Eigensensitivity analysis for symmetric nonviscously damped systems with repeated eigenvalues

An efficient algorithm is derived for computation of eigenvalue and eigenvector derivatives of symmetric nonviscously damped systems with repeated eigenvalues. In the proposed method, the mode shape derivatives of the nonviscously damped systems are divided into a particular solution and a homogeneous solution. A simplified method is given to calculate the particular solution by solving a linear equation with non-singularity coefficients, the method is numerically stable and efficient compared to previous methods since the coefficient matrix is non-singularity and numerically stable. The homogeneous solution are computed by the second order derivative of eigenequation. One numerical example is used to illustrate the validity of the proposed method

Eigensensitivity of damped system with defective multiple eigenvalues

This paper considers the sensitivity of defective multiple eigenvalues of reducible matrix pencil, the average of eigenvalues is proved to be analytic, the derivatives of the average eigenvalues and the corresponding eigenvector matrices are obtained when the generalized eigenvalue is reducible. The sensitivity of defective multiple eigenvalues of a quadratic eigenvalue problem dependent on several parameters are also obtained by the result of generalized eigenvalue problem. The results are useful for investigating structural optimal design, model updating and structural damage detection

Computing eigenpair derivatives of asymmetric damped system by generalized inverse

Many existing approaches for asymmetric damped system are based on the assumption that the eigenvalues are simple or semisimple with separated derivatives. This paper presents a new algorithm for computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of asymmetric damped system. Compared with the existing methods, the algorithm can be applicable to problems whether the repeated eigenvalues have well separated derivatives. In the proposed method, the derivatives of eigenvectors are divided into a particular solution and a homogeneous solution, where the particular solution is constructed by using generalized inverse matrix. The effectiveness of the proposed algorithm is illustrated by one numerical example

Matrix-free time-domain methods for general electromagnetic analysis

Many engineering challenges demand an efficient computational solution of large-scale problems. If a computational method can be made free of matrix solutions, then it has a potential of solving very large scale problems. Among existing computational electromagnetic methods, the explicit finite-difference time-domain (FDTD) method is free of matrix solutions. However, it requires a structured orthogonal grid for space discretization. In this work, we develop a new time-domain method that naturally requires no matrix solution, regardless of whether the discretization is a structured grid or an unstructured mesh. No dual mesh, interpolation, projection and mass lumping are needed. Furthermore, a time-marching scheme is developed to ensure the stability for simulating an unsymmetrical numerical system, while preserving the matrix-free merit of the proposed method. This time-marching scheme is then made unconditionally stable, and hence allowing for the use of an arbitrarily large time step without sacrificing the matrix-free property. Extensive numerical experiments have been carried out on a variety of two- and three-dimensional unstructured meshes and even mixed-element meshes. Correlations with analytical solutions and the results obtained from the time-domain finite-element method have validated the accuracy, matrix-free property, stability, and generality of the proposed method.^ In addition to an extensive development of the proposed method in arbitrary 2- and 3-D unstructured meshes, we have also made a connection between the proposed new method and the classical FDTD method. We have found that the proposed matrix-free method naturally reduces to the FDTD method in an orthogonal grid. It also results in a new patch-based single-grid formulation of the FDTD algorithm. This new formulation not only makes the implementation of the original FDTD much easier, but also reveals a natural rank-1 decomposition of the curl-curl operator. Such a representation leads to an efficient extraction of unstable eigenmodes from fine cells only, from which a fast explicit and unconditionally stable FDTD method is developed. In addition, to efficiently handle multiscale structures, we develop an accurate FDTD subgridding algorithm suitable for arbitrary subgridding settings with arbitrary contrast ratios between the normal gird and the subgrid. Although the resulting system matrix is unsymmetric, we develop a time marching method to overcome the stability problem without sacrificing the matrix-free merit of the original FDTD. This method is general, which is also applicable to other subgridding algorithms whose underlying numerical systems are unsymmetric. The proposed FDTD subgridding algorithm is then further made unconditionally stable, thus permitting the use of a time step independent of space step.^ Last but not the least, the framework of the proposed method can be flexibly extended to solve partial differential equations in other disciplines, which we have demonstrated for thermal analysis

Static and dynamic analysis of linear elastic systems on non-prismatic three dimensional beam elements

Bibliography: leaves 100-103.A computer programme NONPRI, has been developed for the analysis of three dimensional skeletal assemblages consisting of non-prismatic members. It is capable of static and dynamic analysis of structures consisting of members whose constitutive relationship is linear elastic. The finite element formulation is based on the family of quadratic isoparametric finite elements. The three noded space frame element is quite versatile in that it can account for shear as well as flexural 9 axial and torsional deformation effects making it suitable for thin and thick beam analysis and for cases where the axial and torsional deformations are relevant. The element can be degenerated to a truss/frame transition element (3 translational degrees of freedom at each node - rotations ignored) and further degenerated to become a truss element. Furthermore, the element internal node is defined to lie at an arbitrary position inside the element. Thus, this flexibility in the non-prismatic element formulation makes it very powerful in practical analysis problems. An out-of-core solution technique is used for the equations of static analysis bearing in mind the capability for solving large structural systems. An in-core solution technique is used for the equations of dynamic analysis bearing now in mind that these equations represent an iterative process which can otherwise become computationally very expensive

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Topics in multiscale modeling: numerical analysis and applications

We explore several topics in multiscale modeling, with an emphasis on numerical analysis and applications. Throughout Chapters 2 to 4, our investigation is guided by asymptotic calculations and numerical experiments based on spectral methods. In Chapter 2, we present a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, the numerical methodology that we present is based on a spectral method. We use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients in the homogenized equation. Extensions of this method are presented in Chapter 3 and 4, where they are employed for the investigation of the Desai—Zwanzig mean-field model with colored noise and the generalized Langevin dynamics in a periodic potential, respectively. In Chapter 3, we study in particular the effect of colored noise on bifurcations and phase transitions induced by variations of the temperature. In Chapter 4, we investigate the dependence of the effective diffusion coefficient associated with the generalized Langevin equation on the parameters of the equation. In Chapter 5, which is independent from the rest of this thesis, we introduce a novel numerical method for phase-field models with wetting. More specifically, we consider the Cahn—Hilliard equation with a nonlinear wetting boundary condition, and we propose a class of linear, semi-implicit time-stepping schemes for its solution.Open Acces

Aeroelastic instabilities of an airfoil in transitional flow regimes

Cette thèse porte sur l'étude de l'instabilité aéroélastique provenant de l'interaction fluide–structure, dans le cas d'une aile rigide montée sur un ressort en torsion. L'étude est centrée sur le phénomène de flottement dû à un décollement laminaire, et plus précisément sur les oscillations (en torsion) auto-entretenues détectées expérimentalement pour un profil NACA0012 à faible incidence, dans la gamme de nombre de Reynolds dits transitionnels (Re in [10^4 – 10^5]), caractérisé par un décollement de la couche limite initialement laminaire, suivi d'une transition et d'un rattachement. L'objectif principal de la thèse est d'expliquer ce phénomène en se basant sur des concepts d'instabilité. Pour ce faire, différentes approches numériques ont été conduites: des simulations numériques bidimensionnelles et des simulations numériques tridimensionnelles (DNS). Ces approches ont en suite servi de base à des analyses de stabilité linéaire (LSA) autour d'un champ moyen ou d'un champ périodique (analyse de Floquet). Le deuxième objectif vise à explorer les différents scénarios non linéaires qui apparaissent dans cette gamme de Reynolds. La première partie de la thèse est consacrée à la caractérisation de l'écoulement autour de l'aile pour des angles d'incidence fixes. Des simulations temporelles bidimensionnelles montrent l'apparition d'oscillations à haute fréquence associées au détachement tourbillonnaire en aval du profil à partir de Re = 8000. Une analyse de stabilité hydrodynamique (Floquet) est réalisée pour caractériser la transition vers un écoulement tridimensionnel. Des simulations tridimensionnelles sont ensuite réalisées pour Re = 50000 afin de caractériser l'écoulement instantané et moyenné. L'analyse des forces moyennes exercées sur l'aile à incidence fixe permettent de détecter une rigidité aérodynamique négative (rapport moment-incidence) pour la gamme |alpha| 0°), où des solutions chaotiques et quasi-périodiques coexistent pour les mêmes paramètres structuraux, et évolue vers un scénario où les oscillations se font autour de alpha = 0°. La dernière partie de la thèse essaie d'expliquer la déstabilisation des positions d'équilibre non nulles conduisant à un comportement quasi-périodique à l'aide d'analyses LSA autour des champs moyens et périodiques à incidence fixe. Même si ces analyses sont incapables de prédire un mode propre instable, nous concluons que l'inclusion du terme des contraintes de Reynolds dans la dynamique de perturbation de l'écoulement moyen a un effet important.This thesis investigates aeroelastic instability phenomena arising in coupled fluid–structure interactions, considering the flow around a rigid airfoil mounted on a torsion spring. The focus is on the laminar separation flutter phenomenon, namely a self-sustained pitch oscillation detected experimentally on a NACA0012 airfoil in the transitional Reynolds number regime (Re in [10^4 – 10^5]) at low incidences, characterised by a detachment of an initially laminar boundary layer followed by its transition and subsequent reattachment. The main objective of the thesis is to explain this phenomenon in terms of instability concepts. For this, a combination of numerical approaches involving two- and three-dimensional Navier–Stokes simulations—the latter refereed to as Direct Numerical Simulations (DNS)—along with linear stability analyses (LSA) around a mean flow or a periodic flow (Floquet analysis) is employed. A second objective is to numerically explore the different nonlinear scenarios appearing in the low-to-moderate Reynolds number regime. The first part of the thesis is devoted to the characterisation of the fluid flow around the airfoil considering fixed incidences. Two-dimensional time-marching simulations are first employed, showing the emergence of high-frequency vortex shedding oscillations for Re = 8000. A hydrodynamic stability analysis (Floquet) is then employed to characterise the transition to a three-dimensional flow and DNS is eventually used to characterise both instantaneous and averaged flow quantities at Re = 50000. An analysis of the mean forces exerted on a fixed-incidence wing allows to detect a negative aerodynamic stiffness (torque-to-incidence ratio) in the range |alpha| < 2°, indicating a static instability. The second part of the thesis is devoted to the characterisation of the primary instability of the coupled fluid–structure system using LSA around the mean and periodic flow fields. Considering the symmetrical equilibrium position alpha = 0°, the analysis shows the presence of an unstable static mode, in accordance with the existence of a negative aerodynamic stiffness. In the third part of the thesis, the emergence of self-sustained flutter oscillations is investigated via two-dimensional aeroelastic simulations. The investigation shows that the system first transitions towards a pitch oscillation around the nonsymmetrical equilibrium position (alpha > 0°), with coexistence of chaotic and quasi-periodic solutions for the same structural parameters, and subsequently transitions towards a pitch oscillation around the symmetrical position (alpha = 0°) as the Reynolds number increases. In the last part of the thesis, an attempt is made to explain the destabilisation of the nonsymmetrical equilibrium positions leading to a quasi-periodic behaviour using LSA around the mean and periodic flow fields at fixed incidences. Even if these analyses are unable to predict an unstable eigenmode, we conclude that the inclusion of the Reynolds stress term in the mean flow perturbation dynamics has an important effect