Summary
The stability and reflection‐transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one‐dimensional contact‐impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so‐called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact‐free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection‐transmission properties in one‐dimensional contact‐impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples, which demonstrate the stability and reflection‐transmission behavior of the bipenalty method in one‐dimensional contact‐impact and wave propagation problems of homogeneous materials.
A variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated.Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B-splines of high continuity and Bézier elements with a standard C 0 continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures.
KEYWORDSBézier extraction, explicit transient analysis, free vibration, inverse mass matrix, isogeometric analysis, localized Lagrange multipliers Int J Numer Methods Eng. 2019;117:939-966. wileyonlinelibrary.com/journal/nme
The paper is devoted to numerical solution of free vibration problems for elastic bodies of canonical shapes by means of a spline based finite element method (FEM), called Isogeometric Analysis (IGA). It has an advantage that the geometry is described exactly and the approximation of unknown quantities is smooth due to higher-order continuous shape functions. IGA exhibits very convenient convergence rates and small frequency errors for higher frequency spectrum. In this paper, the IGA strategy is used in computation of eigen-frequencies of a block and cylinder as benchmark tests. Results are compared with the standard FEM, the Rayleigh-Ritz method, and available experimental data. The main attention is paid to the comparison of convergence rate, accuracy, and time-consumption of IGA against FEM and also to show a spline order and parameterization effects. In addition, the potential of IGA in Resonant Ultrasound Spectroscopy measurements of elastic properties of general anisotropy solids is discussed.
Abstract.In dynamic transient analysis, recent comprehensive studies have shown that using mass penalty together with standard stiffness penalty, the so-called bipenalty technique, preserves the critical time step in conditionally stable time integration schemes. In this paper, the bipenalty approach is applied in the explicit contact-impact algorithm based on the pre-discretization penalty formulation. The attention is focused on the stability of this algorithm. Specifically, the upper estimation of the stable Courant number on the stiffness and mass penalty is derived based on the simple dynamic system with two degrees-of-freedom. The results are verified by means of the dynamic Signorini problem, which is represented by the motion of a bar that comes into contact with a rigid obstacle.
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