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### Mathematics for innovative technology development

M. Kleiber

President of the Polish Academy of Sciences

Member of the European Research Council

Warsaw, 21.02.2008

Math as backbone of applied science and technology

- Applied math in ERC programme
- 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 apart of mathematics used to model and solve real world problems

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

Mathematics as a key to new technologies

Real Problem

validation of model

modelling

verification of results

Mathematical Model

Computer Simulation

algorithm design and implementation

Mathematics as a key to new technologies

Applied math for innovative technologies:

- used at every level –
- product analysis and design
- process planning
- quality assessment
- life cycle analysis including environmental issues
- distribution and promotional techniques
- …

Mathematics as a key to new technologies

Dr. Claudio BORDIGNON (IT) –medicine (hematology, gene therapy)

Prof. Manuel CASTELLS (ES) – information society, urban sociology

Prof. Paul J. CRUTZEN (NL) – atmospheric chemistry, climatology

Prof. Mathias DEWATRIPONT (BE) – economics, science policy

Dr. Daniel ESTEVE (FR) – physics (quantum electronics, nanoscience)

Prof. Pavel EXNER (CZ) – mathematical physics

Prof. Hans-Joachim FREUND (DE) – physical chemistry, surface physics

Prof. 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) – astrophysics

Prof. Fotis C. KAFATOS (GR) – molecularbiology, biotechnology

Prof. Norbert KROO (HU) – solid-state physics, optics

Dr. Oscar MARIN PARRA (ES) – biology, biomedicine

Lord MAY (UK) – zoology, ecology

Prof. Helga NOWOTNY (AT) – sociology, science policy

Prof. Christiane NÜSSLEIN-VOLHARD (DE) – biochemistry, genetics

Prof. 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 research

Prof. Salvatore SETTIS (IT) – history of art, archeology

Prof. Rolf M. ZINKERNAGEL (CH) – medicine (immunology)

Members of the ERC Scientific CouncilMathematics as a key to new technologies

ERC panel structure:Social Sciences and Humanities

SH1 INDIVIDUALS, INSTITUTIONS AND MARKETS: economics, finance andmanagement.

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.

SH6THE STUDY OF THE HUMAN PAST: archaeology, history and memory.

Mathematics as a key to new technologies

ERC panel structure:Life Sciences

LS1MOLECULAR 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: organphysiology, 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.

Mathematics as a key to new technologies

ERC panel structure:Life Sciences

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 andtreatment 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.

Mathematics as a key to new technologies

ERC panel structure:Physical Sciences and Engineering

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.

Mathematics as a key to new technologies

ERC panel structure:Physical Sciences and Engineering

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.

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 theory
- Algebraic and complex geometry
- Geometry
- Topology
- Lie groups, Lie algebras
- Analysis
- Operator algebras and functional analysis
- ODE and dynamical systems
- Partial differential equations
- Mathematical physics
- Probability and statistics
- Combinatorics
- Mathematical aspects of computer science
- Numerical analysis and scientific computing
- Control theory and optimization
- Application of mathematics in sciences

Mathematics as a key to new technologies

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

- Physical chemistry
- Nanochemistry
- Spectroscopic and spectrometric techniques
- Molecular architecture and Structure
- Surface science
- Analytical chemistry
- Chemical physics
- Chemical instrumentation
- Electrochemistry, electrodialysis, microfluidics
- Combinatorial chemistry
- Method development in chemistry
- Catalysis
- Physical chemistry of biological systems
- Chemical reactions: mechanisms, dynamics, kinetics and catalytic reactions
- Theoretical and computational chemistry
- Radiation chemistry
- Nuclear chemistry
- Photochemistry

Mathematics as a key to new technologies

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

- Computer architecture
- Database management
- Formal methods
- Graphics and image processing
- Human computer interaction and interface
- Informatics and information systems
- Theoretical computer science including quantum information
- Intelligent systems
- Scientific computing
- Modelling tools
- Multimedia
- Parallel and Distributed Computing
- Speech recognition
- Systems and software

Mathematics as a key to new technologies

PE7 SYSTEMS AND COMMUNICATION ENGINEERING: electronic, communication, opticaland systems engineering

- Control engineering
- Electrical and electronic engineering: semiconductors, components, systems
- Simulation engineering and modelling
- Systems engineering, sensorics, actorics, automation
- Micro- and nanoelectronics, optoelectronics
- Communication technology, high-frequency technology
- Signal processing
- Networks
- Man-machine-interfaces
- Robotics

Mathematics as a key to new technologies

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

- Aerospace engineering
- Chemical engineering, technical chemistry
- Civil engineering, maritime/hydraulic engineering, geotechnics, waste treatment
- Computational engineering
- Fluid mechanics, hydraulic-, turbo-, and piston engines
- Energy 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 engineering
- Product design, ergonomics, man-machine interfaces
- Lightweight construction, textile technology
- Industrial bioengineering
- Industrial 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 physics
- Planetary systems sciences
- Interstellar medium
- Formation of stars and planets
- Astrobiology
- Stars and stellar systems
- The Galaxy
- Formation and evolution of galaxies
- Clusters of galaxies and large scale structures
- High energy and particles astronomy – X-rays, cosmic rays, gamma rays, neutrinos
- Relativistic astrophysics
- Dark matter, dark energy
- Gravitational astronomy
- Cosmology
- Space Sciences
- Very large data bases: archiving, handling and analysis
- Instrumentation - telescopes, detectors and techniques
- Solar planetology

Mathematics as a key to new technologies

Further Information

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

Mathematics as a key to new technologies

Discreteelement 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 behaviour
- Contact interaction takes intoaccount friction and cohesion,including the possibility of breakage of cohesive bonds

Mathematics as a key to new technologies

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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)

micro-macro relationships

inverse analysis

Micromechanical constitutive laws

Macroscopic stress-strain relationships

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

DEM simulation of dredging

Model details:

92 000 discrete elements

swing 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/FEM simulation of dredging – example of multiscale modelling

Model details:

48 000 discrete elements

340 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

Map of equivalent stresses

Mathematics as a key to new technologies

Methods of reliability computation

Monte CarloAdaptive Monte CarloImportance Sampling

Simulation

methods

FORM SORM Response Surface Method

Approximation

methods

Mathematics as a key to new technologies

Failure in metal sheet forming processes

Real part (kitchen sink) with breakage

Deformed shape with thickness distribution

Forming Limit Diagram

Results of simulation

Mathematics as a key to new technologies

Deep drawing of a square cup (Numisheet’93)

Minor principal strains

Forming Limit Diagram (FLD)

Major principal strains

Experiment - breakage at 19 mm punch stroke

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

Mathematics as a key to new technologies

Metal sheet forming processes – reliability analysis

Limit state surface – Forming Limit Curve (FLC)

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

Mathematics as a key to new technologies

Results of reliability analysis

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

Proces tłoczenia blach - przykład numeryczny, wyniki

Odchylenie standardowe współczynnika wzmocnienia2 = 0.020

- 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

Mathematics as a key to new technologies

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