CIV-E4010 Finite Element Methods in Civil Engineering

CIV-E4 Fn Elmn Mos n Cl Engnrng Sprng 7 pro V 5 crs MSc/DSc Dparmn of Cl Engnrng Scool of Engnrng Aalo Unrs Jarkko Nrann Asssan Profssor Acam Rsarc Fllo Frs lcr: 4 Tsa Aprl 7

CIV-E4 Fn Elmn Mos n Cl Engnrng Topc crrs Asssans crs Ercss Wb s Maral Fn lmn mos for fnamnal problms n srcral mcancs srcral ngnrng an blng pscs: or applcaons an sofar ools Jarkko Nrann Asssan Profssor Acam Rsarc Fllo; An Nm Snor Rsarc Fllo sng for ks 5 an 6 Srg Kakalo an Vacsla Balobano Docoral Sns Tsas an Trsas 4 n R Fras n R5 ac for orcal assgnmns Monas 4 6 n R66 ac/rrn for compr assgnmns ps://mcorss.aalo.f/cors/.pp?=996 crs sls an assgnmns 7 as pfs n MCorss; T. J. R Hgs: T Fn Elmn Mo; F. Harmann & C. Kaz: Srcral Analss Fn Elmns; A. Öcsnr & M. Mrkl: On-Dmnsonal Fn Elmns; J. N. R: An Inrocon o Fn Elmn Mo; J. N. R: An Inrocon o Nonlnar Fn Elmn Analss CIV-E4 / 7 / Jarkko Nrann

CIV-E4 Fn Elmn Mos n Cl Engnrng Ananc an grang I. Ananc for crs or Torcal Ercs Sssons s no complsor. II. Ananc for Compr Ercs Sssons s complsor for n s grang. III. T fnal gra s bl as a combnaon of amnaon 5% om assgnmns 5% an compr/sofar assgnmns 5%. IV. Passng gra can b ac b abo 4% of oal mamm. V. Eamnaon as n 7: on Ma 6 an n bgnnng of Spmbr. Work loa T nomnal srbon of oal 33 ors 5 crs s as follos: Conac acng 38 % Inpnn sng 6 % crs 8% Rang 8% Ercs classs 9% Hom assgnmns 8% Compr classs 9% Compr assgnmns 8% Eamnaon % Prparaon for amnaon 8% CIV-E4 / 7 / Jarkko Nrann 3

Commrcal fn lmn sofar ampls Commrcal fn lmn analss sofar sall pro a smlaon nronmn faclang all sps n mollng procss: fnng gomr maral aa loangs an bonar conons; coosng lmns msng an solng ssm qaons; 3 salzng an pos-procssng rsls. Som common gnral prpos or mlpscs FEM sofar: Comsol p://.comsol.com/ p://.comsol.com/o/rmal-srss-analss-rbn-saor-bla ps://.comsol.com/rlas/5.a Ana p://.ana.com/ Abaqs p://.smla.com/procs/abaqs_fa.ml Anss p://.anss.com/ Som common srcral ngnrng FEM sofar: Sca p://.sca-onln.com/ sas p://.lsas.com/ RFEM ps://.lbal.com/n/procs/rfm-fa-sofar/a-s-rfm Robo p://.aosk.com/procs/robo-srcral-analss/or CIV-E4 / 7 / Jarkko Nrann 4

CIV-E4 Fn Elmn Mos n Cl Engnrng Conns Wk. Rol of morn fn lmn cnqs n ngnrng analss. Absrac formlaon an accrac of fn lmn mos Wk 3. Fn lmn mos for Krcoff o plas 4. Fn lmn mos for Rssnr Mnln plas Wk 3 5. Fn lmn mos for m pnn problms Wk 4 6. Nonlnars n fn lmn analss Wk 5 7. Fn lmn mos for slls Wk 6 8. Fn lmn mos for braons an bcklng CIV-E4 / 7 / Jarkko Nrann 5

Rol of morn fn lmn cnqs n ngnrng analss s sar som smlaon ampls: Cng procss p://.ana.com/nsgh4.sml Sll folng p://.ana.com/nsgh8.sml Sampng p://.ana.com/sampng.sml Bar braons n fl p://.ana.com/nsgh37.sml Sal sp mas p://.ana.com/nsgh46.sml Fasnr jons p://.ana.com/nsgh5.sml Hmmng p://.ana.com/mmng.sml Comsol rlas 5.: ps://.comsol.com/rlas/5.a

Rol of morn fn lmn cnqs n ngnrng analss Conns. Mollng an compaon n ngnrng sgn an analss. Moaon for compaonal srcral ngnrng arnng ocom A. Unrsanng of man mplcaons of approma nar of compaonal mos n ngnrng sgn an analss B. Rcognzng caracr of compaon an smlaon as a scpln Rfrncs T book : Caprs. CIV-E4 / 7 / Jarkko Nrann 8

A GIMPSE TO THE PREVIOUS COURSES

CIV-E6 Engnrng Compaon an Smlaon Conns. Mollng prncpls an bonar al problms n ngnrng scncs. Bascs of nmrcal ngraon an ffrnaon 3. Basc D fn ffrnc an collocaon mos - bars/ros a ffson spag lcrosacs 4. Enrg mos an basc D fn lmn mos - bars/ros bams a ffson spag lcrosacs 5. Basc D an 3D fn lmn mos - a ffson spag 6. Nmrcal mplmnaon cnqs for fn lmn mos 7. Fn lmn mos for Elr Brnoll bams 8. Fn lmn mos for D an 3D lasc CIV-E6 / 6 / Jarkko Nrann

Mollng prncpls an bonar al problms n ngnrng scncs s sar som smlaon ampls: Cng procss p://.ana.com/nsgh4.sml Sll folng p://.ana.com/nsgh8.sml Sampng p://.ana.com/sampng.sml Bar braons n fl p://.ana.com/nsgh37.sml Sal sp mas p://.ana.com/nsgh46.sml Fasnr jons p://.ana.com/nsgh5.sml Hmmng p://.ana.com/mmng.sml Comsol rlas 5.: ps://.comsol.com/rlas/5.a

Mollng prncpls an bonar al problms n ngnrng scncs Conns. Mollng an compaon n ngnrng sgn an analss. Bonar an nal al problms n ngnrng scncs arnng ocom A. Unrsanng of man mplcaons of approma nar of compaonal mos n ngnrng sgn an analss B. Abl o formla an sol som basc D mol problms Rfrncs cr nos: capr T book: caprs. CIV-E6 / 6 / Jarkko Nrann

. Qsonng compaonal analss Ho ll o compaonal cnqs of ffrn ngnrng fls smla ral lf? CIV-E6 / 6 / Jarkko Nrann 3

. Molng an compaon n ngnrng sgn an analss sp Pscal ngnrng problm sgn crra solon P =? Csomr ns! Dmnsons! as an rglaons! Tm slo! Tcnolog aalabl! Prc rang!... Ho long? Ho ck? Wc maral? Ho man? Wc jons? Ho o consrc?... Ho o g ansrs? CIV-E6 / 6 / Jarkko Nrann 4

. Molng an compaon n ngnrng sgn an analss sp 4D nonlnar all ncls or Pscal ngnrng problm sgn crra Gnral pscomamacal mol solon P =? solon 4D =? + Ialzaon rror P 4D CIV-E6 / 6 / Jarkko Nrann 7

. Molng an compaon n ngnrng sgn an analss sp 4D nonlnar all ncls or Pscal ngnrng problm sgn crra Gnral pscomamacal mol solon P =? solon 4D =? + Ialzaon rror P 4D NONINEAR ANISOTROPIC TIME-DEPENDENT MUTI-PHYSICA CIV-E6 / 6 / Jarkko Nrann 8

. Molng an compaon n ngnrng sgn an analss 4D nonlnar or sp 3D lnar lasc or Kncs Cons mols Knmacs Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol σ b σ Eε ε & BCs solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror 4D 3D 3D INEAR ISOTROPIC TIME-INDEPENDENT CIV-E6 / 6 / Jarkko Nrann 9

. Molng an compaon n ngnrng sgn an analss 3D lnar or Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror sp 3 D aall loa lasc ro E A b N ms smplf psco-mamacal mol N N' b E ' N solon =... + N Molng rror A " 3D " D INEAR ISOTROPIC TIME- INDEPENDENT Han calclaons ork! CIV-E6 / 6 / Jarkko Nrann

. Molng an compaon n ngnrng sgn an analss 4D nonlnar all ncls or sp Pscal ngnrng problm sgn crra Gnral pscomamacal mol Nmrcal mo solon P =? solon 4D =? + Ialzaon rror solon =... + Dscrzaon rror 4D nmrcal_mo4d or; CIV-E6 / 6 / Jarkko Nrann

. Molng an compaon n ngnrng sgn an analss 4D nonlnar all ncls or Pscal ngnrng problm sgn crra Gnral pscomamacal mol solon P =? solon 4D =? + Ialzaon rror sp CIV-E6 / 6 / Jarkko Nrann Nmrcal mo Rlabl & Effcn Applcabl Sabl Accra Cap solon =... + Dscrzaon rror 4D nmrcal_mo4d or;

. Molng an compaon n ngnrng sgn an analss 4D nonlnar all ncls or Pscal ngnrng problm sgn crra Gnral pscomamacal mol solon P =? solon 4D =? + Ialzaon rror sp CIV-E6 / 6 / Jarkko Nrann Nmrcal mo Nr a black bo nor Inapplcabl Unsabl Inaccra Epns solon =... + Dscrzaon rror 4D nmrcal_mo4d or; 3

. Molng an compaon n ngnrng sgn an analss sp 3 3D lnar B&B or Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol Nmrcal mo solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror solon =... + Dscrzaon rror 3D nmrcal_mo3d or; CIV-E6 / 6 / Jarkko Nrann 4

. Molng an compaon n ngnrng sgn an analss Cangs o mos: rfcaon sp 4 Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol Nmrcal mo Obsraons an conclsons solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror solon =... + Dscrzaon rror 3D + Hman rrors CIV-E6 / 6 / Jarkko Nrann 5

. Molng an compaon n ngnrng sgn an analss Cangs o mols: alaon Cangs o mos: rfcaon sp 4 Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol Nmrcal mo Obsraons an conclsons solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror solon =... + Dscrzaon rror 3D + Hman rrors CIV-E6 / 6 / Jarkko Nrann 6

. Molng an compaon n ngnrng sgn an analss Cangs o problm an sgn Cangs o mols: alaon Cangs o mos: rfcaon sp 4 Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol Nmrcal mo Obsraons an conclsons solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror solon =... + Dscrzaon rror 3D + Hman rrors CIV-E6 / 6 / Jarkko Nrann 7

. Molng an compaon n ngnrng sgn an analss Cangs o problm an sgn Cangs o mols: alaon Cangs o mos: rfcaon sp 5 Pscal ngnrng problm sgn crra Gnral pscomamacal mol Smplf pscomamacal mol Nmrcal mo Obsraons an conclsons Accpanc solon P =? solon 4D =? + Ialzaon rror solon 3D =? + Molng rror solon =... + Dscrzaon rror 3D + Hman rrors CIV-E6 / 6 / Jarkko Nrann 8

. Molng an compaon n ngnrng sgn an analss Brak rcs Formla an rror sma for oal rror prsn n a pcal sgn an analss procss n rms of rror rms scrb abo ffrnc bn pscal ral an fnal D nmrcal solon. P " "... CIV-E6 / 6 / Jarkko Nrann 9

BACK TO THE PRESENT COURSE

. Moaon for compaonal srcral ngnrng CIV-E4 / 7 / Jarkko Nrann 3

. Moaon for compaonal srcral ngnrng Wa s common o s acs? CIV-E4 / 7 / Jarkko Nrann 33

. Moaon for compaonal srcral ngnrng Wa s common o s acs? Taln? CIV-E4 / 7 / Jarkko Nrann 34

. Moaon for compaonal srcral ngnrng Wa s common o s acs?? CIV-E4 / 7 / Jarkko Nrann 35

. Moaon for compaonal srcral ngnrng Wa s common o s acs? Taln +? CIV-E4 / 7 / Jarkko Nrann 36

. Moaon for compaonal srcral ngnrng Blng ssmacall on or knolg an sklls for racng op! Yo a aln o js n o ran! - Rcall or Yo ar r! BSc ss! - Rcollc or o! - Rmnsc or cloo! cmsr pscs mcancs BSc mamacs programmng proc sgn cmsr pscs Hg scool mamacs langags bolog pscs Sconar scool mamacs langags sor pscs Prmar scool mamacs mor langag ngls CIV-E4 / 7 / Jarkko Nrann 37

. Moaon for compaonal srcral ngnrng Wa s no common o s acs? Consqncs of ncompnc! CIV-E4 / 7 / Jarkko Nrann 38

. Moaon for compaonal srcral ngnrng Consqncs of ncompnc! CIV-E4 / 7 / Jarkko Nrann 39

.X Connm mcancs n cl ngnrng Blng blocks of bonar al problms n cl ngnrng: Dformaon an moon s fn b connm mcancs concps as Knmacs splacmns an srans Kncs consraon of lnar an anglar momnm 3 Trmonamcs I an II las 4 Cons qaons srsss s. srans T man mamacal ools ar Vcor an nsor algbra an analss Dffrnal ngral an araonal calcls Paral ffrnal qaons D lasc N' D/3D lasc σ σ Eε ε Alogr pscal consraon prncpls.. las of consraon of mass momna an nrg as ll as cons rsponss of marals or or obsr rlaons ar cor b a combnaon of orcal ools abo. b N E ' b A & & BCs BCs CIV-E6 / 6 / Jarkko Nrann 4

.X Connm mcancs n cl ngnrng Mar or maral s compos of parcls from lcrons an aoms p o molcls c can b nr cran assmpons moll as a connm or. Ialzaons of pscs an cmsr ar frr smplf or omognz b or of connm mcancs. CIV-E6 / 6 / Jarkko Nrann 4

.X Connm mcancs n cl ngnrng Connm s a pocal ool spcfc assmpons an fars orlooks parcls p o molclar sz omogn scals of nrs ar larg nog praccal pscal qans of nrs ar connosl ffrnabl mamacal applcapl for all marals gnral Wn connm mcancs a spcrm of pscal pnomna can b s or. Man araons mofcaons or nsons for classcal connm ors s as ll: sconnm-connm pso-connm or Cossra connm gr-orr sran gran conna c. ofn appl o capr mcrosrcral ffcs of granlar marals for nsanc. CIV-E6 / 6 / Jarkko Nrann 4

.X Connm mcancs n cl ngnrng Connm mcancs ss no onl formaon of sols b formaon an flo of a connm corng sols lqs an gass. Engnrng scncs as srcral ngnrng s parclar alorngs of connm mcancs: bars bams plas an slls n lasc plasc scolasc or scoplasc for nsanc. Problms formla n rms of connm mcancs ar ransform b mamacal ools no form of compaonal mcancs: connm mcancs an nmrcal mos corrsponng compr mplmnaons rfrr as nmrcal smlaon ools. CIV-E6 / 6 / Jarkko Nrann 43

QUESTIONS? ANSWERS ECTURE BREAK!

Hom rcs. Spn a copl of ors for carfll rang an makng nos abo a cr sls Scons. an.x from cors CIV-E6 b T book 3 Capr. T book 3: Anras Öcsnr Marks Mrkl: On-Dmnsonal Fn Elmns An Inrocon o FE Mo Sprngr-Vrlag 3 s r man pons from bo sorcs b a f snncs. s mamacal noaons no famlar o o for rcallng m from or arlr s maral or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 45

Hom rcs. Spn a copl of ors for carfll rang an makng nos abo T book 3 Caprs 3 an 4. T book 3: Anras Öcsnr Marks Mrkl: On-Dmnsonal Fn Elmns An Inrocon o FE Mo Sprngr-Vrlag 3 s r man pons from bo Caprs b a f snncs an formla. s mamacal noaons no famlar o o for rcallng m from or arlr s maral or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 46

Absrac formlaon an accrac of fn lmn mos

Absrac formlaon an accrac of fn lmn mos Conns. Absrac mamacal formlaon of fn lmn mos. Basc rror smas an accrac of fn lmn mos arnng ocom A. Abl o formla a fn lmn problm n an absrac mamacal form an nrsanng of rols of basc mamacal concps n formlaon B. Basc knolg an ools for smang an mprong accrac of fn lmn solons Rfrncs T book : Caprs. A.I 4. A4.I. 4 T book : Caprs. 6.4 7. 7. CIV-E4 / 7 / Jarkko Nrann 49

. Moaon for rror analss Ho can a srcral anals oban accra rsls for crcal sgn qans b fn lmn mos? CIV-E4 / 7 / Jarkko Nrann 5

. Moaon for rror analss Conrgng. Ho os accrac of fn lmn appromaon n D mol problm of aall loa bar mpro n rfnng ms? CIV-E4 / 7 / Jarkko Nrann 5

. Moaon for rror analss Conrgng. Ho os accrac of fn lmn appromaon n D mol problm of aall loa bar mpro n rfnng ms - locall - globall? CIV-E4 / 7 / Jarkko Nrann 5

. Moaon for rror analss Conrgng. Ho os accrac of fn lmn appromaon n D mol problm of aall loa bar mpro n rfnng ms - locall - globall? CIV-E4 / 7 / Jarkko Nrann 53

. Absrac mamacal formlaon of fn lmn mos Aall loa lasc bar: T srong form Elr qaons AE' ' AE' b N b A N CIV-E4 / 7 / Jarkko Nrann 54

. Absrac mamacal formlaon of fn lmn mos Aall loa lasc bar: T srong form Elr qaons AE' ' AE' can b rn b n EA' b N AE' ' : N. b A n an absrac form N U CIV-E4 / 7 / Jarkko Nrann 55

. Absrac mamacal formlaon of fn lmn mos Aall loa lasc bar: T srong form Elr qaons AE' ' AE' can b rn b n EA' b N AE' ' : N. b A n an absrac form W an alar fncon sasfng ssnal bonar conon D D ~ an b nong : D an b : b g a problm D N U W CIV-E4 / 7 / Jarkko Nrann 56

. Absrac mamacal formlaon of fn lmn mos Aall loa lasc bar: T srong form Elr qaons AE' ' AE' can b rn b n EA' b N AE' ' : N. n an absrac form W an alar fncon sasfng ssnal bonar conon D D ~ an b nong : D an b : b g a problm ~ b n EA' N EA' D ~ : N. b A D N U W CIV-E4 / 7 / Jarkko Nrann 57

. Absrac mamacal formlaon of fn lmn mos W noaon a ˆ : ' AEˆ' l ˆ : ˆ N ak form araonal form of mol problm of an aall loa bar ' AE ˆ' ˆ N bˆ bˆ CIV-E4 / 7 / Jarkko Nrann 58

. Absrac mamacal formlaon of fn lmn mos W noaon a ˆ : ' AEˆ' l ˆ : ˆ N ak form araonal form of mol problm of an aall loa bar ' AE ˆ' ˆ N racs follong absrac form: Fn a ˆ l ˆ ˆ V a : S V R l : V R bˆ sc a r s a blnar form s a fnconal an S V ar fncon spacs for ral an s fncons rspcl: S ' bˆ S V '. CIV-E4 / 7 / Jarkko Nrann 59

. Absrac mamacal formlaon of fn lmn mos Smmrc blnar form a ˆ a ˆ a : S V R ˆ sasfs conons a c c ˆ ca ˆ ca ˆ ˆ c c R. l : V R nar fnconal l c ˆ cˆ cl ˆ cl ˆ sasfs conons ˆ ˆ. CIV-E4 / 7 / Jarkko Nrann

. Absrac mamacal formlaon of fn lmn mos Smmrc blnar form a ˆ a ˆ a : S V R ˆ sasfs conons a c c ˆ ca ˆ ca ˆ ˆ c c R. l : V R nar fnconal l c ˆ cˆ cl ˆ cl ˆ sasfs conons ˆ Comparson of ak forms for problms U an W : U : a ˆ ' AEˆ' W : a ˆ ' AEˆ' l ˆ ˆ N bˆ a ˆ S V. D l ˆ Rmark. T onl ssnal ffrnc bn s o problm sngs s ssnal Drcl bonar conons of fncon spacs. No a a man far of fn lmn mo n s sanar form Galrkn s mo s conssnc or qal as n W abo bn ral an s fncon spacs. ˆ. ˆ N bˆ S V CIV-E4 / 7 / Jarkko Nrann 6

. Absrac mamacal formlaon of fn lmn mos Fn lmn mo for problm W: Fn a ˆ l ˆ ˆ V. sc a If fn lmn appromaon s sarc n a sbspac of ral fncon spac.. mo s call conformng. V V V V V CIV-E4 / 7 / Jarkko Nrann 63

64 Fn lmn mo for problm W: Fn sc a If fn lmn appromaon s sarc n a sbspac of ral fncon spac.. mo s call conformng.. BAP : ˆ ˆ GO : V a a V a Basc fars of a conformng fn lmn appromaon: Galrkn orogonal an Bs appromaon propr V V V. ˆ ˆ ˆ V l a V V. Absrac mamacal formlaon of fn lmn mos CIV-E4 / 7 / Jarkko Nrann

. Absrac mamacal formlaon of fn lmn mos Fn lmn mo for problm W: Fn a ˆ l ˆ ˆ V. sc a If fn lmn appromaon s sarc n a sbspac of ral fncon spac.. mo s call conformng. Basc fars of a conformng fn lmn appromaon: Galrkn orogonal an Bs appromaon propr GO : BAP : V a a ˆ ˆ V a V. V V Inrpaon of Galrkn orogonal: Error : V s prpnclar o sbspac V rspc o blnar form. a V V V V CIV-E4 / 7 / Jarkko Nrann 65

. Absrac mamacal formlaon of fn lmn mos Fn lmn mo for problm W: Fn a ˆ l ˆ ˆ V. sc a If fn lmn appromaon s sarc n a sbspac of ral fncon spac.. mo s call conformng. Basc fars of a conformng fn lmn appromaon: Galrkn orogonal an Bs appromaon propr GO : BAP : V a a ˆ V V ˆ V a V. V V Inrpaon of Galrkn orogonal: Error : V s prpnclar o sbspac V rspc o blnar form a. Fn lmn appromaon s a projcon of ac solon o sbspac V rspc o blnar form a. 3 Among all fncons of V fncon s on closs o solon rspc o blnar form. a V V CIV-E4 / 7 / Jarkko Nrann 67

. Absrac mamacal formlaon of fn lmn mos Inrpaon of Bs appromaon propr: Fn lmn appromaon s bs solon aalabl n spac rspc o blnar form. V a CIV-E4 / 7 / Jarkko Nrann 68

. Absrac mamacal formlaon of fn lmn mos Inrpaon of Bs appromaon propr: Fn lmn appromaon s bs solon aalabl n spac rspc o blnar form. Consqncs: Pagoras orm an Unrsmaon of nrg a a a a. a V a CIV-E4 / 7 / Jarkko Nrann 69

7 Inrpaon of Bs appromaon propr: Fn lmn appromaon s bs solon aalabl n spac rspc o blnar form.. a a a a a Consqncs: Pagoras orm an Unrsmaon of nrg a V Gomrcal nrpraons ar al n a rgoros mamacal sns f fncon spac s qpp an nnr proc fn b pos blnar form a V V T cor spac or par s no an nnr proc spac. Rmark. Innr proc can b rgar as a gnralzaon of o proc. a V. 4 3 V a a S V a a a S R c V ca c a S V a a S. Absrac mamacal formlaon of fn lmn mos CIV-E4 / 7 / Jarkko Nrann

. Absrac mamacal formlaon of fn lmn mos a Innr proc ncs for spac a naral norm a : a V. V. : V R a Norm. : V R sasfs conons N V N c c V c R N3 V. A cor spac V qpp a norm s call a norm spac. CIV-E4 / 7 / Jarkko Nrann 7

. Absrac mamacal formlaon of fn lmn mos Innr proc ncs for spac a naral norm a a : a V. V. : V R a Norm. : V R sasfs conons N V N c c V c R N3 V. A cor spac V qpp a norm s call a norm spac. Smnorm SN SN SN3 c : V R c sasfs conons V V c R V. CIV-E4 / 7 / Jarkko Nrann 7

. Absrac mamacal formlaon of fn lmn mos Sobol norms an Sobol spacs: For an. nnr proc norm an spac ar fn rspcl as : : : : { R s fn n : R } : H CIV-E4 / 7 / Jarkko Nrann 73

. Absrac mamacal formlaon of fn lmn mos Sobol norms an Sobol spacs: For an. nnr proc norm an spac ar fn rspcl as : H H : ' ' ' ' an nnr proc norm an smnorm rspcl as : : : : { ' ' s fn R : n H ' ' : R } : H CIV-E4 / 7 / Jarkko Nrann 74

75 an nnr proc norm an smnorm rspcl as H H ' ' : ' ' : ' ' ' ' : k H k H. :... ' ' :... ' '... ' ' : k k k k k k k k k k k k H k H an nnr proc norm an smnorm rspcl as. Absrac mamacal formlaon of fn lmn mos : } n s fn { : : : : H Sobol norms an Sobol spacs: For an. nnr proc norm an spac ar fn rspcl as R R : CIV-E4 / 7 / Jarkko Nrann

. Absrac mamacal formlaon of fn lmn mos Sobol spacs ar no fn as follos: H H k : { : { s fn n s fn n k { ' ' }. } / } CIV-E4 / 7 / Jarkko Nrann 76

. Absrac mamacal formlaon of fn lmn mos Sobol spacs ar no fn as follos: H H k Sobol nnr procs an norms can b gnralz o gr mnsons: For an : { : { : : R n n s fn n s fn n : R k { ' ' } }.. / } CIV-E4 / 7 / Jarkko Nrann 77

. Absrac mamacal formlaon of fn lmn mos Sobol spacs ar no fn as follos: Sobol nnr procs an norms can b gnralz o gr mnsons: For an For H H k : { : { : : R n : R m m : :. n s fn s fn n n : R k { ' ' } }.. / } CIV-E4 / 7 / Jarkko Nrann 78

. Absrac mamacal formlaon of fn lmn mos Fn lmn mo for problm W n a compl form: Assmng a b an ar gn fn sc a a a ˆ l ˆ V ˆ l ˆ V ˆ N N ' AEˆ' { H ˆ V bˆ a D ˆ }. V b A N CIV-E4 / 7 / Jarkko Nrann 79

. Absrac mamacal formlaon of fn lmn mos Fn lmn mo for problm W n a compl form: Assmng a b an ar gn fn sc a a l ˆ W lnar lmns r lmn ˆ a ˆ V V ˆ N V { P K K l ˆ N ˆ V ' AEˆ' H bˆ a D ˆ }. { H K P K V nos spac of lnar polnomal fncons fn on manng a fn lmn appromaon s pcs lnar. } K b A K N Kn CIV-E4 / 7 / Jarkko Nrann 8

. Absrac mamacal formlaon of fn lmn mos s consr blnar form of bar problm W: I can b son a A A C a a ˆ ' AE' ' AEˆ' C ˆ V nong consans pnng on problm sng n pracc consans ar rmn b qans. ˆ V E A CIV-E4 / 7 / Jarkko Nrann 8

. Absrac mamacal formlaon of fn lmn mos s consr blnar form of bar problm W: I can b son a A A C a a ˆ ' AE' ' AEˆ' C ˆ V E A ˆ V nong consans pnng on problm sng n pracc consans ar rmn b qans. Conon A mans a blnar form s llpc cors; conon A mans a blnar form s connos bon. Bo of s proprs ar al rspc o spac V or mor prcsl rspc o H norm naral/cosn for problm. CIV-E4 / 7 / Jarkko Nrann 8

. Absrac mamacal formlaon of fn lmn mos s consr blnar form of bar problm W: I can b son a A A C a a ˆ ' AE' ' AEˆ' C nong consans pnng on problm sng n pracc consans ar rmn b qans. Conon A mans a blnar form s llpc cors; conon A mans a blnar form s connos bon. Bo of s proprs ar al rspc o spac V or mor prcsl rspc o H norm naral/cosn for problm. In aon lnar loa fnconal as o b connos bon n orr o oban a nql solabl problm a Mlgram lmma: W a consan c D ˆ V l ˆ ˆ N bˆ a ˆ c ˆ ˆ V. ˆ V E A CIV-E4 / 7 / Jarkko Nrann 83

. Absrac mamacal formlaon of fn lmn mos Brak rcs 5 a Calcla g Calcla norm of fncon sn H. norm of sam fncon. b a ral consan. For c als of fncon a f blongs o spac? a CIV-E4 / 7 / Jarkko Nrann 84

. Basc rror smas an accrac of fn lmn mos T proprs abo conn an cors ogr Galrkn s orogonal mpl man rsls of rror analss for scon orr llpc problms: Frs nrg norm an a C H norm ar qaln V. CIV-E4 / 7 / Jarkko Nrann 85

86 Scon bs appromaon propr BAP gs a prlmnar qala rror sma n nrg norm. V C a H. a a a a V V a a E - BAP. Basc rror smas an accrac of fn lmn mos T proprs abo conn an cors ogr Galrkn s orogonal mpl man rsls of rror analss for scon orr llpc problms: Frs nrg norm an norm ar qaln CIV-E4 / 7 / Jarkko Nrann

87 Scon bs appromaon propr BAP gs a prlmnar qala rror sma n nrg norm Tr llpc an conn g an rror sma n norm of spac c s for s mol problm norm Ca s lmma: H. V C a H. V C E -Ca. a a a a V V a a E - BAP V. Basc rror smas an accrac of fn lmn mos T proprs abo conn an cors ogr Galrkn s orogonal mpl man rsls of rror analss for scon orr llpc problms: Frs nrg norm an norm ar qaln CIV-E4 / 7 / Jarkko Nrann

. Basc rror smas an accrac of fn lmn mos For lnar lmns.. fn lmn spac V { H K P K an rror sma mor qana an on n Ca s lmma can b oban: C c nong sz lng or amr of largs lmn n ms. } E - H- CIV-E4 / 7 / Jarkko Nrann 88

. Basc rror smas an accrac of fn lmn mos For lnar lmns.. fn lmn spac V V { H K P K an rror sma mor qana an on n Ca s lmma can b oban: C c nong sz lng or amr of largs lmn n ms. For gr orr lmns polnomals of orr k qarac cbc. { H K P K k a corrsponng rror sma can b oban b lzng appromaon or: } } E - H- k c. E - H- k k k CIV-E4 / 7 / Jarkko Nrann 89

. Basc rror smas an accrac of fn lmn mos For lnar lmns.. fn lmn spac V V { H K P K an rror sma mor qana an on n Ca s lmma can b oban: C c nong sz lng or amr of largs lmn n ms. For gr orr lmns polnomals of orr k qarac cbc. { H K P K k a corrsponng rror sma can b oban b lzng appromaon or: W cran aonal assmpons an Abn Nsc al cnq: } } E - H- k c. E - H- k k k rror sma can b r ~ k c c. E - - k k k CIV-E4 / 7 / Jarkko Nrann 9

. Basc rror smas an accrac of fn lmn mos Wc ar basc prncpls pro b mos fnamnal rror smas H c c k k k an o o affc accrac of fn lmn mos n consrng onl lmn sz an lmn gr k.. por conrgnc orr as? lmn sz c~ k k k CIV-E4 / 7 / Jarkko Nrann 9

. Basc rror smas an accrac of fn lmn mos Wc ar basc prncpls pro b mos fnamnal rror smas H c c k k k an o o affc accrac of fn lmn mos n consrng onl lmn sz an lmn gr k.. por conrgnc orr as? lmn sz c~ k. Inclng ras n rror norm crass accrac: cf. por of n rg an ss of rror smas H ja. k k CIV-E4 / 7 / Jarkko Nrann 9

. Basc rror smas an accrac of fn lmn mos Wc ar basc prncpls pro b mos fnamnal rror smas H c c k k k an o o affc accrac of fn lmn mos n consrng onl lmn sz an lmn gr k.. por conrgnc orr as? lmn sz c~. Inclng ras n rror norm crass accrac: cf. por of n rg an ss of rror smas H ja.. Rasng lmn gr k ncrass accrac: cf. rror smas lmn grs k = an k = for nsanc. k k k CIV-E4 / 7 / Jarkko Nrann 93

95 an o o affc accrac of fn lmn mos n consrng consans an Sobol-norm of ac solon? Wc ar basc prncpls pro b mos fnamnal rror smas ~ k k k k k k c c c H. Basc rror smas an accrac of fn lmn mos CIV-E4 / 7 / Jarkko Nrann

. Basc rror smas an accrac of fn lmn mos Wc ar basc prncpls pro b mos fnamnal rror smas H c c k k k c~ an o o affc accrac of fn lmn mos n consrng consans an Sobol-norm of ac solon? k 4. T consans n rg an s of nqals ncl conn an llpc consans n form. Incrasng s qon ncrass ol consan nc crasng accrac. C / k k CIV-E4 / 7 / Jarkko Nrann 96

. Basc rror smas an accrac of fn lmn mos Wc ar basc prncpls pro b mos fnamnal rror smas H c c k k k c~ an o o affc accrac of fn lmn mos n consrng consans an Sobol-norm of ac solon? 4. T consans n rg an s of nqals ncl conn an llpc consans n form. Incrasng s qon ncrass ol consan nc crasng accrac. C / 5. T consans n rg an s of nqals also pn on sap of lmns: An longa lmn rngl qarangl raron c. ncrass consan nc crasng accrac. Rgarng accrac an opmal ms consss of qalaral lmns. k k k CIV-E4 / 7 / Jarkko Nrann 97

. Basc rror smas an accrac of fn lmn mos In ral or on canno ak a norm a gr gr an rglar smoonss of ac solon allos: In H an ms ol a c proposs rng rror smas n form H k c c k s s c~ s mn k r r nong mamm al c sll gs a fn smnorm. k s s r CIV-E4 / 7 / Jarkko Nrann 99

. Basc rror smas an accrac of fn lmn mos In ral or on canno ak a norm a gr gr an rglar smoonss of ac solon allos: In H an ms ol a c proposs rng rror smas n form H k c c k s s c~ r nong mamm al c sll gs a fn smnorm. In pracc s mans a rasng lmn orr os no aomacall mpl an opmal ncras n accrac: for a non-smoo ac solon r < k conrgnc orr s rmn b rglar gr: s = r < k. Hnc rgarng accrac g orr lmns can b fll lz onl n os cass n c ac solon s smoo nog. k s s s mn k r r CIV-E4 / 7 / Jarkko Nrann

. Basc rror smas an accrac of fn lmn mos An ampl of opmal conrgnc ra: A qarlaral Rssnr Mnln pla sol b MITC fn lmn mos nform ms rfnmns. CIV-E4 / 7 / Jarkko Nrann

. Basc rror smas an accrac of fn lmn mos An ampl of opmal conrgnc ra 7: conrgnc crs for scon k = an r k = 3 orr ranglar MITC lmns for a problm a smoo ac solon plo n logarmc scals as log log c k k k log. Dflcon =. CIV-E4 / 7 / Jarkko Nrann 3

. Basc rror smas an accrac of fn lmn mos En f rglar of ac solon s problm-pnn an ac solon s nknon som basc mb rls can b gn: Tpcall ac solon s rglar f loang maral paramrs an bonar conons ar rglar an solon oman s con a smoo bonar. T ac solon can b rrglar nar bonars ols n oman or r-nran cornrs an n os pars of oman n c maral paramrs or loangs possss abrp an sp araons. CIV-E4 / 7 / Jarkko Nrann 5

. Basc rror smas an accrac of fn lmn mos Wa ls sol on kno abo rror smas an accrac? 7. In prncpl small sz lo orr lmns sol b s n os pars of oman r ac solon s pc o b rrglar l largr gr orr lmns sol b s n or pars of oman. Aap mos a bn lop for aomac an ra ms rfnmns bas on s prncpls accorng o rror ncaors r from fn lmn appromaon. 8. As an nrg mo a fn lmn mo fns an approma solon bs appromang nrg of ac solon BAP mplng a sanar rror analss os no pro rror smas for sngl pon als as no als. In parclar no als of a fn lmn appromaon o no conc ac solon cp som spcal cass. Hor pon als narall bcom mor an mor accra n rfnng ms an rasng lmn orr. CIV-E4 / 7 / Jarkko Nrann 8

. Basc rror smas an accrac of fn lmn mos An ampl of sprconrgnc pons 7: Spr accra no als c c for a qarlaral Rssnr Mnln pla ranglar scon orr k = MITC lmns: / 4 A Dflcon =. B A B CIV-E4 / 7 / Jarkko Nrann

.X Wak ra In con of Sobol spacs all ras ar consr as so call ak ras. Sanar ras of fncons ar call srong ras: k For nsanc C s a s of fncons fn on a oman ang connos ras p o orr k. C T s of connos fncons fn on a oman s no b. A fncon as a ak ra nr follong conons ar sasf: an ' C C : D r fncon spac s s of nfnl srongl ffrnabl smoo fncons c ar nonzro onl n a compac praccall clos an bon sbs of oman. Rmark. A fncon ang a sanar srong ra also as a ak ra c s qal o srong on l oppos s no ncssarl r. Hnc ak ra s a naral gnralzaon of sanar srong ra. CIV-E4 / 7 / Jarkko Nrann

.X Wak ra Coff rcs. Dr ak ra for fncon H n nral. Dos ol a? Is fncon connos n nral? Dos ol a C? CIV-E4 / 7 / Jarkko Nrann

.Y Cac & Poncar Cac Scarz nqal c s a sfl ool n fn lmn rror analss can b rn for nnr proc n form. W corrsponng norm s nqal can b alrnal rn n form. Cac Scarz nqal s al for an or nnr proc as ll. Poncar Frrcs nqal gs an ppr bon for norm of a fncon H smnorm of sam fncon nlss fncon s f a las b a paral Drcl bonar conon: m V { H D D on a par }. Rmark. In s cas norm an s smnorm ar qaln m V D. D m CIV-E4 / 7 / Jarkko Nrann 3

.Y Cac & Poncar Coff rcs. Pro Poncar Frrcs nqal n a smpl D cas : c V { H } for som c Wa s al of consan c? Hn: Ulz Cac Scarz nqal n proof. R. CIV-E4 / 7 / Jarkko Nrann 4

QUESTIONS? ANSWERS ECTURE BREAK!

Hom rcs. Spn a copl of ors for carfll rang an makng nos abo a cr sls Scons. an. b T book 3 Capr. T book 3: Anras Öcsnr & Marks Mrkl: On-Dmnsonal Fn Elmns An Inrocon o FE Mo Sprngr-Vrlag 3 s r man pons from bo sorcs b a f snncs an/or formla. s mamacal noaons no famlar o o for rcallng m from or arlr s marals or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 6

Hom rcs. s consr a rcall loa sacall rmn bam bnng problm bnng momn an srb loang. Formla ak form of problm n absrac mamacal form. Hn: Us an analog o bar problm of Scon.. Dr ac analcal solon of problm for a b. f f M M f f f f sn / Hn: Ingra an n rmn ngraon consans. B H E b f M M M M M CIV-E4 / 7 / Jarkko Nrann 7

Compr rcs Malab s consr a smpl sppor bam nr a snsoal loang. Wr a fn lmn Malab co of lnar lmns for compng ablang an plong qan M for for ffrn ms szs Hn: Mof an agmn gn mol m-fl. Do o s nrsmaon of nrg prncpl comng r? For for ffrn mss abla an plo rla rror n o norms: M a M M a M M an M / /8 /3 /8. Do o s orcal rror smas comng r? M M Wa s or sma al for rrors ms sz = / o rnnng program for s ms sz? f f sn CIV-E4 / 7 / Jarkko Nrann 8

3 Fn lmn mos for Krcoff o plas

3 Fn lmn mos for Krcoff o plas Conns. Srong an ak forms for Krcoff o plas. Fn lmn mos for Krcoff o plas arnng ocom A. Unrsanng basc fars of Krcoff o pla problm an abl o r basc formlaons rla o problm B. Basc knolg an ools for solng Krcoff o pla problms b fn lmn mos an nonconformng lmns n parclar Rfrncs C T book : Caprs 3.3 5.3. T book : Caprs 5. 5 5.9 CIV-E4 / 7 / Jarkko Nrann

3. Moaon for Krcoff o pla lmn analss Pla srcrs bs bam srcrs ar mos pcal srcral pars n morn srcral ngnrng. Krcoff o pla lmns for nsanc ar classcal b sll common n FEM sofar. CIV-E4 / 7 / Jarkko Nrann

3. Srong an ak forms for Krcoff o plas s consr a n planar pla srcr sbjc o sc a loang an bonar conons a formaon sa of pla can b mol b bnng problm. T basc knmacal assmpons of mnson rcon for a n pla call Krcoff o pla 85 or sorl Krcoff pla.. K normal fbrs of msrfac rman srag rng formaon K normal fbrs of msrfac o no src rng formaon K3 maral pons of msrfac mo n rcal rcon onl K4 normal fbrs of msrfac rman as normals rng formaon z CIV-E4 / 7 / Jarkko Nrann 3

3. Srong an ak forms for Krcoff o plas 5. z z z z z z z z z z z z z Consrng lnar formaons splacmn fl abo mpls srans CIV-E4 / 7 / Jarkko Nrann

3. Srong an ak forms for Krcoff o plas 6. z z z z z z z z z z z z z Consrng lnar formaons splacmn fl abo mpls srans Dfnng bnng momns o spcfng srsss a momn as / / / / / / : : : zz M zz M zz M CIV-E4 / 7 / Jarkko Nrann

3. Srong an ak forms for Krcoff o plas 7. z z z z z z z z z z z z z Consrng lnar formaons splacmn fl abo mpls srans Dfnng bnng momns o spcfng srsss a momn as nrg balanc of prncpl of ral ork can b rn n form r pla s assm o b sbjc o a rcal srb srfac loang onl / / / / / / : : : zz M zz M zz M z z S z S V V z S Az S V V W W b b ε σ ; : / / n z CIV-E4 / 7 / Jarkko Nrann

8 S z S z S S A f A f A zz zz zz : ; / / / / / / 3. Srong an ak forms for Krcoff o plas CIV-E4 / 7 / Jarkko Nrann

9. : ; / / / / / / A f A M M M S S A f A f A zz zz zz S z S z 3. Srong an ak forms for Krcoff o plas CIV-E4 / 7 / Jarkko Nrann

3 Ingraon b pars c n rm for nrnal ral ork gs form A f A s A f A T M Mn M / ; : 3. Srong an ak forms for Krcoff o plas. : ; / / / / / / A f A M M M S S A f A f A zz zz zz S z S z CIV-E4 / 7 / Jarkko Nrann

3 Ingraon b pars c n rm for nrnal ral ork gs form / ; : A f s s s A s A f A s A f A T s Mn n n Mn M n M M Mn M 3. Srong an ak forms for Krcoff o plas. : ; / / / / / / A f A M M M S S A f A f A zz zz zz S z S z CIV-E4 / 7 / Jarkko Nrann

3. Srong an ak forms for Krcoff o plas mplng srong form.. forc balanc M f R K - M CIV-E4 / 7 / Jarkko Nrann 3

3. Srong an ak forms for Krcoff o plas mplng srong form.. forc balanc an bonar conons afr on ngraon b pars for clamp smpl sppor an fr bonars rspcl Mn s Mn n M M c n nnf Mn n M Mn f n nns Mn s M n Q s s c c r ncs an rfr o ss of a bonar angl a a cornr pon c on fr bonar. c f s R n K - M f CIV-E4 / 7 / Jarkko Nrann 33

3. Srong an ak forms for Krcoff o plas mplng srong form.. forc balanc M f R K - M an bonar conons afr on ngraon b pars for clamp smpl sppor an fr bonars rspcl Mn s Mn n M c n nnf Mn r ncs an rfr o ss of a bonar angl a a cornr pon c on fr bonar. T momn qlbrm gs sar forc as Q M. Mn n M n nns c f s Mn s M n Q s s c c n f CIV-E4 / 7 / Jarkko Nrann 34

3. Srong an ak forms for Krcoff o plas Takng no accon lnarl lasc cons rlaons follong assmpons of plan srss sa.. z z z CIV-E4 / 7 / Jarkko Nrann 35

36 Takng no accon lnarl lasc cons rlaons follong assmpons of plan srss sa.. gs srss componns n rms of flcon as. ; E E G z z G z E z z E z E z z E z z z z 3. Srong an ak forms for Krcoff o plas CIV-E4 / 7 / Jarkko Nrann

37 Takng no accon lnarl lasc cons rlaons follong assmpons of plan srss sa.. gs srss componns n rms of flcon as T momn an sar forc componns can b no gn n rms of flcon n form. ; E E G z z G z E z z E z E z z E z z z z 3. Srong an ak forms for Krcoff o plas CIV-E4 / 7 / Jarkko Nrann

38 3 3 3 3 3 3 3 E D D D Q D D Q D M D M D M c mpl a splacmn formlaon for qlbrm qaons an bonar conons no rn o r or. 3. Srong an ak forms for Krcoff o plas CIV-E4 / 7 / Jarkko Nrann

3. Srong an ak forms for Krcoff o plas Assmng consan maral als an consan cknss gs qlbrm as a compac barmonc splacmn formlaon: For a gn loang f : R fn flcon sc a D : R f ; 4 4 4 4 4 4 armonc opraor barmonc opraor corrsponng bonar conons from abo rn n rms of flcon. K - T corrsponng ak form s oban from ral ork prsson abo or as sal b mlplng srong form K-M b a s fncon ngrang or oman an fnall ngrang b pars an collcng rms n fron of s fncons an nr corrsponng ngrals: CIV-E4 / 7 / Jarkko Nrann 4

3. Srong an ak forms for Krcoff o plas fa ˆ M ˆ M n s ˆ M ˆ CIV-E4 / 7 / Jarkko Nrann 4

3. Srong an ak forms for Krcoff o plas fa ˆ M ˆ M n s ˆ M n s ˆ M ˆ M n s ˆ M : ˆ CIV-E4 / 7 / Jarkko Nrann 4

3. Srong an ak forms for Krcoff o plas : D fa ˆ ˆ W b : D M n s ˆ M n s ˆ M n s ˆ / M ˆ M ˆ M n s ˆ M n s ˆ κ : / /. / M : ˆ κ ˆ T D b κ In las qal abo nsor noaon lz for ngraon b pars pas as bn cang o a cor noaon Vog noaon nrocng crar cor κ. CIV-E4 / 7 / Jarkko Nrann 43

3. Srong an ak forms for Krcoff o plas Wak form for Krcoff o pla problm: a clamp pla b sbjc o a srb loang f R. Fn W sc a a ˆ l ˆ ˆ W blnar form loa fnconal an araonal spacs a ˆ l ˆ W { H κ ˆ f ˆ T D b κ n }. CIV-E4 / 7 / Jarkko Nrann 44

3. Srong an ak forms for Krcoff o plas Wak form for Krcoff o pla problm: a clamp pla b sbjc o a srb loang f R. Fn W sc a a ˆ l ˆ ˆ W blnar form loa fnconal an araonal spacs a ˆ l ˆ W { H κ ˆ f ˆ T D b κ n }. Rmark. T araonal spac s onc agan a sbspac of c ll affc ssnall corrsponng fn lmn spacs. H spac CIV-E4 / 7 / Jarkko Nrann 45

3. Srong an ak forms for Krcoff o plas Wak form for Krcoff o pla problm: a clamp pla b sbjc o a srb loang f R. Fn W sc a a ˆ l ˆ ˆ W blnar form loa fnconal an araonal spacs a ˆ l ˆ W { H κ ˆ f ˆ T D b κ n }. H Rmark. T araonal spac s onc agan a sbspac of spac c ll affc ssnall corrsponng fn lmn spacs. Rmark. For smplc pla as bn assm o b fll clamp. Or bonar conons col a bn consr as ll or: nonzro splacmns forcs an momns col a bn prscrb on bonars. CIV-E4 / 7 / Jarkko Nrann 46

3. Srong an ak forms for Krcoff o plas Brak rcs 3 So a blnar form of Krcoff o pla problm s llpc an connos rspc o norm for : a a ˆ κ T κ ˆ D T b D κ b κ...... C H W For c p of als of qans E an qon C / apparng n corrsponng rror smas ll b larg/small? ˆ ˆ W. CIV-E4 / 7 / Jarkko Nrann 47

3. Fn lmn mos for Krcoff o plas Conformng fn lmn mo for Krcoff o pla problm: a clamp pla b sbjc o a srb loa f R. Fn s.. a ˆ l ˆ ˆ W blnar form loa fnconal an araonal spacs a ˆ l ˆ W κ ˆ fˆ T W { H D b κ n }. W CIV-E4 / 7 / Jarkko Nrann 48

3. Fn lmn mos for Krcoff o plas Conformng fn lmn mo for Krcoff o pla problm: a clamp pla b sbjc o a srb loa f R. Fn s.. a W ˆ l ˆ ˆ W blnar form loa fnconal an araonal spacs a ˆ l ˆ W W fˆ { κ ˆ T H D b κ n { C n K P5 K }. W In orr o oban a conformng compabl fn lmn mo rglar rqrmn of orr H as o b sasf. I can flfll b a C connos fn lmn appromaon c las ranglar lmns o pcs ff orr qnc polnomals Argrs rangl 968: }. CIV-E4 / 7 / Jarkko Nrann 49

3. Fn lmn mos for Krcoff o plas T Argrs rangl as grs of from fncon als pls frs an scon ras a cornrs pls normal ras a g mpons c ogr qaran conn across lmns: 3 : C a 3 a 3 K rf a a CIV-E4 / 7 / Jarkko Nrann 5

3. Fn lmn mos for Krcoff o plas T Argrs rangl as grs of from fncon als pls frs an scon ras a cornrs pls normal ras a g mpons c ogr qaran conn across lmns: 4... 9 3 : : C a a 3 a 3 a 3 K rf a a CIV-E4 / 7 / Jarkko Nrann 5

3. Fn lmn mos for Krcoff o plas T Argrs rangl as grs of from fncon als pls frs an scon ras a cornrs pls normal ras a g mpons c ogr qaran conn across lmns: 4... 9... 3 : 8 : C a a : a 3 a 3 a a 3 a 3 K rf a a CIV-E4 / 7 / Jarkko Nrann 5

3. Fn lmn mos for Krcoff o plas T Argrs rangl as grs of from fncon als pls frs an scon ras a cornrs pls normal ras a g mpons c ogr qaran conn across lmns: 4 9... 9...... 3 : 8 : : : C a a n j a a j 3 a 3 a j 3. a 3 a 3 a 3 a 3 K rf a a a CIV-E4 / 7 / Jarkko Nrann 53

3. Fn lmn mos for Krcoff o plas T sap fncons of rfrnc lmn corrsponng o s grs of from ar rmn b a s of nqnss conons for : N j P K N 5 rf j j j j N j N......; CIV-E4 / 7 / Jarkko Nrann 54

55 T sap fncons of rfrnc lmn corrsponng o s grs of from ar rmn b a s of nqnss conons for : 3. Fn lmn mos for Krcoff o plas.......; 5 4 3 9 3 8 4 7 5 6 4 5 3 4 3 3 4 3 9 8 3 7 6 5 4 3 5 5 j j j j j j j j j j j j j j j j j j j j j j rf j j j rf j c c c c c c c c c c c c c c c c c c c c c N j K P N j N j j N K P N... N CIV-E4 / 7 / Jarkko Nrann

56 T fn lmn appromaon s locall of form a a a 3 a a 3 a 3 K rf 3. Fn lmn mos for Krcoff o plas 3 3 3 3 9 8 6 5 5 3 9 9 7 6 6 4 3 3 a n N a n N a n N a N a N a N a N a N a N N n K CIV-E4 / 7 / Jarkko Nrann

57 T fn lmn appromaon s locall of form Rmark. W qaranglar lmns Bognr-Fo-Scm lmn 966 gs a -connos appromaon 6 grs of from: noal als n cornrs 4 ofs; paral ras n cornrs ofs. a a a 3 a a 3 a 3 K rf 3. Fn lmn mos for Krcoff o plas 3 3 3 3 9 8 6 5 5 3 9 9 7 6 6 4 3 3 a n N a n N a n N a N a N a N a N a N a N N n K C / / / CIV-E4 / 7 / Jarkko Nrann

3. Fn lmn mos for Krcoff o plas Nonconformng ncompabl lo orr fn lmn mo for Krcoff o pla problm: T Morl rangl 968 as 6 grs of from fncon als a cornrs an normal ras a g mpons c mpl a fn lmn flcon appromaon pro b lmn s sconnos from lmn o lmn: 4 5 3 6 : : a n j a j 3 j 3. T sap fncons of rfrnc lmn corrsponng o s grs of from ar rmn b a s of nqnss conons for : N P N j j K N c j j j c j j c j c c j3 j...6. j4 c a 3 a 3 a j5 N...6 K rf c a 3 a j6 a CIV-E4 / 7 / Jarkko Nrann 59

3. Fn lmn mos for Krcoff o plas Nonconformng Morl fn lmn mo for Krcoff o pla problm: a clamp pla b sbjc o a srb loa f R. Fn sc a a a l ˆ W W ˆ l ˆ ˆ { fˆ n lm ˆ W K κ ˆ K κ K connos a g mpons m; T ; D b P K; s connos a cornrs c n s c n m } W. CIV-E4 / 7 / Jarkko Nrann 6

3. Fn lmn mos for Krcoff o plas Nonconformng Morl fn lmn mo for Krcoff o pla problm: a clamp pla b sbjc o a srb loa f R. Fn sc a a a W W ˆ l ˆ ˆ l ˆ { fˆ n lm ˆ W K κ ˆ K κ K connos a g mpons m; T ; D b P K; s connos a cornrs c n c } W. Rmark. T flcon appromaon of s lmn Morl rangl 968 s no connos; conn of flcon s sasf onl a cornr nos l conn of normal ra s sasf a g mpons. Hnc ssnal bonar conons ll no b fll sasf: flcon ll b f a cornr nos normal ra a g mpons. n m s CIV-E4 / 7 / Jarkko Nrann 6

3. Fn lmn mos for Krcoff o plas Argrs rangl: conn an llpc of blnar form ogr Galrkn orogonal mpl a sanar rror sma follong Ca s lmma: C W c mpls a mor qana sma assmng a solon smoo nog H c c k 4 5 for k k k 6 Rmark. In orr o fll lz g polnomal orr of s lmn n acal conrgnc ra ac solon of problm sol b rml s rglar. In pracc or r ofn ols a. 5. H s 6 CIV-E4 / 7 / Jarkko Nrann 63

3. Fn lmn mos for Krcoff o plas Ts rsl can b rn n form c. 4 3 an frr n o an sma n a akr norm: c. 4 3 CIV-E4 / 7 / Jarkko Nrann 65

3. Fn lmn mos for Krcoff o plas Ts rsl can b rn n form c. 4 3 an frr n o an sma n a akr norm: c. 4 3 For rglar anor rsl as bn pro 4: For all r ss an sc a for all sasfng H f f. CIV-E4 / 7 / Jarkko Nrann 66

3. Fn lmn mos for Krcoff o plas Ts rsl can b rn n form an frr n o an sma n a akr norm: For rglar anor rsl as bn pro 4: For all r ss an sc a for all sasfng For follong rsls as bn pro 4: c. 4 3 c. 4 3 H f c s f. H f H l ˆ : f I ˆ ; a con. ln.nrpolaor I s.. H b nrprng loang fnconal n form f an f. s I ˆ a H ˆ a CIV-E4 / 7 / Jarkko Nrann 67

3. Fn lmn mos for Krcoff o plas Eampl for conrgnc ra of Morl rangl: Conrgnc orr of Morl lmn appl o a clamp Krcoff o sqar pla sbjc o a nform loang: log log c f log 3 log log. CIV-E4 / 7 / Jarkko Nrann 68

3. Fn lmn mos for Krcoff o plas Eampl for conrgnc ra of Morl rangl: Conrgnc orr of Morl lmn appl o a clamp Krcoff o sqar pla sbjc o a nform loang: log log c f log 3 log log. Rmark. For plong sol b no a N nong nmbr of lmns n ms ols a log / N N log N CIV-E4 / 7 / Jarkko Nrann 69

3. Fn lmn mos for Krcoff o plas Eampl for aap ms rfnmns an rror ncaor lop for Morl lmn 8: Smpl sppor nforml loa Krcoff o pla; rsal bas rror ncaors; aoma Dlana ranglaons. a Enrg rror an ms for raon sp 3. CIV-E4 / 7 / Jarkko Nrann 7

3.X Cr bonars In gnral solon omans cr bonars ar no anl n r ac form n fn lmn formlaons.. gomr of oman s approma sc as problm arabls. In mos cass gomr of oman s mapp bn acal an rfrnc lmn b polnomals: F l : rf l ; F p q p q Pm l. If polnomal orr m of gomr mappng s qal o polnomal orr of lmn m = k ncal or smlar sap fncons of orr k mappng or lmn s call soparamrc l for m < k rm sbparamrc s s an for m > k rm s s sprparamrc. Isoparamrc mappng lmn sap fncons s mos ofn s on; n D cas formla s N l l N. l Rmark. Ingrals n fn lmn forms ar comp or mapp gomr. Hnc a possbl non-consan Jacoban mar of gr orr gomr mappng mpls an ra rm a small or n rror smas. rf rf F F F l l CIV-E4 / 7 / Jarkko Nrann 7

3.X Cr bonars Coff rcs 8 Sol pla bnng problm of a clamp nforml loa crclar pla b sng Comsol Pla Elmns for ffrn mss. Insga conrgnc baor b ablang: flcon bnng momns an sar forc a mpon r = bnng momns an sar forc on bonar r = a. Compar fn lmn appromaon o ac als pro b Krcoff-o pla mol ras s no b a: M r r r fa 6 4 fa 64D 3 r / a r / a Q r r M fr. r fa 6 3 r / a CIV-E4 / 7 / Jarkko Nrann 73

QUESTIONS? ANSWERS ECTURE BREAK!

Hom rcs 3. Spn a copl of ors for carfll rang an makng nos abo cr sls Scons 3. an 3.. s r man pons b a f snncs an/or formla. s mamacal noaons no famlar o o for rcallng m from or arlr s marals or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 75

Hom rcs 3. Dr araonal formlaon or ak form of Krcoff o pla bnng problm sbjc o a bnng sa srfac loang z z z on op an boom srfacs ral ork: Wn W σ : ε V b V V S b sarng onc agan from prncpl of V Hn: No a no no smpl z-rcon s prsn b or rcons of srfac racon cor ar prsn as ll. Hor rmmbr a srfac loang racon ms la o a bnng sa no srcng or mmbran sa c mposs som rsrcons on rcons of loangs. z z S z S. CIV-E4 / 7 / Jarkko Nrann 76

Hom rcs 3.3 Compl raon of srong form for Krcoff o pla problm b akng no accon also mmbran nrg rla o srcng sa b assmng splacmn fl of form z z z : z / z / an n rng on all als for obanng gornng qaons as ll ssnal an naral bonar conons for n pla problm. T raon sol so a mmbran an bnng problms of soropc an lnar plas ar copl c s no r for slls. Hn: For smplc assm a loang nrg s agmn b n-plan srfac racons z z z consan n z-rcon an acng on som pars of laral bonar s fgr appropra rcons. S S z z z S CIV-E4 / 7 / Jarkko Nrann 77

4 Fn lmn mos for Rssnr Mnln plas

4 Fn lmn mos for Rssnr Mnln plas Conns. Srong an ak forms for Rssnr Mnln plas. Fn lmn mos for Rssnr Mnln plas arnng ocom A. Unrsanng basc fars of Rssnr Mnln pla problm an abl o r basc formlaons rla o problm B. Basc knolg an ools for solng Rssnr Mnln pla problms b fn lmn mos lockng fr MITC lmns n parclar Rfrncs T book : Caprs 5. 5.3. 3 T book : Caprs 5.6 8 CIV-E4 / 7 / Jarkko Nrann 8

4. Moaon for Rssnr Mnln pla lmn analss Tr ss som rlabl fn lmns for Rssnr Mnln pla problm ofn mplmn as a par of sll lmns n man FEM sofar. CIV-E4 / 7 / Jarkko Nrann 8

4. Srong an ak forms for Rssnr Mnln plas s consr a n planar pla srcr sbjc o sc a loang a formaon sa of pla can b mol b bnng problm. T basc knmacal assmpons of mnson rcon for a n or moral n or moral ck pla call Rssnr Mnln pla 946 95.. K normal fbrs of msrfac rman srag rng formaon K normal fbrs of msrfac o no src rng formaon K3 maral pons of msrfac mo n rcal rcon onl K4 normal fbrs of msrfac rman as normals rng formaon z CIV-E4 / 7 / Jarkko Nrann 8

4. Srong an ak forms for Rssnr Mnln plas 84. z z z z z z z z z z z z z z z z z z z z z For lnar formaons splacmn fl abo mpls srans CIV-E4 / 7 / Jarkko Nrann

4. Srong an ak forms for Rssnr Mnln plas 85. z z z z z z z z z z z z z z z z z z z z z For lnar formaons splacmn fl abo mpls srans B fnng rslan bnng momns an sar forcs as : : : : : / / / / / / / / / / z z z Q z Q zz M zz M zz M CIV-E4 / 7 / Jarkko Nrann

4. Srong an ak forms for Rssnr Mnln plas nrg balanc of prncpl of ral ork can b rn n form r pla s assm o b sbjc o a rcal srb srfac loang onl / / W W σ : ε V b V S n V V z z S z z Az S S S z CIV-E4 / 7 / Jarkko Nrann 86

87 nrg balanc of prncpl of ral ork can b rn n form r pla s assm o b sbjc o a rcal srb srfac loang onl A f A z z zz zz zz S Az S V V W W z z S S z z z z z S V V : / / / / / / / / / / / / n b ε σ 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

88 nrg balanc of prncpl of ral ork can b rn n form r pla s assm o b sbjc o a rcal srb srfac loang onl : / / / / / / / / / / / / n A f A Q Q M M M A f A z z zz zz zz S Az S V V W W z z S S z z z z z S V V b ε σ 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

89 S z z S z z T S S A f A f A A : / : ; : Q M Ingraon b pars n rm for nrnal ral ork gs nrg balanc n form 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

9 Q Q n Q M Mn Q M : / : ; : A f A s A s S S A f A f A A S z z S z z T Ingraon b pars n rm for nrnal ral ork gs nrg balanc n form 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

9 Q Q n Q M Mn Q M : / : ; : A f A s A s S S A f A f A A S z z S z z T Ingraon b pars n rm for nrnal ral ork gs nrg balanc n form an fnall mpls srong form.. forc an momn balanc an bonar conons for ar clamp sof clamp ar smpl sppor sof smpl sppor an fr bonars rspcl: RM - QM R f Q 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

9 Q Q n Q M Mn Q M : / : ; : A f A s A s S S A f A f A A S z z S z z T Ingraon b pars n rm for nrnal ral ork gs nrg balanc n form an fnall mpls srong form.. forc an momn balanc an bonar conons for ar clamp sof clamp ar smpl sppor sof smpl sppor an fr bonars rspcl: RM - QM R f Q M Q 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

93 Q Q n Q M Mn Q M : / : ; : A f A s A s S S A f A f A A S z z S z z T Ingraon b pars n rm for nrnal ral ork gs nrg balanc n form an fnall mpls srong form.. forc an momn balanc an bonar conons for ar clamp sof clamp ar smpl sppor sof smpl sppor an fr bonars rspcl: RM - QM R f Q M Q 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

4. Srong an ak forms for Rssnr Mnln plas Mn n M Mn n M nnf n s n s Mn s M Mn n M nnss Mn s M c Mn s M nsf nsc nns Q n nsss Q sc n s ss f CIV-E4 / 7 / Jarkko Nrann 94

4. Srong an ak forms for Rssnr Mnln plas Mn n M n s nnf n Mn n M s nnss Mn s M c nsf Mn s M Mn n M Rmark. Frs on sol noc a Rssnr Mnln pla mol prsn abo pros f ffrn ps of bonar conons ras Krcoff o mol onl r Krcoff o mol os no sngs sof an ar ons ras Rssnr Mnln mol os. Scon Rssnr Mnln pla mol s n gnral a br appromaon o 3D lasc problm an Krcoff o mol. Hor for n plas ransrs loangs onl s o mols g farl smlar rsls cp n bonar rgons r rsls can b ffrn bo n qal an n qan. nsc nns Mn s M Q n nsss Q n sc s ss f CIV-E4 / 7 / Jarkko Nrann 96

4. Srong an ak forms for Rssnr Mnln plas A s of mb rls rla o bonar conons an accrac of srss rslans can b gn for coosng spcfc mol appropra bonar conons: Compar o 3D lasc problm Krcoff o mol gs nrlabl rsls nar bonars of solon oman n a sanc clos o cknss of pla l ns oman rsls ar farl goo n man saons. Cornr snglars of Krcoff mol n parclar ar far from corrsponng ons of 3D problm. Alogr Krcoff o mol appromas mc br ar rsons of bonar conons of 3D lasc problm an corrsponng sof ons. CIV-E4 / 7 / Jarkko Nrann 98

4. Srong an ak forms for Rssnr Mnln plas 4 Nar cornrs n obs rnran ons solon of Rssnr Mnln mol caprs solon of corrsponng 3D problm ll for bo ar an sof smpl sppor bonars. CIV-E4 / 7 / Jarkko Nrann

4. Srong an ak forms for Rssnr Mnln plas 4 Nar cornrs n obs rnran ons solon of Rssnr Mnln mol caprs solon of corrsponng 3D problm ll for bo ar an sof smpl sppor bonars. 5 For ar clamp bonar Rssnr Mnln mol os no conc 3D mol nar bonars or. Rmark. For Krcoff o mol cf. Capr 8. plan srss sa z as assm for cons rlaons. Ts assmpon sms o b n conracon fac a knmacal assmpons of mol mpl zro o-of-plan srans for mol c ol rar propos plan sran sa. T sam qson ll ars blo for cons rlaons of Rssnr Mnln mol: knmacal assmpon K mpls zro sran n ransrsal rcon z an nc plan sran sa col b propos. Hor plan srss sa ll b assm for cons rlaons onc agan c can b jsf b orcal mans c ar bon scop of s cors or. CIV-E4 / 7 / Jarkko Nrann 3

4. Srong an ak forms for Rssnr Mnln plas Assmng lnarl lasc cons rlaons of plan srss sa jsf b assmpon s scsson n rmark abo z CIV-E4 / 7 / Jarkko Nrann 4

5 Assmng lnarl lasc cons rlaons of plan srss sa jsf b assmpon s scsson n rmark abo gs srsss n rms of flcon an roaon as. ; E E G z G z G z G z G z z G z E z z E z E z z E z z z z z z 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

6. ; 3 3 G Q G Q E D G M D M D M T momn an sar forc componns can b no gn n rms of flcon an roaon as c mpl a splacmn formlaon for qlbrm qaons an corrsponng bonar conons no rn o r. 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

7. ; 3 3 G Q G Q E D G M D M D M T momn an sar forc componns can b no gn n rms of flcon an roaon as c mpl a splacmn formlaon for qlbrm qaons an corrsponng bonar conons no rn o r. T corrsponng ak form nsa s oban from ral ork prsson abo or as sal b mlplng srong form RM-QM b s fncons ngrang or oman an fnall ngrang b pars: 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

: 4. Srong an ak forms for Rssnr Mnln plas fa ˆ Q A ˆ Q n s ˆ Q A ˆ ˆ CIV-E4 / 7 / Jarkko Nrann 8

: : 4. Srong an ak forms for Rssnr Mnln plas fa ˆ Q A ˆ M ˆ A Q n s ˆ M n ˆ s Q ˆ A M : ˆ A Q A ˆ ˆ Q ˆ A ˆ. CIV-E4 / 7 / Jarkko Nrann 9

s s A A fa A A s A A A s A fa n M n Q M Q Q M n M Q M Q n Q Q ˆ ˆ ˆ : ˆ ˆ ˆ ˆ. ˆ ˆ : ˆ ˆ ˆ : ˆ ˆ ˆ ˆ ˆ : 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

] / ˆ / ˆ / ˆ / ˆ [ : ˆ ˆ; ˆ ˆ ˆ ˆ ˆ ˆ : ˆ ˆ ˆ : ˆ ˆ ˆ ˆ. ˆ ˆ : ˆ ˆ ˆ : ˆ ˆ ˆ ˆ ˆ : s s A G A s s A A fa A A s A A A s A fa T b κ n Q n M κ κ D n M n Q M Q Q M n M Q M Q n Q Q 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

ˆ ˆ ˆ ˆ ˆ ; V W l a Wak form for Rssnr Mnln pla problm: a clamp pla b sbjc o a srb loang. Fn sc a blnar form loa fnconal an araonal spacs V W R f 4. Srong an ak forms for Rssnr Mnln plas. ] [ } ] [ { : } { ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; H H H H W f l G a b T η η V κ D κ CIV-E4 / 7 / Jarkko Nrann

3. ] [ } ] [ { : } { ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; H H H H W f l G a b T η η V κ D κ ˆ ˆ ˆ ˆ ˆ ; V W l a Wak form for Rssnr Mnln pla problm: a clamp pla b sbjc o a srb loang. Fn sc a blnar form loa fnconal an araonal spacs Rmark. For smplc ar clamp bonars ar consr. Or bonar conons col b consr as ll or. Frrmor nonzro splacmns forcs an momns as ll as srfac loas n or rcons or bo forcs cf. Hom rcss col a bn prscrb on bonar. V W R f 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

4 Brak rcs 4. ˆ ˆ ˆ ˆ... ˆ ˆ ˆ ˆ ˆ ;... ; V η η η η η η η κ D η κ η η V η η η η η κ D η κ η η W C G a W G a b T b T Wc p of -pnnc s prsn n rao? C / So a for ac blnar form of Rssnr Mnln pla problm s llpc an connos rspc o norm for consans pnng on cknss or: H 4. Srong an ak forms for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

5 Sanar form fn lmn mo for Rssnr Mnln pla problm: a ar clamp pla b sbjc o a srb loa. Fn sc a blnar form loa fnconal an araonal spacs R f 4. Fn lmn mos for Rssnr Mnln plas }. ] [ ] [ { } { ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; K P C K P C W f l G a k K k K b T η η η V κ D κ ˆ ˆ ˆ ˆ ˆ ; W l a V V V W W CIV-E4 / 7 / Jarkko Nrann

6 Sanar form fn lmn mo for Rssnr Mnln pla problm: a ar clamp pla b sbjc o a srb loa. Fn sc a blnar form loa fnconal an araonal spacs Rmark. Ts sanar form ranglar fn lmn mo s far from opmal: for n plas appromaon conrgs o ac solon rml slol o so-call sar lockng pnomnon c ll b clarf n al blo. 4. Fn lmn mos for Rssnr Mnln plas }. ] [ ] [ { } { ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; K P C K P C W f l G a k K k K b T η η η V κ D κ ˆ ˆ ˆ ˆ ˆ ; W l a V V V W W CIV-E4 / 7 / Jarkko Nrann R f

7 Sar lockng can b ral b rng a sanar rror sma n form c sos onc agan a cors an llpc consans ar prsn n rror sma affcng accrac of appromaon pro b mo. ~ ~ ~ ~ 3 k k k c C W C V η η 4. Fn lmn mos for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

4. Fn lmn mos for Rssnr Mnln plas Sar lockng can b ral b rng a sanar rror sma n form c sos onc agan a cors an llpc consans ar prsn n rror sma affcng accrac of appromaon pro b mo. s consr rao of maral consans n al cf. Brak rcs 4: D 3 E 3 G 6 C ~ ma D G ~ ~ C ~ 3 ~ C ~ k c G k E ~ mn D G η ~ 3 k η W am V CIV-E4 / 7 / Jarkko Nrann 8

4. Fn lmn mos for Rssnr Mnln plas Sar lockng can b ral b rng a sanar rror sma n form c sos onc agan a cors an llpc consans ar prsn n rror sma affcng accrac of appromaon pro b mo. s consr rao of maral consans n al cf. Brak rcs 4: 3 E D ~ C ~ 3 ~ C ~ k c 3 G 6 G E 3 C ~ ma D G ~ ~ mn D G ~ ~ C C ~ ~ ~ qckl for. 3 k η k η W am V CIV-E4 / 7 / Jarkko Nrann 9

4. Fn lmn mos for Rssnr Mnln plas T nar of problm an nc rror sma as ll pns on mnsons of pla cknss paramr c can b clarl sn b ng ak form of problm b sar sffnss gng form T Λ κ ˆ κ ˆ ˆ 6 G 3 g ˆ. CIV-E4 / 7 / Jarkko Nrann

4. Fn lmn mos for Rssnr Mnln plas T nar of problm an nc rror sma as ll pns on mnsons of pla cknss paramr c can b clarl sn b ng ak form of problm b sar sffnss gng form T Λ κ ˆ κ ˆ ˆ 6 Ts sos a n cknss lm n pracc for all n plas coffcn of sar rm n formlaon abo blos p for c mpls a ol sar rm blos p nlss rs of ngran s abl o balanc baor b conrgng o zro as follos: ˆ ˆ. ˆ W ˆ V G 3 g ˆ. CIV-E4 / 7 / Jarkko Nrann

4. Fn lmn mos for Rssnr Mnln plas In cknss lm for all n plas n pracc fn lmn appromaon srs for sasfng Krcoff o conon as ll:. CIV-E4 / 7 / Jarkko Nrann 4

4. Fn lmn mos for Rssnr Mnln plas In cknss lm for all n plas n pracc fn lmn appromaon srs for sasfng Krcoff o conon as ll:. Ts las o nmrcal lockng or c can b sn n compaons as an rml slo conrgnc oars ac solon nnr pla lor conrgnc ra. In lm cas for a clamp pla lnar lmns appromaon locks as fll as possbl: Rmark. T lockng pnomna s no lm o lnar lmns onl. Hor for lo orr lmns lockng ffc aks a lang rol an s ffc can b asl nrsoo b consrng appromaons of flcon an roaon: K P K P K [ P K K K K K K. ] CIV-E4 / 7 / Jarkko Nrann 6

8 } ~ { } ] [ ] [ ] [ { } { ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; 3 3 3 3 K P p b K P b p b b K B K B K P C K P C W f l G a k K k k k K k K b T η η η V R R κ D κ ockng fr MITC fn lmn mo for Rssnr Mnln pla problm 989: a clamp pla b sbjc o a srb loa. Fn sc a rc blnar form loa fnconal an araonal spacs R V V W W 4. Fn lmn mos for Rssnr Mnln plas f ˆ ˆ ˆ ˆ ˆ ; W l a V CIV-E4 / 7 / Jarkko Nrann

9 } ~ { } ] [ ] [ ] [ { } { ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; 3 3 3 3 K P p b K P b p b b K B K B K P C K P C W f l G a k K k k k K k K b T η η η V R R κ D κ ockng fr MITC fn lmn mo for Rssnr Mnln pla problm 989: a clamp pla b sbjc o a srb loa. Fn sc a rc blnar form loa fnconal an araonal spacs R V V W W r nos omognos polnomals of orr k on a rangl K k > bng polnomal orr of lmn s a bbbl spac. ~ P k K 4. Fn lmn mos for Rssnr Mnln plas f ˆ ˆ ˆ ˆ ˆ ; W l a V nonsanar B k K CIV-E4 / 7 / Jarkko Nrann

4. Fn lmn mos for Rssnr Mnln plas T rcon opraor Q G R S R :[ H ] S maps sar forc gn as no a roa Raar Tomas polnomal spac of orr k : S ~ { r Hro; r K [ Pk K] Pk K } CIV-E4 / 7 / Jarkko Nrann 3

4. Fn lmn mos for Rssnr Mnln plas T rcon opraor Q G R S R :[ H ] S maps sar forc gn as no a roa Raar Tomas polnomal spac of orr k : S ~ { r Hro; r K [ Pk K] Pk K } sc a E K R η s E pe R η pk K E η s E η pk pe p P p [ P k k K] K. E K CIV-E4 / 7 / Jarkko Nrann 3

4. Fn lmn mos for Rssnr Mnln plas T rcon opraor Q G R S R :[ H ] S maps sar forc gn as no a roa Raar Tomas polnomal spac of orr k : S sc a E K Rmark. Rcon ssnall appls o sap fncons of roaon snc gran of flcon appromaon alra blongs o projcon spac: R ~ { r Hro; r K [ Pk K] Pk K R η s E pe R η pk K E η s E η pk pe S p [ P p P G R. Rmark. Rc ngraon propos n T book s no nog! Q k k K] K. E K } CIV-E4 / 7 / Jarkko Nrann 3

4. Fn lmn mos for Rssnr Mnln plas T rror analss for MITC pla lmns follos a nonsanar ro: b lzng a Hlmolz composon problm s rn n a m form. Ellpc conon or sabl conon ll b pro n m form b lzng so call Babska Brzz conon c s ofn call nf sp conon. D o rcon an nconssnc rm pops o: a ; ˆ ˆ l ˆ a a ; ˆ ˆ a ; ˆ ˆ ; ˆ ˆ l ˆ G R ˆ. Afr all rror sma of mo s opmal rspc o polnomal orr an rglar of ac solon; cf. Capr.: ˆ R k Q Q c k Q k k k. CIV-E4 / 7 / Jarkko Nrann 35

37 Sablz lnar MITC lmn for Rssnr Mnln pla problm 99: a ar clamp pla b sbjc o a srb loa. Fn sc a sablz blnar form loa fnconal an araonal spacs R f ˆ ˆ / ˆ ˆ ˆ ; G a b T R R κ D κ ˆ ˆ ˆ ˆ ˆ ; W l a V V V W W } ] [ ] [ { } { ˆ ˆ K P C K P C W f l K K η η η V 4. Fn lmn mos for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

38 Sablz lnar MITC lmn for Rssnr Mnln pla problm 99: a ar clamp pla b sbjc o a srb loa. Fn sc a sablz blnar form loa fnconal an araonal spacs T sablzaon paramr for nsanc can b rmn nmrcall. ˆ ˆ / ˆ ˆ ˆ ; G a b T R R κ D κ ˆ ˆ ˆ ˆ ˆ ; W l a V V V W W. } ] [ ] [ { } { ˆ ˆ K P C K P C W f l K K η η η V 4. Fn lmn mos for Rssnr Mnln plas nonsanar CIV-E4 / 7 / Jarkko Nrann R f

39 Elmn sffnss mar. In rms of global sap fncons b nrocng noaon fn lmn appromaons can b rn n form 4. Fn lmn mos for Rssnr Mnln plas j n j j j n j j j n j j CIV-E4 / 7 / Jarkko Nrann

4 Elmn sffnss mar. In rms of global sap fncons b nrocng noaon fn lmn appromaons can b rn n form l n rms of local sap fncons b nrocng noaon local fn lmn appromaons can b rn n form : : n n n K n n n K N N N N N N N ξ ξ ξ N ξ ξ ξ 4. Fn lmn mos for Rssnr Mnln plas j n j j j n j j j n j j ξ CIV-E4 / 7 / Jarkko Nrann

4 For compng lmn sffnss mar s connn o frs compos blnar form no spara pars as r frs rm rprsns bnng nrg an rs of rms rprsn ffrn pars of sar nrg. 4. Fn lmn mos for Rssnr Mnln plas G G G G a b T R R R R κ D κ ˆ / ˆ / ˆ / ˆ / ˆ ˆ ˆ ; CIV-E4 / 7 / Jarkko Nrann

4 For compng lmn sffnss mar s connn o frs compos blnar form no spara pars as r frs rm rprsns bnng nrg an rs of rms rprsn ffrn pars of sar nrg. Accorngl ac local lmn sffnss mar an forc cor can b rn n a block form as 4. Fn lmn mos for Rssnr Mnln plas G G G G a b T R R R R κ D κ ˆ / ˆ / ˆ / ˆ / ˆ ˆ ˆ ; CIV-E4 / 7 / Jarkko Nrann

43. 3 3 3 3 n n n n F F F K K K K K 4. Fn lmn mos for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

44 s consr par of sffnss mar rla o frs sar rm:. 3 3 3 3 n n n n F F F K K K K K 4. Fn lmn mos for Rssnr Mnln plas m K G G a ˆ / ˆ / ˆ ; CIV-E4 / 7 / Jarkko Nrann

45 s consr par of sffnss mar rla o frs sar rm: c corrspons o lmn sffnss mar block rn n rms of global coorna ssm sap fncons c can b ransform o local ons or. 3 3 3 3 n n n n F F F K K K K K 4. Fn lmn mos for Rssnr Mnln plas m K G G a ˆ / ˆ / ˆ ; / K T G Φ Φ K. N Φ Φ CIV-E4 / 7 / Jarkko Nrann

46 T global coorna ssm an nc all calclaons can b ransform o local rfrnc lmn ssm b sng affn mappng r mar T conss of r coornas of lmn onl. 4. Fn lmn mos for Rssnr Mnln plas ξ T F 3 3 : K K rf K F K CIV-E4 / 7 / Jarkko Nrann

47 T global coorna ssm an nc all calclaons can b ransform o local rfrnc lmn ssm b sng affn mappng r mar T conss of r coornas of lmn onl. Hnc b applng cang of arabls ngral ransform as 4. Fn lmn mos for Rssnr Mnln plas ξ T F 3 3 : K K K K rf T K rf K F K CIV-E4 / 7 / Jarkko Nrann

48 T global coorna ssm an nc all calclaons can b ransform o local rfrnc lmn ssm b sng affn mappng r mar T conss of r coornas of lmn onl. Hnc b applng cang of arabls ngral ransform as l b applng can rl gran ransforms as 4. Fn lmn mos for Rssnr Mnln plas ξ T F 3 3 : K K K K rf T T T K rf K F K CIV-E4 / 7 / Jarkko Nrann

49 T global coorna ssm an nc all calclaons can b ransform o local rfrnc lmn ssm b sng affn mappng r mar T conss of r coornas of lmn onl. Hnc b applng cang of arabls ngral ransform as l b applng can rl gran ransforms as an nc sffnss block s rn as 4. Fn lmn mos for Rssnr Mnln plas K K K rf T T T. / K T T T rf G T N T N T K ξ T F 3 3 : K K rf K F K CIV-E4 / 7 / Jarkko Nrann

5 I s ofn sfl o r local lmn gr from of cor n form an accorngl local lmn sap fncon marcs n form. n n n N N N N N N N N N N N 4. Fn lmn mos for Rssnr Mnln plas T n n n CIV-E4 / 7 / Jarkko Nrann

5 I s ofn sfl o r local lmn gr from of cor n form an accorngl local lmn sap fncon marcs n form In s a local fn lmn appromaons can b rn n form an nrs n ffrn pars of sffnss mar an forc cor ll b aomcall poson corrcl n fnal lmn conrbons.. n n n N N N N N N N N N N N 4. Fn lmn mos for Rssnr Mnln plas T n n n K K N N CIV-E4 / 7 / Jarkko Nrann

4. Fn lmn mos for Rssnr Mnln plas W s noaon lmn sffnss mar corrsponng o bnng par of blnar form.. m T T κ ˆ D κ κ ˆ D κ b b K CIV-E4 / 7 / Jarkko Nrann 5

53 W s noaon lmn sffnss mar corrsponng o bnng par of blnar form.. s of form rn n rms of global coorna ssm sap fncons marcs In pracc or s marcs ar ransform o local rfrnc lmn sap fncons sng cnqs prsn abo cf. Hom rcs 4.3. 4. Fn lmn mos for Rssnr Mnln plas : : K b b T b K b T b B D B Φ D Φ K. / / / / 3 3 Φ m K b T b T ˆ ˆ κ D κ κ D κ CIV-E4 / 7 / Jarkko Nrann

54. Q Q Q c T rror sma for sablz lnar MITC rangl s opmal rspc o polnomal orr an rglar of ac solon: 4. Fn lmn mos for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

55. Q Q Q c T rror sma for sablz lnar MITC rangl s opmal rspc o polnomal orr an rglar of ac solon: Fnall l s ak a closr look a rcon opraor n lnar cas: } ] [ ] [ { c b a c b a K P H K K η η η V E s K E s E 3 s 3 4. Fn lmn mos for Rssnr Mnln plas CIV-E4 / 7 / Jarkko Nrann

56. Q Q Q c T rror sma for sablz lnar MITC rangl s opmal rspc o polnomal orr an rglar of ac solon: Fnall l s ak a closr look a rcon opraor n lnar cas: } ~ ] [ ro; { } ] [ ] [ { c a c a K P K P c b a c b a K P H K K K K R r H r S R η η η V 4. Fn lmn mos for Rssnr Mnln plas E s K E s E 3 s 3 CIV-E4 / 7 / Jarkko Nrann

57. Q Q Q c T rror sma for sablz lnar MITC rangl s opmal rspc o polnomal orr an rglar of ac solon: Fnall l s ak a closr look a rcon opraor n lnar cas:. } ~ ] [ ro; { } ] [ ] [ { c a a K E K P p p E p E c a c a K P K P c b a c b a K P H E E E E K K K K s s R R r H r S R η η η V 4. Fn lmn mos for Rssnr Mnln plas E s K E s E 3 s 3 CIV-E4 / 7 / Jarkko Nrann

4. Fn lmn mos for Rssnr Mnln plas Eampl for conrgnc ra of MITC pla lmns 7: Conrgnc crs for scon an r orr ranglar MITC lmns n a cas of nonsmoo ac solon bonar lar of fr bonars cf. Capr.. log log c r r rlog r / k 3. Dflcon =. CIV-E4 / 7 / Jarkko Nrann 58

4. Fn lmn mos for Rssnr Mnln plas Eampl of aap ms rfnmns for scon orr MITC rangls rsal-bas rror ncaors 8 3: Unforml loa Rssnr Mnln pla; sof smpl sppor bonars Dlana ranglaons. a Enrg rror an ms for raon sp. b Enrg rror an ms for raon sp. CIV-E4 / 7 / Jarkko Nrann 59

4.X Cr bams an slls For a cr plan bam crar ras R an srb aal loa b ransrsal loa f an bnng momn m forc an momn qlbrm qaons can b rn n form N' s Q s / R s b s Q' s M ' s Q s N s / R s m s f s s s s. Ts qaons so a crar of bam copls rslan forcs of ffrn rcons l ncrasng crar ras o nfn R a srag plan bam as a rsls copls qaons as follos: N' s b s Q' s f s M ' s Q s s s m s s r aal qlbrm of frs ln aall loa lasc bar cf. Capr can b sol sparal from bam bnng problm of las o ros. An analogos rsl ols for spac bams as ll. s f R m m b CIV-E4 / 7 / Jarkko Nrann 6

4.X Cr bams an slls For pla srcrs corrsponng qaons can b rn n form N Q b f M Q R R m R r mmbran problm of plan lasc an bnng problm cf. Caprs 3 an 4 ar copl o flanss of srcr. Rmark. In gnral sparaon of mmbran an bnng problms ols r onl for soropc an lnar cass l bo ororopc maral proprs an nonlnar baor pcall copl mmbran an bnng problms. Rmark. In sll srcrs mmbran an bnng problms ar copl b crar n a smlar a as n cr bam problm abo rslng n complca qaons. Accorngl sgnng or applng fn lmns mos for sll srcrs s a complca ask no js a sragforar nson of rsls for pla an mmbran srcrs an corrsponng lmns. 3 CIV-E4 / 7 / Jarkko Nrann 6

4.X Cr bams an slls Coff rcs 9 A clnr s clamp n bo ns an loa b o pon loas from oppos rcons n ml of clnr as pc n fgr. Sol problm b sng sll lmns for ffrn ms szs. Tabla conrgnc of mamm flcon as ll as bnng mmbran an sar nrgs for for ffrn ms szs. CIV-E4 / 7 / Jarkko Nrann 6

QUESTIONS? ANSWERS ECTURE BREAK!

Hom rcs 4. Spn a copl of ors for carfll rang an makng nos abo cr sls Scons 4. an 4.. s r man pons b a f snncs an/or formla. s mamacal noaons no famlar o o for rcallng m from or arlr s marals or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 64

Hom rcs 4. Dr araonal formlaon or ak form of Rssnr Mnln pla bnng problm sbjc o a bnng sa srfac loang z z z on op an boom srfacs ral ork: W W : n σ ε V b V V b sarng onc agan from prncpl of Hn: No a no no smpl z-rcon s prsn b or rcons of srfac racon cor ar prsn as ll. Hor rmmbr a srfac loang racon ms la o a bnng sa no srcng or mmbran sa c mposs som rsrcons on rcons of loangs. Compar rsls o Krcoff o pla mol Hom rcs 3.. S V z z S z S. CIV-E4 / 7 / Jarkko Nrann 65

Hom rcs 4.3 s consr a Rssnr Mnln sqar pla bnng problm sol b for ranglar lnar sablz MITC lmns of qal sz. W consan maral als an a consan rcal loang comp loa cor of fn lmn problm b sng rfrnc lmn sap fncons an an affn an nc soparamrc gomr mappng. Calcla -rm rm of sffnss mar rla o sam problm b sng sng rfrnc lmn sap fncons an an affn gomr mappng. Hn: 4 3 rf K K K K K... K... 3n 3n F... K Eplan o assmbl procss s carr o for an. K K F F F... F 3n... F K F... CIV-E4 / 7 / Jarkko Nrann 66

Sofar rcs 4 Comsol s consr a smpl opn floor plan srcr follong famos concp of Dom-Ino Hos of Corbsr 94 s pcr: a 5 mr rcanglar gr of s colmns sppors a qaranglar concr slab of ms 6 mrs C o an rnforcmn; slab s cm ck an colmns a a rcanglar cross scon a s lng of cm. Fn o p of Comsol Pla Elmn b rng manal of sofar. Mol gn srcr b sng Comsol Bam an Pla Elmns; frs a loa alon n b fllng slab b small cars kg pr 5 ms.5 m. For r ffrn frl cosn ms szs abla conrgnc of mos crcal srsss srss rslans of corrsponng fn lmn solons. CIV-E4 / 7 / Jarkko Nrann 67

5 Fn lmn mos for m-pnn problms s sar som smlaon ampls: Cng procss p://.ana.com/nsgh4.sml Bar braons n fl p://.ana.com/nsgh37.sml

5 Fn lmn mos for m-pnn problms Conns. Parabolc problms. Hprbolc problms 3. Tm ngraon algorms arnng ocom A. Abl o r an appl fn lmn mos for m-pnn problms B. Abl o lz an mplmn nmrcal m ngraon scms for FEM Rfrncs T book : Caprs 7. 7.3. A7.I 8 9 T book 3: Capr 3 CIV-E4 / 7 / Jarkko Nrann 7

5. All ncls compaonal mollng? In c kn of saons on as o ak no accon m-pnn pnomna n FE analss? CIV-E4 / 7 / Jarkko Nrann 7

7 Ha ffson concon problm n D: s consr on-mnsonal a ffson. Forr la bls a cons rlaon bn a fl q an mprar T rog rmal conc k as : 3 / or n T D D T k q k q 5. Parabolc problms D mol problm CIV-E4 / 7 / Jarkko Nrann

73 Ha ffson concon problm n D: s consr on-mnsonal a ffson. Forr la bls a cons rlaon bn a fl q an mprar T rog rmal conc k as T frs la of rmonamcs or prncpl of consraon of nrg o saonar sa assmpon mpls follong ffson qaon scrbng problm for a sorc f a capac c an mass ns ρ: f T k T c f q T c T k q 5. Parabolc problms D mol problm CIV-E4 / 7 / Jarkko Nrann

74 Ha ffson concon problm n D: s consr on-mnsonal a ffson. Forr la bls a cons rlaon bn a fl q an mprar T rog rmal conc k as T frs la of rmonamcs or prncpl of consraon of nrg o saonar sa assmpon mpls follong ffson qaon scrbng problm for a sorc f a capac c an mass ns ρ: Rmark. Consolaon of sols follos sam qaon b rplacng mprar b por prssr an coffcns accorngl. f T k T c f q T c T k q 5. Parabolc problms D mol problm CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms D mol problm Elr qaons srong from of nal-bonar al problm ar follong: a c T k T ' ' f T T b q 3 T q T CIV-E4 / 7 / Jarkko Nrann 75

5. Parabolc problms D mol problm Elr qaons srong from of nal-bonar al problm ar follong: 3 T T T mprar nknon fncon of an k f T q T a c T k T rmal conc c a capac mass ns gn maral aa a sppl gn loang aa - oman gn gomrcal aa mprar on bonar gn ssnal Drcl bonar aa a fl on bonar gn naral Nmann bonar aa mprar on bonar gn ssnal Drcl bonar aa T nal mprar gn nal aa ' ' f T T b q q - oman gn aa. CIV-E4 / 7 / Jarkko Nrann 76

77. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl: ' ' ' ' f T k T c f T k T c 5. Parabolc problms ak form CIV-E4 / 7 / Jarkko Nrann

78. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: ' ' ' ' f T k T c f T k T c f T k T c ' ' 5. Parabolc problms ak form CIV-E4 / 7 / Jarkko Nrann

79. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: 3. Ingra b pars lf an s: ' ' ' ' f T k T c f T k T c f T k T c ' ' f kt T c kt kt ' ' ' ' 5. Parabolc problms ak form CIV-E4 / 7 / Jarkko Nrann

8. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: 3. Ingra b pars lf an s: 4. Ulz naral bonar conon b: ' ' ' ' f T k T c f T k T c f T k T c ' ' f kt T c kt kt ' ' ' ' ' q q kt 5. Parabolc problms ak form CIV-E4 / 7 / Jarkko Nrann

8. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: 3. Ingra b pars lf an s: 4. Ulz naral bonar conon b: 5. S a zro ssnal bonar conon a for s fncon: ' ' ' ' f T k T c f T k T c f T k T c ' ' f kt T c kt kt ' ' ' ' ' q q kt 5. Parabolc problms ak form CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms ak form ct kt ' ' f q CIV-E4 / 7 / Jarkko Nrann 8

83 6. Mlpl nal conon 3 b s fncon an maral consans an ngra or oman: 5. Parabolc problms ak form ' ' q f kt T c T c T c T T CIV-E4 / 7 / Jarkko Nrann

84 6. Mlpl nal conon 3 b s fncon an maral consans an ngra or oman: Wak form. Fn sc a sasfs an for all sasfng. 5. Parabolc problms ak form T T T T c T c q f kt T c b ' ' a ' ' q f kt T c T c T c T T CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms ak form Rmark. T solon an s fncon rspcl a o sasf ssnal bonar conons an rglar conons of form T H T T ; n orr o b knmacall amssbl. H CIV-E4 / 7 / Jarkko Nrann 85

86 Rmark. T solon an s fncon rspcl a o sasf ssnal bonar conons an rglar conons of form n orr o b knmacall amssbl. Rmark. Tm-pnnc s sll prsn n formlaon: 5. Parabolc problms ak form ; H T T H T b ' ' a T c T c q f T k T c CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms fn lmns. D solon nral oman no sbnrals lmns nos an lmn sz : n n n CIV-E4 / 7 / Jarkko Nrann 87

88. D solon nral oman no sbnrals lmns nos an lmn sz :. Coos a ral fncon for fn lmn appromaon as a spara sm n n n T j n j j n n 5. Parabolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms fn lmns. D solon nral oman no sbnrals lmns nos an lmn sz : n n. Coos a ral fncon for fn lmn appromaon as a spara sm T sabl local bass fncons n n n n j j j of som polnomal orr no lnar n n CIV-E4 / 7 / Jarkko Nrann 89

5. Parabolc problms fn lmns Ensr a ral fncon sasfs ssnal bonar conons: T T n n CIV-E4 / 7 / Jarkko Nrann 9

9 Ensr a ral fncon sasfs ssnal bonar conons: 3. Coos a s fncon of a smlar form Galrkn mo corrsponng conon: c c c c c n n n T T n n 5. Parabolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

93 Ensr a ral fncon sasfs ssnal bonar conons: 3. Coos a s fncon of a smlar form Galrkn mo corrsponng conon: 4. Insr fncons ral an s no ak form: c c c c c n n n T T n n ' ' a q f kt T c 5. Parabolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

94 Ensr a ral fncon sasfs ssnal bonar conons: 3. Coos a s fncon of a smlar form Galrkn mo corrsponng conon: 4. Insr fncons ral an s no ak form:... ' ' ' ' a n j n j j n j n j j c k c c q f kt T c c c c c c n n n T T n n 5. Parabolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms fn lmns n j j n c j j k j ' ' j f q n CIV-E4 / 7 / Jarkko Nrann 95

96 q f k c n n j j j n j j j ' ' 5. Parabolc problms fn lmns T c c T c T c n j j j b CIV-E4 / 7 / Jarkko Nrann

97 W mass mar sffnss mar compabl for j = n forc cor compabl for = n an splacmn cor nknon for = n j n j j j j j j T c q f f f k K K c M M ' ' M f K M q f k c n n j j j n j j j ' ' 5. Parabolc problms fn lmns T c c T c T c n j j j b CIV-E4 / 7 / Jarkko Nrann

5. Parabolc problms fn lmns s rsls n a smpl qaon ssm M K f CIV-E4 / 7 / Jarkko Nrann 98

5. Parabolc problms fn lmns s rsls n a smpl qaon ssm M K f Rmark. T capac or mass mar M s smmrc an pos-fn. CIV-E4 / 7 / Jarkko Nrann 99

5. Parabolc problms fn lmns s rsls n a smpl qaon ssm M K f Rmark. T capac or mass mar M s smmrc an pos-fn. Rmark. Ts s a smscr formlaon snc m-pnnc s sll connosl prsn n ssm rog m pnn cors an f. CIV-E4 / 7 / Jarkko Nrann 3

5. Parabolc problms fn lmns s rsls n a smpl qaon ssm M K f Rmark. T capac or mass mar M s smmrc an pos-fn. Rmark. Ts s a smscr formlaon snc m-pnnc s sll connosl prsn n ssm rog m pnn cors an f. Rmark. T scrz qaon ssm s a copl ssm of ornar ffrnal qaons of frs orr gn nal al 3 of orgnal problm sng. CIV-E4 / 7 / Jarkko Nrann 3

5. Parabolc problms fn lmns s rsls n a smpl qaon ssm M K Rmark. T capac or mass mar M s smmrc an pos-fn. Rmark. Ts s a smscr formlaon snc m-pnnc s sll connosl prsn n ssm rog m pnn cors an f. Rmark. T scrz qaon ssm s a copl ssm of ornar ffrnal qaons of frs orr gn nal al 3 of orgnal problm sng. Rmark. In a nonlnar cas ssm qaon ol a form or mor gnrall f M K M g. f CIV-E4 / 7 / Jarkko Nrann 3

5. Parabolc problms fn lmns Brak rcs 5 Sol fr oscllaons of a mass-sprng-ampr mkc ssm moll b a lnar scon orr ornar ffrnal qaon consan coffcns n form m c k f Hn: Us ral fncon for rng carasrsc qaon an sngs cass II c^ < 4mk o compl conjga roos nrampng III c^ = 4mk a ral opl roo crcal ampng I c^ > 4mk o snc ral roos orampng.. CIV-E4 / 7 / Jarkko Nrann 33

5. Hprbolc problms s rcall a sac qlbrm or forc balanc for a bo from Elr s las or momnm prncpls qaln gnralzaons of Non s las: Prncpl of lnar momnm can b rn n form b S S R spaal crrn confgraon bo loa b = b srfac racon = mass ns ρ = ρ an spaal loc fl =. 3 follos CIV-E4 / 7 / Jarkko Nrann 34

5. Hprbolc problms s rcall a sac qlbrm or forc balanc for a bo from Elr s las or momnm prncpls qaln gnralzaons of Non s las: Prncpl of lnar momnm can b rn n form b S S spaal crrn confgraon bo loa b = b srfac racon = mass ns ρ = ρ an spaal loc fl =. Cac s la an Gass rgnc orm mpl form from c on obans qaon of moon: b b S S S σ b σ b. σn S R 3 follos CIV-E4 / 7 / Jarkko Nrann 35

5. Hprbolc problms D mol problm Elasonamcs of an aall loa ro: Elr qaons ar no of form a b A E A ' N N ' 3a 3b b CIV-E4 / 7 / Jarkko Nrann 36

5. Hprbolc problms D mol problm Elasonamcs of an aall loa ro: Elr qaons ar no of form a b A N N 3a A cross-sconal ara; lng; E A ' ' b 3b aal splacmn nknon fncon of an E Yong's mols mass ns gn maral aa b N nal splacmn nal loc gn nal aa. - oman gn gomrcal aa aal bo loa gn loang aa aal n pon splacmn gn ssnal/gomrc bonar aa aal n pon forc gn naral/forc bonar aa CIV-E4 / 7 / Jarkko Nrann 37

38. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl: ' ' ' ' b A E A b A E A 5. Hprbolc problms ak form CIV-E4 / 7 / Jarkko Nrann

39. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: ' ' ' ' b A E A b A E A b EA A ' ' 5. Hprbolc problms ak form CIV-E4 / 7 / Jarkko Nrann

3. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: 3. Ingra b pars lf an s: ' ' ' ' b A E A b A E A b EA A ' ' b EA A EA EA ' ' ' ' 5. Hprbolc problms ak form CIV-E4 / 7 / Jarkko Nrann

3. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: 3. Ingra b pars lf an s: 4. Ulz naral bonar conon b: ' ' ' ' b A E A b A E A b EA A ' ' b EA A EA EA ' ' ' ' ' N N EA 5. Hprbolc problms ak form CIV-E4 / 7 / Jarkko Nrann

3. Mlpl ffrnal qaon b a smoo s fncon spcf lar pnng on onl:. Ingra or -oman nral: 3. Ingra b pars lf an s: 4. Ulz naral bonar conon b: 5. S a zro ssnal bonar conon a for s fncon: ' ' ' ' b A E A b A E A b EA A ' ' b EA A EA EA ' ' ' ' ' N N EA 5. Hprbolc problms ak form CIV-E4 / 7 / Jarkko Nrann

5. Hprbolc problms ak form A EA ' ' b N CIV-E4 / 7 / Jarkko Nrann 33

34 6. Mlpl nal conons 3a an 3b b s fncon an maral consans an ngra or oman: 5. Hprbolc problms ak form ' ' N b EA A A A A A CIV-E4 / 7 / Jarkko Nrann

35 6. Mlpl nal conons 3a an 3b b s fncon an maral consans an ngra or oman: Wak form. Fn sc a sasfs an 5. Hprbolc problms ak form ' ' N b EA A. sasfng c b ' ' a A A A A N b EA A A A A A CIV-E4 / 7 / Jarkko Nrann

5. Hprbolc problms ak form Rmark. T solon an s fncon rspcl a o sasf ssnal bonar conons an rglar conons n form H ; n orr o b knmacall amssbl. H CIV-E4 / 7 / Jarkko Nrann 36

37 Rmark. T solon an s fncon rspcl a o sasf ssnal bonar conons an rglar conons n form n orr o b knmacall amssbl. Rmark. Tm-pnnc s sll prsn n formlaon: 5. Hprbolc problms ak form ; H H. sasfng c b ' ' a A A A A N b A E A CIV-E4 / 7 / Jarkko Nrann

5. Hprbolc problms fn lmns. D solon nral oman no sbnrals lmns nos an lmn sz : n n n CIV-E4 / 7 / Jarkko Nrann 38

39. D solon nral oman no sbnrals lmns nos an lmn sz :. Coos a ral fncon for fn lmn appromaon as a spara sm n n n j n j j n n 5. Hprbolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

5. Hprbolc problms fn lmns. D solon nral oman no sbnrals lmns nos an lmn sz : n n. Coos a ral fncon for fn lmn appromaon as a spara sm sabl local bass fncons n n n n j j j of som polnomal orr no lnar n n CIV-E4 / 7 / Jarkko Nrann 3

5. Hprbolc problms fn lmns Ensr a ral fncon sasfs ssnal bonar conons: n n CIV-E4 / 7 / Jarkko Nrann 3

33 Ensr a ral fncon sasfs ssnal bonar conons: 3. Coos a s fncon of a smlar form Galrkn mo corrsponng conon: c c c c c n n n n n 5. Hprbolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

34 Ensr a ral fncon sasfs ssnal bonar conons: 3. Coos a s fncon of a smlar form Galrkn mo corrsponng conon: 4. Insr fncons ral an s no ak form: c c c c c n n n n n ' ' a N b EA A 5. Hprbolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

35 Ensr a ral fncon sasfs ssnal bonar conons: 3. Coos a s fncon of a smlar form Galrkn mo corrsponng conon: 4. Insr fncons ral an s no ak form:... ' ' ' ' a n j n j j n j n j j c EA c A N b EA A c c c c c n n n n n 5. Hprbolc problms fn lmns CIV-E4 / 7 / Jarkko Nrann

5. Hprbolc problms fn lmns n n A j j j j EA ' ' j j b N n CIV-E4 / 7 / Jarkko Nrann 36

5. Hprbolc problms fn lmns n n A j j j j EA ' ' j j b N n b A n A A j j j A CIV-E4 / 7 / Jarkko Nrann 37

38 N b EA A n n j j j n j j j ' ' 5. Hprbolc problms fn lmns A A A A A A A A n j j j n j j j c b CIV-E4 / 7 / Jarkko Nrann

39 N b EA A n n j j j n j j j ' ' 5. Hprbolc problms fn lmns A A A A A A A A n j j j n j j j c b W mass mar sffnss mar compabl for j = n forc cor compabl for = n an splacmn cor nknon for = n CIV-E4 / 7 / Jarkko Nrann

33 s rsls n a smpl qaon ssm f K M 5. Hprbolc problms fn lmns j n j j j j j j A A N b f f EA K K A M M : : : : : ' ' : : : : M M f K M CIV-E4 / 7 / Jarkko Nrann

33 s rsls n a smpl qaon ssm Rmark. T mass mar M s smmrc an pos-fn. f K M 5. Hprbolc problms fn lmns j n j j j j j j A A N b f f EA K K A M M : : : : : ' ' : : : : M M f K M CIV-E4 / 7 / Jarkko Nrann

33 s rsls n a smpl qaon ssm Rmark. T mass mar M s smmrc an pos-fn. Rmark. Ts s a smscr formlaon snc m-pnnc s sll connosl prsn n ssm rog m pnn cors an f. f K M 5. Hprbolc problms fn lmns j n j j j j j j A A N b f f EA K K A M M : : : : : ' ' : : : : M M f K M CIV-E4 / 7 / Jarkko Nrann

5. Hprbolc problms fn lmns Rmark. T scrz qaon ssm s a copl ssm of ornar ffrnal qaons of scon orr gn nal als 3 of orgnal problm sng. CIV-E4 / 7 / Jarkko Nrann 333

5. Hprbolc problms fn lmns Rmark. T scrz qaon ssm s a copl ssm of ornar ffrnal qaons of scon orr gn nal als 3 of orgnal problm sng. Rmark. In a nonlnar cas ssm ol a form M K or mor gnrall M g. f CIV-E4 / 7 / Jarkko Nrann 334

5. Hprbolc problms fn lmns Rmark. T scrz qaon ssm s a copl ssm of ornar ffrnal qaons of scon orr gn nal als 3 of orgnal problm sng. Rmark. In a nonlnar cas ssm ol a form M K or mor gnrall M g. Rmark. In cas of amp braons scos ampng mar C s ncl n ssm as M C K f f. Ofn so call Ralg ampng mar ampng pnomna som consans α an. C M K s s o scrb CIV-E4 / 7 / Jarkko Nrann 335

5.3 Tm ngraon algorms Gnralz rapzoal faml of m ngraon mos for parabolc problms. For nal al problm problm M K f praps mos ll knon an commonl s solon algorms ar mmbrs of gnralz rapzoal faml conssng of m sps follong qaons M K f Rmark. Dffrn als for paramr α g ffrn scms: α = s call forar ffrncs or forar Elr; α = ½ rapzoal rl mpon rl or Crank Ncolson; α = backar ffrncs or backar Elr. Rmark. Dffrn mplmnaons s for mos of s p. CIV-E4 / 7 / Jarkko Nrann 338

5.3 Tm ngraon algorms Gnralz rapzoal m ngraon algorm can b rn as follos: a. b. Sol M for nal al f K Tm sps... for m ncrmn : ~ b. Comp prcor b. Sol from qaon ~ M K f K ~ b.3 Comp : an rrn o b.. CIV-E4 / 7 / Jarkko Nrann 339

5.3 Tm ngraon algorms Gnralz rapzoal m ngraon algorm can b rn as follos: a. b. Sol M for nal f K al Tm sps... for m ncrmn : ~ b. Comp prcor b. Sol from qaon ~ M K f K ~ b.3 Comp an rrn o b.. Rmark. Dffrn als for paramr α g ffrn mo ps: for α = mos s call plc ors mplc l n mplc-plc mos par of K s ra mplcl an par plcl. Rmark. W a agonal M sp b. s sragforar for an plc mo. : CIV-E4 / 7 / Jarkko Nrann 34

34 Nmark faml of m ngraon mos for prbolc an parabolcprbolc problms. For nal al problm problm praps mos l s solon algorms ar mmbrs of Nmark faml conssng of m sps follong qaons l s no 5.3 Tm ngraon algorms f K C M a a a a f K C Ma a : : CIV-E4 / 7 / Jarkko Nrann

5.3 Tm ngraon algorms Nmark faml of m ngraon mos for prbolc an parabolcprbolc problms. For nal al problm problm M C K f praps mos l s solon algorms ar mmbrs of Nmark faml conssng of m sps follong qaons l s no Ma C K f a a a : a : Rmark. Dffrn als for paramr an γ g scms ffrn sabl ncononall sabl cononall sabl an accrac carasrscs. a CIV-E4 / 7 / Jarkko Nrann 34

5.3 Tm ngraon algorms Nmark m ngraon algorm can b rn as follos: a. Sol a ~ ~ Ma b. Sol a for nal a C from M C b.3 Comp an ~ ~ als a K a a an b. Tm sps... for m ncrmn : b. Comp prcors f K a f C~ an rrn o b.. : ~ K CIV-E4 / 7 / Jarkko Nrann 343

5.3 Tm ngraon algorms Rmark. In so call mlsp mos arabls for n m sp ar comp b akng no accon no onl pros sp a f b som of or prcng sps as ll. a a CIV-E4 / 7 / Jarkko Nrann 344

QUESTIONS? ANSWERS ECTURE BREAK!

Hom rcs 5. Spn a copl of ors for carfll rang an makng nos abo a cr sls Scons 5. 5. an 5.3 b T book 3 Capr 3. T book 3: Anras Öcsnr & Marks Mrkl: On-Dmnsonal Fn Elmns An Inrocon o FE Mo Sprngr-Vrlag 3 s r man pons from bo sorcs b a f snncs an/or formla. s mamacal noaons no famlar o o for rcallng m from or arlr s marals or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 346

Hom rcs 5. s consr aall loa ro no a m-pnn loang: aal forc a fr n s gn as N N sn. Calcla local lmn mass mar of corrsponng fn lmn problm sol b lnar lmns b sng rfrnc lmn approac b assmng a ns of maral s kg/m^3. Hn: T rsl s M A 6. Assmbl corrsponng ssm qaons compos of conrbons from o lmns of qal sz. CIV-E4 / 7 / Jarkko Nrann 347

Hom rcs 5.3 Sol fr oscllaons of a mass-sprng ssm mk moll b a lnar scon orr ornar ffrnal qaon consan coffcns n form m k. Hn: So a ac solon of problm s of form cos sn cos k / m an Appl Nmark mo = an γ = ½ for solng problm nmrcall b accomplsng o m sps sasfng conon consans k an m can b frl cosn. / Compar s o sps o ac solon cr b a plo.. CIV-E4 / 7 / Jarkko Nrann 348

Sofar rcs 5 Comsol s consr a smpl opn floor plan srcr follong famos concp of Dom-Ino Hos of Corbsr 94 s pcr: a 5 mr rcanglar gr of s colmns sppors a qaranglar concr slab of ms 6 mrs C o an rnforcmn; slab s cm ck an colmns a a rcanglar cross scon a s lng of cm. Mol gn srcr b sng Comsol Bam an Pla Elmns; a loa alon b assmng a on of colmns fls a rcal splacmn along bam as of a snsoal form n m:. cos s n assm a sam splacmn s fl n orzonal rcon paralll o sor g of slab. Sol m-pnn rspons of srcr for an appropra frl cosn ms sz n cass an. CIV-E4 / 7 / Jarkko Nrann 349

6 Nonlnars n fn lmn analss s sar som smlaon ampls: Sll folng p://.ana.com/nsgh8.sml Sampng p://.ana.com/sampng.sml Sal sp mas p://.ana.com/nsgh46.sml Fasnr jons p://.ana.com/nsgh5.sml Hmmng p://.ana.com/mmng.sml

6 Nonlnars n fn lmn analss Conns. Nonlnar compaonal mcancs. Gomrcal nonlnars n fn lmn analss 3. Maral nonlnars n fn lmn analss 4. narzaon an solon algorms arnng ocom A. Orall an n nrsanng of nonlnars n ngnrng B. Abl o formla an sol smpl mol problms of nonlnar mcancs Rfrncs T book : Capr 4. T book 3: Caprs T book 5: Caprs 3. 3 A.3 J. Bon R.D Woo Nonlnar Connm Mcancs for Fn Elmn Analss: Caprs. 4.3 7.6 CIV-E4 / 7 / Jarkko Nrann 35

6. Ulzng or aong nonlnars Wa can b ac b nrsanng nonlnar pnomna n mcancs an fn lmn mos? CIV-E4 / 7 / Jarkko Nrann 353

6. Ulzng or aong nonlnars Frs of all nrsanng of nar rqrs nrsanng of nonlnars. Hor nonlnar pnomna ar no onl an nrsng rsarc fl for gnral cros. Naml br or mor nnoa procs or srcs can b sgn b aong or plong nonlnars: n marals: composs por mals por concr sap mmor allos nanomarals bomarals larg formaons: bcklng folng form-fnng conacs an nracons: masr-sla conac fl-srcr nracon spcal loangs: mpac loas follor loas ml-pscs an copl problms CIV-E4 / 7 / Jarkko Nrann 355

6. Nonlnar compaonal mcancs To man sorcs of nonlnars ar maral nonlnar an gomrcal nonlnar. Among or n aon o s basc sorcs ffrn ps of nonlnar problms n ngnrng mcancs can b grop as follos: CIV-E4 / 7 / Jarkko Nrann 356

6. Nonlnar compaonal mcancs To man sorcs of nonlnars ar maral nonlnar an gomrcal nonlnar. Among or n aon o s basc sorcs ffrn ps of nonlnar problms n ngnrng mcancs can b grop as follos: Gomrcal nonlnar cabls frams mmbrans pla slls larg splacmns an/or roaons b small srans Fn formaons mal formng r mcancs larg splacmns an/or roaons an larg srans 3 Maral nonlnar polmrs sl concr sol nonlnar/olonal sran-srss cons rspons CIV-E4 / 7 / Jarkko Nrann 357

6. Nonlnar compaonal mcancs In bo lnar an nonlnar connm mcancs formaon an moon of a connm s fn b follong concps: Knmacs: formaons Kncs: qaons of moons an srsss 3 Trmonamcs: nrg nqals 4 Cons baor: knc s. knmac/rmonamc arabls Nmrcal analss as fn lmn mos rfrr as nonlnar compaonal mcancs n appl o nonlnar connm mcancs s a a o smla an analz compl ssms sbjc o compl loangs: A Nonlnar connm mcancs 4 B Nonlnar nrg mos as fn lmn formlaons C Solon cnqs for rslng gornng nonlnar algbrac qaons Rmark. A scrbs a connos ssm B an C s scr appromaons. CIV-E4 / 7 / Jarkko Nrann 36

6. Gomrcal nonlnars n fn lmn analss arg srans n gomrcall nonlnar formaons mpl ffcls n fnng corrc larg sran masrs for connms. On mnsonal sran masrs. s consr a rss mmbr of nal lng an ara A src o a fnal lng l an fnal ara a. Engnrng sran or Cac sran: E l : l l l l For r small cangs n lng s sran masr ns o small sran qan. CIV-E4 / 7 / Jarkko Nrann 36

364 3 Grn sran: 4 Almans sran: l l G ; ; l l A 6. Gomrcal nonlnars n fn lmn analss CIV-E4 / 7 / Jarkko Nrann

365 3 Grn sran: 4 Almans sran: Rmark. For r small cangs n lng onc agan s sran masrs n o small sran qan: l l G ; ; l l A l l l l l l l l l l l l G 6. Gomrcal nonlnars n fn lmn analss CIV-E4 / 7 / Jarkko Nrann

6. Gomrcal nonlnars n fn lmn analss Connm sran masrs. Small sran lnar nsor componns of omnsonal cas ar of form CIV-E4 / 7 / Jarkko Nrann 366

367 Connm sran masrs. Small sran lnar nsor componns of omnsonal cas ar of form l Grn srans g larg sran nonlnar nsor componns n form. Y X Y X X X E Y Y Y E X X X E 6. Gomrcal nonlnars n fn lmn analss CIV-E4 / 7 / Jarkko Nrann

6. Gomrcal nonlnars n fn lmn analss Aall loa gomrcall nonlnar lasc bar problm. s sar from gnral formla of prncpl of ral ork for formabl bos applcabl for bo lnar an nonlnar connm ors: W n V σ : ε V EAεε S S b f V b V W CIV-E4 / 7 / Jarkko Nrann 368

6. Gomrcal nonlnars n fn lmn analss Aall loa gomrcall nonlnar lasc bar problm. s sar from gnral formla of prncpl of ral ork for formabl bos applcabl for bo lnar an nonlnar connm ors: W n V EAεε Nε σ : ε V S b b S N f V b V EAε W CIV-E4 / 7 / Jarkko Nrann 369

6. Gomrcal nonlnars n fn lmn analss Aall loa gomrcall nonlnar lasc bar problm. s sar from gnral formla of prncpl of ral ork for formabl bos applcabl for bo lnar an nonlnar connm ors: W n snc V EAεε Nε σ : ε V ε ' b ' b N ' ' S S an N f V EAε b b V W ε ' ' ' ' ' CIV-E4 / 7 / Jarkko Nrann 37

6. Gomrcal nonlnars n fn lmn analss Ingrang b pars gs form N '' N ' N ' b CIV-E4 / 7 / Jarkko Nrann 37

6. Gomrcal nonlnars n fn lmn analss Ingrang b pars gs form N '' N ' N ' an fnall ffrnal qaon an bonar conons as N '' f ; N EA N or 3 N or. nonl EA ' ' : N ln b N nonl CIV-E4 / 7 / Jarkko Nrann 37

6. Gomrcal nonlnars n fn lmn analss Ingrang b pars gs form N '' an fnall ffrnal qaon an bonar conons as N or 3 N or. c can b compar o lnar cas N ' N ' N '' f ; N EA N' f ; N EA N or 3 N or. ln EA ' ' : N b Rmark. Nonlnar s clarl mpl b nonlnar scon orr aal sran. nonl EA' : N ln ln N nonl CIV-E4 / 7 / Jarkko Nrann 373

6. Gomrcal nonlnars n fn lmn analss Fn lmn formlaon of an aall loa gomrcall nonlnar lasc bar problm. As sal sar from ak form r abo N ' ' a : b EA' N EAε EA ' ' ' ' ' EA' b : l ' CIV-E4 / 7 / Jarkko Nrann 374

375 Fn lmn formlaon of an aall loa gomrcall nonlnar lasc bar problm. As sal sar from ak form r abo an n coos fn lmn ral an s fncons sasfng ssnal bonar conons as sal: : ' ' ' ' : ' ' ' ' ' ' l b EA a EA EA EAε N b N. c c c c c c c c n n n n n n n j n j j n n 6. Gomrcal nonlnars n fn lmn analss CIV-E4 / 7 / Jarkko Nrann

6. Gomrcal nonlnars n fn lmn analss Ts rsls afr a f sps om r n an qaon ssm K f forc cor compabl for = n an splacmn cor nknon for = n as bfor for lnar cas f f f b j CIV-E4 / 7 / Jarkko Nrann 376

6. Gomrcal nonlnars n fn lmn analss Ts rsls afr a f sps om r n an qaon ssm K f forc cor compabl for = n an splacmn cor nknon for = n as bfor for lnar cas f f f b l sffnss mar compabl for j = n no pns on nknons: K K K j j Rmark. T sffnss mar s nonsmmrc l s nrs ar sll concnra n a narro agonal ban formng a ban mar o locall sppor ral an s fncons. CIV-E4 / 7 / Jarkko Nrann 377

6. Gomrcal nonlnars n fn lmn analss B akng a closr look a nrnal nrg bn sffnss mar can b sn as a composon of smmrc lnar par an a copl of nonlnar pars of ffrn orrs: a EA' EA' ' EA ' ' ' 3 ' ' EA ' 3 ' CIV-E4 / 7 / Jarkko Nrann 379

6. Gomrcal nonlnars n fn lmn analss B akng a closr look a nrnal nrg bn sffnss mar can b sn as a composon of smmrc lnar par an a copl of nonlnar pars of ffrn orrs: a K EA' ' EA K K ln EA' K K ' ' ' 3 3 ' nonl ' EA ' ' Rmark. In D an 3D problms nonlnar sffnss par ll b mc mor complca an n crrn D mol problm alog prncpls ar sam. Rmark. In n sall no b nrs n sffnss mar n form abo rar an s lnarzaon so call angnal sffnss mar. 3 CIV-E4 / 7 / Jarkko Nrann 38

6.3 Maral nonlnars n fn lmn analss Aall loa marall nonlnar lasc bar problm. s sar from gnral formla of prncpl of ral ork for formabl bos applcabl for bo lnar an nonlnar connm ors: W n V EAεε Nε σ : ε V S b b T S N f V T b V E Aε E W ln ' A' : N ln N nonl CIV-E4 / 7 / Jarkko Nrann 38

6.3 Maral nonlnars n fn lmn analss Aall loa marall nonlnar lasc bar problm. s sar from gnral formla of prncpl of ral ork for formabl bos applcabl for bo lnar an nonlnar connm ors: W n snc V EAεε Nε σ : ε V ε ' an N ' b b S T S b N f ε ' V T ' b V E Aε E W ln ' A' : N ln N nonl CIV-E4 / 7 / Jarkko Nrann 383

6.3 Maral nonlnars n fn lmn analss Ingrang b pars gs form analog o lnar cas N' N N b an fnall ffrnal qaon an bonar conons N' f ; N E Aε E N or ln ' A' : N ln N nonl 3 N or. CIV-E4 / 7 / Jarkko Nrann 384

6.3 Maral nonlnars n fn lmn analss Ingrang b pars gs form analog o lnar cas N' N N b an fnall ffrnal qaon an bonar conons 3 N' N N f ; or or N E Aε E. c can b compar o lnar cas n c onl normal forc s ffrn o sran-srss rlaon lnar nonl N N E A ε E E A ε Rmark. Nonlnar s clarl mpl b nonlnar aal srss. ln E ln ' A' : N A' : N ln ln ln ' A' : N ln N nonl N nonl CIV-E4 / 7 / Jarkko Nrann 385

6.3 Maral nonlnars n fn lmn analss Fn lmn formlaon of an aall loa marall nonlnar lasc bar problm. As sal sar from ak form r abo N ' a : b N E Aε EA' ' ' E ln ' A' : N b : l ln N nonl CIV-E4 / 7 / Jarkko Nrann 386

387 Fn lmn formlaon of an aall loa marall nonlnar lasc bar problm. As sal sar from ak form r abo an n coos fn lmn ral an s fncons sasfng ssnal bonar conons as sal: : ' ' ' : : ' ' ' l b EA a N N A E Aε E N b N nonl ln ln c c c c c c c c n n n n n n n j n j j n n 6.3 Maral nonlnars n fn lmn analss CIV-E4 / 7 / Jarkko Nrann

6.3 Maral nonlnars n fn lmn analss Ts rsls afr a f sps om r n an qaon ssm K f forc cor compabl for = n an splacmn cor nknon for = n as bfor for lnar cas f f f b l sffnss mar compabl for j = n no pns on nknons: K K K j j Rmark. T sffnss mar s nonsmmrc l s nrs ar sll concnra n a narro agonal ban formng a ban mar o locall sppor ral an s fncons. Rmark. T ssm abo can no b sol b sanar mar nrs mos c s mos sgnfcan ffrnc bn lnar an nonlnar FE mos. CIV-E4 / 7 / Jarkko Nrann 389

6.3 Maral nonlnars n fn lmn analss B akng a closr look a nrnal nrg bn sffnss mar can b sn as a composon of smmrc lnar par an a nonlnar par: a K K ln EA' ' ' K EA' ' EA ' K nonln ' CIV-E4 / 7 / Jarkko Nrann 39

6.3 Maral nonlnars n fn lmn analss B akng a closr look a nrnal nrg bn sffnss mar can b sn as a composon of smmrc lnar par an a nonlnar par: a K K ln EA' ' ' K EA' ' EA ' K nonln ' Rmark. Maral nonlnars can b cas b nonlnar lasc nonlnar nlasc maral baor as laso-plasc sco-plasc c. as ll. Rmark. In parclar nlasc baor mans ffrn loa-rspons pa for loang an nloang pass pa pnnc l for lasc baor s pas ar ncal b oppos rcons. Ts ms b akn no accon n solon algorms. CIV-E4 / 7 / Jarkko Nrann 39

6.4 narzaon an solon algorms Solng a nonlnar qaon ssm a s of nonlnar algbrac qaons as K f can b accompls b sng a Non Rapson raon c nols lnarzaons of qlbrm qaons rqrng rconal ras. Frs loa s gn a scalar mlplr an a rfrnc loa as f f rf. Scon ssm s rn n a gnral form as g : r f : K f rf r nong nrnal forc cor pnng on nknon cor. rf CIV-E4 / 7 / Jarkko Nrann 393

395 Drconal ras apparng n lnarz form ar smpl r angn sffnss mar ll pla an ssnal rol n Non Rapson raon bas on lnarz ssm c mpls Non Rapson raon sps as follos: : f rf g K r g rf k rf k k rf k k k k rf k k k f r f K f r g f K g 6.4 narzaon an solon algorms K CIV-E4 / 7 / Jarkko Nrann

396 Drconal ras apparng n lnarz form ar smpl r angn sffnss mar ll pla an ssnal rol n Non Rapson raon bas on lnarz ssm c mpls Non Rapson raon sps as follos: : rf g g f K r k sop sop k k k k k k rf k k sop k k k k rf k k k k rf k rf k k rf k k k k rf k k k for n ncrmn S for c nl Ira r f r f K f r f K f r g f K g 6.4 narzaon an solon algorms K CIV-E4 / 7 / Jarkko Nrann

6.4 narzaon an solon algorms Coos Incrmn sps k... : a. Coos n loa ncrmn b. Iraon sps...; Coos olranc : b. S b. For comp b.3 Comp b.4 Upa sc a K k k k r Sol Comp r b.5 Cck f conrgnc s ssasf: ; g k ; ofn k ; k k k an s K k k k k g k k k f rf k k r k k f rf If If r s sop k fals rrn back o b.. sop k an sar n nrmn a a. CIV-E4 / 7 / Jarkko Nrann 397

6.4 narzaon an solon algorms Rmark. In so-call mof Non Rapson mo angn sfnss mar s no pa for ac ra sp onl for ac ncrmnal sp. Non Rapson raon Mof Non Rapson raon CIV-E4 / 7 / Jarkko Nrann 398

6.4 narzaon an solon algorms Rmark. In so-call arc lng mos or Rks mos an aonal consran s appl for raons n orr o prn snap-rog nar lm pons from A o A an from B o B n fgr on rg. Non Rapson raon Arc lng mo Rmark. In ra solon algorms appl o problms nolng nlasc nonlnars pa-pnnc of cons mar a o b akn no accon n sp b.4 of Non Rapson raon abo: Anor raon or som or cnq s appl for compng srss ncrmns from sran ncrmns n orr o sa on l srfac an o follo flo rl. CIV-E4 / 7 / Jarkko Nrann 399

QUESTIONS? ANSWERS ECTURE BREAK!

Hom rcs 6. Spn a copl of ors for carfll rang an makng nos abo cr sls Scons 6. 6. 6.3 an 6.4. s r man pons b a f snncs an/or formla. s mamacal noaons no famlar o o for rcallng m from or arlr s marals or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 4

Hom rcs 6. Spn a copl of ors for carfll rang an makng nos abo T book 3 Caprs an. T book 3: Anras Öcsnr & Marks Mrkl: On-Dmnsonal Fn Elmns An Inrocon o FE Mo Sprngr-Vrlag 3 s r man pons from bo caprs b a f snncs an/or formla. s mamacal noaons no famlar o o for rcallng m from or arlr s marals or Wkpa Wolfram MaWorl. CIV-E4 / 7 / Jarkko Nrann 4

Hom rcs 6.3 s consr D mol problm of an aall loa marall nonlnar bar nrnal forc conrbon a E A' ' EA' ' ' an l s s a pcs lnar fn lmn appromaon for fnng an approma solon for problm. Dr local rfrnc lmn sffnss mar corrsponng o form abo. Hn: Frs ma lnar cas as E ol b a consan an r on sffnss mar nrs n rms of sap fncons: K pq ; p Scon nsr maral fncon no nr n form E E E ln '; k N kk. For frr lp s Eampl 3.3. of T book 5 R. CIV-E4 / 7 / Jarkko Nrann 43

Hom rcs 6.4 s consr a logarmc cr of form ln. Dmonsra o nrmnal ra Non Rapson algorm orks for solng a copl of pons from cr b sarng from a knon pon. Hn: Inf cr as an mplcl fn cr ln g rf rf ln sa. Follo raon scm: a. Coos frs k = loa ncrmn gng frs loa ll.. b. Tak raon sps for rmnng ra splacmns an fnall. 5.5 CIV-E4 / 7 / Jarkko Nrann 44

Sofar rcs 6. Comsol Bros Comsol manal Nonlnar Srcral Marals Mol For Agmnng Srcral Mcancs Analss Nonlnar Maral Mols ps://.comsol.com/nonlnar-srcral-marals-mol Wc ons of maral mols col b sfl n cl ngnrng applcaons an for c prposs? Follo nsrcons n ps://.comsol.com/mol/onloa/788/mols.nsm.bar_nckng.pf for solng nckng problm of a sl bar. CIV-E4 / 7 / Jarkko Nrann 45

Sofar rcs 6. Comsol s consr a sl canlr pla lng cknss = / an W = /5 nr a loa an a ln loa a fr n. Compar loa flcon pas of pla follong sanar lnar lasc an gomrcall nonlnar lasc b plong loa agans mamm flcon. CIV-E4 / 7 / Jarkko Nrann 46

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Ingraon b pars Ingraon b pars formla for frs orr nsors or cors: compac noaon n noaon Ingraon b pars formla for scon orr nsors or marcs: compac noaon n noaon j j A j j A A A n s A A σ : A σ A σn s n s j n j s CIV-E4 / 7 / Jarkko Nrann 48