Search results for "Helmholtz equation"
showing 10 items of 44 documents
A Domain Imbedding Method with Distributed Lagrange Multipliers for Acoustic Scattering Problems
2003
The numerical computation of acoustic scattering by bounded twodimensional obstacles is considered. A domain imbedding method with Lagrange multipliers is introduced for the solution of the Helmholtz equation with a second-order absorbing boundary condition. Distributed Lagrange multipliers are used to enforce the Dirichlet boundary condition on the scatterer. The saddle-point problem arising from the conforming finite element discretization is iteratively solved by the GMRES method with a block triangular preconditioner. Numerical experiments are performed with a disc and a semi-open cavity as scatterers.
Full modal analysis of confocal coaxial elliptical waveguides
2000
An efficient method for analysing confocal coaxial elliptical waveguides is presented. Using elliptical coordinates, the differential Helmholtz equation is transformed into a linear matrix eigenvalue problem by means of the method of moments. The expressions of the vector mode functions for the full spectrum of these guides are constructed, including the TEM, TM and TE modes. The convergence of the method is very good, giving an efficient and accurate code. Comparisons with numerical results found in the technical literature validate the presented theory.
Thermodynamic pressure in nonlinear nonequilibrium thermodynamics of dilute nonviscous gases.
2000
In this paper, using extended thermodynamics, we build up a nonlinear theory for a dilute nonviscous gas under heat flux. The fundamental fields are the density, the velocity, the internal energy density, and the heat flux. The constitutive theory is builtup without approximations. We single out the nonlinear complete expressions of the Gibbs equation and of the nonequilibrium pressure. In particular, we determine the complete expressions furnished by the theory for the nonequilibrium pressure tensor and thermodynamic pressure, i.e., the derivative of the nonequilibrium internal specific entropy with respect to the specific volume, times the nonequilibrium temperature. In a second-order app…
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
2009
A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A s…
A damping preconditioner for time-harmonic wave equations in fluid and elastic material
2009
A physical damping is considered as a preconditioning technique for acoustic and elastic wave scattering. The earlier preconditioners for the Helmholtz equation are generalized for elastic materials and three-dimensional domains. An algebraic multigrid method is used in approximating the inverse of damped operators. Several numerical experiments demonstrate the behavior of the method in complicated two-dimensional and three-dimensional domains. peerReviewed
On a topology optimization problem governed by two-dimensional Helmholtz equation
2015
The paper deals with a class of shape/topology optimization problems governed by the Helmholtz equation in 2D. To guarantee the existence of minimizers, the relaxation is necessary. Two numerical methods for solving such problems are proposed and theoretically justified: a direct discretization of the relaxed formulation and a level set parametrization of shapes by means of radial basis functions. Numerical experiments are given.
Controllability method for acoustic scattering with spectral elements
2007
We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improveme…
A fast Fourier transform based direct solver for the Helmholtz problem
2018
This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is O(N log N) operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed…
An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation
2007
A preconditioner defined by an algebraic multigrid cycle for a damped Helmholtz operator is proposed for the Helmholtz equation. This approach is well suited for acoustic scattering problems in complicated computational domains and with varying material properties. The spectral properties of the preconditioned systems and the convergence of the GMRES method are studied with linear, quadratic, and cubic finite element discretizations. Numerical experiments are performed with two-dimensional problems describing acoustic scattering in a cross-section of a car cabin and in a layered medium. Asymptotically the number of iterations grows linearly with respect to the frequency while for lower freq…