Mass scaling

Abstract

The present work deals with innovative numerical methods for the computer simulation of dynamic problems with explicit time integration. The proposed methods aim to increase accuracy as well as reduce the calculation effort. The basis for the development are variationally consistent inertia templates. The term ‘inertia template‘ covers both alternatives to the commonly used diagonal mass matrices and novel reciprocal mass matrices. Reciprocal mass matrices result directly from the variational formulation and are sparse, assemblable, inverse mass matrices, which allow a trivial computation of the acceleration from Newton‘s second law. In the first part of the work, the focus is on the variational consistency of reciprocal mass matrices and the therewith associated minimum requirements on the ansatz spaces. In the second part, the approach is systematically exploited not only to increase the critical time step but also to customize the inertia template to specific needs, like improved low-frequency accuracy. The third part aims at further development and investigation of reciprocal mass matrices to increase their usability for practical applications. The focus is therefore on the development of an efficient time step estimate and the treatment of contact.

Abstract

In this contribution, a novel local, node-based time step estimate for reciprocal mass matrices is proposed. Element-based estimates turn out to be not generally conservative and are consequently inadequate. Therefore, the nodal time step estimate for diagonally lumped mass matrices based on Gershgorin’s theorem is further developed for application to reciprocal mass matrices. Additionally, simplifications of the proposed time step estimate that improve computational efficiency, especially for contact problems with the penalty method, are discussed and evaluated by numerical examples.

Abstract

In this contribution, variationally consistent inertia templates for B-spline and NURBS-based finite elements are proposed as a unified concept for two different purposes: Customization of the template allows construction of masses and reciprocal masses with desired properties like higher-order accuracy or improved dispersion behavior; Inertia scaling allows substantial speed-up for explicit dynamics by increased critical time steps. The derivation of the template is based on a three-field parametrized functional as in previous works of the authors’ group, but with modified primary variables, namely displacement, velocity and mass-specific linear momentum. The latter allows for mass-preservation for non-constant density throughout the domain and is therefore an enhancement to the formulation proposed in Tkachuk and Bischoff (2015). With the focus on B-spline and NURBS-based finite elements, the proposed template provides alternatives to the row-sum- lumped mass matrix, which is only 2nd order accurate independent of the polynomial order. Earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances. Furthermore, higher-order reciprocal masses can be constructed from the template. They are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems. For non-linear problems only small overhead is expected, but this paper focuses on linear problems only and mainly undistorted meshes. Tuning the method towards inertia scaling, a reduction of the maximum eigenfrequency by 25%–40% is obtained in the examples herein, whereas the accuracy is higher than for lumped or consistent mass.

Abstract

Classical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non-diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element-wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed.

Abstract

Um dynamische Finite-Elemente-Berechnungen mit expliziter Zeitintegration zu beschleunigen, ist es in der Praxis üblich, Massenskalierung anzuwenden. Damit soll die kritische Zeitschrittweite erhöht werden ohne maßgeblich an Genauigkeit in den entscheidenden niederfrequenten Moden zu verlieren. Bei der konventionellen Massenskalierung (CMS) wird künstliche Masse zu den Diagonaltermen der diagonalisierten Massenmatrix addiert und die Diagonalform der Massenmatrix bleibt erhalten. CMS wird gewöhnlich auf eine geringe Anzahl kleiner oder steifer Elemente angewandt, deren hohe Eigenfrequenzen die kritische Zeitschrittweite begrenzen. Dabei wird allerdings die Trägheit der Struktur erhöht, was zu unphysikalischen Effekten führen kann. Mit einer sogenannten selektiven Massenskalierung kann wenigstens die translatorische Trägheit (d.h. der Impuls bei gleichmäßiger Anfangsgeschwindigkeit) erhalten werden, allerdings auf Kosten von Nebendiagonaltermen in der Massenmatrix. Diese Methode lässt sich gleichmäßig auf die gesamte Struktur anwenden und hat geringere unphysikalische Nebeneffekte. Bislang werden skalierte Massenmatrizen rein algebraisch, z.B. steifigkeitsproportional konstruiert. Die Auswahl an Skalierungs-Templates ist beschränkt und ihnen liegt keine konsistente Formulierung zugrunde. In diesem Artikel werden variationelle Methoden zur selektiven Massenskalierung vorgestellt, die nicht nur mathematisch konsistent sind, sondern es auch erlauben, gezielt bestimmte Eigenschaften wie die Erhaltung der Trägheit, Reduktion der höchsten Frequenzen und Genauigkeit bei den niedrigen Frequenzen einzustellen. Die Leistungsfähigkeit der Methode wird an numerischen Beispielen mit Tetraeder- und Hexaeder-Elementen demonstriert.

Abstract

The problem of optimal selective mass scaling for linearized elasto-dynamics is discussed. Optimal selective mass scaling should provide solutions for dynamical problems that are close to the ones obtained with a lumped mass matrix, but at much smaller computational costs. It should be equally applicable to all structurally relevant load cases. The three main optimality criteria, namely eigenmode preservation, small number of non-zero entries and good conditioning of the mass matrix are explicitly formulated in the article. An example of optimal mass scaling which relies on redistribution of mass on a global system level is constructed. Alternative local mass scaling strategies are proposed and compared with existing methods using one modal and two transient numerical examples.

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.

Abstract

A new variational method for selective mass scaling is proposed. It is based on a new penalized Hamilton’s principle where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices. In this framework new mass matrices with desired properties can be constructed. It is demonstrated how usage of these non-diagonal mass matrices decreases the maximum frequency of the discretized system and allows for larger steps in explicit time integration. At the same time the lowest eigenfrequencies in the range of interest and global structural response are not significantly changed. Results of numerical experiments for two-dimensional and three-dimensional problems are discussed.