Semi-analytical Finite Element Method Research Articles

Calculation of Cylindrical Corrugated Shells Using a Semi-Analytic Finite Element Method

The method of calculating the shells of revolution for an asymmetric load using a semi-analytic finite element method is considered. An example is given of calculating a corrugated cylindrical tank for wind load. Comparison of meridional and annular forces is performed for a smooth and corrugated shell of identical dimensions.

Investigation of nonlinear ultrasonic guided waves in open waveguides based on perfectly matched layers.

Nonlinear ultrasonic guided waves have been investigated widely in closed waveguides such as plates, pipes, etc. However, the description of nonlinear ultrasonic guided waves remains challenging for open waveguides, as energy may leak into the surrounding medium. In this work, the properties of nonlinear ultrasonic guided waves in open waveguides are investigated. Mathematical framework is first established based on real reciprocity relation and modal expansion with perfectly matched layers. Numerical models are then implemented, including nonlinear semi-analytical finite element (SAFE) method to predict the properties of nonlinear ultrasonic guided waves, and time domain finite element models to simulate the nonlinear guided wave propagation and cross validate the predictions from the nonlinear SAFE method. Two examples, an aluminum plate attached to an elastomer and an aluminum plate with water loaded on one side, are studied to demonstrate the proposed methods and reveal some interesting phenomena that only exist in open waveguides. It is interesting to find out that the amplitude of the attenuated second harmonic wave in immersed waveguides can keep constant with propagation distance, only if the primary wave is non-leaky, which may bring potential non-destructive test applications for underwater inspections. Such a feature is validated by well-designed experiments in one-sidedly immersed plates.

Semianalytic Finite-Element Method in Dynamic Problems of Linear Fracture Mechanics

Semianalytic Finite-Element Method in Dynamic Problems of Linear Fracture Mechanics

Inverse characterization of radius bone using low-frequency ultrasonic guided waves

The success of the cortical bone characterization using the axial transmission is highly dependent on the inversion model used to match experimental data with the true bone properties. Simplified models such as plate or cylinder are typically used in the literature. In our previous work, a more elaborate model based on a bone-like geometry built using semi-analytical finite element (SAFE) method was introduced. The model was successfully implemented in an inverse characterization routine using laboratory-controlled measurements on bone phantoms. Thus, the aim of this work is to apply the proposed inverse scheme on the characterization of ex-vivo radius samples. In order to do so, five radiuses were taken from donors aged between 53 and 88 years old and tested using typical axial transmission configuration. Ultrasonic guided wave modes were excited at low-frequency (100-400 kHz) using a piezoelectric transducer and measured using a laser interferometer at the middle of the diaphysis. The measured velocities were systematically compared to velocities obtained with the SAFE model in order to predict the cortical bone properties. For each sample, four parameters were estimated: (1) Young’s modulus; (2) density; (3) thickness; and (4) outer diameter. The results showed a notable correlation of the thickness and outer diameter with respect to the μCT images of the samples, while a less significant correlation was observed for the Young’s modulus and density with respect to the gray level of the μCT images.

Model-based inversion of ultrasonic guided waves for cortical bone properties

Axial transmission ultrasonography, which uses a set of transmitting and receiving probes placed on the same waveguide’s surface, shows the potential clinical application for cortical bone quality assessment. In this work, a model-based parameter sweep inversion approach has been developed to estimate the thickness and elastic velocities of the cortex from the dispersive axially-transmitted snapshots. The inversion algorithm is formulated in the frequency-phase velocity (f-c) domain. To solve the inverse problem, i.e., to extract bone properties from ultrasound data, a forward modeling has been developed to simulate the f-c dispersion curves given a bone model. A semi-analytical finite element (SAFE) method is used to compute the dispersion curves for a complex structure of a cortical bone plate coupled with overlying soft tissues. A parameter sweep is used to seek within a range of values an optimized solution with the least misfit. The proposed method optimizes the mismatch between the measured and theoretically calculated dispersion curves with a least-square constraint. Numerical and in-vivo experimental data examples are presented to illustrate the technique’s performance.

Stressed Cylinder Dispersion Curves Based on Effective Elastic Constants and SAFE Method

A Semi-Analytical Finite Element (SAFE) formulation is applied to determinethe dispersion curves in homogeneous and isotropic cylindrical waveguides subject touniaxial stress. Bulk waves are required for estimating the guided wave dispersion curvesand acoustoelasticity states a stress dependence of the ultrasound bulk velocities. Therefore,acoustoelasticity influences the wave field of the guided waves. Effective Elastic Constants(EEC) has emerged as a less complex alternative to deal with the acoustoelasticity; allowinga stressed material to be assumed as an unstressed material with EEC which considers thedisturbance linked to the presence of stress. In this approach the isotropic specimen subjectto load is studied by proposing an equivalent stress-free with a modified elasticity matrixwhich terms are the EEC. EEC provides an approximate stress-strain relation facilitating thedetermination of the dispersion curves using the well-studied numerical solution for the stressfreecases reducing the complexity of the numerical implementation. Therefore, a numericalmethod combining the SAFE and EEC is presented as a tool for the dispersion curve generationin stressed cylindrical specimens. The results of this methodology are verified by comparingthem with an approach previously reported in the literature based on SAFE including the fullstrain-displacement relation

An Arnoldi reduction strategy applied to the semi-analytical finite element method to model railway track vibrations

Abstract Modelling the vibrational response of railway tracks is a mandatory step to predict rolling noise. This article proposes a model based on the semi-analytical finite element method (SAFE) taking into account the three main characteristics of railway tracks: infinite length in the longitudinal direction, deformable cross-section, and periodic track support. Although this approach is ideally suited to predict rail vibrations, numerical instabilities undermine its reliability at low frequencies. Consequently, a new approach using the second-order Arnoldi reduction (SOAR) is used, which allows to provide the required numerical stability. Lastly, numerical results are compared with classical railway track behaviours described in the literature, as well as experimental data and 3D finite element simulations.

Analysis of damped guided waves using the method of multiple scales

We analytically investigate the influence of damping on Lamb waves, which are a specific type of guided wave in two-dimensional plates. Considering material attenuation, we suppose that Lamé constants are complex numbers. This leads to the associated wavenumbers being complex, with the imaginary part of the wavenumber being associated with effect of attenuation of the guided wave. In this paper, we show how dispersion curves and attenuation coefficients can be obtained using the self-adjointness and the method of multiple scales (MMS), which is a type of perturbation method. Using the self-adjointness and the MMS, we can calculate the frequency- and wavenumber-dependent attenuation coefficients from the integral values and boundary values of a corresponding eigenfunction with respect to each propagation mode. This analytical method can yield not only dispersion curves but also mode-by-mode attenuation coefficients regardless of the numerical initial values, unlike numerical approaches using the Newton method. Thus, the proposed method can more easily calculate the attenuation coefficients with respect to a particular mode than conventional methods. Furthermore, the results obtained by proposed method were in good agreement with those obtained by the semi-analytical finite element (SAFE) method, which validates the proposed method.

Validity of a Modified Method of Evaluating the Invariant J-integral for Elastoplastic Deformation of Prismatic Solids

The invariance and validity of the evaluation of the J-integral by a modified reaction method in problems of elastoplastic fracture are analyzed based on numerical experiments. Bodies with mode I cracks undergoing elastoplastic deformation under simple static loading without unloading, which corresponds to the deformation theory of plasticity, are considered. To demonstrate the universality of the method of evaluating the J-integral to finite-element schemes, prismatic bodies are considered, which allows using not only the conventional FEM scheme but also the semi-analytic finite-element method (SFEM).

Accurate characterization of 3D dispersion curves and mode shapes of waves propagating in generally anisotropic viscoelastic/elastic plates

We investigate wave propagation in generally anisotropic viscoelastic plates. The generally anisotropic material system has 21 independent complex coefficients in its stiffness matrix when viscoelasticity is taken into account. The dispersion equation is obtained in the form of determinant vanishing in the complex domain based on the straightforward derivation. An accurate root-searching in the dispersion equation for arbitrary ranges of frequencies and complex wavenumbers, however, is very difficult, if not impossible. In this paper, a novel algorithm is introduced to calculate the 3D dispersion curves. An approximate solution in the low attenuation range via the semi-analytical finite element (SAFE) method is also used to compare and validate the introduced algorithm. Using this algorithm, various peculiar wave features are then investigated in details. These include the attenuation jump and branches exchange in viscoelastic model caused by conversion of wave mode shapes, and the veering of dispersion branches in the corresponding elastic medium. The general anisotropic (viscoelastic/elastic) models are further compared with the isotropic ones to identify the similarity and difference on wave features between them. The proposed accurate algorithm along with the observed features should be particularly useful in nondestructive evaluations via waves in viscoelastic/elastic plates and structures.

Analysis of dynamic stress path in inhomogenous subgrade under moving aircraft load

Considering the cut and fill alternating characteristic of airport subgrade in mountainous area, especially in the runway direction, a semi-analytic finite element method was proposed to analyze the dynamic response of inhomogenous subgrade under moving aircraft load. This method can decrease the computational cost effectively by using the Fourier Transform in the transverse direction of the runway to reduce the dimension of the 3D problem and account for the traveling wave effect of moving load. The dynamic responses of a soil element in homogenous and inhomogeneous subgrade subjected to single-wheel aircraft load were analyzed comparatively. The results show that the dynamic stresses generated in soil element are mainly vertical normal stress and in-plane shear stress, forming stress paths with the characteristic of complex principal cyclic stress axis rotation. Due to the traveling wave effect, the dynamic responses of soil are not antisymmetric when the moving load approaches and leaves the observation point. In the homogeneous subgrade, the dynamic stress path in the plane of the two shear stress components is egg-shaped with a tilting angle. As the moving load velocity grows, the long axis of the egg-shaped stress path rotates counterclockwise. In the inhomogeneous subgrade, the dynamic stress is concentrated near the junction of cut and fill areas, and the peak value of the vertical dynamic stress is 1.35 times that in the homogeneous subgrade. Moreover, the stress path near the junction tends to be more prolate and spindle-shaped.

On the Strain–Stress State of Locally Loaded Layered Composite Slabs

Two versions of the semi-analytical finite-element method are developed within the framework of the three-dimensional theory of elasticity. In the first version, the unknown functions are subject to a polynomial finite-element approximation in the planar coordinate X, expanded into Fourier series in the coordinate Y, and are approximated by known polynomials in coordinate Z (thickness). In the second version, linear polynomials are used for the approximation in the coordinate X and the Fourier series are applied in the coordinate Y. The distributions of the unknown functions over the thickness are obtained by analytically solving the appropriate system of differential equations. The reason of developing the two versions of the method is their approximation and arithmetic errors. The stress–strain state of a layered composite slab under local normal and tangential loading is calculated. The site of load application is square with side length equal to the slab thickness. Three types of boundary conditions on a slab bottom surface (free, clamped and resting on an elastic foundation) are considered.

Ultrasonic guided wave focusing in waveguides with constant irregular cross-sections

As essential components of a high-speed railway system, switch rails can be easily damaged by sophisticated operating conditions. Therefore, precise online detection for switch rails is necessary. Methods based on ultrasonic guided waves are ideal candidates for structural integrity of the switch rails, which are natural waveguides with irregular cross-sections. However, energy decentralization in the wave propagation severely restricts detectability. Phased array systems have been developed and implemented to steer and focus acoustic energy in waveguides of ordinary cross-sections such as pipes and plates. This paper proposes a method for ultrasonic guided wave focusing in waveguides with constant irregular cross-sections. We analyzed the characteristics of the guided waves generated by partial loadings based on a semi-analytical finite element method (SAFEM). An algorithm was developed for calculating the amplitude weights and time delays required to modulate excitation signals. Two coefficients were defined to evaluate the wave focusing results, namely the half area coefficient (HAC) and half energy coefficient (HEC). Numerical simulations to verify the proposed method were carried out for a switch rail base with a constant irregular cross-section. The results demonstrate that the guided acoustic beam has been effectively steered to focus at the pre-determined locations with enhanced acoustic wave energy. Furthermore, the influence of various factors on guided wave focusing was studied. Excitation signals of low center frequencies with narrow bandwidths are recommended for ideal focusing results.

A nonlinear grid-search inversion for cortical bone thickness and ultrasonic velocities

Cortical thickness and elasticity are important determinants of bone strength. This study proposes a nonlinear grid-search inversion to estimate the thickness and ultrasonic velocities of long cortical bones from axially-transmitted data. The inversion scheme has been developed in the frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires to systematically examine a set of trial dispersion curves within a pre-defined model space to match the data which minimizes the objective function in a least-squares sense. The theoretical dispersion curves required by the inversion are computed for bilayered bone models using a semi-analytical finite-element method. Its application demonstrates the feasibility of the proposed approach on synthetic data for a 5 mm-thick bone plate, and in-vitro dataset from a bovine femur plate with 2 mm-thick soft-tissue mimic on top. Our results indicate that one can recover the cortical thickness and wave speeds with less than 5% error. The accuracy can be improved by refining the grid; however, the computational cost of the method will increase. In the future, we aim at applying the method to clinical data.

Dispersion curves for Lamb wave propagation in prestressed plates using a semi-analytical finite element analysis.

Acoustoelastic techniques have been recently used to characterize the state of prestress in structures such as plates. The velocity of guided wave modes propagating through plates is sensitive to the magnitude and orientation of the initial state of stress. Dispersion curves for phase velocities of plate guided waves can be computed using the superposition of partial bulk waves (SPBW) method. Here, a semi-analytical finite element (SAFE) method is formulated for the acoustoelastic problem of guided waves in weakly nonlinear elastic plates. The SAFE formulation is shown to provide phase velocity dispersion curve results identical with those provided by the SPBW method for the problem of a plate under a uniaxial and uniform tensile stress. Analytical phase and group velocity dispersion curves are also obtained for a plate with an initial prestress gradient through its thickness using the SAFE method. The magnitude of the prestress gradient is shown to have a significant effect on phase and group velocities of the fundamental and first order Lamb modes, only in certain frequency-thickness regimes.

Semi-analytical discontinuous Galerkin finite element method for the calculation of dispersion properties of guided waves in plates.

The development of reliable guided waves inspection systems is conditioned by an accurate knowledge of their dispersive properties. The semi-analytical finite element method has been proven to be very practical for modeling wave propagation in arbitrary cross-section waveguides. However, when it comes to computations on complex geometries to a given accuracy, it still has a major drawback: the high consumption of resources. Recently, discontinuous Galerkin finite element method (DG-FEM) has been found advantageous over the standard finite element method when applied as well in the frequency domain. In this work, a high-order method for the computation of Lamb mode characteristics in plates is proposed. The problem is discretised using a class of DG-FEM, namely, the interior penalty methods family. The analytical validation is performed through the homogeneous isotropic case with traction-free boundary conditions. Afterwards, functionally graded material plates are analysed and a numerical example is presented. It was found that the obtained results are in good agreement with those found in the literature.

An Ultrasonic Guided Wave Mode Excitation Method in Rails

This paper proposes a new method for exciting a specific mode of guided ultrasonic wave in a rail. The method requires the cross-sectional mode shape to determine excitation direction, excitation coefficient, and excitation position of each mode. The cross-sectional mode shape data of all modes are acquired by the semi-analytical finite element (SAFE) method. The optimum excitation direction is obtained by solving the Euclidean distance between different modes. According to the shape feature of a rail, the cross section of a rail is divided into three segments, including the rail base, the rail web, and the rail head. The excitation coefficients are obtained by analyzing vibration characteristics in these three segments. The covariance matrix of each mode at a specific frequency is established, and the result is used to obtain the best excitation point. The response to an excitation in the rail can then be computed by the SAFE method. It has been verified that the excitation method proposed in this paper can excite a specific ultrasonic-guided wave mode at the low frequency of 200 Hz and the high frequency of 35 kHz, respectively.

Application of Linear Viscoelastic Properties in Semianalytical Finite Element Method with Recursive Time Integration to Analyze Asphalt Pavement Structure

Traditionally, asphalt pavements are considered as linear elastic materials in finite element (FE) method to save computational time for engineering design. However, asphalt mixture exhibits linear viscoelasticity at small strain and low temperature. Therefore, the results derived from the elastic analysis will inevitably lead to discrepancies from reality. Currently, several FE programs have already adopted viscoelasticity, but the high hardware demands and long execution times render them suitable primarily for research purposes. Semianalytical finite element method (SAFEM) was proposed to solve the abovementioned problem. The SAFEM is a three‐dimensional FE algorithm that only requires a two‐dimensional mesh by incorporating the Fourier series in the third dimension, which can significantly reduce the computational time. This paper describes the development of SAFEM to capture the viscoelastic property of asphalt pavements by using a recursive formulation. The formulation is verified by comparison with the commercial FE software ABAQUS. An application example is presented for simulations of creep deformation of the asphalt pavement. The investigation shows that the SAFEM is an efficient tool for pavement engineers to fast and reliably predict asphalt pavement responses; furthermore, the SAFEM provides a flexible, robust platform for the future development in the numerical simulation of asphalt pavements.

A Numerical Study on the Excitation of Guided Waves in Rectangular Plates Using Multiple Point Sources

Ultrasonic guided waves are widely used to inspect and monitor the structural integrity of plates and plate-like structures, such as ship hulls and large storage-tank floors. Recently, ultrasonic guided waves have also been used to remove ice and fouling from ship hulls, wind-turbine blades and aeroplane wings. In these applications, the strength of the sound source must be high for scanning a large area, or to break the bond between ice, fouling and plate substrate. More than one transducer may be used to achieve maximum sound power output. However, multiple sources can interact with each other, and form a sound field in the structure with local constructive and destructive regions. Destructive regions are weak regions and shall be avoided. When multiple transducers are used it is important that they are arranged in a particular way so that the desired wave modes can be excited to cover the whole structure. The objective of this paper is to provide a theoretical basis for generating particular wave mode patterns in finite-width rectangular plates whose length is assumed to be infinitely long with respect to its width and thickness. The wave modes have displacements in both width and thickness directions, and are thus different from the classical Lamb-type wave modes. A two-dimensional semi-analytical finite element (SAFE) method was used to study dispersion characteristics and mode shapes in the plate up to ultrasonic frequencies. The modal analysis provided information on the generation of modes suitable for a particular application. The number of point sources and direction of loading for the excitation of a few representative modes was investigated. Based on the SAFE analysis, a standard finite element modelling package, Abaqus, was used to excite the designed modes in a three-dimensional plate. The generated wave patterns in Abaqus were then compared with mode shapes predicted in the SAFE model. Good agreement was observed between the intended modes calculated in SAFE and the actual, excited modes in Abaqus.

Mode repulsion of ultrasonic guided waves in rails

Accurate computation of dispersion characteristics of guided waves in rails is important during the development of inspection and monitoring systems. Wavenumber versus frequency curves computed by the semi-analytical finite element method exhibit mode repulsion and mode crossing which can be difficult to distinguish. Eigenvalue derivatives, with respect to the wavenumber, are used to investigate these regions. A term causing repulsion between two modes is identified and a condition for two modes to cross is established. In symmetric rail profiles the mode shapes are either symmetric or antisymmetric. Symmetric and antisymmetric modes can cross each other while the modes within the symmetric and antisymmetric families do not appear to cross. The modes can therefore be numbered in the same way that Lamb waves in plates are numbered, making it easier to communicate results. The derivative of the eigenvectors with respect to wavenumber contains the same repulsion term and shows how the mode shapes swop during a repulsion. The introduction of even a small asymmetry appears to lead to repulsion forces that prevent any mode crossings. Measurements on a continuously welded rail track were performed to illustrate a mode repulsion.