A semi-analytical model for dynamic analysis of non-uniform plates

Dynamic properties of the plate structures can be enhanced by introducing discontinuities of different kinds such as using surface-bonded discrete patches or spatially varying the stiffness and mass properties of the plate. Fast and reliable design of such complex structures requires efficient and accurate modeling tools. In this study, a novel semi-analytical model is developed for the dynamic analysis of plates having discrete and/or continuous non-uniformities. Two-dimensional Heaviside unit step functions are utilized to represent the discontinuities. Different from existing numerical methods based on Heaviside functions, a numerical technique is proposed for modeling the discontinuities that are not necessarily aligned with the plate axes. The governing equations are derived using Hamilton's principle and Rayleigh–Ritz method is used for determining the modal variables. The surface-bonded patches are used to demonstrate discrete non-uniformities where variable-stiffness laminates are selected to represent continuous non-uniform structures. Natural frequencies and mode shapes obtained using the proposed method are validated with finite element analyses and the existing results from the literature. The results show that the developed model performs accurately and efficiently.     

Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment

In this paper, we study two species predator–prey Lotka–Volterra type dispersal system with periodic coefficients in two patches, in which both the prey and predator species can disperse between two patches. By utilizing analytic method, sufficient and realistic conditions on permanence and the existence of periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations.     

A new method for the connection of non-conforming NURBS patches in isogeometric analysis

The main objective of isogeometric analysis is the use of one common data set for design and analysis. Geometrical models usually consist of multiple NURBS surface patches with non-matching parametrizations along common edges. The ability to handle non-conforming meshes is essential for isogeometric finite element systems to avoid manipulations of the geometry model. In general two possible strategies exist. The first one is to enhance the weak form of the potential with additional terms to constrain the deformations of adjacent edges to be equal, e.g. with the Lagrange Multiplier method. The second one is to establish a relation between the deformations of adjacent patches. Thus, superfluous degrees of freedom are condensated out of the system and the patches are connected naturally by shared degrees of freedom. No alteration of the weak form is necessary which simplifies the implementation in existing finite element systems. For adjacent edges with hierarchic knot vectors an analytical solution exists. In the general case of non-hierarchic knot vectors it is not possible to find an analytical solution. This contribution presents a new numerical method to establish this relation. A collocation along the connection line links the deformations of adjacent patches. A theoretical derivation of the method is given and the choice of the collocation points is investigated. Numerical studies compare the new method with the Lagrange Multiplier method and show its applicability. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear finite element analysis of functionally graded plates integrated with patches of piezoelectric fiber reinforced composite

In this paper, a nonlinear static finite element analysis of simply supported smart functionally graded (FG) plates in the presence/absence of the thermal environment has been presented. The substrate FG plate is integrated with the patches of piezoelectric fiber reinforced composite (PFRC) material which act as the distributed actuators of the plate. The material properties of the FG substrate plate are assumed to be temperature dependent and graded along the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The derivation of this nonlinear thermo-electro-mechanical coupled finite element model is based on the first order shear deformation theory and the Von Karman type geometric nonlinearity. The numerical solutions of the nonlinear equations of the finite element model are obtained by employing the direct iteration method. The numerical illustrations suggest the potential use of the distributed actuator made of the PFRC material for active control of nonlinear deformations of smart FG structures. The effects of volume fraction index of the FG material of the substrate plates and the locations of the PFRC patches on the control authority of the patches are investigated. Emphasis has also been placed on investigating the effect of variation of piezoelectric fiber orientation angle in the PFRC patches on their actuation capability for counteracting the large deflections of FG plates.     

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.     

Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent

In this paper, we study two species time-delayed predator-prey Lotka-Volterra type dispersal systems with periodic coefficients, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of patches and cannot disperse. Sufficient conditions on the boundedness, permanence and existence of positive periodic solution for this systems are established. The theoretical results are confirmed by a special example and numerical simulations.     

Parameter estimation in a structural acoustic system with fully nonlinear coupling conditions

A methodology for estimating physical parameters in a class of structural acoustic systems is presented. The general model under consideration consists of an interior cavity which is separated from an exterior disturbance by an enclosing elastic structure. Piezoceramic patches are bonded to or embedded in the structure; these can be used both as actuators and sensors in applications ranging from the control of interior noise levels to the determination of structural flaws through nondestructive evaluation techniques. The presence and excitation of the patches, however, changes the geometry and material properties of the structure as well as involves unknown patch parameters, thus necessitating the development of parameter estimation techniques which are applicable in this coupled setting. In developing a framework for approximation, parameter estimation and implementation, strong consideration is given to the fact that the input operator is unbonded due to the discrete nature of the patches. Moreover, the model is weakly nonlinear as a result of the coupling mechanism between the structural vibrations and the interior acoustic dynamics. Within this context, an illustrating model is given, well-posedness and approximation results are discussed and an applicable parameter estimation methodology is presented. The scheme is then illustrated through several numerical examples with simulations modeling a variety of commonly used structural acoustic techniques for system excitation and data collection.     

Boundary Conditions and Multi-Patch Connections in Isogeometric Analysis

Isogeometric analysis is a high-continuity alternative to the standard finite element method. However, for practical application several issues remain to be addressed. This contribution discusses the imposition of Dirichlet boundary conditions as well as the connection between multiple patches. In particular necessary manipulations of the geometrical input data are provided. Dirichlet boundary conditions can be imposed in weak or in strong form. Due to the non-interpolatory characteristics of NURBS surfaces weak imposition of Dirichlet conditions is a viable option which avoids local transformations. The connection of multiple patches can be realized in a weak manner by adding additional terms to the variational equations, for example by the Lagrange multiplier method or the perturbed Lagrangian method. Both base on the idea of multiplying the mutual deformations with an additional unknown to force the deformations on shared edges to be equal. The numerical treatment leads to different sets of equations. In contrast to strong inter-patch connections, where coinciding control points share the same degrees of freedom, weak imposition allows for hanging nodes and therefore local refinement. The theoretical background and issues of implementation are given. Some numerical examples compare error norms for all mentioned methods and demonstrate that in particular cases a reduction of continuity leads to more accurate results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Piezoelectric control of flexible vibrations in rotating beams: An experimental study

Active control of flexible vibrations by distributed piezoelectric actuators and sensors plays an increasing role in engineering, especially in light-weight structures. Exemplarily, in this contribution a rotating beam is studied which can be found in many practical applications, e.g. as robot arms or flexible manipulators in production processes. It has been intensively shown in the literature that it is possible to completely suppress the flexible vibrations by an appropriate distribution of piezoelectric actuation strains. In order to compensate the inertial forces in the considered rotating beam, a complex distribution is obtained, such that a practical realisation would be very extensive. To overcome the problem, a discrete approximation by piezoelectric patches is applied. In order to find an optimal configuration for an experimental setup, and to investigate several control strategies, a numerical simulation model has been implemented based on Bernoulli-Euler beam theory. The numerical results are verified by an experimental set-up, in which 48 piezoelectric patches have been attached on a beam with rectangular hollow cross-section. Each patch can be used either as an actuator or a sensor. Additionally, strain gauges can be used as sensors. For monitoring, acceleration sensors are used. The control system is implemented within a dSpace environment. The results show a significant reduction of the flexible vibrations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.     

Two-Dimensional Riemann Problems for the Compressible Euler System

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model is built via asymptotic analysis, numerical simulation and theoretical analysis. A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process. State-of-the-art numerical methods in numerical simulations are used to discern smallscale structures of the solutions, e.g., semi-hyperbolic patches. Analytical methods are used to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda＇s programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws     

Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented. Supported by CTI Project 6437.1 IWS-IW.     

Bifurcation analysis in an age-structured model of a single species living in two identical patches

In this paper, we consider the age-structured model of a single species living in two identical patches derived in So et al. [J.W.-H. So, J. Wu, X. Zou, Structured population on two patches: modeling dispersal and delay, J. Math. Biol. 43 (2001) 37–51]. We chose a birth function that is frequently used but different from the one used in So et al. which leads to a different structure of the homogeneous equilibria. We investigate the stability of these equilibria and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out for supporting the analytic results.     

An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

Structural health monitoring of hybrid composite structure with piezoelectric patches

Structural health monitoring is performed on a hybrid structure specially designed for simple assembly and disassembly or replacement of components. The net-like structure is composed of composite beams connected with aluminum pads and it is fixed to a frame in selected locations. Modal analysis is performed both experimentally and numerically. The experiment is carried out using impact hammer, accelerometer and spectrum analyzer and the frequency and damping characteristics are thus obtained. The numerical model using finite element method is validated by comparison of the calculated and measured eigenfrequencies and eigenshapes. Furthermore, piezoelectric patches are applied on selected beam. The modal analysis is carried out again using the piezoelectric patches and with some of the composite beams replaced by damaged beam. The variation in modal characteristics between original and damaged configurations is analyzed both experimentally and numerically. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A periodic single species model with intermittent unilateral diffusion in two patches

In this paper, we consider a periodic single species model with intermittent unilateral diffusion in two patches. By using analytic method, Poincare mapping, Lyapunov function approach, sufficient and necessary conditions on the existence, uniqueness and global attractivity of positive periodic solution and the extinction of species for the considered system are established. Two examples and numerical simulations are presented to validate our theoretical results.     

The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.     

Lagrangian coherent structures,transport and chaotic mixing in simple kinematic ocean models

Methods of dynamical system’s theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean—a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle’s scattering and transport. A numerical analysis reveals a non-attracting chaotic invariant set Λ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle’s coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time. The clusters appear due to existence of regions of stability in the phase space which is the physical space in the advection problem.