Harmonic Boundary Research Articles

Boundary excitation of nonlinearly modulated bending strain waves in a metamaterial

A non-linear modulation of bending waves in a metamaterial mass-in-mass lattice model is studied. Various kinds of non-linear wave modulation are obtained using a harmonic boundary excitation. It is shown, that these waves can be described by an asymptotic simplification of the equations of motion resulting in a non-linear modulation equation for the displacements. Exact periodic traveling wave solutions to the equation demonstrate a dependence on the sign of the equation coefficients or the elastic parameters of the original model. Dispersion relation for the carrier part of the solution is obtained as a function of wave number versus frequency, it is found how its solutions relate to the acoustic and optic branches and the band gap. This form of the dispersion relation is further used for a description of numerical results on the wave modulation by a harmonic boundary excitation.

Numerical Solution of the Newtonian Plane Couette Flow with Linear Dynamic Wall Slip

An efficient numerical approach based on weighted-average finite differences is used to solve the Newtonian plane Couette flow with wall slip, obeying a dynamic slip law that generalizes the Navier slip law with the inclusion of a relaxation term. Slip is exhibited only along the fixed lower plate, and the motion is triggered by the motion of the upper plate. Three different cases are considered for the motion of the moving plate, i.e., constant speed, oscillating speed, and a single-period sinusoidal speed. The velocity and the volumetric flow rate are calculated in all cases and comparisons are made with the results of other methods and available results in the literature. The numerical outcomes confirm the damping with time and the lagging effects arising from the Navier and dynamic wall slip conditions and demonstrate the hysteretic behavior of the slip velocity in following the harmonic boundary motion.

Periodic driving shape controls energy transmission.

In the early 2000s, Geniet and Leon [Phys. Rev. Lett. 89, 134102 (2002)0031-900710.1103/PhysRevLett.89.134102] discovered the nonlinear supratransmission (NST) in a medium with a forbidden frequency band gap. It is a process in which nonlinear structures are created by a sinusoidal harmonic boundary condition imposed at a frequency in the band gap. The present study extends this concept and shows that an optimal shape of a periodic nonsinusoidal excitation may induce (or inhibit) energy flow through the lattice below (or above) the NST threshold, demonstrating that nonlinear supratransmission is reliant not only on the driving amplitude but also on its shape. This is evidenced through numerical simulations and mathematical calculations varying the excitation signal shape in the Fermi-Pasta-Ulam case study. Setting the shape parameter to zero recovers the results of the literature in relation to the sinusoidal signal.

Impulse Fluid Simulation

We propose a new incompressible Navier-Stokes solver based on the impulse gauge transformation. The mathematical model of our approach draws from the impulse-velocity formulation of Navier-Stokes equations, which evolves the fluid impulse as an auxiliary variable of the system that can be projected to obtain the incompressible flow velocities at the end of each time step. We solve the impulse-form equations numerically on a Cartesian grid. At the heart of our simulation algorithm is a novel model to treat the impulse stretching and a harmonic boundary treatment to incorporate the surface tension effects accurately. We also build an impulse PIC/FLIP solver to support free-surface fluid simulation. Our impulse solver can naturally produce rich vortical flow details without artificial enhancements. We showcase this feature by using our solver to facilitate a wide range of fluid simulation tasks including smoke, liquid, and surface-tension flow. In addition, we discuss a convenient mechanism in our framework to control the scale and strength of the turbulent effects of fluid.

Analysis of longitudinal oscillations in a vertically moving cable subject to nonclassical boundary conditions

In this paper, we study a model of a flexible hoisting system, in which external disturbances exerted on the boundary can induce large vibrations, and so damage to the performance of the system. The dynamics is described by a wave equation on a slow time-varying spatial domain with a small harmonic boundary excitation at one end of the cable, and a moving mass at the other end. Due to the slow variation of the cable length, a singular perturbation problem arises. By using an averaging method, and an interior layer analysis, many resonance manifolds are detected. Further, a three time-scales perturbation method is used to construct formal asymptotic approximations of the solutions. It turns out that for a given boundary disturbance frequency, many oscillation modes jump up from order ε amplitudes to order ε amplitudes, where ε is a small parameter with 0<ε<<1. Finally, numerical simulations are presented to verify the obtained analytical results.

Localization of travelling and standing waves in a circular membrane coupled to a continuous viscoelastic support

We present an analytical approach for realizing the localization of travelling and standing waves in a circular membrane by means of an interior, continuous, ring-type viscoelastic support when the membrane is subjected to harmonic boundary excitation. This work is a continuation of the wave separation dynamics induced by nonclassical damping in one-dimensional elastic continua. For a two-dimensional azimuthally symmetric configuration, a necessary and sufficient condition is established to realize wave localization, leading to uniformly distributed spring stiffness and damping coefficients for the interior support. A rotating wave localization pattern can also be realized by applying a circumferentially moving boundary excitation. The desired results are shown to depend strongly on the excitation frequency, the circumferential wavenumber, and the radius of the continuous support. For a weakly eccentric configuration with uniformly distributed boundary excitation, non-uniform required distributions of the support properties can be obtained analytically. However, waveform distortion will occur for either non-uniformly distributed boundary excitation with weak support eccentricity or highly eccentric coupling, such that the parameter values of the support have zero crossings on the nodal diameters of the membrane and thus lose physical meaning. The results presented in this work are expected to be useful in the design of mechanical and acoustic systems to localize energy to a sonic bullet by means of the continuous distribution of damping.

Wavenumber-space band clipping in nonlinear periodic structures

In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as an initial condition or as a boundary condition, respectively. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. We then extend the framework to locally resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.

Thermal Characterization of Building Walls Under Random Boundary Conditions

Abstract This work aims to improve the knowledge on dynamic thermophysical characterization of building envelopes by comparing three numerical methods applied on an experimental wall made of masonry brick. The thermal conductivity λ and the thermal capacity ρCp are determined by performing a data fitting optimization between the experimental measurements of the heat flux and the heat flux resulting from these numerical models. The experimental device consists of a thermal box with a controlled ambiance through a radiator linked to a thermostatic bath and placed inside the thermal box, on the opposite side facing the wall. Three different methods were examined: the heat transfer matrix (HTM) analytical method using the HTM, the finite element method (FEM) using comsol multiphysics® software, and the building simulation model (BSM) method using trnsys® Type 56 coupled with Genopt® optimization tool. The reproducibility of the methods was also validated through two other datasets (one random and one harmonic). The obtained results were satisfactory for both λ and for ρCp and for the three studied methods with deviations less than 5% between the results of the different methods. The data logging duration for random boundary conditions was found to be around five days while in harmonic boundary conditions two days were sufficient for the solution to converge.

Steklov Expansion Method for Regularized Harmonic Boundary Value Problems

A new type of meshless method is proposed in this paper to solve regularized Laplacian boundary value problems of the form In this method, solving the Laplacian boundary value problem is starting from finding regularized Steklov eigenpairs which could provide the orthonormal basis of the space of solutions. The solutions are represented by orthogonal regularized Steklov eigenfunctions. Error bounds of Steklov approximations are provided in terms of the traces spaces of solutions. When rectangular domains are considered, explicit formulas for the regularized Steklov eigenpairs are provided. Some numerical experiments are presented to support the efficiency and accuracy of the new method.

An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered.The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.

Steklov approximations of Green’s functions for Laplace equations

Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

Vibration of an Orthotropic Doubly Curved Panel with a Set of Cutouts of Any Configuration Under Mixed Boundary Conditions

Within the framework of a refined model that takes into account transverse shear strains and inertial components, we construct a solution of the problem of steady-state vibration of an orthotropic doubly curved panel with cutouts of any shape, orientations, and location and arbitrary external boundary under mixed harmonic boundary conditions imposed on the outer boundary and the contours of cutouts. The solution is constructed by the indirect method of boundary elements with the help of a sequential approach to the representation of the Green functions. The obtained integral equations are solved by the collocation method.

Transverse Vibration of an Orthotropic Plate with Holes and Inclusions

Within the framework of the refined theory taking into account the transverse shear strains and inertial components, we consider the problem of stationary vibration of an orthotropic plate with holes and absolutely rigid inclusions. The inclusions may have different types of bonds with the plate. We analyze the case of translational motion of the inclusions along the direction of normal to the median surface of the plate. We study various harmonic (in time) boundary conditions on the outer boundary of the plate and on the contours of the holes. On the basis of the indirect method of boundary elements and a sequential approach to the representation of Green’s functions, the boundary-value problem is reduced to a system of integral equations and relations, which is solved by the collocation method. The numerical results are presented for a rectangular plate containing a circular hole and a circular inclusion.

Backstepping-based adaptive error feedback regulator design for one-dimensional reaction-diffusion equation

In this paper, we consider the error feedback regulator problem (EFRP) for one-dimensional reaction-diffusion equation with unknown harmonic boundary disturbance, where the regulated output is anti-collocated with the control. Some auxiliary systems are constructed in order to make the control and the disturbance become collocated, or make the measured tracking error become the output. Then the error feedback adaptive servomechanism is presented, in which the parameters of the disturbance and the reference signals are estimated. Our design is based on the motion planning and the backstepping approaches. In addition, we give a brief description that our approach is applicable to the heat equation with distributed disturbance. It is shown that the proposed adaptive control law regulates the tracking error to zero and keeps the states of all the internal loops bounded.

Uniqueness of the Boundary Value Problem of Harmonic Maps via Harmonic Boundary

We prove the uniqueness of solutions for the boundary value problem of harmonic maps in the setting: given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a regular geodesic ball, there exists a unique harmonic map, which is a limit of a sequence of harmonic maps with finite total energy in the sense of the supremum norm, from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.

Quasi-periodic motions of high-dimensional nonlinear models of a translating beam with a stationary load subsystem under harmonic boundary excitation

Bifurcations and quasi-periodic motions of high-dimensional nonlinear models of a translating beam with a stationary load subsystem under harmonic boundary excitation, where there are combined parametric and forcing excitations, are investigated. It is demonstrated that by adjusting the frequency of the boundary excitation beyond bifurcation points, the nonlinear system exhibits quasi-periodic motion rather than the periodic response reported in an earlier publication. Particular attention is paid to the nonlinear dynamics of models with five and six included trial functions, where quantitative and qualitative results of frequency responses and quasi-periodic motions are significantly different from each other. The nonlinear governing equations of motion of the translating beam are established by using the Newton's second law. The Galerkin method is used to truncate the governing partial differential equation into a set of nonlinear ordinary differential equations. The incremental harmonic balance (IHB) method is used to solve for periodic responses of the high-dimensional models of the translating beam. The Floquet theory along with the precise Hsu's method is used to investigate stability of the periodic responses. The IHB method with two time scales developed earlier is extended to analyze quasi-periodic motion of the nonlinear system with combined parametric and forcing excitations whose spectrum contains uniformly spaced sideband frequencies. Quasi-periodic motion obtained from the IHB method with two time scales is in excellent agreement with that from numerical integration using the fourth-order Runge-Kutta method.

Method for Calculating the Optical Properties of Composites Based on a Transparent Matrix with Residual Porosity and Metal Nanoparticles

A method is proposed for calculating the optical characteristics of composites based on a transparent matrix with residual porosity and metal nanoparticles by solving the transport equation for monochromatic radiation employing a spherical harmonic method and Fresnel boundary conditions. The method is tested by modeling the transport of monochromatic radiation at four wavelengths of practical importance in cyclotrimethylenetrinitramine–aluminum nanoparticle composites with a Rayleigh distribution of pores with respect to radius. It is shown that in the case of small-radius pores the reflectivity increases as the mass fraction of nanoparticles is increased. In the case of pores with large radii, a kink corresponding to complete filling of the pores appears on the dependence of the transmission on the nanoparticle mass fraction. The application of these results to inverse problems in the spectroscopy of lightscattering system is discussed.

Lattice Green function methods for atomistic/continuum coupling: Theory and data-sparse implementation

Flexible harmonic boundary conditions have been proposed by Sinclair in the 1970s in order to overcome spurious effects on atomistic problems due to fixed boundaries. To date this method has never been applied to problems beyond isolated defects, such as dislocations, because it involves dense boundary matrices which become quickly unsuitable to larger problems due to their vast memory requirements. In order to apply the method for larger systems, e.g. arrangements of defects, we propose an implicit approximate representation using hierarchical matrices which have proven efficiency in the context of boundary integral equations while preserving overall accuracy. Despite its simplicity, Sinclair’s staggered method converges rather slowly if the approximate far-field harmonic response and the true nonlinear atomic response differ considerably. Starting from Sinclair’s iteration equation for the harmonic displacements, we derive a discrete variant of the well-known boundary element method (BEM) for the exterior balance equation which is then combined with the fully atomistic problem. To solve the coupled problem we propose a monolithic Newton–Krylov scheme which iterates simultaneously on all unknowns. We outline the superior performance of this method in comparison to other existing methods and to classical clamped boundary conditions with numerical examples. Further, we present guidelines for an efficient implementation into existing molecular dynamics codes.

Separation of traveling and standing waves in a finite dispersive string with partial or continuous viscoelastic foundation

The free and forced vibrations of a linear string with a local spring-damper on a partial elastic foundation, as well as a linear string on a viscoelastic foundation conceptualized as a continuous distribution of springs and dampers, are studied in this paper. Exact, analytical results are obtained for the free and forced response to a harmonic excitation applied at one end of the string. Relations between mode complexity and energy confinement with the dispersion in the string system are examined for the steady-state forced vibration, and numerical methods are applied to simulate the transient evolution of energy propagation. Eigenvalue loci veering and normal mode localization are observed for weakly coupled subsystems, when the foundation stiffness is sufficiently large, for both the spatially symmetric and asymmetric systems. The forced vibration results show that nonproportional damping-induced mode complexity, for which there are co-existing regions of purely traveling waves and standing waves, is attainable for the dispersive string system. However, this wave transition phenomenon depends strongly on the location of the attached discrete spring-damper relative to the foundation and whether the excitation frequency Ω is above or below the cutoff frequency ωc. When Ω<ωc, the wave transition cannot be attained for a string on an elastic foundation, but is possible if the string is on a viscoelastic foundation. Although this study is primarily formulated for a harmonic boundary excitation at one end of the string, generalization of the mode complexity can be deduced for the steady-state forced response of the string-foundation system to synchronous end excitations and is confirmed numerically. This work represents a novel study to understand the wave transitions in a dispersive structural system and lays the groundwork for potentially effective passive vibration control strategies.

Note on uniformly transient graphs

We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have \ell^p independent pure point spectra (for 1 \leq p \leq \infty ). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compactification and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.