Finite element method: Difference between revisions

Revision as of 10:04, 8 November 2023

Revision as of 10:35, 21 November 2023

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==History==

While it is difficult to quote the date of the invention of the finite element method, the method originated from the need to solve complex [[Elasticity (physics)|elasticity]] and [[structural analysis]] problems in [[civil engineering|civil]] and [[aeronautical engineering]].<ref>{{Cite journal |last1=Liu |first1=Wing Kam |last2=Li |first2=Shaofan |last3=Park |first3=Harold S. |date=2022 |title=Eighty Years of the Finite Element Method: Birth, Evolution, and Future |journal=Archives of Computational Methods in Engineering |language=en |volume=29 |issue=6 |pages=4431–4453 |doi=10.1007/s11831-022-09740-9 |s2cid=235794921 |issn=1134-3060|doi-access=free |arxiv=2107.04960 }}</ref> Its development can be traced back to work by [[Alexander Hrennikoff|A. Hrennikoff]]<ref>{{Cite journal |last=Hrennikoff |first=Alexander |title=Solution of problems of elasticity by the framework method |journal=Journal of Applied Mechanics |volume=8 |issue=4 |pages=169–175 |year=1941 |doi=10.1115/1.4009129 |bibcode=1941JAM.....8A.169H }}</ref> and [[Richard Courant|R. Courant]]<ref>{{Cite journal |last=Courant |first=R. |title=Variational methods for the solution of problems of equilibrium and vibrations |journal=Bulletin of the American Mathematical Society |volume=49 |pages=1–23 |year=1943 |doi= 10.1090/s0002-9904-1943-07818-4 | doi-access=free }}</ref> in the early 1940s. Another pioneer was [[Ioannis Argyris]]. In the USSR, the introduction of the practical application of the method is usually connected with the name of [[Leonard Oganesyan]].<ref>{{cite web |url=http://emi.nw.ru/INDEX.html?0/resume/oganesan.htm |title=СПб ЭМИ РАН |website=emi.nw.ru |access-date=17 March 2018|archive-url=https://web.archive.org/web/20150930001741/http://emi.nw.ru/INDEX.html?0%2Fresume%2Foganesan.htm |archive-date=30 September 2015 |url-status=dead}}</ref> It was also independently rediscovered in China by [[Feng Kang]] in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the ''finite difference method based on variation principle''. Although the approaches used by these pioneers are different, they share one essential characteristic: [[Polygon mesh|mesh]] [[discretization]] of a continuous ___domain into a set of discrete sub-domains, usually called elements.

Hrennikoff's work discretizes the ___domain by using a [[Lattice (group)|lattice]] analogy, while Courant's approach divides the ___domain into finite triangular subregions to solve [[Partial differential equation#Linear equations of second order|second order]] [[elliptic partial differential equation]]s that arise from the problem of [[torsion (mechanics)|torsion]] of a [[cylinder (geometry)|cylinder]]. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by [[John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], [[Walther Ritz|Ritz]], and [[Boris Galerkin|Galerkin]].

The finite element method obtained its real impetus in the 1960s and 1970s by the developments of [[John Argyris|J. H. Argyris]] with co-workers at the [[University of Stuttgart]], [[Ray W. Clough|R. W. Clough]] with co-workers at [[University of California, Berkeley|UC Berkeley]], [[Olgierd Zienkiewicz|O. C. Zienkiewicz]] with co-workers [[Ernest Hinton]], [[Bruce Irons (engineer)|Bruce Irons]]<ref>{{Cite journal |last1=Hinton |first1=Ernest |last2=Irons |first2=Bruce |title=Least squares smoothing of experimental data using finite elements |journal=Strain |volume=4 |issue=3 |pages=24–27 |date=July 1968 |doi= 10.1111/j.1475-1305.1968.tb01368.x}}</ref> and others at [[Swansea University]], [[Philippe G. Ciarlet]] at the University of [[Pierre-and-Marie-Curie University|Paris 6]] and [[Richard H. Gallagher|Richard Gallagher]] with co-workers at [[Cornell University]]. Further impetus was provided in these years by available open-source finite element programs. NASA sponsored the original version of [[NASTRAN]]. UC Berkeley made the finite element program SAP IV<ref>{{cite web |title=SAP-IV Software and Manuals |url=http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4 |___location=NISEE e-Library, The Earthquake Engineering Online Archive |access-date=2013-01-24 |archive-date=2013-03-09 |archive-url=https://web.archive.org/web/20130309013628/http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4 |url-status=live }}</ref> widely available. In Norway, the ship classification society Det Norske Veritas (now [[DNV GL]]) developed [[SESAM (FEM)|Sesam]] in 1969 for use in the analysis of ships.<ref>{{cite book |title=Building Trust, The history of DNV 1864-2014 |author1=Gard Paulsen |author2=Håkon With Andersen |author3=John Petter Collett |author4=Iver Tangen Stensrud |date=2014 |publisher=Dinamo Forlag A/S |isbn=978-82-8071-256-1 |___location=Lysaker, Norway |pages=121, 436}}<!-- |access-date=30 June 2015 --></ref> A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by [[Gilbert Strang|Strang]] and [[George Fix|Fix]].<ref>{{cite book |first1=Gilbert |last1=Strang |author-link1=Gilbert Strang |first2=George |last2=Fix |author-link2=George Fix |title=An Analysis of The Finite Element Method |url=https://archive.org/details/analysisoffinite0000stra |url-access=registration |publisher=Prentice Hall |year=1973 |isbn=978-0-13-032946-2}}</ref> The method has since been generalized for the [[numerical analysis|numerical modeling]] of physical systems in a wide variety of [[engineering]] disciplines, e.g., [[electromagnetism]], [[heat transfer]], and [[fluid dynamics]].<ref name="ZienkiewiczTaylor2013">{{cite book |author1=Olek C Zienkiewicz |author2=Robert L Taylor |author3=J.Z. Zhu |title=The Finite Element Method: Its Basis and Fundamentals |url=https://books.google.com/books?id=7UL5Ls9hOF8C |date=31 August 2013 |publisher=Butterworth-Heinemann |isbn=978-0-08-095135-5}}</ref><ref>{{cite book |first1=K.J. |last1=Bathe |author-link1= Klaus-Jürgen Bathe |title=Finite Element Procedures |publisher= Cambridge, MA: Klaus-Jürgen Bathe |year=2006 |isbn= 978-0979004902}}</ref>

==Technical discussion==