Computers and Structures, 80 (2002), 1279-1293.
CONSISTENT TANGENT STIFFNESS
Milan Jirásek and Borek
FOR NONLOCAL DAMAGE MODELS
Swiss Federal Institute of Technology
LSC - ENAC,
This paper deals with the computational analysis of strain localization
problems using nonlocal continuum damage models of the integral type. The
general framework for a consistent derivation of the ``nonlocal''
tangent stiffness is presented. The properties of the tangent stiffness
matrix are discussed and the corresponding assembly procedure is described.
The quadratic rate of convergence of the Newton-Raphson iteration procedure
is demonstrated and the efficiency of the proposed technique is compared
to the standard approach based on the secant or elastic stiffness matrices.
Performance of iterative solvers for the linearized equilibrium equations
for large problems is examined.
Summary and Conclusions
The exact, fully consistent tangent stiffness matrix has been derived for
a class of nonlocal isotropic damage models. Due to the nonlocal interactions,
the stiffness matrix is nonsymmetric and its bandwidth increases
compared to the ``local'' stiffness matrices. The nonlocal character of
the constitutive law makes the element stiffness matrix depend not only
on the degrees of freedom associated with the given element but also on
the degrees of freedom associated with the nodes of all the elements
whose integration points are within the interaction radius. This leads
to nonstandard contributions that must be taken into account in the assembly
procedure. From the computational point of view it is therefore essential
to use an averaging function with a bounded support, in order to keep the
global stiffness matrix sparse.
The general methodology has been applied both to the classical nonlocal
damage model proposed by Pijaudier-Cabot and Bazant (1987) and to
a modified version of that model suitable for quasibrittle materials, in
which the propagation of damage is driven mainly by the tensile strains
and stresses. It has been shown that, despite the seemingly complicated
character of the consistent nonlocal stiffness, the corresponding numerical
algorithms can be implemented into an existing nonlocal finite element
code in a straightforward manner.
The proposed methodology has been tested on three examples, including
a three-dimensional one. It has been demonstrated that the procedure is
robust and the equilibrium iterations converge at a quadratic rate, which
confirms that the linearization is indeed consistent. In terms of efficiency,
the proposed technique is quite competitive, especially when higher
accuracy is required. Another possible area of application is the sensitivity
analysis based on the exact linearization, which is an important ingredient
of various optimization techniques.
For large-scale problems, especially in three dimensions, the performance
of the incremental-iterative solution strategy is substantially increased
by using an indirect, iterative solver, instead of a direct solver based
on the factorization of the stiffness matrix. In the present study, the
Generalized Minimum Residual Method with preconditioning by incomplete
LU decomposition has been successfully adopted.
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