A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

Abstract

SUMMARYAn iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations.

Automatic detection and branch switching methods for steady bifurcation in fluid mechanics

“…[12] or in Ref. [36]. In the latter, a linear solver which is well adapted to ANM is introduced and makes it possible to reduce the computational times.…”

Nonlinear forced vibration of damped plates by an asymptotic numerical method

“…Even if the second member time increases, it is negligible compared to the decomposition time. Then to reduce this computational time, one can use another linear solver which is welladapted to repeated right hand side problems [24], as with the asymptotic numerical method; reduce the number of steps by changing the path parameter definition [25] or use reduction methods to have to decompose a small size matrices [26,27]. These three approaches will be investigated in future works.…”

“…Therefore, the geometric boundary conditions associated to (23) are exactly satisfied by the expansions of u, v and w given by (24).…”

Nonlinear forced vibration of damped plates coupling asymptotic numerical method and reduction models

“…The use of these ANM vectors to obtain a reduced order model has been already applied to define efficient high order iterative correctors [29] or recently linear solvers [30]. In this study, the number of vectors issued from the first step is greater than in the previous study due to the harmonic numbers.…”