Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach

In the present study, we propose a novel multiphysics model that merges two time-dependent problems – the Fluid-Structure Interaction (FSI) and the ultrasonic wave propagation in a fluid-structure domain with a one directional coupling from the FSI problem to the ultrasonic wave propagation problem. This model is referred to as the “eXtended fluid-structure interaction (eXFSI)” problem. This model comprises isothermal, incompressible Navier–Stokes equations with nonlinear elastodynamics using the Saint-Venant Kirchhoff solid model. The ultrasonic wave propagation problem comprises monolithically coupled acoustic and elastic wave equations. To ensure that the fluid and structure domains are conforming, we use the ALE technique. The solution principle for the coupled problem is to first solve the FSI problem and then to solve the wave propagation problem. Accordingly, the boundary conditions for the wave propagation problem are automatically adopted from the FSI problem at each time step. The overall problem is highly nonlinear, which is tackled via a Newton-like method. The model is verified using several alternative domain configurations. To ensure the credibility of the modeling approach, the numerical solution is contrasted against experimental data.     

Mejora de la solución fuertemente acoplada de problemas FSI mediante una aproximación de la matriz tangente de presión

The interaction of fluids with surrounding structures constitutes a classical challenge for the different numerical techniques. The aim of current work is twofold: first we provide a simple theoretical explanation of the problems to be faced in incompressible FSI. Then we introduce and justify a new procedure for the solution of complex fluid-structure interaction problems. Such a new strategy is based on the introduction of an «interface Laplacian» at the coupling boundary. The idea is to consider the dependence between fluid pressure and structural velocity as a non linear problem for which a Quasi-Newton scheme is sought. The new interface term is then proved to be an approximation of the tangent matrix for such non-linear problem. In the derivation of this result we make use exclusively of discrete linear algebra. Finally, we prove the efficiency of the new approach showing its ability to tackle standard benchmark problems.     

Fluid–structure interaction solver for compressible flows with applications to blast loading on thin elastic structures

In this study, we report the development and application of a fluid–structure interaction (FSI) solver for compressible flows with large-scale flow-induced deformation of the structure. The FSI solver utilizes a partitioned approach to strongly couple a sharp interface immersed boundary method-based flow solver with an open-source finite-element structure dynamics solver. The flow solver is based on a higher-order finite-difference method using a Cartesian grid, where it employs the ghost-cell methodology to impose boundary conditions on the immersed boundary. Higher-order accuracy near the immersed boundary is achieved by combining the ghost-cell approach with a weighted least squares error method based on a higher-order approximate polynomial. We present validations for two-dimensional canonical acoustic wave scattering on a rigid cylinder at a low Mach number and for flow past a circular cylinder at a moderate Mach number. The second order spatial accuracy of the flow solver was established in a grid refinement study. The structural solver was validated according to a canonical elastostatics problem. The FSI solver was validated based on comparisons with published measurements and simulations of the large-scale deformation of a thin elastic steel panel subjected to blast loading in a shock tube. The solver correctly predicted the oscillating behavior of the tip of the panel with reasonable fidelity and the computed shock wave propagation was qualitatively consistent with the published results. In order to demonstrate the fidelity of the solver and to investigate the coupled physics of the shock–structure interaction for a thin elastic plate, we employed the solver to simulate a 6.4 kg TNT blast loading on the thin elastic plate. The initial conditions for the blast were taken from previously reported field tests. Using numerical schlieren, the shock front propagation, Mach reflection, and vortex shedding at the tip of the plate were visualized during the impact of the shock wave on the plate. We discuss the coupling between the nonlinear dynamics of the plate and blast loading. The plate oscillates under the influence of blast loading and the restoration of elastic forces. The time-varying displacement of the tip of the plate is the superimposition of two dominant frequencies, which correspond to the first and second modes of the natural frequency of a vibrating plate. The effects of the material properties and length of the plate on the flow-induced deformation are briefly discussed. The proposed FSI solver is a versatile computational tool for simulating the impact of a blast wave on thin elastic structures and the results presented in this study may facilitate the design of thin structures subjected to realistic blast loadings.     

Mathematical analysis of a fluid—solid interaction problem

This paper analyzes a fluid—solid interaction model which describes the interaction between an inviscid fluid and an elastic solid In the model, the linear elastodynamic equations complemented with appropriate interface and boundary conditions are used to describe the wave propagation in the fluid and solid regions, and absorbing boundary conditions are used to minimize unphysical wave reflections. It is shown that the initial boundary value problem of the mathematical model posses a unique global (in time) quasi-strong solution. Regularity of the quasi-strong solution is also obtained under some reasonable assumptions on the data and on the domain.     

Strongly coupling of partitioned fluid–solid interaction solvers using reduced-order models

In this work a powerful technique is described which allows the implicit coupling of partitioned solvers in fluid–structure interaction (FSI) problems. The flow under consideration is governed by the Navier–Stokes equations for incompressible viscous fluids and modeled with the finite volume method. The structure is represented by a finite element formulation. The method allows the use of a black box fluid and structural solver because it builds up a reduced order model of the fluid and structural problem during the coupling process. Each solution of the fluid/structural solver in the coupling process can be seen as a sensitivity response of an applied displacement/pressure mode. The applied modes and their responses are used to build up a reduced-order model. The proposed model is used to predict the unsteady flow fields of a particular flow-induced vibrational phenomenon – a fixed cubic rigid body is submerged in an incompressible fluid flow (water), an elastic plate is attached to the rigid body in the centre of the downstream face, and the vortices, which separate from the corners of the rigid body upstream, generate lift forces which excite continuous oscillations of the elastic plate downstream. The computational results show that a fairly good convergence solution is achieved by using the reduced-order model that is based on only a few displacement and stress modes, which largely reduces the computational cost, compared with traditional approaches. At the same time, comparison of the numerical results of the model with available experimental data validates the methodology and assesses its accuracy.     

Modeling of electrodynamic-mechanical coupling phenomena in smart magnetoelectroelastic materials

The coupling of electric, magnetic and mechanical phenomena may have various reasons. The famous Maxwell equations of electrodynamics describe the interaction of transient magnetic and electric fields. On the constitutive level of dielectric materials, coupling mechanisms are manyfold comprising piezoelectric, magnetostrictive or magnetoelectric effects. Electromagnetically induced specific forces acting at the boundary and within the domain of a dielectric body are, within a continuum mechanics framework, commonly denoted as Maxwell stresses. In transient electromagnetic fields, the Poynting vector gives another contribution to mechanical stresses. First, a system of transient partial differential equations is presented. Introducing scalar and vector potentials for the electromagnetic fields and representing the mechanical strain by displacement fields, seven coupled differential equations govern the boundary value problem, accounting for linear constitutive equations of magnetoelectroelasticity. To reduce the effort of numerical solution, the system of equations is partly decoupled applying generalized forms of Coulomb and Lorenz gauge transformations [1,2]. A weak formulation is given to establish a basis for a finite element solution. The influence of constitutive magnetoelectric coupling on electromagnetic wave propagation is finally demonstrated with a simple one-dimensional example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements

The classical way of solving the time-harmonic linear acousto-elastic wave problem is to discretize the equations with finite elements or finite differences. This approach leads to large-scale indefinite complex-valued linear systems. For these kinds of systems, it is difficult to construct efficient iterative solution methods. That is why we use an alternative approach and solve the time-harmonic problem by controlling the solution of the corresponding time dependent wave equation.In this paper, we use an unsymmetric formulation, where fluid-structure interaction is modeled as a coupling between pressure and displacement. The coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences. After discretization, exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method.     

Lumps and their interaction solutions of a (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.     

The efficient solution of electromagnetic scattering for inhomogeneous media

We consider an electromagnetic scattering problem for inhomogeneous media. In particular, we focus on the numerical computation of the electromagnetic scattered wave generated by the interaction of an electromagnetic plane wave and an inhomogeneity in the corresponding propagation medium. This problem is studied in the VV polarization case, where some special symmetry requirements for the incident wave and for the inhomogeneity are assumed. This problem is reformulated as a Fredholm integral equation of second kind, which is discretized by a linear system having a special form. This allows to compute efficiently an approximate solution of the scattering problem by using iterative techniques for linear systems. Some numerical examples are reported.     

How an added mass matrix estimation may dramatically improve FSI calculations for moving foils

This paper presents a corrected partitioned scheme for investigating fluid–structure interaction (FSI) that may be encountered by lifting devices immersed in heavy fluid such as liquids. The purpose of this model is to counteract the penalizing impact of the added mass effect on the classical partitioned FSI coupling scheme. This work is based on an added mass corrected version of the classical strongly coupled partitioned scheme presented in Song et al. (2013). Results show that this corrected version systematically allows convergence to the coupled solution. The fluid flow model considered here uses a non-stationary potential approach, commonly termed the Panel Method. The advantage of this kind of approach is twofold: first, in restricting itself to a boundary method and, second, in allowing an added mass matrix to be estimated as a post-processing phase. Whereas the classical scheme encounters an acceptable (no numerical oscillation) convergence limit for fluid densities higher than 8 kg/m³ for the considered case, our corrected scheme is not dependent on fluid density and converges with only 6 iterations. This makes it possible to investigate the dynamic behavior of a 2D foil immersed in heavy fluids such as water. For example, it recognizes that frequency shifting may occur as the consequence of a strong added mass effect.     

Analysis of Guided Lamb Wave Propagation (GW) in Honeycomb Sandwich Panels

The paper aims to introduce the guided lamb wave propagation (GW) in a honeycomb sandwich panels to be used in the health monitoring applications. Honeycomb sandwich panels are well-known as lightweight structures with a good stiffness behavior and a wide range of applications in different industries. Due to the complex geometry and complicated boundary conditions in such a structure, the development of analytical solutions for describing the wave propagation and the interaction of waves with damages is hardly possible. Therefore dimensional finite element simulations have been used to model GW for different frequency ranges and different sandwich panels with different geometrical properties. The waves, which are highly dispersive, have been excited by thin piezoelectric patches attached to the surface of the structure. In the first step, the honeycomb panel has been simplified as an orthotropic layered continuum medium. The required material data have been calculated by applying a numerical homogenization method for the honeycomb core layer. The wave propagation has been compared in the homogenized model with the real geometry of a honeycomb sandwich panel. Such calculations of high frequency ultrasonic waves are costly, both in creating a proper finite element model as well as in the required calculation time. In this paper the influence of changes in the geometry of the sandwich panel on the wave propagation is presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Modeling of Ultrasonic Waves in Fractured Rails with an Explicit Approach

Ultrasonic wave propagation in steel rails with explicit identification of flaws is numerically simulated. The problem is to detect a vertical crack in a railhead by applying ultrasonic nondestructive testing techniques. The propagation of elastic waves in the rail profile is simulated for various sizes and positions of the crack. It is shown that the finite-difference grid-characteristic method in the time domain and full-wave simulation can be used to analyze the effectiveness of rail flaw detection by applying ultrasonic nondestructive testing techniques. Full-wave simulation is also used to demonstrate the failure of the widely used echo-mirror method to detect flaws of certain types. It is shown that techniques for practical application of the ultrasonic delta method can be developed using full-wave supercomputer simulation. The study demonstrates a promising potential of geophysical methods as adapted to the analysis of ultrasonic nondestructive testing results.

     

Study of Mode Conversion at Defects in Rope Structures using Ultrasonic Waves

Ultrasonic waves travel in rope structures over long distances as guided waves, allowing for effective health monitoring. In order to localize and characterize defects, an exact knowledge of the propagation, reflection, and transmission properties of the ultrasonic waves is required. These properties can be obtained using the Finite Element Method by modeling a segment of the periodic waveguide with a periodicity condition. The solution of the corresponding eigenvalue problem leads to all propagating modes of the waveguide as well as locally generated evanescent modes. The Boundary Element Method (BEM) is used in combination with the Finite Element Method for characterizing the wave propagation. The mode conversion at discontinuities, such as cracks or notches, can be subsequently described by reflection and transmission coefficients. The simulation results are the corresponding coefficients as a function of frequency and enable the selection of adequate modes for an effective defect detection. Additionally, it is demonstrated that along with the localization of cracks, conclusions about the crack geometry can be made with the help of reflection and transmission coefficients. The reliability and numerical accuracy of the simulation results are verfied by comparison with experimental findings. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explosive instability in nonlinear waves in media with negative viscosity : PMM vol. 38, n 6, 1974, pp. 991–995

The propagation of a wave of a finite amplitude in a medium with a nonlinearity of the second degree and negative viscosity, is examined. It is shown that in a finite time singularities appear in the solution. The exact solution of the Cauchy problem is given for a specific case. Recently the effects of negative viscosity which cause an increase in the energy of the wave motion have been studied intensively in electrodynamics, plasma physics, the Earth's atmosphere, in the theory of the circulation of the oceans and of flow in open channels [1–4], Wave amplification caused by an energy transfer from turbulent to regular motions, is possible in any medium having space-time fluctuations, provided the correlation time is sufficiently small [5, 6]. As the wave amplitude increases, nonlinear effects become important; they have been taken into account in cases where the interaction of a finite number of harmonics [2, 4] and the structure of steady motions have been examined [3].It is shown in this paper that in a medium with negative viscosity and a second degree dynamic nonlinearity, a solution of the Cauchy problem for an arbitrary “good” form of the initial perturbation, exists over a finite time interval. An example of such a solution is given.     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Boundary conditions for wave propagation problems

Wave propagation simulation requires a correct implementation of boundary conditions to avoid numerical instabilities. Similar problems are posed by domain decomposition methods where the aim is to find the correct modeling of physical phenomena across the interfaces separating the subdomains. The technique described here is based on physical grounds since it relies on the fact that the wave equation can be decomposed into incoming and outgoing wave modes at the boundary. The result is a modified wave equation for the boundaries which automatically includes the boundary condition. The boundary treatment is applied to a realistic problem of ultrasonic wave propagation through a vertical interface separating an anelastic solid at the surface. The results show that the method correctly describes the anelastic properties of the Rayleigh wave in the presence of a strong contrast in the material properties.