Modified Helmholtz Equation Research Articles

Methods of fundamental solutions for time‐dependent heat conduction problems

AbstractA time‐dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified Helmholtz equation or the fundamental solution to the heat equation. This paper presents solutions using both formulations in terms of initial and boundary conditions. Such formulations enable calculation of errors and variance, which indicates sensitivities of solutions to uncertainties in initial and boundary conditions. Both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods. Copyright © 2005 John Wiley & Sons, Ltd.

An adaptive fast solver for the modified Helmholtz equation in two dimensions

In this paper, we present a fast multipole-accelerated integral equation method for solving the modified Helmholtz equation Δ u ( x → ) - β 2 u ( x → ) = f ( x → ) in two dimensions. The method is direct, and unlike classical FFT based fast solvers, it allows for adaptive mesh refinement but with comparable amount of work per grid point. When the computational domain is rectangular, Dirichlet, Neumann, periodic, and free-space boundary conditions can be imposed analytically without the need to solve a system of linear equations. Several important features of the algorithm are discussed, including the use of precomputed tables, diagonal translation operators, and lattice sums to impose periodic boundary conditions. Numerical experiments show that, for a wide range of the parameter β, the algorithm is stable and high-order accurate.

A variational meshfree method for solving time-discrete diffusion equations

A meshfree method is developed for solving time-discrete diffusion equations that arise in models in brain research. Important criteria for a suitable method are flexibility with respect to domain geometry and the ability to work with very small moving sources requiring easy refinement possibilities. One part of the work concerns a meshfree discretization of the modified Helmholtz equation based on the related minimization problem and a local least-squares function approximation. In a second part, a node choosing algorithm is presented that moves around randomly distributed nodes for optimizing the node distribution and varying the node density as needed. The method is illustrated by two numerical tests.

Potential–density pairs for axisymmetric galaxies: the influence of scalar fields

We present a formulation for potential-density pairs to describe axisymmetric galaxies in the Newtonian limit of scalar-tensor theories of gravity. The scalar field is described by a modified Helmholtz equation with a source that is coupled to the standard Poisson equation of Newtonian gravity. The net gravitational force is given by two contributions: the standard Newtonian potential plus a term stemming from massive scalar fields. General solutions have been found for axisymmetric systems and the multipole expansion of the Yukawa potential is given. In particular, we have computed potential-density pairs of galactic disks for an exponential profile and their rotation curves.

The modified Helmholtz equation in a semi-strip

We study the modified Helmholtz equation in a semi-strip with Poincaré type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a RH problem cannot be solved in closed form, the Carleman–Vekua method for regularising it is illustrated by analysing in detail a particular case.

Reconstruction of vibroacoustic fields in half-space by using hybrid near-field acoustical holography

In this paper we examine the accuracy and efficiency of reconstructing the vibroacoustic quantities generated by a vibrating structure in half-space by using hybrid near-field acoustic holography (NAH) and modified Helmholtz equation least squares (HELS) formulations. In hybrid NAH, we combine modified HELS with an inverse boundary element method (IBEM) to reconstruct a vibroacoustic field. The main advantage of this approach is that the majority of the input data can be regenerated but not measured, thus the efficiency is greatly enhanced. In modified HELS, we expand the field acoustic pressure in terms of outgoing and incoming spherical waves and specify the corresponding expansion coefficients by solving a system of equations obtained by matching the assumed-form solution to the measured acoustic pressure. Here the system of equations is ill conditioned and Tikhonov regularization is implemented through singular value decomposition (SVD) and the generalized cross-validation (GCV) method. Numerical examples of a dilating and oscillating spheres and finite cylinder are demonstrated. Test results show that hybrid NAH can yield a more accurate reconstruction than does a modified HELS, but a modified HELS is more efficient than is hybrid NAH [Work supported by NSF].

A Hierarchical 3-D Direct Helmholtz Solver by Domain Decomposition and Modified Fourier Method

The paper contains a noniterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the nonhomogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci. Comput., 20 (1999), pp. 2237--2260). This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right-hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. The paper describes in detail the matching algorithm for two boxes which is a basis for the domain decomposition scheme. The hierarchical approach is convenient for parallelization and can minimize the global communication. The algorithm requires O(N3, log,N) operations, where N is the number of grid points in each direction.

Two‐dimensional steady unsaturated flow through embedded elliptical layers

New analytic element solutions are presented for unsaturated, two‐dimensional steady flow in vertical planes that include nonoverlapping impermeable elliptical layers and elliptical inhomogeneities. The hydraulic conductivity, which is represented by an exponential function of the pressure head, differs between the inside and outside of an elliptical inhomogeneity; both the saturated hydraulic conductivity and water retention parameters are allowed to differ between the inside and outside. The Richards equation is transformed, through the Kirchhoff transformation and a second standard transformation, into the modified Helmholtz equation. Analytic element solutions are obtained through separation of variables in elliptical coordinates. The resulting equations for the Kirchhoff potential consist of infinite sums of products of exponentials and modified Mathieu functions. In practical applications the series are truncated but still fulfill the differential equation exactly; boundary conditions are met approximately but up to machine accuracy, provided that enough terms are used. The pressure head, saturation, and flow may be computed analytically at any point in the vadose zone. Examples are given of the shadowing effect of an impermeable elliptical layer in a uniform flow field and funnel‐type flow between two elliptical inhomogeneities. The presented solutions may be applied to study transport processes in vadose zones containing many impermeable elliptical layers or elliptical inhomogeneities.

Reconstruction of vibro-acoustic fields using hybrid nearfield acoustic holography

Hybrid nearfield acoustical holography is utilized to reconstruct the vibro-acoustic fields generated by an arbitrary structure vibrating in an unbounded fluid medium. This hybrid NAH is based on the modified Helmholtz equation least-squares method that expresses the field acoustic pressure in terms of expansions of outgoing and incoming waves. The expansion coefficients are determined by solving an over-determined linear system of equations obtained by matching the assumed-form solution to a finite number of acoustic pressures measured in the field via least squares. The measurements are taken over a conformal surface around the source at close range so that the evanescent waves can be captured. Once the expansion coefficients are specified, enough field acoustic pressures are regenerated on a conformal surface around the source and are taken as input to the Helmholtz integral formulation implemented numerically by the boundary element method. The acoustic pressure and normal velocity on the source surface are reconstructed using singular value decomposition, modified Tikhonov regularization and generalized cross-validation method. This hybrid NAH enables one to obtain a satisfactory reconstruction cost effectively because the majority of the input data are calculated but not measured. Numerical examples of reconstructing the vibro-acoustic quantities on the surfaces of an engine block and a highly elongated cylinder are demonstrated.

The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation

AbstractWe 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.

Analytic Element Modeling of Cylindrical Drains and Cylindrical Inhomogeneities in Steady Two‐Dimensional Unsaturated Flow

The analytic element method, first developed for modeling flow in saturated porous media, is adapted to the solution of the governing equation for steady flow in unsaturated porous media. The governing equation for steady‐state unsaturated flow is made amenable to analytic element solution through transformation with the Kirchhoff integral, representation of the hydraulic conductivity by an exponential function of the pressure head, and use of a coordinate transformation. A number of analytic elements are available for the resulting modified Helmholtz equation; analytic element equations are presented for uniform flow, cylindrical drains, and cylindrical inhomogeneities. The analytic element solution allows for the analytic evaluation of the pressure head, saturation, and Darcy flux at any point in the vadose zone. The applicability of the analytic element method to simulate unsaturated flow is demonstrated by solving for several cases of steady flow in a region containing arbitrarily located cylindrical inhomogeneities, and for flow in a region containing one cylindrical inhomogeneity and two cylindrical drains.

Analytic Element Modeling of Cylindrical Drains and Cylindrical Inhomogeneities in Steady Two-Dimensional Unsaturated Flow

The analytic element method, first developed for modeling flow in saturated porous media, is adapted to the solution of the governing equation for steady flow in unsaturated porous media. The governing equation for steady-state unsaturated flow is made amenable to analytic element solution through transformation with the Kirchhoff integral, representation of the hydraulic conductivity by an exponential function of the pressure head, and use of a coordinate transformation. A number of analytic elements are available for the resulting modified Helmholtz equation; analytic element equations are presented for uniform flow, cylindrical drains, and cylindrical inhomogeneities. The analytic element solution allows for the analytic evaluation of the pressure head, saturation, and Darcy flux at any point in the vadose zone. The applicability of the analytic element method to simulate unsaturated flow is demonstrated by solving for several cases of steady flow in a region containing arbitrarily located cylindrical inhomogeneities, and for flow in a region containing one cylindrical inhomogeneity and two cylindrical drains.

A mesh free approach using radial basis functions and parallel domain decomposition for solving three‐dimensional diffusion equations

AbstractThe analysis of transient heat conduction problems in large, complex computational domains is a problem of interest in many technological applications including electronic cooling, encapsulation using functionally graded composite materials, and cryogenics. In many of these applications, the domains may be multiply connected and contain moving boundaries making it desirable to consider meshless methods of analysis. The method of fundamental solutions along with a parallel domain decomposition method is developed for the solution of three‐dimensional parabolic differential equations. In the current approach, time is discretized using the generalized trapezoidal rule transforming the original parabolic partial differential equation into a sequence of non‐homogeneous modified Helmholtz equations. An approximate particular solution is derived using polyharmonic splines. Interfacial conditions between subdomains are satisfied using a Schwarz Neumann–Neumann iteration scheme. Outside of the first time step where zero initial flux is assumed, the initial estimates for the interfacial flux is given from the converged solution obtained during the previous time step. This significantly reduces the number of iterations required to meet the convergence criterion. The accuracy of the method of fundamental solutions approach is demonstrated through two benchmark problems. The parallel efficiency of the domain decomposition method is evaluated by considering cases with 8, 27, and 64 subdomains. Copyright 2004 © John Wiley & Sons, Ltd.

Hybrid near-field acoustic holography.

Hybrid near-field acoustical holography (NAH) is developed for reconstructing acoustic radiation from an arbitrary object in a cost-effective manner. This hybrid NAH is derived from a modified Helmholtz equation least squares (HELS) formula that expands the acoustic pressure in terms of outgoing and incoming waves. The expansion coefficients are determined by solving an overdetermined linear system of equations obtained by matching the assumed-form solution to measured acoustic pressures through the least squares. Measurements are taken over a conformal surface around a source at close range so that the evanescent waves can be captured. Next, the modified HELS is utilized to regenerate as much acoustic pressures on the conformal surface as necessary and take them as input to the Helmholtz integral formulation implemented numerically by boundary element method (BEM). The acoustic pressures and normal velocities on the source surface are reconstructed by using a modified Tikhnov regularization (TR) with its regularization parameter determined by generalized cross validation (GCV) method. Results demonstrate that this hybrid NAH combines the advantages of HELS and inverse BEM. This is because a majority of the input data are regenerated but not measured, thus the efficiency of reconstruction is greatly enhanced. Meanwhile, the accuracy of reconstruction is ensured by the Helmholtz integral theory and modified TR together with GCV method, provided that HELS converges fast enough on the measurement surface. Numerical examples of reconstructing acoustic quantities on the surface of a simplified engine block are demonstrated. [Work supported by NSF.]

Convergence of the method of fundamental solutions for solving the boundary value problem of modified Helmholtz equation

In this paper we derive the rate of convergence for the method of fundamental methods in solving the boundary value problem of a homogeneous modified Helmholtz equation. By using the Fourier sine expansion to find particular solutions we establish the convergence results for solving the boundary value problem of a non-homogeneous equation.

Laplace-transform analytic element solution of transient flow in porous media

A Laplace-transform analytic element method (LT-AEM) is described for the solution of transient flow problems in porous media. Following Laplace transformation of the original flow problem, the analytic element method (AEM) is used to solve the resultant time-independent modified Helmholtz equation, and the solution is inverted numerically back into the time domain. The solution is entirely general, retaining the mathematical elegance and computational efficiency of the AEM while being amenable to parallel computation. It is especially well suited for problems in which a solution is required at a limited number of points in space–time, and for problems involving materials with sharply contrasting hydraulic properties. We illustrate the LT-AEM on transient flow through a uniform confined aquifer with a circular inclusion of contrasting hydraulic conductivity and specific storage. Our results compare well with published analytical solutions in the special case of radial flow.

Reconstruction of acoustic radiation from vibrating objects in a half space using hybrid nearfield acoustical holography

Hybrid nearfield acoustical holography (NAH) [Wu, J. Acoust. Soc. Am. 113, 2252 (2003)] is utilized to reconstruct the vibro-acoustic fields generated by vibrating structures in half space. These hybrid formulations are derived from a modified Helmholtz equation least-squares method, where the field acoustic pressure is expanded in terms of both outgoing and incoming spherical waves, and the Helmholtz integral formulations implemented by the boundary element method (BEM). To overcome the ill-conditioning difficulties in the resultant matrix equations, regularization techniques such as the method of Tikhonov and generalized cross validation are employed. The main advantage of this hybrid approach is that satisfactory reconstruction can be obtained with relatively few acoustic pressure measurements taken over a conformal surface around the source. This is because a large portion of the input data is regenerated by the modified HELS, but not actually measured. Hence, the efficiency of reconstruction is enhanced. If the same number of measurements is used in an inverse BEM code, the reconstructed vibro-acoustic quantities on the source surface may be distorted. Numerical examples of a dilating and an oscillating sphere and a cylinder with spherical endcaps in half space are demonstrated. [Work supported by NSF.]

Modelling self-gravitating macroscopic media in general relativity: Solution to Szekeres' model of gravitational quadrupole

A model for the static weak-field macroscopic medium is analyzed and the equation for the macroscopic gravitational potential is derived. This is a biharmonic equation which is a non-trivial generalization of the Poisson equation of Newtonian gravity. In case of the strong gravitational quadrupole polarization it essentially holds inside a macroscopic matter source. Outside the source the gravitational potential fades away exponentially. The equation is equivalent to a system of the Poisson equation and the nonhomogeneous modified Helmholtz equations. The general solution to this system is obtained by using Green's function method and it does not have a limit to Newtonian gravity. In case of the insignificant gravitational quadrupole polarization the equation for macroscopic gravitational potential becomes the Poisson equation with the matter density renormalized by the factor including the value of the quadrupole gravitational polarization of the source. The general solution to this equation obtained by using Green's function method has a limit to Newtonian gravity.

The gravitational polarization in general relativity: solution to Szekeres' model of quadrupole polarization

A model for the static weak-field macroscopic medium is analysed and the equation for the macroscopic gravitational potential is derived. This is a biharmonic equation which is a non-trivial generalization of the Poisson equation of Newtonian gravity. In the case of strong gravitational quadrupole polarization, it essentially holds inside a macroscopic matter source. Outside the source the gravitational potential fades away exponentially. The equation is equivalent to a system of the Poisson equation and the non-homogeneous modified Helmholtz equations. The general solution to this system is obtained by using the Green function method and it is not limited to Newtonian gravity. In the case of insignificant gravitational quadrupole polarization, the equation for macroscopic gravitational potential becomes the Poisson equation with the matter density renormalized by a factor including the value of the quadrupole gravitational polarization of the source. The general solution to this equation obtained by using the Green function method is limited to Newtonian gravity.

Hydrodynamic analysis of wave interactions with a moored floating breakwater using the element-free Galerkin method

This paper deals with a numerical investigation of incident wave interactions with a moored pontoon-type floating breakwater. The element-free Galerkin method, in which only nodal data are required to analyze the problem, is employed to solve the diffraction and radiation boundary value problems addressed by the modified Helmholtz equation. The numerical model includes the hydrodynamic and mooring analyses, and it is validated by previous numerical and experimental results. Using the numerical model, we are able to assess the hydrodynamic performance of a moored pontoon-type floating breakwater in regular waves. Numerical results are presented to show the effects of wave conditions and mooring system configuration. This paper also presents the simple forms of stiffness coefficients of a slack mooring line. The influence of mooring line condition on the performance of a floating breakwater is highlighted. Key words: moored floating breakwater, element-free Galerkin method, mooring line condition.