Contact algorithms for parallel computations

Vladislav Yastrebov

Contact : Vladislav YASTREBOV

Abstract

Contact is one of the most complicated problems in mechanics. Consequently numerical treatment of contact problems in the framework of the Finite Element method is also a very complicated task due to many reasons: non-smooth mesh surface, asymmetric stiffness matrix, nonexistence of smooth potentials, etc. Many methods to treat the contact have been proposed in last two decades, more and more developments appear nowadays. The interest to the subject is growing up because of increasing number of tribological analyses which can be performed numerically. However most of the proposed methods are often restricted to sufficiently small contact problems, whereas industry requires more and more elaborated analyses of very complicated and very large (from a computational point of view) contact problems including large deformations, nonlinear materials and frictional contact. Nowadays the limits on the mesh size in the Finite Element Analysis are largely extended by powerful parallelization methods and affordable parallel computers. However the contact often presents a bottleneck in resolution process. In the light of such changes an improvement of existing algorithms and methods is highly desirable.

Numerical treatment of contact problem can be divided into two stages: contact detection and contact resolution. The first stage represents a big challenge in parallel treatment of contact and often it is responsible for the significant increase of CPU time. For very large problems, the detection time can overpass significantly the time needed for parallel resolution of the problem. A new technique - grid detection method - has been proposed to manage with contact detection in case of known and unknown a priori (e.g., self-contact) master-slave discretizations. Carried out tests demonstrated that the gain in time for the new method reaches 160,000 times in comparison to a simple all-to-all method. The proposed method has been efficiently parallelized and integrated in the parallel computational loop of the finite element code ZéBuLoN.

Another very specific problem in contact mechanics is a rigorous description of contact geometry. This task has been achieved using a new tensor based mathematical apparatus proposed in the thesis. The precise geometrical description allows to retain quadratic convergence rate and to incorporate in the finite element code many geometry related features.

Four different contact resolution methods have been implemented in the code: penalty method, Lagrange multipliers method, augmented Lagrangian method and a special technique to get fast and precise solutions for cases when one of contacting solids is flat and rigid. These methods have been compared in different cases.

Actually the final steps for contact parallelization have been being implemented. The principle challenge consists in the fact that parallel techniques based on the Schur complement are not suitable to deal with asymmetric matrix rising from frictional contact problem.

 

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