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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
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
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)
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
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)
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
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.
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
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
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
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.
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
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
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
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!
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
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);
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
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
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
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)
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
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)
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
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 ???
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
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”).
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
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”
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
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 .
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
International Challenge
... and for rectangular domains (Fourier 1807):
But existence of solutions of Laplace equation onarbitrary domains appears hopeless !
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
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)
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
Birth of Numerical Methods for PDE’s
Finite difference methods:
Variational methods ⇒ Ritz-Galerkin ⇒ FE methods:
Both in 1908; both in Gottingen !
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
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. . .
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
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
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
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
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
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?
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
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
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
Some Chladni Figures Computed by Ritz
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
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
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
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]
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
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)
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
Galerkin in his seminal 1915 paper cites the work
of Ritz, Bubnov and Timoshenko . . .
· · ·
· · ·
and calls what is known today as the Galerkinmethod:
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
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))
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
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.”
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
First Finite Element Solution by Courant
∫ ∫
(∇u)2 + 2u −→ min
with u = 0 on outer boundary, and u = c , unknown constanton the inner boundary.
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
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.
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
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.