martin j. gander before ritz … · · 2012-03-20johann bernoulli (1696), ... oder die...

Euler, Schwarz,Ritz, Galerkin

Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

From Euler, Schwarz, Ritz and Galerkin

to Modern Computing

Martin J. [email protected]

University of Geneva

RADON Colloquium, November, 2011

In collaboration with Gerhard Wanner


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Brachystochrone(βραχυς = short, χρoνoς = time)

Johann Bernoulli (1696), challenge to his brother Jacob:

“Datis in plano verticali duobus punctis A & B,

assignare Mobili M viam AMB, per quam gravitate

sua descendens, & moveri incipiens a puncto A,

brevissimo tempore perveniat ad alterum punctum

B.”

dxdx

dydydsds

A

BM

x

y

See already Galilei (1638)


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Mathematical Formulation

Letter of de l’Hopital to Joh. Bernoulli, June 15th, 1696:

Ce probleme me paroist des plus curieux et des

plus jolis que l’on ait encore propose et je serois

bien aise de m’y appliquer ; mais pour cela il seroit

necessaire que vous me l’envoyassiez reduit a la

mathematique pure, car le phisique m’embarasse

. . .

Time for passing through a small arc length ds: dJ = dsv.

Speed (Galilei): v =√2gy

Need to find y(x) with y(a) = A, y(b) = B such that

J(y) =

∫ b

a

√

dx2 + dy2√2gy

=

∫ b

a

√

1 + p2√2gy

dx = min (p =dy

dx)


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Euler’s TreatmentEuler (1744): general variational problem

J(y) =

∫ b

a

Z (x , y , p) dx = min (p =dy

dx)

Theorem (Euler 1744)

The optimal solution satisfies the differential equation

N − d

dxP = 0 where N =

∂Z

∂y, P =

∂Z

∂p

Proof.


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Joseph Louis de LagrangeAugust 12th, 1755: Ludovico de la Grange Tournier (19years old) writes to Vir amplissime atque celeberrime L. Euler

September 6th, 1755: Euler replies to Vir praestantissime

atque excellentissime Lagrange with an enthusiastic letterIdea of Lagrange: suppose y(x) is solution, and add anarbitrary variation εδy(x). Then

J(ε) =

∫ b

a

Z (x , y + εδy , p + εδp) dx

must increase in all directions, i.e. its derivative with respectto ǫ must be zero for ǫ = 0:

∂J(ε)

∂ε|ε=0 =

∫ b

a

(N · δy + P · δp) dx = 0.

Since δp is the derivative of δy , we integrate by parts:∫ b

a

(N − d

dxP) · δy · dx = 0


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Central Highway of Variational Calculus

Since δy is arbitrary, we conclude from

∫ b

a

(N − d

dxP) · δy · dx = 0

that for all x

N − d

dxP = 0

Central Highway of Variational Calculus:

1. J(y) −→ min

2. dJ(y+ǫv)dε

|ε=0!= 0: weak form

3. Integration by parts, arbitrary variation: strong form

Connects the Lagrangian of a mechanical system (differenceof potential and kinetic energy) to the differential equationsof its motion. This later led to Hamiltonian mechanics.


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Gravitation of a Complicated BodyNewton (Principia 1687): inverse square law for celestialbodies, f proportional to 1

r2

r

f

(x , y , z)

(ξ, η, ζ)

With f = (f1, f2, f3) (see also Euler 1749) we get

f1 ≈x − ξ

r3, r :=

√

(x − ξ)2 + (y − η)2 + (z − ζ)2

Laplace (1785): What if the celestial body is not a point ?

f1 =

∫∫∫

ρ(ξ, η, ζ)x − ξ

r3dξ dη dζ


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Potential FunctionIdea of Laplace: Introduce the potential function

u =

∫∫∫

ρ(ξ, η, ζ)1

rdξ dη dζ

Taking a derivative with respect x , we obtain

∂

∂x

1

r= −x − ξ

r3=⇒ f = −

(

∂u

∂x,∂u

∂y,∂u

∂z

)

since e.g.

f1 =

∫∫∫

ρ(ξ, η, ζ)x − ξ

r3dξ dη dζ

Differentiating once more, we obtain ∂∂x

x−ξr3

= r3−3(x−ξ)2rr6

∆u =∂2u

∂x2+∂2u

∂y2+∂2u

∂z2= 0!


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Laplace’s Equation

Laplace’s equation was also discovered independantly in:

theory of stationary heat transfer (Fourier 1822);

theory of magnetism (Gauss and Weber in Gottingen1839);

theory of electric fields (W. Thomson, later Lord Kelvin1847, Liouville 1847);

conformal mappings (Gauss 1825);

irrotational fluid motion in 2D (Helmholtz 1858)

in complex analysis (Cauchy 1825, Riemann Thesis1851);


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Riemann Mapping TheoremRiemann (Thesis 1851 in Gottingen):“Eine vollkommen in sich abgeschlossene mathematische Theorie,

welche . . . fortschreitet, ohne zu scheiden, ob es sich um die

Schwerkraft, oder die Electricitat, oder den Magnetismus, oder

das Gleichgewicht der Warme handelt.”

TheoremIf f (z) = u(x , y) + iv(x , y) is holomorph in Ω, then

∆u = uxx + uyy = 0 and ∆v = vxx + vyy = 0.

Proof:

df (z)

dz=

limh→0f (z+h)−f (z)

h= ∂f

∂x

limh→0f (z+ih)−f (z)

ih=−i ∂f

∂y

=⇒ ux+ivx = −iuy+vy


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Conformal maps

The Jacobian of such a function satisfies(

∂u∂x

∂u∂y

∂v∂x

∂v∂y

)

=

(

∂u∂x

∂u∂y

−∂u∂y

∂u∂x

)

= ||(ux , uy )|| ·(

cosφ sinφ− sinφ cosφ

)

“ ... und ihre entsprechenden kleinsten Theile ahnlich sind;”

(Thesis §21)


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Riemann Mapping Theorem“Zwei gegebene einfach zusammenhangende Flachen konnen stets

so aufeinander bezogen werden, dass jedem Punkte der einen ein

mit ihm stetig fortruckender Punkt entspricht...;”

(drawing M. Gutknecht 18.12.1975)


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Idea of Riemann’s Proof

Find f which maps Ω to the unit disk and z0 to 0: set

f (z) = (z − z0)eg(z), g = u + iv =⇒ z0 only zero

In order to arrive on the boundary of the disk

|f (z)| = 1, z ∈ ∂Ω =⇒ u(z) = −log |z − z0|, z ∈ ∂Ω.

Once harmonic u with this boundary condition is found,construct v with the Cauchy-Riemann equations.Question: Does such a u exist ???


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Riemann’s Audacious “Proof”

Riemann 1857, Werke p. 97:

“Hierzu kann in vielen Fallen . . . ein Princip dienen,

welches Dirichlet zur Losung dieser Aufgabe fur

eine der Laplace’schen Differentialgleichung

genugende Function . . . in seinen Vorlesungen . . .

seit einer Reihe von Jahren zu geben pflegt.”

Idea: For all functions defined on a given domain Ω with theprescribed boundary values, the integral

J(u) =

∫∫

Ω

1

2

(

u2x + u2y)

dx dy is always > 0.

Choose among these functions the one for which thisintegral is minimal !(see citation; from here originates the name “DirichletPrinciple” and “Dirichlet boundary conditions”).


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

But is the Dirichlet Principle Correct?

Weierstrass’s Critique (1869, Werke 2, p. 49):

∫ 1

−1(x · y ′)2 dx → min y(−1) = a, y(1) = b.

=⇒ y = a+b2 + b−a

2

arctan xǫ

arctan 1ǫ

“Die Dirichlet’sche Schlussweise fuhrt also in dem

betrachteten Falle offenbar zu einem falschen

Resultat.”

Riemann’s Answer to Weierstrass: “... meineExistenztheoreme sind trotzdem richtig”. (see F. Klein)

Helmholtz: “Fur uns Physiker bleibt das DirichletschePrinzip ein Beweis”


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

International ChallengeFind harmonic functions ∆u = 0 on any domain Ω withprescribed boundary conditions u = g for (x , y) ∈ ∂Ω.

Solution easy for circular domain (Poisson 1815) . . .

u(r , φ) =1

2π

∫ 2π

0

1− r2

1− 2r cos(φ− ψ) + r2f (ψ) dψ .


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

International Challenge

... and for rectangular domains (Fourier 1807):

But existence of solutions of Laplace equation onarbitrary domains appears hopeless !


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Proof without Dirichlet PrincipleH.A. Schwarz (1870, Crelle 74, 1872)Solve alternatingly in subdomains Ω1 and Ω2 which arerectangles or discs. . .

replacements

Ω1 Ω2Γ1Γ2

∂Ω

D. Hilbert (Gottingen 1901, Annalen 1904, CrelleJ. 1905): Rehabilitation of Dirichlet Principle:

“... eine besondere Kraftleistung beweisender Mathematik,...” (F. KleinEntw. Math. 19. Jahrh., p. 266).

“Mittlerweile war das verachtete und scheintote DirichletschePrinzip durch Hilbert wieder zum Leben erweckt worden;...”(Hurwitz-Courant Funktionentheorie, Springer Grundlehren 3,

p. 392)


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Birth of Numerical Methods for PDE’s

Finite difference methods:

Variational methods ⇒ Ritz-Galerkin ⇒ FE methods:

Both in 1908; both in Gottingen !


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Ritz: Vaillant Prize 1907

Ritz had worked with many such problems in his thesis,where he tried to explain the Balmer series in spectroscopy(1902); it therefore appeared to him that he had goodchances to succeed in this competition.

But: Hadamard will win the price. . .


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Chladni FiguresErnst Florens Friedrich Chladni (1787): Entdeckung uberdie Theorie des Klangs, Leipzig.

Chladni figures correspond to eigenpairs of the Bilaplacian

∆2w = λw in Ω := (−1, 1)2


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Key Idea of Walther Ritz (1909)“Das wesentliche der neuen Methode besteht darin, dassnicht von den Differentialgleichungen und Randbedingungendes Problems, sondern direkt vom Prinzip der kleinsten

Wirkung ausgegangen wird, aus welchem ja durch Variationjene Gleichungen und Bedingungen gewonnen werdenkonnen.”

J(w) :=

∫ 1

−1

∫ 1

−1

[

(

∂2w

∂x2

)2

+

(

∂2w

∂y2

)2

+2µ∂2w

∂x2∂2w

∂y2+2(1−µ)

(

∂2w

∂x∂y

)2]

Idea: approximate w by

ws :=

s∑

m=0

s∑

n=0

Amnum(x)un(y)

and minimize J(ws) as a function of a = (Amn) to get

Ka = λa


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Problems at the Time of Ritz

1) How to compute K ?

“Verwendet man als Annaherung der Funktion . . . ”“Begnugt man sich mit vier genauen Ziffern. . . ”“Mit einer Genauigkeit von mindestens 2 Prozent. . . ”

One of the matrices obtained by Ritz:

2) How to solve the eigenvalue problem Ka = λa?


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Convergence of the Eigenvalue

Eigenvalue approximations obtained with this algorithm forthe first eigenvalue:

13.95, 12.14, 12.66, 12.40, 12.50, 12.45, 12.47, . . .

Eigenvalue approximations from the original Ritz matrix,results when calculating in full precision and results whenusing the exact model:

12.47 12.49 12.49379.85 379.14 379.34

1579.79 1556.84 1559.282887.06 2899.82 2899.935969.67 5957.80 5961.3214204.92 14233.73 14235.30


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Some Chladni Figures Computed by Ritz


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

S.P. Timoshenko (1878–1972)Timoshenko was the first to realize the importance of Ritz’invention for applications (1913):“Nous ne nous arreterons plus sur le cote mathematique de cette

question: un ouvrage remarquable du savant suisse, M. Walter

Ritz, a ete consacre a ce sujet. En ramenant l’integration des

equations a la recherche des integrales, M. W. Ritz a montre que

pour une classe tres vaste de problemes, en augmentant le nombre

de parametres a1, a2, a3,. . . , on arrive a la solution exacte du

probleme. Pour le cycle de problemes dont nous nous occuperons

dans la suite, il n’existe pas de pareille demonstration, mais

l’application de la methode approximative aux problemes pour

lesquels on possede deja des solutions exactes, montre que la

methode donne de tres bons resultats et pratiquement on n’a pas

besoin de chercher plus de deux approximations”

schweizarskogo utshenogo Walthera Ritza


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Ivan Bubnov (1872-1919)

Bubnov was a Russian submarineengineer and constructor

Worked at the PolytechnicalInstitute of St. Petersburg (withGalerkin, Krylov, Timoshenko)

Work motivated by Timoshenko’sapplication of Ritz’ method tostudy the stability of plates andbeams

Structural Mechanics of Shipbuilding[Part concerning the theory of shells]


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Boris Grigoryevich Galerkin (1871-1945) Studies in the Mechanics Department of

St. Petersburg Technological Institute

Worked for Russian Steam-LocomotiveUnion and China Far East Railway

Arrested in 1905 for political activities,imprisoned for 1.5 years.

Devote life to science in prison.

Visited Switzerland (among otherEuropean countries) for scientific reasonsin 1909.

Beams and Plates: Series solution of some problems inelastic equilibrium of rods and plates (Petrograd, 1915)


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Galerkin in his seminal 1915 paper cites the work

of Ritz, Bubnov and Timoshenko . . .

· · ·

· · ·

and calls what is known today as the Galerkinmethod:


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Hurwitz and Courant: the Birth of FEMWhile Russian scientists immediately used Ritz’ method tosolve many difficult problems, pure mathematicians fromGottingen had little interest:

Hurwitz and Courant (1922): Funktionentheorie(footnote, which disappeared in the second edition (1925))


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Richard Courant (1888-1972)

Variational Methods for the Solution of Problems ofEquilibrium and Vibrations (Richard Courant, addressdelivered before the meeting of the AMS, May 3rd, 1941)

“But only the spectacular success ofWalther Ritz and its tragic circum-stances caught the general interest.In two publications of 1908 and 1909,Ritz, conscious of his imminent deathfrom consumption, gave a masterlyaccount of the theory, and at thesame time applied his method to thecalculation of the nodal lines of vi-brating plates, a problem of classicalphysics that previously had not beensatisfactorily treated.”


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

First Finite Element Solution by Courant

∫ ∫

(∇u)2 + 2u −→ min

with u = 0 on outer boundary, and u = c , unknown constanton the inner boundary.


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

The Name Finite Element Method

The term Finite Element Method was then coined by RayClough in:

Ray W. Clough: The finite element method in plane stressanalysis, Proc ASCE Conf Electron Computat, Pittsburg,PA, 1960

Based on joint work with Jon Turner from Boeing onstructural dynamics, and this work led to the first publisheddescription of the finite element method, without the nameyet, in

N. J. Turner and R. W. Clough and H. C. Martin andL. J. Topp: Stiffness and Deflection analysis of complexstructures, J. Aero. Sci., Vol. 23, pp. 805–23, 1956.


Martin J. Gander

Before Ritz

Brachystochrone

Euler

Lagrange

Laplace

Riemann

Schwarz

Runge/Ritz

Ritz

Vaillant Prize

Chladni Figures

Ritz Method

Results

Road to FEM

Timoshenko

Bubnov

Galerkin

Courant

Clough

ConcludingExample

Concluding Example: Apartment in Montreal

−20 −15 −10 −5 0 5 10 15 20 25

0 2 4 6 8 10 120

1

2

3

4

5

0 2 4 6 8 10 12

0

1

2

3

4

5

x

y

From Euler, Ritz and Galerkin to Modern Computing,with Gerhard Wanner, SIREV, 2011.

martin j. gander before ritz … · · 2012-03-20johann bernoulli (1696), ... oder die...

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