Semi-analytical Finite Element Method Research Articles

Ultrasonic guided waves in bone system with degradation

This paper investigates the feasibility of using ultrasonic guided waves for assessing the cortical bone and hence detect conditions such as osteoporosis. Guided wave propagation in bone systems modeled as multi-layered tubular structures consisting of anisotropic bone filled with viscous marrow and surrounded by tissue is studied using the Semi Analytical Finite Element (SAFE) method. Effects of changes to cortical bone thickness and mechanical properties are investigated. An attempt is also made to consider bone anisotropy in the models.The results, validated by experiments with bone phantoms, show that material and geometric condition strongly impacts the velocity of guided waves supported in the bone system. Identification of optimal guided wave modes for practical assessment is also discussed.

Ultrasonic guided waves in elliptical annular cylinders.

This paper studies the influence of cross-section ovalness or ellipticity on lower order axisymmetric guided wave modes in thin pipes. The second longitudinal mode L(0,2) and the fundamental torsional mode T(0,1) are studied, as these are of interest to current pipe inspection. The semi-analytical finite element (FE) method is mainly used, with three-dimensional FE simulations for visualization and cross-validation of results. The studies reveal that even a small degree of ovalness can affect mode shapes and velocities. The effect is more pronounced on the L(0,2) mode than on T(0,1) and this may be important for practical inspection applications.

Low frequency axisymmetric longitudinal guided waves in eccentric annular cylinders.

This paper studies the effect of axially uniform eccentricity on the modal structures and velocities of the lower order axisymmetric guided wave mode L(0,2) in circular tubes or pipes. The semi-analytical finite element method is mainly used, supported by fully three-dimensional finite element models and validated using experiments. The studies show that even a small eccentricity in the pipe can cause a loss in the L(0,2) mode axisymmetry, leading to its confinement in the thinned side of the pipe cross-section and also a reduction in mode velocities. The physics of this phenomenon is related to the feature-guiding and mode confinement effects noted in recent years in the literature, particularly studies on waveguides with local cross-section variations and curvature.

Numerical investigations of ultrasound wave propagating in long bones using a poroelastic model

Ultrasonic responses probed from an axial transmission test (ATT) may provide useful information about material and structural properties of cortical bone. For the mathematical modeling of ultrasonic wave propagation in long bones, most of studies assumed an (visco-)elastic behavior for cortical bone tissue by neglecting the interstitial pressure in the pores presented within this material. Here, a functionally graded anisotropic poroelastic model is proposed for describing the behavior of long bones in the ultrasonic frequency range. The simulation of time-domain wave propagation can efficiently be carried out by using a semi-analytical finite element method. The proposed model allows us investigate the influence of the presence of the pores, as well as their distribution in a bone layer on the speed of sound propagated in a cortical bone layer coupled with the marrow and the soft tissue. The effects of emitted signal’s frequency will also be examined.

Solving of spatial problem of non-stationary heat conduction based on semi-analytical finite element method

Based on the semi-analytical finite element method it is developed solution for solving of spatial problem of non-stationary heat conduction for prismatic bodies of complex shape cross-section. The basis of the initial equation is correlations of spatial unsteady heat conduction problem in curvilinear coordinates in differential and variational formulations. The formula for determining the matrix elements of thermal conductivity and heat capacity based on semi-analytical finite element method are obtained based on the presentation of temperature distribution along the coordinate by x 3 Mikhlin polynomials allowing to use effective algorithm of the iterations block of upper relaxation systems for linear algebraic equations and implement efficient algorithm solution for solution system of differential equations in time. The value of semi-analytical finite element method and algorithms implemented in the form of problem-oriented subsystems for computer modeling of unsteady thermal processes are obtained. The reliability of results is substantiated by the interpretation of test cases with analytical and numerical results.

Transient analysis of leaky Lamb waves with a semi-analytical finite element method

We previously formulated a semi-analytical finite element technique for Lamb waves in a plate surrounded by fluids and investigated the dispersion curves and wave structures for leaky Lamb waves. Herein, this technique is extended to the calculation of transient responses both in a plate and in fluids for dynamic loading on the plate surface. To gain fundamental insights into guided wave inspection for a water-filled pipe or tank, guided waves generated upon transient loading of a flat plate water-loaded on one side were analyzed. The results show that a quasi-Scholte mode propagating at the plate-water interface is useful for the long-range inspection of a water-loaded plate because of its non-attenuation and minimal dispersion; moreover, this mode has superior generation efficiency in the low-frequency range, while it is localized near the plate-water interface at higher frequencies.

On the study of vibration of a supported railway rail using the semi-analytical finite element method

An improved rail vibration model based on the semi-analytical finite element (SAFE) method has been developed, which allows multiple layers of support and the accurate shape of the rail cross-section to be considered in the modeling. Rail supports are treated as a continuous layer of equivalent springs connected to the rail foot. Three different assumed support conditions corresponding to no support; rail pads; and rail pads, sleepers and ballast have been modeled. By determining the track parameters from a field test, this model is demonstrated to be able to determine the dispersion relations and the forced responses of both free and variously supported rails. Further, the effect of rail damping including frequency-dependent viscous damping and frequency-independent structural damping on rail vibration responses is investigated. The effect of the adopted discretization strategy on the calculation is also studied. Model results are compared with predictions from the Timoshenko beam-based rail models with the same support conditions. The results show that track support dynamic stiffness can be incorporated into the developed SAFE rail model which can then be used to represent the real track situation at a site; the rail support can significantly affect rail dispersion and frequency responses below 1kHz; the Timoshenko beam-based rail models provide an inadequate prediction of rail vibration response, particularly in the lateral direction, if response at specific frequencies is required; the rail damping has limited impact on rail vibration responses; and the number of elements has insignificant effect on the calculated rail responses and wave dispersion relations, especially away from the excitation point.

Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods

Among the numerous techniques of non-destructive evaluation, elastic guided waves are of particular interest to evaluate defects inside industrial and civil elongated structures owing to their ability to propagate over long distances. However for guiding structures buried in large solid media, waves can be strongly attenuated along the guide axis due to the energy radiation into the surrounding medium, usually considered as unbounded. Hence, searching the less attenuated modes becomes necessary in order to maximize the inspection distance. In the numerical modeling of embedded waveguides, the main difficulty is to account for the unbounded section. This paper presents a numerical approach combining a semi-analytical finite element method and a perfectly matched layer (PML) technique to compute the so-called trapped and leaky modes in three-dimensional embedded elastic waveguides of arbitrary cross-section. Two kinds of PML, namely the Cartesian PML and the radial PML, are considered. In order to understand the various spectral objects obtained by the method, the PML parameters effects upon the eigenvalue spectrum are highlighted through analytical studies and numerical experiments. Then, dispersion curves are computed for test cases taken from the literature in order to validate the approach.

Guided wave propagation and scattering in pipeworks comprising elbows: Theoretical and experimental results

Elastic guided waves (GW) are used to inspect pipeworks in various industries. Modelling tools for simulating GW inspection are necessary to understand complex scattering phenomena occurring at specific features (welds, elbows, junctions...). In pipeworks, straight pipes coexist with elbows. GW propagation in the former cases is well-known, but is less documented in the latter. Their scattering at junction of straight and curved pipes constitutes a complex phenomenon. When a curved part is joined to two straight parts, these phenomena couple and give rise to even more complex wave structures. In a previous work, the SemiAnalytic Finite Element method extended to curvilinear coordinates was used to handle GW propagation in elbows, combined with a mode matching method to predict their scattering at the junction with a straight pipe. Here, a pipework comprising an arbitrary number of elbows of finite length and of different curvature linking straight pipes is considered. A modal scattering matrix is built by cascading local scattering and propagation matrices. The overall formulation only requires meshing the pipe section to compute both the modal solutions and the integrals resulting from the mode-matching method for computing local scattering matrices. Numerical predictions using this approach are studied and compared to experiments.

Application of semi-analytical finite element method coupled with infinite element for analysis of asphalt pavement structural response

A specific computational program SAFEM was developed based on semi-analytical finite element (FE) method for analysis of asphalt pavement structural responses under static loads. The reliability and efficiency of this FE program was proved by comparison with the general commercial FE software ABAQUS. In order to further reduce the computational time without decrease of the accuracy, the infinite element was added to this program. The results of the finite-infinite element coupling analysis were compared with those of finite element analysis derived from the verified FE program. The study shows that finite-infinite element coupling analysis has higher reliability and efficiency.

Semi-analytical FEM Analysis and Experimental Measurement for Propagating Guided Waves Modes in Rail

Acoustical guided waves modal analysis in rail is the basis of long distance guided wave inspection and rail vibration and noises control. In particular the various propagating guided waves in the rail are identified in terms of their propagation wavenumber coefficients and their corresponding deformed shapes. The semi-analytical finite element method is used to formulate the governing equation for guided waves propagating in elastic waveguides of arbitrary cross-section on the basis of the virtual work principle, wavenumber dispersion curves and deformed shapes are extracted by solving governing equation, which provide the basis for selecting mode and frequency during application. The propagating guided waves wavenumber dispersion curves and deformed shaped under 0-8 k Hz frequency range for CHN60 type free rail are obtained using semi-analytical finite element method, and the characteristics of the eight based guided waves mode in rail are discussed. Instrumented hammer experimental study and modal analysis using sever vibration accelerometers mounted on rail are carried out to verify the vertically vibrating mode and horizontally vibrating mode guided waves wavenumber dispersion coefficient. Modeling and experimental investigations of lateral and vertical excitation in railhead shows that good agreement between numerical calculation and experiment.

J0420203 半解析的有限要素法による漏洩ラム波の過渡応答解析

J0420203 半解析的有限要素法による漏洩ラム波の過渡応答解析

616 有限幅を有するCFRP接着構造におけるラム波を用いた剥がれ検知

Debonding in a skin-stringer panel is one of the main problems in actual application of carbon fiber reinforced plastic (CFRP) composite. Although an ultrasonic structural health monitoring technique using mode conversions of Lamb waves can evaluate this defect effectively in infinite-width CFRP laminates, this method cannot be applied directly to the bonded structure with a finite-width stringer. Hence, in this research, the mode conversion behavior in a simple finite-width CFRP laminate with an artificial delamination was investigated as a basic study. The theoretical dispersion curves of Lamb waves in the finite-width laminate were calculated by a semi-analytical finite element method. Then, the experiments were conducted at various frequencies with an acousto-ultrasonic system based on macro fiber composite actuators and fiber Bragg grating sensors for the investigation of Lamb wave propagation behavior in the finite-width laminate. The results indicated that mode conversion occurs in the delaminated area of the finite-width CFRP laminate. Therefore, the mode conversion of Lamb waves has a possibility to detect delamination damage in finite-width composite structures.

Effect of Dimensional Degradation on Ultrasonic Guided Waves in Bone System

This paper investigates the feasibility of using ultrasonic guided waves for assessing the mechanical properties of human bone towards realizing a safe and economical non-invasive method for the detection of conditions such as osteoporosis. Effects of dimensional (changes in thickness) degradation in the cortical bone on guided wave characteristics are studied using numerical simulation and analysis. Guided wave propagation in tubular multi-layered waveguides mimicking the bone system is studied using the Semi Analytical Finite Element (SAFE) method. Uniform tri-layered structures consisting of bone-like tubes filled with marrow and surrounded by tissue are considered for the models. The results show that geometric condition strongly impacts the velocity of guided waves supported in the bone system. The impact of underlying assumptions in these models and identification of suitable guided wave modes for practical assessment of bone condition are also discussed.

A semianalytical finite element analysis of electromagnetic propagation in stratified media

ABSTRACTA semianalytical finite element method (FEM) is proposed for analyzing propagation of electromagnetic waves in stratified media. Conventional finite elements are used to discretize the cross‐section of a layered structure, and a Riccati equation‐based integration scheme is used to handle the longitudinal direction with a very high precision. This semianalytical FEM is easy to implement. Numerical examples demonstrate that the proposed method has high efficiency and accuracy. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:15–18, 2015

Numerical investigation of leaky modes in helical structural waveguides embedded into a solid medium

Helical multi-wire cables are widely used in bridges (suspended or prestressed) and anchored retaining wall constructions. Such structures can be damaged or degraded due to corrosion and fatigue. Non destructive evaluation techniques are required to reveal defects inside cable structures. Among these numerous techniques, elastic guided waves are of potential interest owing to their ability to propagate over long distances. However in civil engineering, cables are often buried or grouted in large solid media that can be considered as unbounded. Waves can strongly attenuate along the guide axis due to the energy leakage into the surrounding medium, which reduces the propagating distance. This energy leakage can be enhanced in helical structures, which further complicates their inspection. Searching modes with low attenuation becomes necessary. The goal of this work is to propose a numerical approach to compute modes in embedded helical structures, combining the so-called semi analytical finite element method and a radial perfectly matched layer technique. Two types of radial perfectly matched layer, centered and off-centered, are considered. Both are implemented in a twisting coordinate system which preserves translational invariance. The centered configuration is validated thanks to the twisted cylinder test case. The effect of twist on the eigenspectrum is briefly discussed. Then, an embedded helical wire of circular cross-section is considered. The off-centered configuration is shown to give the same results as the centered one. The effect of twist on modal attenuation is investigated. Finally, computations are performed for a seven-wire strand embedded into concrete, widely used in civil engineering cables.

Symmetric low-frequency feature-guided ultrasonic waves in thin plates with transverse bends

Recent research by the authors shows that bends in plates can act as features that can concentrate and guide ultrasonic energy along their axis. At low frequencies, two feature-guided modes are identified when the bent plate is subjected to ‘in-plane’ or axial excitation applied uniformly along a through-thickness line bisecting the bent edge. Of these, the slower mode has properties similar to the A0 (fundamental antisymmetric) Lamb mode in flat plates. This paper focuses on the faster bend-guided mode that is similar to the S0 (fundamental symmetric) Lamb mode in flat plates. Using 3D finite element (FE) simulation validated with experiments, this mode is shown to be more strongly generated in smaller angle bends. Features of the mode including velocity, attenuation and modal structure are considered in detailed studies. Results are discussed in light of simple modal studies using the Semi Analytical Finite Element (SAFE) method, suggesting a relationship of bend-guided waves to modes of curved bars.

Reciprocity principle for scattered fields from discontinuities in waveguides

This study investigates the scattering of guided waves from a discontinuity exploiting the principle of reciprocity in elastodynamics, written in a form that applies to waveguides. The coefficients of reflection and transmission for an arbitrary mode can be derived as long as the principle of reciprocity is satisfied at the discontinuity. Two elastodynamic states are related by the reciprocity. One is the response of the waveguide in the presence of the discontinuity, with the scattered fields expressed as a superposition of wave modes. The other state is the response of the waveguide in the absence of the discontinuity oscillating according to an arbitrary mode. The semi-analytical finite element method is applied to derive the needed dispersion relation and wave mode shapes. An application to a solid cylinder with a symmetric double change of cross-section is presented. This model is assumed to be representative of a damaged rod. The coefficients of reflection and transmission of longitudinal waves are investigated for selected values of notch length and varying depth.

Finite element computation of trapped and leaky elastic waves in open stratified waveguides

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.

Calculation of leaky Lamb waves with a semi-analytical finite element method

A semi-analytical finite element method (SAFE) has been widely used for calculating dispersion curves and mode shapes of guided waves as well as transient waves in a bar like structures. Although guided wave inspection is often conducted for water-loaded plates and pipes, most of the SAFE techniques have not been extended to a plate with leaky media. This study describes leaky Lamb wave calculation with the SAFE. We formulated a new solution using a feature that a single Lamb wave mode generates a harmonic plane wave in leaky media. Dispersion curves obtained with the SAFE agreed well with the previous theoretical studies, which represents that the SAFE calculation was conducted with sufficient accuracy. Moreover, we discussed dispersion curves, attenuation curves, and displacement distributions for total transmission modes and leaky plate modes in a single side and both two side water-loaded plate.