Error Estimation

Selection of a reliable algorithm for the error assessment is probably the most crucial part of the adaptive strategy for softening problems. The error estimate should be based on an energy norm in order to capture correctly the energy dissipation typical for materials with softening. Furthermore, the error analysis should account for the incremental character of the nonlinear analysis.

In this work, a residual-based error estimation developed initially for linear problems [5] and later extended to nonlinear problems [22][23][24] is employed. It features several advantages. The nonlinear version inherits all the properties of the linear counterpart taking into account both the material and geometrical nonlinearity. Its efficiency does not depend on super-convergent properties, that have been proved only for linear problems, but relies on the a-priori convergence analysis of the finite element method. Contrary to classical error estimators of residual type, the computation of flux jumps and associated flux splitting procedure is avoided. Moreover, the algorithm can be easily applied to problems discretized by general unstructured meshes, even with different element types. This makes it very suitable for application in general nonlinear adaptive analysis. The most significant deficiency of this error estimator is its prohibitively large costs, especially in nonlinear analysis.

The discrete equilibrium provided by the finite element method on the discretization of the characteristic size can be written as

where stands for the vector of internal forces associated with the vector of nodal displacement and is the discretized external force term. The error of the solution is generally unknown because it is impossible to obtain the exact solution . However the exact solution can be approximated by a more accurate finite element solution (reference solution) associated with a new discretization of characteristic size . Since the interpolation space corresponding to the new (reference) discretization is much richer than the original one, the reference error is a good approximation of . The graphical representation of the reference error and its relation to the residuum given by

is illustrated in Figure 1. However, evaluation of (or ) by equilibrating the internal and external forces on the fine discretization

is computationally much more demanding than the original (coarse) problem (due to the much larger number of degrees of freedom in the global reference problem) and should be therefore avoided.

The basic idea of residual-based error estimators is to approximate the reference error by solving a set of local reference low-cost problems. Since the elements of coarse mesh form the most natural partitioning of the original problem the local reference problems are typically related to individual elements of the original mesh. The solution of elementary problems requires proper boundary conditions. This is usually achieved by prescribing equilibrated set of fluxes around each element of the coarse mesh. However, the procedure of evaluation of the flux jumps across the interface of neighbouring elements and the consequent flux splitting ensuring the equilibrium is computationally very expensive. This can be overcome by imposing the flux (traction of the original problem) on the outer boundary and by prescribing homogeneous Dirichlet conditions for the error along the interior (inter-element) boundary of each elementary problem. However, the choice of such artificial boundary conditions implies underestimation of the reference error because the error is generally nonzero along the inter-element boundaries. This effect can be alleviated by considering another set of local problems with boundary not coinciding with the inter-element interfaces. In these problems, the homogeneous Dirichlet condition for the error and the original tractions are again applied as boundary conditions. The resulting approximation of the reference error is then obtained by combining contributions from both sets of problems.

The additional set of local problems is associated with a new partitioning of the original problem related to the nodes of the coarse mesh. The disjoint subdomains (covering in a union the whole domain) are formed by adjacent parts of elements sharing the particular node as it is depicted in Figure 2 on the right. The two sets of local problems are referred to as element patches and nodal patches. The local reference problems are discretized in a compatible way to form a global reference mesh as it is shown in Figure 2.

The estimates of the error on the element patch and on the nodal patch (for the sake of clarity, the subscript is omitted in all the error terms thereafter) are obtained by solving Eq. (10) with appropriate boundary conditions on each element and nodal patch. Each local problem is a nonlinear problem with the same nonlocal constitutive model as the original global problem. However, these nonlinear problems are much easier to solve because is assumed to be a good initial approximation of . Note that due to the nonlocal interactions the consistent tangent stiffness matrix is usually not available and it is necessary to use the secant or elastic stiffness matrix to recover the equilibrium. Note also that the local element patches overlap the nodal patches. Therefore the error estimation procedure accounts partially for nonlocal interactions between adjacent elements of the original problem. However, as the size of the elements decreases during the adaptive refinement process below the material characteristic length, more and more nonlocal interactions are neglected due to the truncation of the bell-shaped weight function on the patch boundary.

The square of the energy norm of the element and nodal patch error can be computed on each element of the reference mesh as

The question is how to interpret . For linear problems, it is simply the elastic stiffness matrix. For nonlinear problems, tangent stiffness matrix may be employed providing that it is positive definite. However this is not the case for materials with strain softening. Moreover, as mentioned above, when considering the nonlocal regularization, the consistent tangent stiffness is likely to be not available. Also, due to the nonlocal interactions, the tangent stiffness matrix would be different on the same reference element shared by element and nodal patches, which would complicate the error orthogonalization discussed in the next paragraph. Therefore the elastic stiffness matrix has been adopted as in this study.

It is important to point out, that the resulting approximation of the reference error cannot be gathered simply by summing the contributions from element and nodal patches because and are generally not orthogonal

If the orthogonality condition is enforced on each element of the reference mesh, the total reference error reads

(13) |

and the corresponding energy norm is given by

(14) |

The local estimates can be assembled to build up the global estimate covering the whole domain by summing the contributions from individual elements of the reference mesh. Since these elements are disjoint the norm of the global error estimate can be easily computed using the Pythagoras theorem

(15) |

Assuming the orthogonality of and , the global relative error of the solution is then assessed by

where is the norm of the coarse solution expressed in the same energy norm as the error.

*Daniel Rypl
2005-12-03*