Abstract
We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. We propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach, i.e., nonlinear analogues to the Finite Element Tearing and Interconnecting (FETI) solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. Moreover, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is illustrated on two academic test cases, namely a water diffusion problem and a nonlinear thermal behavior.
Highlights
We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion
Let us cite in the case of overlapping decompositions, the (Restricted) Additive Schwarz methods (RAS/AS) [1,2] and, in the absence of overlap, the Finite Element Tearing and Interconnecting (FETI) [3] and the Balancing Domain Decomposition (BDD) [4]
The coarse propagator can be of additive form or implemented as a projector. It can be replaced by a set of well chosen continuity conditions, like in the FETI-DP [5] or BDDC [6] methods
Summary
We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. The hope when using these methods is that letting the subdomains undergo independent nonlinear evolution provides a better estimation of the state of the structure which should limit the number of outer Newton iterations and the communications associated with the global tangent solver. These methods try to avoid useless computations on subdomains associated with linear evolutions which have no influence on the global convergence, allowing the CPUs to idle and reduce their energy consumption.
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