Analysis of Traveling Wave Responses of Structures

In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the [Formula: see text] norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic.

Realistic physics-based 3D earthquake simulation for source-to-structure wave propagation consists of a powerful numerical tool for seismic response prediction of critical structures submitted to high safety standards. Structural response considering soil–structure interaction (SSI) is usually estimated by Finite Element Method (FEM) approach, as it is considered as the most flexible numerical approach for nonlinear structural dynamics. However, current engineering practice considers seismic input motion as vertically incident plane waves, despite the fact that this assumption excludes wave passage effects for large infrastructures and surface waves appearing from possible local basin effects. In this framework, a realistic input excitation needs to be defined as an input excitation of the FEM model, accounting for (i) a realistic dynamic excitation, (ii) wave propagation path in the regional scale, and (iii) local site-effects. The Domain Reduction Method (DRM), which allows for the imposition of a 3D complex incident wave field as an input to the SSI model, is adapted and examined here in a Spectral Element Method (SEM) – FEM weak coupling approach. The weak coupling is verified at first for a canonical case-study and for an increasing complexity of the dynamic excitation: (i) double-couple point-source and (ii) extended fault. An optimization approach, based on the decimation of SEM output signal, is then examined in order to decrease the computational burden by maintaining the same accuracy of the final solution. The SEM-FEM weak coupling is then used to study the SSI problem, where the impact of the reduced domain size on structural response is examined at first. The current study shows that a reduced domain of dimension greater than , where is the maximum wavelength, is sufficient for a proper representation of structural response. For a fixed size model, structural and soil responses are then examined for a hypothetical case-study.

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

The propagation of elastic transient waves in a multi–layered medium subjected to in–plane loadings is investigated in this study. One of the objectives of this study is to develop an effective analytical method for determining transient full–field solutions in the layered medium. A matrix method is developed by expanding the matrix solution obtained directly from the boundary–value problem in the integral transform domain into a power series of the phase–related reflection and transmission matrix which characterizes the multiple reflections and transmissions of all waves in every layer. The transient response of the multi–layered medium is decomposed into infinite wave groups in which the waves are either reflected by, or transmitted through, the interfaces. The connection between the proposed matrix method and the generalized ray method is established for the layered medium in the transform domain. The matrix representation of the solution enables us to calculate the transient response of the layered medium without tracing the ray path manually. The obtained analytical solution can easily be applied to numerical calculations. The double inverse transform is performed based on Cagniard9s method and the theoretical transient solution for a layered half–space subjected to an in–plane dynamic force will be presented in part II. An experimental set–up that simulates the plane stress condition for a layered half–space was established to obtain the dynamic displacement response. The experimental result agrees well with the theoretical solution. The proposed methodology in this study can be extended to solve more complicated problems such as waves propagating in three–dimensional space.

Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach

In this paper, a spectral finite element model (SFEM) is developed for an n-layered elastic beam and subsequently used to investigate its dynamic response and wave propagation characteristics. Each layer of the beam is idealized by a Timoshenko beam, in which shear deformation as well as rotational inertia are included. This higher order theory is critical to capture high frequency response of the multi-layered beam structures. Semi-analytical solutions were determined for the governing equations in order to construct the SFEM. Our frequency predictions were validated by the results of two and three-layer beams in the literature and good correlations were achieved. Fewer elements were used in our SFEM compared to conventional finite element based approaches, which substantially benefits the ultrasonic frequency simulations. Wave propagation responses were calculated for a two-layer beam, in which a notch in the top layer was assumed to represent the damage case. Wave reflection from the notch was observed to indicate the existence of damage. This newly developed SFEM can be served as a platform to conduct comprehensive simulations in order to capture wave propagation characteristics in multi-layered beam structures.

The conventional modal methods become inefficient when expanded to the high frequency region. This leads to a large number of degrees of freedom (DOF) even for one-dimension structures. Meanwhile, a short computational time is also the requirement of on-line structural health monitoring. This paper presented an application of B-spline wavelet on interval (BSWI) finite element for one-dimension elastic wave propagation problems. By using central difference method in time domain, numerical results involving wave propagations in crack free rod and beam were obtained and compared with results calculated by classical finite element method. Numerical examples validated the good performance of BSWI. BSWI could provide a accurate and efficient solution for wave propagation problems. The proposed method provides a good reference for structural health monitoring.

SummaryShear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.

Modified Hermitian cubic spline wavelet on interval finite element for wave propagation and load identification

The design of structures and machineries in present days are based on optimizing of multi-objectives such as maximum strength, maximum life, minimum weight and minimum cost. Due to this flexiblity they allow having a very high level of stresses. This leads to development of cracks in their elements. Due to long-term service many engineering structures may have structural defects such as cracks. So it is very much essential to know the property of structures and response of such structures in various cases. In this article the natural frequencies and mode shapes of an uncracked and cracked cantilever Timoshenko beam is studied by using finite element method (FEM) and MATLAB programme. The effect of crack on the natural frequencies of the uncracked and cracked Timoshenko beam is studied.

The effect of liquid on the propagation of waves in a micropolar elastic layer with voids immersed in an infinite liquid has been investigated. In case I the frequency equations for symmetric and antisymmetric vibrations have been obtained and some special cases have been deduced. Case II is concerned with wave propagation in a micropolar elastic solid with voids. Reflection of micropolar elastic waves at the free surface of a micropolar elastic body with voids is discussed. In case III, propagation of Rayleigh waves in a micropolar elastic body with voids and particle motion for Rayleigh waves are discussed. Numerical computations of the amplitude ratios for different reflected waves have been performed and plotted against the angle of emergence. The frequency equation of Rayleigh wave propagation is obtained and phase velocity as a function of Poisson’s ratio, frequency ratio, non-dimensional micropolar constant and normalized wave number have been plotted graphically for a particular model.

In numerical simulation of soil media, the problem of wave propagation is of a great importance. Proposed in this paper is a new formulation of infinite elements which can be used in numerical simulation of wave propagation problems in infinite domains. In numerical simulation of wave propagation, the finite elements are used to model the near field, whereas the infinite elements are used to represent the behaviour of the far field. Formulations and various implementation aspects of the proposed infinite elements are illustrated. The accuracy and efficiency of the proposed approach is considered by comparing the obtained results with analytical and other numerical results. For better explanation, a couple of examples were analyzed such as one dimensional wave propagation problems arising from the Heaviside step function and impulse functions. In order to get a more complete insight, two dimensional wave propagations in soil medium were considered and the results are presented accordingly. Finally, a soil layer subjected to seismic excitation was analysed. The new infinite element was developed using the User Programmable Features of the ANSYS software, which enables creating new elements within the robust ANSYS core algorithm. The main advantage of the proposed infinite elements is that they can be used directly within a finite framework with minor modifications such as the Jacobian matrix and added absorbing properties. The performed analysis using the infinite elements shows very promising results and provides a good tool for simulation of boundary conditions.

This work presents a new method, the SOR_FEM, spurious oscillation reduction-FEM, that uses analytical solutions of simple problems to find numerical solutions of more complex problems to reduce the spurious oscillations that occur due to solution discontinuity. With the proposed method it is possible to find numerical solutions without spurious oscillations of problems submitted to impulsive point sources without using any approximation for the Dirac delta function. To validate the proposed method, linear problems of heat diffusion and wave propagation in homogeneous and heterogeneous media are solved with the Finite Element Method (FEM).

To simulate or design a reasonably large system, fast and simple models are necessary. To verify the design versus the specifications, more detailed (and costly) calculations can be performed and final adjustments made. In wave propagation problems, circuit analogs provide a powerful, yet simple, means of computing the desired response of the system, such as reflection or transmission coefficients. The reason circuit analog models are good for wave propagation problems, is that they are exact for one-dimensional wave propagation, regardless of whether we consider acoustic or electromagnetic waves. Typically, wave propagation through homogeneous media is modeled as a transmission line with propagation constant β and characteristic impedance Z, whereas obstacles such as thin sheets are modeled as lumped elements. If the sheets are lossless, the circuit models contain only reactive elements such as capacitors and inductors. Modeling complex wave propagation problems with circuit analogs was to a large extent developed in conjunction with the development of radar technology during the Second World War. Many of the results from this very productive era are collected in the Radiation Laboratory Series and related literature, in particular (Collin, 1991; 1992; Marcuvitz, 1951; Schwinger & Saxon, 1968). Further development has been provided by research on frequency selective structures (Munk, 2000; 2003). In recent years, the circuit analogs have even been used in an inverse fashion: by observing that wave propagation through a material with negative refractive index could be modeled as a transmission line with distributed series capacitance and shunt inductance, i.e., the dual of the standard transmission line, the most successful realization of negative refractive index material is actually made by synthesizing this kind of transmission line using lumped elements (Caloz & Itoh, 2004; Eleftheriades et al., 2002). This chapter is organized as follows. In Section 2 we show that propagation of electromagnetic waves in any material, regardless how complicated, boils down to an eigenvalue problem which can be solved analytically for isotropic media, and numerically for arbitrary media. From this eigenvalue problem, the propagation constant and characteristic impedance can be derived, which generates a transmission line model. In Section 3, we show how sheets with or without periodic patterns can be modeled as lumped elements connected by transmission lines representing propagation in the surrounding medium. The lumped elements can be given a firm definition and physical interpretation in the low frequency limit, and in Section 4 we show how these low frequency properties

This paper presents a numerical study on the dynamic behaviours of an elastic visco-plastic steel beam under repeated mass impacts, with a finite element method. The numerical approach is done through a three-dimensional finite element model and a repeated impact scenario adopted in the numerical simulation is proposed. The study comprises repeated impacts under identical initial conditions, whereby the plastic deformation and resulting residual stresses due to previous impacts are included. The impact process is described through simulations involving the local contact behaviour and structural response, to get a better understanding into the mechanism of repeated impacts. Impact parameters related to the accumulation of plastic deformation, progression of residual stress, wave propagation, and among others are presented and analyzed in detail. From the numerical results, it is found that for current impact system, the impact parameters vary monotonously with impact number, and tend to a saturation state. In addition, the effects of viscoplasticity and impact position on the repeated impact responses are discussed. It is shown that repeated impact behaviours are mainly depend on the initial impact energy, material property and impact number, and the effect of impact position on structural resistance to repeated impacts is negligible.