General Wave Propagation Problems Research Articles

Numerical Solution of the Wave Propagation Problem in a Plate

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.

Dispersion Reduction for the Wave Propagation Problems Using a Coupled “FE-Meshfree” Triangular Element

As is known to all, there always exists the numerical dispersive effects of the standard finite element (FE) for the wave propagation problems and the corresponding FE solutions are usually unreliable in relatively high frequency range. In this work, a coupled “FE-Meshfree” element based on triangular mesh is introduced to reduce the dispersion effects for wave propagation problems. In this coupled element, the standard FE nodal shape functions are combined with the meshfree nodal shape functions to give a new hybrid nodal shape functions. As a result, both the individual advantages of the FE technique and the meshfree technique are strengthened by the present hybrid method. Through the dispersion analysis for the wave propagation problems, it is found that this coupled “FE-Meshfree” element could significantly reduce the numerical dispersive effects and it also have a higher tolerance to the mesh distortion than the other existing elements, hence the present method is quite promising to handle the general wave propagation problems in practical engineering application.

A wave packet enriched finite element for electroelastic wave propagation problems

A two dimensional finite element (FE) is developed by enriching the conventional Lagrange interpolation functions with local element domain wave functions, for accurate solution of general wave propagation problems in piezoelastic media. The enrichment functions, applied to both displacement and electric potential fields, satisfy the partition of unity condition, and vanish at the nodes. The formulation is developed using the extended Hamilton’s principle for piezoelastic solids, considering two-way electromechanical coupling. The enriched FE is shown to perform equally well in terms of computational efficiency and accuracy for broadband impact induced electroelastic wave to narrowband ultrasonic guided wave propagation problems, unlike the other available elements. The solution for impact in a piezoelectric plate is shown to alleviate the spurious undulations in both velocity and electric potential fields, which are encountered in the conventional FE solutions. For the problem of high frequency Lamb wave actuation and sensing in a thin plate bonded with piezoelectric actuator/sensor patches, the element shows significant improvement in the computational efficiency over the conventional FE. Further, the free edge effect of steep gradients in the shear stress distribution at the actuator-plate interface is accurately captured by the proposed element using much fewer degrees of freedom than the conventional FE.

Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems

AbstractWe analyse the convergence of finite element discretizations of time-harmonic wave propagation problems. We propose a general methodology to derive stability conditions and error estimates that are explicit with respect to the wavenumber $k$. This methodology is formally based on an expansion of the solution in powers of $k$, which permits to split the solution into a regular, but oscillating part, and another component that is rough, but behaves nicely when the wavenumber increases. The method is developed in its full generality and is illustrated by three particular cases: the elastodynamic system, the convected Helmholtz equation and the acoustic Helmholtz equation in homogeneous and heterogeneous media. Numerical experiments are provided, which confirm that the stability conditions and error estimates are sharp.

An enriched finite element method for general wave propagation problems using local element domain harmonic enrichment functions

We present an enriched finite element (FE) formulation applicable for general wave propagation problems in one- and two-dimensional domains, using local element domain spatial harmonic enrichment functions which satisfy the partition of unity condition. It allows prescription of boundary conditions in the same way as in the conventional FE method. The method is assessed for different classes of wave propagation problems such as impact and high frequency-guided wave propagation in bars and plates, and surface and body wave propagation in semi-infinite solid media for which the classical FE method either fails to yield accurate results or is prohibitively expensive. It is shown that the present formulation gives accurate solutions to the former and shows significant improvement in computational efficiency for the latter category of problems. The performance is also assessed against other special FEs such as the spectral FE and a recently proposed enriched FE with global harmonic basis functions.

Transient implicit wave propagation dynamics with overlapping finite elements

We present novel overlapping finite elements used with the Bathe time integration method to solve transient wave propagation problems. The solution scheme shows two important properties that have been difficult to achieve in the numerical solution of general wave propagations: monotonic convergence of calculated solutions with decreasing time step size and a solution accuracy almost independent of the direction of wave propagation through the mesh. The proposed scheme can be efficiently used with irregular meshes. These properties make the scheme (the combined spatial and temporal discretizations) promising to solve general wave propagation problems in complex geometries involving multiple waves. A dispersion analysis is given and various example problems are solved to illustrate the performance of the solution scheme.

Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media

Solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering. The design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration. In this paper, we focus on multi-dimensional wave problems and consider linear second-order wave equation in heterogeneous media. We develop and analyze an LDG method, in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy. Compatible high order energy conserving time integrators are also proposed. The optimal error estimates and the energy conserving property are proved for the semi-discrete methods. Our numerical experiments demonstrate optimal rates of convergence, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property.

Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation

The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuˇska & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems.

Nonuniform distortionless transmission lines

Heavside's classical condition for distortionless signalling, g/c=r/l, is generalized to nonuniform transmission lines. The resulting class of nonuniform lines possesses several interesting properties, which are investigated in some detail. For instance, there exists a preferred direction for distortionless signalling, a phenomenon not observed with uniform lines. We also show that closed-form solutions to quite general wave propagation problems can be derived directly in the time domain.

Wave propagation across ion cyclotron resonance harmonic layers

Wave propagation across ion cyclotron resonance layers is investigated by use of a boundary layer analysis for a perpendicularly stratified, weakly inhomogeneous plasma. Corresponding to the situation in many plasmas of interest, it is assumed that the wave frequency is much less than the ion plasma frequency, but that the ratio of ion plasma frequency to wave frequency times the ratio of ion thermal speed to the speed of light is small. Nontrivial effects of the cyclotron resonance appear in the boundary layer and modify the current density from the cold plasma limit. These corrections suggest the breakdown of the geometrical optics limit. Wave equations for the field valid within an arbitrary harmonic resonance layer are derived, as well as for the two ion hybrid resonance case. Analytical proofs of reciprocity relations are given for the general wave propagation problems.