Discrete Helmholtz Equation Research Articles

Inverse source problem for discrete Helmholtz equation

Abstract We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice Z d , d ⩾ 1 . We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.

Wave propagation on hexagonal lattices

Abstract We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.

Exterior diffraction problems for a triangular lattice

Exterior Dirichlet problems for two-dimensional (2D) lattice waves on the infinite triangular lattice are considered. Namely, we study Dirichlet problems for the 2D discrete Helmholtz equation in a plane with a hole. New results are obtained for the existence and uniqueness of the solution in the case of the real wave number without passing to a complex wave number. Besides, Green’s representation formula for the solution is derived with the help of difference potentials. To demonstrate the results, we propose a method for numerical calculation.

Half-plane diffraction problems on a triangular lattice

We investigate thin-slit diffraction problems for two-dimensional lattice waves. Namely, Dirichlet problems for the two-dimensional discrete Helmholtz equation on the triangular lattice in a half-plane are studied. Using the notion of the radiating solution, we prove the existence and uniqueness of a solution for the real wave number $$k\in (0,3)\backslash \{2\sqrt{2}\}$$ without passing to the complex wave number. Besides, an exact representation formula for the solution is derived. Here, we develop a numerical calculation method and demonstrate by example the effectiveness of our approach related to the propagation of wavefronts in metamaterials through two small openings.

Surface wave across crack-tip in a lattice model.

For a triangular lattice of particles, with nearest neighbour interactions and a traction free boundary, there exists a surface wave band for out-of-plane motion. Interactions between such a surface wave and stationary crack tip in mode III are investigated. The discrete Helmholtz equation for scattered waves, that incorporates an anisotropy parameter for unequal spring constants in horizontal versus slant directions, is solved exactly. The coefficient of transmission from one crack face to another, as well as that owing to reflection on the same face, is obtained in a closed form; the same leads to an estimate of energy fraction of incident wave that is leaked at the crack tip via bulk waves. It is found, in terms of surface wave band, that the transmission coefficient attains its maximum magnitude above the mid-band while the energy leak is minimum at the upper-band limit. Besides surface wave propagation across crack tip, surface wave excitation due to incident bulk wave is also discussed. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Diffraction by a Dirichlet right angle on a discrete planar lattice

A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of the work by A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020).

Diffraction problems for two-dimensional lattice waves in a quadrant

Motivated by applications of recent interest related to propagation problems in the left-handed 2D inductor–capacitor metamaterial and standard 2D inductor–capacitor lattice with monochromatic inputs along the left and bottom boundary of a rectangular slab, we address the problem of wave diffraction on the 2D square lattice in a quadrant. The peculiar structure allows us to consider problems on half-plane, consequently, we study Dirichlet problems for the two-dimensional discrete Helmholtz equation in a half-plane and a quadrant. In view of the existence and uniqueness of solution, we provide new results for the real wave number k∈(0,22)∖{2} without passing to the complex wave number and derive an exact representation formula for solutions. For this purpose, we use the notion of the radiating solution and propose sufficient conditions for the given boundary data at infinity.

On block triangular preconditioned iteration methods for solving the Helmholtz equation

To further improve the efficiency of solving the Helmholtz equation in heterogeneous media with large wavenumber, the Krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two-by-two linear system derived from the discrete Helmholtz equation. We further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods. The superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods. We also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber, roughly. Furthermore, the computational wave-fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm.

Critical Review of Preconditioners for Helmholtz Equation and their Spectral Analysis

Objectives: The study reviews the iterative solvers for solving Helmholtz equation. It also surveys the preconditioners for Krylov subspace methods for discrete Helmholtz equation. Numerical experiments based conclusions are drawn, after a thorough comparison of various preconditioners. Methods/Statistical Analysis: Five point finite difference scheme for two-dimensional Helmholtz equation is employed to obtain discrete analogue of Helmholtz equation. Krylov subspace method for is considered for iterations. Matrix based preconditioner ILU with variety of fill-in and operator based preconditioner complex shifted Laplace preconditioner (CSLP) are tested. Different shifts are proposed and experimented in CSLP. Optimization of shifts is recommended on basis of results. Spectral analysis is performed to validate results. Findings: For small wave number problem, ILU type preconditioned performs better and is fairly comparable with CSLP. However, increasing wave number makes unaffordable in terms of computational cost. The pure imaginary shift (0,1) and complex shift (1, / 4 π ) costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly. Approximate inversion may affect iterations, but will save solve time for large problems. Extensive spectral analysis, validating results, highlights this work. Application/Improvements: Dependency upon wave number is decreased with this recommended choice of shifts in CSLP. The number of iterations increases at very low rate when wave number is increased at rate of 2. Keywords: Helmholtz Equation, Iterative Methods, Krylov Subspace Methods, Preconditioners, Spectral Analysis

Exterior diffraction problems for two-dimensional square lattice

We study exterior Dirichlet problems for the two-dimensional discrete Helmholtz equation with the real wave number $$k\in (0,2\sqrt{2})\backslash \{2\}$$ . The investigation is carried out without passing to the complex wave number. Similarly to the continuum theory, we use the notion of radiating solution. Then, the unique solvability result and Green’s representation formula are obtained with the help of difference potentials. Finally, we proposed a method for numerical calculation. Efficiency of our approach is demonstrated on an example related to the propagation problems in the left-handed 2D inductor–capacitor metamaterial with a hole.

An improved sweeping domain decomposition preconditioner for the Helmholtz equation

In this paper we generalize and improve a recently developed domain decomposition preconditioner for the iterative solution of discretized Helmholtz equations. We introduce an improved method for transmission at the internal boundaries using perfectly matched layers. Simultaneous forward and backward sweeps are introduced, thereby improving the possibilities for parallellization. Finally, the method is combined with an outer two-grid iteration. The method is studied theoretically and with numerical examples. It is shown that the modifications lead to substantial decreases in computation time and memory use, so that computation times become comparable to that of the fastests methods currently in the literature for problems with up to 108 degrees of freedom.

Edge diffraction on triangular and hexagonal lattices: Existence, uniqueness, and finite section

The diffraction of lattice waves by two distinct types of defects, namely a rigid constraint and a crack, in the triangular and hexagonal (honeycomb) lattice models is analyzed. A semi-analytical solution of the discrete Helmholtz equation is provided everywhere in the lattice using the lattice Green’s function, when the length of the defect is finite, in all four problems. For the case of a semi-infinite defect, each problem involves the inversion of a Toeplitz operator on ℓp (1≤p≤∞), a truncation of which appears in the finite defect problem. It is shown that the symbol of the Toeplitz operator satisfies the Krein conditions and is continuous on the unit circle in the complex plane. The existence and uniqueness of solution of the discrete Wiener–Hopf equation in ℓp, corresponding to each of the four problems, follow and the method of finite section is applicable. It is established that the solution of each of the four discrete Sommerfeld problems is unique in ℓ2 and for the case of finite defect the displacement field, near any one tip, is approximated by its semi-infinite counterpart. The paper includes several graphical illustrations supplementary to the mathematical analysis presented and the main results established.

Diffraction of waves on triangular lattice by a semi-infinite rigid constraint and crack

An exact solution is provided for a discrete analog of each of the two Sommerfeld diffraction problems using a triangular lattice model, with nearest neighbor interactions, deforming in the anti-plane direction. The discrete Helmholtz equation with time-harmonic data prescribed on semi-infinite row(s) of lattice sites is solved using the discrete Wiener–Hopf method. An asymptotic expression of the exact solution, given in the form of a contour integral, has been obtained in far field using the standard approximation of the diffraction integrals based on the method of stationary phase. In case of the rigid constraint diffraction, the displacement of particles on the semi-infinite row complementing the constrained lattice sites, as well as on the adjacent row, is presented in closed form as a discrete convolution. For the crack diffraction problem, the length of both types of slant bonds on the semi-infinite row complementing the crack, as well as the crack opening displacement, is given in a similar form but in terms of the Fourier coefficients, of the associated Wiener–Hopf kernel, which are not available in closed form. The displacement field associated with the scattered waves at sites far from the defect tip, as well as few sites near the tip, is compared graphically with that of a numerical solution on a finite grid. Both discrete Sommerfeld problems are naturally relevant to their continuous counterparts, involving the traditional Helmholtz equation, that model the diffraction of electromagnetic and acoustic waves by a semi-infinite screen, based on 7-point discretization on a triangular grid.

Diffraction of waves on square lattice by semi-infinite rigid constraint

The problem of diffraction of a time harmonic lattice wave in a two-dimensional square lattice, by a semi-infinite rigid constraint, is investigated as a discrete analogue of diffraction by a Sommerfeld ‘soft’ half plane. The discrete Helmholtz equation, with input data prescribed on a semi-infinite row of lattice sites, is solved exactly using the discrete Wiener–Hopf method. The far-field asymptotic approximation of exact solution is provided. The scattered wave, in far field, is compared with a numerical solution of the problem for a set of frequencies in the pass band. The low frequency approximation of the exact solution is derived and it coincides with the Sommerfeld’s solution in its integral form. The results and discussion associated with the discrete Sommerfeld problem are relevant to numerical methods based on a 5-point discretization of the two-dimensional Helmholtz equation. In addition to the mechanics of waves in lattices, other physical applications of the latter concern the scattering of an E-polarized electromagnetic wave by a conducting half plane as well as its acoustic counterpart.

Diffraction of Waves on Square Lattice by Semi-Infinite Crack

In this paper, a discrete analogue of Sommerfeld half-plane diffraction is investigated. The two-dimensional problem of diffraction on a square lattice, of a plane (transverse) wave by a semi-infinite crack, is solved. The discrete Wiener--Hopf method has been used to obtain the exact solution of the discrete Helmholtz equation, with input data prescribed on the crack boundary sites due to a time harmonic incident wave. It is established that there exists a unique saddle point for the diffraction integral and its properties are characterized. An asymptotic approximation of the solution in the far field is provided and, for some values of the frequency it is compared with the numerical solution, of the diffraction problem, using a finite grid. A low frequency approximation of the solution in integral form recovers the classical continuum solution. At sufficiently large frequency in the pass band, the effects due to discreteness and anisotropy appear. The analysis is relevant to a 5-point discretization bas...

Preconditioned Krylov Subspace Methods for Sixth Order Compact Approximations of the Helmholtz Equation

We consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann, and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and lower order preconditioned Krylov subspace methodology. The resulting systems of finite-difference equations are solved by different preconditioned Krylov subspace-based methods. In the analysis of the lower order preconditioning developed here, we introduce the term “kth order preconditioned matrix” in addition to the commonly used “an optimal preconditioner.” The necessity of the new criterion is justified by the fact that the condition number of the preconditioned matrix in some of our test problems improves with the decrease of the grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the construction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination of the separation of variables technique and fast Fourier transform (FFT) type methods. The resulting numerical methods allow efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach.

Efficient Scalable Algorithms for Solving Dense Linear Systems with Hierarchically Semiseparable Structures

Hierarchically semiseparable (HSS) matrix techniques are emerging in constructing superfast direct solvers for both dense and sparse linear systems. Here, we develop a set of novel parallel algorithms for key HSS operations that are used for solving large linear systems. These are parallel rank-revealing QR factorization, HSS constructions with hierarchical compression, ULV HSS factorization, and HSS solutions. The HSS tree-based parallelism is fully exploited at the coarse level. The \textttBLACS and \textttScaLAPACK libraries are used to facilitate the parallel dense kernel operations at the fine-grained level. We appply our new solvers for discretized Helmholtz equations for multifrequency seismic imaging and iteratively solve time-harmonic seismic inverse boundary value problems. In particular, we use the HSS algorithms to solve the dense Schur complement systems associated with the root separator of the separator tree obtained from nested dissection of the graph of discretized Helmholtz equations. We...

An implicit finite-difference operator for the Helmholtz equation

We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly.

Diffraction on the Two-Dimensional Square Lattice

We solve the thin-slit diffraction problem for two-dimensional lattice waves. More precisely, for the discrete Helmholtz equation on the semi-infinite square lattice with data prescribed on the left boundary (the aperture), we use lattice Green's functions and a discrete Sommerfeld outgoing radiation condition to derive the exact solution everywhere in the lattice. The solution is a discrete convolution that can be evaluated in closed form for the wave number $k=2$. For other wave numbers, we give a recursive algorithm for computing the convolution kernel.

Rapid Solution of the Wave Equation in Unbounded Domains

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubich's convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior.