Wave-based Method Research Articles

Wave propagation in a porous composite beam: Porosity determination, location and quantification

A wave-based method is developed to quantify the defect due to porosity and also to locate the porous regions, in a composite beam-type structure. Wave propagation problem for a porous laminated composite beam is modeled using spectral finite element method (SFEM), based on the modified rule of mixture approach, which is used to include the effect of porosity on the stiffness and density of the composite beam structure. The material properties are obtained from the modified rule of mixture model, which are used in a conventional SFEM to develop a new model for solving wave propagation problems in porous laminated composite beam. The influence of the porosity content on the group speed and also the effect of variation in theses parameters on the time responses are studied first, in the forward problem. The change in the time responses with the change in the porosity of the structure is used as a parameter to find the porosity content in a composite beam. The actual measured response from a structure and the numerically obtained time responses are used for the estimation of porosity, by solving a nonlinear optimization problem. The effect of the length of the porous region (in the propagation direction), on the time responses, is studied. The damage force indicator technique is used to locate the porous region in a beam and also to find its length, using the measured wave propagation responses.

Measurement of Viscoelastic Properties of In Vivo Swine Myocardium Using Lamb Wave Dispersion Ultrasound Vibrometry (LDUV)

Viscoelastic properties of the myocardium are important for normal cardiac function and may be altered by disease. Thus, quantification of these properties may aid with evaluation of the health of the heart. Lamb wave dispersion ultrasound vibrometry (LDUV) is a shear wave-based method that uses wave velocity dispersion to measure the underlying viscoelastic material properties of soft tissue with plate-like geometries. We tested this method in eight pigs in an open-chest preparation. A mechanical actuator was used to create harmonic, propagating mechanical waves in the myocardial wall. The motion was tracked using a high frame rate acquisition sequence, typically 2500 Hz. The velocities of wave propagation were measured over the 50-400 Hz frequency range in 50 Hz increments. Data were acquired over several cardiac cycles. Dispersion curves were fit with a viscoelastic, anti-symmetric Lamb wave model to obtain estimates of the shear elasticity, μ(1), and viscosity, μ(2) as defined by the Kelvin-Voigt rheological model. The sensitivity of the Lamb wave model was also studied using simulated data. We demonstrated that wave velocity measurements and Lamb wave theory allow one to estimate the variation of viscoelastic moduli of the myocardial walls in vivo throughout the course of the cardiac cycle.

A fast wave-based hybrid method for interactive acoustic simulation in large and complex environments

Modern acoustic applications require fast and accurate simulation in large and complex environments. Current methods are rather limited to very simple environments or not able to perform fast interactive simulations. Our objective is to develop an innovative and practical method to perform real-time or near real-time accurate interactive simulation in large and complex environments, such as urban environments up to one kilometer. A new method, the embedded hybrid method using immersed boundary within k-space PSTD has been proposed and developed to achieve the objective. Its performance, accuracy, parallel hardware acceleration, and capabilities to handle all kinds of environmental complexities have been rigorously validated and verified by a number of examples. It shows that with the utilization of modern many-core GPUs, the method can perform real-time or near real-time accurate interactive simulation in large environments of hundreds to thousands of meters with all kinds of complexities including (1) inhomogeneous and absorptive medium, (2) curved and arbitrarily-shaped objects, (3) and moving sources, receivers, and objects, and time-varying medium.

Investigation of Sensor Placement in Lamb Wave-Based SHM Method

In this paper the investigation of a structural health monitoring method for thin-walled parts of structures is presented. The concept is based on the guided elastic wave propagation phenomena. This type of waves can be used in order to obtain information about structure condition and possibly damaged areas. Guided elastic waves can travel in the medium with relatively low attenuation, therefore they enable monitoring of extensive parts of structures. In this way it is possible to detect small defects in their early stage of growth. It is essential because undetected damage can endanger integrity of a structure. In reported investigation piezoelectric transducer was used to excite guided waves in chosen specimens. Dispersion of guided waves results in changes of velocity with the wave frequency, therefore a narrowband signal was used. Measurement of the wave field was realized using laser scanning vibrometer that registered the velocity responses at points belonging to a defined mesh. An artificial discontinuity was introduced to the specimen. The goals of the investigation was to detect it and find optimal sensor placement for this task. Determination of the optimal placement of sensors is a very challenging mission. In conducted investigation laser vibrometer was used to facilitate the task. The chosen mesh of measuring points was the basis for the investigation. The purpose was to consider various configuration of piezoelectric sensors. Instead of using vast amount of piezoelectric sensors the earlier mentioned laser vibrometer was used to gather the necessary data from wave propagation. The signals gather by this non-contact method for the considered network were input to the damage detection algorithm. Damage detection algorithm was based on a procedure that seeks in the signals the damage-reflected waves. Knowing the wave velocity in considered material the damage position can be estimated.

A Wave Based Method for the efficient solution of the 2D poroelastic Biot equations

The vibro-acoustic behaviour of poroelastic materials is often formulated as boundary value problems based on the continuum mechanics Biot’s theory expressed as two coupled partial differential equations. This paper presents an extension of the Wave Based Method (WBM), a numerical technique to solve these vibro-acoustic boundary value problems in a computationally efficient manner. At present, the Finite Element Method (FEM) is the most commonly used prediction technique to deal with these Biot equations, but suffers from the disadvantage that the system matrices have to be recalculated for each frequency of interest due to the frequency-dependent equation parameters. This harms the inherent effectiveness of the FEM. Additionally, due to the discretisation into a large number of small finite elements and the high number of unknowns per node, the computational efforts involved practically restrict the use of FEM to low-frequency applications. The method discussed in this paper is based on an indirect Trefftz approach. Exact solutions of the three coupled waves, supported by Biot’s equations, are used as basis functions in a solution expansion to approximate the field variables in a poroelastic boundary value problem. This approach leads to smaller systems of equations, enabling an efficient solution at higher frequencies.

A Trefftz based method for solving Helmholtz problems in semi-infinite domains

The wave based method (WBM), which is based on an indirect Trefftz approach, is a deterministic prediction method posed as an alternative to the element-based methods. It uses wave functions, which are exact solutions of the underlying differential equation, to describe the dynamic field variables. In this way, it can avoid the pollution errors associated with the polynomial element-based approximations. As a consequence, a dense element discretization is no longer required, yielding a smaller numerical system. The resulting enhanced computational efficiency of the WBM as compared to the element-based methods has been proven for the analysis of both bounded and unbounded acoustic problems. This paper extends the applicability of the WBM to semi-infinite domains. An appropriate function set is proposed, together with a calculation procedure for both semi-infinite radiation and scattering problems, and transmission or diffraction problems containing a rigid baffle. The resulting technique is validated on two numerical examples.

Vibro-acoustic design sensitivity analysis using the wave-based method

Conventional element-based methods, such as the finite element method (FEM) and boundary element method (BEM), require mesh refinements at higher frequencies in order to converge. Therefore, their applications are limited to low frequencies. Compared to element-based methods, the wave-based method (WBM) adopts exact solutions of the governing differential equation instead of simple polynomials to describe the dynamic response variables within the subdomains. As such, the WBM does not require a finer division of subdomains as the frequency increases in order to exhibit high computational efficiency. In this paper, the design sensitivity formulation of a semi-coupled structural–acoustic problem is implemented using the WBM. Here, the results of structural harmonic analyses are imported as the boundary conditions for the acoustic domain, which consists of multiple wave-based subdomains. The cross-sectional area of each beam element is considered as a sizing design variable. Then, the adjoint variable method (AVM) is used to efficiently compute the sensitivity. The adjoint variable is obtained from structural reanalysis using an adjoint load composed of the system matrix and an evaluation of the wave functions of each boundary. The proposed sensitivity formulation is subsequently applied to a two-dimensional (2D) vehicle model. Finally, the sensitivity results are compared to the finite difference sensitivity results, which show good agreement.

Modelling of sound propagation to three-dimensional urban courtyards using the extended fourier PSTD method

Noise from road traffic propagates to acoustically shielded areas as roadside courtyard by multiple reflection and diffraction paths in a complex three-dimensional (3D) environment. The computation of noise levels and assessment of candidate noise mitigation measures for these areas has up to now been based upon two-dimensional (2D) geometrical assumptions. Here, a recently developed efficient wave-based method, the extended Fourier pseudospectral time-domain (PSTD) method, is used to investigate the necessity of a 3D model. For frequencies up to 500 Hz and low traffic velocities of 30 km/h and 50 km/h, a road traffic noise configuration of an urban street canyon with or without cross streets and a closed roadside courtyard is compared to the 2D configuration as studied previously. It can be concluded that the contribution of distant sources is overpredicted by the 2D configuration. As noise mitigation measures, additional façade absorption, façade screens and roof screens have been studied. Results show that the 2D configuration underpredicts the effect of façade mitigation measures, by maximum 1.5 dB(A) for the absorption case and 4.4 dB(A) for the screens case. The effect of roof screens is overpredicted up to 1.7 dB(A). Given these deviations and the found deviations between the 3D configurations of street canyon with and without cross streets, the need for a 3D model can be concluded to be strongly configuration dependent. The 3D model is finally used to investigate the effect of a façade opening to the courtyard, which could lead to up to 10 dB(A) higher noise levels as compared to the noise propagating over the roof level and may prohibit the use of these courtyards as quiet areas. Absorption in the façade opening can significantly limit this negative effect.

On the analysis of vibro-acoustic systems in the mid-frequency range using a hybrid deterministic-statistical approach

At present, several numerical prediction techniques are available for the steady-state dynamic analysis of vibro-acoustic problems. Most of these techniques can be classified as either deterministic or statistical approaches. The use of the deterministic methods, like the Finite Element Method (FEM), is limited to the so-called low-frequency range due to the decreasing wavelengths and subsequent increasing model sizes and computational loads for increasing frequency. The underlying assumptions for the statistical methods, like the Statistical Energy Analysis (SEA), limit their use to the so-called high-frequency range. In-between the low- and high-frequency ranges there is a relatively wide mid-frequency-range, in which some of the subsystems fulfill the requirements for the statistical approach and some others do not (yet). Therefore, Shorter and Langley developed a deterministic-statistical framework combining FEM and SEA models. However, this method is limited in use due to the computational load of the FEM models. This paper proposes a framework for coupling Trefftz-based deterministic models with statistical SEA models. Instead of the FEM, the recently developed and computationally more efficient Wave Based Method (WBM) is being used. The presented method is verified in a numerical example.

Spline-based boundaries: A first step towards generic geometric domain descriptions for efficient mid-frequency acoustic analysis using the Wave Based Method

The application of numerical simulation techniques for the analysis and optimization of the acoustic behavior of all kinds of products has become very important in almost every phase of a design process. The large computational burden associated with the Finite Element Method (FEM) limits its applicability to low-frequency problems. Recently, the Wave Based Method (WBM) was proposed as an efficient alternative to the element based methods. This method is based on an indirect Trefftz approach, using an expansion of exact solutions of the governing differential equation to describe the dynamic field variables. An important disadvantage of the WBM is the limited geometrical flexibility as compared to the element based techniques. This paper aims to alleviate the geometrical restrictions by using B-splines for the efficient description of curved edges. The introduction of B-splines within the WBM requires an adaptation of the numerical integration procedure used to evaluate the weighted residual formulation. To this end, different types of numerical integration techniques are studied: the Gauss–Legendre and the Romberg integration procedure. A comparative study with the finite element method and the original WBM indicates that the application of B-splines and the adapted numerical integration procedure leads to accurate and computationally affordable WB models.

A wave-based anisotropic quadrangulation method

This paper proposes a new method for remeshing a surface into anisotropically sized quads. The basic idea is to construct a special standing wave on the surface to generate the global quadrilateral structure. This wave based quadrangulation method is capable of controlling the quad size in two directions and precisely aligning the quads with feature lines. Similar to the previous methods, we augment the input surface with a vector field to guide the quad orientation. The anisotropic size control is achieved by using two size fields on the surface. In order to reduce singularity points, the size fields are optimized by a new curl minimization method. The experimental results show that the proposed method can successfully handle various quadrangulation requirements and complex shapes, which is difficult for the existing state-of-the-art methods.

A wave-based anisotropic quadrangulation method

This paper proposes a new method for remeshing a surface into anisotropically sized quads. The basic idea is to construct a special standing wave on the surface to generate the global quadrilateral structure. This wave based quadrangulation method is capable of controlling the quad size in two directions and precisely aligning the quads with feature lines. Similar to the previous methods, we augment the input surface with a vector field to guide the quad orientation. The anisotropic size control is achieved by using two size fields on the surface. In order to reduce singularity points, the size fields are optimized by a new curl minimization method. The experimental results show that the proposed method can successfully handle various quadrangulation requirements and complex shapes, which is difficult for the existing state-of-the-art methods.

A multi-level wave based numerical modelling framework for the steady-state dynamic analysis of bounded Helmholtz problems with multiple inclusions

The Wave Based Method (WBM) is an alternative numerical prediction method for both interior and exterior steady-state dynamic problems, which is based on an indirect Trefftz approach. It applies wave functions, which are exact solutions of the governing differential equation, to describe the dynamic field variables. The smaller system of equations and the absence of pollution errors make the WBM very suitable for the treatment of Helmholtz problems in the mid-frequency range, where element-based methods are no longer feasible due to the associated computational costs. A sufficient condition for convergence of the method is the convexity of the considered problem domain. As a result, only problems of moderate geometrical complexity can be considered and some geometrical features cannot be handled at all. In this paper, these limitations are alleviated through the development of a general modelling framework based on existing WBM methodologies which allows for the efficient introduction of inclusion configurations in bounded WBM models for problems governed by one or more Helmholtz equations. The feasibility and efficiency of the method is illustrated by means of numerical verification studies in which the methodology is applied to two types of dynamic problems. On the one hand, a single Helmholtz equation associated with the steady-state dynamic behaviour of acoustic cavities is studied. On the other hand, the framework is applied to the solution of the Navier system of partial differential equations that describe the elastodynamic response of two-dimensional perforated solids.

Development of 2.5 dimensional wave-based numerical analysis method and its application to road traffic noise prediction

Development of 2.5 dimensional wave-based numerical analysis method and its application to road traffic noise prediction

Reprint of: Physics-Based Computational Methods for Aero-Acoustics

This paper provides a review of several physics-based methods developed for predicting sound propagation in flows. The basic principle of physics-based methods is to incorporate some known properties of the underlying physics into the numerical model. For instance, instead of using standard polynomials or Chebyshev polynomials to interpolate the solution, physics-based methods generally use canonical solutions such as Green's functions or plane waves to construct a local description of the solution. The methods described in this paper include the Green's function discretisation, the partition of unity finite element method and the wave-based discontinuous Galerkin method. The principles of these methods are described and their performance and shortcomings are discussed. A key issue that emerges in several of these methods is that the canonical solutions are only valid for uniform coefficients, while the methods are intended to be used with strongly inhomogeneous propagation media.

Physics-based computational methods for aero-acoustics

This paper provides a review of several physics-based methods developed for predicting sound propagation in flows. The basic principle of physics-based methods is to incorporate some known properties of the underlying physics into the numerical model. For instance, instead of using standard polynomials or Chebyshev polynomials to interpolate the solution, physics-based methods generally use canonical solutions such as Green’s functions or plane waves to construct a local description of the solution. The methods described in this paper include the Green’s function discretisation, the partition of unity finite element method and the wave-based discontinuous Galerkin method. The principles of these methods are described and their performance and shortcomings are discussed. A key issue that emerges in several of these methods is that the canonical solutions are only valid for uniform coefficients, while the methods are intended to be used with strongly inhomogeneous propagation media.

Highly parallel rigorous simulations of phase-shift masks with a generalized eigen-oscillation spectral element method

We apply a newly developed parallel generalized eigen-oscillation spectral element method (GeSEM) for rigorous simulations of 2-D phase-shift masks (PSMs). The GeSEM combines highly parallel Schwarz-domain decomposition iterations and an eigen-oscillation-based spectral method to model high-frequency oscillatory electromagnetic fields in PSMs with dispersive chrome materials in the PSMs. The performance of the GeSEM has been compared to the popular plane wave-based waveguide method for 2-D masks. The numerical results have clearly demonstrated the GeSEM's advantages in modeling the effects of nonperiodic structures such as optical images near mask edges, and its speedup through parallel implementations, which makes the simulation of a large-scale mask possible in whole-chip mask modeling.

A wave-based finite element analysis for acoustic transmission in fluid-filled elastic waveguides

This paper describes an original numerical prediction technique developed for the analysis of coupled vibro-acoustic problems in fluid waveguides. Specifically it is a wave-based method that adopts a spectral element approach. Unlike the conventional element-based methods, this technique uses wave functions that satisfy the governing equations to describe the dynamic variables exactly. One advantage is that fine domain discretizations, used by element-based methods near the fluid-structure interface typically, are no longer required. Hence the resulting model sizes are much smaller than element-based methods yielding a more time-efficient prediction technique that may allow handling of mid-frequency applications. Another advantage is that dispersion relations between propagating and excitation wavenumbers are easily obtained, and an example to show this, as well as a discussion on how the wave-based prediction technique can be used for two coupled vibro-acoustic problems: a cavity with a non-reflecting boundary and a silencer duct problem, is presented. Its beneficial characteristics compared to element-based methods are demonstrated through the validation study and transmission loss examples.

A novel modelling approach for sound propagation analysis in a multiple scatterer environment

In the past decades element‐based numerical modelling techniques have become a commonly used and invaluable tool for the analysis of acoustic radiation and scattering problems. However, the pollution errors associated with the element discretisation inherent to these methods increase with the increasing size of the problem domain. As a result, the applicability of these methods for radiation and scattering problems in which the source and receiver positions are located far from each other is often prohibited. The Wave Based Method (WBM) is an alternative deterministic prediction method for the analysis of steady‐state acoustic problems. It is based on an indirect Trefftz approach in that wave functions, which are exact solutions of the underlying differential equation, are used to describe the dynamic response. The enhanced computational efficiency of the WBM as compared to the element based methods has been shown already for the analysis of both finite and (semi‐)infinite acoustic problems. This paper introduces a novel WBM‐based methodology to model the acoustic source‐receiver transfer path functions in a multiple scatterer environment. A sound propagation validation case illustrates the potential of the proposed approach.

Bloch wave-based calculation of imaging properties of high-resolution scanning confocal electron microscopy

An efficient, Bloch wave-based method is presented for simulation of high-resolution scanning confocal electron microscopy (SCEM) images. The latter are predicted to have coherent nature, i.e. to exhibit atomic contrast reversals depending on the lens defocus settings and sample thickness. The optimal defocus settings are suggested and the 3D imaging capabilities of SCEM are analyzed in detail. In particular, by monitoring average image intensity as a function of the probe focus depth, it should be possible to accurately measure the depth of a heavy-atom layer embedded in a light-element matrix.