1. No category

An introduction to Hybrid High-Order methods
Linear elasticity
Daniele A. Di Pietro
I3M, University of Montpellier
A.A. 2014–2015
1 / 56
Outline
1 Continuous setting
2 Discrete problem and stability
3 Error analysis
4 Numerical examples
2 / 56
Outline
1 Continuous setting
2 Discrete problem and stability
3 Error analysis
4 Numerical examples
3 / 56
Continuous setting I
Let Ω Ă Rd , d ě 2, be a bounded connected polygonal domain
The linear elasticity problem reads: Find u : Ω Ñ Rd s.t.
´∇¨σpuq “ f
u“0
in Ω,
on BΩ
where, for real Lam´e parameters 0 ď λ ď `8 and 0 ă µ ă `8,
σpuq “ 2µ∇s u ` λp∇¨uqI d
Here, σpuq and ∇s u are, respectively, the stress and strain tensors
4 / 56
Continuous setting II
Assume f P L2 pΩqd and set U :“ H01 pΩqd
The weak formulation reads: Find u P U s.t.
apu, vq “ pf , vq
@v P U ,
with bilinear form
apu, vq :“ 2µp∇s u, ∇s vq ` λp∇¨u, ∇¨vq
Equivalently, we have the variational formulation
"
*
1
inf Jen pvq :“ apv, vq ´ pf , vq
vPU
2
5 / 56
Continuous setting III
Lemma (Korn’s first inequality)
For all functions v P H01 pΩqd , it holds that
}∇v}2 ď 2}∇s v}2 .
Let v P Cc8 pΩqd . We have
ż
ż
|∇s v|2 “
2
Ω
ż
ż
|∇v|2 `
Ω
∇v¨∇v T “
Ω
ż
|∇v|2 `
Ω
|∇¨v|2 ě }∇v}2
Ω
The conclusion follows from the density of Cc8 pΩqd in H01 pΩqd
6 / 56
Continuous setting IV
Combining Korn’s and Poincar´e’s inequalities, we obtain
}v} À }∇s v},
and well-posedness follows from Lax–Milgram’s Lemma
Similar results can be proved for more general boundary conditions
The mechanical engineering terminology for boundary conditions is
Dirichlet Ð displacement
Neumann Ð traction
7 / 56
Quasi-incompressible limit I
Lemma (A priori estimate)
Let Ω Ă R2 be a bounded convex polygonal domain. Then, there is
CΩ ą 0 only depending on Ω s.t.
}u}H 2 pΩqd ` }λ∇¨u}H 1 pΩq ď CΩ }f }L2 pΩqd .
This result is basically linked to Cattabriga’s regularity for Stokes
8 / 56
Quasi-incompressible limit II
Let us introduce the pressure
p :“ λ∇¨u
We can reformulate the linear elasticity problem as
2µp∇s u, ∇s vq ` p∇¨v, pq “ pf , vq
p∇¨u, qq “ λ´1 pp, qq
@v P H01 pΩqd ,
@q P L20 pΩq
For λ Ñ `8, u thus solves a Stokes problem since
∇¨u Ñ 0
Numerically, we need to approximate non-trivial solenoidal fields
9 / 56
Outline
1 Continuous setting
2 Discrete problem and stability
3 Error analysis
4 Numerical examples
10 / 56
DOFs and reduction map I
••
•• ••
•• •• ••
••
•••• ••
•
•• •
•• ••
••
•• •• ••
••
•• ••
••
••
•• ••
•• ••
•• ••
•••• ••
••
•• •• ••
••
••
••
•• •• ••
••
k=2
••
k=1
•
•• •
k=0
Figure: U kT for k P t0, 1, 2u
For all k ě 0 and all T P Th , we define the local space of DOFs
#
U kT
:“
Pkd pT qd
+
ą
ˆ
Pkd´1 pF qd
F PFT
The global space is obtained enforcing single-valuedness at interfaces
+
#
U kh
:“
ą
T PTh
Pkd pT qd
#
+
ą
ˆ
Pkd´1 pF qd
F PFh
11 / 56
DOFs and reduction map II
For a collection of DOFs v h P U kh , we use the underlined notation
˘
`
v h “ pv T qT PTh , pv F qF PFh
For all T P Th , v T P U kT denotes its restriction to U kT s.t.
˘
`
v T “ v T , pv F qF PFT
12 / 56
DOFs and reduction map III
We define the local reduction map I kT : H 1 pT qd Ñ U kT s.t.
`
˘
I kT : v ÞÑ πTk v, pπFk vqF PFT
Finally, the global reduction map I kh : H 1 pΩqd Ñ U kh is s.t.
pI kh vq|T “ I kT pv |T q
@T P Th
13 / 56
Rigid body motions
Applied to vector fields, the operator ∇s yields strains
Consistently, the kernel RMpΩq of ∇s contains rigid-body motions
In d “ 3, RMpΩq has dimension 6 and one can prove that
(
RMpΩq :“ v P H 1 pΩq3 | Dα, ω P R3 , vpxq “ α ` ω b x ,
where we recall that
i
ω b x “ ω1
x1
j
ω2
x2
k ω3 “ pω2 x3 ´ x2 ω3 qi ` pω3 x1 ´ ω1 x3 qj ` pω1 x2 ´ ω2 x1 qk
x3 The space of planar rigid-body motions has dimension 3
14 / 56
Displacement reconstruction I
Let T P Th . The local displacement reconstruction operator
pk`1
: U kT Ñ Pk`1
pT qd
T
d
`
˘
is s.t., for all v T “ v T , pv F qF PFT P U kT and w P Pk`1
pT qd ,
d
p∇s pk`1
T v T , ∇s wqT “ ´pv T , ∇¨∇s wqT `
ÿ
pv F , ∇s wnT F qF ,
F PFT
Rigid-body motions are prescribed from v T setting
ż
ż
pk`1
vT “
T
T
ż
∇ss pk`1
vT “
T
vT ,
T
T
ÿ ż 1
pnT F bv F ´v F bnT F q
F 2
F PF
T
15 / 56
Displacement reconstruction II
k
Lemma (Approximation properties for pk`1
T IT )
There exists C ą 0 independent of hT s.t., for all v P H k`2 pT qd ,
k`1 k
}v ´ pk`1
I kT v}T ` hT }∇pv ´ pT
I T vq}T
T
1{2
3{2
k`1 k
` hT }v ´ pk`1
I kT v}BT ` hT }∇pv ´ pT
I T vq}BT ď Chk`2
}v}H k`2 pT qd .
T
T
16 / 56
Displacement reconstruction III
pT qd ,
By definition of pTk`1 and of I kT , we have, for all w P Pk`1
d
ÿ
p∇s pk`1
I kT v, ∇s wqT “ ´pπTk v, ∇¨∇s wqT `
pπFk v, ∇s wnT F qF
T
F PFT
“ ´pv, ∇¨∇s wqT `
ÿ
pv, ∇s wnT F qF
F PFT
Integrating by parts the right-hand side yields
p∇s pTk`1 I kT v ´ ∇s v, ∇s wqT “ 0
@w P Pk`1
pT qd
d
This orthogonality (Euler) condition implies that
k
}∇s ppk`1
T I T v´vq}T “
inf
wPPk`1
pT qd
d
}∇s pw´vq}T À hk`1
T }v}H k`2 pT qd
17 / 56
Displacement reconstruction IV
The closure conditions for the local problem yield for the translations
ż
ż
ż
k`1 k
k
πT v “
v
pT I T v “
T
T
T
Similarly, for the rotations we have
ż
ÿ ż 1
∇ss ppTk`1 I kT vq “
pnT F b πFk v ´ πFk v b nT F q
2
T
F
F PFT
ż
ÿ ż 1
pnT F b v ´ v b nT F q “
“
∇ss v
F 2
T
F PF
T
k
Hence, pk`1
T I T v and v have the same rigid-body motions
18 / 56
Displacement reconstruction V
Using the local Poincar´e’s and Korn’s inequalities, we have
}pTk`1 I kT v ´ v}T ` hT }∇ppk`1
I kT v ´ vq}T À hT }∇s ppk`1
I kT v ´ vq}T
T
T
À hk`1
}v}H k`2 pT qd
T
Boundary terms can be estimated as for Poisson
19 / 56
Stabilization I
Define, for T P Th , the stabilization bilinear form sT as
ÿ
k
k
sT puT , v T q :“
pk`1
h´1
pk`1
T v T ´ v F qqF ,
T uT ´ uF q, πF pp
F pπF pp
F PFT
pk`1
pT qd s.t.
with displacement reconstruction p
: U kT Ñ Pk`1
T
d
@v T P U kT ,
k`1
k k`1
pk`1
p
T v T :“ v T ` ppT v T ´ πT pT v T q
We next investigate the stability and consistency properties of sT
20 / 56
Stabilization II
Lemma (Stabilization)
Assume k ě 1. There is η ą 0 independent of h, µ, and λ such that, for
all T P Th and all v T P U kT , the following stability property holds:
2
´1
η}v T }2ε,T ď }∇s pk`1
}v T }2ε,T ,
T v T }T ` sT pv T , v T q ď η
where the discrete strain norm }¨}ε,T is s.t.
ÿ
2
}v T }2ε,T :“ }∇s v T }2T `
h´1
F }v F }F .
F PFT
Moreover, for all v P H k`2 pT qd , we have the approximation property
sT pI kT v, I kT vq {2 À hk`1
T }v}H k`2 pT qd .
1
21 / 56
Stabilization III
We only detail the proof of stability (boundedness is similar)
2
}v T }2ε,T À }∇s pk`1
T v T }T ` sT pv T , v T q
Taking w “ v T in the definition of pk`1
T , we infer that
ÿ
}∇s v T }2T “ p∇s pk`1
pv T ´ v F , ∇s v T nT F qF
T v T , ∇s v T qT `
F PFT
1
2
2
2
2
ď }∇s pk`1
T v T }T ` }∇s v T }T ` NB Ctr |v T |ε,BT
2
As a result, we have
2
2
}∇s v T }2T À }∇s pk`1
T v T }T ` |v T |ε,BT
22 / 56
Stabilization IV
Additionally, for all F P FT ,
´1{2
hF
}v F ´ v T }F
´1{2
p k`1
}v F ´ πFk p
v T }F ` hF
T
´1{2
p k`1
}πFk pv F ´ p
v T q}F ` hF
T
´1{2
k`1
p Tk`1 v T q}F ` Ctr h´1
}πFk pv F ´ p
v T ´ πTk pk`1
v T }T
F }pT
T
ď hF
“ hF
“ hF
´1{2
p k`1
}πFk p
v T ´ v T }F
T
´1{2
}πFk pp
pk`1
v T ´ v T q}F
T
For any function w P H 1 pT qd , writing
"ż
wRM “ |T |´1
d
T
*
"ż
*
w ` |T |´1
∇
w
px ´ xT q,
ss
d
T
we observe that πTk wRM “ wRM since k ě 1, whence we get
}w´πTk w}T “ }pw´wRM q´πTk pw´wRM q}T ď }w´wRM }T À hT }∇s w}T
23 / 56
Stabilization V
Applying this inequality to the rightmost term we infer
k`1
k k`1
´2
Ctr h´1
Ctr CK }∇s pk`1
F }pT v T ´ πT pT v T }T ď %
T v T }T ,
so that
´1{2
hF
´1{2
}v F ´ v T }F À hF
k`1
pk`1
}πFk pv F ´ p
T v T q}F ` }∇s pT v T }T
Squaring and summing over F P FT leads to
2
|v T |2ε,BT À sT pv T , v T q ` NB }∇s pk`1
T v T }T
The first inequality follows recalling that we had proven
2
2
}∇s v T }2T À }∇s pk`1
T v T }T ` |v T |ε,BT
24 / 56
Stabilization VI
Let now v P H k`2 pT qd . For all T P Th and all F P FT we have
´1{2
hF
k
k
}πFk pp
pk`1
T I T v ´ πF vq}F
´1{2
k
}p
pk`1
T I T v ´ v}F
´1{2
k
k`1 k
k
}ppk`1
T I T v ´ vq ´ πT ppT I T v ´ vq}F
´1{2
k
´1
k`1 k
}pk`1
T I T v ´ v}F ` Ctr hF }pT I T v ´ v}T
ď hF
“ hF
À hF
À hk`1
T }v}H k`2 pT qd
Using the approximation properties of pk`1
the conclusion follows
T
25 / 56
Divergence reconstruction
We define the local local discrete divergence operator
DTk : U kT Ñ Pkd pT q
`
˘
s.t., for all v T “ v T , pv F qF PFT P U kT and all q P Pkd pT q,
pDTk v T , qqT :“ ´pv T , ∇qqT `
ÿ
pv F ¨nT F , qqF
F PFT
By construction, we have the following commuting diagram:
U pT q
∇¨
I kT
U kT
L2 pT q
πTk
DTk
Pkd pT q
26 / 56
Discrete problem
We define the local bilinear form aT on U kT ˆ U kT as
(
k`1
aT puT , v T q :“ 2µ p∇s pk`1
T uT , ∇s pT v T qT ` sT puT , v T q
` λpDTk uT , DTk v T q
The discrete problem reads: Find uh P U kh,0 s.t.
ah puh , v h q :“
ÿ
T PTh
aT puT , v T q “
ÿ
pf , v T qT
@v h P U kh,0
T PTh
where displacement boundary conditions are enforced setting
!
)
`
˘
U kh,0 :“ v h “ pv T qT PTh , pv F qF PFh P U kh | v F ” 0 @F P Fhb
27 / 56
Outline
1 Continuous setting
2 Discrete problem and stability
3 Error analysis
4 Numerical examples
28 / 56
Energy error analysis I
p h , where u
p h P U kh,0 is s.t.
We bound the error uh ´ u
p h “ ppπTk uqT PTh , pπFk uqF PFh q
u
We measure the error in the energy norm s.t., for all v h P U kh ,
}v h }2en,h :“ ah pv h , v h q
Using stability, this will also yield a bound in the strain norm since
p2µηq}v h }2ε,h ď }v h }2en,h
29 / 56
Energy error analysis II
Theorem (Convergence)
Assume k ě 1 and the additional regularity
u P H k`2 pTh qd and ∇¨u P H k`1 pTh q.
Then, there exists C ą 0 independent of h, µ, and λ s.t.
˘
`
1
p h }en,h ď Chk`1 looooooooooooooooooooooooomooooooooooooooooooooooooon
p2µq {2 }uh ´ u
p2µq}u}H k`2 pTh qd ` λ}∇¨u}H k`1 pTh q .
Bpu,kq
30 / 56
Energy error analysis III
Under usual regularity for Ω, the above estimate is locking-free
For d “ 2 and Ω convex, it holds (cf. [Brenner and Sung, 1992])
Bpu, 0q “ µ}u}H 2 pΩqd ` λ}∇¨u}H 1 pΩq ď Cµ }f },
with Cµ ą 0 depending on Ω and µ but independent of λ
More generally, for k ě 1, we need the regularity shift
Bpu, kq ď Cµ }f }H k pΩqd
31 / 56
Energy error analysis IV
For all v h P U kh,0 , we observe that
#
}v h }2en,h “ ah pv h , v h q ď
sup
wh PU k
h,0
ah pv h , wh q
}wh }ε,h
+
ˆ }v h }ε,h
Thus, using stability, we have that
1
p2µηq {2 }v h }en,h ď
sup
wh PU k
h,0 ,}w h }ε,h “1
ah pv h , wh q
32 / 56
Energy error analysis V
p h q and using the discrete problem yields
Making above v h “ puh ´ u
1
p h }en,h ď
p2µηq {2 }uh ´ u
sup
v h PU k
h,0 ,}w h }ε,h “1
Eh pwh q,
with consistency error
uh , wh q
Eh pwh q :“ lh pwh q ´ ah pp
We next bound Eh pwh q for a generic wh P U kh,0 s.t. }wh }ε,h “ 1
33 / 56
Energy error analysis VI
Let an element T P Th be fixed
q T :“ pk`1
p T and using the definition of pk`1
Setting u
T , we infer
T u
p T , ∇s pk`1
q T , ∇s wT qT `
p∇s pk`1
u
wT qT “ p∇s u
T
T
ÿ
q T n T F , w F ´ w T qF
p∇s u
F PFT
“ p∇s u, ∇s wT qT `
ÿ
q T n T F , w F ´ w T qF
p∇s u
F PFT
Similarly, using the definition of DTk and the commuting property,
p T , DTk wT qT “ pπTk p∇¨uq, DTk wT qT
pDTk u
ÿ
“ p∇¨u, ∇¨wT qT `
pπTk p∇¨uq, pwF ´ wT q¨nT F qF
F PFT
34 / 56
Energy error analysis VII
Replacing the above expressions we obtain
aT pp
uT , wT q “ pσ, ∇s wT qT ` p2µqsT pp
uT , wT q
ÿ
q T ` λI d πTk p∇¨uq, wF ´ wT qF
`
p2µ∇s u
F PFT
Using f “ ´∇¨σ and integrating by parts element-wise, we infer
#
+
ÿ
ÿ
lh pwh q “
pσ, ∇s wT qT ´
pσnT F , wT ´ wF qF
T PTh
F PFT
35 / 56
Energy error analysis VIII
We thus rewrite the consistency error Eh pwh q as
Eh pwh q “
ÿ
ÿ
q T qnT F , wF ´ wT qF
p∇s pu ´ u
p2µq
T PTh
F PFT
looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon
T1
ÿ
πTk p∇¨uqqnT F , wF
`
λpp∇¨u ´
´ wT qF ´ p2µqsh pp
uh , wh q
F PFT
looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon looooooooooomooooooooooon
T2
T3
36 / 56
Energy error analysis IX
k
Using the approximation properties of pk`1
T I T , we have
|T1 | À p2µqhk`1 }u}H k`2 pTh qd }wh }ε,h
Similarly, we have for T2 using the approximation properties of πTk ,
|T2 | À λhk`1 }∇¨u}H k`1 pTh q }wh }ε,h
To estimate T3 , we use the consistency of sh to infer
p h q {2 sh pwh , wh q {2 À p2µqhk`1 }u}H k`2 pTh qd }wh }ε,h
|T3 | ď p2µqsh pp
uh , u
1
1
Using these bounds to estimate Eh pwh q the conclusion follows
37 / 56
L2 -error displacement for the potential I
Assumption (Elliptic regularity)
For all g P L2 pΩqd , the unique solution of
´∇¨ς “ g
in Ω,
ς “ 2µ∇s z ` λp∇¨zqI d
in Ω,
z“0
on BΩ,
satisfies the a priori estimate
p2µq}z}H 2 pΩqd ` λ}∇¨z}H 1 pΩq ď Cµ }g}.
38 / 56
L2 -error displacement for the potential II
Theorem (L2 -error estimate for the displacement)
Let eh P Pkd pTh qd be s.t.
eh|T :“ uT ´ πTk u
@T P Th .
Then, elliptic regularity for Ω and provided that
u P H k`2 pTh qd and ∇¨u P H k`1 pTh q,
it holds with Cą0 depending on Ω, µ, and % but independent of λ and h,
}eh } ď Chk`2 Bpu, kq.
39 / 56
L2 -error displacement for the potential III
p h P U kh,0 the energy error estimate yields
With eh :“ uh ´ u
}eh }ε,h ` sh peh , eh q {2 À hk`1 Bpu, kq
1
Consider the auxiliary problem with g “ eh and solution z and ς
Integrating by parts element-wise and since eh|T “ eT , we infer that
}eh }2 “ ´
ÿ
peT , ∇¨ς qT
T PTh
+
#
ÿ
“
T PTh
p∇s eT , ς qT `
ÿ
peF ´ eT , ς nT F qF
F PFT
40 / 56
L2 -error displacement for the potential IV
Let
ph :“ ppπTk zqT PTh , pπFk zqF PFh q P U kh,0
z
We have
ph q “ ah puh , z
ph q´ah pp
ph q “ lh pp
ph q “ Eh pp
ah peh , z
uh , z
z h q´ah pp
uh , z
zhq
Therefore, we can decompose }eh }2 as follows:
$
,
.
ÿ &
ÿ
p h q ` Eh pp
}eh }2 “
p∇s eT , ς qT `
peF ´ eT , ς nT F qF ´ ah peh , z
zh q
%
T PTh
F PFT
looooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooon looooomooooon
T1
T2
41 / 56
L2 -error displacement for the potential V
For all T P Th , we have, letting S kT :“ p2µq∇s pk`1
` λI d DTk ,
T
#
ÿ
ph q “
ah peh , z
+
p∇s eT , S kT uT qT
T PTh
ÿ
`
peF ´
eT , S kT uT nT F qF
F PFT
ph q
` p2µqsh peh , z
pT
Plugging this expression into T1 we obtain with δ T :“ ς ´ S kT z
|T1 | ď }eh }2ε,h ` sh peh , eh q
(1{2
+1{2
#
ˆ
ÿ “
}δ T }2T ` hT }δ T }2BT
‰
ph q
` p2µq2 sT pp
zh, z
T PTh
Therefore,
`
˘
|T1 | À hk`2 Bpu, kq }z}H 2 pΩqd ` λ}∇¨z}H 1 pΩq À hk`2 Bpu, kq}eh }
42 / 56
L2 -error displacement for the potential VI
Consider now T2 . Adding pσ, ∇s zq ´ pf , zq “ 0 and since
ÿ
lh pp
zhq “
pf , πTk zqT ,
T PTh
we have the following decomposition:
T2 “
)
ÿ !
pT qT ´ λpDTk u
p T , DTk z
pT q
p T , ∇s pk`1
pσ, ∇s zqT ´ p2µqp∇s pk`1
u
z
T
T
T PTh
loooooooooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooooooooon
T2,1
ÿ
ÿ
pT q `
´
pf , πTk z ´ zqT
p2µqsT pp
uT , z
T PTh
T PTh
loooooooooooooomoooooooooooooon looooooooooooomooooooooooooon
T2,2
T2,3
43 / 56
L2 -error displacement for the potential VII
Using orthogonality, we infer
#
T2,1 “
ÿ
ph qqT
p2µqp∇s pu ´ pk`1
I kT uq, ∇s pz ´ pk`1
z
T
T
T PTh
+
p h qT
` λp∇¨u ´ πTk p∇¨uq, ∇¨z ´ DTk z
k
k
Thus, using the approximation properties of pk`1
T I T and πT ,
`
˘
|T2,1 | À hk`2 Bpu, kq }z}H 2 pΩqd ` λ}∇¨z}H 1 pΩq
Furthermore, consistency of sT yields
|T2,2 | À hk`2 Bpu, kq}z}H 2 pΩqd
44 / 56
L2 -error displacement for the potential VIII
Finally, since πTk is self-adjoint and since k ě 1, we infer that
pf , πTk z ´ zqT “ pπTk f ´ f , zqT “ pπTk f ´ f , z ´ πT1 zqT ,
whence,
|T2,3 | À hk`2 }f }H k pΩqd }z}H 2 pΩqd
Using the estimates for T2,1 , T2,2 and T2,3 with elliptic regularity,
|T2 | À hk`2 Bpu, kq}eh }
The convergence estimates results from the bound for T1 and T2
45 / 56
Outline
1 Continuous setting
2 Discrete problem and stability
3 Error analysis
4 Numerical examples
46 / 56
Numerical validation I
We consider the following exact solution:
`
˘
u sinpπx1 q sinpπx2 q ` p2λq´1 x1 , cospπx1 q cospπx2 q ` p2λq´1 x2
The right-hand side is
`
˘
f “ 2π 2 sinpπx1 q sinpπx2 q, 2π 2 cospπx1 q cospπx2 q
The solution u has vanishing divergence in the limit λ Ñ `8
47 / 56
Numerical validation II
Figure: Meshes for the numerical example
48 / 56
Numerical validation III
k“1
k“2
k“3
10´1
10´1
10´2
10´2
10´3
10´3
10´3
1.89
10´4
10
´4
2.65
3.54
10´5
10´5
2.97
10´6
10´5
3.93
10´7
10´6
10´7
5.17
4.85
10´9
10´7
´3
´2
10
10
10´2
3.02
4.22
4.28
4.95
10´8
k“4
10
´2.2
10
´2
´1.8
10
10
´2.5
10´2
10´1.5
10´3
10´3
10´4
10´4
10´5 3.75
2.91
10´6
10´8
10´6
3.98
10´7
4.97
10
5.99
10´10
10´3
10´2
´8
10´9
4.85
10´5
2.99
10´7
4.08
10´9
10´2.2
10´2
10´1.8
10´11
5.05
6
5.94
10´2.5
10´2
10´1.5
Figure: Energy (above) and displacement (below) errors vs. h for λ “ 1
49 / 56
Numerical validation IV
k“1
k“2
k“3
10´1
k“4
10´1
10´2
10´2
10´3
10
10´5
2.98
10´6
´4
10´5
3.94
10´7
10´3
10´3
1.93
10´4
10´5
10´6
10´7
5.15
4.79
10´9
10´7
´3
´2
10
3.04
4.26
4.3
4.97
10´8
2.69
3.55
10
10
´2.2
10
´2
´1.8
10
10
´2.5
10´2
10´1.5
10´2
10´3
10´3
10´4
10´4
10´5
2.92
3.86
10´5
2.99
´7
4.07
10´6
10´6
3.98
10´8
10´7
4.98
10
4.91
10´9
5.98
10´8
10´10
10´3
10´2
10´2.2
10´2
10´1.8
10´11
5.04
5.99
5.57
10´2.5
10´2
10´1.5
Figure: Energy (above) and displacement (below) errors vs. h for λ “ 1000
50 / 56
Numerical validation V
101
101
100
k
k
k
k
100
10
Figure:
τass
τsol
2
“1
“2
“3
“4
10´1
10
3
4
10
k
k
k
k
“1
“2
“3
“4
102
103
104
vs. cardpFh q for the triangular (left) and hexagonal (right) mesh families
51 / 56
Numerical validation VI
10´1
10´1
10´3
10´3
´0.84
´1.04
10´5
10´5
´1.09
´1.24
10´7
10´9
k
k
k
k
“1
“2
“3
“4
10´1
´1.73
10´7
´2.29
10´9
100
101
102
10´2
10´2
10´2
´4
10´4
10
´1.64
“1
“2
“3
“4
10´1
´1.99
100
101
102
´1.17
´1.3
10´6
k
k
k
k
10´6
´1.46
10´8
10´10
10´12
´1.66
k
k
k
k
“1
“2
“3
“4
10´1
´2.19
´2.77
100
101
10´8
10´10
102
10´12
10´2
k
k
k
k
“1
“2
“3
“4
10´1
´1.97
´2.32
100
101
102
Figure: Energy (above) and displacement (below) error vs. τtot (s) for the triangular
and hexagonal mesh families
52 / 56
44
A
F
16
Cook’s membrane test case I
48
Figure: Cook’s membrane test case (µ “ 0.375, λ “ 7.5 ¨ 106 )
53 / 56
Cook’s membrane test case II
Figure: Deformed configuration for the coarsest (cardpTh q “ 22), intermediate
(cardpTh q “ 280) and finest (cardpTh q “ 4192) hexagonal meshes, k “ 1. The color
represents the magnitude of the displacement field.
54 / 56
Cook’s membrane test case III
u
qh,2 pxA q
16.6
´7.25
´7.3
qh,1 pxA q
u
Tria1
Kers1
Hexa1
Tria2
Kers2
Hexa2
16.7
Tria1
Kers1
Hexa1
Tria2
Kers2
Hexa2
´7.35
16.5
´7.4
102
103
cardpFh q
104
102
103
cardpFh q
104
Figure: Vertical (left) and horizontal (right) displacement at A
55 / 56
References
Brenner, S. C. and Sung, L.-Y. (1992).
Linear finite element methods for planar linear elasticity.
Math. Comp., 59(200):321–338.
Di Pietro, D. A. and Ern, A. (2012).
Mathematical Aspects of Discontinuous Galerkin Methods, volume 69 of Math´
ematiques & Applications.
Springer, Berlin Heidelberg.
Di Pietro, D. A. and Ern, A. (2015).
A hybrid high-order locking-free method for linear elasticity on general meshes.
Comput. Meth. Appl. Mech. Engrg., 283:1–21.
56 / 56