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Numerical analysis of a locking-free mixedfem for a bending moment formulation of
Reissner-Mindlin plates
LOURENCO BEIRAO DA VEIGA1 , DAVID MORA2 , RODOLFO RODRIGUEZ3
1 Dipartimento di Matematica, Universit a degli Studi di Milano.2 Departamento de Matem atica, Universidad del Bıo Bıo.
3 Departamento de Ingenierıa Matem atica, Universidad de Concepci on.
Sesi on Especial de An alisis Num erico
XX Congreso de Matem atica Capricornio COMCA 2010
Universidad de Tarapac a, Arica, Chile
4–6 de Agosto de 2010.
Contents
• The model problem
• The continuous formulation
• Galerkin formulation
• Numerical tests
Mixed FEM for Reissner-Mindlin plates. – 2 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The model problem.
Given g ∈ L2(Ω), find β, γ and w
−div(C(ε(β))) − γ = 0 in Ω,
−div γ = g in Ω,
γ =κ
t2(∇w − β) in Ω,
w = 0, β = 0 on ∂Ω,
• w is the transverse displacement ,
• β = (β1, β2) are the rotations ,
• γ is the shear stress ,
• t is the thickness,
• κ := Ek/2(1 + ν) is the shear modulus (k is a correction factor),
• Cτ := E12(1−ν2) ((1 − ν)τ + νtr(τ )I) ,
• we restrict our analysis to convex plates.
Mixed FEM for Reissner-Mindlin plates. – 3 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The model problem. (cont.)
We introduce as a new unknown the bending moment σ = (σij)1≤i,j≤2, defined by
σ := C(ε(β)).
• C−1τ :=
12(1 − ν2)
E
(1
(1 − ν)τ −
ν
(1 − ν2)tr(τ )I
).
We rewrite the equation above as:
C−1σ = ∇β −
1
2(∇β − (∇β)t) = ∇β − rJ,
• r := − 12 rotβ,
• J :=
0 1
−1 0
.
Mixed FEM for Reissner-Mindlin plates. – 4 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The continuous formulation.
Continuous mixed problem
Find ((σ, γ), (β, r, w)) ∈ H × Q such that
∫
Ω
C−1σ : τ +
t2
κ
∫
Ω
γ · ξ +
∫
Ω
β · (div τ + ξ) +
∫
Ω
r(τ12 − τ21) +
∫
Ω
wdiv ξ = 0
∫
Ω
η · (div σ + γ) +
∫
Ω
s(σ12 − σ21) +
∫
Ω
vdiv γ = −
∫
Ω
gv,
for all ((τ , ξ), (η, s, v)) ∈ H × Q, where
H := H(div; Ω) × H(div ; Ω),
Q := [L2(Ω)]2 × L2(Ω) × L2(Ω),
with
H(div; Ω) := τ ∈ [L2(Ω)]2×2 : div τ ∈ [L2(Ω)]2,
and
H(div ; Ω) := ξ ∈ [L2(Ω)]2 : div ξ ∈ L2(Ω).
Mixed FEM for Reissner-Mindlin plates. – 5 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The continuous formulation. (cont.)
Continuous mixed problem
a((σ, γ), (τ , ξ)) + b((τ , ξ), (β, r, w)) = 0 ∀(τ , ξ) ∈ H,
b((σ, γ), (η, s, v)) = −
∫
Ω
gv ∀(η, s, v) ∈ Q,
where
a((σ, γ), (τ , ξ)) :=
∫
Ω
C−1σ : τ +
t2
κ
∫
Ω
γ · ξ,
b((τ , ξ), (η, s, v)) :=
∫
Ω
η · (div τ + ξ) +
∫
Ω
s(τ12 − τ21) +
∫
Ω
vdiv ξ.
In the analysis we will utilize the following t-dependent norm for the space H
‖(τ , ξ)‖H := ‖τ‖0,Ω + ‖div τ + ξ‖0,Ω + t‖ξ‖0,Ω + ‖div ξ‖0,Ω,
while for the space Q, we will use
‖(η, s, v)‖Q := ‖η‖0,Ω + ‖s‖0,Ω + ‖v‖0,Ω.
Mixed FEM for Reissner-Mindlin plates. – 6 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The continuous formulation. (cont.)
Additional regularity
Proposition 1 Suppose that Ω is a convex polygon and g ∈ L2(Ω). Then, there
exists a constant C , independent of t and g, such that a
‖w‖2,Ω+‖β‖2,Ω+‖γ‖H(div ;Ω)+t‖γ‖1,Ω+‖σ‖1,Ω+t‖div σ‖1,Ω+‖r‖1,Ω ≤ C‖g‖0,Ω.
aD. N. ARNOLD AND R. S. FALK, A uniformly accurate finite element method for the Reissner-Mindlin plate,
SIAM J. Numer. Anal., 26 (1989) 1276–1290.
Mixed FEM for Reissner-Mindlin plates. – 7 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The continuous formulation. (cont.)
Ellipticity in the kernel
V := (τ , ξ) ∈ H : ξ + div τ = 0, τ = τt and div ξ = 0 in Ω.
Lemma 1 There exists C > 0, independent of t, such that
a((τ , ξ), (τ , ξ)) ≥ C‖(τ , ξ)‖2H ∀(τ , ξ) ∈ V.
Proof . Given (τ , ξ) ∈ V , using tr(τ )2 ≤ 2(τ : τ ) ∀τ ∈ [L2(Ω)]2×2, we obtain
a((τ , ξ), (τ , ξ)) ≥12(1 − ν)
E‖τ‖2
0,Ω +t2
κ‖ξ‖2
0,Ω.
Since ‖div τ + ξ‖0,Ω = 0 and ‖div ξ‖0,Ω = 0, we get
a((τ , ξ), (τ , ξ)) ≥ C(‖τ‖2
0,Ω + ‖div τ + ξ‖20,Ω + t2‖ξ‖2
0,Ω + ‖div ξ‖20,Ω
),
⇒ a((τ , ξ), (τ , ξ)) ≥ C‖(τ , ξ)‖2H,
Mixed FEM for Reissner-Mindlin plates. – 8 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
The continuous formulation. (cont.)
Inf-Sup condition
Lemma 2 There exists C > 0, independent of t, such that
sup(τ ,ξ)∈H(τ ,ξ) 6=0
|b((τ , ξ), (η, s, v))|
‖(τ , ξ)‖H≥ C‖(η, s, v)‖Q ∀(η, s, v) ∈ Q.
Well-posedness
Theorem 2 There exists a unique ((σ, γ), (β, r, w)) ∈ H × Q solution of the
continuous mixed problem and
‖((σ, γ), (β, r, w))‖H×Q ≤ C‖g‖0,Ω,
where C is independent of t.
Mixed FEM for Reissner-Mindlin plates. – 9 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation.
• Thh>0: regular family of triangulations of the polygonal region Ω.
• hT : diameter of the triangle T ∈ Th.
• h := maxhT : T ∈ Th.
• Ω =⋃T : T ∈ Th.
Finite elements subspaces
Hγh := ξh ∈ H(div ; Ω) : ξh|T ∈ RT0(T ), ∀T ∈ Th,
Qwh := vh ∈ L2(Ω) : vh|T ∈ P0(T ), ∀T ∈ Th,
Mixed FEM for Reissner-Mindlin plates. – 10 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation. (cont.)
We consider the unique polynomial bT ∈ P3(T ) that vanishes on ∂T and is
normalized by∫
TbT = 1.
B(Th) := τ ∈ H(div ; Ω) : (τi1, τi2)|T ∈ spancurl(bT ), i = 1, 2,∀T ∈ Th ,
where curl v := (∂2v,−∂1v).
Hσ
h := τh ∈ H(div ; Ω) : τh|T ∈ [RT0(T )t]2, ∀T ∈ Th ⊕ B(Th),
Qβh := ηh ∈ [L2(Ω)]2 : ηh|T ∈ [P0(T )]2, ∀T ∈ Th,
Qrh :=
sh ∈ H1(Ω) : sh|T ∈ P1(T ), ∀T ∈ Th
,
Note that Hσ
h × Qβh × Qr
h correspond to the PEERSa finite elements.aD. N. ARNOLD, F. BREZZI AND J. DOUGLAS, PEERS: A new mixed finite element for the plane elasticity,
Japan J. Appl. Math., 1 (1984) 347–367.
Mixed FEM for Reissner-Mindlin plates. – 11 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation. (cont.)
Discrete mixed problem:
Find ((σh, γh), (βh, rh, wh)) ∈ Hh × Qh such that
a((σh, γh), (τh, ξh)) + b((τh, ξh), (βh, rh, wh)) = 0 ∀(τh, ξh) ∈ Hh,
b((σh, γh), (ηh, sh, vh)) = −
∫
Ω
gvh ∀(ηh, sh, vh) ∈ Qh.
Hh := Hσ
h × Hγh ,
Qh := Qβh × Qr
h × Qwh .
Mixed FEM for Reissner-Mindlin plates. – 12 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation. (cont.)
Discrete kernel
Vh :=
(τh, ξh) ∈ Hh :
∫
Ω
ηh · (div τh + ξh) +
∫
Ω
sh(τ12h − τ21h)
+
∫
Ω
vhdiv ξh = 0 ∀(ηh, sh, vh) ∈ Qh
.
Let (τh, ξh) ∈ Vh. Taking (0, 0, vh) ∈ Qh and using that (div ξh)|T is a constant,
since vh|T is also a constant, we conclude that div ξh = 0 in Ω.
Now, taking (ηh, 0, 0) ∈ Qh, since div τh = 0 in Ω ∀τh ∈ B(Th), we have that
(div τh)|T is a constant vector. Since div ξh = 0, we have that ξh|T is also a
constant vector. Therefore, since ηh|T is also a constant vector, we conclude that
(div τh + ξh) = 0 in Ω. Thus, we obtain
Vh =
(τh, ξh) ∈ Hh : ξh + div τh = 0,
∫
Ω
sh(τ12h − τ21h) = 0 ∀sh ∈ Qrh
and div ξh = 0 in Ω
.
Mixed FEM for Reissner-Mindlin plates. – 13 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation. (cont.)
Ellipticity in the discrete kernel
Lemma 3 There exists C > 0 such that
a((τh, ξh), (τh, ξh)) ≥ C‖(τh, ξh)‖2H ∀(τh, ξh) ∈ Vh,
where the constant C is independent of h and t.
Discrete inf-Sup condition
Lemma 4 There exists C > 0, independent of h and t, such that
sup(τh,ξh)∈Hh
(τh,ξh) 6=0
|b((τh, ξh), (ηh, sh, vh))|
‖(τh, ξh)‖H≥ C‖(ηh, sh, vh)‖Q ∀(ηh, sh, vh) ∈ Qh.
Mixed FEM for Reissner-Mindlin plates. – 14 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation. (cont.)
Theorem 3 There exists a unique ((σh, γh), (βh, rh, wh)) ∈ Hh × Qh solution of
the discrete mixed problem. Moreover, there exist C, C > 0, independent of h and t,
such that
‖((σh, γh), (βh, rh, wh))‖H×Q ≤ C‖g‖0,Ω,
and
‖((σ, γ), (β, r, w)) − ((σh, γh), (βh,rh, wh))‖H×Q
≤ C inf(τh,ξh)∈Hh
(ηh,sh,vh)∈Qh
‖((σ, γ),(β, r, w)) − ((τh, ξh), (ηh, sh, vh))‖H×Q,
where ((σ, γ), (β, r, w)) ∈ H × Q is the unique solution of the continuous mixed
problem.
Mixed FEM for Reissner-Mindlin plates. – 15 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Galerkin formulation. (cont.)
Rate of convergence
Theorem 4 Let ((σ, γ), (β, r, w)) ∈ H × Q and
((σh, γh), (βh, rh, wh)) ∈ Hh ×Qh be the unique solutions of the continuous and
discrete mixed problem, respectively. If g ∈ H1(Ω), then,
‖((σ, γ), (β, r, w)) − ((σh, γh), (βh, rh, wh))‖H×Q ≤ Ch‖g‖1,Ω.
Mixed FEM for Reissner-Mindlin plates. – 16 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests.
• Isotropic and homogeneous plate.
• Ω := (0, 1) × (0, 1).
• t = 0.001.
• E = 1, ν = 0.30 and k = 5/6.
Figure 1: Square plate: uniform meshes.
Mixed FEM for Reissner-Mindlin plates. – 17 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Choosing the load g as:
g(x, y) =E
12(1 − ν2)
[12y(y − 1)(5x2 − 5x + 1)
(2y2(y − 1)2
+x(x − 1)(5y2 − 5y + 1))
+ 12x(x − 1)(5y2 − 5y + 1)(2x2(x − 1)2
+y(y − 1)(5x2 − 5x + 1))]
,
so that
w(x, y) =1
3x3(x − 1)3y3(y − 1)3
−2t2
5(1 − ν)
[y3(y − 1)3x(x − 1)(5x2 − 5x + 1)
+ x3(x − 1)3y(y − 1)(5y2 − 5y + 1)],
β1(x, y) =y3(y − 1)3x2(x − 1)2(2x − 1),
β2(x, y) =x3(x − 1)3y2(y − 1)2(2y − 1).
Mixed FEM for Reissner-Mindlin plates. – 18 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
e(σ) := ‖σ − σh‖0,Ω, e(γ) := ‖γ − γh‖t,H(div ;Ω),
e(β) := ‖β − βh‖0,Ω, e(r) := ‖r − rh‖0,Ω, e(w) := ‖w − wh‖0,Ω,
rc(·) := −2log(e(·)/e′(·))
log(N/N ′),
where N and N ′ denote the degrees of freedom of two consecutive triangulations with
errors e and e′.
Mixed FEM for Reissner-Mindlin plates. – 19 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Table 1: Errors and experimental rates of convergence for variables σ and γ, computed
on uniform meshes.
N e(σ) rc(σ) e(γ) rc(γ)
1345 0.40270e-04 – 0.31715e-02 –
5249 0.19649e-04 1.054 0.15876e-02 1.016
20737 0.09760e-04 1.019 0.07942e-02 1.008
82433 0.04868e-04 1.008 0.03971e-02 1.004
328705 0.02431e-04 1.004 0.01986e-02 1.002
Mixed FEM for Reissner-Mindlin plates. – 20 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Table 2: Errors and experimental rates of convergence for variables β, r and w, com-
puted on uniform meshes.
N e(β) rc(β) e(r) rc(r) e(w) rc(w)
1345 0.39713e-04 – 0.87462e-04 – 0.66226e-05 –
5249 0.18189e-04 1.147 0.39217e-04 1.178 0.27707e-05 1.280
20737 0.08884e-04 1.043 0.15009e-04 1.398 0.13136e-05 1.086
82433 0.04416e-04 1.013 0.05491e-04 1.457 0.06478e-05 1.025
328705 0.02205e-04 1.004 0.01991e-04 1.466 0.03228e-05 1.007
Mixed FEM for Reissner-Mindlin plates. – 21 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
1000 10000 100000 1e+06
e
degrees of freedom N
σ
♦♦
♦♦
♦
♦γ
++
++
+
+β
r
××
××
×
×w
h
⋆⋆
⋆⋆
⋆ ⋆
Mixed FEM for Reissner-Mindlin plates. – 22 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Figure 2: Approximate transverse displacement (left) and first component of the rotation
vector (right).
Mixed FEM for Reissner-Mindlin plates. – 23 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Figure 3: Approximate shear vector: first component (left) and second component
(right).
Mixed FEM for Reissner-Mindlin plates. – 24 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Figure 4: Approximate bending moment: σ11h(left) and σ12h
(right).
Mixed FEM for Reissner-Mindlin plates. – 25 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Numerical tests. (cont.)
Figure 5: Approximate bending moment: σ21h(left) and σ22h
(right).
Mixed FEM for Reissner-Mindlin plates. – 26 – BEIRAO DA VEIGA, MORA, RODRIGUEZ
Many thanks for your attention.
Mixed FEM for Reissner-Mindlin plates. – 27 – BEIRAO DA VEIGA, MORA, RODRIGUEZ