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Register a new userEliminating the fictitious frequency problem in BEM solutions of the external Helmholtz equation
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The problem of the fictitious frequency spectrum resulting from numerical implementations of the boundary element method for the exterior Helmholtz problem is revisited. When the ordinary 3D free space Greens function is replaced by a modified Greens function, it is shown that these fictitious frequencies do not necessarily have to correspond to the internal resonance frequency of the object. Together with a recently developed fully desingularized boundary element method that confers superior numerical accuracy, a simple and practical way is proposed for detecting and avoiding these fictitious solutions. The concepts are illustrated with examples of a scattering wave on a rigid sphere.
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Assigning boundary conditions, such as acoustic impedance, to the frequency domain thermoviscous wave equations (TWE), derived from the linearized Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which yields a discrete set of complex eigenfunctions and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain, via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs on an unstructured staggered grid arrangement. Only the momentum equation is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. The iHS is finally closed via assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure can be carried out independently for different frequencies, making it embarrassingly parallel, and able to return the complete broadband complex impedance distribution at the IBs in any desired frequency range to arbitrary numerical precision. The iHS approach is first validated against Rotts theory for viscous rectangular and circular ducts. The impedance of a toy porous cavity with a complex geometry is then reconstructed and validated with companion fully compressible unstructured Navier-Stokes simulations resolving the cavity geometry. Verification against one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC) is also carried out. Finally, results from a preliminary analysis of a thermoacoustically unstable cavity are presented.
Assigning homogeneous boundary conditions, such as acoustic impedance, to the thermoviscous wave equations (TWE) derived by transforming the linearized Navier-Stokes equations (LNSE) to the frequency domain yields a so-called Helmholtz solver, whose output is a discrete set of complex eigenfunction and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs derived on an unstructured grid with staggered grid arrangement. The momentum equation only is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. One closure condition considered for the iHS is the assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure is carried out independently for each frequency in order to return the complete broadband complex impedance distribution at the IBs in any desired frequency range. The iHS approach is first validated against Rotts theory for both inviscid and viscous, rectangular and circular ducts. The impedance of a geometrically complex toy cavity is then reconstructed and verified against companion full compressible unstructured Navier-Stokes simulations resolving the cavity geometry and one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC). The iHS methodology is also shown to capture thermoacoustic effects, with reconstructed impedance values quantitatively in agreement with thermoacoustic growth rates.
We present concise, computationally efficient formulas for several quantities of interest -- including absorbed and scattered power, optical force (radiation pressure), and torque -- in scattering calculations performed using the boundary-element method (BEM) [also known as the method of moments (MOM)]. Our formulas compute the quantities of interest textit{directly} from the BEM surface currents with no need ever to compute the scattered electromagnetic fields. We derive our new formulas and demonstrate their effectiveness by computing power, force, and torque in a number of example geometries. Free, open-source software implementations of our formulas are available for download online.
We develop an approach to solving numerically the time-dependent Schrodinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin expansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.
We present a new axially symmetric monochromatic free-space solution to the Klein-Gordon equation propagating with a superluminal group velocity and show that it gives rise to an imaginary part of the causal propagator outside the light cone. We address the question about causality of the spacelike paths and argue that the signal with a well-defined wavefront formed by the superluminal modes would propagate in vacuum with the light speed.
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