共查询到20条相似文献，搜索用时 15 毫秒
1.
P. Gamallo R. J. Astley 《International journal for numerical methods in engineering》2007,71(4):406-432
Trefftz methods for the numerical solution of partial differential equations (PDEs) on a given domain involve trial functions which are defined in subdomains, are generally discontinuous, and are solutions of the governing PDE (or its adjoint) within each subdomain. The boundary conditions and matching conditions between subdomains must be enforced separately. An interesting novel result presented in this paper is that the least-squares method (LSM) and the ultraweak variational formulation, two methods already established for solving the Helmholtz equation, can be derived in the framework of the Trefftz-type methods. In the first case, the boundary conditions and interelement continuity are enforced by means of a least-squares procedure. In the second, a Galerkin-type weighted residual method is used. Another goal of the work is to assess the relative efficiency of each method for solving shortwave problems in acoustics and to study the stability of each method. The numerical performance of each scheme is assessed with reference to two 2-D test problems; acoustic propagation in an uniform soft-walled duct, and propagation in an L-shaped domain, the latter involving singular behaviour at a sharp corner. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
2.
Tomi Huttunen Jari P. Kaipio Peter Monk 《International journal for numerical methods in engineering》2004,61(7):1072-1092
We investigate the feasibility of using the perfectly matched layer (PML) as an absorbing boundary condition for the ultra weak variational formulation (UWVF) of the 3D Helmholtz equation. The PML is derived using complex stretching of the spatial variables. This leads to a modified Helmholtz equation for which the UWVF can be derived. In the standard discrete UWVF, the approximating subspace is constructed from local solutions of the Helmholtz equation. In previous studies plane wave basis functions have been advocated because they simplify the building of the UWVF matrices. For the PML domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections. The method is validated by numerical experiments. In comparison to a low‐order absorbing boundary condition, the PML shows superior performance. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
3.
G. Gabard P. Gamallo T. Huttunen 《International journal for numerical methods in engineering》2011,85(3):380-402
Several numerical methods using non‐polynomial interpolation have been proposed for wave propagation problems at high frequencies. The common feature of these methods is that in each element, the solution is approximated by a set of local solutions. They can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation. There are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated. The similarities and discrepancies between several non‐polynomial numerical methods are discussed in the context of the Helmholtz equation. The present comparison is concerned with the ultra‐weak variational formulation (UWVF), the least‐squares method (LSM) and the discontinuous Galerkin method with numerical flux (DGM). An analysis in terms of Trefftz methods provides an interesting insight into the properties of these methods. Second, the UWVF and the LSM are reformulated in a similar fashion to that of the DGM. This offers a unified framework to understand the properties of several non‐polynomial methods. Numerical results are also presented to put in perspective the relative accuracy of the methods. The numerical accuracies of the methods are compared with the interpolation errors of the wave bases. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
4.
A. Aimi M. Diligenti C. Guardasoni I. Mazzieri S. Panizzi 《International journal for numerical methods in engineering》2009,80(9):1196-1240
In this paper we consider Dirichlet or Neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential. Starting from a natural energy identity, a space–time weak formulation for 1D integral problems is briefly introduced, and continuity and coerciveness properties of the related bilinear form are proved. Then, a theoretical analysis of an extension of the introduced formulation for 2D problems is proposed, pointing out the novelty with respect to existing literature results. At last, various numerical simulations will be presented and discussed, showing unconditional stability of the space–time Galerkin boundary element method applied to the energetic weak problem. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
5.
Louis Kovalevsky Pierre Gosselet 《International journal for numerical methods in engineering》2016,107(11):903-922
The variational theory of complex rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz‐like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared with classical polynomial approaches, but the resulting system is prone to be ill‐conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework, and it traces back the ill‐conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
6.
Dalei Wang Radek Tezaur Jari Toivanen Charbel Farhat 《International journal for numerical methods in engineering》2012,89(4):403-417
The efficient finite element discretization of the Helmholtz equation becomes challenging in the medium frequency regime because of numerical dispersion, or what is often referred to in the literature as the pollution effect. A number of FEMs with plane wave basis functions have been proposed to alleviate this effect, and improve on the unsatisfactory preasymptotic convergence of the polynomial FEM. These include the partition of unity method, the ultra‐weak variational formulation, and the discontinuous enrichment method. A previous comparative study of the performance of such methods focused on the first two aforementioned methods only. By contrast, this paper provides an overview of all three methods and compares several aspects of their performance for an acoustic scattering benchmark problem in the medium frequency regime. It is found that the discontinuous enrichment method outperforms both the partition of unity method and the ultra‐weak variational formulation by a significant margin. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
7.
Charbel Farhat Radek Tezaur Jari Toivanen 《International journal for numerical methods in engineering》2009,78(13):1513-1531
A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
8.
Leopoldo P. Franca Antonini P. Macedo 《International journal for numerical methods in engineering》1998,43(1):23-32
A two-level finite element method is introduced and its application to the Helmholtz equation is considered. The method retains the desirable features of the Galerkin method enriched with residual-free bubbles, while it is not limited to discretizations using elements with simple geometry. The method can be applied to other equations and to irregular-shaped domains. © 1998 John Wiley & Sons, Ltd. 相似文献
9.
G. Gabard O. Dazel 《International journal for numerical methods in engineering》2015,104(12):1115-1138
Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
10.
Numerical simulation of complex three‐dimensional flow through the spiral casing has been accomplished using a finite element method. An explicit Eulerian velocity correction scheme has been deployed to solve the Reynolds average Navier–Stokes equations. The simulation has been performed to describe the flow in high Reynolds number (106) regime. For spatial discretization, a streamline upwind Petrov Galerkin technique has been used. The velocity field and the pressure distribution inside the spiral casing corroborate the results available in literature. The flow structure reveals the fact that very strong secondary flow is evolved on the cross‐stream planes. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
11.
Daniel Appelö Siyang Wang 《International journal for numerical methods in engineering》2019,119(7):618-638
We consider wave propagation in a coupled fluid-solid region separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic wave equation for the displacement in the solid. At the fluid solid interface, we impose suitable interface conditions to couple the two equations. We use a recently developed energy-based discontinuous Galerkin method to discretize the governing equations in space. Both energy conserving and upwind numerical fluxes are derived to impose the interface conditions. The highlights of the developed scheme include provable energy stability and high order accuracy. We present numerical experiments to illustrate the accuracy property and robustness of the developed scheme. 相似文献
12.
Radek Tezaur Charbel Farhat 《International journal for numerical methods in engineering》2006,66(5):796-815
Recently, a discontinuous Galerkin finite element method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of Helmholtz problems in the mid‐frequency regime. In this paper, this method is extended to three dimensions and several new elements are proposed. Computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher‐order Galerkin method. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
13.
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
14.
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
15.
L. L. THOMPSON P. M. PINSKY 《International journal for numerical methods in engineering》1996,39(10):1635-1657
A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method. 相似文献
16.
H. G. Aksoy E. Şenocak 《International journal for numerical methods in engineering》2011,88(7):673-692
Modeling of discontinuities (shock waves, crack surfaces, etc.) in solid mechanics is one of the major research areas in modeling the mechanical behavior of materials. Among the numerical methods, the discontinuous Galerkin method (DGM) poses some advantages in solving these problems. In this study, a novel formulation for DGM is derived for elastostatics based on the peridynamic theory. Derivation of the proposed formulation is presented. Numerical analyses are performed for different problems, and the numerical results are compared to that of the known exact solutions of the problems. The proposed weak formulation is stable and coercive. Peridynamic discontinuous Galerkin formulation is found to be robust and successful in modeling elastostatic problems. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
17.
Dalei Wang Radek Tezaur Charbel Farhat 《International journal for numerical methods in engineering》2014,99(4):263-289
Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
18.
G. Gabard 《International journal for numerical methods in engineering》2006,66(3):549-569
Several special finite element methods have been proposed to solve Helmholtz problems in the mid‐frequency regime, such as the Partition of Unity Method, the Ultra Weak Variational Formulation and the Discontinuous Enrichment Method. The first main purpose of this paper is to present a discontinuous Galerkin method with plane waves (which is a variant of the Discontinuous Enrichment Method) to solve the displacement‐based acoustic equation. The use of the displacement variable is often necessary in the context of fluid–structure interactions. A well‐known issue with this model is the presence of spurious vortical modes when one uses standard finite elements such as Lagrange elements. This problem, also known as the locking phenomenon, is observed with several other vector based equations such as incompressible elasticity and electromagnetism. So this paper also aims at assessing if the special finite element methods suffer from the locking phenomenon in the context of the displacement acoustic equation. The discontinuous Galerkin method presented in this paper is shown to be very accurate and stable, i.e. no spurious modes are observed. The optimal choice of the various parameters are discussed with regards to numerical accuracy and conditioning. Some interesting properties of the mixed displacement–pressure formulation are also presented. Furthermore, the use of the Partition of Unity Method is also presented, but it is found that spurious vortical modes may appear with this method. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
19.
Konstantinos T. Panourgias John A. Ekaterinaris 《International journal for numerical methods in engineering》2017,112(11):1687-1711
The discontinuous Galerkin FEM is used for the numerical solution of the three‐dimensional Maxwell equations. Control of errors in the numerical level for the divergence‐free constraint of the magnetic field can be obtained through the use of divergence‐free vector bases. In this work, the so‐called perfectly hyperbolic formulation of the Maxwell equations is used to retain both divergence‐free magnetic field and in the presence of charges to satisfy the Gauss constraint for the electric field at the numerical level. For both approaches, it is found that higher‐order approximations have favorable effect on the preservation of the divergence constraints and that the perfectly hyperbolic formulations retains these errors to a lower level. It is shown that high‐order accuracy in space and time is achieved in unstructured meshes using implicit time marching. For nonuniform meshes, local resolution refinement is used using p‐type adaptivity to ensure accurate electromagnetic wave propagation. Thus, the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated. 相似文献
20.