Computational Physics

13.4 Error estimators 435

first derivatives of the displacement fields). A more accurate solution would lead
to continuous stresses and this can be achieved by some suitable averaging of the
stress components at the mesh points. To be specific, the nodal stress at a mesh
pointpwould be given by

σp=

1

∑

elemswelem

∑

elems

welemσelem (13.47)

where the stresses on the right hand side (in the sum) are the result of the finite
element calculation; the weightswelemmay be taken equal or related to the surface
area of the elements sharing the vertexp. The error is then the difference between the
‘old’ stresses resulting from the calculation and the improved values based on the
recipe above. We shall refer to the ‘old’ stress, resulting from the FEM calculation,
as the FEM stress.
The question arises how the weightswelemcan be chosen optimally. One answer
to this question is provided by theprojection method[7– 10 ]. In this method we
seek acontinuous, piecewise linear stress field, which deviates to a minimal extent
from the FEM stress. The deviation can be defined as theL 2 -norm of the difference
between the FEM stress and the continuous stressσCwhich is a piecewise linear
FEM-type expansion, based on the valuesσσσp:

=

∫



(σC−σFEM)T(σC−σFEM)d. (13.48)

We write the continuous stress within a particular triangle(a,b,c), as usual, in the
form

σC=σaξa+σbξb+σcξc, (13.49)

whereσaetc. are the values of the stresses at the three vertices (as the stress is
continuous, it must be single-valued at the mesh points). The optimal approximation
of the actual stress is defined by those values ofσCat the vertices for which the
deviation is minimal. This directly leads to the condition

∂

[σC]

∂σp

= 2

∫



(

∂σC

∂σp

)T

(σC−σFEM)d=0. (13.50)

As the continuous stress field is a linear function of the values at the mesh points,
we immediately obtain

∑

q

∫



ξpξqσqd−

∫



ξpσFEM,pd=0. (13.51)

This expression needs some explanation. For the pointp, the pointsqrun overp
and all its neighbours. The functionsξpandξqare defined within the same triangle;