On the inverse problem of soil profile reconstruction: a comparison of time-domain approaches

We discuss a one-dimensional inverse material profile reconstruction problem that arises in layered media underlain by a rigid bottom, when total wavefield surficial measurements are used to guide the reconstruction. To tackle the problem, we adopt the systematic framework of PDE-constrained optimization and construct an augmented misfit functional that is further endowed by a regularization scheme. We report on a comparison of spatial regularization schemes such as Tikhonov and total variation against a temporal scheme that treats the model parameters as time-dependent. We study numerically the effects of inexact initial estimates, data noise, and regularization parameter choices for all three schemes, and report inverted profiles for the modulus, and for simultaneous inversion of both the modulus and viscous damping. Our numerical experiments demonstrate comparable or superior performance of the time-dependent regularization over the Tikhonov and total variation schemes for both smooth and sharp target profiles, albeit at increased computational cost. Support for this work was provided by the US National Science Foundation under grant awards ATM-0325125 and CMS-0348484.     

Probabilistic inverse problem and system uncertainties for damage detection in piezoelectrics

This paper provides a probabilistic formulation to design a monitoring setup for damage detection in piezoelectric plates, solving a model-based identification inverse problem (IP). The IP algorithm consists on the minimization of a cost functional defined as the quadratic-difference between experimental and trial measurements simulated by the finite element method. The motivation of this work comes from the necessity for a more rational design criteria applied to damage monitoring of piezoelectric materials. In addition, it is very important for the solving of the inverse problem to take into account the random nature of the system to be solved in order to obtain accurate and reliable solutions. In this direction, two investigations are considered. For the first, the experimental measurements are simulated combining a finite element and a Monte Carlo analysis, both validated with already published results. Then, an uncertainty analysis is used to obtain the statistical distribution of the simulated experimental measurements, while a sensitivity analysis is employed to find out the influence of the uncertainties in the model parameters related to the measurement noise. Upon the study of the measurements, they are used as the input for the damage identification IP which produces the location and extension of a defect inside a piezoelectric plate. For the second investigation, a probabilistic IP approach is developed to determine the statistical distribution and sensitivities of the IP solutions. This novel approach combines the Monte Carlo and the IP algorithm, considering the trial measurements as random. In conclusion, the analysis demonstrates that in order to improve the quality of the damage characterization, only a few material parameters have to be controlled at the experimental stage. It is important to note that this is not an experimental study, however, it can be considered as a first step to design a rational damage identification experimental device, controlling the variables that increase the noise level and decrease the accuracy of the IP solution.     

Comparison of the born iterative method and tarantola's method for an electromagnetic time-domain inverse problem

Two methods of solving the nonlinear two-dimensional electromagnetic inverse scattering problem in the time domain are considered. These are the Born iterative method and the method originally proposed by Tarantola for the seismic reflection inverse problems. The former is based on Born-type iterations on an integral equation, whereby at each iteration the problem is linearized, and its solution is found via a regularized optimization. The latter also uses an iterative method to solve the nonlinear system of equations. Although it linearizes the problem at each stage as well, no optimization is carried out at each iteration; rather the problem as a whole is posed as a (regularized) optimization. Each method is described briefly and its computational complexity is analyzed. Tarantola's method is shown to have a lower numerical complexity compared to the Born iterative method for each iteration, but in the examples considered, required more iterations to converge. Both methods perform well when inverting a smooth profile; however, the Born iterative method gave better results in resolving localized point scatterers.     

Phosphorescence lifetime imaging in turbid media: the inverse problem and experimental image reconstruction

Three-dimensional phosphorescence lifetime imaging is a novel method for the mapping of oxygen concentration in biological tissues. We present reconstruction techniques for recovering phosphorescent objects in highly scattering media based on the telegraph equation and two regularization methods, i.e., the Tikhonov-Phillips regularization and the maximum entropy method. Theoretical results are experimentally validated, and the reconstructed images of phosphorescent objects rendering oxygen maps in a layer are presented.     

Interpretation of the solution to the inverse problem for the positive function and the reconstruction of neutron spectra

The present paper treats the problem of reconstruction of a positive function S(E > 0) from its measured functionals $\text{Y}{}_{\mathrm{n}}\text{= ?}\text{d}\text{EA}{}_{\mathrm{n}}\text{(E)S(E)}$. The reconstruction method is based on the Tikhonov's regularization of the problem as well as on the priori information that S(E) is positive. In detail are considered the statistical properties of the reconstructed function $\text{S}\text{?}\text{(E)}$.     

An exact analytical solution of the inverse problem for a plasma cylinder expanding in a compressible medium

A strict analytical solution of the wave equation with cylindrical symmetry in a region with mobile boundaries was obtained by the method of inverse problems with allowance for the interaction of nonlinear arguments. The proposed method is universal and applicable to solving both inverse and direct problems for arbitrary values of the initial radius and displacements. The solution describes the near wave field of an expanding plasma piston, including the field formed in the initial moments of a pulsed expansion process. The solution gives exact values of a given pressure and velocity wave profile at a fixed point of the wave zone in the initial moment, as well as particular finite values of the pressure and velocity at a mobile boundary of the expanding plasma piston at the moments of time approaching zero. The solution is obtained with allowance for additional nonlinear conditions.     

Finite element methods for the analysis of softening plastic hinges in beams and frames

This paper presents new finite element methods for the analysis of localized failures in plastic beams and frames in the form of plastic hinges. The hinges are modeled as discontinuities of the generalized displacements of the underlying Timoshenko beam/rod theory. Hinges accounting for a discontinuity in the transversal and longitudinal displacements and the rotation field are developed in this context. A multi-scale framework is considered in the incorporation of the dissipative effects of these discontinuities in the large-scale problem of a beam and a general frame. A localized softening cohesive law relating these generalized displacements with the stress resultants acting at the level of the cross section is effectively introduced in the frame response. The resulting models, referred to as localized models, are then able to capture the localized dissipation observed in the localized failures of these structural members, avoiding altogether the inconsistencies observed for classical models in the stress resultants with strain softening. The constructive approach followed in the development of these models leads naturally to the formulation of enhanced strain finite elements for their numerical approximation. In this context, we develop new finite elements incorporating the singular strains associated to the plastic hinges at the element level. A careful analysis is presented so the resulting finite elements avoid the phenomenon of stress locking, that is, an overstiff response in the softening of the hinge, not allowing for the full release of the stress. The accurate approximation of the kinematics of the hinges requires a strain enhancement linking the jumps in the deflection and the rotation fields, given the coupled definition of the transverse shear strain in these two fields. Different enhanced strain elements, involving different base finite elements and different enhancement strategies, are considered and analyzed in detail. Their performance are then compared in several representative numerical simulations. These analyses identify optimally enhanced finite elements for the accurate modeling the localized failures observed in common framed structures.Financial support for this research has been provided by the ONR under grant no. N00014-00-1-0306 with UC Berkeley. This support is gratefully acknowledged.     

NOSER: An algorithm for solving the inverse conductivity problem

The inverse conductivity problem is the mathematical problem that must be solved in order for electrical impedance tomography systems to be able to make images. Here we show how this inverse conductivity problem is related to a number of other inverse problem. We then explain the workings of an algorithm that we have used to make images from electrical impedance data measured on the boundary of a circle in two dimensions. This algorithm is based on the method of least squares. It takes one step of a Newton's method, using a constant conductivity as an initial guess. Most of the calculations can therefore be done analytically. The resulting code is named NOSER, for Newton's One-Step Error Reconstructor. It provides a reconstruction with 496 degrees of freedom. The code does not reproduce the conductivity accurately (unless it differs very little from a constant), but it yields useful images. This is illustrated by images reconstructed from numerical and experimental data, including data from a human chest.     

Investigation of the direct and inverse piezoeffect in the dynamic problem of electroelasticity for an unbounded medium with a tunnel opening

Summary An effective approach to the investigation of antiplane electroacoustic wave fields in an unbounded piezoelectric medium weakened by a tunnel opening with the given system of surface electrodes is suggested. The approach is based on the method of boundary integral equations. It is assumed that the medium excitation occurs due to the effect of the given electrodes electric potential differences (inverse piezoeffect). The solution of the boundary-value problem of electroelasticity is also considered in the case of the direct piezoeffect when as a result of diffraction on the opening of monochromatic shear waves the generation of electric potential difference takes place on electrode coverings. An approximate scheme of numerical realization of the solvable system of singular integro-differential equations of the second kind with discontinuous kernels is given. There are presented some examples.     

A boundary element formulation for the inverse elastostatics problem (iesp) of flaw detection

A boundary integral formulation is presented for the detection of flaws in planar structural members from the displacement measurements given at some boundary locations and the applied loading. Such inverse problems usually start with an initial guess for the flaw location and size and proceed towards the final configuration in a sequence of iterative steps. A finite element formulation will require a remeshing of the object corresponding to the revised flaw configuration in each iteration making the procedure computationally expensive and cumbersome. No such remeshing is required for the boundary element approach. The inverse problem is written as an optimization problem with the objective function being the sum of the squares of the differences between the measured displacements and the computed displacements for the assumed flaw configuration. The geometric condition that the flaw lies within the domain of the object is imposed using the internal penalty function approach in which the objective function is augmented by the constraint using a penalty parameter. A first-order regularization procedure is also implemented to modify the objective function in order to minimize the numerical fluctuations that may be caused in the numerical procedure due to errors in the experimental measurements for displacements. The flaw configuration is defined in terms of geometric parameters and the sensitivities with respect to these parameters are obtained in the boundary element framework using the implicit differentiation approach. A series of numerical examples involving the detection of circular and elliptical flaws of various sizes and orientations are solved using the present approach. Good predictions of the flaw shape and location are obtained.     

Delamination detection in laminated composite beams using the empirical mode decomposition energy damage index

Fiber reinforced plastic composite (FRPC) beams have been widely used as effective stand-alone structural elements and/or reinforcing agents in various engineering applications. Such lightweight, stiff and strong structural elements may suffer, however, from inter-ply defects (referred to as delaminations), arising from various causes and factors. A delamination in such structural components could propagate into severely large damage zone, thus compromising the structural integrity. Therefore, detection of delamination at an early stage is of paramount importance.     

Hamilton体系中Timoshenko梁冲击问题的描述和求解

针对T im oshenko梁的冲击问题,在H am ilton体系中通过直接求解H am ilton矩阵的本征值问题并利用共轭辛正交关系,给出了以位移和应力表示的全状态向量解的一般表达式。根据边界条件给出了频率方程和广义模态函数向量。从H am ilton正则方程出发证明了广义模态函数向量的正交性,并给出了动响应及数值结果。从文中可以看出在H am ilton体系中结构弹性动力学问题的描述和求解的一般方法。通过比较本文结果与已有结果可以看出本文方法的正确性和可行性。     

An inverse problem approach to stiffness mapping for early detection of breast cancer: tissue phantom experiments

Early detection of breast cancer will continue to be crucial in improving patient survival rates. Our ultimate goal is to develop an electro-mechanical device to automate and refine the manual breast exam process, and use inverse techniques to generate a tissue stiffness map of the breast tissue. We have previously presented computational simulations of the stiffness mapping approach, which employs static indentations of the tissue and measurements of surface displacements. In this paper, we report on experimental validation of the technique with tissue phantom experiments. We tested 12 tissue phantom samples without simulated tumours and 14 tissue phantom samples with simulated tumours. Our stiffness mapping approach correctly identified all 26 samples.     

A conforming unified finite element formulation for the vibration of thick beams and frames

Beams and frames are common features in many engineering structures and in this paper an approach is given to model their dynamic behaviour adequately. Whilst the eigen‐frequencies of continuous systems comprising of slender beams can be identified, in most cases of practical interest, by means of Euler or Timoshenko beam theory, for structures comprising of thick beam models this is not necessarily true since such idealizations constrain the cross‐sections to remain planar. This paper suggests an alternative approach by means of a unified fully conforming plane stress rectangular finite element which is believed to allow for more realistic representation of the shear effects and hence the strain field around the joints of such structures. The usefulness and functionality of this improved numerical approach is explored via comparison against a non‐conforming two‐dimensional plate as well as one‐dimensional Euler–Bernoulli and Timoshenko finite element formulations corresponding to a variety of beam aspect ratios representing the structures of a rotor and a portal frame. The idealization is shown to be particularly advantageous for simulating the effects of shear distortion where beams join at right angles and the transverse forces in one member interact with the extensional forces of the adjoining structure. Copyright © 2004 John Wiley & Sons, Ltd.     

The method of fundamental solutions for the inverse space-dependent heat source problem

In this study, the inverse heat source problem in which the heat source is space-dependent is treated. The method proposed in Yan et al. [The method of fundamental solutions for the inverse heat source problem. Eng Anal Boundary Elem 2008;32:216–22] where the heat source is considered to be only time-dependent, is modified so that it can be applied to only space-dependent problems. We have used a new transformation to simplify the problem.     

An inverse problem in estimating the relaxation time for nanoscale phonon radiative transfer problem

An inverse nanoscale phonon radiative transfer problem is solved in this study by using conjugate gradient method (CGM) to estimate the unknown frequency‐ and temperature‐dependent relaxation time, based on the simulated phonon intensity measurements. The CGM in dealing with the present integro‐differential governing equations is not as straightforward as for the normal differential equations; special treatments are needed to overcome the difficulties. Results obtained in this inverse analysis will be justified based on the numerical experiments where two different unknown distributions of relaxation time are to be estimated. Finally, it is shown that the reliable frequency and temperature‐dependent relaxation time can be obtained with CGM. Copyright © 2007 John Wiley & Sons, Ltd.     

The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation

In this article, a meshless numerical method for solving the inverse source problem of the space-fractional diffusion equation is proposed. The numerical solution is approximated using the fundamental solution of the space-fractional diffusion equation as a basis function. Since the resulting matrix equation is extremely ill-conditioned, a regularized solution is obtained by adopting the Tikhonov regularization scheme, in which the choice of the regularization parameter is based on generalized cross-validation criterion. Two typical numerical examples are given to verify the efficiency and accuracy of the proposed method.     

Two-dimensional inverse problem of diffusion tomography: the approach applicable for small inclusions

A method for solving the two-dimensional inverse problems of optical diffusion tomography is proposed. The method is especially designed for the imaging of small inclusions embedded in the backgrounds of strongly scattering media. Numerical simulations show that the results are stable with respect to external noise at the boundary of the sample. The location of an inclusion is obtained with an accuracy of the order of several photon transport mean-free paths in the medium in cases both with and without noise in the scattering data used for the solution of the inverse problem.     

The hybrid Laplace transform/finite element method applied to the quasi-static and dynamic analysis of viscoelastic Timoshenko beams

The quasi-static and dynamic responses of a linear viscoelastic beam are solved numerically by using the hybrid Laplace transform/finite element method. In the analysis, the Timoshenko beam theory, which includes the transverse shear and rotatory inertia effect and conventional beam theory, are used to solve this problem. The temperature field is assumed to be constant and homogeneous and that the relaxation modulus has the form of the Prony series. In the hybrid method, the Laplace transform with respect to time is applied to the coupled equations and the finite element model is developed by applying Hamilton's variational principle without any integral transformation. The numerical results of quasi-static and dynamic responses for the models of Maxwell fluid and three parameter solid types are presented and discussed.