Mat-1.600 Laskennallisen tieteen ja
tekniikan seminaari
16.2.2004 14.15
U322
Marcus Rüter, Hannover
Error-controlled
adaptive finite element methods in large strain hyperelasticity
and fracture mechanics
In this presentation, three different strategies for
goal-oriented a posteriori error estimates are derived and compared
numerically in terms of their accuracy and reliability. The error
estimators
are capable of controlling the error obtained while approximately
evaluating a given arbitrary linear or nonlinear functional using
the finite element method. In this contribution, these functionals
typically
represent quantities of engineering interest arising in the theories of
(in)compressible finite hyperelasticity and elastic fracture mechanics.
As a point of departure, an abstract framework for goal-oriented a
posteriori error
estimation -- based on the solution of an auxiliary linear dual problem
-- is
established. In what follows, the abstract model problems are specified
by the
boundary value problems of finite hyperelasticity. On the constitutive
side,
fully spatial stress tensors within classical Newtonian mechanics and
their dual counterparts, namely fully material stress tensors within
Eshelbian mechanics, are established. Benefit of the so-called Eshelby
stress
tensors can be derived within the theory of elastic fracture mechanics
in
order to derive the $J$-integral as a fracture criterion. Since
quantities
of interest that rely on stress tensors -- such as the $J$-integral and
mean
stresses in subdomains -- are generally nonlinear, the corresponding
linearizations are presented.
The linearizations are required in order to control the error of these
functionals by means of goal-oriented a posteriori error estimators.
The
first of the three presented strategies for a posteriori error
estimation
rests upon the solution of Neumann problems on the element level
subjected to
so-called equilibrated residuals eventually leading to residual
error estimators. Secondly, hierarchical error estimators based on the
solutions of local Dirichlet problems on element patches are presented.
The third
strategy is based on averaging techniques in the sense that the
approximate gradient
fields, that appear in the associated error representations, are
recovered. As it
turns out, however, only by the first strategy guaranteed upper error
bounds are
obtained under certain assumptions, whereas the latter two strategies
yield remarkably
sharp error estimates with the side effect that no error bounds can be
achieved.
Finally, illustrative numerical examples are presented, and the
convergence behavior
of the estimated errors and the associated effectivity indices are
comparatively
studied.