#### Transcript Non-linear dynamic analysis of shells with frictional

Mathematics for innovative technology development M. Kleiber President of the Polish Academy of Sciences Member 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 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 Computer Simulation verification of results modelling Mathematical Model 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 Members of the ERC Scientific Council 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) – molecular biology, 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) Mathematics as a key to new technologies ERC panel structure: Social Sciences and Humanities 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. Mathematics as a key to new technologies ERC panel structure: Life Sciences 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. 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 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. 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, chemical theory, 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, optical and 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 and control, 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 at http://erc.europa.eu Mathematics as a key to new technologies Discrete element method – main assumptions Material represented by a collection of particles of different shapes, in the presented formulation spheres (3D) or discs (2D) are used (similar to P. Cundall´s formulation) Rigid discrete elements, deformable contact (deformation is localized in discontinuities) Adequate contact laws yield desired macroscopic material behaviour Contact interaction takes into account friction and cohesion, including the possibility of breakage of cohesive bonds Mathematics as a key to new technologies Micro-macro relationships Fn s Rn un kn e micro-macro relationships compression un < 0 tension RT fn e fT inverse analysis fn kT uT s Micromechanical constitutive laws Macroscopic stress-strain relationships Parameters of micromechanical model: kn , kT , Rn , RT Macroscopic material properties: E , , s c , s t Determination of the relationship between micro- and macroscopic parameters Homogenization, averaging procedures Simulation of standard laboratory tests (unconfined compression, Brazilian test) 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: Analysis details: 35 000 discrete elements, 20 hours CPU (Xeon 3.4 GHz) experiment: 7500 N 2D simulation: 5500 N (force/20mm, 20 mm – spacing between passes of cutting tools) 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 steps 30 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 steps 16 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 Carlo Simulation methods x2 Adaptive Monte Carlo Importance Sampling u2 f u2 G (u) = 0 f n ( u,0,I ) = const f n ( u,0,I ) = const u* f X ( x ) = const. G (u) = 0 g(x) = 0 u1 0 u1 0 s s 0 x1 s FORM v n v n u2 u2 Approximation ( u,0,I ) = const methods Response Surface Method SORM f ~ v u2 f f n u* * f v* R s G (u) = 0 2 1 u1 l(u) = 0 3 4 v n = sv ( ~ v) G (u) = 0 0 5 v ) – vn = 0 Gv(v) = f v ( ~ region of most contribution to probability integral u1 0 s Mathematics as a key to new technologies s u1 Failure in metal sheet forming processes Real part (kitchen sink) with breakage Deformed shape with thickness distribution Results of simulation Mathematics as a key to new technologies Forming Limit Diagram 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 Results of reliability analysis 4,00E-01 0,38 3,50E-01 3,00E-01 Pro bability o f failure n2 0,331 2,50E-01 0,193 2,00E-01 n1 1,50E-01 0,136 1,00E-01 0,0483 5,00E-02 0,00442 0,00E+00 0,00 0,0169 1,00 2,00 3,00 4,00 0,000369 5,00 6,00 S a fe ty Ma rg in Probability of failure in function of the safety margin for two different hardening coefficients 0,0000954 7,26E-07 7,00 8,00 Proces tłoczenia blach - przykład numeryczny, wyniki MPO(FORM) d min [mm] [%] 16 7.44 0.00010 3.718 39 0.00010 3.723 54 3.721 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 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 Pf MPO(SORM) Metoda Monte Carlo Adaptacyjna Klasyczna N N Skok N Pf N 1000 1000 2000000 Odchylenie standardowe współczynnika wzmocnienia s2 = 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 7000 500