Semi-analytical Finite Element Formulation Research Articles

Full wave simulation of arterial response under acoustic radiation force

With the ultimate goal of estimating arterial viscoelasticity using shear wave elastography, this paper presents a practical methodology to simulate the response of a human carotid artery under acoustic radiation force (ARF). The artery is idealized as a nearly incompressible viscoelastic hollow cylinder submerged in incompressible, inviscid fluid. For this idealization, we develop a multi-step methodology for efficient computation of three-dimensional response under complex ARF excitation, while capturing the fluid-structure interaction between the arterial wall and the surrounding fluid. The specific steps include (a) performing dimensional reduction through semi-analytical finite element formulation, (b) efficient finite element discretization using traditional and recent techniques. The computational efficiency is further enhanced by utilizing (c) modal superposition, followed by, where appropriate, (d) impulse response function. In addition to developing the methodology, convergence analysis is performed for a typical arterial geometry, leading to recommendations on various discretization parameters. At the end, the computational effort is shown to be several orders of magnitude less than the traditional, fully three-dimensional analysis using finite element methods, leading to a practical yet accurate simulation of arterial response under ARF excitations.

Excitation and Propagation of Guided Waves in Multilayer Hollow Cylinders Using PWAS Transducers: A Theoretical and Experimental Study

In this paper, the excitation and propagation of guided waves in multilayer hollow cylinders with piezoelectric wafer active sensor (PWAS) transducers were modeled with the normal mode expansion (NME) method using the semi-analytical finite element (SAFE) formulation. The theoretical development of SAFE for hollow cylindrical structures was introduced and used to obtain guided-wave mode shapes and dispersion curves of multilayer hollow cylinders. The SAFE discretization was applied across the thickness. The layers present in the cylinder were modeled by grouping the elements in the region corresponding to the respective layers. Each finite element region was given the property of the layer that it represented. The number of elements in a layer was determined through convergence studies. The PWAS excitation effect, introduced using the ideal-bonding assumption, was represented by a line-force acting on the PWAS boundary. The SAFE-NME solution obtained in the wavenumber domain was resolved in the physical domain through inverse Fourier transform and residue theorem. Experimental validation of theoretical prediction was performed by comparison with scanning laser Doppler vibrometer (SLDV) measurements from a “6-inch schedule-40” pipe of 11 mm thickness installed with a 7-mm square PWAS transducer for wave excitation. Numerical prediction of the guided wave propagation emanating from the PWAS was first performed and wavefront visualization was obtained. An SLDV area scan of the guided waves generated by the PWAS was then performed and compared with numerical predictions. A good match between experiment and prediction was observed.

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.

Defect imaging in layered composite plates and honeycomb sandwich structures using sparse piezoelectric transducers network

Carbon fiber reinforced polymer (CFRP) plates and honeycomb composite sandwich structures (HCSS) are widely used in the aerospace industry as they exhibit excellent strength-to-weight ratio, stiffness, toughness, corrosion resistance, etc. Nevertheless, defects, such as face sheet delamination or core-sheet debonding, may appear due to the impact forces or thermo-mechanical aging and can degrade these properties. Structural health monitoring (SHM) based on the use of guided elastic waves (GW) is regarded as a promising solution to detect such defects, and consequently to reduce maintenance costs and to extend structure service time. GWs propagate over large distances while being sensitive to structural inhomogeneities. Here, a SHM system prototype is proposed. It relies on a sparse grid of piezoelectric transducers distributed over the structure, used for both actuating and sensing GWs. Defect imaging is performed by means of the correlation based algorithm, the so-called Excitelet. It computes correlation coefficients between the theoretical and experimental GW signals for each pixel on the image representing the region of interest of the structure. The theoretical signals for CFRP are computed using a model based on a 2D semi-analytical finite element formulation. Analytical prediction of theoretical signals for the HCSS being intractable, a homogenization model is applied to the honeycomb core to replace it by an equivalent orthotropic plate preserving the same modelling approach. Defect imaging results are presented for both structures, namely a CFRP plate and a HCSS. The resolution of the corresponding images can be related to the wavelength of the inspecting mode.

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

CALCULATING DISPERSION RELATIONS FOR WAVEGUIDE IMMERSED IN PERFECT FLUID

Conventional finite element method (FEM) is capable of obtaining wave solutions, but large discretized structures at high frequency require high computational resources, the computational domain can be reduced by combining FEM with analytical assumption for guided wave. Semi Analytical Finite Element (SAFE) formulation for immersed waveguide in perfect fluid is used for acquiring propagating wave modes as dynamic equilibrium states. Modes are solutions to eigenvalue problem and provide with important characteristic features of the guided waves – phase velocity, attenuation, wave structure, etc. The effect of surrounding leaky medium is modeled via traction boundary condition, which is based on assumption of the continuity of stresses at solid-fluid interface. The boundary condition causes wave attenuation due to energy leakage into outer medium. The derivation of the eigen-problem takes into account complex wavenumbers of leaky wave in fluid and guided wave in a three-dimensional waveguide. Linearization procedure for solving nonlinear eigenvalue problem is used. Dispersion relations for immersed waveguide with Rayleigh damping are obtained. The limits of applications of Rayleigh damping and convergence analysis of immersed waveguide model are discussed.

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.

Numerical modeling of waveguides accounting for translational invariance and rotational symmetry

The analysis of high-frequency wave propagation in arbitrarily shaped waveguides requires specific numerical methods. A widely spread technique is the so-called semi-analytical finite element (SAFE) formulation. This formulation enables to account for the translational invariance of waveguide problems and leads to a two-dimensional modal problem reduced on the cross-section. Despite this, solving the problem can still be computationally demanding. In order to further reduce the size of the modal problem, this paper presents a SAFE method for waveguides of rotationally symmetric cross-sections. Such structures are encountered in many applications. Typical examples are bars of circular cross-section, regular polygons, and multiwire cables. Numerical results show that the computational effort required for solving the SAFE modal problem is tremendously reduced by accounting for rotational symmetry.

Elastic wave simulation based on modal excitation in 3D medium

Paper describes a simulation of an elastic wave in homogenous isotropic waveguide with generic cross-section using semi-analytical finite element (SAFE) formulation. The wave is considered a travelling displacement field in the waveguide as a result to forcing excitation and is expressed via modes’ superposition. The solutions for modes are obtained solving SAFE governing eigen problem. Attenuation of the medium in SAFE framework differently from prior researchers is simulated via Rayleigh damping. It is shown, that severe damping is not properly supported by the SAFE formulation and revision for properly accepting linear viscosity is needed.

Wave dispersion in thin-walled orthotropic waveguides using the first order shear deformation theory

The aim of this paper is to extract the dispersion parameters, i.e. phase velocity, energy velocity and attenuation, of orthotropic thin-walled waveguides with generic cross-section. To this end, a semi-analytical finite element (SAFE) formulation is presented, which is based on the Reissner–Mindlin theory of curved shells.Complex axial wavenumbers and mode shapes of guided wave modes are extracted from a second-order polynomial eigenvalue problem, while the energy velocity is post-processed using the computed eigensolutions and SAFE operators.Different numerical examples are proposed, for which the obtained results are in very good agreement with those computed using other well-stated SAFE formulations.

Nonlinear ultrasonic waves for structural monitoring: Thermal stress measurement and guided wave management

This presentation will cover two aspects of ultrasonic nonlinear wave propagation in solids. First, a new model is proposed to justify the existence of wave nonlinearities in constrained solids subjected to thermal excursions. This problem is solved on the basis of the interatomic potential of the solid that indicates a “residual” strain energy, due to the prevented thermal expansion, that is at least cubic as a function of strain. This study finds applications in the monitoring of thermal stresses in buckling-prone structures, such as continuously welded railroad tracks and pipelines. Experimental tests conducted on railroad tracks with realistic support will be also presented. Second, in the case of waveguides the efficiency of nonlinear ultrasonic testing based on higher-harmonic generation strongly relies on the correct identification of favorable combinations of primary and resonant double-harmonic nonlinear wave modes. This presentation will identify these combinations of wave modes in complex waveguides by extending the classical Semi-Analytical Finite Element formulation to the nonlinear regime, and implementing it into a highly flexible commercial Finite Element code. The proposed algorithm is benchmarked for four case-studies, including a railroad track, a viscoelastic plate, a composite quasi-isotropic laminate, and a reinforced concrete slab.

On nonlinear thermal buckling analysis of cylindrical shells

In this article, the nonlinear buckling behavior of imperfect cylinders made of isotropic, composite and functionally graded materials is studied. A continuum-based semi-analytical finite element formulation is introduced to study the nonlinear behavior of cylinders under thermal loads. A method is proposed to implement the initial geometric imperfection of the cylinder by transformation of structure due to deformation gradients. The influences of geometrical parameters, different materials and imperfection factors are investigated on pre- and post-buckling paths. A comparison is made between the classical von Kármán-based and continuum-based approaches to ensure the validation of the results and to study the applicability of the von-Kármán approximation.

Three-dimensional modeling of elastic guided waves excited by arbitrary sources in viscoelastic multilayered plates

This paper provides a modal solution for the three-dimensional modeling of Lamb and SH waves excited by sources of arbitrary shape. This solution is applicable to elastic and viscoelastic plates, in the far-field as well as in the near-field regions, under the assumption of transverse isotropy about the thickness direction. The theoretical developments are conducted based on a semi-analytical finite element formulation. This formulation yields a one-dimensional modal problem, fast from a computational point of view, and allows to readily handle heterogeneous materials having depth-varying properties (multilayered, piecewise or continuously varying, functionally graded). The modal solution is shown to be expressed in terms of Hankel functions of multiple order thanks to a proper application of inverse transforms and Cauchy residue calculus. The link between the proposed formulation and a fully analytical approach is discussed. The solution of this paper is then successfully compared to literature results and degenerates to the point source case. Formulas are presented to calculate point source excitabilities from lines sources. These formulas remain valid for non-propagating modes, viscoelastic materials and account for the near-field contribution. Finally, the example of a viscoelastic bilayer waveguide excited by a rectangular source is considered in order to check the theoretical results.

Computation of leaky guided waves dispersion spectrum using vibroacoustic analyses and the Matrix Pencil Method: A validation study for immersed rectangular waveguides

The paper aims at validating a recently proposed Semi Analytical Finite Element (SAFE) formulation coupled with a 2.5D Boundary Element Method (2.5D BEM) for the extraction of dispersion data in immersed waveguides of generic cross-section. To this end, three-dimensional vibroacoustic analyses are carried out on two waveguides of square and rectangular cross-section immersed in water using the commercial Finite Element software Abaqus/Explicit. Real wavenumber and attenuation dispersive data are extracted by means of a modified Matrix Pencil Method. It is demonstrated that the results obtained using the two techniques are in very good agreement.

Ultrasonic leaky guided waves in fluid-coupled generic waveguides: hybrid finite-boundary element dispersion analysis and experimental validation

A numerical procedure is presented for the computation of dispersive parameters in elastic mechanical waveguides of generic cross-section immersed in non-viscous fluids. The method uses a semi-analytical finite element formulation to describe the solid waveguide, while a two-and-a-half dimensional boundary element method is used to represent the unbounded surrounding fluid. Leaky and trapped guided wave modes are found on the appropriate Riemann sheets by enforcing the generalized Snell–Descartes law along the fluid-structure interface and solving a nonlinear eigenvalue problem. The method is validated experimentally by extracting the frequency-real wavenumber dispersion curves of a rectangular bar and a thin angle aluminum bar via a two-dimensional Fast Fourier Transform. In both cases, a very good agreement is observed between the numerical and the experimental solutions.

Development of a semi-analytical nonlinear finite element formulation for cylindrical shells

This article introduces a new semi-analytical nonlinear finite element formulation for thin cylinders according to a continuum-based approach. A comparison between the continuum-based approach and the classical approach for the buckling behavior of isotropic and orthotropic perfect cylinders validates the results. The classical approach is defined according to thin shell theories based on the von Karman approximation. A mathematical modeling for geometry imperfection of cylinders is derived according to a continuum-based approach whose results are compared with the results of the classical approach for imperfect cylinders. The influence of neglecting some nonlinear terms in the classical approach for perfect and imperfect cylinders on the buckling path is investigated. In the buckling analysis, two methods, i.e. the perturbation and load disturbance methods, which undertake to switch to the post-buckling path, are compared to each other.

A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary cross-section

The paper presents a Semi-Analytical Finite Element (SAFE) formulation coupled with a 2.5D Boundary Element Method (BEM) for the computation of the dispersion properties of viscoelastic waveguides with arbitrary cross-section and embedded in unbounded isotropic viscoelastic media. Attenuation of guided modes is described through the imaginary component of the axial wavenumber, which accounts for material damping, introduced via linear viscoelastic constitutive relations, as well as energy loss due to radiation of bulk waves in the surrounding media. Energy radiation is accounted in the SAFE model by introducing an equivalent dynamic stiffness matrix for the surrounding medium, which is derived from a regularized 2.5D boundary element formulation.The resulting dispersive wave equation is configured as a nonlinear eigenvalue problem in the complex axial wavenumber. The eigenvalue problem is reduced to a linear one inside a chosen contour in the complex plane of the axial wavenumber by using a contour integral method. Poles of leaky and evanescent modes are obtained by choosing appropriately the phase of the wavenumbers normal to the interface in compliance with the nature of the waves in the surrounding medium. Finally, the obtained eigensolutions are post-processed to compute the energy velocity and the radiated wavefield in the surrounding domain.The reliability of the method is first validated on existing results for waveguides of circular cross sections embedded in elastic and viscoelastic media. Next, the potential of the proposed numerical framework is shown by computing the dispersion properties for a square steel bar embedded in grout and for an H-shaped steel pile embedded in soil.

Mechanical modeling of helical structures accounting for translational invariance. Part 2 : Guided wave propagation under axial loads

This paper corresponds to the second part of a study that aims at modeling helical structures accounting for translational invariance. In the Part 1 of this paper, the static behavior has been addressed using a helical homogenization approach which provides the stress state corresponding to axial loads. The latter is considered as a prestressed state, for elastic wave propagation analysis in helical waveguides, which is the subject of the Part 2 of this paper. Non destructive testing of springs and multi-wire strands is a potential application of the proposed model. Accounting for translational invariance, the elastodynamic equations of prestressed helical structures yield a 2D problem posed on the cross-section, corresponding to a so-called semi-analytical finite element (SAFE) formulation. For helical springs, the numerical model is validated with an analytical solution corresponding to a Timoshenko beam approximation. It is shown that the influence of the prestressed state is significant at low frequencies. Finally, a seven-wire strand subjected to axial loads is considered. The computed dispersion curves are compared to experimental data. Good agreement is obtained for the first compressional-like modes and their veering central frequency.

Characterization of the elastic moduli in composite plates via dispersive guided waves data and genetic algorithms

In this study, an inverse procedure based on stress guided waves is proposed for the characterization of the elastic moduli of composite plates. The characterization is carried out via genetic algorithms by minimizing the discrepancy between experimental and numerical group delay curves for different directions of propagation along the plate. Experimentally, for a given distance source–receiver, the group delay curves are obtained by processing the guided waves time–transient signals via a time–frequency transform. For the same distance, a fast and reliable semi-analytical finite element formulation is used for the forward computation of the group delay curves. Here, pseudo-experimental data, generated by means of the semi-analytical finite element model for an assumed known set of elastic coefficients, are used to test the reliability of the proposed procedure. The results obtained for three different plates are promising. Since semi-analytical finite element formulations can also handle plates with unif...

Modeling of nonlinear guided waves and applications to structural health monitoring

Research efforts on nonlinear guided wave propagation have increased dramatically in the last few decades because of the large sensitivity of nonlinear waves to structural condition (defects, quasi-static loads, instability conditions, etc...). However, the mathematical framework governing the nonlinear guided wave phenomena becomes extremely challenging in the case of waveguides that are complex in either materials (damping, anisotropy, heterogeneous, etc...) or geometry (multilayers, geometric periodicity, etc…). The present work develops predictions of nonlinear second-harmonic generation in complex waveguides by extending the classical Semi-Analytical Finite Element formulation to the nonlinear regime, and implementing it into a highly flexible, yet very powerful, commercial Finite Element code. Results are presented for the following cases: a railroad track, a viscoelastic plate, a composite quasi-isotropic laminate, and a reinforced concrete slab. In these cases, favorable combinations of primary wave modes and resonant double-harmonic nonlinear wave modes are identified. Knowledge of such combinations is important to the implementation of structural monitoring systems for these structures based on higher-harmonic wave generation. The presentation will also present a specific application of nonlinear guided waves for the monitoring of thermal stresses in rail tracks to prevent buckling.