Keywords: non-linear computations, parallel strategies, algorithms, large scale problems, load balancing.

The joint use of parallel computers and powerful algorithms is necessary to strongly reduce the numerical costs of complex non linear Finite Element simulations [1]. To reach this goal, we have created a parallel environment language that eases the development of parallel algorithms either at the programming level or at the user level. It is based on the development environment of the Finite Element code CAST3M. On the other hand, we use a parallel approach that is well suited to the simulation of a large class of non-linear problems and which allows a good balancing of the workload between the tasks without using dynamic load-balancing. This property is obtained by using two domain decompositions, each using the mechanical properties of the different sub-problems to be solved.

The developed parallel language, which is based on an object-based virtual shared memory system, offers the user the vision of a unique and global address space over the individual memories [2]. It ensures the data coherence and hides data exchanges between processors and a great part of the sequential code can be reused. This system frees the programmer from parallel programming intricacies (management of data, coherence of data,...) and lets him focus on the program design, the most critical aspect for the application efficiency. The propounded system can be implemented on most parallel computers as it is developed with machine-independent programming techniques and it is important to notice that the different concepts can be used in other object-based parallel languages. Moreover the object-based shared virtual system allows two levels of parallelization: at the programming level and at the user level.

Non-linear problems are usually solved by means of NEWTON methods and lead mainly to compute two types of sub-problems. The proposed parallel strategy uses the mechanical properties of these sub-problems. On one hand, a domain decomposition technique with a direct resolution of the condensed problem is proposed to solve the linear global problems, in order to be compatible with the BFGS type convergence speed-up. One can also use a parallel direct solver associated to a "nested dissection" ordering approach which limits the fill-in effect in the factorization of the matrix [3]. In fact, this strategy is similar to a decomposition domain technique and gives good numerical results on shared memory computers. On the other hand, it is nearly impossible to predict the space evolution of the CPU time spent to integrate the constitutive laws. Therefore, in order to have a well-balanced load, without communication, we propose the use of a second domain decomposition. An optimisation of the communications between the two domain decompositions is necessary to obtain good performances. The implementation of this strategy is carried out starting from the parallel user language of CAST3M.

An extension of the previous strategy to solving problems associating frictional contact and material non-linearities is also proposed in case the contact zones are small compared to the dimensions of the structure. To solve the global problem associated in the search of solutions verifying the kinematic conditions and equilibrium equation, which contains non-linearities relations through contact conditions involving friction, we propose to use an iterative algorithm based on the LATIN method [4]. For this strategy the equations governing contact and friction are expressed as an explicit way, as the solution is sought at a given time. Furthermore, it is important to notice that the NEWTON type algorithm uses a constant stiffness. Therefore the problem can be condensed on the interface where contact is defined and the various non-linearities are solved on their definition domains which have not the same dimensions.

Numerical examples, in the case of large scale problems (non-linear material behaviour, geometrical non-linearities, frictional contact) are presented to validate the propounded parallel approach.

1

A.K. Noor, S.L. Venneri, D.B. Paul, M.A. Hopkins, "Structure technology for future aerospace systems", Computer & Structures, 74, 507-519, 2000. doi:10.1016/S0045-7949(99)00067-X

2

J.Y. Cognard, F. Thomas, P. Verpeaux, "An integrated approach to solving mechanical problems on parallel computers", Advances in Engineering Software, 31, 885-899, 2000. doi:10.1016/S0965-9978(00)00063-6

3

M.T. Heath and P. Raghavan, "A cartesian parallel nested dissection algorithm", SIAM J. Matrix Anal. Appl., 16, 235-253, 1995. doi:10.1137/S0895479892238270

4

P. Ladevèze, "Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation", Springer Verlag, 1999.

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