Helmholtz Kernel Research Articles

Recursive analytical quadrature for the close evaluation of Laplace and Helmholtz layer potentials over flat boundary elements in [formula omitted

A method for the analytical evaluation of singular and nearly singular layer potentials arising in the collocation boundary element method for the Laplace and Helmholtz equation is developed for flat boundary elements with polynomial shape functions. The method is based on dimension-reduction via the divergence theorem and a Recursive scheme for evaluating the resulting line Integrals for Polynomial Elements (RIPE). It is used to evaluate single layer, double layer, adjoint double layer, and hypersingular potentials, for both the Laplace and the Helmholtz kernels. It naturally supports nearly singular, singular, and hypersingular integrals under a single framework. The developed recursive algorithm allows accurate evaluation of layer potentials associated with O(p2) density functions used in a pth order boundary element in O(p3) time for the Laplace case.

Nonlinear thermo-mechanical stress-driven modeling of nano arches augmented by higher order double-scaled kernel

Modeling the nonlocal behavior of small-scale structures is almost controversial. The recently developed approach, known as the stress-driven nonlocal model, in which the stress is the input of nonlocal integral, has been proved to pose as a remedy for the ill-posing behavior of strain-driven solutions. Though it is still in its early stages, stress-driven modeling of the nonlinear behavior of nano-scaled problems needs further thought. The coupled displacement field, in particular, makes nano-arches more challenging. The stress-driven formulation, enhanced by constitutive boundary conditions, reveals the thermo-mechanical buckling and post-buckling features of shallow nano-arches for the first time in this research. Furthermore, the single-scaled Helmholtz kernel is replaced with the more appropriate double-scaled kernel known as the bi-Helmholtz averaging nonlocal kernel. Higher-order continuity of the bi-Helmholtz kernel and, hence, constitutive boundary conditions and higher-order differential equations corresponding to the stress-driven scheme hold the key to efficiency. The arch is subjected to uniform temperature and radial pressure at the same time. To demonstrate the response of the nano-arch during buckling and post-buckling, the nonlinear equilibrium path, whether for mechanical or thermal loading, is given for the first time by prevailing on the challenges of the higher-order problems. Moreover, it would be beneficial if one could have the chance to recognize how and when the arch buckles. These valuable data are available in this study to give the limiting parameters at the moment of buckling, including limiting temperature, nonlocal parameter, and geometry.

A Directional Equispaced Interpolation-Based Fast Multipole Method for Oscillatory Kernels

Fast multipole methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from boundary integral equations (BIEs) in acoustics or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm: directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach and show how symmetries can be exploited in the Fourier domain. We also describe the algorithmic design of defmm, well-suited for the BIE nonuniform particle distributions, and present performance optimizations on one CPU core. Finally, we exhibit important performance gains on all test cases for defmm over a state-of-the-art FMM library for oscillatory kernels.

An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

Implementation of a Photovoltaic System Model Using FAIR Digital Objects

Implementation of a Photovoltaic System Model Using FAIR Digital Objects

Realizing FAIR Digital Objects for the German Helmholtz Association of Research Centres

Realizing FAIR Digital Objects for the German Helmholtz Association of Research Centres

Nonlocal gradient integral models with a bi-Helmholtz averaging kernel for functionally graded beams

In this paper, we develop the bi-Helmholtz averaging kernel-based variationally nonlocal strain and stress gradient (NStrainG and NStressG) integral models to study the size-dependent bending of functionally graded (FG) beams. The integral constitutive relation is transformed into a higher-order differential form while four non-standard constitutive boundary conditions are derived in detail. The Laplace transform technique is utilized to derive closed-form solutions. Interestingly, we find that, unlike the Helmholtz kernel case, the local stress contribution must be included in the bi-Helmholtz kernel-based NSrainG model to avoid the conflict between equilibrium and constitutive requirements. In the numerical simulations, a series of examples are presented to study the size-effects on the bending deformation of the beams with various boundary types. Numerical results show that the bi-Helmholtz kernel-based nonlocal gradient theories are capable of describing both softening and stiffening size-effects in small structures by suitably tuning the nonlocal and gradient relevant parameters involved in nonlocal gradient models. Moreover, the effect of FG parameter is also presented.

A New Directional Algebraic Fast Multipole Method Based Iterative Solver for the Lippmann-Schwinger Equation Accelerated with HODLR Preconditioner

We present a fast iterative solver for scattering problems in 2D, where a penetrable object with compact support is considered. By representing the scattered field as a volume potential in terms of the Green's function, we arrive at the Lippmann-Schwinger equation in integral form, which is then discretized using an appropriate quadrature technique. The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method (DAFMM). The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel. And the construction of low-rank factorizations of the appropriate low-rank matrix sub-blocks is based on our new Nested Cross Approximation (NCA)~\cite{ arXiv:2203.14832 [math.NA]}. The advantage of our new NCA is that the search space of so-called far-field pivots is smaller than that of the existing NCAs. Another significant contribution of this work is the use of HODLR based direct solver as a preconditioner to further accelerate the iterative solver. In one of our numerical experiments, the iterative solver does not converge without a preconditioner. We show that the HODLR preconditioner is capable of solving problems that the iterative solver can not. Another noteworthy contribution of this article is that we perform a comparative study of the HODLR based fast direct solver, DAFMM based fast iterative solver, and HODLR preconditioned DAFMM based fast iterative solver for the discretized Lippmann-Schwinger problem. To the best of our knowledge, this work is one of the first to provide a systematic study and comparison of these different solvers for various problem sizes and contrast functions. In the spirit of reproducible computational science, the implementation of the algorithms developed in this article is made available at \url{https://github.com/vaishna77/Lippmann_Schwinger_Solver}.

Corrected trapezoidal rules for boundary integral equations in three dimensions

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nyström method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nyström method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, \(O(h^{3})\) convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-point correction stencil leads to an \(O(h^5)\) scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann–Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.

Nonlinear flexure mechanics of beams: stress gradient and nonlocal integral theory

In order to study the intrinsic size-effects, the stress gradient theory is implemented to a nano-scale beam model in nonlinear flexure. The nonlocal integral elasticity model is considered as a suitable counterpart to examine the softening behavior of nano-beams. Reissner variational principle is extended consistent with the stress gradient theory and applied to establish the differential, constitutive and boundary conditions of a nano-sized beam in nonlinear flexure. The nonlinear integro-differential and boundary conditions of inflected beams in the framework of the nonlocal integral elasticity are determined utilizing the total elastic strain energy formulation. A practical series solution approach in terms of Chebyshev polynomials is introduced to appropriately estimate the kinematic and kinetic field variables. A softening structural behavior is observed in the flexure of the stress gradient and the nonlocal beam in terms of the characteristic parameter and the smaller-is-softer phenomenon is, therefore, confirmed. The flexural response associated with the stress gradient theory is demonstrated to be in excellent agreement with the counterpart results of the nonlocal elasticity model equipped with the Helmholtz kernel function. The nonlocal elasticity theory endowed with the Error kernel function is illustrated to underestimate the flexural results of the stress gradient beam model. Detected numerical benchmark can be efficiently exploited for structural design and optimization of pioneering nano-engineering devices broadly utilized in advanced nano-electro-mechanical systems.

Free high-frequency vibrations of nonlocally elastic beam with varying cross-section area

In the framework of the two-phase theory of nonlocal elasticity we study free high-frequency vibrations of nanobeams with varying cross section. Considering the Helmholtz kernel in the constitutive equation, we reduce the original integro-differential equation to an equivalent six-order differential equation with variable coefficients and derive the pair of additional boundary conditions accounting for the nonlocal edge effects. Applying WKB method, a solution of the boundary-value problem is constructed in the form of superposition of the outer expansion and edge effect integrals with high variability along the beam axis. As an example, a clamped nanobeam with a shape matching to the shape of a slab waveguide in a photonic crystal beam laser is considered. The derived relations for natural frequencies and performed calculations revealed strong effect of both the nonlocal model fraction and the cross-section variability on the correction to natural frequencies calculated within the classical beam theory

Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity

Free high-frequency longitudinal vibrations of an inhomogeneous nanosized rod are studied on the basis of the nonlocal theory of elasticity. The upper part of spectrum with the wavelength comparable to the internal characteristic dimension of a nanorod is examined. An equations in the integral form with the Helmholtz kernel, incorporating both local and nonlocal phases, is used as the constitutive one. The original integro-differential equation is reduced to the forth-order differential equation with variable coefficients, the pair of additional boundary conditions being deduced. UsingWKB-method, a solution of the boundaryvalue problem is constructed in the form of the superposition of a main solution and edge effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model which allows estimating the upper part of spectrum of eigen-frequencies. Considering the nanorod with a variable cross-section area, we revealed a fair convergence of eigen-frequencies found in the framework of two models when the local fraction in the two-phase model vanishes.

Sparsity-Aware Precorrected Tensor Train Algorithm for Fast Solution of 2-D Scattering Problems and Current Flow Modeling on Unstructured Meshes

Acceleration of the method of moments (MoM) solution of the volume integral equation (VIE) on unstructured meshes is performed using a precorrected tensor train (P-TT) algorithm. The elements of the MoM’s unstructured mesh are projected onto a regular Cartesian grid. This enables representation of the MoM matrix as the Toeplitz matrix of point-to-point interactions pre- and post-multiplied by sparse matrices projecting MoM’s basis and testing functions on the Cartesian grid. The Toeplitz matrix is subsequently cast into the form of a multidimensional tensor. The latter is decomposed into the product of smaller dimensional matrices also known as tensor train (TT). TT allows to store Toeplitz matrix in $O(\log N)$ memory for VIE with the Laplace kernel and in $O(N \log N)$ memory for VIE with the Helmholtz kernel. Unlike the FFT-based fast algorithms, the P-TT method enables further memory reduction due to the sparsity of sources on the Cartesian grid. This sparsity pattern can be accounted for in construction of TT cores. It further reduces memory use as well as CPU time required for the multiplication of the Toeplitz matrix with a vector. It is shown that upon sufficient sparsity in representation of the sources on the grid, the P-TT algorithm can outperform both in CPU time and memory its FFT-based counterpart known as the precorrected FFT algorithm.

Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions

We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX (‘GIGAQBX’) and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion (‘QBX’) using local expansions formed via the Fast Multipole Method (FMM). Target-specific expansions serve to lower the cost of the formation and evaluation of QBX local expansions, reducing the associated computational effort from O((p+1)2) to O(p+1) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a ‘global’ QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for ‘local’ QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels.

Stress-driven nonlocal integral model for Timoshenko elastic nano-beams

Size-dependent structural behavior of inflected Timoshenko elastic nano-beams is investigated by nonlocal continuum mechanics. An innovative stress-driven integral model of elasticity is conceived by swapping source and output fields of Eringen's strain-driven theory. Unlike Eringen's model, the stress-driven nonlocal integral formulation leads to well-posed structural problems. Solution uniqueness and continuous dependence on data are thus ensured. Selected case-studies of technical interest are examined and exact nonlocal solutions of Timoshenko nano-beams are provided, detecting thus also new benchmarks for numerical analyses. The contributed results are compared with those obtained by the gradient theory of elasticity and by the differential constitutive model consequent (not equivalent) to Eringen's strain-driven theory equipped with Helmholtz's kernel.

Longitudinal vibrations of nano-rods by stress-driven integral elasticity

ABSTRACTIn elastic continuous structures defined on bounded domains, Eringen strain-driven integral model leads to ill-posed elastostatic problems since the constitutive stress field, got by convoluting the elastic strain field with an averaging kernel, conflicts with equilibrium requirements. The innovative stress-driven integral model, recently proposed in literature, is instead able to overcome inconsistencies of Eringen theory. In the present paper, size-dependent free vibrations of nano-rods are investigated by adopting the stress-driven nonlocal model equipped with Helmholtz kernel. Natural frequencies and mode shapes of nano-rods are established by an effective analytical solution strategy and are compared with those obtained by strain gradient theory.

Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion

We introduce a quadrature scheme—QBKIX —for the ubiquitous high-order accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracy at all distances to the domain boundary as well as on the boundary itself. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no analytic expansion nor addition theorem is required. Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. Our work builds upon quadrature by expansion, which approximates the potential by an analytic expansion in the neighborhood of each expansion center. In contrast, we use a sum of fundamental solutions lying on a ring enclosing the neighborhood, and solve a small dense linear system for their coefficients to match the potential on a smaller concentric ring. We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier (elastostatic) kernels in two dimensions (2D) using adaptive, panel-based boundary quadratures on smooth and corner domains. Advantages of the algorithm include its relative simplicity of implementation, immediate extension to new kernels, dimension-independence (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM.

Theory and implementation of [formula omitted]-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

In this work, we study the accuracy and efficiency of hierarchical matrix (H-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard H-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. H2-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard H-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the H-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.

Approximation of the high-frequency Helmholtz kernel by nested directional interpolation: error analysis

We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and postmultiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity.

Optimization of numerical orbitals using the Helmholtz kernel.

We present an integration scheme for optimizing the orbitals in numerical electronic structure calculations on general molecules. The orbital optimization is performed by integrating the Helmholtz kernel in the double bubble and cube basis, where bubbles represent the steep part of the functions in the vicinity of the nuclei, whereas the remaining cube part is expanded on an equidistant three-dimensional grid. The bubbles' part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kinds. The angular part of the bubble functions can be integrated analytically, whereas the radial part is integrated numerically. The cube part is integrated using a similar method as we previously implemented for numerically integrating two-electron potentials. The behavior of the integrand of the auxiliary dimension introduced by the integral transformation of the Helmholtz kernel has also been investigated. The correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations on H2, H2O, and CO. The obtained energies are compared with reference values in the literature showing that an accuracy of 10-4 to 10-7 Eh can be obtained with our approach.