Resumen de Innovative mathematical and numerical models for studying the deformation of shells during industrial forming processes with the finite element method
The doctoral thesis "Innovative mathematical and numerical models for studying the deformation of shells during industrial forming processes with the Finite Element Method" aims to contribute to the development of finite element methods for the analysis of stamping processes, a problematic area with a clear industrial application.
To achieve the proposed objectives, the first part of this thesis covers the solid-shell elements. This type of element is attractive for the simulation of forming processes, since any type of three-dimensional constitutive law can be formulated without the need to consider any additional conjecture. Additionally, the contact of both sides can be easily treated.
This work first presents the development of a triangular prismatic solid-sheet element, for the analysis of thick and thin sheets with capacity for large deformations. This element is in total Lagrangian formulation, and uses neighboring elements to compute a field of quadratic displacements. In the original formulation, a modified right Cauchy tensor was obtained; however, in this work, the formulation is extended obtaining a modified strain gradient, which allows the concepts of push-forward and pull-back to be used. These concepts provide a mathematically consistent method for the definition of temporary derivatives of tensors and, therefore, can be used, for example, to work with elasto-plasticity.
This work continues with the development of the contact formulation used, a methodology found in the bibliography on computational contact mechanics for implicit simulations. This formulation consists of an exact integration of the contact interface using mortar methods, which allows obtaining the most consistent integration possible between the integration domains, as well as the most exact possible solution. The most notable contribution of this work is the consideration of dual augmented Lagrange multipliers as an optimization method. To solve the system of equations, a semi-smooth Newton method is considered, which consists of an active set strategy, also extensible in the case of friction problems. The formulation is functional for both frictionless and friction problems, which is essential for simulating stamping processes. This frictional formulation is framed in traditional friction models, such as Coulomb friction, but the development presented can be extended to any type of friction model.
The remaining necessary component for the simulation of industrial processes are the constitutive models. In this work, this is materialized in the formulation of plasticity considered. These constitutive models will be considered plasticity models for large deformations, with an arbitrary combination of creep surfaces and plastic potentials: the so-called non-associative models. To calculate the tangent tensor corresponding to these general laws, numerical implementations based on perturbation methods have been considered.
Another fundamental contribution of this work is the development of techniques for adaptive remeshing, of which different approaches will be presented. On the one hand, metric-based techniques, including the level-set and Hessian approaches. These techniques are general-purpose and can be considered in both structural problems and fluid mechanics problems. On the other hand, the SPR error estimation method, more conventional than the previous ones, is presented. In this area, the contribution of this work consists in the estimation of error using the Hessian and SPR techniques for the application to numerical contact problems.
