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 !