298 research outputs found
Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM
AbstractThe paper addresses the properties of finite element solutions for the Helmholtz equation. The h-version of the finite element method with piecewise linear approximation is applied to a one-dimensional model problem. New results are shown on stability and error estimation of the discrete model. In all propositions, assumptions are made on the magnitude of hk only, where k is the wavelength and h is the stepwidth of the FE-mesh. Previous analytical results had been shown with the assumption that k2h is small. For medium and high wavenumber, these results do not cover the meshsizes that are applied in practical applications. The main estimate reveals that the error in H1-norm of discrete solutions for the Helmholtz equation is polluted when k2h is not small. The error is then not quasioptimal; i.e., the relation of the FE-error to the error of best approximation generally depends on the wavenumber k. It is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution. In the result of this analysis, thorough and rigorous understanding of error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. The h-p-version of the FEM is studied in Part II
Efficiency of Higher Order Finite Elements for the Analysis of Seismic Wave Propagation
The analysis of wave propagation problems in linear damped media must take
into account both propagation features and attenuation process. To perform
accurate numerical investigations by the finite differences or finite element
method, one must consider a specific problem known as the numerical dispersion
of waves. Numerical dispersion may increase the numerical error during the
propagation process as the wave velocity (phase and group) depends on the
features of the numerical model. In this paper, the numerical modelling of wave
propagation by the finite element method is thus analyzed and dis-cussed for
linear constitutive laws. Numerical dispersion is analyzed herein through 1D
computations investigating the accuracy of higher order 15-node finite elements
towards numerical dispersion. Concerning the numerical analy-sis of wave
attenuation, a rheological interpretation of the classical Rayleigh assumption
has for instance been previously proposed in this journal
A semi-analytical scheme for highly oscillatory integrals over tetrahedra
This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and Díez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.This paper details a semi-analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the Partition of Unity Method (PUM) enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non-oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared to standard Gauss-Legendre quadratures.Peer ReviewedPostprint (author's final draft
Efficient implementation of high-order finite elements for Helmholtz problems
Computational modeling remains key to the acoustic design of various applications, but it is constrained by the cost of solving large Helmholtz problems at high frequencies. This paper presents an efficient implementation of the high-order Finite Element Method for tackling large-scale engineering problems arising in acoustics. A key feature of the proposed method is the ability to select automatically the order of interpolation in each element so as to obtain a target accuracy while minimising the cost. This is achieved using a simple local a priori error indicator. For simulations involving several frequencies, the use of hierarchic shape functions leads to an efficient strategy to accelerate the assembly of the finite element model. The intrinsic performance of the high-order FEM for 3D Helmholtz problem is assessed and an error indicator is devised to select the polynomial order in each element. A realistic 3D application is presented in detail to demonstrate the reduction in computational costs and the robustness of the a priori error indicator. For this test case the proposed method accelerates the simulation by an order of magnitude and requires less than a quarter of the memory needed by the standard FEM
On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning
This paper studies and analyzes a preconditioned Krylov solver for Helmholtz
problems that are formulated with absorbing boundary layers based on complex
coordinate stretching. The preconditioner problem is a Helmholtz problem where
not only the coordinates in the absorbing layer have an imaginary part, but
also the coordinates in the interior region. This results into a preconditioner
problem that is invertible with a multigrid cycle. We give a numerical analysis
based on the eigenvalues and evaluate the performance with several numerical
experiments. The method is an alternative to the complex shifted Laplacian and
it gives a comparable performance for the studied model problems
A Simple Numerical Absorbing Layer Method in Elastodynamics
The numerical analysis of elastic wave propagation in unbounded media may be
difficult to handle due to spurious waves reflected at the model artificial
boundaries. Several sophisticated techniques such as nonreflecting boundary
conditions, infinite elements or absorbing layers (e.g. Perfectly Matched
Layers) lead to an important reduction of such spurious reflections. In this
Note, a simple and efficient absorbing layer method is proposed in the
framework of the Finite Element Method. This method considers Rayleigh/Caughey
damping in the absorbing layer and its principle is presented first. The
efficiency of the method is then shown through 1D Finite Element simulations
considering homogeneous and heterogeneous damping in the absorbing layer. 2D
models are considered afterwards to assess the efficiency of the absorbing
layer method for various wave types (surface waves, body waves) and incidences
(normal to grazing). The method is shown to be efficient for different types of
elastic waves and may thus be used for various elastodynamic problems in
unbounded domains
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
A linearised hp-finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners
We propose a new computational framework for the treatment of acousto-magneto-mechanical coupling that arises in low-frequency electro-magneto-mechanical systems such as MRI scanners. Our transient Newton-Raphson strategy involves the solution of a monolithic system obtained from the linearisation of the coupled system of equations. Moreover, this framework, in the case of excitation from static and harmonic current sources, allows us to propose a simple linearised system and rigorously motivate a single-step strategy for understanding the response of systems under different frequencies of excitation. Motivated by the need to solve industrial problems rapidly, we restrict ourselves to solving problems consisting of axisymmetric geometries and current sources. Our treatment also discusses in detail the computational requirements for the solution of these coupled problems on unbounded domains and the accurate discretisation of the fields using hp-finite elements. We include a set of academic and industrially relevant examples to benchmark and illustrate our approach
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