一个改进的Reissner-Mindlin 矩形元

4

10ACTAMATHEMATICAEAPPLICATAESINICA

Vol.26No.4

Oct.,2003

Reissner-Mindlin

(

100080)

(

420002)

(

100080)

Reissner-Mindlin

Reissner-Mindlin

1

Reissner-Mindlin

[1]

.

“

”(

[1,2]

:

(

).

[1,3,4]

:

).,

[2,5−9]

,

Arnold-Falk

[5,6]

“

”

(

“

”

).

[10]

Reissner-Mindlin

Raviart-Thomas

Stokes-Inf-Sup

.[1,12]

,[10][10]

[1]

,-Helmholtz-[11]

[5]

.

2001

9

26

2003

6

4

630

26

[10].

[1,12]

-[5,6,10]

Inf-Sup

,

Stokes-Helmholtz-

“

”

L2

[10]

Reissner-Mindlin[13]

Sobolev:Hm(D),m≥0,

L2(D)=H0(D):(·,·)0,D,󰀒·󰀒0,D.D=Ω

1H0(Ω)H−1(Ω),

C()t

EuclideanT

󰀁󰀔∂v1∂v2∂v1∂v∂vT2+,rotv=−,rotv=∂v∂x∂y∂y,∂x∂x−∂y.

󰀒·󰀒m,D,|·|m,D.

D(m≥1)0,D󰀒·󰀒1(|·|1),󰀒·󰀒−1.h•

󰀔∂v∂v󰀁T

󰀉v=∂x,∂y,divv=

󰀂

L2(Ω)

󰀃2

f,t>0,

󰀂1󰀃2

θ=(θ1,θ2)∈H0(Ω),

Ω⊂IR2.Reissner-Mindlin

1

w∈H0(Ω)γ=(γ1,γ2)∈

⎧

⎪⎨a(θ,η)−(γ,η)=0,(γ,󰀉v)=(f,v),⎪⎩

λ−1t2(γ,s)−(󰀉w−θ,s)=0,

󰀂1󰀃2

∀η∈H0(Ω),

1∀v∈H0(Ω),

󰀂󰀃2

∀s∈L2(Ω),

(1.1)

󰀏󰀒󰀆

∂θ2󰀇∂η1󰀆∂θ1∂θ2󰀇∂η2E∂θ1

+ν+ν+a(θ,η)=

12(1−ν2)Ω∂x∂y∂x∂x∂y∂y󰀆󰀇󰀆󰀇󰀓1−ν∂θ1∂θ2∂η1∂η2

+++,

2∂y∂xy∂x(1.2)

λ=

eκ

,κ

2(1+ν)

eYoung’s

νPoisson

󰀂1󰀃2

Θ=H0(Ω),

[1]

1

W=H0(Ω),

󰀂󰀃2

Q=L2(Ω).

(1.3)

:

a(η,η)≥C1󰀒η󰀒21,

sup

(η,v)∈Θ×W

∀η∈Θ,∀s∈Q,

(1.4)(1.5)

(󰀉v−η,s)

≥C󰀒s󰀒H,

󰀒η󰀒1+󰀒v󰀒1

󰀄󰀂󰀃2󰀅H=s∈H−1(Ω);divs∈H−1(Ω),

󰀒·󰀒H=󰀒·󰀒−1+󰀒div·󰀒−1.

(1.1)

(θ,w,γ),

󰀒θ󰀒2+󰀒w󰀒2+󰀒γ󰀒+󰀒u󰀒2+󰀒p󰀒1+t󰀒p󰀒2≤C󰀒f󰀒,(1.6)

4

Reissner-Mindlin

631

Helmholtz-Ωf∈L2(Ω),1

u∈H0(Ω),p∈H1(Ω)/R.

λ=1.

[1,5];γ=󰀉u+rotp

2

Ω

Ch

Ω

[13]

.hK

h=suphK.

K∈Ch

Eh

nE,

τE.

Γh=Eh∪∂Ω.E∈Γh

vE∈Eh

|E|,

[v],i.e.,

[v]=v|K1−v|K2,

E=K1∩K2.K

2L(x,K),2L(y,K),K

(xK,yK).

󰀔󰀁T

φK(x,y):=−(x−xK),(y−yK),BK(x,y):=(x−xK)2−(y−yK)2,R(K):=span{1,x,y,BK(x,y)},

󰀅󰀄

S(K):=span(1,0)T,(0,1)T,φK.

4L(x,k)L(y,K)

(2.1)(2.2)(2.3)(2.4)

󰀅

,1≤i≤4,

󰀄󰀅

(x󰀐i,y󰀐i)∈(1,1),(−1,1),(−1,−1),(1,−1).

Q1(K):=

󰀄[L(x,K)+󰀐xi(x−xK)][L(y,K)+󰀐yi(y−yK)]

P1(K)=span{1,x,y}.

2.1

󰀄󰀅

Vh=v∈H1(Ω);v|K∈Q1(K),∀K∈Ch,

(2.5)

Θh:=(Vh∩W)×(Vh∩W),

󰀏󰀒

Wh=v∈L2(Ω);v|K∈R(K),∀K∈Ch,[v]=0,∀E∈Eh,

E

󰀏󰀓v=0,∀E∈∂Ω,

󰀅󰀄E

Qh=s∈Q;s|K∈S(K),∀K∈Ch,

󰀎:=|󰀉v|2󰀒v󰀒2h0,K,

K∈Ch

(2.6)(2.7)

󰀒·󰀒h

Wh

2.1[16]

.

w∈W∩H2(Ω),w󰀑∈Wh

󰀏󰀏w󰀑=w,∀E∈Γh,

[13,14]

󰀉h.

EE

(2.8)

w󰀑

󰀒w−w󰀑󰀒+h󰀒w−w󰀑󰀒h≤Ch2󰀒w󰀒2,

󰀇1/2󰀆󰀎

2m−22hK|w−w󰀑|m,K+󰀒w󰀑󰀒h≤C|w|1,

K∈Ch

(2.9)

m=0,1.

(2.10)

632

26

2.2

γ∈Qh,

K∈Ch,

Ei∈∂K,1≤i≤4,

divγ|K

=0,

γ•nE|E

=

,

∀E∈∂K.2.3

rotVh⊂Qh,󰀉hWh⊂Qh,

(rotv,󰀉hw)=0,

∀v∈Vh,∀w∈Wh.

v∈Vh.

K∈Ch,

4v=1

󰀎4Lvi󰀂L󰀃󰀂y󰀐󰀃(x,k)L(y,K)(x,K)+x󰀐i(x−xK)L(y,K)+i(y−yK),

i=1

vi,1≤i≤4

v

󰀈

4L(x,k)L(y,K)rotv=

LvL3+v4)

󰀉

(x,K)(−v1−v2+(y,K)(v1−v2−v3󰀈

+v4)

󰀉+(v1−v2+v3−v4)

−(yx−−yxK)

K

.(2.12),

󰀉hWh⊂Qh

(2.13).

(rotv,󰀉󰀎󰀏

hw)=

rotv•n∂K.K∈Ch

∂K

wrotv•n∂K=

∂τ∂v∂K

(

v∈Vh

rotv•n∂K|∂K=

Wh

(2.6)

(2.13).

2.4[12,16](Cl´ement-∈H2(Ω),

)

qq󰀑∈Vh

󰀆󰀎

h2K

m−2

|q−

q󰀑|2󰀇1/2

m,K

+|q󰀑|1≤C󰀒q󰀒2,m=0,1,K∈Ch

󰀒q−q󰀑󰀒+h|q−q󰀑|1≤Ch2󰀒q󰀒2.

2.2

(1.1)

(θh,wh,γh)∈Θh×Wh×Qh

Lh󰀔(θh,γvh,s󰀁

h,wh);(ηh,h)=(f,vh),

∀(ηh,vh,sh)∈Θh×Wh×Qh,

Lh󰀔(θ,w,γ);(η,v,s)󰀁

=a(θ,η)+(󰀉hw−θ,s)+(󰀉hv−η,γ)

−󰀎(t2+h2K)(γ,s)0,K.

K∈Ch

(2.11)

(2.12)(2.13)

(2.14)

(2.15)

(2.16)),

(2.17)(2.18)

(2.19)

(2.20)

4

Reissner-Mindlin

633

󰀒s󰀒2t,h

2.1

:=

󰀎

K∈Ch

222

h2K󰀒s󰀒0,K+t󰀒s󰀒.

(2.21)

(θ,w,γ)∈Θh×Wh×Qh,

sup

(η,v,s)∈Θh×Wh×Qh

󰀅󰀄Lh((θ,w,γ);(η,v,s))

≥C󰀒θ󰀒1+󰀒w󰀒h+󰀒γ󰀒t,h.

󰀒η󰀒1+󰀒v󰀒h+󰀒s󰀒t,h

(2.22)

(θ,w,γ)∈Θh×Wh×Qh,

(η∗,v∗,s∗)=(θ,w,−γ+δ󰀉hw)∈Θh×Wh×Qh,

(2.23)

δ>0

󰀎󰀔󰀁2

Lh(θ,w,γ);(η∗,v∗,s∗)=a(θ,θ)+δ󰀒w󰀒2−δ(󰀉w,θ)+(t2+h2hhK)󰀒γ󰀒0,K

−δ

󰀎

K∈Ch

K∈Ch

(t2+h2K)(γ,󰀉hw)0,K.

(2.24)

2|a󰀒b|≤a2/ε+εb2,

a∈R,b∈R,ε>0,

󰀔󰀁󰀆󰀃δ󰀇δ󰀂∗∗∗22

1−ε(tLh(θ,w,γ);(η,v,s)≥C1−++h)󰀒w󰀒2󰀒θ󰀒21h

22󰀆δ󰀇󰀎22+1−(t+h2K)󰀒γ󰀒0,K.2ε

K∈Ch

(2.25)

0<δ0<ε<

1

,

t2+h2(2.27)

󰀔󰀅󰀁󰀄22Lh(θ,w,γ);(η∗,v∗,s∗)≥C󰀒θ󰀒2+󰀒w󰀒+󰀒γ󰀒1ht,h,

(2.28)

󰀅󰀄

󰀒η∗󰀒1+󰀒v∗󰀒h+󰀒s∗󰀒t,h≤C󰀒θ󰀒1+󰀒w󰀒h+󰀒γ󰀒t,h,

(2.29)

Lh((θ,w,γ);(η,v,s))Lh((θ,w,γ);(η∗,v∗,s∗))

sup≥∗󰀒η󰀒1+󰀒v∗󰀒h+󰀒s∗󰀒t,h(η,v,s)∈Θh×Wh×Qh󰀒η󰀒1+󰀒v󰀒h+󰀒s󰀒t,h

󰀅󰀄

≥C󰀒θ󰀒1+󰀒w󰀒h+󰀒γ󰀒t,h.

(2.30)

634

26

2.1

Θh×Wh×Qh,

(θ,w,γ),(θh,wh,γh)

(1.1),(2.19)

(ηh,vh,sh)∈

󰀏󰀎󰀒󰀔󰀁2

Lh(θ−θh,w−wh,γ−γh);(ηh,vh,sh)=−hK(γ,sh)0,K+

󰀓

γ•n∂Kvh.(2.31)

K∈Ch

∂K

(ηh,vh,sh)∈Θh×Wh×Qh,󰀏

(f,vh)=−(divγ,vh)=(󰀉hvh,γ)−

󰀎L󰀎

∂K

γ•n∂Kvh,h󰀔(θ,w,γ);(η󰀁

K∈Ch

h,vh,sh)=(γ,󰀉hvh)−

h2K(γ,sh)0,K,

K∈Ch

(2.32),(2.33),(2.31).

2.2(θ,w,γ),(θh,wh,γh)

(1.1),(2.19)

󰀒θ−θh󰀒1+󰀒w−wh󰀒h+󰀒γ−γh󰀒t,h≤Ch󰀒f󰀒.

󰀑θ∈Θh

θ∈󰀂H2(Ω)

󰀃2

[13]

,

󰀒󰀑θ−θ󰀒+h|󰀑θ−θ|1≤Ch2󰀒θ󰀒2.

w󰀑∈Wh

w

(

2.1).

γ=󰀉u+rotp,

u∈W∩H2(Ω),

p∈H2(Ω)/R.

γ󰀑=󰀉hu󰀑+rotp󰀑,

u󰀑∈Wh,

p󰀑∈Vh,

p󰀑

Cl´ement-(

2.4).γ󰀑∈Qh(

2.3)

ξθ=󰀑θ−θh,ξw=w󰀑−wh,ξγ=γ󰀑−γh,ρθ=θ󰀑

−θ,ρw=w󰀑−w,

ργ=γ󰀑−γ.

2.1,

2.1

C󰀒

󰀒ξθ󰀒1+󰀒ξw󰀒h+󰀒ξγ

󰀒θt,h≤,ξw,ξγ);(η,v,s))(η,v,s)∈Θsup

Lh((ξh×Wh×Qh

󰀒η󰀒1+󰀒v󰀒h+󰀒s󰀒t,h

=

sup

Lh((ρθ,ρw,ργ);(η,v,s))+R(γ,v,s)(η,v,s)∈Θh×Wh×Qh

󰀒η󰀒1+󰀒v󰀒h+󰀒s󰀒t,h

,

󰀏R(γ,v,s)=󰀎󰀆

−

h2K(γ,s)0,K

+

K∈Ch

∂K

γ•n∂Kv󰀇

.

(2.32)(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

4

Reissner-Mindlin

635

󰀔󰀁

Lh(ρθ,ρw,ργ);(η,v,s)=a(ρθ,η)+(󰀉hρw−ρθ,s)+(󰀉hv−η,ργ)

󰀎γ

−(t2+h2K)(ρ,s)0,K,

K∈Ch

(2.40)

(󰀉hρw−ρθ,s)=(−ρθ,s)≤Ch󰀒θ󰀒2󰀒s󰀒t,h,

(2.41)

(󰀉hρw,s)=0(

2.1,2.2

),

󰀎󰀏

K∈Ch

∂K

γ•n∂Kv+(󰀉hv−η,ργ)

󰀔󰀁

γ•n∂Kv+󰀉hv,rot(p−p󰀑)∂u

v∂n∂K

(2.42)

[13,16]

󰀎󰀏󰀁󰀔

󰀑)−(rotη,p−p󰀑)+=󰀉hv−η,󰀉h(u−u

󰀁󰀔

󰀑)−(rotη,p−p󰀑)+=󰀉hv−η,󰀉h(u−u≤Ch󰀒u󰀒2+󰀒p󰀒1

K∈Ch

󰀎󰀏

∂K

󰀄󰀅󰀄

󰀒v󰀒h+󰀒η󰀒1,

:

󰀅

K∈Ch

∂K

(2.13),(2.36)

󰀎󰀏

K∈Ch

∂K

∂u

v≤Ch󰀒u󰀒2󰀒v󰀒h,∂n∂K

(2.43)

(2.9),(2.10),(2.17)(2.18)

󰀎󰀅󰀄γ−(t2+h2)(ρ,s)≤Ch󰀒u󰀒+t󰀒p󰀒+󰀒p󰀒0,K221󰀒s󰀒t,h,K

(2.44)(2.45)(2.46)

K∈Ch

−

󰀎

h2K(γ,s)0,K≤Ch󰀒γ󰀒󰀒s󰀒t,h,

(

K∈Ch

a(ρθ,η)≤Ch󰀒θ󰀒2󰀒η󰀒1

(2.35)).

(2.38),(1.6),(2.41),(2.42)(2.44)–(2.46)

󰀒ξθ󰀒1+󰀒ξw󰀒h+󰀒ξγ󰀒t,h≤Ch󰀒f󰀒.

(2.47)

(1.6)

(2.47)

󰀄󰀅

󰀒θ−θh󰀒1+󰀒w−wh󰀒h+󰀒γ−γh󰀒t,h≤C󰀒ρθ󰀒1+󰀒ρw󰀒h+󰀒ργ󰀒t,h+h󰀒f󰀒,

(2.48)

󰀒ργ󰀒t,h:

󰀅󰀄

t󰀒ργ󰀒≤Ch󰀒u󰀒2+t󰀒p󰀒2,

󰀇1/2󰀆󰀎󰀅󰀄2γ2

hK󰀒ρ󰀒0,K≤Ch󰀒u󰀒1+󰀒p󰀒1,

K∈Ch

󰀒ργ󰀒t,h

(2.49)(2.50)(2.51)

󰀅󰀄

≤Ch󰀒u󰀒2+󰀒p󰀒1+t󰀒p󰀒2.

636

26

1

L2-

Ph:η∈Θh→Phη∈Qh

∀s∈S(K),∀K∈Ch.

(Phη,s)0,K=(η,s)0,K,

(2.19)

(2.52)

(θh,wh)∈Θh×Wh

a(θh,η)+

󰀎

K∈Ch

−1

(t2+h2(󰀉hwh−Phθh,󰀉hv−Phη)0,K=(f,v),K)

∀(η,v)∈Θh×Wh,

(2.53)

−1

γh|=(t2+h2(󰀉hwh−Phθh),K)

K

∀K∈Ch.(2.54)

2

[15]

:

(2.55)

󰀒θ−θh󰀒+󰀒w−wh󰀒≤Ch2󰀒f󰀒.

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ANIMPROVEDREISSNER-MINDLIN

RECTANGULARNONCONFORMINGELEMENT

DUANHuoyuan

CHENXing

(AcademyofMathematicsandSystemSciences,ChineseAcademyofSciences,Beijing100080)

LIShaoyou

(DepartmentofAppliedMathematics,XiangtanUniversity,Xiangtan420002)

LIANGGuoping

(FeijianCorporation,Beijing100080)

AbstractFortheReissner-Mindlinplateproblem,animprovedrectangularnonconform-ingmixedfiniteelementmethodisanalyzed.Conformingbilinearelementisusedfortherotation,andnonconformingrotatedQrotforthetransversedisplacement,andconstantele-1

mentenrichedbygivenfunctionsisusedfortheshearstress.Itisshownthatthismethodhasauniformstabilityandauniformoptimalerrorbound,withrespecttotheplate-thickness.KeywordsReissner-Mindlinplate,mixedfiniteelementmethod