Nonhomogeneous Helmholtz Equation Research Articles

Efficient Finite Difference Approaches for Solving Initial Boundary Value Problems in Helmholtz Partial Differential Equations

This study presents numerical solutions for initial boundary value problems of homogeneous and nonhomogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes’ robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.

Far-field analytical solutions of the non-homogeneous Helmholtz and wave equations for spatially non-localized sources.

Non-localized impulsive sources are ubiquitous in underwater acoustic applications. However, analytical expressions of their acoustic field are usually not available. In this work, far-field analytical solutions of the non-homogeneous scalar Helmholtz and wave equations are developed for a class of spatially extended impulsive sources. The derived expressions can serve as benchmarks to verify the accuracy of numerical solvers.

Nonlinear scattering by non-Hermitian multilayers with saturation effects.

We theoretically investigate the optical properties of a one-dimensional non-Hermitian dispersive layered system with saturable gain and loss. We solve the nonhomogeneous Helmholtz equation perturbatively by applying the modified transfer matrix method and we obtain closed-form expressions for the reflection or transmission coefficients for TM incident waves. The nonreciprocity of the scattering process can be directly inferred from the analysis of the obtained expressions. It is shown that by tuning the parameters of the layers we can effectively control the impact of nonlinearity on the scattering characteristics of the non-Hermitian layered structure. In particular, we investigate the asymmetric and nonreciprocal characteristics of the reflectance and transmittance of multilayered parity-time (PT)-symmetric slab. We demonstrate that incident electromagnetic wave may effectively tunnel through the PT-symmetric multilayered structures with zero reflection. The effect of nonlinearity to the scattering matrix eigenvalues is systematically examined.

A conjugate gradient method for distributed optimal control problems with nonhomogeneous Helmholtz equation

A conjugate gradient algorithm with strong Wolfe-Powell line search for distributed optimal control problem is proposed. The optimal system has been discussed in [1]. The proposed algorithm is employed to solve the problem in infinite dimensional function space. With low complexity, it is suitable for large-scale problem. The sufficient descent condition of conjugate gradient and the existence of iterative step are proved. The algorithm also has global convergence property and linear convergence rate. At last, numerical experiments are presented to illustrate the efficiency and the convergence rate of the proposed algorithm.

Generalized plane wave discontinuous Galerkin methods for nonhomogeneous Helmholtz equations with variable wave numbers

ABSTRACT In this paper we are concerned with numerical method for nonhomogeneous Helmholtz equations with variable wave numbers. We derive generalized plane wave basis functions for three-dimensional homogeneous Helmholtz equations with variable wave numbers. Then, by combining the local spectral element method, we design a generalized plane wave discontinuous Galerkin method for the discretization of such nonhomogeneous Helmholtz equations (in both dimensions two and three dimensions). We show that the approximation solutions generated by the proposed discretization method yield error estimates with high accuracies. Numerical results indicate that the resulting approximate solutions generated by the new method possess high accuracy and verify the validity of the theoretical results.

Solution of Nonhomogeneous Helmholtz Equation with Variable Coefficient Using Boundary Domain Integral Method

The Boundary Domain Integral Method (BDIM) is applied to the solution of the nonhomogeneous Helmholtz equation with variable coefficient. The analytical formulas for the integrals over the individual boundaries and domain integrals are used to increase the accuracy of the numerical approach. Comparisons of the developed BDIM with the analytical solutions for the homogeneous Helmholtz equation with constant coefficient and the nonhomogeneous Helmholtz equation with variable coefficient are given.

A Two-Steps Method Based on Plane Wave for Nonhomogeneous Helmholtz Equations in Inhomogeneous Media

A Two-Steps Method Based on Plane Wave for Nonhomogeneous Helmholtz Equations in Inhomogeneous Media

Comparisons of three kinds of plane wave methods for the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers

In this paper we are concerned with some plane wave discretization methods of the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers. We design two new variants of the variational theory of complex rays and the ultra weak variational formulation for the discretization of these types of equations, respectively. The well posedness of the approximate solutions generated by the two methods is derived. Moreover, based on the PWLS–LSFE method introduced in Hu and Yuan (2018), we extend these two methods (VTCR method and UWVF method) combined with local spectral element to discretize nonhomogeneous Helmholtz equation and Maxwell’s equations. The numerical results show that the resulting approximate solution generated by the UWVF method is clearly more accurate than that generated by the VTCR method.

A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations

In this paper we are concerned with plane wave discretizations of nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. To this end, we design a plane wave method combined with local spectral elements for the discretization of such nonhomogeneous equations. This method contains two steps: we first solve a series of nonhomogeneous local problems on auxiliary smooth subdomains by the spectral element method, and then apply the plane wave method to the discretization of the resulting (locally homogeneous) residue problem on the global solution domain. We derive error estimates of the approximate solutions generated by this method. The numerical results show that the resulting approximate solutions possess high accuracy.

An implementation of QSAOR iterative method for non-homogeneous Helmholtz equations

This paper aims to show the usefulness of the quarter-sweep acceler-ated over relaxation (QSAOR) method by implementing the quarter-sweep approximation equation based on finite difference (FD) to solvetwo-dimensional (2D) Helmholtz equations compared to full-sweep ac-celerated over relaxation (FSAOR) and half sweep accelerated over re-laxation (HSAOR) methods. The formulation and implementation ofthe QSAOR, HSAOR and FSAOR methods are also presented. Somenumerical tests were carried out to illustrate that the QSAOR methodis superior to HSAOR and FSAOR methods.

Time-Harmonic Wavefields of “Complex Sources” and Their Sources in the Real Space

The paper deals with the complexified Green function of the 3D Helmholtz equation in a free space, which is of interest as an exact solution demonstrating the Gaussian beam behavior. This function involves a square root and satisfies a nonhomogeneous Helmholtz equation, the right-hand side of which depends on a cut and on the branch of a root. An explicit representation of this generalized function is studied. Bibliography: 21 titles.

A “Complex Source” in the 2D Real Space

The paper concerns a complexified Green’s function g* for the 2D Helmholtz equation, which is studied as a nonparaxial model of a Gaussian beam. The function g* satisfies a certain nonhomogeneous Helmholtz equation in the real space with a source distribution dependent on a choice of branch of a certain complex square root. Various choices of branch cut are discussed, and the corresponding source distribution is calculated. Bibliography: 13 titles.

A pseudo-kinetic approach for Helmholtz equation

A lattice Boltzmann type pseudo-kineticmodel for a non-homogeneous Helmholtz equation is derived in this paper. Numerical results for some model problems show the robustness and efficiency of this lattice Boltzmann type pseudo-kinetic scheme. The computation at each site is determined only by local parameters, and can be easily adapted to solve multiple scattering problems with many scatterers or wave propagation in nonhomogeneous medium without increasing the computational cost.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

AbstractFourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

Analytical modeling of squeeze film damping for rectangular elastic plates using Green's functions

An analytical method for the solution of squeeze film damping based on Green's function to the nonlinear Reynolds equation considering an elastic plate is presented. This allows calculating the stiffness and damping forces rapidly for various boundary conditions. The elastic plate velocity is applied to the nonlinear Reynolds equation as a forcing term. The nonlinear Reynolds equation is divided into multiple linear nonhomogeneous Helmholtz equations, which then can be solvable using the presented approach. Approximate mode shapes of a rectangular elastic plate are used, enabling the calculation of the damping ratio and frequency shift for the linear case, as well as the complex resistant pressure, for both linear and nonlinear cases.

On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity

A simple but robust scale invariance argument earlier used to derive flow rules for macroscopic kinematic hardening plasticity from the configuration of single slip is now applied to consider yielding of fiber reinforced composites as first pioneered, on a purely phenomenological basis, by Tony Spencer and his co-workers. It is shown that the approach results to simpler constitutive equations than those obtained by the phenomenological methodology of the standard representation theorems for isotropic functions. The argument is then extended to obtain simplified but physically-based constitutive equations for gradient plasticity and gradient elasticity as such type of approaches have been extensively adopted in the recent literature to interpret interesting phenomena at the micron scale, not captured by classical theories. For gradient plasticity, the scale invariance argument involves two steps: first the algebraic elimination of the orientation tensors defining crystal slip between the microscopic expressions for the back (dislocation) stress and plastic strain rate; and, second, a maximization procedure for the rate of plastic work (maximum dissipation or entropy production). For gradient elasticity the algebraic elimination of the lattice orientation vectors between the microscopic expressions for the lattice stress and lattice strain suffices, as no dissipation occurs in this case. As an application of gradient theory, we consider, in particular, the size effect in strain gradient-dependent plastic layers and the removal of elastic singularities at screw dislocation lines and crack tips under a shear mode loading are considered. It is shown that a number of insightful results are obtained which are in accord with experimental observations and/or elaborate numerical simulations. This is largely due to the fact that the method used provides simple expressions for the strain/stress field which turns out to be governed by a non-homogeneous Helmholtz equation for which analytical solutions are available.

Finite element analysis of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles

A numerical investigation is presented of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles. The equation determining the acoustic displacement in response to the electrostriction caused by the pump and Stokes waves reduces to the non-homogeneous Helmholtz equation for fibers with a uniform acoustic velocity profile. In this special case the acoustic displacement and subsequently the Brillouin gain are calculated using a Green's function. These results are then used to validate a finite-element solution of the same equation. This finite element method is then used to analyze a standard large mode area fiber as well as fibers incorporating four different acoustic velocity profiles with 5% variation in the acoustic velocity across the core. The profiles which suppress the peak Brillouin gain most effectively exhibit a maximum acoustic gradient near the midpoint between the center and boundary of the fiber core. These profiles produce 11 dB of suppression relative to standard large mode area fibers.

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.

The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods

The treatment of domain integrals has been a topic of interest almost since the inception of the boundary element method (BEM). Proponents of meshless methods such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM) have typically pointed out that these meshless methods obviate the need for an interior discretization. Hence, the DRM and MRM maintain one of the biggest advantages of the BEM, namely, the boundary-only discretization. On the other hand, other researchers maintain that classical domain integration with an interior discretization is more robust. However, the discretization of the domain in complex multiply-connected geometries remains problematic. In this research, three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries are compared. The methods include the DRM, classical cell-based domain integration, and a novel auxiliary domain method. The auxiliary domain method allows the evaluation of the domain integral by constructing an approximately C 1 extension of the domain integrand into the complement of the multiply-connected domain. This approach combines the robustness and accuracy of direct domain integral evaluation while, at the same time, allowing for a relatively simple interior discretization. Comparisons are made between these three methods of domain integral evaluation in terms of speed and accuracy.

Diffraction of a directionally spread wave group by a cylinder

The problem of diffraction of a directionally spread focused wave group by a bottom-seated circular cylinder is considered from the viewpoint of second-order perturbation theory. After applying the time Fourier transform and separation of vertical variable the resulting two-dimensional non-homogeneous Helmholtz equations are solved numerically using finite differences. The detailed formulation of the second-order radiation condition is presented. Numerical solutions of the problem are obtained for JONSWAP amplitude spectra for the incoming wave group with various types of directional spreading. The results are compared with the corresponding results for a unidirectional wave group of the same amplitude spectrum. Finally we discuss the applicability of the averaged spreading angle concept for practical applications.