We present for the first time an **approach** for identifying the geometric and material characteristics of layered compos- ite structures through an inverse **wave** and **finite** **element** **approach**. More specifically, this Non-Destructive Evaluation (NDE) **approach** is able to recover the thickness, density, as well as all independent mechanical characteristics such as the tensile and shear moduli for each layer of the composite structure under investigation. This is achieved through multi-frequency single shot measurements. It is emphasized that the success of the **approach** is independent of the employed excitation frequency regime, meaning that both structural dynamics and ultrasound frequency spectra can be employed. It is demonstrated that more efficient convergence of the identification process is attained closer to the bending-to-shear transition range of the layered structure. Since a **full** FE description is employed for the periodic composite, the presented **approach** is able to account for structures of arbitrary complexity. The procedure is ap- plied to a sandwich panel with composite facesheets and results are compared with two **wave**-based characterization techniques: the Inhomogeneous **Wave** Correlation method and the Transition Frequency Characterization method. Numerical simulations and experimental results are presented to verify the robustness of the proposed method. Keywords: Structural identification, Non-Destructive Evaluation, **Finite** Elements, **Wave** Propagation, Layered Structures, Ultrasound

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It can be shown that the damping coecient required for **full** absorption of energy depends on the angle of incidence of the incident **wave**. Since waves with dierent angles may hit the boundary, a local boundary with a certain damping coecient always reects a portion of the energy of the incident **wave**. Additional problems arise when divergent surface waves reach the local boundary. Since the phase velocity of these waves depends on frequency, a frequency-independent damper is required to fully absorb the energy. The eects of reections of local boundaries can be reduced by increasing the distance between the boundary and the range of concern. However, it should be noted that, depending on the dimensions of the model and software capabilities as well as the extent of stress zone, an increase in such distance is not always feasible. Even with probable and feasible options, the process is always associated with excessive time for analysis and requires large core storage.

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Due to bending waves doesn't travel from the impact point to the edge of the beam and back during the predicted contact duration, we have a **wave**-controlled **approach** might be adequate because in that case, the laminate behaves in a small mass impact. The present **finite** **element** **approach** taking into account the **full** dynamic behaviors of the beam and boundary conditions provides a good result in comparison with a **wave** propagation method for infinite composites because the wavefront never reaches the edges of the beam. Throughout the present **finite** **element** analysis and **wave** propagation method, it can be seen that the contact force during impact event is proportional to the bending stiffness but the deflection at midplane of laminated composites is reversed. The bending stiffness affects so much on the wavefront by impact of laminate. That is, the wavefront in x- direction on stacking sequence [0/30/0/-30] S and [0/30/0/-30] 2S

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Other authors have proposed a different technique to analyse **wave** propagation in periodic structures. The term “periodic structures” indicates structures which exhibit characteristics that repeat periodically in one, two or three-dimensions. In other words, a periodic structure can be considered as an assemblage of identical elements, called cells or periods, which are coupled to each other on all sides and corners by identical junctions. This characteristic is indeed observable in many real systems. Examples include railway tracks, flat or curved panels regularly supported, such as stringer stiffened panels, fluid filled pipes with flanges, acoustical ducts, rail structures, car tyres, composite plates or shells etc. The use of the structures pe- riodicity can be exploited to solve dynamic problem with reduced computational cost. Many previous works have concerned periodic structures. One of the classical books where the math- ematics of **wave** propagation in periodic structures has been discussed is that of Brillouin [10]. More recent works, [11–15], have studied the problem of defining the dynamical behaviour of periodic structures considering the properties of its characteristic waves. These works consider structures with cross–sectional properties that are uniform along one direction. Much attention is focused on uniform plates and shells with identical stiffeners at regular intervals. In partic- ular in [12] a **finite** **element** **approach** is used for obtaining the dispersion curve of a periodic structure. The main idea is to relate the displacements, q, and the forces, f, at the left and the right hand–side of a periodic **element** of the structure by the propagation constant λ in the following way:

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Most of the containment structures in Korea being made of PSC, the seismic damage assessment system performs on PSC containment structures, adopting a special nonlinear PSC shell **finite** **element** using layered **approach** and appropriate inelastic material models. The accuracy of the system to describe correctly the behaviour of PSC containment structures is validated by means of comparative study between analytical and experimental results at **element** and structural levels. Introducing appropriate inelastic material models and including also a nonlinear prestressed tendon model, the special PSC shell **element** improves the reinforced concrete shell **element** using a four- node quadrilateral thin flat shell **finite** **element** with 6 DOFs per node proposed by Kim et al. (2001) [2].

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A finite element modeling of the human lumbar unit including the spinal cord.. An Approach to Lumbar Vertebra Biomechanical Analysis Using the Finite Element Modeling Based on CT Images.[r]

In fact, the interaction between the nuclear explosion blast **wave** and tank is a typical fluid-structure dynamic problem. But until recently when we analyze its physical course, the force of the blast **wave** is loaded to the tank as a foreknown force. Usually, we divide the region of action into straight reflection area, oblique reflection area and composition **wave** reflection area, and then compute the overpressure on different part of the tank. Such method could not reflect the dynamic course of blast **wave** and tank. MSC.Dytran is one of the most famous **finite** **element** analysis software which has great power in simulating fluid-structure interaction. The Lagrange mesh and the Euler mesh are joined in one analysis model, and the interaction takes place in their boundary.

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Dispersive behaviour of guided waves in laminated plates of **finite** thickness has been studied extensively in recent years. It has been shown that dispersive modal propagation behavior is strongly influenced by the anisotropic properties of each lamina or layer and the stacking sequence of the layers. Several investigators have successfully used comparison of experimental results with modeling predictions to determine the anisotropic elastic constants of individual layers in composite plates. References to these works can be found in the symposium volumes edited by Datta et al. (1990) and Kinra et al. (1994). Both free and fluid-loaded plates have been considered. Chimenti (1997) has published a comprehensive review of guided waves in composite plates and their use for material characterization. Also, Datta (2000) has given a detailed review of the theory of guided waves in composite plates and shells. Although a vast body of work on guided ultrasonic waves in layered plates and shells now exists, relatively few studies have dealt with scattering of these waves by cracks or delaminations. To our knowledge, most of these studies have been confined to the problems of horizontally polarized shear (SH) waves and plane strain (two-dimensional) waves.

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Thermomechanical behaviour of laminated structures have been conducted on various topics. In [2], the elastodynamic response of a polymeric laminate subjected to a discrete range of tem- peratures at a constant relative humidity is studied, with damping and dynamic longitudinal elastic modulus presented as a function of temperature dependent. In [3, 4], the temperature dependent elastic constant and dynamic shear properties of an epoxy resin and its carbon fibre-reinforced composite are presented. More recently, the effect of high temperature on the thermomechanical response of various composite structures, such as multi-layered plates and shells [5], glass epoxy composites [6, 7] and carbon fibre epoxy composites [8], [9], has been extensively assessed. More- over, temperature dependent **wave** based detection of structural damage has been an extensive field of study over the recent years. In [10, 11] baseline subtraction **approach** is used to predict temper- ature e ff ect on guided **wave** signal and to optimally enhance the long term stability of the signal. The **approach** is extended in [12, 13] to reduce the number of baseline measurements to be used. Pitch-catch **approach** is used to numerically predict and experimentally measure the e ff ect of low [14] as well as moderately [15] and extremely [16] elevated temperature on the lamb **wave** re- sponse in sandwich panels and aluminium plates respectively. This is extended in [17] to cover a wider range of temperature in a large frequency range. Semi analytical **finite** **element** (SAFE) model is developed in [18] to predict guided **wave** response under varying temperature in plate. More recently co-integration technique is developed to control the effect of varying temperature in damage detection of structure based on spectral lines analysis [19, 20], wavelet decomposition [21] and direct Lamb **wave** responses [22]. To the best of the authors’ knowledge, FE based com- putational scheme is quite limited in this field of research and the investigation of thermal effect on **wave** interaction in complex and arbitrarily layered composite structures is almost in-existent in the open literature.

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While the forces and the moments are matters of interest, none of the above mentioned papers had studied the parameters of the force. Battjes [9] developed expressions based on the linear theory for intersecting short- crested waves. Clark [10] solved the three dimensional **wave** problems for off-shore structures using frequency domain method. In the same year, Liao [11] developed the solitary **wave** equations using **finite** process method. Cao [12] studied the solitary waves generated by ship motion. Cao, et al [13] analyzed the solitary waves generated by submarines and moving objects. Johnson [14] simulated the impact of **wave** on solid boundaries. Takikawa [15] used the model developed by Washisu to study the **wave** effect on floating bodies.

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Figure 2 shows the histories of contact force and deflection for homogeneous and layered system with various thicknesses of film and substrate obtained from the present **finite** **element** analysis at velocity 10m/s. From Figure 2, the maximum contact forces for homogeneous and layered system occur at around 15μs and 255μs after the initial impact, respectively. In order to verify this coded **finite** **element** program, the present **finite** **element** analysis is compared with an energy balance model that the maximum contact force and the contact duration can be estimated, and **wave** propagation method that the maximum contact force can be predicted simply as shown in Table 2.

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Although there are some design guidances as shown above, some major shortcomings have been recognized by the researchers. As pointed out by Izzuddin et al [6][7], the prescriptive nature of the tying force requirements, deemed sufficient for the avoidance of disproportionate collapse yet unrelated to real structural performance, and the exclusion of ductility considerations at all levels of the provisions made the provisions unsafe. On the other hand, the alternative notional member removal provisions are more performance-based, but these are applied with conventional design checks, and hence they ignore the beneficial effects of such nonlinear phenomena as compressive arching and catenary actions. This in turn can lead to the prediction of an unrealistically large damage area exceeding the prescribed limits, thus forcing the member to be designed as a key **element** when this may be unnecessary. Therefore, more detailed research toward the progressive collapse of multi-storey building is timely. However, as mentioned above, the research on the behavior of the progressive collapse of high rise building is quite limited due to the limited research tools. **Full** scale test of this type of problem is not possible due to its high cost. A 3-D **Finite** **element** model is definitely a best option. However, due to the geometric complexity of multi-storey building and poor preprocessing functions of current general purpose **finite** **element** packages, no **full** scale 3-D **finite** **element** model for investigating progressive collapse has been built so far there is also little research toward the modeling of the structural behavior of multi-story buildings under sudden column loss. Most of the modeling techniques mentioned in section 1 are either the simplified models based on current design guidance or two dimensional models, which could not accurately monitor the overall structural behavior of the whole building.

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NOMENCLATURE Zeroth Order Tensors A material constant associated with elastic a dimensionless material b dimensionless Cv specific heat e specific internal energy 6 weighting parameter e[r]

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In this paper, we study the controllability problem of the semi-discrete internally controlled one-dimensional **wave** equation with the ﬁnite **element** method. We derive the observability inequality and prove the exact controllability for the semi-discrete internally controlled **wave** equation, with the controls taken from a ﬁnite dimensional space.

However, as far as we know, there has not been any report that the POD method is used to reduce the number of unknowns in the classical FE method for the D viscoelastic **wave** equation. Therefore, in this article, we devote ourselves to building an optimized FE extrapolating (OFEE) method that includes very few unknowns but maintains desired accuracy via the POD method, analyzing the existence, stability, and convergence of the OFEE solutions and verifying the eﬃciency and feasibility of the OFEE method by some numerical simulations.

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Tyre noise is becoming a significant source of traffic noise [1] and understanding the vibrational behaviour of a tyre is thus becoming more important. At high frequencies, where FE models become impractically large, knowledge of the **wave** properties of a structure is of great value. **Wave** approaches can then be used. In this paper, the WFE method [2,3] is applied to model the free and forced vibration of a tyre. The formulation is first reviewed. **Element** matrices of a short section of a uniform structure are post-processed to yield the **wave** behaviour. The matrices can be obtained from a conventional FE method and a commercial package. This is in contrast to the spectral **finite** **element** method [4]. A short section of an unloaded tyre is modelled in ANSYS. Complete structural details can be included, which may be difficult in analytical approaches (e.g. [5]).

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Figure 3.29 shows similar dispersion curves for the predominantly extensional (i=3) and flexural nearfield (i=4) **wave** modes. All the i=3,4 types **wave** numbers are complex conjugate pairs below the bifurcation points. Similar bifurcations as in Figure 3.28 can be seen but only the i=3 **wave** modes cut-on as the i=4 modes are the flexural nearfield waves. For example, for the i=3,4 (n=0) **wave** mode, complex conjugate wavenumbers bifurcate to two purely imaginary wavenumbers at Ω = 0.96 and the i=3 type wavenumber becomes purely real at Ω = 1.0 . The WFE results are seen to be less accurate especially at around the bifurcations. The curve which is not close to the bifurcation shows better agreement even if kR is larger than that at the bifurcation. The large errors around the bifurcations are believed to be due to both the FE discretisation and geometrical approximation errors. Although the contribution of each cause is not clear, it is believed that such approximations associated with the WFE modelling dominate the error around the bifurcation points.

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Rotations of the stack about a vertical axis were not accounted for in the finite element model: yet these could be important for the coupling of rotational ground noise to horizontal no[r]

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The ultrasonic propagation speed could be measured by using two types of configuration which is the through transmission method and pulse – echo method which will only take effect if the ultrasonic **wave** able to transmitted through the pipe wall into the liquid medium [10]. This system uses ultrasound to detect the changes of acoustic impedance (Z) which is closely related to density (ρ) of the medium. This can be a useful descriptor to identify the complex ratio of sound pressure to particle velocity which is analogous to electrical impedance. The acoustic equivalent to this relation is given by below equation [11, 12].

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Sanjay Aloni [2] in his work **finite** **element** analysis **approach** is used to modify existing rear axle of tractor trolley. Fatigue failure of the rear axle **finite** **element** model was predicted after the dynamic load was imposed on it. For analysis, a 6.0 ton 2 wheeler tractor trolley i.e. semitrailer manufactured by Awachat Industries Ltd., Wardha is selected. The **finite** **element** analysis of existing rear axle of tractor trolley revealed the stresses distribution on rear axle. So, an effort is made to modify the design of existing rear axle along with change of material so that advantage of weight reduction along with safe stress can be obtained

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