Universität Karlsruhe (TH) - am Institut für Baustatik

B
Remarks on patch test and stability
Assuming infinitesimal deformations δv e e =1, 2..., numel we examine conditions to fulfil the
patch test and stability conditions for the discrete problem based on the presented mixed
formulation. The described FE–method can be interpreted as a ¯B − approach.
B.1 The patch test
Introduce
¯B := N ε F T −1 G (50)
the material part of the element stiffness matrix can be reformulated as follows
∫
k m = G T ĤG= ¯B T C ¯B dA . (51)
Now consider an element displacement vector δv, such that
B δv ≡ constant (52)
(Ω e)
for each element and formulate
⎡ ∫
¯B δv = N ε F T −1 G δv = [ ] ⎡
1, Ñε ⎣ 1/A ⎤
e 1 0
0 f T −1 ⎦
⎢
⎣
(Ω e)
∫
(Ω e)
B δv dA
Ñ T σ B δv dA
⎤
⎥
⎦
(53)
which yields
¯B δv = ε 0 ≡ constant (54)
if the following two conditions hold:
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(i) (Ω B δv dA = B(ξ = η =0)δv = e) ε0 . This is fulfilled for the shear part with
assumed strains, but not with the standard bilinear finite element interpolation of the
displacement field along with (5) 3 . In this context we refer to the investigations for a
linear plate [33].
A e
∫
(ii) ∫ (Ω e) ÑT σ dA = 0 , which is the case for the present interpolation.
Thus ε 0 ≡ constant yields with C = constant for an arbitrary patch of elements to a constant
stress state.
B.2 Stability of the discrete problem
Here the numerical stability of the presented mixed hybrid shell element with 20 nodal degrees
of freedom based on the Hu–Washizu functional is discussed. We assume that C is positive
definite and that the geometric contribution to the tangential stiffness does not introduce
instabilities. We denote by ker[B] the null space of B. Recall that ker[B] consists all nodal
infinitesimal rigid body motions, i.e. a vector δv R in ker[B] satisfies
B δv R = 0 ⇔ δv R = nodal rigid body motion (55)
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