1Preacutesentation de lrsquoenseignement speacutecialiseacute laquo eacuteleacutements finis raquo ndash S3733-S3735

S Basseville G Cailletaud T Coupez F Feyel M Kern Anneacutee 2005

Jour | 9 00ndash10 30 10 45ndash12 30 | 13 45ndash15 30 15 45ndash17 30Lundi | Introduction Eleacutements finis | PC Preacutesentation21 nov | (GC FF) en 1D (MK) | (MK) des projets (GC MK)

| |Mardi | Preacutesentations Meacutethode des | suite Formulation22 nov | des logiciels (GC MK) Elt finis (MK) | (MK) matricielle (GC)

| |Mercredi | Estimation Technologie des | Maillage Projet23 nov | drsquoerreur a priori(MK) eacuteleacutements finis (GC) | (FF) (tuteurs)

| |Jeudi | Maillage erreur a Formulations | Projet Projet

24 nov | posteriori (TC) mixtes (TC) | (tuteurs) (tuteurs)| |

Vendredi | Reacutesolution de Inteacutegration | Projet Projet25 nov | systegravemes (FF) temporelle (SB) | (tuteurs) (tuteurs)

| |

2

Introduction

bull GC Points de repegraveres historiques (FEM) Points de repegravere sur le marcheacute actuel (FEA) Exemples drsquoapplications

bull FF Les physiques

3

bull 1950ndash1960 les pionniers lrsquoimpulsion vient de lrsquoindustrie aeacuteronautiqueaux US

bull 1955 Argyris theacuteoregravemes de lrsquoeacutenergie [1] Turneret al le triangle agrave troisnœuds [18] Clough la deacutenomination drsquolaquoeacuteleacutements finisraquo [6]

bull 1960ndash1970 deacutebut des deacuteveloppements systeacutematiques rapprochemententre les approches ingeacutenieur et les formulations matheacutematiques

bull Connection BoeingndashBerkeley Ecole europeacuteenne (Liegravege Swansea) [9 19]bull 1964 Gallagher premiegravere revue ([10])bull 1967 Zienkiewicz et Cheung premier livre pour les ingeacutenieurs [19]

extension au geacutenie civilbull Deacuteveloppement de nouveaux types drsquoeacuteleacutements finis inteacutegration numeacute-

rique fonctions de forme notamment par Ironset al [12] formulationsmixtes (reacutesumeacute dans [16])

bull 1970ndash1980 consolidation des acquis bonne compreacutehension desressorts theacuteoriques deacutebut de lrsquoexploitation commerciale laquoTriraquo parmi lesnombreux deacuteveloppements de la preacuteceacutedente deacutecade

bull Extension agrave drsquoautres domaines que la meacutecanique [14]bull Deacutebut du controcircle drsquoerreur eacuteleacutements de plaque et de coques performantsbull Tentatives de deacuteveloppement drsquoautres meacutethodes (eacutequations inteacutegrales

devenues ensuite laquoeacuteleacutements frontiegraveresraquo (BEM))

4

bull 1980ndash1990 la maturiteacute divulgation de la meacutethode en dehors deslaboratoires et des grands groupes industriels

bull Arriveacutee des stations puis des PC interfaces utilisateurs efficacesbull Des livres de reacutefeacuterence en franccedilais [8] refleacutetant lrsquoeacutecole ameacutericaine cocircte

Est [2] cocircte Ouest [11] une encyclopeacutedie [13]bull 1990ndash2000 lrsquointeacutegration dans le processus de fabrication lien renforceacute

avec la CA0bull Drsquoautres livres de reacutefeacuterence en franccedilais [3 5] ou non [7] rappelant

lrsquoaspect laquoingeacutenieurraquo [15] (MSC-Nastran)bull Le non-lineacuteaire [17 4] le dynamique (codes explicites pour le crash)

lrsquoarriveacutee de lrsquooptimisation (conception parameacutetrique)bull Les meacutethodes alternatives (meacutethodes sans maillage par exemple)bull La maturiteacute du calcul parallegravelebull 2000ndash le bizeness rachats-regroupementsbull Le multindashphysique la reacutealiteacute virtuelle (lunettes casques )bull Diminuer les coucircts augmenter la fiabiliteacute (ISO9000)

5

Quelques exemples des codes de calcul par eacuteleacutements finisDeacutemarche commercialebull Nastran 3000 personnes 45000 licencesbull Ansys (fondeacute en 1970) 600 personnes 8600 sites 130000

licences acadeacutemiquesbull Abaqus 300 personnes racheteacute en 2005 par Dassault System

(Catia)bull Algor la conception parameacutetriquebull Samcef 50 personnes 1500 licencesbull Radiossbull Pam-crashet albull LS-Dyna crash 11 sept 2002+bull Fluentbull Forge2Forge3 (CEMEF)

Autres deacutemarchesbull ModulefFreeFEM++FEAPbull CastemCode_Asterbull Zebulon

2

Introduction

bull GC Points de repegraveres historiques (FEM) Points de repegravere sur le marcheacute actuel (FEA) Exemples drsquoapplications

bull FF Les physiques

3

bull 1950ndash1960 les pionniers lrsquoimpulsion vient de lrsquoindustrie aeacuteronautiqueaux US

bull 1955 Argyris theacuteoregravemes de lrsquoeacutenergie [1] Turneret al le triangle agrave troisnœuds [18] Clough la deacutenomination drsquolaquoeacuteleacutements finisraquo [6]

bull 1960ndash1970 deacutebut des deacuteveloppements systeacutematiques rapprochemententre les approches ingeacutenieur et les formulations matheacutematiques

bull Connection BoeingndashBerkeley Ecole europeacuteenne (Liegravege Swansea) [9 19]bull 1964 Gallagher premiegravere revue ([10])bull 1967 Zienkiewicz et Cheung premier livre pour les ingeacutenieurs [19]

extension au geacutenie civilbull Deacuteveloppement de nouveaux types drsquoeacuteleacutements finis inteacutegration numeacute-

rique fonctions de forme notamment par Ironset al [12] formulationsmixtes (reacutesumeacute dans [16])

bull 1970ndash1980 consolidation des acquis bonne compreacutehension desressorts theacuteoriques deacutebut de lrsquoexploitation commerciale laquoTriraquo parmi lesnombreux deacuteveloppements de la preacuteceacutedente deacutecade

bull Extension agrave drsquoautres domaines que la meacutecanique [14]bull Deacutebut du controcircle drsquoerreur eacuteleacutements de plaque et de coques performantsbull Tentatives de deacuteveloppement drsquoautres meacutethodes (eacutequations inteacutegrales

devenues ensuite laquoeacuteleacutements frontiegraveresraquo (BEM))

4

bull 1980ndash1990 la maturiteacute divulgation de la meacutethode en dehors deslaboratoires et des grands groupes industriels

bull Arriveacutee des stations puis des PC interfaces utilisateurs efficacesbull Des livres de reacutefeacuterence en franccedilais [8] refleacutetant lrsquoeacutecole ameacutericaine cocircte

Est [2] cocircte Ouest [11] une encyclopeacutedie [13]bull 1990ndash2000 lrsquointeacutegration dans le processus de fabrication lien renforceacute

avec la CA0bull Drsquoautres livres de reacutefeacuterence en franccedilais [3 5] ou non [7] rappelant

lrsquoaspect laquoingeacutenieurraquo [15] (MSC-Nastran)bull Le non-lineacuteaire [17 4] le dynamique (codes explicites pour le crash)

lrsquoarriveacutee de lrsquooptimisation (conception parameacutetrique)bull Les meacutethodes alternatives (meacutethodes sans maillage par exemple)bull La maturiteacute du calcul parallegravelebull 2000ndash le bizeness rachats-regroupementsbull Le multindashphysique la reacutealiteacute virtuelle (lunettes casques )bull Diminuer les coucircts augmenter la fiabiliteacute (ISO9000)

5

Quelques exemples des codes de calcul par eacuteleacutements finisDeacutemarche commercialebull Nastran 3000 personnes 45000 licencesbull Ansys (fondeacute en 1970) 600 personnes 8600 sites 130000

licences acadeacutemiquesbull Abaqus 300 personnes racheteacute en 2005 par Dassault System

(Catia)bull Algor la conception parameacutetriquebull Samcef 50 personnes 1500 licencesbull Radiossbull Pam-crashet albull LS-Dyna crash 11 sept 2002+bull Fluentbull Forge2Forge3 (CEMEF)

Autres deacutemarchesbull ModulefFreeFEM++FEAPbull CastemCode_Asterbull Zebulon

3

bull 1950ndash1960 les pionniers lrsquoimpulsion vient de lrsquoindustrie aeacuteronautiqueaux US

bull 1955 Argyris theacuteoregravemes de lrsquoeacutenergie [1] Turneret al le triangle agrave troisnœuds [18] Clough la deacutenomination drsquolaquoeacuteleacutements finisraquo [6]

bull 1960ndash1970 deacutebut des deacuteveloppements systeacutematiques rapprochemententre les approches ingeacutenieur et les formulations matheacutematiques

bull Connection BoeingndashBerkeley Ecole europeacuteenne (Liegravege Swansea) [9 19]bull 1964 Gallagher premiegravere revue ([10])bull 1967 Zienkiewicz et Cheung premier livre pour les ingeacutenieurs [19]

extension au geacutenie civilbull Deacuteveloppement de nouveaux types drsquoeacuteleacutements finis inteacutegration numeacute-

rique fonctions de forme notamment par Ironset al [12] formulationsmixtes (reacutesumeacute dans [16])

bull 1970ndash1980 consolidation des acquis bonne compreacutehension desressorts theacuteoriques deacutebut de lrsquoexploitation commerciale laquoTriraquo parmi lesnombreux deacuteveloppements de la preacuteceacutedente deacutecade

bull Extension agrave drsquoautres domaines que la meacutecanique [14]bull Deacutebut du controcircle drsquoerreur eacuteleacutements de plaque et de coques performantsbull Tentatives de deacuteveloppement drsquoautres meacutethodes (eacutequations inteacutegrales

devenues ensuite laquoeacuteleacutements frontiegraveresraquo (BEM))

4

bull 1980ndash1990 la maturiteacute divulgation de la meacutethode en dehors deslaboratoires et des grands groupes industriels

bull Arriveacutee des stations puis des PC interfaces utilisateurs efficacesbull Des livres de reacutefeacuterence en franccedilais [8] refleacutetant lrsquoeacutecole ameacutericaine cocircte

Est [2] cocircte Ouest [11] une encyclopeacutedie [13]bull 1990ndash2000 lrsquointeacutegration dans le processus de fabrication lien renforceacute

avec la CA0bull Drsquoautres livres de reacutefeacuterence en franccedilais [3 5] ou non [7] rappelant

lrsquoaspect laquoingeacutenieurraquo [15] (MSC-Nastran)bull Le non-lineacuteaire [17 4] le dynamique (codes explicites pour le crash)

lrsquoarriveacutee de lrsquooptimisation (conception parameacutetrique)bull Les meacutethodes alternatives (meacutethodes sans maillage par exemple)bull La maturiteacute du calcul parallegravelebull 2000ndash le bizeness rachats-regroupementsbull Le multindashphysique la reacutealiteacute virtuelle (lunettes casques )bull Diminuer les coucircts augmenter la fiabiliteacute (ISO9000)

5

Quelques exemples des codes de calcul par eacuteleacutements finisDeacutemarche commercialebull Nastran 3000 personnes 45000 licencesbull Ansys (fondeacute en 1970) 600 personnes 8600 sites 130000

licences acadeacutemiquesbull Abaqus 300 personnes racheteacute en 2005 par Dassault System

(Catia)bull Algor la conception parameacutetriquebull Samcef 50 personnes 1500 licencesbull Radiossbull Pam-crashet albull LS-Dyna crash 11 sept 2002+bull Fluentbull Forge2Forge3 (CEMEF)

Autres deacutemarchesbull ModulefFreeFEM++FEAPbull CastemCode_Asterbull Zebulon

4

bull 1980ndash1990 la maturiteacute divulgation de la meacutethode en dehors deslaboratoires et des grands groupes industriels

bull Arriveacutee des stations puis des PC interfaces utilisateurs efficacesbull Des livres de reacutefeacuterence en franccedilais [8] refleacutetant lrsquoeacutecole ameacutericaine cocircte

Est [2] cocircte Ouest [11] une encyclopeacutedie [13]bull 1990ndash2000 lrsquointeacutegration dans le processus de fabrication lien renforceacute

avec la CA0bull Drsquoautres livres de reacutefeacuterence en franccedilais [3 5] ou non [7] rappelant

lrsquoaspect laquoingeacutenieurraquo [15] (MSC-Nastran)bull Le non-lineacuteaire [17 4] le dynamique (codes explicites pour le crash)

lrsquoarriveacutee de lrsquooptimisation (conception parameacutetrique)bull Les meacutethodes alternatives (meacutethodes sans maillage par exemple)bull La maturiteacute du calcul parallegravelebull 2000ndash le bizeness rachats-regroupementsbull Le multindashphysique la reacutealiteacute virtuelle (lunettes casques )bull Diminuer les coucircts augmenter la fiabiliteacute (ISO9000)

5

Quelques exemples des codes de calcul par eacuteleacutements finisDeacutemarche commercialebull Nastran 3000 personnes 45000 licencesbull Ansys (fondeacute en 1970) 600 personnes 8600 sites 130000

licences acadeacutemiquesbull Abaqus 300 personnes racheteacute en 2005 par Dassault System

(Catia)bull Algor la conception parameacutetriquebull Samcef 50 personnes 1500 licencesbull Radiossbull Pam-crashet albull LS-Dyna crash 11 sept 2002+bull Fluentbull Forge2Forge3 (CEMEF)

Autres deacutemarchesbull ModulefFreeFEM++FEAPbull CastemCode_Asterbull Zebulon

5

Quelques exemples des codes de calcul par eacuteleacutements finisDeacutemarche commercialebull Nastran 3000 personnes 45000 licencesbull Ansys (fondeacute en 1970) 600 personnes 8600 sites 130000

licences acadeacutemiquesbull Abaqus 300 personnes racheteacute en 2005 par Dassault System

(Catia)bull Algor la conception parameacutetriquebull Samcef 50 personnes 1500 licencesbull Radiossbull Pam-crashet albull LS-Dyna crash 11 sept 2002+bull Fluentbull Forge2Forge3 (CEMEF)

Autres deacutemarchesbull ModulefFreeFEM++FEAPbull CastemCode_Asterbull Zebulon

6

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10

Quelques exemples drsquoutilisation

bull Automobilebull Aeacuteronautiquebull Electronique produits grands publicbull Industries meacutecaniquesbull Meacutedicalbull Caoutchouc

FE marketInfos sur le marcheacute

11

AUTOMOBILE

Abaqus et lrsquoautomobile

Engine durability Transmission performance Seal integrity Component design Powertrain bending Noise ampvibration Gasket analysis Thermal cycling Sheet metal forming Forging analysis Mechanisms analysis Assembly(bolt loading) (drsquoapregraveshttp wwwabaquscom)

12

AERONAUTIQUE

Crash-test drsquoun tronccedilon de Boeing 737 par la NASA

Static dynamic and coupled acoustic-structural analysis of aircraft frames Simulations of large deployable spacestructures such as solar sails space radars and reflector antennas Simulating the performance of various aircraftcomponents such as bulkhead under pressurization wing panel buckling and crack propagation in the fuselage Bladecontainment evaluations and bird strike simulations Thermomechanical simulation of aircraft engines and rocket motorsunder different operating conditions Verification of turbine blade designs Simulation of various aircraft mechanismssuch as landing gears wing flaps and cargo doors Peen forming and creep age forming of large aircraft componentssuch as wing skins Design of explosion resistant components and ballistic damage resistant armors Simulating theeffects of underwater explosions on ships and submarines Structural analysis of submarine frames to predict bucklingmodes Attenuation of noise generated by underwater vehicles Design and simulation of critical components such as anozzles piezoelectric motors self-lubricating hinges and various bearings Nonlinear analysis and damage assessmentsof structural components Optimization of structural design details considering compression buckling analysis Crackpropagation analysis in pressurized fuselage panels Manufacturing applications such as forming analyses of rivets

13

ELECTRONIQUE PRODUITS GRAND PUBLIC

Calcul thermique Calcul meacutecanique

http wwwalgorcom

Drop testing of diverse products such as consumer electronics and fluid-filled containers Pressurization and top-loading of packaging containers Development and manufacturing of new materials such as paper and film Product-human body interactions Simulation of media transport Design andvalidation of acoustic properties of audio equipment Thermomechanical simulation of electronic products Thermal analysis of chip packaging Designof consumer appliances such as washers and dryers (drsquoapregraveshttp wwwabaquscom)

14

CONSTRUCTIONMECANIQUE

15

Cuve de centrale nucleacuteaire

Dynamic and acoustic analysis of rotating machinery such as pumps motors and compressors Thermomechanical and fatigue analysis of powergeneration equipment such as boilers turbines and heat exchangers Design of reliable rollover and falling object protective systems (ROPS and FOPS) Simulated operation of tools such as drills grinders and jackhammers under impact and other severe loading conditions Crash analysis of rollingstock such as locomotives and wagons Static and dynamic analysis of piping systems under various loads such as thermal pressure seismic andloads Structural durability and fatigue analysis of farm and construction equipment Life predictions of various mechanisms such as transmissions andindustrial brakes and clutches Thermomechanical simulations of material processing operations such as rolling sintering extrusion casting and drawing Manufacturing of thermoplastic and polymeric materials (drsquoapregraveshttp wwwabaquscom)

15

Cuve de centrale nucleacuteaire

Dynamic and acoustic analysis of rotating machinery such as pumps motors and compressors Thermomechanical and fatigue analysis of powergeneration equipment such as boilers turbines and heat exchangers Design of reliable rollover and falling object protective systems (ROPS and FOPS) Simulated operation of tools such as drills grinders and jackhammers under impact and other severe loading conditions Crash analysis of rollingstock such as locomotives and wagons Static and dynamic analysis of piping systems under various loads such as thermal pressure seismic andloads Structural durability and fatigue analysis of farm and construction equipment Life predictions of various mechanisms such as transmissions andindustrial brakes and clutches Thermomechanical simulations of material processing operations such as rolling sintering extrusion casting and drawing Manufacturing of thermoplastic and polymeric materials (drsquoapregraveshttp wwwabaquscom)

16

BIOMEDICAL

Finite element analysis for the design of Nitinol medical devices Nuno Rebelo (Hibbitt Karlsson amp Sorensen (West) Inc Fremont CA) and MichaelPerry (PEMSTAR Pacific Consultants Mt View CA)

http wwwpacificconsultantscomDocumentsfeanitinolpdf Manufacturing of different types of medical devices Design of mechanical connectors commonly used in all types of medical devices Extending

patentability enhancing patent protection and obtaining regulatory (FDA and CE mark) approval Design of dental implants and hip knee and otherprosthetic implants Wear and impact simulation to improve durability and reliability of implants Heart valve design Assessment and prediction ofinjuries Design and deployment of vascular stents catheters minimally invasive surgical equipment drug delivery systems etc Soft tissue and jointmodeling Simulating the physiological response of body systems (drsquoapregraveshttp wwwabaquscom)

17

CAOUTCHOUC ET JOINTS

OA Olatunbosun AM Burke Univ Birmingham UK Tire Science and Technology Conference Akron Ohio Avril 2000http tiresocietyorgpdfbooklet_combined00pdf

Bushings Sealing systems Engine mounts Sporting goods Vibration suppression systems Thermomechanical analysis of rubber products Tire modeling and performance simulation o Tire inflation foot print rolling braking and cornering simulations under different road conditions o Impactanalysis o Modeling belt stresses o Tread wear simulation o Coupled acoustic-structural tire analysis (drsquoapregraveshttp wwwabaquscom)

18

Calculs de magneacutetisme

Doc support EMASwwwansoftcom

19

ANSYS

bull Ansys et le secteur aeacuterospatial Simulating Rocket Start Up DesignSpace Lightens the Load for Aerospace Design Keeping 747sFlight-Worthy Optimizing an Underwater Robot Arm DesignSpace Minimizes Weight of Aircraft Cargo Door Support StructureUsing FEA to Simulate a Space Mission

bull Ansys et lrsquoautomobile Bowles Fluidics Corp - building a better windshield washer nozzle that dominates the auto industry DuPont- ANSYS software helps DuPont eliminate rattle from Porsche Boxster manifold Eaton Corp - steering to big gains paradigm shiftsby optimizing designs Fisher Electric Motor Technology Inc - designing an electric motor for a pickup truck Hendrickson TrailerSuspension Systems - over-the-road semi trailers ride better with analysis redesign of suspension system Larry Hedrick Motorsports -design optimization in the fast lane with DesignSpace from ANSYS Inc Lola Cars Ltd - using ANSYS structural analysis to speedrace car development LuK - engineers use ANSYS design analysis and optimization software to conduct fatigue tests to guaranteecomponent safety LuK - reducing costs of ensuring excellent clutch performance through the use of simulation MAN AG - boostingbus body structural integrity through optimization and verification Melroe Co - optimizing Bobcat equipment with finite elementanalysis software from ANSYS Inc

bull Ansys et le biomeacutedical BLOODSIM - European consortium develops blood simulation tool to fight heart disease Crystal MedicalTechnology - building a better artificial tooth DEKA Research amp Development Corp - developing an improved version of a medicalstent design DePuy Inc - finding ways to decrease stress between the upper and lower articulating surfaces of an artificial knee jointKings University and Others - ANSYS Saves UK and US Researchers Time and Money Constructing Patient-Specific Brain ModelsMitroflow International - improving the performance and longevity of implanted heart valves Technical University of Hamburg-Harburg- German University Avoids Costly Testing with ANSYS

bull Ansys et lrsquoeacutelectronique Compaq Computer Corporation - helping to solve solder problems in Compaqrsquos next-generation AlphaServer Genus Ion Technology Division - implanting a speed-up in ion machines Genus Ion Technology Division -using ANSYS for virtualprototyping Motorola - using the viscoplastic analysis capabilities of ANSYS to optimize the manufacture of hybrid power modulesMotorola - helping Motorola solve resonance problems with ANSYS analysis experimental verification Symbol Technologies Inc -leveraging concurrent engineering in developing scanners Symbol Technologies Inc - developing a computer rugged enough to survivea six foot drop onto concrete Turk Elektrik Endustrisi - redesigning a washing machine motor for Siemens AG

20

Encore quelques liens

NB cours + elements + finis = 383000 (94100 reacuteponses en 2004)finite + element + course= 4680000 (1080000 reacuteponses en 2004)

bull Un cours agrave lrsquoUCL (Belgique)

bull Un cours agrave Marseille

bull Les cours de Carlos Felippa (Boulder) Introduction Non linear Advanced

bull Les plus gros calculs du monde(10 milliards drsquoeacutequations)

21Reacutefeacuterences

[1] JH Argyris and S Kelsey Energy theorem and structural analysisAircraft Engrg 26-27 1954-1955

[2] KJ BatheFinite element procedures in engineering analysis Prentice Hall Inc 1982

[3] JL Batoz and G DhattModeacutelisation des structures par eacuteleacutements finis ImdashIII Hermegraves 1991

[4] T Belytschko WK Liu and Brian MoranNonlinear Finite Elements for Continua and Structures Wiley 2000

[5] PG Ciarlet and JL LionsHandbook of Numerical Analysis Finite Element Methods (Part 1) Numerical Methods for Solids (Part 2)North Holland 1995

[6] RW Clough The finite element method in plane stress analysis InProc 2nd ASCE Conf on Electronic Computation pages 345ndash378Pittsburg 1960

[7] MA Crisfield Nonlinear Finite Element Analysis of Solids and Structures Wiley 1991

[8] G Dhatt and G TouzotUne preacutesentation de la meacutethode des eacuteleacutements finis Maloine 1981

[9] BM Fraejis de Veubeke Upper and lower bounds in matrix structural analysis instress analysis oc zienkiewicz and g hollister edwiley pp145ndash1971965Int J Numer Meth Engng 52 287ndash342 1965ndash2001

[10] RH GallaguerA correlation study of methods of matrix structural analysis Pergamon 1964

[11] TJR HughesThe finite element method Linear static and dynamic finite element analysis PrenticendashHall Inc 1987

[12] B Irons and S AhmadTechniques of finite elements EllisndashHorwood 1980

[13] H Kardestuncer editorFinite Element Handbook Mc Graw Hill 1987

[14] HC Martin and GF CareyIntroduction to finite element analysis McGrawndashHill 1973

[15] R Mc NealFinite Element their design and performance Marcel Dekker 1993

[16] THH Pian Some notes on the early history of hybrid stress finite element methodInt J Numer Meth Engng 47 419ndash425 2000

[17] JC Simo and TRJ HughesComputational Inelasticity Springer Verlag 1997

[18] MJ Turner RW Clough HC Martin and LJ Topp Stiffness and deflection analysis of complex structuresJ Aero Sci 23 805ndash8241956

[19] OC Zienkiewicz and YK CheungThe finite element method in Engineering Science McGrawndashHill 1967