1、BEM course May 13,2008(typicalBVP-L.ppt)123 4(s,x)eK(s,x)iK(x)(s,x)(s)(s)BKdBjf=(x)(s)(x)(s)(s)BdbBajf=211(),()xsxsOO-(x)(s)ab5a6Analytical methods for solving Laplace problems with circular holesLebedev,Skalskaya and Uyand,1979,“Work problem in applied mathematics”,Dover PublicationsChen and Weng,2
2、001,“Torsion of a circular compound bar with imperfect interface”,ASME Journal of Applied MechanicsHonein,Honein and Hermann,1992,“On two circular inclusions in harmonic problem”,Quarterly of Applied Mathematics789102(x)(s,x)(s)(s)(s,x)(s)(s),xBBuTudBUtdBDp=-蝌0(s,x)(s)(s)(s,x)(s)(s),xcBBTudBUtdBD=-蝌
3、ss(s,x)ln x sln(s,x)(s,x)(s)(s)UrUTut=-=nnxDxcDxDxcD1112111(,;,)ln()cos(),(s,x)1(,;,)ln()cos(),immemmURRmRm RURURmRmrq r fqfrq r frqfrr=-=-0101(s)(cossin),s(s)(cossin),sMnnnMnnnuaanbnBtppnqnBqqqq=+=+13-101020246810UHs,xL2121(x)(s),sx(s,x)(s)(x),xsiiiiiiabUab=1(sx),sx12(s,x)12(xs),xs2Ur-=-101020-0.4-
4、0.20.20.4THs,xLs1,sx2(s,x)1,xs2T=-14-20-15-10-505101520-20-15-10-505101520-20-15-10-505101520-20-15-10-505101520Ros(,)Rq=x(,)r f=iUeUr111(,;,)ln()cos(),(s,x)1(,;,)ln()cos(),immemmURRmRm RURURmRmrq r fqfrq r frqfrr=-=-x(,)r f=151617(,)r f1819zxzx-(s,x)1(s,x)(s,x)cos()cos()2UULrpzxzxrrf抖=-+-+抖(s,x)1(s
5、,x)(s,x)cos()cos()2TTMrpzxzxrrf抖=-+-+抖zx=(s,x)(s,x)ULrr=(s,x)(s,x)TMrr=(x)2(s,x)(s)(s)(s,x)(s)(s),x(x)2(s,x)(s)(s)(s,x)(s)(s),xBBBBuMudBLtdBDuMudBLtdBDrrffpp=-=-蝌蝌ntnttn2021012N禳镲镲镲镲镲镲镲镲=睚镲镲镲镲镲镲镲镲铪tttttM U tT u=000101011101NNNNNN轾犏犏犏=犏犏犏犏臌UUUUUUUUUULLMMOML0B1B220111111101122222011333330112222()()()(
6、)()()()()()()()()()()()()()()()ccsMcMspkpkpkpkpkccsMcMspkpkpkpkpkccsMcMspkpkpkpkpkpkccsMcMspkMpkMpkMpkMpkUUUUUUUUUUUUUUUUUUUUfffffffffffffffffff轾=犏臌ULLLMMMOMML20112121212121()()()()()()MccsMcMspkMpkMpkMpkMpkMUUUUUffffff+轾犏犏犏犏犏犏犏犏犏犏犏犏臌L 011TkkkkkkMMppqpq=tL1f2f3f2Mf21Mf+230(s,x)(s)(s,x)(s)(s)BTuUtdB=
7、-24252627281u=0u=12.5a=21.0a=1u=1u=0u=02.0R=aa290un=1u=0u=0un=02.0R=aaa1u=0u=0un=1u=02.0R=aaa30-2.5-2-1.5-1-0.500.511.522.5-2.5-2-1.5-1-0.500.511.522.5-1.5-1-0.500.511.522.533.5-2.5-2-1.5-1-0.500.511.522.5-1.5-1-0.500.511.522.533.5-2.5-2-1.5-1-0.500.511.522.531090180270360D egree()0123R el ati ve er
8、ror of fl ux on the sm al l ci rcl e(%)B EM-B EPO 2D (N=21)Present m ethod(M=10)Trefftz m ethod(NT=21)M FS (NM=21)(a1=3.0,a2=0.7)32048121620Term s of Fouri er seri es(M)101112131415Parseval s sum048121620Term s of Fouri er seri es(M)22.42.83.23.6Parseval s sum22222001()2()Mnnnfdaabpqqpp=+B&3300.20.4
9、0.60.811.21.41.61.8200.20.40.60.811.21.41.61.823400.20.40.60.811.21.41.61.8200.20.40.60.811.21.41.61.823500.20.40.60.811.21.41.61.8200.20.40.60.811.21.41.61.8236371u=-1u=2laa1u=1u=-1u=1u=-1u=1u=-38-10-8-6-4-20246810-10-8-6-4-20246810-10-8-6-4-20246810-10-8-6-4-2024681039-10-8-6-4-20246810-10-8-6-4-2
10、02468104041vgvg2laa42-14-12-10-8-6-4-2024-10-8-6-4-20246810-14-12-10-8-6-4-2024-10-8-6-4-20246810180g=o135g=o43442()0,xxDj=jsincoskkkkxynjqq=-kB22cos,siniiiixbybNNpp=2 kNpaaabqR45-2-1.5-1-0.500.511.52-2-1.5-1-0.500.511.5246-2-1.5-1-0.500.511.52-2-1.5-1-0.500.511.5247-2-1.5-1-0.500.511.52-2-1.5-1-0.5
11、00.511.524849tmsw2()0,sw xxD=()sinsw xntqm=swww=+kB500123456(i n r adi ans)-202468z /(a r o u n d h o le w it h r a d iu s a1)d/a1=0.01d/a1=0.1d/a1=2.0si ngl e hol e510123456(i n r adi ans)-202468z /(a r o u n d h o le w it h r a d iu s a1)d/a1=0.01d/a1=0.1d/a1=2.0520123456(i n r adi ans)-2024681 0z
12、 /(a r o u n d h o le w it h r a d iu s a1)d/a1=0.01d/a1=0.1d/a1=2.0530123456(i n r adi ans)-2024681 0z /(a r o u n d h o le w it h r a d iu s a1)M=10M=20M=30M=401a2a540sinwRq=0Rd2a1asinwnq=0Rd2a1a5509 01 8 02 7 03 6 0-2024R0=7.5R0=15.0R0=30.009 01 8 02 7 03 6 0-2024R0=7.5R0=15.0R0=30.05603060901201
13、50R adi us R002468z at =45 degreesEqui val ent di spl acem ent approachEqui val ent stress approach0sinwRq=sinwnq=0t=0t=0R570123456(i n radi ans)-202468z/(around hol e w i th radi us a1)d/a1=2.0d/a1=0.1d/a1=0.011a2a3ayxtmdd580123456(i n radi ans)-2-1012z/(around hol e w i th radi us a1)d/a1=2.0d/a1=
14、0.1d/a1=0.01xy1a2a3atmdd590123456(i n radi ans)-20246810z/(around hol e w i th radi us a1)d/a1=2.0d/a1=0.1d/a1=0.011a2a3axytmdd60610u=1u=1B2B0u=1un=1B2Bhhaa62-10-8-6-4-20246810-10-8-6-4-20-10-8-6-4-20246810-10-8-6-4-2063-10-8-6-4-20246810-10-8-6-4-20-10-8-6-4-20246810-10-8-6-4-2064652r1rauspecified=
15、uspecified=66221111112221212222111212222222222211121121222212222lncossin2ln()cos()sin2lncossin2ln()cos()sin2lncossin2ln()cos()sinMMMaaaaaaaaaaaaaaaaaaaaaaaaaaappfpfprpfpfrrppfpfprpfpfrrppfpfprpfprr+LLLLMMMOMMMOL2111111111222121111111121222222211111112112122221112ln()cos()sin2lncossin2ln()cos()sin2ln
16、cossin2ln()cos()sin2lncosMMMMaaaaaaaaaaaaaaaaaaaaaaaaaaaaafrrppfpfppfpfrrppfpfppfpfrrppfpfppf+LLLLLMMMOMMMOL1,01,11,11,1,2,02,12,12,2,221sinMMMMMppqpqppqpqapf+轾犏禳犏镲镲犏镲镲犏镲犏镲镲犏镲镲犏镲犏镲镲犏镲镲犏镲犏镲镲犏镲镲犏镲镲睚犏镲镲犏镲镲犏镲镲犏镲犏镲镲犏镲镲犏镲犏镲犏犏犏犏 铪犏犏犏臌犏MML 1,01,11,11,1,2,02,12,12,2,MMMMaababaabab禳镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲=睚镲镲镲镲
17、镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲镲铪镲镲TMMa a1 1=1.0,influence matrix=1.0,influence matrix U is singularis singular1a67 =UtT u(s,x)(s,x)mUUc=+12 a cp+112(ln)aacp轾+犏犏臌LMO112lnaap轾犏犏臌LMO =UtT u=wtvu =UtT u 轾轾犏犏=犏犏犏犏臌臌UTtuwv1a680.511.522.53R adi us a100.10.20.30.40.51Present m ethodA ddi ng a CH EEF poi ng(5.0,5.0)A ddi
18、ng a ri gi d body term (c=1.0)1a697071727374sxs(,),x(,)Rzzqr f=xsln xsln zz-=-xxxsxsss1sss1ln()ln()(1)ln()ln(1)ln()()mmzzzzzzzzzzm z=-=-=+-=-()x1111s1111()()()()cos()immmimmimmmmzeemm zm Rem Rm Rffqqrrrqf-=-邋邋sxzz-111(,;,)ln()cos(),(s,x)1(,;,)ln()cos(),immemmURRmRm RURURmRmrq r fqfrq r frqfrr=-=-lnR
19、22311ln(1)(1)111()231mmxdxxxdxxxxxxm=-=-=-+-=-+=-蝌LL0k 75uspecifiedn=uspecified=1a=1a=UtT u =UtT uL tMu =UtT uL tMu20u=20u=20u=76aaaa771u=-1u=1u=(s;x,x)ln xsln xsU=-1s(s;x,x)0BU=0u=1u=1B2B2B1B1Bxsxrr2B782(x)(s;x,x)(s)(s)(s;x,x)(s)(s),xBBuTudBUtdBDp=-蝌0(s;x,x)(s)(s)(s;x,x)(s)(s),xcBBTudBUtdBD=-蝌11111(,;,)ln()cos()1ln()cos(),(s;x,x)1(,;,)ln()cos()1ln()cos(),immmmemmmmURRmm RRmRmURURmmRmRmrq r f r fqfrqfrrrq r f r frqfrrqfrrr=-+-=-+-79
