Frequency domain Bernstein-Bézier finite element solver for modelling short waves in elastodynamics
Introduction
Computer modelling of elastic wave propagation and scattering is an effective tool for predicting and testing in a wide variety of practical applications, including traffic-induced vibrations from roads and railways, seismic induced vibrations, seismic inversion, design of foundation elements, geophysical exploration, nondestructive evaluation and structural engineering.
The Finite Element Method (FEM) is a popular discretisation method commonly adopted for the numerical solution of wave problems. It is favoured over other methods, owing to its ability in accurately handling complex geometries, material heterogeneity and anisotropy. When typically piecewise linear FEs are used, around ten grid points per wavelength are needed to ensure a satisfactory resolution of the wave pattern. It is noteworthy that this rule of thumbnail of ten nodal points per wavelength enables to control the local approximation error [1]. However, due to the pollution effect in the mid or high frequency regimes [1], [2], [3], [4], even more than ten nodal points are required to achieve an acceptable level of accuracy, and thus the procedure becomes prohibitively expensive and less effective.
The past few decades have seen many attempts to design robust numerical schemes alleviating the pollution phenomenon. A substantial improvement was made in developing wave based, or Trefftz, methods enabling to achieve engineering accuracy with a significant reduction in the computational effort. The common idea in these approaches is to incorporate the oscillatory behaviour of the solution into the approximate space by including plane waves or Bessel functions. Examples of such methods, among others, are the Partition of Unity Finite Element Method (PUFEM) [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], the Generalized Finite Element Method (GFEM) [15], [16], the Ultra Weak Variational Formulation (UWVF) [17], [18], [19], the least-squares method [20], the Discontinuous Enrichment Method (DEM) [21], [22] and the wave-based discontinuous Galerkin method [23], [24], [25]. It is worth noticing that all the aforementioned methods (except PUFEM, GFEM and DEM) share the same discontinuous Galerkin framework and differ only in the used numerical fluxes to ensure continuity between elements [26].
Wave based methods generally make use of the plane wave basis decomposition by pressure and shear waves in the approximation of time harmonic elastic wave problems. Perrey-Debain et al. [27] proposed the partition of unity boundary element method. Huttunen et al. [28], [29] developed the UWVF in two and three space dimensions. Also, Zhang et al. [30] extended the DEM to frequency domain elastic wave computations. Note that inter-element continuity in the UWVF and DEM is, respectively, enforced weakly by means of numerical fluxes and Lagrange multipliers. In References [31], [32], [33], [34], [35] and [36], PUFEM was successfully applied to time harmonic elastic wave problems, in two and three space dimensions. More recently, Yuan and Liu [37] proposed another approach in the discontinuous Galerkin setting based on plane wave basis and local spectral elements.
Higher-order polynomials such as those used in the -version of FEM and spectral elements are less vulnerable to the pollution effect [3], [4], [38], [39]. In the context of Helmholtz problems, Petersen et al. [40] assessed the efficiency of various -FEM shape functions including Lagrange Gauss-Lobatto, integrated Legendre and Bernstein polynomials. They pointed out the advantage of high-order polynomials in reducing the pollution error and the good performance of Bernstein-Bézier FEs in conjunction with Krylov subspace solvers. Further, Lieu et al. [41] compared the performance of -FEM, with Lobatto polynomials, and the wave-based discontinuous Galerkin method on various benchmarks. Similarly, El Kacimi et al. [42] compared the performance of the Bernstein-Bézier Finite Element method (BBFEM) and PUFEM. It was concluded from these comparative studies that high-order polynomial methods in combination with static condensation are able to deliver comparable or, in some cases, even superior performance. On the other hand, high order -discontinuous Galerkin methods have been demonstrated as competitive discretisation schemes (see, e.g., [43], [44] and the references cited therein), due to their flexibility in dealing with -adaptivity and general shaped elements, and their ability in achieving the exponential convergence rate of spectral techniques.
Another alternative to cope with the pollution error is the use of high-order isogeometric elements. The Isogeometric Analysis (IGA) [45] has gained popularity in recent years, due to a number of advantages afforded by spline basis functions, such as exact representation of geometries, higher order continuity and improved convergence properties. This methodology was successfully applied in a number of applications, as e.g. acoustic [46], [47], [48], [49], elastodynamic [50], [51], [52], [53] and electromagnetic [54], [55], [56] waves. Peake et al. [57], [58] extended the IGA concept within the framework of boundary elements for exterior acoustic problems, using the partition of unity in two and three space dimensions. They reported superior convergence compared to the partition of unity boundary element method. More recently, Ayala et al. [59] studied the performance of the enriched isogeometric collocation method in time-harmonic acoustics. They found improved convergence compared to the isogeometric collocation method. Willberg et al. [50] assessed the performance of isogeometric elements with non-uniform rational B-splines (NURBS) against high order SEM and -FEM in Lamb wave propagation analysis. They concluded that higher order schemes deliver much improved accuracy, with a significant reduction in the total number of degrees of freedom (DoF), compared to conventional FEMs. Moreover, they pointed out that isogeometric elements enable to achieve high convergence rates.
In contrast to standard low order FEMs, where most of the CPU time is spent in the solution process, the computational burden of high order methods shifts to the evaluation of the element matrices and assembling (see, e.g., [60]). Therefore, the design of efficient quadrature rules is crucial to alleviate this major drawback. On tensor product elements, sum factorisation dating back to Orszag [61], has proven fundamental in the efficient implementation of SEMs. This technique relies mainly on a tensor-product structure of the shape functions and enables to achieve near optimal complexity. For simplicial elements, Karniadakis and Sherwin [62] designed special bases preserving a tensor product property. Recently, Ainsworth et al. [63] have shown that the Bernstein-Bézier basis naturally possesses the tensorial property needed for the sum factorisation method. They proposed an algorithm to set up the element mass and stiffness matrices with optimal complexity. Kirby and Thinh [64] developed low complexity matrix-free algorithm with Bernstein polynomials for applying the local finite element operators.
The main focus of this paper is to extend BBFEM to accurately solve time-harmonic elastic wave problems. As one of the key ingredients of the method, particular attention is paid to the computation of the element mass and stiffness matrices. Here, analytical rules are used for affine triangular elements and the sum factorisation method together with Stroud quadrature [76] is applied for elements with curved edges, where the geometry of such elements is interpolated via the linear blending map of Gordon and Hall [65], [66], [67]. The applicability of BBFEM with a variable polynomial order, using a simple a priori indicator, is also investigated throughout a benchmark dealing with the transmission of elastic waves.
The remainder of this paper is organized as follows. Section 2 introduces the model problem and its weak form. The Bernstein-Bézier FE approximation of the governing equations is presented in Section 3. Section 4 gives a description of the analytical and quadrature rules used for evaluating element integrals. Section 5 is devoted to numerical results. Finally, some concluding remarks are made in Section 6.
Notation
The following notation will be used throughout this paper. Each point in is identified by its components relative to the Cartesian vector system denoted , i.e. . The dot product of two vectors and in is a scalar given by . The double dot product of two second-order tensors and in is a scalar given by . The double dot product of a fourth-order tensor and a second-order tensor is a second order tensor given by . We denote the scalar product either in or by , that is, and , where the notation ’’ refers to the complex conjugate. The induced norms in or will be denoted by .
We also denote the usual inner product on the complex-valued Sobolev space , where , by , with . We keep for simplicity the same notation for the inner products on the space of vector and tensor valued functions and , that is,Likewise, for a given , the inner products on and are denoted by . We further introduce the induced norm in and the semi-norm in , where , is the gradient operator, and the superscript ’’ denotes the transpose.
Standard multi-index notation will be used. For and , we set , , and . If such that , i.e., , , . We denote by the multi-index whose the th entry is unity and remaining entries are zero.
Section snippets
Mathematical model
Let be a bounded Lipschitz domain in , consisting of a solid, homogeneous, isotropic and linear elastic material. We denote by its boundary, and the outward unit normal and tangent vectors to . In the absence of a source, the time harmonic Navier equation reads as [68]Herein, is the angular frequency, is the material density, is the displacement field and is the divergence operator. The Cauchy stress tensor is linearly related to the infinitesimal
Bernstein-Bézier FE approximation
Let be a conforming partition of the domain into triangular elements such that , where is an approximation of and is the mesh size of given by , with . Each is the image of the triangular master element defined byi.e. , where is a suitable reference map. For , let be the finite dimensional approximation space defined bywhere is the space of
Computation of element integrals
This section is devoted to the computation strategies of the element matrices and right hand side vectors of the stated problem. For ease of presentation, we suppose the polynomial order to be uniform.
Numerical results
In this section, numerical results are presented to assess the performance of the proposed approach on various two dimensional benchmark problems. The BBFEM is efficiently implemented via static condensation such that the internal DoF are eliminated at the element level. This leads to a condensed linear system involving only DoF attached the mesh skeleton. Once the solution related to the vertex and edge modes is computed, the internal DoF can be recovered at the post-processing stage by
Conclusions
In this paper, BBFEM has been extended to efficiently solve time-harmonic elastic wave problems, using unstructured triangular mesh grids. Key aspects of the method rely on the use of analytical rules to set up the local FE matrices for affine elements, while the tensorial construction of Bernstein polynomials amenable to sum factorisation is exploited for curved elements to speed up the assembly process. These yield a notable saving in terms of computational cost and hence afford high order
References (81)
- et al.
Finite-element solution of the Helmholtz-equation with high wavenumber. Part I: the -version of the FEM
Computers and Mathematics with Applications.
(1995) - et al.
The partition of unity finite element method: basic theory and applications
Computer Methods in Applied Mechanics and Engineering.
(1996) - et al.
Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed
Computer Methods in Applied Mechanics and Engineering.
(2005) - et al.
Locally enriched finite elements for the Helmholtz equation in two dimensions
Computers & structures.
(2010) - et al.
The generalized finite element method for Helmholtz equation: theory, computation, and open problems
Computer Methods in Applied Mechanics and Engineering.
(2006) - et al.
The generalized finite element method for Helmholtz equation. part II: effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment
Computer Methods in Applied Mechanics and Engineering.
(2008) - et al.
Computation aspects of the ultra weak variational formulation
Journal of Computational Physics.
(2002) - et al.
A least-squares method for the helmholtz equation
Computer Methods in Applied Mechanics and Engineering.
(1999) - et al.
The discontinuous enrichment method
Computer Methods in Applied Mechanics and Engineering.
(2001) - et al.
A discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid-frequency regime
Computer Methods in Applied Mechanics and Engineering.
(2003)
Discontinuous galerkin methods with plane waves for time-harmonic problems
Journal of Computational Physics.
Numerical analysis of two plane wave finite element schemes based on the partition of unity method for elastic wave scattering
Computers and Structures.
Wavelet based ILU preconditioners for the numerical solution by PUFEM of high frequency elastic wave scattering
Journal of Computational Physics.
High-order finite elements for the solution of helmholtz problems
Computers & Structures.
Assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics
Computer Methods in Applied Mechanics and Engineering.
A comparison of high-order polynomial and wave-based methods for helmholtz problems
Journal of Computational Physics.
Bernstein-Bézier based finite elements for efficient solution of short wave problems
Computer Methods in Applied Mechanics and Engineering.
High-order discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes
Comput Methods Appl Mech Eng
Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement
Computer Methods in Applied Mechanics and Engineering.
Isogeometric finite element analysis of interior acoustic problems
Applied Acoustics.
A performance study of NURBS-based isogeometric analysis for interior two-dimensional time-harmonic acoustics
Computer Methods in Applied Mechanics and Engineering.
Pollution studies for high order isogeometric analysis and finite element for acoustic problems
Computer Methods in Applied Mechanics and Engineering.
Comparison of different higher order finite element schemes for the simulation of lamb waves
Computer Methods in Applied Mechanics and Engineering.
Accurate solutions of wave propagation problems under impact loading by the standard, spectral and isogeometric high-order finite elements. comparative study of accuracy of different space-discretization techniques
Finite Elements in Analysis and Design.
Isogeometric numerical dispersion analysis for two-dimensional elastic wave propagation
Comput Methods Appl Mech Eng
Extended isogeometric boundary element method (XIBEM) for two-dimensional helmholtz problems
Computer Methods in Applied Mechanics and Engineering.
Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems
Computer Methods in Applied Mechanics and Engineering.
Enriched isogeometric collocation for two-dimensionaltime-harmonic acoustics
Computer Methods in Applied Mechanics and Engineering.
A spectral finite element approach to modeling soft solids excited with high-frequency harmonic loads
Computer methods in applied mechanics and engineering.
Spectral methods for problems in complex geometries
Journal of Computational Physics.
Enhanced conformal perfectly matched layers for bernstein-bézier finite element modelling of short wave scattering
Computer Methods in Applied Mechanics and Engineering.
Multifrontal parallel distributed symmetric and unsymmetric solvers
Computer Methods in Applied Mechanics and Engineering.
Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation
Int. J. Numer. Methods Eng
Finite element solution of the Helmholtz equation with high wavenumber. part II: the -version of the FEM
SIAM Journal on Numerical Analysis.
Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?
SIAM Review.
The partition of unity method
International Journal for Numerical Methods in Engineering.
Short wave modelling using special finite elements
Journal of Computational Acoustics.
An improved partition of unity finite element model for diffraction problems
International Journal for Numerical Methods in Engineering.
Modelling of short wave diffraction problems using approximating systems of plane waves
International Journal for Numerical Methods in Engineering.
Plane wave basis for wave scattering in three dimensions
Communications in Numerical Methods in Engineering.
Cited by (3)
Bernstein–Bézier H(curl) -Conforming Finite Elements for Time-Harmonic Electromagnetic Scattering Problems
2023, Journal of Scientific ComputingAiry stress function for proposed thermoelastic triangular elements
2023, Journal of Engineering MathematicsUsing Chimera Grids to Describe Boundaries of Complex Shape
2022, Smart Innovation, Systems and Technologies