A Virtual Element Method for Contact Modeling and Dynamics

Decreasing resources and limited energy results in a greater demand for virtual development processes and efficient product development. This trend points out the importance of digitalization and the subsequent need for efficient and accurate numerical prediction methods for product development. Due to their flexibility, numerical methods are gradually and steadily replacing physical tests in industrial product developments. The finite element method is perhaps the most well-known and widely used numerical method in industry and science. Increasing computer capabilities and further developments of these methods in recent years have increased the amount of application fields, including civil, automotive, naval, space and geo-technical engineering. However, along with complex geometries the spatial discretization of the domain emerges as a very time consuming step. Due to the fact that the classical finite element method is restricted to basic regular shaped element topologies, a more general choice of element shapes would give more flexibility. Within mesh-based methods, polygonal methods are a helpful alternative and showed great performance in engineering and science. However, most of these methods seem to need more computational effort and beside the aforementioned advantage of flexible element shapes, disadvantages appear as well. A relatively new method, the virtual element method, promises great numerical properties and can be seen as a generalization of the classical finite element method. All new methods need to be investigated for different applications in engineering and science before they can be applied commercially. This work deals with the application of the virtual element method to dynamic and elastoplastic material behavior. To deal with elastic and plastic incompressibility, a mixed virtual element formulation is presented as well. As a further development, the virtual element method is used to model three dimensional contact with different contact discretizations. A new projection algorithm is developed to manipulate the mesh at the contact interface, such that a very simple and efficient node-to-node contact formulation can be used. Various numerical examples for all aforementioned applications are performed, including benchmark problems such as the classical patch test. For comparison purposes, different finite element formulations are also adopted. As a final example, all models, including plasticity, dynamics and contact, are coupled to model mechanical impact.Eine Verringerung von Ressourcen und die damit einhergehende Energieknappheit f ¨uhren zu einem erh¨ohten Bedarf an virtuellen Entwicklungsprozessen und effizienter Produktentwicklung. Dieser Trend verdeutlicht die Bedeutung der Digitalisierung und den daraus resultierenden Bedarf an effizienten und hoch genauen numerischen Vorhersagemethoden f ¨ur die Produktentwicklung. Aufgrund ihrer Flexibilit¨at und mit steigenden Rechnerkapazit¨aten ersetzen numerische Methoden allm¨ahlich und stetig physikalische Tests in der industriellen Produktentwicklung. Die Finite Elemente Methode ist vielleicht die bekannteste und am weitesten verbreitete numerische Methode in Industrie und Wissenschaft. Durch die zunehmenden Rechnerkapazit ¨aten und die Weiterentwicklung dieser Methoden in den letzten Jahren hat sich die Zahl der Anwendungsbereiche vergr¨oßert. Numerische Methoden werden unter anderem im Bauwesen, im Automobilbau, in der Schifffahrt, in der Luft- und Raumfahrt und in der Geotechnik eingesetzt. Bei komplexen Geometrien erweist sich jedoch die r¨aumliche Diskretisierung des Gebiets als ein sehr zeitaufw¨andiger Prozess. Da die klassische Finite Elemente Methode auf einfache, regelm¨aßig geformte Elementgeometrien beschr¨ankt ist, w¨urde eine allgemeinere Auswahl von Elementgeometrien mehr Flexibilit¨at bieten. Innerhalb der netzbasierten Methoden sind polygonale Methoden eine hilfreiche Alternative und haben sich bereits in Industrie und Wissenschaft bew¨ahrt. Allerdings scheinen die meisten dieser Methoden einen h¨oheren Rechenaufwand zu erfordern, und neben dem bereits erw¨ahnten Vorteil der flexiblen Elementgeometrien treten auch gewisse Nachteile auf. Eine relativ neue Methode, die Virtuelle Elemente Methode, verspricht gute numerische Eigenschaften und kann als eine Verallgemeinerung der klassischen Finite Elemente Methode angesehen werden. Wie bei allen neuen Methoden m¨ussen auch hier verschiedene Anwendungen in der Industrie und Wissenschaft untersucht werden, bevor die Methode kommerziell eingesetzt werden kann. Diese Arbeit befasst sich mit der Anwendung der Methode der virtuellen Elemente auf dynamisches und elasto-plastisches Materialverhalten. Um elastische und plastische Inkompressibilit¨at zu behandeln, wird auch eine gemischte virtuelle Elementformulierung vorgestellt. In einem weiteren Schritt wird die Virtuelle Elemente Methode zur Modellierung dreidimensionaler Kontaktprobleme mit verschiedenen Kontaktdiskretisierungen verwendet. Es wird ein neuer Projektionsalgorithmus vorgestellt, welcher das Netz an der Kontaktschnittstelle so manipuliert, dass eine sehr einfache und effiziente Knoten-zu-Knoten Kontaktformulierung verwendet werden kann. Es werden verschiedene numerische Beispiele f ¨ur alle oben genannten Anwendungen behandelt, darunter auch Benchmark-Probleme wie der klassische Patch-Test. Um einen geeigneten Vergleich durchzuf¨uhren, werden die entwickelten Formulierungen mit verschiedene Finite Elemente Formulierungen verglichen. Als letztes Beispiel werden alle Modelle, einschließlich Plastizit¨at, Dynamik und Kontakt, gekoppelt, um einen mechanischen Stoß zu modellieren

Local Projection-Based Stabilized Mixed Finite Element Methods for Kirchhoff Plate Bending Problems

Based on stress-deflection variational formulation, we propose a family of local projection-based stabilized mixed finite element methods for Kirchhoff plate bending problems. According to the error equations, we obtain the error estimates of the approximation to stress tensor in energy norm. And by duality argument, error estimates of the approximation to deflection in H1-norm are achieved. Then we design an a posteriori error estimator which is closely related to the equilibrium equation, constitutive equation, and nonconformity of the finite element spaces. With the help of Zienkiewicz-Guzmán-Neilan element spaces, we prove the reliability of the a posteriori error estimator. And the efficiency of the a posteriori error estimator is proved by standard bubble function argument

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

Direct computation of asymptotic states for ratcheting prevention in elastoplasticity

This thesis summarizes three contributions to improve design accounting elastoplasticity under cyclic loads. (i) The development of a direct method to compute the asymptotic steady-state solution in ideal elastoplasticity. Validation examples show that the method is fast and accurate. The performance ranges from one to two orders of magnitude higher than incremental analysis. (ii) The performance allowed the development of a direct strategy to identify the structural ratchet-limit. The procedure applies to periodical loads and has no limitations on the number of applied loads. (iii) The upgrade of the asymptotic solution method with nonlinear kinematic hardening, which is required to model the Baushinger effect and material ratcheting. To the author knowledge, this is the first direct method for steadystate solution with this capability. Comparison with step-by-step solutions shows a increase in performance of one order of magnitude, at least.Esta tese descreve três contribuições relacioadas a estruturas elastoplasticas sob carregamento cíclico. (i) O desenvolvimento de um método direto para determinação de resposta assintótica em elastoplasticidade ideal. Exemplos de validação demonstram que o método é rápido e preciso. O incremento de desempenho registrado foi de uma a duas ordens de grandeza superior à integração incremental. (ii) O desempenho permitiu o desenvolvimento de uma estratégia direta para identificação do limite de ratcheting estrutural. O procedimento é aplicável para cargas cíclicas e não possui limitação no número de cargas aplicadas. (iii) A extensão do método de resposta assintótica contemplando encruamento não linear que é requerido para modelagem do efeito Baushinger e ratcheting material. De conhecimento do autor, este é o primeiro método direto com tal capacidade. A comparação com análise incremental demonstra um aumento de desempenho de ao menos uma ordem de grandeza

Finite element pressure stabilizations for incompressible flow problems

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis

Stabilized Multiscale Nonconforming Finite Element Method for the Stationary Navier-Stokes Equations

We consider a stabilized multiscale nonconforming finite element method for the two-dimensional stationary incompressible Navier-Stokes problem. This method is based on the enrichment of the standard polynomial space for the velocity component with multiscale function and the nonconforming lowest equal-order finite element pair. Stability and existence uniqueness of the numerical solution are established, optimal-order error estimates are also presented. Finally, some numerical results are presented to validate the performance of the proposed method

A mixed finite element method for solving coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term

This paper is concerned with the numerical approximation of the solution of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term using a mixed finite element method. The Raviart-Thomas mixed finite element method is one of the most prominent techniques to discretize the second-order wave equations; therefore, we apply this scheme for space discretization. Furthermore, an L2-in-space error estimate is presented for this mixed finite element approximation. Finally, the efficiency of the method is verified by a numerical example. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd

Two-Grid Method for Burgers’ Equation by a New Mixed Finite Element Scheme

In this article, we present two-grid stable mixed finite element method for the 2D Burgers’ equation approximated by the -P1 pair which satisfies the inf–sup condition. This method consists in dealing with the nonlinear system on a coarse mesh with width H and the linear system on a fine mesh with width h << H by using Crank–Nicolson time-discretization scheme. Our results show that if we choose H2 = h this method can achieve asymptotically optimal approximation. Error estimates are derived in detail. Finally, numerical experiments show the efficiency of our proposed method and justify the theoretical results

Investigation of the use of meshfree methods for haptic thermal management of design and simulation of MEMS

This thesis presents a novel approach of using haptic sensing technology combined with virtual environment (VE) for the thermal management of Micro-Electro-Mechanical-Systems (MEMS) design. The goal is to reduce the development cycle by avoiding the costly iterative prototyping procedure. In this regard, we use haptic feedback with virtua lprototyping along with an immersing environment. We also aim to improve the productivity and capability of the designer to better grasp the phenomena operating at the micro-scale level, as well as to augment computational steering through haptic channels. To validate the concept of haptic thermal management, we have implemented a demonstrator with a user friendly interface which allows to intuitively "feel" the temperature field through our concept of haptic texturing. The temperature field in a simple MEMS component is modeled using finite element methods (FEM) or finite difference method (FDM) and the user is able to feel thermal expansion using a combination of different haptic feedback. In haptic application, the force rendering loop needs to be updated at a frequency of 1Khz in order to maintain continuity in the user perception. When using FEM or FDM for our three-dimensional model, the computational cost increases rapidly as the mesh size is reduced to ensure accuracy. Hence, it constrains the complexity of the physical model to approximate temperature or stress field solution. It would also be difficult to generate or refine the mesh in real time for CAD process. In order to circumvent the limitations due to the use of conventional mesh-based techniques and to avoid the bothersome task of generating and refining the mesh, we investigate the potential of meshfree methods in the context of our haptic application. We review and compare the different meshfree formulations against FEM mesh based technique. We have implemented the different methods for benchmarking thermal conduction and elastic problems. The main work of this thesis is to determine the relevance of the meshfree option in terms of flexibility of design and computational charge for haptic physical model

Asynchronous Stabilisation and Assembly Techniques for Additive Multigrid

Multigrid solvers are among the best solvers in the world, but once applied in the real world there are issues they must overcome. Many multigrid phases exhibit low concurrency. Mesh and matrix assembly are challenging to parallelise and introduce algorithmic latency. Dynamically adaptive codes exacerbate these issues. Multigrid codes require the computation of a cascade of matrices and dynamic adaptivity means these matrices are recomputed throughout the solve. Existing methods to compute the matrices are expensive and delay the solve. Non- trivial material parameters further increase the cost of accurate equation integration. We propose to assemble all matrix equations as stencils in a delayed element-wise fashion. Early multigrid iterations use cheap geometric approximations and more accurate updated stencil integrations are computed in parallel with the multigrid cycles. New stencil integrations are evaluated lazily and asynchronously fed to the solver once they become available. They do not delay multigrid iterations. We deploy stencil integrations as parallel tasks that are picked up by cores that would otherwise be idle. Coarse grid solves in multiplicative multigrid also exhibit limited concurrency. Small coarse mesh sizes correspond to small computational workload and require costly synchronisation steps. This acts as a bottleneck and delays solver iterations. Additive multigrid avoids this restriction, but becomes unstable for non-trivial material parameters as additive coarse grid levels tend to overcorrect. This leads to oscillations. We propose a new additive variant, adAFAC-x, with a stabilisation parameter that damps coarse grid corrections to remove oscillations. Per-level we solve an additional equation that produces an auxiliary correction. The auxiliary correction can be computed additively to the rest of the solve and uses ideas similar to smoothed aggregation multigrid to anticipate overcorrections. Pipelining techniques allow adAFAC-x to be written using single-touch semantics on a dynamically adaptive mesh