danarTi 4
proeqtis reziume
proeqtis dasaxeleba
kerZowarmoebuliani diferencialuri gantolebebi
hiperzedapirebze: garsTa Teoriis gantolebebi da
maqsvelis sistema
•
kvlevis kategoria
1. fundamenturi kvleva
•
samecniero kvlevis
mimarTulebis kodebi
I II
III
3-101
3-102
3-111
•
proeqtis menejeri da
samecn. xelmZRvaneli
duduCava rolandi
•
wamyvani organizacia
a. razmaZis maTematikis instituti
SemoTavazebuli proeqti Seicavs or Temas:
A.
drekad garsTa amocanebi. B. maqsvelis sistema.
A
. drekad garsTa amocanebi. garsebi da maTgan Sedgenili detalebi TavianTi
SesamCnevi simsubuqisa da deformaciis mimarT gansakuTrebuli mdgradobis gamo
xSirad gamoiyeneba mTel rig Tanamedrove konstruqciebSi. kargad gaTvlil garss
SeuZlia masalis minimaluri xarjvis pirobebSi gauZlos did datvirTvas rRvevis
mimarT. am mizezis gamo garsebi warmoadgenen sasurvel konstruqciul elementebs
rogorc bunebriv, ise xelovnur konstruqciebSi. garsebis gamoyenebis mudmivad
mzrdi tendencia gansazRvravs organzomilebiani garsebis modelebis rogorc
fundamenturi kvlevis, aseve ricxviTi modelirebis aucileblobas. samganzomile-
biani drekadobis Teoria ufro naklebadaa damuSavebuli. amasTan Zalze Txeli
samganzomilebiani sxeulis sruli modelireba aRemateba yvelaze mZlavri
Tanamedrove kompiuterebisa da ricxviTi meTodebis SesaZleblobebs.
progresis miuxedavad drekadi garsebis maTematikuri Teoria Sorsaa
saboloo srulyofisagan. jer kidev Riad aris darCenili metad mniSvnelovani sa-
kiTxi Tu rogor gamoviyvanoT organzomilebiani garsebis mkacri maTematikuri
Teoria ufro zusti samganzomilebiani modelisgan. arsebobs aqamde SemoTavaze-
bul garsTa modelebis grZeli sia: koseras (1909), goldenvaizeris (1961), nagdis
(1963), vekuas (1965), novoJilovis (1970), koiteris (1970) da mravali sxva. bolo 2
aTwleulSi maTematikosebis sanCes-palensias, siarles, lodsis, miaras da sxvaTa
ZalisxmeviT moxda garsTa Teoriis mniSvnelovani ganviTareba, ramac arsebiTi
gavlena moaxdina Cvens kvlevaze: moxda wrfivi organzomilebiani garsTa Teoriis
dafuZneba asimptoturi analizis safuZvelze rodesac „sisqe“
ε
iTvleba mcire
parametrad. Aam midgomiT moxda garsTa Teoriis sakmaod
zusti klasifikacia
„moqnili“, „membranis“ da „ganzogadoebuli membranis“ tipis garsebisTvis.
meore miRweva, romelic daedo safuZvlad warmodgenil proeqts, aris hiper-
zedapirze diferencialur gantolebaTa aRricxva, romelic iyenebs garemomcveli
sivrcis dekartul koordinatebs. operatorebi am aRricxvaSi Caiwerebian giunte-
risa da stoqsis mxebi warmoebulebis meSveobiT, erTeulovan gare normalis
monawileobiT. klasikuri operatorebis (zedapiruli gradienti, zedapiruli
divergencia, laplas-beltramis da sxva) miRebuli warmodgenebi gacilebiT
martivia vidre hiperzedapirze klasikuri rimanis metrikuli tenzorebiT
warmodgena (ix. r. duduCava, d. mitrea & m. mitreas Sromebi). am aRricxvis
meSveobiT aseve gamoyvanili iyo lames operatori hiperzedapirebisaTvis. misi
mTavari nawili emTxveva lames operators evklides sivrceSi, Tuki kerZo warmo-
ebulebs SevcvliT giunteris warmoebulebiT. Tumca miRebul gantolebaSi ar
monawileobs simrudis tenzori da is ar SeiZleba gamoyenebul iqnas garsebisTvis.
proeqtis pirveli nawilis kvlevis mizania energiis funqcionalis
minimizaciis meTodis gamoyenebiT garsisaTvis gantolebis Cawera giunteris
warmoebulebSi da misi asimptoturi analizi. miRebuli Sedegebis gamoyeneba
„moqnili“, „membranis“ da „ganzogadoebuli membranis“ klasis garsebis gantoleba-
Ta gamosayvanad. Cven imedi gvaqvs miviRoT amonaxsnTa arsebobisa da erTaderTo-
2
bis Teoremebi, agreTve davamtkicoT garsebisaTvis miRebuli gantolebebisTvis
garkveuli aproqsimaciuli sqemebis krebadoba da mdgradoba.
B.
maqsvelis sistema. maqsvelis sistema da helmholcis gantoleba warmoadgenen
yvelaze gavrcelebul talRis gantolebebs romlebic gvxdebian fizikasa da
sainJinro mecnierebebSi. isini aRweren mraval movlenas yoveldRiuri cxovrebi-
dan. warmodgenili Tema Sedgeba ori nawilisagan:
•
B.
1. maqsvelis
sistema bianizotropul garemoSi, romelic moicavs
specialur SemTxvevas-qiralur garemos.
•
B.
2. helmholcis gantoleba.
B.
1. maqsvelis sistema bianizotropul garemoSi. Cengma da kongma [CK, Ko1] 1968
wels Semoitanes bianizotropuli garemos cneba, romelic aRiwereba yvelaze
zogadi wrfivi ZiriTadi damokidebulebiT moqmed eleqtrul da magnitur velebsa
da maT induqciebs Soris. bianizotropuli sxeuli erTdroulad imuxteba da
magnitdeba eleqtrul an magnitur velSi moTavsebisas. es efeqti Teoriulad
nawinaswarmetyvebeli iyo landausa da lifSicis mier 1957 wels. mogvianebiT
aRmoCnda, rom aseTi efeqti SeuZlia warmoaCinos mravali saxis kristals.
bianizitropuli garemos kerZo SemTxvevaa qiraluri garemo, romelic xasiaTdeba
anizotropiiT inversiis mimarT. qiraluroba axasiaTebs, magaliTad, kristalTa 65
jgufs. bolo wlebSi SeiniSneba sul ufro mzardi interesi aseTi masalebis
mimarT praqtikaSi maTi farTo danergvis mizniT.
miuxedavad imisa, rom eleqtromagnituri velebi Seiswavleba mraval SromaSi,
aSkarad igrZnoba im SromaTa ukmarisoba, romlebic moZRvnilia eleqtromagnituri
velebis da bianizotropul/qiralur garemos urTierTqmedebasTan. es miuTiTebs
kvlevis obieqtis sirTuleze da Cveni mizania CavataroT Semdegi modeluri
fizikuri amocanis mkacri maTematikuri analizi:
samganzomilebian sivrcis
SemosazRvruli are Sevsebulia bianizotropuli garemoTi.
xolo sivrcis darCe-
nili nawili ki
izotropuli an qiraluri garemoTi. Midealurad gamtari metalu-
ri eleqtrodebi moTavsebulia sazRvrze
.
Mda bianizotropul garemoSi. moiTxove-
ba aRvweroT am sistemis gare
eleqtromagnituri velTan urTierTqmedebiT war-
moqmnili eleqtromagnituri veli.
Cven davamtkicebT am amocanis amoxsnis arsebobasa da erTaderTobas, Sevis-
wavliT amonaxsnis regularobas, miviRebT amonaxsnis asimptotikuri gaSlas Cad-
gmuli zedapiris sazRvris maxloblobaSi.
CavwerT maqsvelis gantolebebs mrudwi-
rul sivrce-drois koordinatebSi giunter/stoqsis warmoebulebis meSveobiT.
B.2.
helmholcis gantolebebi. araelifsuroba SesamCnevad arTulebs maqsvelis
gantolebaTa sistemebis kvleva. meores mxriv es sistema SeiZleba Caiweros ro-
gorc helmholcis gantolebaTa sistema eleqtruli da magnituri velebis mimarT,
romelTa divergencia nulis tolia. am meTodiT maqsvelis gantolebaTa sisteme-
bis kvleva SeiZleba dayvanili iqnas helmholcis gantolebis kvlevaze. amitomac
SedarebiT martivi helmholcis gantoleba lideria matematikuri modelirebis
sxva gantolebebTan SedarebiT kvlevaTa raodenobisa da dafinansebis mxriv.
mravali fundamenturi amocana helmholcis gantolebisaTvis gamoukvlevelia.
aseT amocanaTa ricxvs ganekuTvneba sasazRvro amocanebi or ganzomilebian areeb-
Si kuTxovani da ukuqcevis wertilebiT. aqamde am tipis amocanebi Seiswavleboda
mxolod modeluri laplasisa da lames gantolebebisTvis.
helmholcis gantolebebis sasazRvro amocanebis Sesaswavlad aseT areebSi
SemoTavazebuli konformuli asaxvis meTodi. amocana daiyvaneba Sesabamis sasazRv-
ro amocanamde erTeulovan wreSi da miRebuli sasazRvro amocana Seiswavleba
kompleqsuri analizis meTodebiTa da fsevdodiferencialuri operatorebis
teqnikis gamoyenebiT. es midgoma aprobirebulia laplasisa da lames
gantolebisaTvis, Tumca helmholcis gantolebas gaaCnia Taviseburebebi romlebic
ar iCenen Tavs laplasis operatoris SemTxvevaSi.
damtkicebuli iqneba kuTxuri da ukuqcevis wertilebis mqone areebSi
helmholcis gantolebebisaTvis sasazRvro amocanebis amonaxsnis arseboba da
erTaderToba. aseve Seswavlili iqneba amonaxsnis regularobis sakiTxi.