , 0 0 7 , 0 0 8 , 0 0

S a fe ty M a rg i n

Pro

bab

ilit

y o

f fa

ilu

re

n 2

n 1

Odchylenie standardowe współczynnika wzmocnienia 2 = 0.020

[mm] [%] P f N P f

N N N

16 7.44 0.00010 3.718 39 0.00010 3.723 54 3.721 1000 2000000

17 5.50 0.0044 2.622 29 0.00401 2.651 44

18 3.77 0.0485 1.660 29 0.0476 1.669 44 1.670 1000 7000

19 2.14 0.192 0.869 40 0.182 0.907 55

20 0.77 0.373 0.324 23 0.395 0.267 38 0.342 500

Metoda Monte CarloAdaptacyjna Klasyczna

Skok d min MPO(FORM)  MPO(SORM)

• Porównanie z metodami symulacyjnymi potwierdza dobrą dokładność

wyników otrzymanych metodą powierzchni odpowiedzi

• Metoda powierzchni odpowiedzi wymaga znacznie mniejszej liczby

symulacji (jest znacznie efektywniejsza obliczeniowo)

• Dla małych wartości Pf metoda adaptacyjna jest efektywniejsza niż

klasyczna metoda Monte Carlo

Proces tłoczenia blach - przykład numeryczny, wyniki

Mathematics as a key to new technologies