Academic literature on the topic 'Hybrid variational formulation'

Abstract. A hybrid variational ensemble data assimilation has been developed on top of the HIRLAM variational data assimilation. It provides the possibility of applying a flow-dependent background error covariance model during the data assimilation at the same time as full rank characteristics of the variational data assimilation are preserved. The hybrid formulation is based on an augmentation of the assimilation control variable with localised weights to be assigned to a set of ensemble member perturbations (deviations from the ensemble mean). The flow-dependency of the hybrid assimilation is demonstrated in single simulated observation impact studies and the improved performance of the hybrid assimilation in comparison with pure 3-dimensional variational as well as pure ensemble assimilation is also proven in real observation assimilation experiments. The performance of the hybrid assimilation is comparable to the performance of the 4-dimensional variational data assimilation. The sensitivity to various parameters of the hybrid assimilation scheme and the sensitivity to the applied ensemble generation techniques are also examined. In particular, the inclusion of ensemble perturbations with a lagged validity time has been examined with encouraging results.

A scattering process can be a natural process or a process carried out in a laboratory. The scattering of particles from targets has resulted in important discoveries in physics. We discuss various scattering theories of electrons and positrons and their applications to elastic scattering, resonances, photoabsorption, excitation, and solar and stellar atmospheres. Among the most commonly employed approaches are the Kohn variational principle, close-coupling approximation, method of polarized orbitals, R-matrix formulation, and hybrid theory. In every formulation, an attempt is made to include exchange, long-range and short-range correlations, and to make the approach variationally correct. The present formulation, namely, hybrid theory, which is discussed in greater detail compared to other approximations, includes exchange, long-range correlations, and short-range correlations at the same time, and is variationally correct. It was applied to calculate the phase shifts for elastic scattering, the resonance parameters of two-electron systems, photoabsorption in two-electron systems, excitation of atomic hydrogen by an electron and positron impact, and to study the opacity of the Sun’s atmosphere. Calculations of polarizabilities, Rydberg states, and bound states of atoms are also discussed.

Elastic large deformation analysis based on the hybrid natural element method (HNEM) is presented in this paper. The natural neighbor interpolation is adopted to construct the shape functions for the HNEM. The incremental formulation of Hellinger–Reissner variational principle is used to derive discrete system of incremental equations under the total Lagrangian formulation. And the Newton-Raphson iteration is applied to solve these incremental equations. Compared with the natural element method (NEM), the HNEM can directly obtain nodal stresses of higher precision, which will bring advantage in the iteration process and improve computational efficiency in solving elastic large deformation problems. Some numerical examples demonstrate the validity of the HNEM for elastic large deformation problems.

Abstract Recent numerical weather prediction systems have significantly improved medium-range forecasts by implementing hybrid background error covariance, for which climatological (static) and ensemble-based (flow-dependent) error covariance are combined. While the hybrid approach has been investigated mainly in variational systems, this study aims at exploring methods for implementing the hybrid approach for the local ensemble transform Kalman filter (LETKF). Following Kretschmer et al., the present study constructed hybrid background error covariance by adding collections of climatological perturbations to the forecast ensemble. In addition, this study proposes a new localization method that attenuates the ensemble perturbation (Z-localization) instead of inflating observation error variance (R-localization). A series of experiments with a simplified global atmospheric model revealed that the hybrid LETKF resulted in smaller forecast errors than the LETKF, especially in sparsely observed regions. Due to the larger ensemble enabled by the hybrid approach, optimal localization length scales for the hybrid LETKF were larger than those for the LETKF. With the LETKF, the Z-localization resulted in similar forecast errors as the R-localization. However, Z-localization has an advantage in enabling us to apply different localization scales for flow-dependent perturbation and climatological static perturbations with the hybrid LETKF. The optimal localization for climatological perturbations was slightly larger than that for flow-dependent perturbations. This study also proposes optimal eigendecomposition (OED) ETKF formulation to reduce computational costs. The computational expense of the OED ETKF formulation became significantly smaller than that of standard ETKF formulations as the number of climatological perturbations was increased beyond a few hundred.

Abstract A hybrid formulation of the 2.5D elastodynamic scattering problem combining the finite-element method and boundary integral equation method is presented and validated. The formulation of the 2.5D boundary integral equation method that is used was presented in detail by Papageorgiou and Pei (1998) and is an extension of the discrete wavenumber boundary integral equation method originally proposed by Kawase (1988) for 2D scattering problems. Modeling of the wave field in the domain of the scatterer is based on the variational principle of virtual displacements, and discretization of the domain is accomplished using the finite-element method. The formulation may be used to study the wave field in models of sedimentary deposits (e.g., valleys) or topography (e.g., canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The hybrid method exploits the versatility of the finite element method for modeling the scatterer and the effectiveness of the boundary integral equation method for taking care of the radiation condition in the half-space. The advantage of the 2.5D formulation is that it provides the means for calculations of 3D wave fields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations.

Dans cette thèse, une nouvelle approche est développée pour les problèmes de viscoplasticité et de dynamique non linéaire. En particulier, les équations variationnelles sont élaborées selon le principe de Helligner-Reissner, de sorte que les champs de contrainte et de déplacement apparaissent comme des champs inconnus sous la forme faible. Trois nouveaux éléments finis sont développés. Le premier élément fini est formulé pour le problème axisymétrique, dans lequel le champ de contraintes est approximé par des polynômes d’ordre inférieur tels que des fonctions linéaires. Cette approche donne des solutions précises spécifiquement dans les problèmes incompressibles et rigides. De plus, un élément fini de flexion de membrane et de plaque est nouvellement conçu en discrétisant le champ de contraintes en utilisant l’espace vectoriel de Raviart-Thomas d’ordre le plus bas RT0. Cette approche garantit la continuité du champ de contraintes sur tout un domaine discret, ce qui est un avantage significatif dans la méthode numérique, notamment pour les problèmes de propagation des ondes. Les développements sont effectués pour le comportement constitutif visco-plastique des matériaux, où les équations d’évolution correspondantes sont obtenues en faisant appel au principe de dissipation maximale. Pour résoudre les équations d’équilibre dynamique, des schémas de conservation et de décroissance de l’énergie sont formulés en conséquence. Le schéma de conservation de l’énergie est inconditionnellement stable, car il peut préserver l’énergie totale d’un système donné sous une vibration libre, tandis que le schéma décroissant peut dissiper des modes de vibration à plus haute fréquence. La dernière partie de cette thèse présente les procédures d’upscaling du comportement des matériaux visco-plastiques. Plus précisément, la mise à l’échelle est effectuée par une méthode d’identification stochastique via une mise à jour baysienne en utilisant le filtre de Gauss-Markov-Kalman pour l’assimilation des propriétés importantes des matériaux dans les régimes élastique et inélastique
In this thesis, a novel approach is developed for visco-plasticity and nonlinear dynamics problems. In particular, variational equations are elaborated following the Helligner-Reissner principle, so that both stress and displacement fields appear as unknown fields in the weak form. Three novel finite elements are developed. The first finite element is formulated for the axisymmetric problem, in which the stress field is approximated by low-order polynomials such as linear functions. This approach yields accurate solutions specifically in incompressible and stiff problems. In addition, a membrane and plate bending finite element are newly designed by discretizing the stress field using the lowest order Raviart-Thomas vector space RT0. This approach guarantees the continuity of the stress field over an entire discrete domain, which is a significant advantage in the numerical method, especially for the wave propagation problems. The developments are carried out for the viscoplastic constitutive behavior of materials, where the corresponding evolution equations are obtained by appealing to the principle of maximum dissipation. To solve the dynamic equilibrium equations, energy conserving and decaying schemes are formulated correspondingly. The energy conserving scheme is unconditional stable, since it can preserve the total energy of a given system under a free vibration, while the decaying scheme can dissipate higher frequency vibration modes. The last part of this thesis presents procedures for upscaling of the visco-plastic material behavior. Specifically, the upscaling is performed by stochastic identification method via Baysian updating using the Gauss-Markov-Kalman filter for assimilation of important material properties in the elastic and inelastic regimes

Dans ces travaux de thèse on s’intéresse au développement d’un outil numérique pour résoudre le problème de conduction instationnaire avec changement de phase dans un milieu composite constitué d’une mousse de graphite infiltrée par un matériau à changement de phase tel que le sel, dans le contexte du stockage de l’énergie thermique solaire.Au chapitre 1, on commence par présenter le modèle sur lequel on va travailler. Il estséparé en trois sous-parties : un problème de conduction de chaleur dans la mousse, un problème de changement de phase dans les pores remplis de sel et une condition de résistance thermique de contact entre les deux matériaux qui est traduite par une discontinuité du champ de température.Au chapitre 2, on étudie le problème stationnaire de conduction thermique dans un milieu composite avec résistance de contact. Ceci permet de se focaliser sur la plus grande difficulté présente dans le problème qui est le traitement de la condition de saut à l’interface.Deux méthodes d’éléments finis sont proposées pour résoudre ce problème : une méthode basée sur les éléments finis Lagrange P1 et une méthode hybride-duale utilisant les éléments finis Raviart-Thomas d’ordre 0 et P0. L’analyse numérique des deux méthodes est effectuée et les résultats de tests numériques attestent des efficacités des deux méthodes [10]. Les matériaux à changement de phase qu’on étudie dans le cadre de cette thèse sont des matériaux pures, par conséquent le changement de phase s’effectue en une valeur de température fixe qui est la température de fusion. Ceci est modélisé par un saut dans la fonction fraction liquide et par conséquent dans la fonction enthalpie du matériau. Cette discontinuité représente une difficulté numérique supplémentaire qu’on propose de surmonter en introduisant un intervalle de régularisation autour de la température de fusion.Cette procédure est présentée dans le chapitre 3 où une étude analytique et numérique montre que l’erreur sur la température se comporte comme " en dehors de la zone de mélange, où " est la largeur de l’intervalle de régularisation. Cependant, à l’intérieur l’erreur se comporte comme p " et on montre que cette estimation est optimale. Cette diminution de vitesse de convergence est due à l’énergie qui reste bloquée dans la zone de mélange [58].Dans le chapitre 4 on présente quatre des schémas les plus utilisés pour le traitement de la non-linearité due au changement de phase: mise à jour du terme source, linéarisation de l’enthalpie, la capacité thermique apparente et le schéma de Chernoff. Différents tests numériques sont réalisés afin de tester et comparer ces quatre méthodes pour différents types de problèmes. Les résultats montrent que le schéma de linéarisation de l’enthalpie est le plus précis à chaque pas de temps tans dis que le schéma de la capacité thermique apparente donne de meilleurs résultats au bout d’un certain temps de calcul. Cela indique que si l’on s’intéresse aux états transitoires du matériaux le premier schéma est lemeilleur choix. Cependant, si l’on s’intéresse au comportement thermique asymptotique du matériau le second schéma est plus adapté. Les résultats montrent également que le schéma de Chernoff est le plus rapide parmi les quatre schémas en terme de temps de calcul et donne des résultats comparables à ceux des deux plus précis.Enfin, dans le chapitre 5 on utilise le schéma de Chernoff avec la méthode d’éléments finis hybride-duale Raviart-Thomas d’ordre 0 et P0 pour résoudre le problème non-linéaire de conduction thermique dans un milieu composite réel avec matériau à changement de phase. Le but étant de déterminer si un matériau composite avec une distribution uniforme de pores est assimilable à un matériau à changement de phase homogènes avec des propriétés thermo-physiques équivalentes. Pour toutes les expériences numériques exposées dans ce manuscrit on a utilisé le logiciel libre d’éléments finis FreeFem++ [41]
In this thesis we aim to develop a numerical tool that allow to solve the unsteady heatconduction problem in a composite media with a graphite foam matrix infiltrated witha phase change material such as salt, in the framework of latent heat thermal energystorage.In chapter 1, we start by explaining the model that we are studying which is separated in three sub-parts : a heat conduction problem in the foam, a phase change problem in the pores of the foam which are filled with salt and a contact resistance condition at the interface between both materials which results in a jump in the temperature field.In chapter 2, we study the steady heat conduction problem in a composite media withcontact resistance. This allow to focus on the main difficulty here which is the treatment of the thermal contact resistance at the interface between the carbon foam and the salt. Two Finite element methods are proposed in order to solve this problem : a finite element method based on Lagrange P1 and a hybrid dual finite element method using the lowest order Raviart-Thomas elements for the heat flux and P0 for the temperature. The numerical analysis of both methods is conducted and numerical examples are given to assert the analytic results. The work presented in this chapter has been published in the Journal of Scientific Computing [10].The phase change materials that we study here are mainly pure materials and as a consequence the change in phase occurs at a single point, the melting temperature. This introduces a jump in the liquid fraction and consequently in the enthalpy. This discontinuity represents an additional numerical difficulty that we propose to overcome by introducing a smoothing interval around the melting temperature. This is explained in chapter 3 where an analytical and numerical study shows that the error on the temperature behaves like " outside of the mushy zone, where _ is the width of the smoothing interval. However, inside the error behaves like p " and we prove that this estimation is optimal due to the energy trapped in the mushy zone. This chapter has been published in Communications in Mathematical Sciences [58].The next step is to determine a suitable time discretization scheme that allow to handle the non-linearity introduced by the phase change. For this purpose we present in chapter 4 four of the most used numerical schemes to solve the non-linear phase change problem : the update source method, the enthalpy linearization method, the apparent heat capacity method and the Chernoff method. Various numerical tests are conducted in order to test and compare these methods for various types of problems. Results show that the enthalpy linearization is the most accurate at each time step while the apparent heat capacity gives better results after a given time. This indicates that if we are interestedin the transitory states the first scheme is the best choice. However, if we are interested in the asymptotic thermal behavior of the material the second scheme is better. Results also show that the Chernoff scheme is the fastest in term of calculation time and gives comparable results to the one given by the first two methods.Finally, in chapter 5 we use the Chernoff method combined with the hybrid-dual finiteelement method with P0 and the lowest order Raviart-Thomas elements to solve thenon-linear heat conduction problem in a realistic composite media with a phase change material. Numerical simulations are realised using 2D-cuts of X-ray images of two real graphite matrix foams infiltrated with a salt. The aim of these simulations is to determine if the studied composite materials could be assimilated to an equivalent homogeneous phase change material with equivalent thermo-physical properties. For all simulationsconducted in this work we used the free finite element software FreeFem++ [41]

Le travail que nous présentons dans cette thèse repose sur l'étude mathématique et numérique de formulations mixtes et hybrides en magnétostatique tridimensionnelle, discrétisées par des éléments finis appropriés. Ce travail est fait dans le cadre de matériaux isotropes où l'on n'a besoin d'aucune condition au bord de la région magnétique, le problème extérieur étant résolu par une méthode intégrale. Le premier chapitre est consacré aux aspects théoriques d'une première formulation variationnelle mixte en champ magnétique et potentiel vecteur. L'existence et l'unicité de la solution de cette formulation y sont montrées. Après discrétisation, le système matriciel obtenu est alors résolu par la méthode d'Uzawa. Le second chapitre présente une méthode de pénalisation du système matriciel et de bons résultats ont été obtenus pour le Problème 13 du Team Work Shop. Nous avons également comparé cette technique de pénalisation du système matriciel avec celle du problème continu et nous avons conclu que notre technique est plus avantageuse que cette dernière. Le troisième chapitre expose la technique d'arbre en facettes, technique implémentée pour assurer l'unicité du potentiel vecteur. Le quatrième chapitre introduit les multiplicateurs de Lagrange sur les interfaces afin de relaxe les deux contraintes de continuité de la composante tangentielle du champ magnétique et de la composante normale de l'induction. Deux méthodes de résolution sont utilisées: la méthode d'Uzawa et la méthode de condensation statique. De bons résultats numériques sont obtenus. Finalement, une annexe présente un préconditionnement polynômial de la matrice de masse du système matriciel et un code est implémentée pour un domaine à perméabilité constante
We present a mathematical and a numerical study of mixed and hybrid formulations for magnetostatics 3D in an unbounded domain, discretised with well-chosen finite elements. This study deals with isotrope materials without any condition on the boundary of the magnetic domain, the external problem is solved by an integral method. The first chapter is devoted to the theoritical sights of a first mixed formulation where the magnetic field and the vector potentiel are the unknowns. The existence and the unicity of the solution are established. After the discretization, the matrix system is solved by Uzawa's method. The second chapter presents a perturbation method of the linear system. Good results are obtained for the Problem 13 of the Team Work Shop. We also compared these results with those obtained when the continued problem is perturbed. We, then, deduce that our method is more advantageous than the latter. The third chapter sets out the tree-cotree technic in order to impose the unicity of the vector potential. The fourth chapter introduces the lagrange multipliers used to relax continuity constraints: the continuity of the tangential component of magnetic field and the continuity of the normal component of the induction. Linear systems obtained have block diagonal matrix. Uzawa's method and static condensation method are used to solve these systems. Good results with good accuracy are obtained. Finally, an appendix relates the purpose of a polynomial preconditioning and a code is implemented for a domain having a constant permeability

Dans une 1ere partie, on construit des formulations variationnelles associées aux équations de Maxwell résonantes. Les équations dégénèrent dans le domaine, entraînant la non-unicité et la singularité des solutions. L’ajout de viscosité permet de les désingulariser, et par un procédé d’absorption limite, lorsque ce paramètre de viscosité tend vers zéro, on identifie la solution physique. Mais la dégénérescence sépare le problème à la limite en deux équations sur des domaines différents couplées par leur interface, le long de laquelle les solutions explosent. Ce travail caractérise la solution limite de manière explicite comme solution d’une formulation bien posée, ce qui permet d’approcher numériquement la solution physique des équations de Maxwell résonantes. L’étude est motivée par la modélisation de résonances hybrides dans un plasma de fusion. Une 2nde partie concerne les méthodes numériques de décomposition de domaine (DDM). En présence de coins et de points de croisement, lorsqu’on utilise un mailleur automatique par exemple, il est nécessaire de traiter ces points pour obtenir des conditions d’absorption (ABC) ou de transmission (TC) d’ordre supérieur à 1. Nous définissons des ABC d’ordre 2 pour l’équation de Helmholtz sur un domaine à coins, avec en vue des TC traitant les points de croisement. Chaque algorithme présenté est lié à une énergie décroissante et converge
In 1st part, variational formulations associated with resonant Maxwell equations are constructed. The equations degenerate in the domain, leading to the non-unicity and singularity of the solutions. Adding viscosity desingularizes the equations, and a limiting absorption process, when this viscosity parameter goes to zero, allows to identify the physical solution. The degeneracy separates the problem at the limit into two equations on different domains coupled by their interface, along which the solutions blow up. This work explicitly characterizes the limit solution as a solution of a well-posed formulation, which allows the numerical approximation of the physical solution to the resonant Maxwell equations. The study is motivated by the modeling of hybrid resonances in fusion plasma. A 2nd part concerns numerical domain decomposition methods (DDM). In the presence of corners and cross points, when using an automatic mesher for example, it is necessary to treat these points to obtain absorption (ABC) or transmission (TC) conditions of order higher than 1. We define ABCs of order 2 for the Helmholtz equation on a polygonal domain, with the further intention of deriving TCs treating cross points. Each algorithm presented is endowed with a decreasing energy and is convergent

Chapter 6 summarizes the main contributions of the book, as well as topics for future research. From an empirical point of view, the book has shown that four distinct macro-typologies of verb movement can be identified which can be predicted on the basis of independent morphological properties of the languages under investigation, thus casting new light on the long-debated issue of the interplay between ‘rich’ morphology and verb movement. From a methodological point of view, the volume has shown the importance of formulating analyses which are not language-specific but have a wider empirical basis. From a theoretical perspective, it has underlined the advantages of using a hybrid minimalist-cartographic framework. Questions for future research include whether it is possible to extend and adapt the present approach to other language families, such as Germanic, and investigating whether the proposed approach makes the correct predictions once diachronic variation too is taken into account.

Welfare-to-work services have been a key area of experimentation in quasi-marketised public service delivery. This chapter develops and extends a conceptual framework for unpacking variation in the formulation of quasi-markets and finds that there are important differences in the type of market adopted by two British employment support schemes: Work Programme and Work Choice. A novel quasi-experimental analysis is then used to investigate the implications of the alternate market formulations for those with health conditions and disabilities, by comparing the employment and earning outcomes for a matched group of participants on the two schemes. The findings suggest that the promises of innovation and performance improvement allied to the provider-directed Work Programme are not met. Employment and earnings outcomes are significantly and sizeably lower for the Work Programme than for Work Choice. The hybrid market position of Work Choice – which leans towards a provider-directed arrangement but retains important levers for both the state- and user- preferences – emerges as an important mediator of programme participant experiences.

Abstract In the present investigation the incremental Hellinger-Reissner variational principle, a hybrid-strain based formulation and an updated Lagrangian formulation are adopted to derive explicit expressions for element stiffness matrices of various flat triangular shell finite elements. These elements are developed for application to the analysis of thin and thick shell structures undergoing large geometrically nonlinear deformation at finite strain. Correct representations of finite rotations and the specifically chosen strain field maintain appropriate rigid body motions and prevent shear locking phenomena. Consideration of thickness updating and a finite strain formulation relaxes any plane stress assumptions and enables the analyst to deal with three-dimensional constitutive models. Two numerical examples are presented to demonstrate the performance of the elements.

Abstract An assumed-stress hybrid 4-node plate element is developed based on the Hellinger-Reissner variational principle modified with a generalized least-squares operator for accurate vibration and wave propagation response of Reissner-Mindlin plates. The least-squares operator is proportional to a weighted integral of a differential operator acting on the residual of the steady-state equations of motion for Reissner-Mindlin plates. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent framework for enhancing the accuracy of mixed Reissner-Mindlin plate elements that have no shear locking or spurious modes. Improved methods are designed such that the complex wave-number finite element dispersion relations closely match the analytical relations for all wave angle directions. For uniform meshes, optimal methods are designed to achieve zero dispersion error along given wave directions. Comparisons of finite element dispersion relations demonstrate the superiority of the new hybrid least-squares plate element over the underlying hybrid element, and standard Galerkin elements based on selectively reduced integration. Numerical experiments validate these conclusions.

Responses of geometrically nonlinear shell structures under combined conservative and non-conservative loads are investigated and presented in this paper. The shell structures are discretized by the finite element method and represented by the hybrid strain based three node flat triangular shell elements that were developed previously by the authors. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are employed. Features such as large or small strain deformation, finite rotation, updated thickness so as to account for the “thinning effect” due to large strain deformation, and inclusion or exclusion of the mid-surface director field are incorporated in the finite element formulation. Representative results of two examples are included to demonstrate the capability, accuracy and efficiency of the computational strategy proposed.

Abstract Mechanisms are inherently constrained devices. Combining flexibility with mechanisms usually requires using Lagrange multipliers to handle the constraints. The added algebraic or numerical tedium, associated with the Lagrange multipliers, is well documented. Presented in this paper is a technique for obtaining the minimal set of hybrid parameter differential equations for a constrained device. That is, the set of equations that inherently incorporate the constraints. The technique illustrated in this paper is a recently developed hybrid parameter multiple body (HPMB) system modeling methodology. The variational nature of the methodology allows rigorous equation formulation providing not only the complete nonlinear, hybrid differential equations, but also the boundary conditions. The methodology is formulated in the constraint-free subspace of the system’s configuration space, thus Lagrange multipliers are not needed for constrained systems, regardless of the constraint type (holonomic or nonholonomic). To evince the utility of the method, a flexible four bar mechanism is modeled. Particularly, the inversion of the slider crank found in the quick return mechanism. A comparison of Hamilton’s principle and the described technique, as they are applied to the mechanism, is included. It is shown that the same equations result from either method, but the new technique is much more concise, more efficiently handles the constraints, and requires less algebraic tedium.

The investigation reported in this paper is concerned with the development of efficient, conceptually simpler, mathematically rigorous and physically meaningful three-dimensional (3D) finite elements for solid modelling. A mixed variational principle based on the hybrid strain formulation has been adopted for the derivation of element stiffness matrices of two lower order tetrahedral finite elements. Explicit expressions for element matrices have been derived with a combination of hand manipulation and computer algebraic package, MAPLE. Each of the two finite elements has four nodes. Every one of the latter has six degrees of freedom (DOF). These include three translational and three rotational DOF. Element performance is evaluated with benchmark problems. For brevity, only two benchmark problems are included in this paper. It is shown numerically that the results converge to the true solution and have superior accuracy compared with those previously published in the literature. Mathematically, the elements being proposed are simple and frame invariant. Computationally, they are very efficient compared with higher order tetrahedral finite elements and other lower order tetrahedral finite elements.

In this paper, the sling load dynamics of an aerial vehicle carrying a payload are investigated by employing three formulations of the governing equations. They are the hybrid formulation where the system exists in either a taut cable or slack cable configuration, with appropriate treatment of the transition between the two; the linear complementarity problem (LCP) formulation where the cable constraints are imposed as linear complementarity conditions and finally, the lumped parameter formulation where the cable is modelled with a series of spring-mass elements. The hybrid and LCP formulations neglect the elasticity of the cable while the lumped parameter model explicitly accounts for the elastic properties of the cable, albeit in a discrete way. The importance of the incorporation of elastic properties of the cable on the system is investigated for the variation in solution space of the payload. The three formulations are compared numerically, for information on the computational cost, motion of the payload, and tension profile, for several aerial maneuvers, including an aggressive obstacle avoidance with a window clearance flight.

Abstract The application of stabilized finite element methods to model the vibration of elastic plates coupled with an acoustic fluid medium is considered. New stabilized methods based on the Hellinger-Reissner variational principle with a generalized least-squares modification are developed which yield improvement in accuracy over the Galerkin and Galerkin Generalized Least Squares (GGLS) finite element methods for both in vacuo and acoustic fluid-loaded Reissner-Mindlin plates. Through judicious selection of design parameters this formulation provides a consistent framework for enhancing the accuracy of mixed Reissner-Mindlin plate elements. Combined with stabilization methods for the acoustic fluid, the method presents a new framework for accurate modeling of acoustic fluid-loaded structures. The technique of complex wave-number dispersion analysis is used to examine the accuracy of the discretized system in the representation of free-waves for fluid-loaded plates. The influence of different finite element approximations for the fluid-loaded plate system are examined and clarified. Improved methods are designed such that the finite element dispersion relations closely match each branch of the complex wavenumber loci for fluid-loaded plates. Comparisons of finite element dispersion relations demonstrate the superiority of the hybrid least-squares (HLS) plate elements combined with stabilized methods for the fluid over standard Galerkin methods with mixed interpolation and shear projection (MITC4) and GGLS methods.