scholarly journals Homogenization of the wave equation with non-uniformly oscillating coefficients

2022 ◽  
pp. 108128652110650
Author(s):  
Danial P. Shahraki ◽  
Bojan B. Guzina
Keyword(s):  
Wave Equation ◽  
Wave Motion ◽  
Shear Waves ◽  
Low Frequency ◽  
Wave Dispersion ◽  
Second Order ◽  
Frequency Wave ◽  
Cell Functions ◽  
Varying Coefficients ◽  
Periodic Microstructure

The focus of our work is a dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g., functionally graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in [Formula: see text], [Formula: see text], within the framework of multiple scales expansion. When either [Formula: see text] or [Formula: see text], this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in one dimension by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

Effects of director rotation relaxation on viscoelastic wave dispersion in nematic elastomer beams

2018 ◽  
Vol 24 (4) ◽  
pp. 1103-1115 ◽  
Author(s):  
Dong Zhao ◽  
Ying Liu
Keyword(s):  
Wave Motion ◽  
Relaxation Times ◽  
Low Frequency ◽  
Wave Dispersion ◽  
Frequency Limit ◽  
Anisotropic Parameter ◽  
Acoustic Design ◽  
Phase Velocities ◽  
Nematic Elastomer ◽  
Viscoelastic Wave

In this paper, the transverse wave dispersion in a nematic elastomer (NE) Timoshenko beam is studied by considering anisotropy and viscoelasticity of NEs in the low frequency limit. Firstly, the characteristic equations of wave motion in an NE beam are derived, and then numerically solved to obtain the corresponding phase velocities and attenuation factors. The influences of anisotropic parameter, director rotation and rubber relaxation times on the wave dispersion in an NE beam are discussed. Results show that unlike the situation in general isotropic viscoelastic beam, non-classical viscoelastic wave dispersion is found in NE beams. Geometric dispersion is restrained with the vanishing of cut-off frequencies for shear waves due to director rotation relaxation of NEs. This unique property promises prospective applications of NE beams in optic or acoustic design.

Laboratory Wave Modelling for Floating Structures in Shallow Water

2006 ◽  
Author(s):  
Carl Trygve Stansberg
Keyword(s):  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
Wave Groups ◽  
Large Wave ◽  
Frequency Wave ◽  
Wave Modelling ◽  
Floating Structures ◽  
Wave Basin ◽  
Wave Drift Forces

The analysis of moored floating vessels in shallow water requires special attention, when compared to similar problems in deep water. In particular, low-frequency wave drift forces need to be studied. Model testing is essential in validation of numerical prediction tools for these problems. Wave-group induced low-frequency wave components is an important part of the problem. Their reproduction in laboratories needs special attention. In general, two types of low-frequency waves are present: “bound” waves following the wave groups, and “free” waves propagating with their own speed. The former is included in second-order numerical codes for floater is included in second-order numerical codes for floaters, while the latter is normally not. Therefore, identification and possible reduction of the free components is of interest. A practical way to do this in a large wave basin is described in this paper. Results from generation of bi-chromatic waves without and with correction are presented. Corrected results show a clear reduction of the free wave component.

Side-by-side FLNG and shuttle tanker linear and second order low frequency wave induced dynamics

2016 ◽  
Vol 111 ◽  
pp. 234-253 ◽  
Author(s):  
João Pessoa ◽  
Nuno Fonseca ◽  
C. Guedes Soares
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Frequency Wave ◽  
Wave Induced

Approximation of Second-Order Low-Frequency Wave Loading in Multi-Directional Waves

2010 ◽  
Author(s):  
Flavia C. Rezende ◽  
Xiao-bo Chen
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Wave Loads ◽  
Wave Loading ◽  
Frequency Wave ◽  
New Developments ◽  
Directional Waves ◽  
Novel Method ◽  
The Difference ◽  
Accurate Scheme

Further to the studies by Chen & Rezende (OMAE2009) on the quadratic transfer function (QTF) of low-frequency wave loading in which the QTF is developed by the series expansion associated with the difference-frequency up to the order-Δω2, new formulations have been developed in order to take into account the effect of interactions between waves of different headings. It provides a novel method to evaluate the low-frequency second-order wave loads in a more accurate than usual order-Δω approximation (often called Newman approximation) and more efficient way comparing to the computation of complete QTF in multi-directional waves. New developments including numerical results of different components of QTF are presented here. Furthermore, the time-series reconstruction of excitation loads by quadruple sums in the motion simulation of mooring systems is analyzed and a new efficient and accurate scheme using only a triple sum is demonstrated.

Efficient Computations of Second-Order Low-Frequency Wave Load

2009 ◽  
Author(s):  
Xiao-Bo Chen ◽  
Fla´via Rezende
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Wave Loads ◽  
Wave Loading ◽  
Wave Load ◽  
Frequency Wave ◽  
New Developments ◽  
Novel Method ◽  
The Difference ◽  
New Formulation

As the main source of resonant excitations to most offshore moored systems like floating LNG terminals, the low-frequency wave loading is the critical input to motion simulations which are important for the design. Further to the analysis presented by Chen & Duan (2007) and Chen & Rezende (2008) on the quadratic transfer function (QTF) of low-frequency wave loading, the new formulation of QTF is developed by the series expansion of the second-order wave loading with respect to the difference-frequency upto the order-2. It provides a novel method to evaluate the low-frequency second-order wave loads in a more accurate than usual order-0 approximation (often called Newman approximation) and more efficient way comparing to the computation of complete QTF. New developments including numerical results of different components of QTF are presented here. Furthermore, the time-series reconstruction of excitation loads in the motion simulation of mooring systems is analyzed and a new efficient and accurate scheme is demonstrated.

Investigation of Nonlinear Difference-Frequency Wave Excitation on a Semisubmersible Offshore-Wind Platform With Bichromatic-Wave CFD Simulations

2021 ◽  
Author(s):  
Lu Wang ◽  
Amy Robertson ◽  
Jason Jonkman ◽  
Yi-Hsiang Yu ◽  
Arjen Koop ◽  
...  
Keyword(s):  
Potential Flow ◽  
Low Frequency ◽  
Flow Theory ◽  
Second Order ◽  
Difference Frequency ◽  
Offshore Wind ◽  
Viscous Drag ◽  
Cfd Simulations ◽  
Frequency Wave ◽  
Potential Flow Theory

Abstract The natural surge and pitch frequencies of semisubmersible offshore wind platforms are typically designed to be below the wave frequencies to avoid direct excitation. However, surge or pitch resonance can be excited by the nonlinear low-frequency loads generated by irregular incident waves. Second-order potential-flow models with added Morison drag have been found to underpredict this low-frequency excitation and response. As part of the OC6 project1, the authors performed computational fluid dynamics (CFD) simulations to enable a better understanding of the low-frequency loads and the limitations of lower-fidelity models. The focus of this paper is to set up a computationally cost-effective CFD simulation of a fixed semisubmersible platform to investigate nonlinear difference-frequency loads and establish the corresponding uncertainty in the results. Because of the high computing cost, CFD simulations of irregular waves can be challenging. Instead, simulations were performed with bichromatic waves having a shorter repeat period. A preliminary comparison with quadratic transfer functions from second-order potential-flow theory shows that CFD models consistently predict higher nonlinear wave loads at the difference frequency, likely because of flow separation and viscous drag not accounted for in potential-flow theory.

Mean- and Low-Frequency Wave Forces on Semisubmersibles

1982 ◽  
Vol 22 (04) ◽  
pp. 563-572
Author(s):  
J.A. Pinkster
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Irregular Waves ◽  
Wave Forces ◽  
Frequency Wave ◽  
First Order ◽  
Wave Drift ◽  
Frequency Components ◽  
Wave Drift Forces ◽  
Drift Forces

Abstract Mean- and low-frequency wave drift forces on moored structures are important with respect to low-frequency motions and peak mooring loads. This paper addresses prediction of these forces on semisubmersible-type structures by use of computations based on three-dimensional (3D) potential theory. The discussion includes a computational method based on direct integration of pressure on the wetted part of the hull of arbitrarily shaped structures. Results of computations of horizontal drift forces on a six-column semisubmersible are compared with model tests in regular and irregular waves. The mean vertical drift forces on a submerged horizontal cylinder obtained from model tests also are compared with results of computations. On the basis of these comparisons, we conclude that wave drift forces on semisubmersible-type structures in conditions of waves without current can be predicted with reasonable accuracy by means of computations based on potential theory. Introduction Stationary vessels floating or submerged in irregular waves are subjected to large first-order wave forces and moments that are linearly proportional to the wave height and that contain the same frequencies as the waves. They also are subjected to small second-order mean- and low- frequency wave forces and moments that are proportional to the square of the wave height. Frequencies of second-order low-frequency components are associated with the frequencies of wave groups occurring in irregular waves.First-order wave forces and moments cause the well-known first-order motions with wave frequencies. First-order wave forces and motions have been investigated for several decades. As a result of these investigations, methods have been developed to predict these forces and moments with reasonable accuracy for many different vessel shapes.For semisubmersibles, which consist of a number of relatively slender elements such as columns, floaters, and bracings, computation methods have been developed to determine the hydrodynamic loads on those elements without accounting for interaction effects between the elements. For the first-order wave loads and motion problem, these computations give accurate results.This paper deals with the mean- and low-frequency second-order wave forces acting on stationary vessels in regular and irregular waves in general and presents a method to predict these forces on the basis of computations.The importance of mean- and low-frequency wave drift forces, from the point of view of motion behavior and mooring loads on vessels moored at point of view of motion behavior and mooring loads on vessels moored at sea, has been recognized only within the last few years. Verhagen and Van Sluijs, Hsu and Blenkarn, and Remery and Hermans showed that the low-frequency components of wave drift forces in irregular waves-even though relatively small in magnitude-can excite large-amplitude low- frequency horizontal motions in moored structures. It was shown for irregular waves that the drift forces contain components with frequencies coinciding with the natural frequencies of the horizontal motions of moored vessels. Combined with minimal damping of low-frequency horizontal motions of moored structures, this leads to large-amplitude resonant behavior of the motions (Fig. 1). Remery and Hermans established that low-frequency components in drift forces are associated with the frequencies of wave groups present in an irregular wave train.The vertical components of the second-order forces sometimes are called suction forces. SPEJ p. 563

Accuracy Effects of Low Frequency Wave Loads on Ships Offshore to LNG Marine Terminal Design

2004 ◽  
Author(s):  
Monica J. Holboke ◽  
Robert G. Grant
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
Surface Boundary ◽  
Wave Loads ◽  
Wave Load ◽  
Frequency Wave ◽  
First Order ◽  
Terminal Design

This paper presents the results of a two-body analysis for a moored ship sheltered by a breakwater in shallow water with and without free surface forcing in the low frequency wave load calculation. The low frequency wave loads are determined by second order interactions from the first order. The free surface forcing term arises from the free surface boundary condition, which is trivial to first order but is not at second order. We demonstrate in the frequency domain the importance of this term in a two-body analysis. Additionally, we show how inaccurate calculations of the off-diagonal terms of the Quadratic Transfer Function can translate to over or under prediction of low frequency wave loads on moored ships sheltered by breakwaters in shallow water. Low frequency wave load accuracy has direct consequence for LNG marine terminal design. Generally, LNG marine terminals are sited in sheltered harbors, however increasingly they are being proposed in offshore locations where they will require protection from persistent waves and swells. Since breakwaters typically cost twice as much as the rest of the marine facilities, it is important to optimize their size, orientation and location. In a previous paper we described this optimization process [1], which identified a key step to be the transforming of waves just offshore the breakwater into wave loads on the moored ships. The ability to do this step accurately is of critical importance because if the loads are too large, the breakwater will be larger and more expensive than necessary and if the loads are too small, the terminal will experience excessive downtime and loss of revenue.

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