Anne-Kathrin Schäuble

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

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 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.