Title: Theoretical foundations of the finite element method
Abstract: The finite element method is nowadays the most general and one of the most powerful tools for the analysis of structures. It is also a general mathematical technique and the main concern of the paper is to present it in this light. Functional Analysis is used as the ideal frame for a general abstract formulation. The ability to predict convergence to the exact solution of a sequence of approximate solutions obtained from patterns of finite elements with decreasing size is fundamental in the application of the method. In case conformity between elements is obtained, the finite element method is a particular case of Ritz's method, so that convergence can be ensured as far as completeness is achieved. A general completeness criterion is justified in the paper. Such criterion requires that the field components and all their derivatives, of order not higher than the highest order of derivative entering into the energy density expression, can take up any constant value within the element. It is finally proved that such criterion is also a general convergence criterion, i.e. a sufficient condition for convergence even if conformity is not achieved. Me;тoд кoнe;чнoгo злe;Me;нтa яbлтe;тcя b нacтoящe;e; bpe;Mя нaибoлe;e; oбщиM и oдниM из Moщныч cпocoбob pacчe;тa кoнcтpy;кций. Oн oкaзыbae;тcя тaкжe; oбщиM Maтe;Maтичe;cкиM Me;тoдoM. Глabнoй зaдaчe;й paбoты яbляe;тcя пpe;дcтabлe;ниe; иMe;ннo этoгo acпe;ктa пpoблe;Mы. Иcпoльзy;e;тcя фy;нкциoнaльный aнaлиз кaк идe;aльнoe; opy;диe; для oбщe;й aбcтpaктнoй фopMy;лиpobки. OcнobныM ддя пpиMe;нe;ния этoгo Me;тoдa яbляe;тcя oпpe;дe;лe;ниe; cчoдиMocти к тoчнoMy; pe;щe;нии пocлe;дobaтe;льнocти пpиближe;нныч pe;шe;ний, пoлy;чe;нныч нa ocнobe; Moдe;лe;й кoнe;чныч элe;Me;нтob, пpи y;Me;ньшaнии paзMe;pa. Для cлy;чaя cooтbe;тcтbия Me;чдy; элe;Me;нтaMи, Me;тoд кoнe;чнoгo элe;Me;нтa oкaзыbae;тcя чacтныM cлy;чae;M Me;тoдa Pитцa, тaк чтo cчoдиMocть Moжнo гapaнтиpobaть нacтoлькo нacкoлькo дocтигнy;тa пoлнoтa. b paбoтe; oбcнabыae;тcя oбщий кpитe;pий пoлнoты. Этoт кpитe;pий тpe;бy;e;т, чтoбы кoMпoнe;нты пoля и bce; ич пpoизboдныe;, пopядкa нe; bышe; cтapшe;й пpoизboднoй, bчoдящe;й b bыpaжe;ниe; для плoтнocти энe;pгии, Moгли пpиниMaть кaкoe; либo пocтoяннoe; знaчe;ниe; b пpe;дe;лaч элe;Me;нтa. Haкoнe;ц дoкaзыbae;тcя чтo тaкoй кpитe;pий йbляe;тcя тaкжe; oбщиM кpитe;pиe;M cчoдиMocти, тo e;cть дocтaтoчныM y;cлoиe;M cчoдиMocти, дaжe; e;cли нe; дocтигae;тcя cooтbe;тcтbия.
Publication Year: 1968
Publication Date: 1968-10-01
Language: en
Type: article
Indexed In: ['crossref']
Access and Citation
Cited By Count: 117
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot