Omar Laghrouche

SHORT WAVE MODELLING USING SPECIAL FINITE ELEMENTS

The solutions to the Helmholtz equation in the plane are approximated by systems of plane waves. The aim is to develop finite elements capable of containing many wavelengths and therefore simulating problems with large wave numbers without refining the mesh to satisfy the traditional requirement of about ten nodal points per wavelength. At each node of the meshed domain, the wave potential is written as a combination of plane waves propagating in many possible directions. The resulting element matrices contain oscillatory functions and are evaluated using high order Gauss-Legendre integration. These finite elements are used to solve wave problems such as a diffracted potential from a cylinder. Many wavelengths are contained in a single finite element and the number of parameters in the problem is greatly reduced.

Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering.

Classical finite–element and boundary–element formulations for the Helmholtz equation are presented, and their limitations with respect to the number of variables needed to model a wavelength are explained. A new type of approximation for the potential is described in which the usual finite–element and boundary–element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions evenly distributed on the unit sphere. Compared with standard piecewise polynomial approximation, the plane–wave basis is shown to give considerable reduction in computational complexity. In practical terms, it is concluded that the frequency for which accurate results can be obtained, using these new techniques, can be up to 60 times higher than that of the conventional finite–element method, and 10 to 15 times higher than that of the conventional boundary–element method.

A numerical integration scheme for special finite elements for the Helmholtz equation

The theory for integrating the element matrices for rectangular, triangular and quadrilateral finite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration abscissae and weights are made available. The results are compared with those obtained using large numbers of Gauss-Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives confidence in the procedures, and show that large savings in computation time can be achieved.

Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements

We consider a two-dimensional wave diffraction problem from a closed body such that the complex progressive wave potential satisfies the Sommerfield condition and the Helmholtz equation. We are interested in the case where the wavelength is much smaller than any other length dimensions of the problem. We introduce new mapped wave envelope infinite elements to model the potential in the far field, and test them for some simple Dirichlet boundary condition problems. They are used in conjuction with wave envelope finite elements developed earlier [1] to model the potential in the near field. An iterative procedure is used in which an initial estimate of the phase is iteratively improved. The iteration scheme, by which the wave envelope and phase are recovered, is described in detail. Copyright

Railway cuttings and embankments: Experimental and numerical studies of ground vibration.

Railway track support conditions affect ground-borne vibration generation and propagation. Therefore this paper presents a combined experimental and numerical study into high speed rail vibrations for tracks on three types of support: a cutting, an embankment and an at grade section. Firstly, an experimental campaign is undertaken where vibrations and in-situ soil properties are measured at three Belgian rail sites. A finite element model is then developed to recreate the complex ground topology at each site. A validation is performed and it is found that although the at-grade and embankment cases show a correlation with the experimental results, the cutting case is more challenging to replicate. Despite this, each site is then analysed to determine the effect of earthworks profile on ground vibrations, with both the near and far fields being investigated. It is found that different earthwork profiles generate strongly differing ground-borne vibration characteristics, with the embankment profile generating lower vibration levels in comparison to the cutting and at-grade cases. Therefore it is concluded that it is important to consider earthwork profiles when undertaking vibration assessments.

Time-independent hybrid enrichment for finite element solution of transient conduction-radiation in diffusive grey media

We investigate the effectiveness of the partition-of-unity finite element method for transient conduction-radiation problems in diffusive grey media. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media. The coupled equations are integrated in time using a semi-implicit method in the finite element framework. We show that for the considered problems, a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method. It is illustrated that this approach can be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls. The performance of the proposed partition-of-unity method is analyzed on several test examples for transient conduction-radiation problems in two space dimensions.

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 P 1 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 P 1 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.

Wavelet based ILU preconditioners for the numerical solution by PUFEM of high frequency elastic wave scattering

Daubechies family of wavelets combined to the Incomplete Lower-Upper (ILU) factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the Partition of Unity Finite Element Method (PUFEM). After applying the discrete wavelet transform (DWT) to each dense block in the final matrix and the known right-hand side, due to the local enrichment by pressure (P) and shear (S) plane waves, the resulting linear system is solved by the restarted Generalized Minimum RESidual method (GMRES) with ILU preconditioners allowing fill-in elements in the L and U matrix factors. A reordering algorithm of the vertices based on Gibbs method is also introduced. It leads to a significant reduction of the bandwidth of the wave based Finite Element (FE) matrix and enables the ILU preconditioners to be more effective. To study the performance of the proposed preconditioners, a problem of a horizontal S plane wave scattered by a rigid circular body in an infinite elastic medium is considered. The numerical tests show the good performance of the DWT based ILU preconditioners in improving the rate of convergence of GMRES, for high numbers of approximating P and S plane waves, on coarse mesh grids containing multi-wavelength sized elements. Moreover, the Haar DWT enhances the scalability with respect to the problem size, when the number of the nodal points increases, of the ILU preconditioner which uses the threshold strategy in the control of the fill-in elements. Despite the high level of the conditioning, a good accuracy may be achieved for a discretization level of around 1.9 degrees of freedom per S wavelength, which is far below the rule of thumb of 10 nodal points per wavelength, adopted for the conventional FE.

Nonlinear stress analysis of plates under thermo-mechanical loads

The aim of the present work is to carry out a simplified mathematical modelling for nonlinear stress analysis of plates under temperature changes and mechanical transverse loads. The material properties of the plate are proposed to be temperature-dependent. The geometrically nonlinear plate theory is employed to understand the stress distribution due to thermo-mechanical loads. A set of coupled nonlinear partial differential equations are solved using harmonic series expansion to find the static responses. Two boundary conditions are considered for simply supported plates, namely, movable edges and immovable edges. The accuracy of the results is checked by comparing with the output of other solution methods.

A comparison of NRBCs for PUFEM in 2D Helmholtz problems at high wave numbers

In this work, exact and approximate non-reflecting boundary conditions (NRBCs) are implemented with the Partition of Unity Finite Element Method (PUFEM) to solve short wave scattering problems governed by the Helmholtz equation in two dimensions. By short wave problems, we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer. Various NRBCs are implemented and a comparison of their performance is carried out based on the accuracy of the results, ease of implementation and computational cost. The aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number.