A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established

Structure-preserving space-time discretization in a mixed framework for multi-field problems in large strain elasticity

The present work deals with the design of structure-preserving numerical methods in the field of nonlinear elastodynamics with an extension to multi-field problems. A new approach to the design of energy-momentum (EM) consistent time-stepping schemes for nonlinear elastodynamics is proposed. Moreover, we extend the formalism to multi-field problems

Discretisation techniques for large deformation computational contact elastodynamics

The present thesis deals with large deformation contact problems of flexible bodies in the field of nonlinear elastodynamics. Special emphasis will be placed on a consistent spatial and temporal discretization. For the spatial discretization of the underlying bodies, the finite element method will be used. For the contact boundaries the collocation type node-to-surface method as well as the variationally consistent Mortar method will be applied

A novel mixed and energy‐momentum consistent framework for coupled nonlinear thermo‐electro‐elastodynamics

A novel mixed framework and energy-momentum consistent integration scheme in the field of coupled nonlinear thermo-electro-elastodynamics is proposed. The mixed environment is primarily based on a framework for elastodynamics in the case of polyconvex strain energy functions. For this elastodynamic framework, the properties of the so-called tensor cross product are exploited to derive a mixed formulation via a Hu-Washizu type extension of the strain energy function. Afterwards, a general path to incorporate nonpotential problems for mixed formulations is demonstrated. To this end, the strong form of the mixed framework is derived and supplemented with the energy balance as well as Maxwell\u27s equations neglecting magnetic and time dependent effects. By additionally choosing an appropriate energy function, this procedure leads to a fully coupled thermo-electro-elastodynamic formulation which benefits from the properties of the underlying mixed framework. In addition, the proposed mixed framework facilitates the design of a new energy-momentum consistent time integration scheme by employing discrete derivatives in the sense of Gonzalez. A one-step integration scheme of second-order accuracy is obtained which is shown to be stable even for large time steps. Eventually, the performance of the novel formulation is demonstrated in several numerical examples

International Conference on Nonlinear Differential Equations and Applications

Dear Participants, Colleagues and Friends It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA). This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics. The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area

Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC

The present work addresses the design of structure-preserving numerical methods that emanate from the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism. Novel energy-momentum (EM) consistent time-stepping schemes in the realm of molecular dynamics are proposed. Moreover, the GENERIC-based structure-preserving numerical methods are extended to the context of large-strain thermoelasticity and thermo-viscoelasticity

Modelling Seismic Wave Propagation for Geophysical Imaging

International audienceThe Earth is an heterogeneous complex media from the mineral composition scale (10−6m) to the global scale ( 106m). The reconstruction of its structure is a quite challenging problem because sampling methodologies are mainly indirect as potential methods (Günther et al., 2006; Rücker et al., 2006), diffusive methods (Cognon, 1971; Druskin & Knizhnerman, 1988; Goldman & Stover, 1983; Hohmann, 1988; Kuo & Cho, 1980; Oristaglio & Hohmann, 1984) or propagation methods (Alterman & Karal, 1968; Bolt & Smith, 1976; Dablain, 1986; Kelly et al., 1976; Levander, 1988; Marfurt, 1984; Virieux, 1986). Seismic waves belong to the last category. We shall concentrate in this chapter on the forward problem which will be at the heart of any inverse problem for imaging the Earth. The forward problem is dedicated to the estimation of seismic wavefields when one knows the medium properties while the inverse problem is devoted to the estimation of medium properties from recorded seismic wavefields

Robuste Berechnungsverfahren zur nichtlinearen dynamischen Analyse von Balken- und Schalenstrukturen

Gegenwärtige und zukünftige dynamisch beanspruchte, schlanke Strukturen aus mehrschichtig verbundenen, hyperelastischen Werkstoffen, z. B. Windenergieanlagen und Hubschrauber usw., sind sehr komplex. Eine genaue Untersuchung im Zeitbereich erfordert den Einsatz von Methoden, die kinematische, geometrische sowie, bis zu einem gewissen Grad, materielle Nichtlinearitäten berücksichtigen sollten. Daher könnten Simulationen mit Beachtung von großen Verschiebungen, Drehungen und Verzerrungen nötig sein, um das mechanische Verhalten akkurat zu vorhersagen zu vermögen. Zunächst werden die Bewegungsgleichungen räumlich diskretisiert. Dann werden die zum Teil diskretisierten Gleichungen mittels eines Integrationsverfahrens zeitlich diskretisiert. Solche diskreten Gleichungen sind sehr steif, sodass sich die Berechnung der langzeitigen Lösung erschwert. Darüber hinaus ist die Einführung von Nebenbedingungen oft nötig, um komplexere Strukturen aufstellen zu können, wodurch sich die Komplexität erhöht wird und unerwünschte Eigenschaften noch verschärft werden. Um Robustheit zu gewinnen, sollen Berechnungsverfahren hergeleitet werden, die die zugrunde legende Physik in gewissem Maße erhalten können und gleichzeitig den hochfrequenten Anteil der Lösung unterdrücken können. Die Erfüllung dieser Anforderungen stellt sich als sehr herausfordernd dar. Das Hauptziel dieser Arbeit liegt an der Entwicklung von Berechnungsverfahren zur Vertiefung des Verständnises des dynamischen Verhaltens von Balken- und Schalenstrukturen. Um dieses Ziel zu erreichen, wird ein umfassender Ansatz vorgeschlagen. Dieser besteht aus: i) Einer auf Direktoren basierenden, Finite-Elemente-Formulierung für den geometrisch exakten Balken mit allgemeinen Querschnittseigenschaften; ii) einer auf Direktoren basierenden, Finite-Elemente-Formulierung für die Kontinuumsmechanik-basierte Schale aus mehrschichtig verbundenen, hyperelastischen Werkstoffen; iii) einer vereinheitlichten Beschreibung von Starrkörpern, Balken und Schalen und deren Kopplung mittels kinematischer Nebenbedingungen; und, iv) einem robusten Integrationsverfahren basierend auf dem gemittelten Vektorfeld. Des Weiteren wird Folgendes ebenfalls vorgeschlagen: v) Die Partikularisierung der Hauptgeodätenanalyse zur nichtlinearen Identifikation von Bewegungsmoden an Balkenstrukturen; und, vi) ein neues konservatives/dissipatives Integrationsverfahren für allgemeine nichtlineare mechanische Systeme basierend auf optimierten Modifizierungen höherer Ordnung, die die Defizite der Mittelpunktsregel beheben. Die sehr gute Leistung des vorgeschlagenen Ansatzes wird durch mehrere Beispiele unterschiedlicher Komplexität nachgewiesen.Existing and new slender structures made of hyperelastic multilayer composite materials subject to highly dynamic loads, e.g., wind turbines, helicopters, cars, speedboats or submarines inter alia, are very complex. Their dynamic analysis requires fully nonlinear formulations, at least from the kinematic and geometric point of view, and also to some extent from the material point of view. Thus, simulations in time-domain involving large displacements, rotations and strains could be necessary to predict their mechanical behavior accurately. Numerical procedures to carry out such simulations rely firstly on the partial discretization in space of the governing equations, for instance with finite elements. These semi discrete equations are further discretized in time with an integration scheme. The resulting discrete equations are in fact very stiff and therefore, the computation of the long-term behavior could be problematic. In many applications, the introduction of constraints is also necessary for rendering more complex structures. Besides introducing a new level of complexity, this can sharpen conditioning problems already present in the fully discrete problem. Additionally, we also require procedures able to annihilate the unwanted unresolved high-frequency content without upsetting of the underlying physics. However, the simultaneous satisfaction of all these requirements is a very challenging task. The main objective of this work is to provide means intended for helping to understand further the nonlinear dynamics of beam and shell structures made of hyperelastic multilayer composite materials subject to highly dynamic loads. To accomplish this main goal, we propose a unifying computational approach that relies on: i) a director-based finite-element formulation for geometrically exact beams with general cross-section properties; ii) a director-based finite-element formulation for solid-degenerate shells made of hyperelastic multilayer composite materials; iii) a unifying description of rigid bodies, geometrically exact beams and solid-degenerate shells and their combination with kinematic pairs, which avoids inherently the necessity of rotational degrees of freedom; and, iv) a robust integration scheme based on the average vector field. Additionally, we propose: v) the particularization of the principal geodesic analysis to identify motion patters exhibited by beam structures in a purely nonlinear setting; and, vi) a new conservative/dissipative integration method for general nonlinear mechanical systems, which relies on high-order correction terms that optimally modify the midpoint rule. Moreover, the excellent numerical performance of the proposed unifying framework and procedures is illustrated by means of a good number of examples with different difficulty levels