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

International audienceAn 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

Simulation of stretch forming with intermediate heat treatments of aircraft skins - a physically based modeling approach

In the aerospace industry stretch forming is often used to produce skin parts. During stretch forming a sheet is clamped at two sides and stretched over a die, such that the sheet gets the shape of the die. However for complex shapes it is necessary to use expensive intermediate heat-treatments in between, in order to avoid Lüders lines and still achieve large deformations. To optimize this process FEM simulations are performed. The accuracy of finite element analysis depends largely on the material models that describe the work hardening during stretching and residual stresses and work hardening reduction during heat treatments due to recovery and particle coarsening. In this paper, a physically based material modeling approach used to simulate the stretch forming with intermediate heat treatments and its predictive capabilities is verified. The work hardening effect during stretching is calculated using the dislocation density based Nes model and the particle coarsening and static recovery effects are modeled with simple expressions based on physical observations. For comparison the simulations are also performed with a phenomenological approach of work hardening using a power law. The Vegter yield function is used to account for the anisotropic and biaxial behavior of the aluminum sheet. A leading edge skin part, made of AA 2024 has been chosen for the study. The strains in the part have been measured and are used for validation of the simulations. From the used FEM model and the experimental results, satisfactory results are obtained for the simulation of stretching of aircraft skins with intermediate heat treatments and it is concluded that the physics based material modeling gives better results