Non-linear dynamic analysis of shells with frictional

Report
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

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