TY - CHAP A1 - Wolff, Sebastian ED - Gürlebeck, Klaus ED - Könke, Carsten T1 - NODALLY INTEGRATED FINITE ELEMENTS N2 - Nodal integration of finite elements has been investigated recently. Compared with full integration it shows better convergence when applied to incompressible media, allows easier remeshing and highly reduces the number of material evaluation points thus improving efficiency. Furthermore, understanding it may help to create new integration schemes in meshless methods as well. The new integration technique requires a nodally averaged deformation gradient. For the tetrahedral element it is possible to formulate a nodal strain which passes the patch test. On the downside, it introduces non-physical low energy modes. Most of these "spurious modes" are local deformation maps of neighbouring elements. Present stabilization schemes rely on adding a stabilizing potential to the strain energy. The stabilization is discussed within this article. Its drawbacks are easily identified within numerical experiments: Nonlinear material laws are not well represented. Plastic strains may often be underestimated. Geometrically nonlinear stabilization greatly reduces computational efficiency. The article reinterpretes nodal integration in terms of imposing a nonconforming C0-continuous strain field on the structure. By doing so, the origins of the spurious modes are discussed and two methods are presented that solve this problem. First, a geometric constraint is formulated and solved using a mixed formulation of Hu-Washizu type. This assumption leads to a consistent representation of the strain energy while eliminating spurious modes. The solution is exact, but only of theoretical interest since it produces global support. Second, an integration scheme is presented that approximates the stabilization criterion. The latter leads to a highly efficient scheme. It can even be extended to other finite element types such as hexahedrals. Numerical efficiency, convergence behaviour and stability of the new method is validated using linear tetrahedral and hexahedral elements. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Architektur KW - Computerunterstütztes Verfahren KW - Computer Science Models in Engineering; Multiscale and Multiphysical Models; Scientific Computing Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-29028 UR - http://euklid.bauing.uni-weimar.de/ikm2009/paper.html SN - 1611-4086 ER - TY - THES A1 - Wolff, Sebastian T1 - Implementation und Test eines Optimierungsverfahrens zur Loesung nichtlinearer Gleichungen der Strukturmechanik T1 - Implementation and Test of an Optimization Method Solving Nonlinear Equations of Structural Mechanics N2 - In displacement oriented methods of structural mechanics may static and dynamic equilibrium conditions lead to large coupled nonlinear systems of equations. In many cases they are solved iteratively utilizing derivatives of Newton's method. Alternatively, the equations may be expressed in terms of the Karush-Kuhn-Tucker conditions of an optimization problem and, therefore, may be solved using methods of mathematical programming. To begin with, the work deals with the fundamentals of the formulation as optimization problem. In particular, the requirements of material nonlinearity and contact situations are analyzed. Proximately, an algorithm is implemented which utilizes the usually sparse structure of the Hessian matrix, whereby particularly the convergence behaviour is analyzed and adjusted. The implementation was tested using examples from statics and dynamics of large systems. The results are verified considering the accuracy comparing alternative solutions (e.g. explicit methods). The potential areas of application is shown and the efficiency of the method is evaluated. N2 - In weggroeßenorientierten Verfahren der Strukturmechanik fuehren die statischen oder dynamischen Gleichgewichtsbedingungen auf große gekoppelte nichtlineare Gleichungssysteme. In vielen Faellen werden diese Gleichungssysteme iterativ auf der Grundlage des Newton'schen Naeherungsverfahrens gelost. Die Gleichungen koennen alternativ als Karush-Kuhn-Tucker-Bedingungen eines Optimierungsproblems aufgefasst, und daher mit Verfahren der mathematischen Optimierung geloest werden. Die Arbeit beschaeftigt sich zunaechst mit den Grundlagen der Problemformulierung als Optimierungsaufgabe, um dabei speziell die Anforderungen aus Werkstoffnichtlinearitaet und Kontakt untersuchen. In weiterer Folge ist ein Optimierungsalgorithmus innerhalb der Softwareumgebung SLang zu implementieren, der die in der Strukturmechanik typische schwach besetzte Struktur der Hessematrix ausnutzt. Dabei ist insbesondere das Konvergenzverhalten des Algorithmus zu untersuchen und moeglichst gut einzustellen. Die Implementation soll anhand von Beispielen aus der Statik und Dynamik großer Systeme getestet und die Resultate hinsichtlich ihrer Genauigkeit anhand von Alternativloesungen (z.B. aus expliziten Verfahren) verifiziert werden. Die potenziellen Anwendungsgebiete des entwickelten Algorithmus sind aufzuzeigen, und die Effizienz des Verfahrens ist zu bewerten. KW - Nichtlineare Optimierung KW - Nichtlineare Finite-Elemente-Methode KW - Kontaktmechanik KW - Kontaktkraft KW - Dynamik Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20111215-7271 N1 - Der Volltext-Zugang wurde im Zusammenhang mit der Klärung urheberrechtlicher Fragen mit sofortiger Wirkung gesperrt. ER -