Numerical Projection Methods Research Articles

Efficient and accurate numerical-projection of electromagnetic multipoles for scattering objects

In this paper, we develop an efficient and accurate procedure of electromagnetic multipole decomposition by using the Lebedev and Gaussian quadrature methods to perform the numerical integration. Firstly, we briefly review the principles of multipole decomposition, highlighting two numerical projection methods including surface and volume integration. Secondly, we discuss the Lebedev and Gaussian quadrature methods, provide a detailed recipe to select the quadrature points and the corresponding weighting factor, and illustrate the integration accuracy and numerical efficiency (that is, with very few sampling points) using a unit sphere surface and regular tetrahedron. In the demonstrations of an isotropic dielectric nanosphere, a symmetric scatterer, and an anisotropic nanosphere, we perform multipole decomposition and validate our numerical projection procedure. The obtained results from our procedure are all consistent with those from Mie theory, symmetry constraints, and finite element simulations.Graphical

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations.

Numerical projection methods are elaborated for the calculation of eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with selected rotational and parity quantum numbers employing shifted explicitly correlated Gaussian functions, which are, in general, not eigenfunctions of the total angular momentum and parity operators. The increased computational cost of numerically projecting the basis functions onto the irreducible representations of the three dimensional rotation-inversion group is the price to pay for the increased flexibility of the basis functions. This increased flexibility allowed us to achieve a substantial improvement for the variational upper bound to the Pauli-allowed ground-state energy of the molecular ion treated as an explicit five-particle system. We compare our pre-Born-Oppenheimer result obtained for this molecular ion with rotational-vibrational calculations carried out on a potential energy surface.

Holographic characterization of therapeutic array transducers with assessment of measurement limitations and robustness

Acoustic holography measurements and corresponding numerical projection methods have recently gained acceptance for characterizing the fields generated by high intensity therapeutic ultrasound (HITU) applications. To facilitate the standardization of such measurement-based simulation methods, it is important to understand how practical measurement challenges and limitations impact the results. Toward this end, holography measurements were acquired and analyzed for characterization of two therapeutic array transducers, each with 256 elements. For different focusing configurations, measurements comprised 2D holography scans as well as 1D scans used to independently quantify the focal lobe. The fields projected from holograms were compared to independent 1D scans to evaluate the impacts of hydrophone directivity, non-orthogonality of the axes of the positioner used to move the hydrophone, and overall mechanical stability of measurements. Results demonstrate that directivity effects exceeded those expected based on hydrophone size and altered focal pressures on the order of 10%. Non-orthogonality of the positioner shifted apparent focal locations by measurable distances, which could be confused with small shifts in device fixturing. However, holographic reconstruction of the ultrasound field structure near the focal lobe was robust with respect to directivity and non-orthogonality, which enables compensation by relative calibration of source output levels. [Funding support by NIH R01-EB025187, R01-EB007643, and P01-DK043881.]

Riccati-based Boundary Feedback Stabilization of Incompressible Navier--Stokes Flows

In this article a boundary feedback stabilization approach for incompressible Navier--Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. The nested iteration resul...

Holography and numerical projection methods for characterizing the three-dimensional acoustic fields of arrays in continuous-wave and transient regimes

The use of projection methods is increasingly accepted as a standard way of characterizing the 3D fields generated by medical ultrasound sources. When combined with hydrophone measurements of pressure amplitude and phase over a surface transverse to the wave propagation, numerical projection can be used to reconstruct 3D fields that account for operational details and imperfections of the source. Here, we use holography measurements to characterize the fields generated by two array transducers with different geometries and modes of operation. First, a seven-element, high-power therapy transducer is characterized in the continuous-wave regime using holography measurements and nonlinear forward-projection calculations. Second, a C5-2 imaging probe (Philips Healthcare) with 128 elements is characterized in the transient regime using holography measurements and linear projection calculations. Results from the numerical projections for both sources are compared with independent hydrophone measurements of select waveforms, including shocked focal waveforms for the therapy transducer. Accurate 3D field representations have been confirmed, though a notable sensitivity to hydrophone calibrations is revealed. Uncertainties associated with this approach are discussed toward the development of holography measurements combined with numerical projections as a standard metrological tool. [Work supported by NIH EB007643, P01DK043881, R01DK092197 and T32DK007779, NSBRI through NASA NCC 9-58, and RFBR 13-02-00183.]

Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context

This paper derives conditions under which a stable solution to the least-squares linear estimation problem for multifractional random fields can be obtained. The observation model is defined in terms of a multifractional pseudodifferential equation. The weak-sense and strong-sense formulations of this problem are studied through the theory of fractional Sobolev spaces of variable order, and the spectral theory of multifractional pseudodifferential operators and their parametrix. The Theory of Reproducing Kernel Hilbert Spaces is also applied to define a stable solution to the direct and inverse estimation problems. Numerical projection methods are proposed based on the construction of orthogonal bases of these spaces. Indeed, projection into such bases leads to a regularization, removing the ill-posed nature of the estimation problem. A simulation study is developed to illustrate the estimation results derived. Some open research lines in relation to the extension of the derived results to the multifractal process context are also discussed.

Nonlinear dynamics of flow in piping trees using projection methods

PurposeSimple methods for the steady‐state analysis of a flow network are readily available, but the dynamic behavior of a large‐scale flow network is difficult to study due to the complex differential‐algebraic equation system resulting from its modeling. It is the aim of this paper to present two simple methods for the dynamic analysis of large‐scale flow networks and to demonstrate their use by examining the dynamics of a self‐similar branching tree network.Design/methodology/approachTwo numerical projection methods are proposed for one‐dimensional dynamic analysis of large piping networks. Both are extensions of that suggested by Chorin for the nonlinear differential‐algebraic system resulting from the Navier‐Stokes equations. Each numerical algorithm is discussed and verified for turbulent flow in a nonlinear, self‐similar, branching tree network with constant friction factor for which an exact solution is available.FindingsThe dynamics of this network are calculated for more realistic friction factors and described as system parameters are varied. Self‐excited oscillations due to laminar‐turbulent transition are found for some parameter values and dynamic component behavior is observed in the network which is not observable in components apart from it.Practical implicationsIt is shown that the dynamics of a flow network can exhibit unexpected behavior, reinforcing the need for simple methods to perform dynamic analysis.Originality/valueThis paper presents two numerical projection schemes for dynamic analysis of large‐scale flow networks to aid in their study and design.

Projection method for the Hankel transform

The numerical projection method for the zero-order Hankel transform inversion for the case of data given on a finite interval has been developed. The justification of the convergence of the projection method has been done for the general case that also includes the sine-Fourier transform inversion. An inequality on the norms of the zeroth-order Laguerre functions has been proved. The efficiency of the method was illustrated with the test data and for the problem of cylindrical distribution function calculation using the melt surface layer diffraction data.

In-depth resolution of the magneto-optical Kerr effect in ferromagnetic multilayers

How does one determine the magnetization state and hysteresis loop corresponding to one of the ferromagnetic (FM) layers located at a given depth in a stack of FM/non-FM layers by means of the magneto-optical Kerr effect? For this purpose the representation of the Kerr effect in the complex rotation-ellipticity plane is introduced. A depth sensitivity function controlling the Kerr effect is defined and its dependence on the photon energy and angle of incidence is studied. A general way to determine the in-depth location of the FM layer, from which the Kerr signal originates, is proposed. In the case of a FM bilayer structure, previous proposed solutions are discussed within a unified formalism. For a system with three or more FM layers two approaches are proposed to extract selectively the magneto-optical signals originating at individual FM layers: the parallel Kerr vector and cascade numerical projection methods. These methods are successively checked experimentally on simple multilayer structures. Finally, on the basis of the developed approaches a readout solution for multivalued magneto-optical recording in a four-storage-layer structure is proposed.

Numerical Projection Method For Inverse Fourier Transform And Its Application

Numerical projection method of the Fourier transform inversion from data given on a finite interval is proposed. It is based on an expansion of the solution into a series of eigenfunctions of the Fourier transform. The number of terms of the expansion depends on the length of the data interval. Convergence of the solution of the method is proved. The projection method for the case of the sine Fourier transform and the set of the odd Hermite functions being its eigenfunctions are examined and applied to numerical Fourier filtering.

Justification of the projection method for solving axisymmetric problems of diffraction by locally inhomogeneous bodies

THE PROOF of convergence in the appropriate functional spaces, of the approximate solutions to the exact solution, is discussed for an incomplete numerical projection method of Galerkin type, for axisymmetric problems of the excitation of metal bodies surrounded by a layer of plasma. The cases of two different polarizations, and of ideal and impedance boundary conditions on the metal surface, are distinguished.