Zusammenfassung
In this paper, a new approach is proposed to improve efficiency of the integration procedure for mortar integrals within finite element mortar methods for contact. Appropriate approaches subdivide polygonal integration segments into triangular integration cells where well-established quadrature rules can be applied for numerical integration. Here, a subdivision of segments into quadrilateral integration cells is proposed and investigated in detail. By this procedure, the numerical effort is decreased because the number of integration cells is smaller and less quadrature points are needed. In all the aforementioned methods, necessary projections of integration points result in rational polynomials in the integrand. Thus, an exact numerical integration is impossible. Using quadrilateral integration cells additionally involves non-constant Jacobian determinants which further increases the polynomial degree of the integrand. Numerical experiments indicate, that the resulting increase in the error is small enough to be acceptable in consideration of the gained speed-up.BibTeX
Wilking, C. (2017).
Effiziente Integration und verbesserte Kontaktspannungen für duale Mortar-Formulierungen. Doktorarbeit, Bericht Nr. 66, Institut für Baustatik und Baudynamik, Universität Stuttgart.
https://doi.org/10.18419/opus-9244 Zusammenfassung
This work deals with computer simulations of contact problems using the finite element method. Two modifications are proposed for the mortar method, which is the method applied to discretise the contact. The first modification concerns the numerical calculation of so-called contact integrals. For the corresponding integration in general cases polynomial integrands have to be integrated over polygonal areas. In order to use ordinary numerical quadratures the polygonal areas are usually subdivided into triangular integration cells. In this work an alternative subdivision into quadrilateral integration cells is suggested, which yields less integrations points. With the numerical experiments described in this work it is shown that due to this reduction the numerical effort is decreased considerably without deteriorating the accuracy of integration significantly. The second modification improves the contact stresses of the dual mortar method. This method uses dual functions to approximate the Lagrange multiplier field, yielding the advantage that the dual mortar method is more efficient than the standard mortar method. However, the contact stresses of the dual mortar method are less accurate than the contact stresses of the standard mortar method. In this work a modification of the contact stresses based on an L2 projection is presented for the dual mortar method. Numerical experiments show that with the introduced L2 projection the accuracy of the contact stress of the dual mortar method is improved and comparable to the accuracy of the standard mortar method.BibTeX
Tkachuk, A., Wohlmuth, B., & Bischoff, M. (2013). Hybrid-mixed discretization of elasto-dynamic contact problems using consistent singular mass matrices.
International Journal for Numerical Methods in Engineering,
94, 473–493.
https://doi.org/10.1002/nme.4457 Zusammenfassung
An alternative spatial semi-discretization of dynamic contact based on a modified Hamilton’s principle is proposed. The modified Hamilton’s principle uses displacement, velocity and momentum as variables, which allows their independent spatial discretization. Along with a local static condensation for velocity and momentum, it leads to an approach with a hybrid-mixed consistent mass matrix. An attractive feature of such a formulation is the possibility to construct hybrid singular mass matrices with zero components at those nodes where contact is collocated. This improves numerical stability of the semi-discrete problem: the differential index of the underlying differential-algebraic system is reduced from 3 to 1, and spurious oscillations in the contact pressure, which are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Results of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis.BibTeX
Mangold, O., Prohl, R., Tkachuk, A., & Trickov, V. (2012). Reduction of Numerical Sensitivities in Crash Simulations on HPC-Computers (HPC-10).
Wolfgang E. Nagel, Dietmar B. Kröner, Michael M. Resch, (Eds.): High Performance Computing in Science and Engineering ’11. Transactions of the High Performance Computing Center Stuttgart. Springer, 631–636.
https://doi.org/10.1007/978-3-642-23869-7_46 Zusammenfassung
For practical application in engineering numerical simulations are required to be reliable and reproducible. Unfortunately crash simulations are highly complex and nonlinear and small changes in the initial state can produce big changes in the results. This is caused partially by physical instabilities and partially by numerical instabilities. Aim of the project is to identify the numerical sensitivities in crash simulations and suggest methods to reduce the scatter of the results.BibTeX
Cichosz, T. (2012).
Stabile und konsistente Kontaktmodellierung in Raum und Zeit. Doktorarbeit, Bericht Nr. 58, Institut für Baustatik und Baudynamik, Universität Stuttgart.
https://doi.org/10.18419/opus-495 Zusammenfassung
Die vorliegende Arbeit befasst sich mit verschiedenen Aspekten der Diskretisierung von Kontaktvorgängen in Raum und Zeit. Im Hinblick auf eine stabile und konsistente Modellierung werden bestehende Verfahren verglichen und Verbesserungen erarbeitet.
Schwerpunkt der räumlichen Untersuchungen ist die Weiterentwicklung der in Hartmann u. a. (2007) und Hartmann und Ramm (2008) vorgestellten Kontaktdiskretisierung, die auf der dualen Mortar-Methode (Wohlmuth 2000, 2001) basiert. Durch die Verwendung der Methode der Lagrange’schen Multiplikatoren erfüllt diese Formulierung die Nichtdurchdringungsbedingung exakt. Gleichzeitig erlaubt die Diskretisierung der Multiplikatoren mit dualen Formfunktionen die einfache Kondensation der zusätzlichen Unbekannten aus dem resultierenden Gleichungssystem. Somit wird der übliche Nachteil der Methode der Lagrange’schen Multiplikatoren vermieden.
Mit der herkömmlichen Definition der dualen Formfunktionen können am Rand des Kontaktbereichs inkonsistente Mortar-Matrizen entstehen. Als Folge dessen resultieren unphysikalische Werte für die Knotenklaffung und fehlerhaft übertragene Kontaktkräfte. Zur Korrektur dieses Verhaltens wird in dieser Arbeit eine modifizierte Definition der Mortar-Matrizen vorgeschlagen. Damit die Modifikation nicht die Konditionierung des resultierenden Gleichungssystems verschlechtert, wird zusätzlich eine Wichtungsprozedur für die modifizierten Mortar-Matrizen vorgestellt. Als Ergebnis ist in allen Fällen eine konsistente Übertragung der Kontaktkraft und eine konsistente Berechnung der Normalklaffung möglich, ohne dabei die Konditionierung zu beeinträchtigen.
Die Betrachtungen zur zeitlichen Diskretisierung analysieren zunächst den Einfluss von Kontaktereignissen auf die Eigenschaften der dynamischen Strukturantwort. Beruhend auf den gewonnenen Erkenntnissen wird anschließend eine möglichst optimale zeitliche Kontaktdiskretisierung formuliert. Diese ist mit einer Strategie nach Kane u. a. (1999) energetisch stabil. Durch die Erweiterung einer Idee von Deuflhard u. a. (2008) auf Probleme mit großen Deformationen werden Oszillationen in der Kontaktkraft vermieden. Die Modifikation der Geschwindigkeit in einer Nachlaufrechnung stellt physikalisch sinnvolle Kontaktgeschwindigkeiten sicher. Darüber hinaus wird der Kontakt energieerhaltend modelliert, ohne die Nichtdurchdringungsbedingung zu verletzen. Hierzu kommt das Energie-Korrekturkraft-Verfahren zum Einsatz, das eine im Rahmen der vorliegenden Arbeit formulierte Weiterentwicklung des Konzepts von Armero und Petocz (1998) darstellt. Außer mit dem präsentierten Verfahren ist eine energieerhaltende Kontaktbehandlung bei gleichzeitiger exakter Erfüllung der Nichtdurchdringungsbedingung nur mit der „Velocity-Update-Method“ (Laursen und Love 2002) möglich. Im Gegensatz zu dieser gibt das Energie-Korrekturkraft-Verfahren die dissipierte Energie jedoch nicht ausschließlich in kinetischer Form zurück. Stattdessen bestimmt die Systemantwort, wie die Energie-Korrekturkraft die Gesamtenergie vergrößert.
Anhand von numerischen Experimenten werden abschließend die untersuchten Verfahren bewertet. Zusätzlich wird die Leistungsfähigkeit der entwickelten Methoden demonstriert.BibTeX
Tkachuk, A., & Bischoff, M. (2011). Buckling under Contact Constraints as a Source of Scatter in Car Crash Simulations. II International Conference on Computational Contact Mechanics 15-17 June 2011, Hannover, Germany.
BibTeX
Cichosz, T., & Bischoff, M. (2011). Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers.
Computer Methods in Applied Mechanics and Engineering,
200, 1317–1332.
https://doi.org/10.1016/j.cma.2010.11.004 Zusammenfassung
Computational modelling of contact problems raises two basic questions: Which method should be used to enforce the contact conditions and how should this method be discretised? The most popular enforcement methods are the Lagrange multiplier method, the penalty method and combinations of these two. A frequently used discretisation method is the so called node-to-segment approach. However, this approach might lead to problems like jumps in contact forces, loss of convergence or failure to pass the patch test. Thus in the last few years, several segment-to-segment contact algorithms based on the mortar method were proposed.
Combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations. On the other hand, a Lagrange multiplier mortar method requires additional unknowns. Hence, condensation of the Lagrange multipliers is desirable to preserve the initial size of the system of equations. This can be achieved by interpolating the Lagrange multipliers with so-called dual shape functions.
Discretising two contacting bodies leads to opposed contact surface representations of finite element edges, called slave and master elements, respectively. In current versions of dual Lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element (instead of the entire element) belongs to the contact area. We present a modified definition of the dual shape functions in such slave elements. The basic idea is to construct dual shape functions that fulfill the so-called biorthogonality condition within the contact area. This leads to consistent mortar matrices also in the boundary region. To avoid ill-conditioning of the stiffness matrix, the modified mortar matrices are weighted with appropriate weighting factors. In doing so, the corresponding modified Lagrange multiplier nodal values are of the same order as the unmodified ones. Various examples demonstrate the performance of the modified mortar contact algorithm.BibTeX
Tkachuk, A. (2010). A contact-stabilized Newmark method for coupled dynamical thermo-elastic problem. 3rd International Conference on Nonlinear Dynamic, September 21-24, 2010, Kharkov, Ukraine.
Zusammenfassung
A Lagrange multipliers formulation for dynamical frictionless thermo-elastic contact problem is considered. Thermal deformations and dependency of contact thermal resistance on contact pressure are assumed to be the only two coupling effects. Application of standard Newmark method to the problem may lead to spurious numerical oscillations of contact pressures and heat fluxes, inaccurate or divergent solutions. A modification of the Newmark method is proposed where contact contributions are integrated non-monolithically with backward Euler. Elimination of spurious numerical oscillations is shown in a numerical example.BibTeX
Hartmann, S., & Ramm, E. (2008). A mortar based contact formulation for non-linear dynamics using dual Lagrange multipliers.
Finite Elements in Analysis and Design,
44, 245–258.
https://doi.org/10.1016/j.finel.2007.11.018 Zusammenfassung
Many existing algorithms for the analysis of large deformation contact problems use a so-called node-to-segment approach to discretize the contact interface between dissimilar meshes. It is well known, that this discretization strategy may lead to problems like loss of convergence or jumps in the contact forces. Additionally it is popular to use penalty methods to satisfy the contact constraints. This necessitates a user defined penalty parameter the choice of which is somehow arbitrary, problem dependent and might influence the accuracy of the analysis.
In this work, a frictionless segment-to-segment contact formulation is presented that does not require any user defined parameter to handle the non-linearity of the contact conditions. The approach is based on the mortar method enforcing the compatibility condition along the contact interface in a weak integral sense. The application of dual spaces for the interpolation of the Lagrange multiplier allows for a nodal decoupling of the contact constraints. A local basis transformation in combination with a primal¿dual active set strategy enables the exact enforcement of the contact constraints via prescribed incremental boundary conditions. Due to the biorthogonality condition of the basis functions the Lagrange multipliers can be locally eliminated. A static condensation leads to a reduced system of equations to be solved solely for the unknown nodal displacements. Thus the size of the system of equations remains constant during the whole calculation. The discrete Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the analysis of dynamic contact problems the proposed contact description is combined with the implicit Generalized Energy-Momentum Method. Several numerical examples illustrate the performance of the suggested contact formulation.BibTeX
Hartmann, S., Brunssen, S., Wohlmuth, B., & Ramm, E. (2007). Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy.
International Journal for Numerical Methods in Engineering,
70, 883–912.
https://doi.org/10.1002/nme.1894 Zusammenfassung
The efficient modelling of three-dimensional contact problems is still a challenge in non-linear implicit structural analysis. We use a primal-dual active set strategy, based on dual Lagrange multipliers to handle the non-linearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to achieve a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way.
For the application to thin-walled structures we adapt a three-dimensional non-linear shell formulation including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface-oriented shell element, which allows the application of contact conditions directly to nodes lying on the contact surface. Shell typical locking phenomena are treated with the enhanced-assumed-strain-method and the assumed-natural-strain-method.
The discretization in time is done with the implicit Generalized-alpha method and the Generalized Energy-Momentum Method to compare the development of energies within a frictionless contact description.
In order to conserve the total energy within the discretized frictionless contact framework, we follow an approach from Laursen and Love, who introduced a discrete contact velocity to update the velocity field in a post-processing step.
Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.BibTeX
Hartmann, S. (2007).
Kontaktanalyse dünnwandiger Strukturen bei großen Deformationen. Doktorarbeit, Bericht Nr. 49, Institut für Baustatik und Baudynamik, Universität Stuttgart.
https://doi.org/10.18419/opus-263 Zusammenfassung
The present thesis is concerned with the numerical simulation of contact problems of thin-walled structures using the finite element method. A mortar-based contact formulation is presented and combined with suitable strategies for the discretization in space and time.
In view of a useful coupling with the element independent contact description, a trilinear surface oriented hybrid shell element is derived on the basis of the 7-parameter shell model by Büchter and Ramm (1992). Additionally, a trilinear geometric nonlinear brick element based on the principle of Hu-Washizu is devised. Numerical tests demonstrate the performance of both element formulations.
For the discretization in time, two implicit time integration algorithms are used. In addition to the existing "Generalized-α"-Method especially the "Generalized-Energy-Momentum-Method" is applied. The latter is proven to be unconditionally stable in all performed numerical analyses.
The essential part of this thesis is the extension of the mortar contact formulation presented by Hüeber and Wohlmuth (2005) to the geometrically nonlinear regime. Introducing continuously approximated Lagrange Multipliers, physically representing the contact pressure, the geometric impenetrability condition is formulated in a weak, integral sense. Using dual shape functions (Wohlmuth (2000)) for the interpolation of the Lagrange Multipliers allows for a nodal decoupling of the geometric constraints. The combination with an active set strategy results in an algorithm which allows for elimination of the discrete nodal values of the Lagrange Multipliers. They can be easily recovered from the displacements in a variational consistent way. In contrast to many other formulations, the resulting contact algorithm combines two main advantages: Only the discrete nodal displacements appear as primal unknowns, thereby the size of the system of equations to be solved remains constant; there is no need for any user defined paramters like a penalty parameter.
Detailed numerical analyses of dynamic contact problems illustrate the necessity of additional, algorithmic energy-conserving strategies. The "Velocity-Update"-method by Laursen and Love (2002) is characterized by the fact that it guarantees the exact conservation of energy while simultaneously satisfying the geometric impenetrability condition. This method is revised according to the presented contact formulation and generalized for combination with the "Generalized-Energy- Momentum-Method".
Numerical examples are investigated to analyze and judge the effectiveness of the proposed solution strategy.BibTeX
