José A. González

This work presents a novel stabilized bipenalty formulation for explicit contact‐impact analysis, aiming to enhance traditional bipenalty methods and resolve issues associated with a large mass penalty parameter. To address this challenge, two key contributions are introduced. First, a new formulation integrates the bipenalty method with a predictor‐corrector scheme, enabling a more accurate penetration estimation by dividing the computational process into prediction and correction phases. This separation ensures that the mass penalty affects only the contact forces, thereby eliminating undesirable mass effects on internal forces when a large mass penalty parameter is used. Second, new criteria tailored to the predictor‐corrector scheme are proposed for two types of contact problems: the flexible‐rigid case and the flexible‐flexible case. Unlike traditional bipenalty methods, which rely on stability conditions for explicit time integrators, the proposed criteria focus on enforcing the kinematic constraints. As a result, any predicted penetration is eliminated during the correction phase, leading to zero penalty energy. Stability analysis confirms that the computation of the gap within the correction phase maintains stability. Contact impact examples are performed in 1D, 2D, and 3D and demonstrate that the proposed method provides improved stability and superior performance compared to the penalty and traditional bipenalty methods for various contact scenarios, including extremely large penalty parameter cases.

A general bipenalty formulation with a parameter selection criterion is proposed for explicit contact‐impact finite element analysis. In order to model the contact phenomenon using penalties, it is necessary to introduce an artificial contact stiffness that can increase the maximum natural frequency of the system. When a large stiffness penalty parameter is used, the stability condition forces to reduce the time step size, increasing at the same time the computational demands. A solution to avoid the small time step size requirement and possible instabilities is to include a mass penalty in addition to the stiffness penalty term, using a technique known as the bipenalty method. However, this method presents some other challenges, mainly related to the parameter selection criterion and its final accuracy. In this work, two important improvements for the bipenalty method are presented. (i) An element‐type and dimension‐independent criterion for the selection of penalty mass and stiffness parameters. (ii) A modified predictor‐corrector scheme, specifically designed for the bipenalty method, that improves the accuracy of contact‐impact problems. Different 1D, 2D, and 3D contact‐impact examples are used to illustrate the stability, applicability to general contact cases, and reduction of the zigzag effect and spurious oscillations exhibited by the proposed methods.

New methods are presented for the direct computation of higher‐order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three‐field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher‐order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total‐mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.