Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
The Story of Variational Principle
and van Gogh
Jingzhe Pan
Department of Engineering
University of Leicester
My team
Ding Lifeng
Huang Ruoyu
Li Fan
Wang Ying
Ahmad Ashhab
Ch’ng Heok Ngee
Chris Westwood
Sasan Kiani
Sergei Kucherenko
Wang Donglian
Leonard Euler (1707-1783)
“Since the fabric of the universe is most perfect,
and is the work of a most wise Creator, nothing
whatsoever takes place in the universe in which
some relation of maximum and minimum does not
appear.”
Leonard Euler (1707-1783)
The Gateway Arch in St. Louis
Two means to find the CatenaryFirst – the direct method
The shape balances T, F and
Two means to find the CatenarySecond – the variational method
The shape makes h as small as possible.
“…...from the agreement between the two
solutions we secure the very highest satisfaction.”
Leonard Euler (1707-1783)
The finite element method (1960-70)
The shape makes h as small as possible.
Solving complicated problems
The finite element method
The shape makes h as small as possible.
The finite element method
The shape makes the total potential energy
as small as possible.
Light travels between two given points along the
path of shortest time - Fermat's principle.
Nature is thrifty in all its actions - Maupertuis’
least action principle.
An elastic structure deforms to minimise its total
potential energy.
……….
In 1980, Needleman and Rice developed a variational
principle for diffusional crack growth
In 1990, Cocks and Pan used it for crack propagation in
polycrystalline materials
It turned out that this variational principle can be
modified to model ceramic processing.
It also turned out that the ceramists had
never heard of a variational principle !
www.portlandpottery.com
A long list of engineering products is produced
by sintering.
Industrial SinteringRidgway Powdered Metals Inc.
SV
gbgb
a
RhM
dt
da
5
2
32
This gave us an opportunity to establish a
unified sintering theory.
r
jg
b
jg
b
02
Lr
r
rjgb Lrjgb
SV
gbgb
a
RhM
dt
da
5
2
32
tR
hM
R
aSV
gbgb 4
6
192
This gave us an opportunity to establish a
unified sintering theory.
duP
dt
dEdV
M
C
V
2
1
0
r
jg
b
jg
b
02
Lr
r
rjgb Lrjgb
SV
gbgb
a
RhM
dt
da
5
2
32
tR
hM
R
aSV
gbgb 4
6
192
duP
dt
dEdV
M
C
V
2
1
0
This gave us an opportunity to establish a
unified sintering theory.
In sintering, solid particles consolidate
Department of Materials Science and Engineering
Carnegie Mellon University
http://neon.materials.cmu.edu/
Computer simulated sintering
- microstructure evolution
Computer simulated sintering deformation
- non-uniform initial density
- Layered systems
L=200h
h
D0=0.45
D0=0.64
Computer simulated sintering deformation
• coatings
• fuel cells
• microelectronic films
Computer simulated warping of a bilayer
Now the ceramists have got it, and
we are moving on ………
Orthopedic surgery (from Wikipedia)
The ideal fixation:
Force is gradually transferred from
the plate to the bone as the bone
heals.
This requires the stiffness of the plate
decreases in a controlled and
predictable manner.
Biodegradable fixation (PolyMax®)
made of poly(L-co-D,L-lactide) 70/30
Solution – using biodegradable polymers
98 weeks
52 weeks
In vivo poly(L-lactide) implant(from Gutwald et al. Injury, Int. J. Care Injured 33 (2002) S-B-4-16)
State of the art – trial and error
Model prediction Experiment by Grizzi et al.
Variational modelling of polymer degradation
Variational orthopedic surgeons ?
www.st-joseph.org/news/surgnav.htm
What next ?
25,000-17,000 BC
Chauvet cave, Ardeche valley, France
1550 BC
Fisherman, National Archeological Museum Athens
410 BC
Meidias, vase, British Museum, London
100 AD
Baker and his wife. Museo Archeologico Nazionale, Naples
800 AD
The Virgin Mary and St Joseph on the Way to Bethlehem, Castelseprio
1304
Giotto, Lamentation. Scrovegni Chapel, Padua
1495
Leonardo da Vinci, Last Supper
1503
Leonardo da Vinci, Mona Lisa, Louvre
1633
Anthony van Dyck, Charles I. Buckingham Palace
1785
Mari Louise Elisabeth Vigee-Lebrun, The Artist and her Daughter. Louvre
1814
John Constable, Stour Valley and Dedham Church. Museum of Fine Arts, Boston
1863
Edouard Manet, Lucheon on the Grass, Musee d’Orsay, Paris
1872
Claude Monet, Impression – Sunrise. Musee Marmottan, Paris
1889
Vincent van Gogh, Starry Night
1888
Vincent van Gogh, Starry Night Over the Rhone
1909
Pablo Picasso, Three Women, State Hermitage, St Petersburg
1929
Georges Braque, Le Gueridon. Philips Collection, Washington DC
1964
Cy Twombly, Untitled 1964. Saatchi Collection, London
1984
Georg Baselitz, Mocking, Waddington Gallery, London
Mathematical models should be
using a broad brush
much less details
right to the point !