Euler, Lagrange, Ritz, Brachystochrone Euler Lagrange Galerkin, - PowerPoint PPT Presentation

Walther Ritz Martin J. Gander Before Ritz Euler, Lagrange, Ritz, Brachystochrone Euler Lagrange Galerkin, Courant, Clough: Ritz Chladni Figures On the Road to the Finite Element Method Ritz Method Results Road to FEM Timoshenko Martin J. Gander Bubnov Galerkin Courant martin.gander@unige.ch Clough Summary University of Geneva October, 2011 In collaboration with Gerhard Wanner

Walther Ritz Brachystochrone Martin J. Gander ( βραχυς =short, χρ o ν o ς =time) Before Ritz Johann Bernoulli (1696) , challenge to his brother Jacob: Brachystochrone Euler Lagrange “Datis in plano verticali duobus punctis A & B, Ritz assignare Mobili M viam AMB, per quam gravitate Chladni Figures Ritz Method sua descendens, & moveri incipiens a puncto A, Results Road to FEM brevissimo tempore perveniat ad alterum punctum Timoshenko B.” Bubnov Galerkin Courant Clough A x Summary dx dx dy dy ds ds M B y See already Galilei (1638)

Walther Ritz Mathematical Formulation Martin J. Gander Before Ritz Letter of de l’Hˆ opital to Joh. Bernoulli, June 15th, 1696: Brachystochrone Euler Ce probleme me paroist des plus curieux et des Lagrange Ritz plus jolis que l’on ait encore propos´ e et je serois Chladni Figures Ritz Method bien aise de m’y appliquer ; mais pour cela il seroit Results necessaire que vous me l’envoyassiez reduit ` a la Road to FEM Timoshenko math´ ematique pure, car le phisique m’embarasse Bubnov Galerkin . . . Courant Clough Time for passing through a small arc length ds : dJ = ds v . Summary Speed (Galilei): v = √ 2 gy Need to find y ( x ) with y ( a ) = A , y ( b ) = B such that � b � b dx 2 + dy 2 � � 1 + p 2 ( p = dy J ( y ) = √ 2 gy = √ 2 gy dx = min dx ) a a

Walther Ritz Euler’s Treatment Martin J. Gander Euler (1744) : general variational problem Before Ritz � b Brachystochrone ( p = dy Euler J ( y ) = Z ( x , y , p ) dx = min dx ) Lagrange a Ritz Chladni Figures Ritz Method Results Theorem (Euler 1744) Road to FEM The optimal solution satisfies the differential equation Timoshenko Bubnov Galerkin Courant N − d N = ∂ Z P = ∂ Z Clough dx P = 0 where ∂ y , Summary ∂ p Proof.

Walther Ritz Joseph Louis de Lagrange Martin J. Gander August 12th, 1755: Ludovico de la Grange Tournier (19 years old) writes to Vir amplissime atque celeberrime L. Euler Before Ritz Brachystochrone September 6th, 1755: Euler replies to Vir praestantissime Euler Lagrange atque excellentissime Lagrange with an enthusiastic letter Ritz Idea of Lagrange: suppose y ( x ) is solution, and add an Chladni Figures Ritz Method arbitrary variation εδ y ( x ). Then Results Road to FEM � b Timoshenko Bubnov J ( ε ) = Z ( x , y + εδ y , p + εδ p ) dx Galerkin Courant a Clough must increase in all directions , i.e. its derivative with respect Summary to ǫ must be zero for ǫ = 0: � b ∂ J ( ε ) | ε =0 = ( N · δ y + P · δ p ) dx = 0 . ∂ε a Since δ p is the derivative of δ y , we integrate by parts: � b ( N − d dx P ) · δ y · dx = 0 a

Walther Ritz Central Highway of Variational Calculus Martin J. Gander Before Ritz Since δ y is arbitrary, we conclude from Brachystochrone Euler � b Lagrange ( N − d dx P ) · δ y · dx = 0 Ritz Chladni Figures a Ritz Method Results that for all x Road to FEM N − d Timoshenko dx P = 0 Bubnov Galerkin Courant Central Highway of Variational Calculus: Clough Summary 1. J ( y ) − → min dJ ( y + ǫ v ) ! 2. | ε =0 = 0: weak form d ε 3. Integration by parts, arbitrary variation: strong form Connects the Lagrangian of a mechanical system (difference of potential and kinetic energy) to the differential equations of its motion. This later led to Hamiltonian mechanics.

Walther Ritz Chladni Figures Martin J. Gander Ernst Florens Friedrich Chladni (1787): Entdeckung ¨ uber die Theorie des Klangs, Leipzig. Before Ritz Brachystochrone Euler Lagrange Ritz Chladni Figures Ritz Method Results Road to FEM Timoshenko Bubnov Galerkin Courant Clough Summary Chladni figures correspond to eigenpairs of the Bilaplacian ∆ 2 w = λ w in Ω := ( − 1 , 1) 2

Walther Ritz Key Idea of Walther Ritz (1909) Martin J. Gander “Das wesentliche der neuen Methode besteht darin, dass nicht von den Differentialgleichungen und Randbedingungen Before Ritz Brachystochrone des Problems, sondern direkt vom Prinzip der kleinsten Euler Lagrange Wirkung ausgegangen wird, aus welchem ja durch Variation Ritz jene Gleichungen und Bedingungen gewonnen werden Chladni Figures Ritz Method k¨ onnen.” Results Road to FEM Timoshenko � ∂ 2 w Bubnov � 1 � 1 � 2 2 2 � � ∂ 2 w � ∂ 2 w +2 µ∂ 2 w ∂ 2 w � � � Galerkin J ( w ):= + ∂ y 2 +2(1 − µ ) Courant ∂ x 2 ∂ y 2 ∂ x 2 ∂ x ∂ y Clough − 1 − 1 Summary Idea: approximate w by s s � � w s := A mn u m ( x ) u n ( y ) m =0 n =0 and minimize J ( w s ) as a function of a = ( A mn ) to get K a = λ a

Walther Ritz Problems at the Time of Ritz Martin J. Gander Before Ritz 1) How to compute K ? Brachystochrone Euler “Verwendet man als Ann¨ aherung der Funktion . . . ” Lagrange Ritz “Begn¨ ugt man sich mit vier genauen Ziffern. . . ” Chladni Figures “Mit einer Genauigkeit von mindestens 2 Prozent. . . ” Ritz Method Results Road to FEM One of the matrices obtained by Ritz: Timoshenko Bubnov Galerkin Courant Clough Summary 2) How to solve the eigenvalue problem K a = λ a?

Walther Ritz Convergence of the Eigenvalue Martin J. Gander Eigenvalue approximations obtained with this algorithm for Before Ritz the first eigenvalue: Brachystochrone Euler Lagrange 13 . 95 , 12 . 14 , 12 . 66 , 12 . 40 , 12 . 50 , 12 . 45 , 12 . 47 , . . . Ritz Chladni Figures Ritz Method Results Road to FEM Eigenvalue approximations from the original Ritz matrix, Timoshenko Bubnov results when calculating in full precision and results when Galerkin Courant using the exact model: Clough Summary 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

Walther Ritz Some Chladni Figures Computed by Ritz Martin J. Gander Before Ritz Brachystochrone Euler Lagrange Ritz Chladni Figures Ritz Method Results Road to FEM Timoshenko Bubnov Galerkin Courant Clough Summary

Walther Ritz S.P. Timoshenko (1878–1972) Martin J. Gander Timoshenko was the first to realize the importance of Ritz’ Before Ritz invention for applications (1913): Brachystochrone “Nous ne nous arrˆ eterons plus sur le cˆ ot´ e math´ ematique de cette Euler Lagrange question: un ouvrage remarquable du savant suisse, M. Walter Ritz Chladni Figures Ritz, a ´ et´ e consacr´ e ` a ce sujet. En ramenant l’int´ egration des Ritz Method Results ´ equations ` a la recherche des int´ egrales, M. W. Ritz a montr´ e que Road to FEM pour une classe tr` es vaste de probl` emes, en augmentant le nombre Timoshenko Bubnov de param` etres a 1 , a 2 , a 3 ,. . . , on arrive ` a la solution exacte du Galerkin Courant probl` eme. Pour le cycle de probl` emes dont nous nous occuperons Clough dans la suite, il n’existe pas de pareille d´ emonstration, mais Summary 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

Walther Ritz Ivan Bubnov (1872-1919) Martin J. Gander ◮ Bubnov was a Russian submarine Before Ritz Brachystochrone engineer and constructor Euler Lagrange ◮ Worked at the Polytechnical Ritz Institute of St. Petersburg (with Chladni Figures Ritz Method Galerkin, Krylov, Timoshenko) Results Road to FEM ◮ Work motivated by Timoshenko’s Timoshenko Bubnov application of Ritz’ method to Galerkin Courant study the stability of plates and Clough Summary beams Structural Mechanics of Shipbuilding [Part concerning the theory of shells]

Walther Ritz Boris Grigoryevich Galerkin (1871-1945) Martin J. Gander ◮ Studies in the Mechanics Department of Before Ritz St. Petersburg Technological Institute Brachystochrone Euler ◮ Worked for Russian Steam-Locomotive Lagrange Union and China Far East Railway Ritz Chladni Figures Ritz Method ◮ Arrested in 1905 for political activities, Results imprisoned for 1.5 years. Road to FEM Timoshenko ◮ Devote life to science in prison. Bubnov Galerkin Courant ◮ Visited Switzerland (among other Clough European countries) for scientific reasons Summary in 1909. Beams and Plates: Series solution of some problems in elastic equilibrium of rods and plates (Petrograd, 1915)

Walther Ritz Galerkin in his seminal 1915 paper cites the work Martin J. Gander of Ritz, Bubnov and Timoshenko . . . Before Ritz Brachystochrone Euler Lagrange · · · Ritz Chladni Figures Ritz Method · · · Results Road to FEM Timoshenko Bubnov Galerkin Courant Clough Summary and calls what is known today as the Galerkin method: