1,720,983 research outputs found
The partition of unity finite element method for short wave acoustic propagation on non-uniform potential flows
No full textA novel numerical method is proposed for modelling time-harmonic acoustic propagation of short wavelength disturbances on non-uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non-uniform. Many wavelengths can be included within a single element leading to ultra-sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid-based schemes. Numerical results for lined uniform two-dimensional ducts and for non-uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the pollution effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated
A comparison of two Trefftz-type methods: the ultraweak variational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems
No full textTrefftz 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
The partition of unity finite element method for short wave acoustic propagation on non-uniform potential flows
No full textA comparison of two wave element methods for the Helmholtz problem
No full textIn comparison with low-order finite element methods (FEMs), the use of oscillatory basis functions has been shown to reduce the computational complexity associated with the numerical approximation of Helmholtz problems at high wave numbers. We compare two different wave element methods for the 2D Helmholtz problems. The methods chosen for this study are the partition of unity FEM (PUFEM) and the ultra-weak variational formulation (UWVF). In both methods, the local approximation of wave field is computed using a set of plane waves for constructing the basis functions. However, the methods are based on different variational formulations; the PUFEM basis also includes a polynomial component, whereas the UWVF basis consists purely of plane waves. As model problems we investigate propagating and evanescent wave modes in a duct with rigid walls and singular eigenmodes in an L-shaped domain. Results show a good performance of both methods for the modes in the duct, but only a satisfactory accuracy was obtained in the case of the singular field. On the other hand, both the methods can suffer from the ill-conditioning of the resulting matrix system
Finite element method and local active control of enclosed sound fields (Paper 80931)
No full textIn this paper we analyse, from a mathematical point of wiew an active noise control problem for a dissipative acoustic cavity. This problem consists of determining the amplitudes, phases and positions of several monopole sources (speakers) to minimize de noise level at some points due to a primary harmonic source, for example, the vibration of an elastic structure in contact with the fluid. The model of the system is the Helmholtz equation and the Active Control problem is set in the framework of mathematical optimal control theory of distributed systems. We propose a finite element method to obtain approximated solutions
Approximation of a structural acoustic vibration problem by hexahedral finite elements
No full textA finite-element method to compute elastoacoustic vibration modes in 3D problems on hexahedral meshes is analysed. It is based on displacement formulations for solid and fluid domains. In order to avoid spurious modes, the discretization consists of lowest order hexahedral Raviart–Thomas elements for the former coupled with classical trilinear isoparametric hexahedral elements for the latter. The kinematic constraint is weakly imposed and the meshes on the fluid and solid domains do not need to match on the common interface. Basic interpolation results are proved for the lowest order hexahedral Raviart–Thomas elements. These results are used to prove convergence of the coupled finite-element method, non-existence of spurious modes and optimal-order error estimates for eigenfunctions and eigenvalues, under the assumption that the meshes on the fluid domain are asymptotically parallelepiped. Numerical results showing sufficiency and necessity of this hypothesis are reported
- …
