Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 312

This proves (3.45) with C(h) = ch , and it follows from the preceding investiga-
tions that the P 1 element locks.
On the other hand, it is clear that the construction (3.53) of pairs (θ, w) with
B(θ, w) 0 can be approximated by piecewise polynomials of degree 2, and then

the locking is avoided.

=

We consider now the remedies for the low degree elements discussed in
Remark 3.12 and start with a mixed method with a penalty term. The weak solution
(θ, w) of the variational problem with the internal stored energy (3.51) and a load
(f, w) : = 0 fwdx is given by

(θ ,ψ ) + t − 2 (w − θ,v − ψ) = (f, v) for ψ, v ∈ H 0 . ( 3 . 56 )

= t − 2 (w − θ) ∈ L 2 leads to the saddle

The introduction of the shear term γ :

point formulation

(θ ,ψ ) + (v − ψ, γ ) = (f,v) ψ,v ∈ H 0 ,

( 3 . 57 )

(w − θ,η) − t 2 (γ, η) =

0 η ∈ L 2 .

{ (ψ, v) ∈ (H 0 ) 2 ; v − ψ =

The ellipticity on the kernel 0 } is obtained by applying

Friedrichs' inequality

1
2 | ψ |

1
2 | ψ |

1
2 | ψ |

c
2 ψ

1
2 | ψ |

c

ψ

2 v

2
0

2
1

2
1

2
1

2
0

2
1

2

= + ≥ + = +

0 .( 3 . 58 )

Given η ∈ L 2 , we define ρ(x) : = x(b − x) , and an appropriate pair (ψ, v) for

verifying the inf-sup condition is given by

0 η(ξ)dξ
0 ρ(ξ)dξ

A : = ,

( 3 . 59 )

x

η(ξ)dξ − A x
0

v(x) : = ρ(ξ)dξ , ψ(x) : =− Aρ(x).

0

By Theorem III.4.11 the saddle point problem is stable.
We consider the discretization of (3.57). The finite element spaces for θ and
w are the same as above. Specifically, let θ h ,w h ∈ M

1

0 , 0 , i.e., they are piecewise
linear functions on a partition of the interval [0 ,b ]. Now the shear terms are chosen
as piecewise constant functions, i.e., γ h ∈ M

0 , and the finite element equations

are

(θ h ,ψ ) + (v − ψ, γ h ) = (f,v)

1

ψ,v ∈ M

0 , 0 ,

( 3 . 60 )

(w − θ h ,η) − t 2 (γ h ,η) =

0 .

η ∈ M

0

The inf-sup condition can obviously be established in the same way as for (3.57),
and the ellipticity can be verified also with minor changes since w is piecewise
constant. The mixed method is stable. Thus locking is now eliminated.