Abstract
Singular and selectively-scaled mass matricesare useful for finite element modeling of numerous problems of structural dynamics, for example for low velocity impact, deep drawing and drop test simulations. Singular mass matrices allow significant reduction of spurious temporal oscillations of contact pressure. The application of selective mass scaling in the context of explicit dynamics reduces the computational costs without substantial loss in accuracy. Known methods for singular and selectively-scaled mass matrices rely on special quadrature rules or algebraic manipulations applied on the standard mass matrices. This thesis is dedicated to variationally rigorous derivation and analysis of these alternative matrices. The theoretical basis of this thesis is a novel parametric HAMILTON’s principle with independent variables for displacement, velocity and momentum. The numerical basis is hybrid-mixed discretization of the novel mixed principle and skillful tuning of ansatz spaces and free parameters. The qualities of novel mass matrices are thoroughly analyzed by various tests and benchmarks.
The thesis has three main parts. In the first part of the thesis, the essential fundamentals and notations are introduced. This includes the basic continuum mechanics, the local form of an initial boundary value problem for an elasto-dynamic contact problem and its treatment with finite elements. In addition, an extension of the central difference method to non-diagonal mass matrices and a theoretical estimate of speed-up with selective mass scaling is given. Besides, a motivation for implementation of alternative mass matrices is given.
In the second part of the thesis, the novel variational approach for elasto-dynamic problems is presented. The corner stone of the thesis is the derivation of the novel penalized HAMILTON’s principle and an extension of the modified HAMILTON’s principle for small sliding unilateral contact. These formulations are discretized in space with the BUBNOV-GALERKIN approach. As a result, families of singular and selectively-scaled mass matrices are obtained. The correspondingshape functions are builtfor several families of finite elements. These families include truss and TIMOSHENKO beam elements for one-dimensional problems, as well as solid elements for two and three dimensions. Shape functions for singular mass matrices are derived for quadratic and cubic elements. Selectively-scaled mass matrices are given for elements up to the order three.
In the third part of the thesis, the novel mass matrices are analyzed and an outlook for future work is given. Propagation of harmonic waves, free and forced vibrations and impact problems are used for evaluation of the new mass matrices. First, the propagation of harmonic waves is studied with the help of a FOURIER analysis applied to the semi-discretized equation of motion. This analysis results in a set of dispersion relations. Comparison of the analytical expressions for discrete dispersion relations with the corresponding continuous ones allows efficient error estimation. In this way, the proposed truss and beam elements are analyzed. Secondly, eigenvalue problems are solved for two-and three-dimensional problems. The error in the lowest frequencies (modes) and in the whole spectrum is computed. Thirdly, spectral response curves for forced vibrations are obtained for the new mass matrices in ranges of interest. These curves are compared with the ones obtained with consistent mass matrices via the frequency response assurance criterion. The values of the frequency response assurance criterion indicate the error for linear problems. Finally, several transient examples are solved with singular and scaled mass matrices. These examples confirm expected superiority of singular mass matrices for impact problems, i.e. spurious temporal oscillations of contact pressures are significantly reduced. Variational selective mass scaling reduces computational cost of explicit dynamic simulations. In the outlook, possible developments regarding new element types, alternative weak forms and several multi-physic applications are proposed.
As by-product of this thesis, patch tests for inertia terms, an overview of parametric and non-parametric variational principles of elasto-dynamics and a derivation of the penalized HAMILTON’s principle with a semi-inverse method can be noted. Besides, the topic of finite element technology for mass matrices is posed. This can open new horizons for evolving branches of computational dynamics such as drop test and car crash simulations, phononic crystals and devices.
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