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We present a general technique to solve one-dimensional time-dependent Schrödinger-type equations. A mapped infinite elements approach is used to eliminate spurious reflections of outgoing wave packets from the boundaries of the interval of interest. This procedure leads to more precise solutions because the space coordinates are discretized to approximate the solution in the entire physical domain. We show its utility giving numerical results on three typical examples: a bounded short-range potential; the unbounded potential of a particle in a dc field; and a Hamiltonian with an intrinsic time dependence like the Hamiltonian of a charged particle in an ac electromagnetic field.

In this paper a Control Volume Finite Element Method for harmonic acoustic problems is presented. A dispersion analysis for control volume constructed on Q1 finite elements is compared to Galerkin FEM. The spatial convergence is also given in an eigenfrequency determination process for a cavity. The application for exterior acoustic problems is also studied by dividing the whole field into inner and outer domains using a fictitious boundary. A control volume formulation is used to compute the inner field of the truncated problem, and several approaches are combined to describe the outer field behavior on the outside of the fictitious boundary. The task of coupling is easily implemented through the balance of local flux through polygonal volumes. A two-dimensional configuration with a circular interface demonstrates the validity of this approach.

In this article we present an algorithm for the three-dimensional numerical simulation of the sound spectrum and the propagation of acoustic radiation inside and around long slender hollow objects. The fluid inside and close to the object is meshed by Lagrangian tetrahedral finite elements. To obtain results in the far field of the object, complex conjugated Astley-Leis infinite elements are used. To apply these infinite elements the finite element domain is meshed either in a spherical or an ellipsoidal shape. Advantages and disadvantages of both shapes regarding the form of the object are discussed in this article. The formulation leads to a quadratic eigenvalue problem with real, large and nonsymmetric matrices. An eigenvalue search algorithm is implemented to concentrate on the computation of the interior eigenmodes. This algorithm is based on a linearization of the quadratic problem in a state space formulation. The search algorithm uses a complex shift to efficiently extract the relevant eigenvalues only.

Finite element analysis of transient acoustic phenomena on unbounded exterior domains is very common in engineering analysis. In these problems there is a common need to compute the acoustic pressure at points outside of the acoustic mesh, since meshing to points of interest is impractical in many scenarios. In aeroacoustic calculations, for example, the acoustic pressure may be required at tens or hundreds of meters from the structure. In these cases, a method is needed for post-processing the acoustic results to compute the response at far-field points. In this paper, we compare two methods for computing far-field acoustic pressures, one derived directly from the infinite element solution, and the other from the transient version of the Kirchhoff integral. We show that the infinite element approach alleviates the large storage requirements that are typical of Kirchhoff integral and related procedures, and also does not suffer from loss of accuracy that is an inherent part of computing numerical derivatives in the Kirchhoff integral. In order to further speed up and streamline the process of computing the acoustic response at points outside of the mesh, we also address the nonlinear iterative procedure needed for locating parametric coordinates within the host infinite element of far-field points, the parallelization of the overall process, linear solver requirements, and system stability considerations.

This paper presents three-dimensional (3D) infinite elements for the multi-layered elastodynamics problems. There are three types of elements, namely horizontal, vertical and corner elements. They have been developed using wave functions in function spaces that can simulate real wave problems properly. The elements are extended forms of the axisymmetric infinite elements developed previously. Since these elements can simulate multiple layers and multiple wave numbers, the response of a 3D general structural system considering soil-structure interaction effect can be determined effectively. Numerical analyses are carried out for a rigid massless disk and square footings on the surface of various layer conditions for verification purposes. The calculated results are compared with existing analytical and numerical data and they were found to be in good agreement.

The problem of a half-space with cavity under vertically incident waves was solved by many researchers using different approaches. However, substantially different solutions were obtained, partially due to the difference in the method of formulation, and partially due to the lack of complete identical data for use in analysis. In this paper, the finite/infinite element approach has been adopted to study the two-dimensional response of an elastic half-space containing a buried, unlined, infinitely long cylindrical of circular shape subjected to harmonic P and SV waves. First, the analysis procedure based on the finite and infinite elements is summarized. Second, considerations in preparing the finite element mesh to ensure the accuracy and convergence of the solution are presented. Next, the validity of the procedure of solution is verified for some intuitive, fundamental cases. Finally, the problems solved by previous researchers with identical or assumed data will be re-solved, along with discussions on the discrepancies existing among the three solutions. One feature with the finite/infinite element approach is that it is simple and straightforward, involving less assumptions and mathematical operations, whose reliability has been verified in solving various soil vibration problems. The fact that the present solutions are in close agreement to those by Luco and De Barros (1994) for all the cases studied indicates that the latter is the most reliable one among the existing theories.

An assumption of collinear straight coastlines in the far field, traditionally used in wave models for harbors, is a limitation to wave simulations in the domain extending to infinity, since this assumption is invalid for most real coastlines. By alleviating this limitation based on the geometric-optics approximation, functions of these coastal wave-deformation models are extended to be able to predict wave patterns around the semi-infinite breakwater and convex and concave coasts, such as bulkheads with discontinuous alignment. Mapped infinite elements are also formulated for exterior wave problems. A basic concept in developing is that the true decaying property of scattered waves, i.e. with the modes of r^-1/2, r^-3/2,…, where r is the radial distance, must be represented directly in infinite elements. To do so, a complementary element is introduced to the infinite element. Infinite element integrals can be expressed explicitly because of analytical integrability due to weak singularity in the infinite mapping. Three test problems for coastal wave-deformation models are solved combined with the cubic infinite element with r^-1/2 and r^-3/2 decays; this combination technique is found to be effective and accurate for problems of wave scattering.

Tunnel–medium interaction problems are one of the important problems in engineering. Because it has a high risk in terms of reliability, tunnel structures should be modeled in detail. The modeling of tunnel structures under static and dynamical loads is a difficult problem in engineering because of consists of a lot of conditions and effects, such as heterogeneous medium, soil layers, porosity, soil–structure interaction, groundwater, earthquake, etc. In this paper, aims to investigate the tunnel–medium interaction problems for nonlinear static and dynamic analyses. This study includes nonlinear static and dynamic analyses for layered porous semi-infinite viscoelastic medium with twin tunnels. The constitutive property of each layer of medium is considered in bilinear stress–strain relation with uniform porosity and Kelvin–Voigt viscoelastic property. The considered study is solved via the finite element method within the two-dimensional (2D) model. Layered medium is modeled as finite and infinite elements. In the solution process, the incremental force method is implemented and, for each load step, finite element equations are solved according to the bilinear stress–strain relation. In nonlinear dynamic analysis, the dynamic loads are divided by a certain finite number and applied incrementally depending on the time-dependent load function. At each load step, the final displacement, velocity and acceleration of that load step, obtained as a result of applying the Newmark $\beta$ method procedure, are assigned as the starting value of the next load step according to the bilinear stress–strain relation. Influences of porosity and position of tunnels on the nonlinear static and dynamic deflections of the system are investigated. Also, differences between linear and nonlinear responses are compared and discussed.

Exterior acoustic problems occur in a wide range of applications, making the finite element analysis of such problems a common practice in the engineering community. Various methods for truncating infinite exterior domains have been developed, including absorbing boundary conditions, infinite elements, and more recently, perfectly matched layers (PML). PML are gaining popularity due to their generality, ease of implementation, and effectiveness as an absorbing boundary condition. PML formulations have been developed in Cartesian, cylindrical, and spherical geometries, but not ellipsoidal. In addition, the parallel solution of PML formulations with iterative solvers for the solution of the Helmholtz equation, and how this compares with more traditional strategies such as infinite elements, has not been adequately investigated.

In this paper, we present a parallel, ellipsoidal PML formulation for acoustic Helmholtz problems. To faciliate the meshing process, the ellipsoidal PML layer is generated with an on-the-fly mesh extrusion. Though the complex stretching is defined along ellipsoidal contours, we modify the Jacobian to include an additional mapping back to Cartesian coordinates in the weak formulation of the finite element equations. This allows the equations to be solved in Cartesian coordinates, which is more compatible with existing finite element software, but without the necessity of dealing with corners in the PML formulation. Herein we also compare the conditioning and performance of the PML Helmholtz problem with infinite element approach that is based on high order basis functions. On a set of representative exterior acoustic examples, we show that high order infinite element basis functions lead to an increasing number of Helmholtz solver iterations, whereas for PML the number of iterations remains constant for the same level of accuracy. This provides an additional advantage of PML over the infinite element approach.