From Euler, Schwarz, Ritz and Galerkin Brachystochrone Euler - - PowerPoint PPT Presentation

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

Concluding Example

From Euler, Schwarz, Ritz and Galerkin to Modern Computing

Martin J. Gander martin.gander@unige.ch

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

Concluding Example

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.”

dx dx dy dy ds ds A B M 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

Concluding Example

Mathematical Formulation

Letter of de l’Hˆ

• pital to Joh. Bernoulli, June 15th, 1696:

Ce probleme me paroist des plus curieux et des plus jolis que l’on ait encore propos´ e et je serois bien aise de m’y appliquer ; mais pour cela il seroit necessaire que vous me l’envoyassiez reduit ` a la math´ ematique pure, car le phisique m’embarasse . . . Time for passing through a small arc length ds: dJ = ds

v .

Speed (Galilei): v = √2gy Need to find y(x) with y(a) = A, y(b) = B such that J(y) = b

a

• dx2 + dy 2

√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

Concluding Example

Euler’s Treatment

Euler (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 dx P = 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

Concluding Example

Joseph Louis de Lagrange

August 12th, 1755: Ludovico de la Grange Tournier (19 years old) writes to Vir amplissime atque celeberrime L. Euler September 6th, 1755: Euler replies to Vir praestantissime atque excellentissime Lagrange with an enthusiastic letter Idea of Lagrange: suppose y(x) is solution, and add an arbitrary 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 respect to 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 dx P) · 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

Concluding Example

Central Highway of Variational Calculus

Since δy is arbitrary, we conclude from b

a

(N − d dx P) · δy · dx = 0 that for all x N − d dx P = 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 (difference

• f potential and kinetic energy) to the differential equations
• f 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

Concluding Example

Gravitation of a Complicated Body

Newton (Principia 1687): inverse square law for celestial bodies, 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 r 3 , 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

r 3 dξ 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

Concluding Example

Potential Function

Idea of Laplace: Introduce the potential function u =

• ρ(ξ, η, ζ)1

r dξ dη dζ Taking a derivative with respect x, we obtain ∂ ∂x 1 r = −x − ξ r 3 = ⇒ f = − ∂u ∂x , ∂u ∂y , ∂u ∂z

• since e.g.

f1 =

• ρ(ξ, η, ζ)x − ξ

r 3 dξ dη dζ Differentiating once more, we obtain

∂ ∂x x−ξ r3 = r3−3(x−ξ)2r r6

∆u = ∂2u ∂x2 + ∂2u ∂y 2 + ∂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

Concluding Example

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 G¨

• ttingen

1839);

◮ theory of electric fields (W. Thomson, later Lord Kelvin

1847, Liouville 1847);

◮ conformal mappings (Gauss 1825); ◮ irrotational fluid motion in 2D (Helmholtz 1858) ◮ in complex analysis (Cauchy 1825, Riemann Thesis

1851);

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

Concluding Example

Riemann Mapping Theorem

Riemann (Thesis 1851 in G¨

• ttingen):

“Eine vollkommen in sich abgeschlossene mathematische Theorie, welche . . . fortschreitet, ohne zu scheiden, ob es sich um die Schwerkraft, oder die Electricit¨ at, oder den Magnetismus, oder das Gleichgewicht der W¨ arme handelt.”

Theorem

If 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→0

f (z+h)−f (z) h

= ∂f

∂x

limh→0

f (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

Concluding Example

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)|| ·

cos φ sin φ − sin φ cos φ

• “ ... 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

Concluding Example

Riemann Mapping Theorem

“Zwei gegebene einfach zusammenh¨ angende Fl¨ achen k¨

• nnen stets

so aufeinander bezogen werden, dass jedem Punkte der einen ein mit ihm stetig fortr¨ uckender 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

Concluding Example

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

Concluding Example

Riemann’s Audacious “Proof”

Riemann 1857, Werke p. 97: “Hierzu kann in vielen F¨ allen . . . ein Princip dienen, welches Dirichlet zur L¨

• sung dieser Aufgabe f¨

ur eine der Laplace’schen Differentialgleichung gen¨ ugende Function . . . in seinen Vorlesungen . . . seit einer Reihe von Jahren zu geben pflegt.” Idea: For all functions defined on a given domain Ω with the prescribed boundary values, the integral J(u) =

• Ω

1 2

• u2

x + u2 y

• dx dy

is always > 0. Choose among these functions the one for which this integral is minimal ! (see citation; from here originates the name “Dirichlet Principle” 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

Concluding Example

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+b

2

+ b−a

2 arctan x

ǫ

arctan 1

ǫ

“Die Dirichlet’sche Schlussweise f¨ uhrt also in dem betrachteten Falle offenbar zu einem falschen Resultat.” Riemann’s Answer to Weierstrass: “... meine Existenztheoreme sind trotzdem richtig”. (see F. Klein) Helmholtz: “F¨ ur uns Physiker bleibt das Dirichletsche Prinzip 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

Concluding Example

International Challenge

Find harmonic functions ∆u = 0 on any domain Ω with prescribed boundary conditions u = g for (x, y) ∈ ∂Ω. Solution easy for circular domain (Poisson 1815) . . . u(r, φ) = 1 2π 2π 1 − r 2 1 − 2r cos(φ − ψ) + r 2 f (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

Concluding Example

International Challenge

... and for rectangular domains (Fourier 1807): But existence of solutions of Laplace equation on arbitrary 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

Concluding Example

Proof without Dirichlet Principle

H.A. Schwarz (1870, Crelle 74, 1872) Solve alternatingly in subdomains Ω1 and Ω2 which are rectangles or discs. . . Ω1 Ω2 Γ1 Γ2 ∂Ω

• D. Hilbert (G¨
• ttingen 1901, Annalen 1904, Crelle
• J. 1905): Rehabilitation of Dirichlet Principle:

“... eine besondere Kraftleistung beweisender Mathematik, ...” (F. KleinEntw. Math. 19. Jahrh., p. 266). “Mittlerweile war das verachtete und scheintote Dirichletsche Prinzip 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

Concluding Example

Birth of Numerical Methods for PDE’s

Finite difference methods: Variational methods ⇒ Ritz-Galerkin ⇒ FE methods: Both in 1908; both in G¨

• ttingen !

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

Concluding Example

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 good chances 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

Concluding Example

Chladni Figures

Ernst Florens Friedrich Chladni (1787): Entdeckung ¨ uber die 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

Concluding Example

Key Idea of Walther Ritz (1909)

“Das wesentliche der neuen Methode besteht darin, dass nicht von den Differentialgleichungen und Randbedingungen des Problems, sondern direkt vom Prinzip der kleinsten Wirkung ausgegangen wird, aus welchem ja durch Variation jene Gleichungen und Bedingungen gewonnen werden k¨

• nnen.”

J(w):= 1

−1

1

−1

• ∂2w

∂x2

• 2

+ ∂2w ∂y 2

• 2

+2µ∂2w ∂x2 ∂2w ∂y 2 +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

Concluding Example

Problems at the Time of Ritz

1) How to compute K ? “Verwendet man als Ann¨ aherung der Funktion . . . ” “Begn¨ ugt 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

Concluding Example

Convergence of the Eigenvalue

Eigenvalue approximations obtained with this algorithm for the 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 when using the exact model: 12.47 12.49 12.49 379.85 379.14 379.34 1579.79 1556.84 1559.28 2887.06 2899.82 2899.93 5969.67 5957.80 5961.32 14204.92 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

Concluding Example

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

Concluding Example

S.P. Timoshenko (1878–1972)

Timoshenko was the first to realize the importance of Ritz’ invention for applications (1913):

“Nous ne nous arrˆ eterons plus sur le cˆ

• t´

e math´ ematique de cette question: un ouvrage remarquable du savant suisse, M. Walter Ritz, a ´ et´ e consacr´ e ` a ce sujet. En ramenant l’int´ egration des ´ equations ` a la recherche des int´ egrales, M. W. Ritz a montr´ e que pour une classe tr` es vaste de probl` emes, en augmentant le nombre de param` etres a1, a2, a3,. . . , on arrive ` a la solution exacte du probl`

• eme. Pour le cycle de probl`

emes dont nous nous occuperons dans la suite, il n’existe pas de pareille d´ emonstration, mais l’application de la m´ ethode approximative aux probl` emes pour lesquels on poss` ede d´ ej` a des solutions exactes, montre que la m´ ethode donne de tr` es bons r´ esultats 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

Concluding Example

Ivan Bubnov (1872-1919)

◮ Bubnov was a Russian submarine

engineer and constructor

◮ Worked at the Polytechnical

Institute of St. Petersburg (with Galerkin, Krylov, Timoshenko)

◮ Work motivated by Timoshenko’s

application of Ritz’ method to study the stability of plates and beams 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

Concluding Example

Boris Grigoryevich Galerkin (1871-1945)

◮ Studies in the Mechanics Department of

• St. Petersburg Technological Institute

◮ Worked for Russian Steam-Locomotive

Union 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 other

European countries) for scientific reasons in 1909.

Beams and Plates: Series solution of some problems in elastic 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

Concluding Example

Galerkin in his seminal 1915 paper cites the work

• f Ritz, Bubnov and Timoshenko . . .

· · · · · ·

and calls what is known today as the Galerkin method:

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

Concluding Example

Hurwitz and Courant: the Birth of FEM

While Russian scientists immediately used Ritz’ method to solve many difficult problems, pure mathematicians from G¨

• ttingen 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

Concluding Example

Richard Courant (1888-1972)

Variational Methods for the Solution of Problems of Equilibrium and Vibrations (Richard Courant, address delivered before the meeting of the AMS, May 3rd, 1941) “But only the spectacular success of Walther Ritz and its tragic circum- stances caught the general interest. In two publications of 1908 and 1909, Ritz, conscious of his imminent death from consumption, gave a masterly account of the theory, and at the same time applied his method to the calculation of the nodal lines of vi- brating plates, a problem of classical physics that previously had not been satisfactorily 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

Concluding Example

First Finite Element Solution by Courant

(∇u)2 + 2u − → min with u = 0 on outer boundary, and u = c, unknown constant

• n 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

Concluding Example

The Name Finite Element Method

The term Finite Element Method was then coined by Ray Clough in: Ray W. Clough: The finite element method in plane stress analysis, Proc ASCE Conf Electron Computat, Pittsburg, PA, 1960 Based on joint work with Jon Turner from Boeing on structural dynamics, and this work led to the first published description of the finite element method, without the name yet, in

• N. J. Turner and R. W. Clough and H. C. Martin and
• L. J. Topp: Stiffness and Deflection analysis of complex

structures, 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

Concluding Example

Concluding Example: Apartment in Montreal

−20 −15 −10 −5 5 10 15 20 25 2 4 6 8 10 12 1 2 3 4 5

2 4 6 8 10 12 1 2 3 4 5 x y

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