Semi-analytical Finite Element Method Research Articles

Semiаnalytical finite elements method efficiency in the geometrically nonlinear elastic-plastic problems

Particular interest, among the variety of objects considered using analytical and numerical methods, are bodies of revolution with complex shape and cross-sectional structure. The selected geometric class is used as natural structure components in the construction and various fields of mechanical engineering. The sufficiently high prevalence of these forms in the construction and machine-building sectors on the one hand, and the possibility of a significant simplification math equestions by taking into account their geometric features on the other, is attracting increasing attention of researchers.It is known that today the finite element method (FEM) is the most powerful tool for analyzing the problems of building mechanics and deformable solid mechanics. Over the past few years, the dimension of FEM models has grown dramatically, driven by increased demands for accuracy and reliability of results. In addition, the difficulties of studying the behavior of structures in the presence of dynamic loads are many times increased in comparison with static analysis. To overcome these problems, in many cases, it introduce additional hypotheses, which, as a rule, narrow the class of objects and processes under study, but can significantly improve efficiency and significantly reduce the duration of the calculation. Semi-analytical finite element method (SAFEM) is one of such approaches that is widely used for solving problems whose objects are prismatic and rotational bodies. High efficiency of SAFEM for a certain range of objects was demonstrated in the field of static analysis, continuous mechanics of fracture under creep conditions, processes of nonlinear deformation of reinforced concrete structures. The analysis of the results obtained by domestic and foreign scientists on this issue shows that most analytical and numerical methods of scientific research are usually oriented to geometrically nonlinear problems at static load. The veracity and effectiveness of the semi-analytical finite element method in the problems of geometrically nonlinear elastoplastic deformation of axisymmetric structures under dynamic loads is considered. The capabilities of methodology are demonstrated by examples of numerical modeling of the stress-strain state of building structures with large linear strains and analysis of technological processes of pulsed metal processing.

A semianalytical finite element method for stress and deformation analyses of bi-directional functionally graded truncated conical shells

On the basis of Reissner’s mixed variational theorem, the authors develop a semianalytical finite annular prism method (FAPM) for three-dimensional (3D) stress and deformation analyses of bi-directional functionally graded (FG) truncated conical shells with various boundary conditions subjected to either uniformly or sinusoidally distributed loads. The material properties of the FG-truncated conical shell are assumed to obey a bi-directional power-law distribution of the volume fractions of the constituents through the meridian–thickness surface, the effective material properties of which are estimated using the rule of mixtures. Implementation of the current FAPMs shows their solutions converge rapidly and that the convergent solutions are in excellent agreement with the 3D solutions available in the literature.

Aerothermoelastic analysis of imperfect FG cylindrical shells in supersonic flow

The supersonic panel flutter of functionally graded cylindrical shells under thermal loadings is investigated by considering the effects of various axisymmetric and asymmetric geometric imperfections. The problem is formulated using a nonlinear first-order shear deformation theory of shells with the first-order aerodynamic piston theory. The temperature is assumed to vary in the thickness direction according to the steady-state heat conduction equation. The semi-analytical finite element method (FEM), based on the field consistency redistribution approach is used to obtain discretized nonlinear aeroelastic equations. In the process of FEM, the imperfection functions are also discretized and efficiently approximated by the Hermitian polynomials in order to facilitate the required integrations. For analysis purpose, the deformations and stresses induced by the temperature rise are first computed by solving the nonlinear static aerothermoelastic equations using the Newton-Raphson method. The pseudo-arclength continuation method is also employed to detect the possible snap-through of the shell. The linearized stability equation about the equilibrium state is then used for determining the flutter boundaries. Numerous parametric studies are conducted to examine the effects of temperature rise and imperfections on the flutter boundaries, which show considerable effects of imperfections on changing the trend of variation of the flutter pressure with temperature.

Nonlinear Inversion of Ultrasonic Dispersion Curves for Cortical Bone Thickness and Elastic Velocities.

In this study, a nonlinear grid-search inversion has been developed to estimate the thickness and elastic velocities of long cortical bones, which are important determinants of bone strength, from axially-transmitted ultrasonic data. The inversion scheme is formulated in the dispersive frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve overlying soft tissue thickness, cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires systematic examination of a large set of trial dispersion-curve solutions within a pre-defined model space to match the data with minimum cost in a least-squares sense. The theoretical dispersion curves required to solve the inverse problem are computed for bilayered bone models using a semi-analytical finite-element method. The feasibility of the proposed approach was demonstrated by the numerically simulated data for a 1 mm soft tissue-5 mm bone bilayer and ex-vivo data from a bovine femur plate with an overlying 2 mm-thick soft-tissue mimic. The bootstrap method was employed to evaluate the inversion uncertainty and stability. Our results have shown that the cortical thickness and wave speeds could be recovered with fair accuracy.

Hybrid SAFE-GMM approach for predictive modeling of guided wave propagation in layered media

Dispersion curve, and displacement modeshapes of multilayer structures can be obtained using Semi-Analytical Finite Element (SAFE) method. Stress mode shapes calculated from SAFE were discontinuous at the interface, because SAFE method formulation does not apply stress continuity at the interface. A case study of 1 mm aluminum-1 mm steel double-layer plate is presented, showing the SAFE stress predictions at the interface. The results observed were discontinuity of out-of-plane stress modeshapes at the interface. Mesh refinement was performed at the interface to study the convergence of stress mode shapes at the interface, yet the stress discontinuity problem was not solved. A fine mesh at the interface was created by using variable mesh techniques. This also did not provide continuous stress mode shape at the interface. Therefore, in this paper, a novel hybrid SAFE-GMM (Global Matrix Method) approach was used to obtain stress mode shapes accurately and efficiently. GMM develops the displacement and stress equations for individual layers in a multilayered structure and assembles a global matrix by applying the boundary and interface continuity conditions. A case of practical importance, CFRP strengthened concrete structure was analyzed using the SAFE-GMM approach. The drawback of the SAFE technique to this case is also presented, and it shows that the hybrid SAFE-GMM approach gave the interface stresses accurately.

Investigation of guided wave properties of anisotropic composite laminates using a semi-analytical finite element method

Composite materials have been widely used in various applications, and guided waves are often used as a non-destructive testing tool to inspect defects or damage in composite laminates which are assemblies of multiple composite layers. These can have anisotropic material properties and arbitrary fibre orientation such that the guided wave properties of Lamb and shear-horizontal modes are direction-dependent. The group velocity of each wave mode is therefore to be considered with a component parallel to the wave propagation direction and a component perpendicular to the wave propagation direction. In this article, a semi-analytical finite element method is developed to model composite laminates with arbitrary fibre orientation and anisotropic material properties in each layer. Galerkin's principle is used to derive the weak forms of the governing equations, and an energy velocity formulation is used to calculate the parallel and perpendicular energy velocities. The finite element solutions are compared with available analytical and numerical solutions in the literature for forward waves, and excellent agreement is demonstrated. On this basis, the guided wave properties of backward waves have also been investigated. It is well understood that in an isotropic plate, the energy velocity of a backward wave is directed opposite to the phase velocity. However, in a composite laminate, the energy velocity of a backward wave is normally not exactly opposed to (180° out of phase with) the phase velocity but exhibits a skew angle. The angular dependences of wave properties of the forward and backward waves are investigated in this article.

On the identification of the elastic properties of composites by ultrasonic guided waves and optimization algorithm

This paper examines the potential of ultrasonic guided waves to identify the elastic properties of fiber-reinforced composite laminates widely used in light-weight structures. Due to variability in manufacturing processes, complexities involved in the constituent materials, or degradation in service, a non-destructive tool for the characterization of the composite’s properties can be extremely useful. A common assumption is the necessity for multiple wave propagation directions to fully characterize the properties. This paper examines the potential for property characterization by a single wave propagation direction. It proposes a property inversion scheme based on matching phase velocity dispersion curves of relevant guided modes by means of a Simulated Annealing optimization algorithm and a Semi-Analytical Finite Element method to solve the forward problem. Proof-of-principle numerical studies are presented to demonstrate the potential of each selected wave mode to provide identification of several properties, including the lamina elastic constants and the laminate effective constants. It is shown that the complex stress and strain profiles generated by the waves, aided by the anisotropy of the composite, create interesting “coupling” effects that ultimately enable accurate identification of several elastic properties away from the wave propagation direction.

Numerical Study on Ultrasonic Guided Waves for the Inspection of Polygonal Drill Pipes.

The polygonal drill pipe is one of the most critical yet weakest part in a high-torque drill machine. The inspection of a polygonal drill pipe to avoid its failure and thus to ensure safe operation of the drilling machine is of great importance. However, the current most frequently used ultrasonic inspection method is time-consuming and inefficient when dealing with a polygonal drill pipe, which is normally up to several meters. There is an urgent need to develop an efficient method to inspect polygonal drill pipes. In this paper, an ultrasonic guided wave technique is proposed to inspect polygonal drill pipes. Dispersion curves of polygonal drill pipes are firstly derived by using the semi-analytical finite element method. The ALID (absorbing layer using increasing damping) technique is applied to eliminate unwanted boundary reflections. The propagation characteristics of ultrasonic guided waves in normal, symmetrically damaged, and asymmetrically damaged polygonal drill pipes are studied. The results have shown that the ultrasonic guided wave technique is a promising and effective method for the inspection of polygonal drill pipes.

Mode analysis method for the computation of guided wave dispersion in metal sheet

Mode analysis method based on finite element theory is proposed to plot dispersion curves in metal sheet. This method is implemented by adopting COMSOL software to numerically solve and plot the dispersion curves. Compared with results obtained by semi-analytical finite element method, the calculating results are basically consistent and the modal caused by mode blending is obtained, which proves the mode analysis method is effective. Finally, the wave structure characteristics of various models are analysed, this provides a basis for the application of ultrasonic guided wave testing.

Research on a Rail Defect Location Method Based on a Single Mode Extraction Algorithm

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.

Acoustoelastic guided wave propagation in axial stressed arbitrary cross-section

This paper investigates the effect of axial stress on acoustoelastic guided wave propagation in arbitrary cross-section. Many researchers have investigated the effect of stress on guided wave propagation in arbitrary cross-section. However, the effect of high-order elastic constants in guided waves propagation of such structures is not as well understood. This paper deals with an acoustoelastic theory combined with semi-analytical finite element (AE-SAFE) method for modeling stressed wave propagation. The dispersive solutions are obtained in terms of phase velocities, group velocities and velocity sensitivities to stress variations in complex arbitrary cross-section. These results can be utilized to select the appropriate points and are significant for stress measure in arbitrary cross-section at high frequency region. The AE-SAFE method is applied to three examples, including plates, load bearing stringer and rail track.

Propagating, Evanescent, and Complex Wavenumber Guided Waves in High-Performance Composites.

The study of propagating, evanescent and complex wavenumbers of guided waves (GWs) in high-performance composites using a stable and robust semi-analytical finite element (SAFE) method is presented. To facilitate understanding of the wavenumber trajectories, an incremental material change study is performed moving gradually from isotropic aluminum alloy to carbon fiber reinforced polymer (CFRP) composites. The SAFE results for an isotropic aluminum alloy plate are compared with the exact analytical solutions, which shows that N = 20 SAFE elements across the thickness provides <0.5% error in the highest evanescent wavenumber for the given frequency-wavenumber range. The material change study reveals that reducing the transverse and shear moduli moves the wavenumber solution towards one similar to composite material. The comparison of the propagating, evanescent and complex wavenumber trajectories between composites and aluminum alloy show that antisymmetric imaginary Lamb wave modes always exist in composites although they may not exist in isotropic aluminum alloy at some frequencies. The wavenumber trajectories for a unidirectional CFRP plate show that the range of real wavenumber is much smaller than in the isotropic aluminum alloy. For laminated CFRP composite plates (e.g., unidirectional, off-axis, transverse, cross-ply and quasi-isotropic laminates), the quasi Lamb wave and shear horizontal (SH) wave trajectories are also identified and discussed. The imaginary SH wave trajectories in laminated composites are distorted due to the presence of ±45 plies. The convergence study of the SAFE method in various CFRP laminates indicates that sufficient accuracy can always be achieved by increasing the number of SAFE elements. Future work will address the stress-continuity between composite layers.

Estimation of rail properties using semi-analytical finite element models and guided wave ultrasound measurements

Guided wave based monitoring systems require accurate knowledge of mode propagation characteristics such as wavenumber and group velocity dispersion curves. These characteristics may be computed numerically for a rail provided that the material and geometric properties of the rail are known. Generally, the rail properties are not known with sufficient accuracy and these properties also change due to temperature, rail wear and rail grinding. An automated procedure is proposed to estimate material and geometric properties of a rail by finding the properties which, when input into a Semi-Analytical Finite Element (SAFE) model, accurately reproduce measured dispersion characteristics. Pulse-echo measurements were performed and spectrograms show the reflections from aluminothermic welds of three modes of propagation. The SAFE method was used to solve the forward problem of predicting the dispersion characteristics for specified rail properties. Dispersion curves are computed for different combinations of Poisson’s ratio and three geometric parameters. These dispersion curves are scaled to cover a range of longitudinal speeds of sound of the rail material. A technique is developed to determine which SAFE model provided the best fit to the experimental measurements. The technique does not require knowledge of the distances to the reflectors; rather these distances are estimated as part of the proposed procedure. A SAFE model with the estimated rail parameters produced dispersion curves and distances in very good agreement with the measured spectrograms. In addition, the estimated mean geometric parameters agreed with the measured profile of the rail head.

Effect of intracortical bone properties on the phase velocity and cut-off frequency of low-frequency guided wave modes (20-85 kHz).

The assessment of intracortical bone properties is of interest since early-stage osteoporosis is associated with resorption in the endosteal region. However, understanding the interaction between ultrasonic guided waves and the cortical bone structure remains challenging. The purpose of this work is to investigate the effect of intracortical bone properties on the ultrasonic response obtained at low-frequency (<100 kHz) using an axial transmission configuration. The semi-analytical finite element method was used to simulate the propagation of guided waves in a waveguide with realistic geometry and material properties. An array of 20 receivers was used to calculate the phase velocity and cut-off frequency of the excited modes using the two-dimensional Fourier transform. The results show that the position of the emitter around the circumference of the bone is an important parameter to control since it can lead to variations of up to 10 dB in the amplitude of the transmitted modes. The cut-off frequency of the high order modes was, however, only slightly affected by the circumferential position of the emitter, and was sensitive mainly to the axial shear modulus. The phase velocity and cut-off frequency in the 20-85 kHz range are promising parameters for the assessment of intracortical properties.

A numerical method for modeling ultrasonic guided waves in thin-walled waveguides coupled to fluids

The paper presents a hybrid numerical method for the computation of the dispersion characteristics of ultrasonic guided waves propagating in isotropic and viscoelastic thin-walled waveguides in contact with fluids along their inner or outer wall. To this end, the solid waveguide is modeled by means of a Semi-Analytical Finite Element method implementing the first order shear deformation theory, while the effects of the fluid are taken into account by means of a regularized 2.5D Boundary Element Method with Helmholtz-type kernels. By coupling the two methods along the mid-surface of the solid waveguide, the dispersion relations of the elastic-acoustic system are obtained as the solution of a nonlinear eigenvalue problem in the complex axial wavenumbers for any fixed circular frequency. Two different case studies are presented: an oil-filled pipe and a water-loaded steel rectangular waveguide.

Guided wave propagation in buried and immersed fluid-filled pipes: Application of the semi analytic finite element method

Developing an understanding of guided wave propagation in multi-layered systems has important applications in non-destructive evaluation. This article presents a general weighted residual formulation for guided wave propagation in fluid-filled pipes buried in an elastic solid, or immersed in a quiescent fluid. A one-dimensional semi-analytic finite element (SAFE) approach is combined with a perfectly matched layer (PML), to compute dispersion curves for different pipe applications. The speed and accuracy of this approach are compared against the scaled boundary finite element method (SBFEM) and it is shown that for a rod immersed in a liquid the two methods provide very similar computational speeds. The speed and accuracy of the model is then investigated for immersed and buried fluid-filled pipes, and it is shown that when discretising the PML no advantage in computational speed is found when using either quadratic or higher order spectral elements, provided the number of degrees of freedom in the PML is equivalent. Accordingly, it is shown that the SAFE-PML method is capable of obtaining accurately the modal characteristics of buried and immersed fluid-filled pipes, with computational speeds comparable to the SBFEM approach.

Guided wave propagation in cylindrical ducts with elastic walls enclosing a fluid moving with a uniform velocity

Theoretical models for elastic wave propagation in fluid filled ducts normally neglects mean fluid flow. However, in many engineering applications the velocity of the fluid may influence the modal characteristics of the duct, for example in gas pipelines, turbomachinery applications and ventilation systems. Accordingly, the influence of a mean uniform fluid flow on acoustically driven duct wall vibration is analysed here for a cylindrical geometry. The semi-analytic finite element method is used to couple the elastodynamic wave equation for the duct wall to the convected wave equation for sound propagation in a uniform fluid flow. A one dimensional finite element approach is described and this is used to find the coupled eigenmodes for the duct. Under certain conditions, a uniform mean flow is seen to significantly affect the phase speed for different eigenmodes, and it is shown that this may cause energy to transfer from the fluid to the surrounding wall at frequencies much lower than those seen without mean flow. This behaviour has the potential to increase sound radiation from ducts at lower frequencies when mean flow is present.

Determining the Parameters of Linear Fracture Mechanics in Dynamic Problems Based on Semianalytical Finite-Element Method

The results of studying the reliability and efficiency of the semianalytical finite-element method for determining the parameters of linear fracture mechanics in dynamic problems are presented. The effect of approximation of a crack by special finite elements on the accuracy of the calculated parameters of fracture mechanics is analyzed. The direct determination of the stress intensity factor (SIF) is compared to the energy approach based on the J-integral. The reliability of the method of reactions in dynamical problems for different schemes of time integration of the equations of motion is studied. As tests, the dynamical SIFs are determined for problems of mixed-mode fracture. The capabilities of the approach are demonstrated by studying the dynamic deformation of a high-pressure vessel with a ring crack.

Selective generation of ultrasonic guided waves in a bi-dimensional waveguide

Non-destructive testing and structural health monitoring systems based on ultrasonic guided waves propagation are particularly used in civil engineering or aerospace applications. Guided waves are commonly employed as they propagate through large distances and can inspect the entire cross-section of the structure. In order to optimize the sensitivity to a specific damage type, it is often preferable to generate a carefully selected pure mode. Although single-mode generation has been achieved for Lamb waves in infinite plate-like structures, such generation is much harder in a rectangular bar since less conventional modes propagate in finite cross-section waveguides. This article presents a general methodology for mode selective generation in a finite cross-section waveguide, using multiple transducers. Obtaining modal identification through conventional spatial Fourier transform on a longitudinal scan has proven to be inconvenient for waveguides with a two-dimensional cross-section. An alternative technique is proposed, consisting in the decomposition over the modal basis of the three displacement components measured across the bar width at the bar surface. The methodology is applied to the single-mode generation within an aluminum bar instrumented with eight piezoelectric transducers bonded to the surface. The modal basis is obtained with a semi-analytical finite element method. Numerical simulations and experiments using a three-dimensional laser Doppler vibrometer are conducted in order to validate the methodology.

Comparison of quasi-Rayleigh waves and Rayleigh waves, and clarifying the cut-off frequency of quasi-Rayleigh waves

The Partial Wave Method is unique in that it establishes a foundation on which various elastodynamic guided waves can be compared. In this paper, the method is used to compare quasi-Rayleigh waves and Rayleigh waves, and investigate the eccentricities of the Partial Wave Method at phase velocities equal to the Rayleigh wave speed. The comparison results in the definition of two types of quasi-Rayleigh waves and an explanation for quasi-Rayleigh wave behavior reported in the literature at frequencies that do not satisfy Viktorov’s quasi-Rayleigh wave condition. These conclusions are also verified by the superposition of A0 and S0 mode wave-structures calculated using the semi-analytical finite element method.