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

In guided wave inspection, dispersion curves and wave structures are indispensable for designing inspection tools and analyzing waveforms. This report describes solutions of the dispersion curves and wave structures with a semi-analytical finite element method and discusses the applications of the dispersion curves and wave structures in guided wave inspection. Formularization of the semi-analytical finite element method revealed that external dynamic loading should be distributed in a shape similar to the wave structure of the desired mode and enabled the analysis of excitation potentials of desired modes with an angle beam transducer. Using a semi-analytical finite element analysis, guided wave tools for inspecting bottom-edges of rails were developed and gave good results for practical use.

Modeling lamb wave propagation in visco-elastic composite plates using a fifth-order plate theory

The enormous potential of Guided waves have a significant use for non-destructive evaluation(NDE) and structural health monitoring(SHM). It is common to use different solution approaches to predict dispersion curves depending on the material type, isotropic, transversely isotropic, or orthotropic, of the medium in which the wave propagates, shape of the waveguide, etc. Inability to model waveguides of arbitrary cross-section by existing matrix methods has led to development of Semi-Analytical Finite Element(SAFE) Method. In this study, rapid and accurate prediction of the propagation characteristics for guided waves in a wide range of structures has been performed by implementation of SAFE method. Damped and undamped waveguides of different shapes can be modelled. Wave propagation characteristics in both isotropic and anisotropic composite structures can also be studied. To benchmark the solution, a case study using Transfer Matrix Method (TMM) for composite plate has been evaluated. It was found that the element discretizaton plays an important role in accurate prediction of modes in SAFE method, neglecting which leads to loss of higher order modes.

Using a low frequency range in long-range inspection of a plate with Lamb waves, directivity of Lamb waves becomes weak and Lamb wave propagation is affected by the both sides of a plate. Thus, dispersion curves for a plate with a finite width should be used in nondestructive testing for it. Since dispersion curves for a plate with a finite width cannot be analytically obtained, guided wave calculation technique developed by authors using a semi-analytical finite element method is useful. In this study, therefore, theoretical dispersion curves for a plate with a finite width were derived using the calculation technique, and the characteristic of dispersion curves and the relationship between dispersion curves and vibration distributions were described. And also, the theoretical dispersion curves and vibration distributions obtained by the calculation were verified by experiments in which vibration distributions were obtained by measuring waveforms at many points with a laser interferometer

Chapter 8 - Guided-wave excitation in aerospace composites

Dispersion analysis of guided waves in the finned tube using the semi-analytical finite element method

On the basis of a semi-analytical finite element method, an effective approach has been developed for studying transient processes of dynamic deformation of three-dimensional heterogeneous bodies of revolution and prismatic bodies of complex shape and structure by the action of time-and space-varying pulsed stressing with allowance for the plastic properties of the material and time-varying contact interaction conditions. New types of finite elements have been created, on the basis of which computational relationships of the semi-analytical finite element method (SAFEM) for problems of dynamics have been constructed. Modified relationships of the Newmark method have been obtained, which have been formulated for the amplitude subsystems of SAFEM. Effective block iteration algorithms for the solution of large systems of nonlinear equations of SAFEM have been developed and realized.

The main types of connections are welded, riveted, and bolted. Bolted connections have many advantages over other types of connections, namely: high manufacturability, reliability, and speed of installation work, so we will further consider various approaches to modeling bolted connections. One of the most crucial and important points in the design of a structure is the calculation of the connection elements of the components, since the reliability of the structure as a whole depends on the accuracy and reliability of the calculation results. For a wide range of tasks, the finite element method (FEM) is often used in calculations; this numerical method has proven to be one of the most common and versatile. When calculating elements of assemblies such as bolts, rivets, etc., simplified design schemes are usually used, but this approach does not allow for a complete analysis of the distribution of forces. For a complete analysis, it is necessary to create detailed spatial models, which in turn leads to the need to solve systems of high-order equations. To increase the efficiency of the FEM, it is necessary to combine it with the method of distribution of unknowns, as a result, the resulting approach is called the semi-analytical finite element method (SFEM). The bolt, bolt head, and nut were modeled by spatial elements, and the bolt tension was modeled by a load from uniform heating. The obtained results of the stress-strain state using the finite element method and the semi-analytical finite element method allow us to conclude that there is a slight difference in the stress distribution in the spatial formulation of the FEM and SFEM with experimental data and allowed us to identify a rational approach to modeling bolt tension.

This paper proposes a rail defect location method based on a single mode extraction algorithm (SMEA) of ultrasonic guided waves. Simulation analysis and verification were conducted. The dispersion curves of a CHN60 rail were obtained using the semi-analytical finite element method, and the modal data of the guided waves were determined. According to the inverse transformation of the excitation response algorithm, modal identification under low-frequency and high-frequency excitation was realized, and the vibration displacements at other positions of a rail were successfully predicted. Furthermore, an SMEA for guided waves is proposed, through which the single extraction results of four modes were successfully obtained when the rail was excited along different excitation directions at a frequency of 200 Hz. In addition, the SMEA was applied to defect location detection, and the single reflection mode waveform of the defect was extracted. Based on the group velocity of the mode and its propagation time, the distance between the defect and the excitation point was measured, and the defect location was predicted as a result. Moreover, the SMEA was applied to locate the railhead defect. The detection mode, the frequency, and the excitation method Were selected through the dispersion curves and modal identification results, and a series of signals of the sampling nodes were obtained using the three-dimensional finite element software ANSYS. The distance between the defect and the excitation point was calculated using the SMEA result. When compared with the structure of the simulated model, the errors obtained were all less than 0.5 m, proving the efficacy of this method in precisely locating rail defects, thus providing an innovated solution for rail defect location.

The guided wave has been widely employed to characterize thin plates and layered media. The dispersion curves of phase and group velocities are essential for the quantitative application of guided waves. The technique using leaky Lamb wave (LLW) is one of the excellent methods to obtain dispersion curves. In the present work, a fully automated system for the measurement of backward radiation of LLW has been developed. The specimen moves in two dimensional planes as well as in angular rotation. The signals of backward radiation of LLW were measured from an elastic plate in which specific modes of Lamb wave were strongly generated. Phase velocity of the corresponding modes was determined from the incident angle. The generated Lamb waves propagated forward and backward with the leakage of energy into water. Backward radiated LLW was detected by the same transducer and its frequency components were analyzed to extract the related information to the dispersion curves. The dispersion curves of phase velocity were measured by varying the incident angle. Moving the specimen in the linear direction of LLW propagation, group velocity was determined by measuring the transit time shift in the rf waveform.

The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.

Purpose. To compare the efficiency of using finite elements with variable and averaged mechanical and geometric parameters and to investigate the convergence of results obtained by the semi-analytical finite element method (SAFEM) using Fourier series and polynomials with the results obtained by the finite element method (FEM). The methods. The construction and development of an algorithm for studying the stress-strain state of spatial bodies with variable and averaged mechanical and geometric parameters were carried out based on SAFEM. Findings. Solvability indicators of SAFEM were obtained for calculating nodal reactions and stiffness matrix coefficients of finite elements with variable and averaged mechanical and geometric parameters. Numerical convergence studies of results obtained using SAFEM with Fourier series, polynomials, and the finite element method were conducted for a benchmark example, which was the Boussinesq problem for a half-space subjected to a concentrated force. The results indicate that the convergence of the investigated coordinate functions in the considered problem is of the first order. The originality. The obtained solvability indicators of SAFEM for calculating nodal reactions and stiffness matrix coefficients of finite elements with variable and averaged mechanical and geometric parameters allow for the study of various classes of problems. Numerical convergence studies using Fourier series, polynomials, and the finite element method were conducted for the benchmark example. Practical implementation. The practical significance lies in the development of a methodology for determining the stress-strain state of relevant spatial elements of structures with variable and averaged mechanical and geometric parameters subjected to arbitrarily distributed spatial loads.

The semi-analytical finite element method (SAFEM) has an advantage in computational efficiency since it only requires meshing in the cross-section while substituting meshing with interpolations of Fourier series along the third dimension. This study aims to develop this method for viscoelastic analysis further. More specifically, use the Generalized Maxwell model to describe the stress-strain relation, and the state variables approach to achieve the viscoelastic iteration. Then a program based on this method for simulating the BBR tests was developed, and its accuracy was verified by comparing with the test results of different asphalt binders. Besides, the comparison with the 3D finite element model in ABAQUS indicated the efficiency of SAFEM. Altogether, the modeling approach has some value for the widespread application of SAFEM in structural modeling.

Conventional processing algorithms for acoustic logging data, based on propagation analysis of guided modes, have sufficient limitations when underground formation is anisotropic or borehole cross-section is essentially non-circular, as there is no analytical estimates for such cases. In this study, normal modes eigenvectors are calculated numerically with SAFE method and used for accurate approximation of measurement data. We demonstrate that residual functional between approximate and observed data has a strong dependence on elastic waves velocities and Thomsen parameters of TI media. The gradient of residual functional is calculated by adjoint-state method in order to solve inverse problem of elastic properties determination.

To confirm the reliability of the results obtained on the basis of the semi-analytical finite element method using the approaches suggested in [1-2, 4-6], a wide range of test problems is considered. Within the framework of the elastic approach, the spatial problem of an unevenly loaded shell, the middle surface of which has the shape of an elliptical paraboloid, is considered. The validation of the reliability of solutions obtained on the basis of the semi-analytical finite element method for physically nonlinear problems is carried out using the example of elastic-plastic calculation of an unevenly heated cube, the physical and mechanical characteristics of the material of which depend on temperature. To substantiate the reliability of the results obtained when solving problems of large plastic deformations, the elastic-plastic deformation of a parallelepiped between plane-parallel plates in the absence of contact friction is considered. The efficiency of application of the semi-analytical finite element method to the calculation of curvilinear inhomogeneous prismatic objects is shown. The solution of the control problems of the theory of elasticity, thermoelasticity and thermoplasticity, as well as the problems of shape change, makes it possible to draw a conclusion about the reliability of the results of the study of the selected class of objects on the basis of the developed methodology and the applied software package that implements it.