Sakurai-Sugiura Method Research Articles

A numerical study on the relationship between local defect resonance and sub-harmonic resonance for closed cracks

Abstract The sub-harmonic resonance in closed cracks is caused by resonant properties of the system and the contact of crack surfaces. Local defect resonance is considered a potential resonant phenomenon near cracks. This study compares local resonance and sub-harmonic resonance in a surface-breaking crack using numerical simulations. The authors previously simulated nonlinear sub-harmonic resonance using the harmonic balance-boundary element method. The investigation of local defect resonance utilizes both the boundary element method and the Sakurai-Sugiura method. The numerical results indicate a strong relation between the vibration fields of sub-harmonic frequency components and local defect resonance.

A Boundary Element Method (BEM) Solver for Low Frequency Room Modes

Room modes are known to be problematic in small critical listening environments. They degrade the acoustic quality at low frequencies, producing peaks and nulls in the frequency domain and ringing in the time domain. The Finite Element Method (FEM) is currently the easiest way to predict such resonances for arbitrarily shaped rooms. This solves for mode frequencies and shapes, as well as Q-factors and decay rates. Such 'eigenfrequency' solvers are commonplace in FEM, but FEM has the disadvantage of needing to mesh the entire air volume in the room. The Boundary Element Method (BEM) avoids this and only requires a simple boundary mesh, but solution of its eigenfrequency problems is much more challenging and appears in only a few academic papers. Here we transfer those approaches to Room Acoustics. We implement the block Sakurai-Sugiura method, which uses a contour integral in the complex frequency plane to convert the BEM eigenfrequency problem, which is usually non-linear in wavenumber, into a standard linear eigenfrequency problem that is straightforward to solve. The method is demonstrated through application to a cuboid room and an irregularly shaped room, both with impedance boundary conditions. Results are validated against FEM and discussed.

A topology optimization of open acoustic waveguides based on a scattering matrix method

This study presents a topology optimization scheme for realizing a bound state in the continuum along an open acoustic waveguide comprising a periodic array of elastic materials. First, we formulate the periodic problem as a system of linear algebraic equations using a scattering matrix associated with a single unit structure of the waveguide. The scattering matrix is numerically constructed using the boundary element method. Subsequently, we employ the Sakurai–Sugiura method to determine resonant frequencies and the Floquet wavenumbers by solving a nonlinear eigenvalue problem for the linear system. We design the shape and topology of the unit elastic material such that the periodic structure has a real resonant wavenumber at a given frequency by minimizing the imaginary part of the resonant wavenumber. The proposed topology optimization scheme is based on a level-set method with a novel topological derivative. We demonstrate a numerical example of the proposed topology optimization and show that it realizes a bound state in the continuum through some numerical experiments.

The tropical scaling for the polynomial eigenvalue problem solved by a contour integral method

AbstractThe contour integral method is a class of efficient methods for computing partial eigenvalues of polynomial eigenvalue problem (PEP). Among them, the recently developed Sakurai–Sugiura method with Rayleigh–Ritz projection (SS‐RR) method has received much attention for its effectiveness. However, the SS‐RR method may suffer from numerical instability when the coefficient matrices of the projected PEP vary widely in norm. To improve the numerical stability, we incorporate the tropical scaling technique into the SS‐RR method and establish upper bounds for the backward error of an approximate eigenpair of the original eigenvalue problem. These bounds shed light on mechanism that the tropical scaling improves the numerical stability of the original SS‐RR method. Numerical experiments show that the actual backward errors can be successfully reduced by scaling and the bounds can predict well the errors occurring before and after scaling.

Computing Group Velocities and Group-Velocity Dispersions of Optical Fibers Through Automatic Differentiation of Explicit Forms of Propagation Constants

We present a method to compute differential coefficients of the first and second orders, for propagation constants of nondegenerate and degenerate modes, propagating in an optical fiber to calculate dispersion characteristics of group velocities (GV) and group-velocity dispersions (GVD). This method automatically differentiates explicit forms of eigenvalues extracted from determinants of a generalized eigenvalue problem. This problem is converted from a nonlinear eigenvalue problem derived from the Trefftz method (e.g., transfer matrix and multipole methods) by the Sakurai-Sugiura method. We compute the differential coefficients of the second order of degenerate modes without computing eigenvectors and their first order differential coefficients in the generalized eigenvalue problems. Therefore, in parametric optimization, we can compute the differential coefficients of all propagating modes without remodeling the waveguide cross-section, as analyzing the whole cross-section avoids the symmetric conditions derived from the numerical model. Computed results with the proposed method for step-index and holey fibers validate the method and its effectiveness. In particular, our results show that our method computes the GV and GVD more accurately than computation using numerical differentiation.

Backward error bounds for polynomial eigenvalue problem solved by a Rayleigh–Ritz type contour integral-based eigensolver

The contour integral-based eigensolvers have attracted much attention in recent years. In this paper, we consider solving a polynomial eigenvalue problem (PEP) by a contour integral-based eigensolver named the Sakurai–Sugiura method with Rayleigh–Ritz projection (SS–RR method). We derive a backward error bound of PEP solved by the SS–RR method. This bound can be used to show the accuracy of the computed approximate eigenpairs of PEP. The accuracy of the derived bounds is demonstrated by several examples.

Stability of boundary element methods for the two dimensional wave equation in time domain revisited

This study considers the stability of time domain BEMs for the wave equation in 2D. We show that the question of stability of time domain BEMs is reduced to a nonlinear eigenvalue problem related to frequency domain integral equations. We propose to solve this non-linear eigenvalue problem numerically with the Sakurai-Sugiura method. After validating this approach numerically in the exterior Dirichlet problem, we proceed to transmission problems in which we find that some time domain counterparts of “resonance-free” integral equations in frequency domain lead to instability. We finally show that the proposed stability analysis helps to reformulate these equations to obtain stable numerical schemes.

Hybrid Trefftz Finite-Element Method for Analyzing the Eigenmodes of Optical Fibers

We present a hybrid Trefftz finite-element method (HTFEM) for analyzing the eigenmodes of optical fibers. Using this method, waveguide problems that include open regions can be analyzed without requiring absorbing boundary conditions or utilizing artificial absorbers to enclose the region under analysis because the open region is treated as a special “Trefftz element.” In addition, using a Trefftz element for the homogeneous core region and utilizing the Sakurai–Sugiura method to solve the nonlinear eigenvalue problems resulting from the HTFEM, we are able to obtain the degenerate effective indices of waveguides accurately without imposing symmetry conditions. We examine the convergence behavior of the effective indices of optical fibers computed using the HTFEM and compare the results with those obtained from analytic solutions or other numerical methods. Numerical examples demonstrate the validity of the formulation and the usefulness of eigenmode analyses using the HTFEM.

Contour integral method for obtaining the self-energy matrices of electrodes in electron transport calculations

We propose an efficient computational method for evaluating the self-energy matrices of electrodes to study ballistic electron transport properties in nanoscale systems. To reduce the high computational cost incurred in large systems, a contour integral eigensolver based on the Sakurai-Sugiura method combined with the shifted biconjugate gradient method is developed to solve exponential-type eigenvalue problem for complex wave vectors. A remarkable feature of the proposed algorithm is that the numerical procedure is very similar to that of conventional band structure calculations. We implement the developed method in the framework of the real-space higher-order finite difference scheme with nonlocal pseudopotentials. Numerical tests for a wide variety of materials validate the robustness, accuracy, and efficiency of the proposed method. As an illustration of the method, we present the electron transport property of the free-standing silicene with the line defect originating from the reversed buckled phases.

Performance evaluation of the Sakurai-Sugiura method with a block Krylov subspace linear solver for large dense Hermitian-definite generalized eigenvalue problems

Performance evaluation of the Sakurai-Sugiura method with a block Krylov subspace linear solver for large dense Hermitian-definite generalized eigenvalue problems

Efficient Calculation of Electronic Structure Using O(N) Density Functional Theory.

We propose an efficient way to calculate the electronic structure of large systems by combining a large-scale first-principles density functional theory code, Conquest, and an efficient interior eigenproblem solver, the Sakurai-Sugiura method. The electronic Hamiltonian and charge density of large systems are obtained by Conquest, and the eigenstates of the Hamiltonians are then obtained by the Sakurai-Sugiura method. Applications to a hydrated DNA system and adsorbed P2 molecules and Ge hut clusters on large Si substrates demonstrate the applicability of this combination on systems with 10,000+ atoms with high accuracy and efficiency.

Finding Propagation Constants of Leaky and Degenerate Modes Using Simultaneous Transcendental Equations of Holey Optical Fibers

This paper presents a simple solution to simultaneous transcendental equations of propagation constants for leaky and degenerate modes. Using a block version of the Sakurai–Sugiura method (SSM) and a criterion for distinguishing physical solutions from spurious ones, we solved a nonlinear matrix equation formulated by the multipole method in a two-dimensional cylindrical coordinate system. Numerical examples showed that we can obtain accurate propagation constants and extract field distributions of leaky and degenerate modes of holey fibers without imposing symmetry. In addition, the condition numbers of the eigenvalues are efficient criteria for distinguishing solutions computed by the block version of the SSM.

English

One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP.

A FEAST algorithm with oblique projection for generalized eigenvalue problems

The contour integral-based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non-Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non-Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design-efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

Improving the numerical stability of the Sakurai-Sugiura method for quadratic eigenvalue problems

Improving the numerical stability of the Sakurai-Sugiura method for quadratic eigenvalue problems

Periodic band structure calculation by the Sakurai–Sugiura method with a fast direct solver for the boundary element method with the fast multipole representation

In this paper, we present a numerical method for periodic band structure calculation, which is associated with eigenvalue problems for periodic problems, using the boundary element method (BEM). In the BEM, the eigenvalue problems are converted into non-linear eigenvalue problems, which are not tractable with conventional eigensolvers. In the present study, to solve non-linear eigenvalue problems, the block Sakurai–Sugiura (SS) method, which can convert non-linear eigenvalue problems into generalised eigenvalue problems, is utilised. A fast direct solver for the BEM with a fast multipole representation is employed in the algorithm of the block SS method since algebraic equations need to be solved for multiple right-hand sides in the block SS method. We conduct several numerical experiments related to phononic structures to confirm the validity and efficiency of the proposed method. We confirm that the proposed method can calculate the band structure of the phononic structures, and the computational time with the proposed method is less than that with a conventional FEM-based eigensolvers with triangular linear elements even for relatively small problems.

Isospin symmetry breaking and large-scale shell-model calculations with the Sakurai-Sugiura method

Recently isospin symmetry breaking for mass 60-70 region has been investigated based on large-scale shell-model calculations in terms of mirror energy differences (MED), Coulomb energy differences (CED) and triplet energy differences (TED). Behind these investigations, we have encountered a subtle problem in numerical calculations for odd-odd N = Z nuclei with large-scale shell-model calculations. Here we focus on how to solve this subtle problem by the Sakurai-Sugiura (SS) method, which has been recently proposed as a new diagonalization method and has been successfully applied to nuclear shell-model calculations. The isospin symmetry breaking is one of the current topics in nuclear structure physics. An asymmetry be- tween spectra of isospin analogue states with mirror pair nuclei has been extensively investigated in the upper sd− and the lower fp −shell regions (1), and in mass 60-70 region (2-6). This asymmetry, that is, isospin symmetry breaking arises partly due to the Coulomb force and partly due to the strong nucleon-nucleon interaction. To analyze this isospin symmetry breaking, mirror en- ergy differences (MED) (7), Coulomb energy differences (CED)(8) and triplet energy differences(TED) (7) have been discussed. The MED is a measure of charge sym- metry breaking in an effective interaction and is defined as

Efficient Parameter Estimation and Implementation of a Contour Integral-Based Eigensolver

We consider an eigensolver for computing eigenvalues in a given domain and the corresponding eigenvectors of large-scale matrix pencils. The Sakurai-Sugiura (SS) method is an eigensolver based on complex moments given by contour integrals of matrix inverses with several shift points. This method has good parallel scalability, and is suitable for massively parallel computing environments. The SS method has several parameters, and the choice of these parameters is crucial for achieving high accuracy and good parallel performance. We discuss some numerical properties of the method, and present efficient parameter estimation techniques. We demonstrate the efficiency of our method with numerical experiments.

Eigenvalue analysis for acoustic problem in 3D by boundary element method with the block Sakurai–Sugiura method

Abstract This paper presents accurate numerical solutions for nonlinear eigenvalue analysis of three-dimensional acoustic cavities by boundary element method (BEM). To solve the nonlinear eigenvalue problem (NEP) formulated by BEM, we employ a contour integral method, called block Sakurai–Sugiura (SS) method, by which the NEP is converted to a standard linear eigenvalue problem and the dimension of eigenspace is reduced. The block version adopted in present work can also extract eigenvalues whose multiplicity is larger than one, but for the complex connected region which includes a internal closed boundary, the methodology yields fictitious eigenvalues. The application of the technique is demonstrated through the eigenvalue calculation of sphere with unique homogenous boundary conditions, cube with mixed boundary conditions and a complex connected region formed by cubic boundary and spherical boundary, however, the fictitious eigenvalues can be identified by Burton–Miller's method. These numerical results are supported by appropriate convergence study and comparisons with close form.

Analysis of Band Structure for 2D Acoustic Phononic Structure by BEM and the Block SS Method

By its very basis, the block Sakurai-Sugiura (SS) method may be ap- plied to solving nonlinear eigenvalue problem formulated by boundary integral equation. In this paper, the methodology of BEM combined with the block SS method is applied to the computation of band structures of phononic structures for acoustic problems. Since the conventional boundary integral equation yields the real fictitious eigenfrequencies when it is applied to an exterior problem of scatters, Burton-Miller's method is employed to remove the real fictitious eigenfrequen- cies resulted by the boundary of the scatterers. The application of the method is demonstrated through analyses of band structures of homogenous and composite structures.