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1.
Arnaud Deraemaeker Ivo Babu&#x;ka Philippe Bouillard 《International journal for numerical methods in engineering》1999,46(4):471-499
For high wave numbers, the Helmholtz equation suffers the so‐called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p‐method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual‐free bubble [3,4] is effective in one dimension but not in higher dimensions. The least‐square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss‐cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
2.
Gang Wang Guodong Zeng Xiangyang Cui Shizhe Feng 《International journal for numerical methods in engineering》2019,120(4):473-497
This paper reports a detailed analysis on the numerical dispersion error in solving one-, two-, and three-dimensional acoustic problems governed by the Helmholtz equation using the gradient weighted finite element method (GW-FEM) in comparison with the standard FEM and the modified methods presented in the literatures. The discretized system equations derived based on the gradient weighted operation corresponding to the considered method are first briefed. The discrete dispersion relationships relating the exact and numerical wave numbers defined in different dimensions are then formulated, which will be further used to investigate the dispersion effect mainly caused by the approximation of field variables. The influence of nondimensional wave number and wave propagation angle on the dispersion error is detailedly studied. Comparisons are made with the classical FEM and high-performance algorithms. Results of both theoretical and numerical experiments show that the present method can effectively reduce the pollution effect in computational acoustics owning to its crucial effectiveness in handing the dispersion error in the discrete numerical model. 相似文献
3.
Assad A. Oberai Peter M. Pinsky 《International journal for numerical methods in engineering》2000,49(3):399-419
A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
4.
Eran Grosu Isaac Harari 《International journal for numerical methods in engineering》2009,78(11):1261-1291
5.
V. Lacroix Ph. Bouillard P. Villon 《International journal for numerical methods in engineering》2003,57(15):2131-2146
Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly for medium frequencies. We have thus formulated a new numerical method better suited to the acoustical problem: the element‐free Galerkin method (EFGM) improved by appropriate basis functions computed by a defect correction approach. One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct the basis of the approximation with terms that are suited to the problem which has to be solved. Acoustical problems, in cavities Ω with boundary T, are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x,y) is a complex variable, it can always be expressed as a function of cosθ(x,y) and sinθ(x,y) where θ(x,y) is the phase of the wave in each point (x,y). If the exact distribution θ(x,y) of the phase is known and if a meshless basis {1, cosθ(x,y), sinθ (x,y) } is used, then the exact solution of the acoustic problem can be obtained. Obviously, in real‐life cases, the distribution of the phase is unknown. The aim of our work is to resolve, as a first step, the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p(x,y). As a second step, from p(x,y) we compute the distribution of the phase θ(x,y) and we introduce it in the meshless basis in order to compute a second approximated pressure field p(x,y). From p(x,y), a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion, concerning the pressure or the phase, is obtained. So, an iterative defect‐correction type meshless method has been developed to compute the pressure field in Ω. This work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time. We will also compare the performance of this method with the classical finite element method. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
6.
S. Suleau Ph. Bouillard 《International journal for numerical methods in engineering》2000,47(6):1169-1188
The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined meshes are used, leading to unacceptable resolution times. The paper presents an application of the element‐free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element‐free Galerkin method solution is much more accurate than the finite element one. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
7.
This paper demonstrates the application of a meshfree least square-based finite difference (LSFD) method for analysis of metallic waveguides. The waveguide problem is an eigenvalue problem that is governed by the Helmholtz equation. The second order derivatives in the Helmholtz equation are explicitly approximated by the LSFD formulations. TM modes and TE modes are calculated for some metallic waveguides with different cross-sectional shapes. Numerical examples show that the LSFD method is a very efficient meshfree method for waveguide analysis with complex domains. 相似文献
8.
G. Gabard R. J. Astley M. Ben Tahar 《International journal for numerical methods in engineering》2005,63(7):947-973
The dispersion properties of finite element models for aeroacoustic propagation based on the convected scalar Helmholtz equation and on the Galbrun equation are examined. The current study focusses on the effect of the mean flow on the dispersion and amplitude errors present in the discrete numerical solutions. A general two‐dimensional dispersion analysis is presented for the discrete problem on a regular unbounded mesh, and results are presented for the particular case of one‐dimensional acoustic propagation in which the wave direction is aligned with the mean flow. The magnitude and sign of the mean flow is shown to have a significant effect on the accuracy of the numerical schemes. Quadratic Helmholtz elements in particular are shown to be much less effective for downstream—as opposed to upstream—propagation, even when the effect of wave shortening or elongation due to the mean flow is taken into account. These trends are also observed in solutions obtained for simple test problems on finite meshes. A similar analysis of two‐dimensional propagation is presented in an accompanying article. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
9.
G. Gabard R. J. Astley M. Ben Tahar 《International journal for numerical methods in engineering》2005,63(7):974-987
This is the second of two articles that focus on the dispersion properties of finite element models for acoustic propagation on mean flows. We consider finite element methods based on linear potential theory in which the acoustic disturbance is modelled by the convected Helmholtz equation, and also those based on a mixed Galbrun formulation in which acoustic pressure and Lagrangian displacement are used as discrete variables. The current paper focuses on the effects of numerical anisotropy which are associated with the orientation of the propagating wave to the mean flow and to the grid axes. Conditions which produce aliasing error in the Helmholtz formulation are of particular interest. The 9‐noded Lagrangian element is shown to be superior to the more commonly used 8‐noded serendipity element. In the case of the Galbrun elements, the current analysis indicates that isotropic meshes generally reduce numerical error of triangular elements and that higher order mixed quadrilaterals are generally less effective than an equivalent mesh of lower order triangles. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
10.
Joan Baiges Ramon Codina 《International journal for numerical methods in engineering》2013,93(6):664-684
In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
11.
Lindaura Maria Steffens Núria Parés Pedro Díez 《International journal for numerical methods in engineering》2011,86(10):1197-1224
An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in Steffens and D'?ez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400). In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post‐processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in Steffens and Díez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400) is based in a polynomial least‐squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
12.
D. Dreyer O. von Estorff 《International journal for numerical methods in engineering》2003,58(6):933-953
An optimized version of the so‐called mapped wave envelope elements, also known as Astley–Leis elements, is introduced. These elements extend to infinity in one dimension and therefore provide an approach to the simulation of exterior acoustical problems in both frequency and time domains. Their formulation is improved significantly through the proper choice of polynomial bases in the direction of radiation. In particular, certain Jacobi polynomials are identified which behave well with respect to conditioning of the system matrices. As a consequence, the size of the finite element discretization may reduce considerably without any loss of accuracy. In addition, the new polynomial bases lead to superior performance of the infinite elements in conjunction with iterative solvers. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
13.
Alessio Alexiadis 《International journal for numerical methods in engineering》2014,100(10):713-719
In this work, a hybrid numerical approach, which combines elements of SPH and coarse‐grained molecular dynamics, is used to investigate the effect of various flow conditions to deformable and breakable shell‐structures such as capsules, vesicles or biological cells. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
14.
Prashanth Nadukandi Eugenio Oñate Julio Garcia 《International journal for numerical methods in engineering》2011,86(1):18-46
We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f( x )+Δφ+ξ2φ=0. The scheme consists of taking the alpha‐interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha‐interpolation method (J. Comput. Appl. Math. 1982; 8 (1):15–19) and in 2D making the choice α=0.5 we recover the generalized fourth‐order compact Padé approximation (J. Comput. Phys. 1995; 119 :252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163 :343–358) (therein using the parameter γ=2). We follow (SIAM Rev. 2000; 42 (3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128 :325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi‐stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128 :325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((ξ?)4), where ξ, ? represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L2 norm, the H1 semi‐norm and the l∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
15.
X. Antoine 《International journal for numerical methods in engineering》2002,54(7):1021-1041
This paper describes a new formulation to solve the acoustic scattering problem by a non‐convex structure. The presented hybrid algorithms connect the on‐surface radiation condition method to accelerate the calculations with a two‐dimensional finite element method to treat the non‐convex part of the domain. This allows the problem of truncating the computational domain to be overcome. The variational formulations associated with a Dirichlet or a Neumann boundary condition are considered. Finally, we give some numerical results on a model test case to illustrate the improvement of accuracy of the new algorithm compared to a pure OSRC approach. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
16.
Vincent Darrigrand Ángel Rodríguez‐Rozas Ignacio Muga David Pardo Albert Romkes Serge Prudhomme 《International journal for numerical methods in engineering》2018,113(1):22-42
In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method. 相似文献
17.
Guilhem Mollon 《International journal for numerical methods in engineering》2016,108(12):1477-1497
In this paper, a multibody meshfree framework is proposed for the simulation of granular materials undergoing deformations at the grain scale. This framework is based on an implicit solving of the mechanical problem based on a weak form written on the domain defined by all the grains composing the granular sample. Several technical choices, related to the displacement field interpolation, to the contact modelling, and to the integration scheme used to solve the dynamic equations, are explained in details. A first implementation is proposed, under the acronym Multibody ELement‐free Open code for DYnamic simulation (MELODY), and is made available for free download. Two numerical examples are provided to show the convergence and the capability of the method. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
18.
A. Quaglino R. Krause 《International journal for numerical methods in engineering》2018,114(6):581-597
We propose a novel finite‐element method for polygonal meshes. The resulting scheme is hp‐adaptive, where h and p are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis‐aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices. 相似文献
19.
A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=4Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results. 相似文献
20.
T. Rabczuk J. Eibl 《International journal for numerical methods in engineering》2003,56(10):1421-1444
The simulation of concrete fragmentation under explosive loading by a meshfree Lagrangian method, the smooth particle hydrodynamics method (SPH) is described. Two improvements regarding the completeness of the SPH‐method are examined, first a normalization developed by Johnson and Beissel (NSPH) and second a moving least square (MLS) approach as modified by Scheffer (MLSPH). The SPH‐Code is implemented in FORTRAN 90 and parallelized with MPI. A macroscopic constitutive law with isotropic damage for fracture and fragmentation for concrete is implemented in the SPH‐Code. It is shown that the SPH‐method is able to simulate the fracture and fragmentation of concrete slabs under contact detonation. The numerical results from the different SPH‐methods are compared with the data from tests. The good agreement between calculation and experiment suggests that the SPH‐program can predict the correct maximum pressure as well as the damage of the concrete slabs. Finally the fragment distributions of the tests and the numerical calculations are compared. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献