Numerical Inversion Methods Research Articles

Comparative study for a new solution to the inverse heat conduction problem

AbstractThe solution we proposed for the inverse problem1 is evaluated. The study is based on Raynaud and Beck's paper,2 which provides test cases and simulation results for some common numerical inverse methods.

Excitation Force Identification of Rotating Machines Using Operational Rotor/Stator Amplitude Data and Analytical Synthesized Transfer Functions

Hydraulic excitation forces of rotating machinery are normally determined by direct measurement techniques like strain gages, load cells, etc. This paper presents an indirect method, in which frequency response transfer functions are analytically generated, using linear rotor/stator models. Inverse transfer function matrices are multiplied with operational vibration data to yield fourier transformed operational excitation forces. Analytical excitation techniques and numerical inversion methods of the system transfer function matrix are evaluated. External error sources and guidelines for an error sensitivity analysis of the predicted forces are described. Experimental verification is presented on a large horizontal centrifugal pump, with reasonable results. Typical application is shown on multistage hydro carbon and boilerfeed pumps.

The Effect of Discontinuities on the Order of Convergence of Numerical Inversion Methods

We study numerical methods for a one-dimensional inverse problem in reflection seismology. An impulsive wave in pressure is applied to the surface of a piecewise-smooth stratified elastic half-space and the resulting particle velocity is measured at the surface. The characteristic impedance of the medium is to be determined as a function of travel time. It has been shown recently that if the impulse response is measured appropriately, then several numerical methods to solve this inverse problem approximately are uniformly second-order convergent for piecewise-smooth media with discontinuities occurring only on the grid, i.e., at integer multiples of the mesh width. Piecewise-constant examples show that if the discontinuities do not occur on the grid, uniform convergence is lost near travel times, either where discontinuities occur or which correspond to multiple reflections. In intervals bounded away from these points, convergence appears to be uniform and at least second order. We present an example here of a piecewise-polynomial medium with a discontinuity off the grid for which at best first-order convergence is obtained at any point past the discontinuity. We conclude that unless the location of the discontinuities is either known a priori or determined as part of the solution to the inverse problem, no numerical method for this inverse problem can be expected to converge beyond the first discontinuity any faster than first order. This conclusion carries over to many related reflection inverse problems, including inverse problems for the wave equation with unknown variable velocity.

Methodology for Comparison of Inverse Heat Conduction Methods

The inverse heat conduction problem involves the calculation of the surface heat flux from transient measured temperatures inside solids. The deviation of the estimated heat flux from the true heat flux due to stabilization procedures is called the deterministic bias. This paper defines two test problems that show the tradeoff between deterministic bias and sensitivity to measurement errors of inverse methods. For a linear problem, with the statistical assumptions of additive and uncorrelated errors having constant variance and zero mean, the second test case gives the standard deviation of the estimated heat flux. A methodology for the quantitative comparison of deterministic bias and standard deviation of inverse methods is proposed. Four numerical inverse methods are compared.

Numerical solution of hydraulic models based on the axially-dispersed plug flow model by Laplace transforms

The problem of solving hydraulic models based on the axially-dispersed plug flow model which are applicable for the mathematical modelling of different flow-through systems both in chemical analysis (e.g., chromatography, flow injection analysis) and chemical industry (e.g., different tubular reactors) is discussed. Methods for numerical inversion of the model solution in the Laplace domain by expanding it into series of orthogonal functions are compared. Best results with respect to precision and consumption of computation time are given by the methods employing Chebyshov polynomials of the first kind and Fourier sine series. These methods were found to be better in these respects than some other frequently used numerical inversion methods.

Convergence of Numerical Inversion Methods for Discontinuous Impedance Profiles

We study numerical methods for the one-dimensional inverse problem of reflection seismology. An impulsive plane wave in pressure is applied to the surface of a piecewise smooth stratified elastic half space and the resulting particle velocity is measured at the surface. The characteristic impedance of the medium is to be recovered as a function of travel time. We consider two methods for the numerical solution of this inverse problem: a discretization of the Gopinath–Sondhi integral equation and a downward continuation method. Both methods are exact for so-called Goupillaud layered media. Both use second order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium. A formally second order accurate equation based on the theory of geometrical acoustics is used to recover the impedance. By analyzing the discretization of the Gopinath–Sondhi integral equation, we prove that if the impulse response is measured by sampling the step response or ramp response appropriately, then these two methods are second order convergent for piecewise smooth media with discontinuities occurring only at integer multiples of the mesh width. We conclude with an example showing that if the discontinuities occur at noninteger multiples of the mesh width, convergence need not be second order.

Velocity dependence of rotationally inelastic collisions: 7Li*2(A 1Σ)+Ne, Ar, and Xe

We present velocity-selected measurements of the cross sections for rotationally inelastic collisions of 7Li*2(A 1Σ) with Xe, Ar, and Ne, with various pairs of initial and final rotational levels, ji and jf: Li*2(vi=9, ji) +X→Li*2(vf=9, jf=ji+Δ)+X. For each target gas measurements were made with ji=8, 22, and 42 for approximately eight values of Δ with an overall range of −20 to +24; a total of 69 cross sections were measured. Velocity selection is accomplished by a Doppler-based laser-selection technique combined with further numerical inversion methods. We achieve a measurable relative velocity range of ≂0.5→3×105 cm/s, corresponding to kinetic energies of 120 to 3500 K. In contrast to the few previous similar studies, our data show a dramatic and varied velocity dependence. We have observed a few cross sections which increase substantially with velocity, some which rise faster than v−1r at low velocity, some that fall like v−2r at high velocity, and numerous examples of peaks in endoergic cross sections near threshold. We have seen two distinct types of rotationally adiabatic reduction in certain cross sections: ‖Δ‖=2 in Li*2–Xe and ji=42, Δ≲−16, in all three systems, particularly in Li*2–Ne. The velocity dependent behavior of these cross sections is sensitive to the long-range and short-range interaction potential, respectively. Adiabatic effects in the latter case result in the existence of classical ‘‘dynamical threshold velocities’’ for these large −Δ collisions. Some analytical results of a quantitative investigation into these rotationally adiabatic effects are also given.

Exciton migration with end detectors: calculation of time-dependent luminescence

We present explicit plots of the time dependence of the detector luminescence intensity in surface quenching observations. They are obtained with the aid of numerical inversion methods applied to exact consequences of a master-equation theory of exciton migration. The existing disparity in reported values of the singlet diffusion constant in aromatic hydro-carbon crystals is one of the main issues addressed. It is shown how the theory can be used to extract the exciton diffusion constant without prior knowledge of the (detector) trapping rates, by combining information obtained from steady-state (yield) and time-resolved experiments.

Mathematical properties of polydisperse systems, 4. Calculation of molecular weight distribution from light scattering on gaussian polymer coils. Purely numerical inversion methods

AbstractThe molecular weight distribution (MWD) of a high polymer is calculated from a weakly perturbed Zimm‐plot of the classical light scattering on dilute solutions of Gaussian polymer coils (theta state). A typical Zimm‐plot is simulated corresponding to the measurements of high accuracy as would be obtained by using the laser photometer described by Hack and Meyerhoff. The accuracy as published by these authors for small dissymmetries is used. Two inversion procedures are described in detail and tested, both of them avoiding the use of an empirical formula for the Laplace image of the calculated MWD. A complete numerical analysis of the results is given. The results are compared with those of the previous paper.

Delta methods for numerical inversion of the laplace transform, and their convergence

Delta methods for numerical inversion of the laplace transform, and their convergence

Projectile impact on an infinite, viscoplastic plate

An analysis is presented for the problem of a rigid/viscoplastic infinite plate subjected to normal projectile impact. The solution is first obtained in Laplace transform space for the finite clamped plate and then limiting conditions and numerical inversion methods are used to get the solution for the infinite plate in the time domain. The results are compared with experimental data for projectile impact on mild steel and aluminum alloy plates. The theory presents a useful analytical method for the evaluation of the response of large plates composed of strain rate dependent material subjected to projectile impact.

Numerical inversion of rational Laplace transforms

Effective methods for numerical inversion of any rational Laplace transform are described. The methods are based on a stable algorithm for the numerical solution of a homogeneous system of 1st-order differential equations.