An efficient adaptive strategy to master the global quality of viscoplastic analysis
Introduction
Nowadays, the industry requires the resolution of highly complex problems: 3D problems, non-linear material behaviours, contacts, large deformations, etc. Ahead of solution of such problems, it is necessary to take into account various economical aspects, which are very important: computation time, time necessary to prepare the computation (especially the meshing of complex structures). For a given resolution algorithm, adaptive computation allows us to minimise the computational costs while obtaining a prescribed accuracy by mastering the parameters of the computation (size and type of elements, time increment length). This has been done first in case of linear problems like elasticity [1], [2], [3]. And, the proposed methods have been extended to some non-linear time-dependent problems [4], [5], [6], [7]. Regarding the latter problems, all the adaptive computations proposed are based on the incremental resolution method. Although a non-incremental method, called LATIN method [8], [9], allows us to treat problems of non-linear material behaviours, contacts, or large deformations.
As the incremental method is not the only approach to solve non-linear time-dependent problems, we can wonder if it is the most suitable method to build an adaptive strategy characterised by a short computation time and robust error control. In this work, we address this issue by considering the use of the LATIN method to build a non-incremental adaptive strategy. To control conveniently the numerical approximations, we studied the computation of viscoplastic structures with standard constitutive equations [10], on the assumption of small displacements and for quasi-static loading. But numerous other non-linear time-dependant problems could have been studied to build a similar strategy.
The LATIN method is an iterative method to improve approximate solution defined on all the structure and on all the history studied. In the case studied, as well as in Newton’s method, the goal of the iterations is to treat the non-linearities. But here it is realised on the whole history. A discrete representation of the solution is obviously used. In our approach, this representation is built on a base of space functions computed by employing the finite element method. On this base, the components of an approximate solution are scalar time functions.
To reduce the computational cost, we have proposed an adaptive strategy, in which the improvement of the solution representation is due to the improvement of the treatment of the non-linear equations. A coarse discretisation must be chosen at the beginning of the adaptive strategy. An initial solution is provided through an elastic computation on the interval studied. According to the flaws in the elastic approximation, an error indicator provides a level of non-linearity for the studied problem. In the case of significant non-linearity, LATIN iterations are initiated with the coarse time and space representation, and this representation then becomes more precise over the course of the iterations.
Errors in constitutive equation, especially dissipation error [9], provide a global error estimator, which characterises the distance between an admissible solution and the exact solution. We propose an analytical partition of the global error into indicators associated with each error contributions (time, space, and iteration defaults). Hence, the goal of the main piloting criteria is chosen to obtain a balanced contribution of each kind of error indicator during the computation. The computation is stopped when the global error satisfies the prescribed quality.
The layout of this paper is the following: A brief analysis of existing adaptive strategies enables us to discuss the drawbacks and advantages of the incremental approach. The equations of the viscoplastic reference problem are introduced in Section 2 together with the definition of the dissipation error estimator. A straightforward LATIN algorithm is proposed to solve the reference problem in Section 3. At the end of this section, an example of time and space representation is given. Section 4 outlines some error indicators used for the piloting criteria. And at last, the non-incremental adaptive strategy proposed is described in Section 5, where numerical examples also show the reliability and the efficiency of our approach.
Section snippets
Adaptive strategy for incremental resolution
By adaptive strategy, we exactly mean the way to organise the numerical resolution, the error control procedure and the parameter adaptation. It could be a mesh adaptation or a time discretisation adaptation for example (Fig. 1). The difficulties encountered to build such strategy are common to numerous time-dependent problems. Hence, we extend our analysis from viscoplastic or plastic problems to dynamic problems. Throughout the studied strategy, the resolution method is an incremental one. We
The reference problem
We suppose that the structure is a domain . On a part of the boundary , we suppose that the imposed displacement field is (Fig. 6). On the complementary part , density of forces is imposed. Moreover, is submitted to a density of body forces .
In viscoplasticity, the value of the stress at t is a function of the history of the strain at the instant t, and this function may be expressed at each current point of the structure , by the relation:where A is an
A non-incremental resolution
In this section, we present the numerical algorithm that is used to approximately solve the reference problem. It is built in the framework of the LATIN method introduced by Ladevèze in 1985 [8], [9]. In the first part of this section, we present a base-line algorithm common to other implementations of the method. This algorithm is completely defined without any discretisation. Then, we describe the way we use a discretisation to build an efficient and robust algorithm. An example is presented
Error indicators for adaptation
One of the main advantages of the LATIN is to enable to control the errors after each iteration (Fig. 15). This is due to the computation on of the admissible solution.
In order to simplify the notations, we are going to use s and sh instead of sn+1 and sh,n+1, respectively. There are numerous error sources, which all contribute to the global error estimator eref. The difficulty is to separate these contributions through error indicators. These indicators must be efficient enough to
The non-incremental adaptive strategy
In this section, we present the adaptive strategy based on the non-incremental algorithm and the error indicators introduced above. The main idea is to utilise the solution improvement process, which is obtained through the LATIN method. In the course of the iterations, the admissible solution satisfies the non-linear equations to a greater extent. However, in the beginning, the computation is very far from the solution. We therefore do not need to use fine discretisation to represent the first
Conclusion
The LATIN method is well suited for adaptive computations based on error control, because it allows an effective improvement in the numerical solution over the entire time interval.
By balancing the error contributions during the computation, we obtain lower computation cost. We can say that the proposed strategy allows to adapt the computational effort during the iteration in comparison with the convergence defaults.
Thanks to the partition of the global error into error indicator for each kind
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