High-order finite elements for the solution of Helmholtz problems

In this paper, two high-order finite element models are investigated for the solution of two-dimensional wave problems governed by the Helmholtz equation. Plane wave enriched finite elements, developed in the Partition of Unity Finite Element Method (PUFEM), and high-order Lagrangian-polynomial based finite elements are considered. In the latter model, the Chebyshev-Gauss-Lobatto nodal distribution is adopted and the approach is often referred to as the Spectral Element Method (SEM). The two strategies, PUFEM and SEM, were developed separately and the current study provides data on how they compare for solving short wave problems, in which the characteristic dimension is a multiple of the wavelength. The considered test examples include wave scattering by a rigid circular cylinder, evanescent wave cases and propagation of waves in a duct with rigid walls. The two approaches are assessed in terms of accuracy for increasing SEM order and PUFEM enrichment. The conditioning, discretization level, total number of storage locations and total number of non-zero entries are also compared

Settlement behaviour of hybrid asphalt-ballast railway tracks

The use of structural asphalt layers inside ballasted railway tracks is attractive because it can increase track bending stiffness. Therefore, for the first time, this paper investigates the long-term settlement characteristics of asphaltic track in the presence of a subgrade stiffness transition zone. Phased load cyclic compression laboratory tests are performed on a large-scale hybrid asphalt-ballast track, supported by subgrade with varying stiffness. It is found that an asphaltic layer acts as a bridge to shield the subgrade from high stresses. It is also found that the asphalt reduces track settlement, and is particularly effective when subgrade stiffness is low

An enriched finite element model with q-refinement for radiative boundary layers in glass cooling

Radiative cooling in glass manufacturing is simulated using the partition of unity finite element method. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary simplified P1 approximation for the radiation in non-grey semitransparent media. To integrate the coupled equations in time we consider a linearly implicit scheme in the finite element framework. A class of hyperbolic enrichment functions is proposed to resolve boundary layers near the enclosure walls. Using an industrial electromagnetic spectrum, the proposed method shows an immense reduction in the number of degrees of freedom required to achieve a certain accuracy compared to the conventional h -version finite element method. Furthermore the method shows a stable behaviour in treating the boundary layers which is shown by studying the solution close to the domain boundaries. The time integration choice is essential to implement a q -refinement procedure introduced in the current study. The enrichment is refined with respect to the steepness of the solution gradient near the domain boundary in the first few time steps and is shown to lead to a further significant reduction on top of what is already achieved with the enrichment. The performance of the proposed method is analysed for glass annealing in two enclosures where the simplified P1 approximation solution with the partition of unity method, the conventional finite element method and the finite difference method are compared to each other and to the full radiative heat transfer as well as the canonical Rosseland model

Study of railway track stiffness modification by polyurethane reinforcement of the ballast

This paper presents the measured results of full-scale testing of railway track under laboratory conditions to examine the effect on the track stiffness when the ballast is reinforced using a urethane cross-linked polymer (polyurethane). The tests are performed in the GRAFT I (Geopavement and Railways Accelerated Fatigue Testing) facility and show that the track stiffness can be significantly enhanced by application of the polymer. The track stiffness is measured at various stages during cyclic loading and compared to the formation stiffness, which is determined prior to testing using plate load tests. The results indicate that the track stiffness increased by approximately 40–50% based on the measured results and from the previously published GRAFT I settlement model. The track stiffness was monitored during loading for a maximum of 500,000 load cycles. The paper concludes by presenting and commenting on, the application of the technique to a real site where the Falling Weight Deflectometer was used before and after polymer treatment to determine the dynamic sleeper support stiffness. The very challenging site conditions are highlighted, in particular the water logged nature of the site, and comment made on the effect of the water on polymer installation. The results of the FWD measurements indicate that a good increase in overall track stiffness was measured. These results are consistent with the laboratory tests which are performed on a different soil and use a different measurement technique and hence confirm that regardless of the soil and measurement system track stiffness increases are observed using this technique

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems

Enriched finite elements for initial-value problem of transverse electromagnetic waves in time domain

This paper proposes a partition of unity enrichment scheme for the solution of the electromagnetic wave equation in the time domain. A discretization scheme in time is implemented to render implicit solutions of systems of equations possible. The scheme allows for calculation of the field values at different time steps in an iterative fashion. The spatial grid is partitioned into a finite number of elements with intrinsic shape functions to form the bases of solution. Furthermore, each finite element degree of freedom is expanded into a sum of a slowly varying term and a combination of highly oscillatory functions. The combination consists of plane waves propagating in multiple directions, with a fixed frequency. This significantly reduces the number of degrees of freedom required to discretize the unknown field, without compromising on the accuracy or allowed tolerance in the errors, as compared to that of other enriched FEM approaches. Also, this considerably reduces the computational costs in terms of memory and processing time. Parametric studies, presented herein, confirm the robustness and efficiency of the proposed method and the advantages compared to another enrichment method

Mixed enrichment for the finite element method in heterogeneous media

Problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand, different enriched finite element methods such as the partition of unity, which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work, we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context, the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials

Non-linear soil behaviour on high speed rail lines

This paper gives new insights into non-linear subgrade behaviour on high speed railway track dynamics. First, a novel semi-analytical model is developed which allows for soil stiffness and damping to dynamically change as a function of strain. The model uses analytical expressions for the railroad track, coupled to a thin-layer element formulation for the ground. Material non-linearity is accounted for using a ‘linear equivalent’ approach which iteratively updates the soil material properties. It is validated using published datasets and in-situ field data. Four case studies are used to investigate non-linear behaviour, each with contrasting subgrade characteristics. Considering an 18 tonne axle load, the critical velocity is significantly lower than the linear case, and rail deflections are up to 30% higher. Furthermore, at speeds close-to, but below the non-linear critical velocity, dynamic amplification is highly sensitive to small increases in train speed. These findings are dependent upon soil material properties, and are important for railway track-earthwork designers because often 70% of the linear critical velocity is used as a design limit. This work shows that designs close to this limit may be still at risk of high dynamic effects, particularly if line speed is increased in the future

Settlement behaviour of hybrid asphalt-ballast railway tracks

The use of structural asphalt layers inside ballasted railway tracks is attractive because it can increase track bending stiffness. Therefore, for the first time, this paper investigates the long-term settlement characteristics of asphaltic track in the presence of a subgrade stiffness transition zone. Phased load cyclic compression laboratory tests are performed on a large-scale hybrid asphalt-ballast track, supported by subgrade with varying stiffness. It is found that an asphaltic layer acts as a bridge to shield the subgrade from high stresses. It is also found that the asphalt reduces track settlement, and is particularly effective when subgrade stiffness is low