Free-space Problem Research Articles

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.

Complete Radiation Boundary Conditions for Maxwell’s Equations

We describe the construction, analysis, and implementation of arbitrary-order local radiation boundary condition sequences for Maxwell’s equations. In particular we use the complete radiation boundary conditions which implicitly apply uniformly accurate exponentially convergent rational approximants to the exact radiation boundary conditions. Numerical experiments for waveguide and free space problems using high- order discontinuous Galerkin spatial discretizations are presented.

A High‐Order Integral Equation‐Based Solver for the Time‐Dependent Schrödinger Equation

AbstractWe introduce a numerical method for the solution of the time‐dependent Schrödinger equation with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver applicable to simulations of light‐matter interaction.The primary computational challenge in using the Volterra formulation is the application of a spacetime history dependent integral operator. This may be accomplished by projecting the solution onto a set of Fourier modes, and updating their coefficients from one time step to the next by a simple recurrence. In the periodic case, the modes are those of the usual Fourier series, and the fast Fourier transform (FFT) is used to alternate between physical and frequency domain grids. In the free space case, the oscillatory behavior of the spectral Green's function leads us to use a set of complex‐frequency Fourier modes obtained by discretizing a contour deformation of the inverse Fourier transform, and we develop a corresponding fast transform based on the FFT.Our approach is related to pseudospectral methods, but applied to an integral rather than the usual differential formulation. This has several advantages: it avoids the need for artificial boundary conditions, admits simple, inexpensive, high‐order implicit time marching schemes, and naturally includes time‐dependent potentials. We present examples in one and two dimensions showing spectral accuracy in space and eighth‐order accuracy in time for both periodic and free space problems. © 2020 Wiley Periodicals LLC.

Novel Extension of SBR–PO Method for Solving Electrically Large and Complex Electromagnetic Scattering Problem in Half-Space

To extend hybrid high-frequency shooting and bouncing rays–physical optics (SBR–PO) method to the modeling of electromagnetic (EM) scattering from an object in half-space, this paper focuses on addressing two key issues: 1) how to accurately obtain the induced electric currents on the object surface by taking into account the EM interactions between the object and half-space and 2) how to accurately calculate far-fields in half-space configuration. First, by introducing virtual interface of half-space, the proposed method avoids physically modeling the underlying ground, and only induced electric currents on the object are effectively taken into account in the SBR process. Second, the closed-form half-space Green’s function is used to calculate scattered far-fields to avoid the time-consuming evaluation of the Sommerfeld integrals. The computational errors are only from the approximation of induced electric currents with the same level of the SBR–PO method for the solution of the free-space problem, so the proposed method is more accurate than the conventional SBR–PO technique. Besides, performing the proposed method on the graphics processing unit platform to further accelerate the ray-tracing process makes the method even more powerful to solve the extra-large-scale scattering problems in the half-space.

Trimers in the resonant (2+1)-fermion problem on a narrow Feshbach resonance: Crossover from Efimovian to hydrogenoid spectrum

We study the quantum three-body free space problem of two same-spin-state fermions of mass $m$ interacting with a different particle of mass $M$, on an infinitely narrow Feshbach resonance with infinite s-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio $\alpha=m/M$. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within {\sl each} sector of fixed total angular momentum $l$. For $\alpha$ increasing from 0, we find that the trimer states first appear at the $l$-dependent Efimovian threshold $\alpha_c^{(l)}$, where the Efimov exponent $s$ vanishes, and that the {\sl entire} trimer spectrum (starting from the ground trimer state) is geometric for $\alpha$ tending to $\alpha_c^{(l)}$ from above, with a global energy scale that has a finite and non-zero limit. For further increasing values of $\alpha$, the least bound trimer states still form a geometric spectrum, with an energy ratio $\exp(2\pi/|s|)$ that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.

Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide

The Lippmann–Schwinger integral equation describes the scattering of acoustic waves from an inhomogeneous medium. For scattering problems in free space, Vainikko proposed a fast spectral solution method exploiting the convolution structure of this equation's integral operator and the fast Fourier transform. Although the integral operator of the Lippmann–Schwinger integral equation for scattering in a planar three-dimensional waveguide is not a convolution, we show in this paper that the separable structure of the kernel allows to construct fast spectral collocation methods. The numerical analysis of this method requires smooth material parameters; for discontinuous materials there is no theoretical convergence statement. Therefore, we construct a Galerkin variant of Vainikko's method avoiding this drawback. For several distant scattering objects inside the three-dimensional waveguide this discretization technique would lead to a computational domain consisting of one large box containing all scatterers and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects.

Lifting line theory for wing-in-ground effect in proximity to a free surface

Although it has some limitations in applications, the classical Prandtl lifting line theory remains a standard methodology for evaluating lifting problems in free space. It is of theoretical interest in revealing lifting mechanisms. It is therefore, interesting to generalize the classical lifting line theory to cases more general than just the free space problem. In this article, we present the Prandtl lifting line theory for wing-in-ground effect (WIG) near a free surface. While, the fundamental methodology being similar to the classical lifting line theory, it turns out that the difficulty lies in finding the three-dimensional Green’s function for the system of horseshoe vortices operating over the deformable free surface. Linear free surface boundary conditions are applied to deal with the two-dimensional lifting problem solved by the singularity distribution method and the three-dimensional correction found by placing a system of horseshoe vortices on the wing. This approach was validated against published data. Excellent agreement is found among results obtained from this study, experiments and numerical simulations. Extensive numerical examples are carried out to show the features of lift coefficients in the vicinity of a free surface. As expected, the free surface can be represented by a rigid wall for the case of high velocity. Finally, the free surface effect on WIG is discussed.

Nonrelativistic conformal field theory and trapped atoms: Virial theorems and the state-operator correspondence in three dimensions

The field theory of nonrelativistic fermions interacting via contact interactions can be used to calculate the properties of few-body systems of cold atoms confined in harmonic traps. The state-operator correspondence of nonrelativistic conformal field theory (NRCFT) shows that the energy eigenvalues (in oscillator units) of $N$ harmonically trapped fermions can be calculated from the scaling dimensions of $N$-fermion operators in the NRCFT. They are also in one-to-one correspondence with zero energy, scale-invariant solutions to the $N$-body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the $SL(2,R)$ algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.

Fast Direct Solution of Method of Moments Linear System

A novel algorithm, the compressed block decomposition (CBD), is presented for highly accelerated direct (noniterative) method of moments (MoM) solution of electromagnetic scattering and radiation problems. The algorithm is based on a block-wise subdivision of the MoM impedance matrix. Impedance matrix subblocks corresponding to distant subregions of the problem geometry are not calculated directly, but approximated in a compressed form. Subsequently, the matrix is decomposed preserving the compression. Examples are presented of typical problems in the range of 5000 to 70000 unknowns. The total execution time for the largest problem is about 1 h and 20 min for a single excitation vector. The main strength of the method is for problems with multiple excitation vectors (monostatic RCS computations) due to the negligible extra cost for each new excitation. For radiation and scattering problems in free space, the numerical complexity of the algorithm is shown to be N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and the storage requirements scale with N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> .

Earth surface effects on active faults: An eigenvalue asymptotic analysis

We study in this paper an eigenvalue problem (of Steklov type), modeling slow slip events (such as silent earthquakes, or earthquake nucleation phases) occurring on geological faults. We focus here on a half space formulation with traction free boundary condition: this simulates the earth surface where displacements take place and can be picked up by GPS measurements. We construct an appropriate functional framework attached to a formulation suitable for the half space setting. We perform an asymptotic analysis of the solution with respect to the depth of the fault. Starting from an integral representation for the displacement field, we prove that the differences between the eigenvalues and eigenfunctions attached to the half space problem and those attached to the free space problem, is of the order of d - 2 , where d is a depth parameter: intuitively, this was expected as this is also the order of decay of the derivative of the Green's function for our problem. We actually prove faster decay in case of symmetric faults. For all faults, we rigorously obtain a very useful asymptotic formula for the surface displacement, whose dominant part involves a so called seismic moment. We also provide results pertaining to the analysis of the multiplicity of the first eigenvalue in the line segment fault case. Finally we explain how we derived our numerical method for solving for dislocations on faults in the half plane. It involves integral equations combining regular and Hadamard's hypersingular integration kernels.

A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations

In this paper, a new multilevel Green's function interpolation method (MLGFIM) is presented to solve integral equations for large-scale electrostatic problems. In MLGFIM, the problem domain is first divided into multilevel cubes. Next, the peer-level Green's function interpolation technique is employed, and then, a new lower-to-upper-level Green's function interpolation technique is devised. They are used with the multilevel discretization to speed up the matrix-vector multiplications in the iterative solution in which a computational complexity of O(N) is achieved. The MLGFIM is used to extract the capacitances encountered in radio frequency integrated circuits (RFICs) and microelectromechanical systems. Moreover, to demonstrate its efficiency both in simulation speed and memory storage requirement, MLGFIM is compared with FastCap for free space problems and applied to extract capacitances from multilayered structures. For problems with 375,000 unknowns, the proposed method only requires 343 MB of computer memory.

An efficient model for three-dimensional surface wave simulations: Part I: Free space problems

An efficient numerical scheme for simulations of fully nonlinear non-breaking surface water waves in 3D is presented. The water depth is either shallow, finite or infinite. The method is based on a fast, rapidly converging, iterative algorithm to compute the Dirichlet to Neumann operator. This is evaluated by expanding the operator as a sum of global convolution terms and local integrals with kernels that decay quickly in space. The global terms are computed very quickly via FFT. The local terms are evaluated by numerical integration. Analytical integration of the linear part of the prognostic equations in Fourier space is obtained to machine precision. The remaining nonlinear components are integrated forward in time using an RK-scheme combined with a special step size control technique. This yields a very stable and accurate time marching procedure. Zeros-padding in the spectral space represents the anti-aliasing strategy. The method requires no smoothing. Illustration through examples show that the total energy is well conserved during the numerical simulations. The scheme is stable and accurate, even for very long time simulations of very steep wave events. The scheme is easily parallelizable. It propagates for example a Stokes wave of slope 0.2985 with a phase shift error of about 0.3° after 1000 periods of propagation.

Explicit finite-difference time-domain method with perfectly matched layers for transient low-frequency eddy-current analysis

The finite-difference time-domain (FDTD) method is a robust algorithm for the solution of high-frequency electromagnetic problems, but, unfortunately, the method has no counterpart in eddy-current analysis. Hence, the extension of FDTD to power frequencies is an attractive theoretical problem that could give rise to an entirely new methodology for low-frequency electromagnetics. In this paper, we introduce a general explicit FDTD algorithm for transient eddy-current problems. We perform a theoretical investigation of a nonstandard difference scheme, including the issues of stability and consistency. The study provides a class of explicit methods with varying properties and computational complexity. From these, we chose the DuFort-Frankel algorithm for its simplicity and efficiency. However, the most intricate issue for the application of an explicit scheme is the solution of the open boundary problem. Unlike conventional integral equation or forced truncation approaches, the free-space problem is successfully treated by using a specially designed perfectly matched layer (PML) for eddy currents. The explicit scheme within the conductors is, finally, efficiently combined with the free space-PML equations via the interface conditions.

A finite element method for approximating electromagnetic scattering from a conducting object

We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell's equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincare-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.

Fast inhomogeneous plane wave algorithm for the analysis of electromagnetic scattering

The fast inhomogeneous plane wave algorithm has been developed to accelerate the solution of three‐dimensional electromagnetic scattering problems in free space. By expanding the kernel of the Green's function using the Weyl identity and choosing a proper steepest descent path, the diagonalization of the translation matrix is achieved after the interpolation and extrapolation techniques are applied. The proposed algorithm is implemented on top of the scalable multipole engine, a portable implementation of the dynamic multilevel fast multipole algorithm for distributed‐memory computers. The computational time per matrix vector multiplication is reduced to O(NlogN) and the memory requirement is reduced to O(N), where N is the number of unknowns in the discretized integral equation. The algorithm is validated by applying it to the solution of the electromagnetic scattering from the perfect electric conducting scatterers. This approach can be easily extended to more general problems with complicated Green's function expressed in terms of the plane wave spectral integrals, such as the ones encountered in the multilayered medium studies.

Computation of transient radiation in semi-infinite regions based on exact nonreflecting boundary conditions and mixed time integration

Transient radiation in a semi-infinite region, bounded by a planar infinite baffle with a local acoustic source is considered. The numerical simulation of the transient radiation problem requires an artificial boundary Γ, here chosen to be a hemisphere, which separates the computational region from the surrounding unbounded acoustic medium. Inside the computational region we use a semidiscrete finite element method. On Γ, we apply the exact nonreflecting boundary condition (NRBC) first derived by Grote and Keller for the free-space problem. Since the problem is symmetric about the infinite planar surface, in order to satisfy the rigid baffle condition it is sufficient to restrict the indices in the spherical harmonic expansion which defines the NRBC and scale the radial harmonics which drive auxiliary equations on the boundary. The Fourier expansion in the circumferential angle appearing in the NRBC may be used to efficiently model axisymmetric problems in two dimensions. A new mixed explicit-implicit time integration method which retains the efficiency of explicit pressure field updates without the need for diagonal matrices in the auxiliary equations on Γ is presented. Here, the interior finite element equations are integrated explicitly in time while the auxiliary equations are integrated implicitly. The result is a very natural and highly efficient algorithm for large-scale wave propagation analysis. Numerical examples of fully transient radiation resulting from a piston transducer mounted in an infinite planar baffle are compared to analytical solutions to demonstrate the accuracy of the mixed time integration method with the NRBC for the half-space problem.

Cauchy–Characteristic Evolution and Waveforms

We investigate a new methodology for the computation of waves generated by isolated sources. This approach consists of a global spacetime evolution algorithm based on a Cauchy initial-value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior; we match the two schemes at their common interface. The characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval, so that our numerical solution extends to infinity and accurately models the free-space problem. The matching interface need not be situated far from the sources, the wavefronts may have arbitrary nonspherical geometry, and strong nonlinearity may be present in both the interior and the exterior regions. Stability and second-order convergence of the algorithms (to the exact solution of the infinite-domain problem) are established numerically in three space dimensions. The matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature. For linear problems, matching outperformed the local radiation conditions chosen for testing, and was about as accurate (for the same grid resolution) as the exact nonlocal conditions. However, since the computational cost of the nonlocal conditions is many times that of matching, this algorithm may be used with higher grid resolutions, yielding a significantly higher final accuracy. For strongly nonlinear problems, matching was significantly more accurate than all other methods tested. This seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field, while matching consistently takes nonlinearity into account in both interior and exterior regions.

Polynomial Time-Marching for Nonreflecting Boundary Problems

The newly developed polynomial time-marching technique has been successfully extended to nonperiodic boundary condition cases. In this paper, a special non-periodic boundary condition, nonreflecting or absorbing boundary condition, is incorporated into the pseudospectral polynomial time-marching scheme. Thus, this accurate and stable time-dependent PDE solver can be applied to some open domain or free space problems. The balanced overall spectral accuracy is illustrated by some numerical experiments in the one-dimensional case. The error goes to zero at a rate faster than many fixed orders of the finite-difference scheme. The order of the absorbing boundary approximation is addressed in one-dimension. Also, in the two-dimensional case, a 2nd-order absorbing approximation has been incorporated into the polynomial time-marching scheme with Chebyshev collocation in space. Comparison with the previous finite-difference implementation indicates that the high stability and efficiency of the polynomial time-marching remains. The overall accuracy is mainly limited by the 2nd-order absorbing approximation.

Calculation of unbounded field problems in free space by a 3D FEM/BEM-hybrid approach

In recent times electromagnetic scattering calculations are often done by local methods such as Finite Elements or Finite Differences with approximate local absorbing boundary conditions because of...

A Network-Oriented Approach to the Teaching of Electromagnetics

In this paper, electromagnetic theory is presented by expanding upon ideas that electrical engineering students already possess. The equivalent network concept, well known for lumped circuits (ordinary differential equations), is generalized to electromagnetic fields in free space (partial differential equations). The network, rigorously equivalent to Maxwell's equations, is suitable for all problems in free space. One illustrative antenna problem is presented and its relevance to lumped network calculations is also shown.