Benchmark problems for defect size and shape determination in eddy-current nondestructive evaluation

A set of four benchmark problems is presented for verification of theoretical calculations of defect size and shape in eddy-current nondestructive evaluation. The benchmark problems are based on careful measurements of the change in coil impedance as a function of frequency for a circular air-cored coil which is scanned along the axis of an electrodischarge machined slot in a thick aluminum alloy plate. Slots of (i) semi-elliptical, and (ii) double-peaked profiles are considered. Deviations from ideal coil behavior are identified and corrected so that the final impedance data and experimental parameters can be directly used to verify theoretical inversion algorithms.     

Dual regularization in non-linear inverse scattering problems

A new method in the theory of non-linear ill-posed problems is adapted and applied to various one-dimensional inverse scattering problems of electromagnetic subsurface diagnostics of permittivity inhomogeneities. Based on the developed theory, solving algorithms have been worked out and applied in problems of low-frequency diagnostics of conductivity profile in geomagnetic exploration, microwave monitoring of water diffusion in soil and reflectometry diagnostics of inhomogeneities in multilayer structures of X-ray optics – covering the scale range from nanometers to kilometres. Results demonstrate new possibilities of developed approach in these applications.     

A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems

The two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley & Sons, Ltd.     

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the -Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

Weight-adaptive isogeometric analysis for solving elastodynamic problems based on space-time discretization approach

In this paper, a novel adaptive isogeometric analysis (IGA) is introduced and its application in the numerical solution of two-dimensional elastodynamic problems based on the space-time discretization (STD) approach is studied. In the STD approach, the time is considered as an additional dimension and is discretized the same as the spatial domain. The weights of control points play the main role in the proposed method. In the conventional IGA, the same set of weights is used in the modeling of geometric and solution spaces. The idea is to define two groups of weights: geometric and solution weights. Geometric weights are known and can be determined based on the position of control points, but the solution weights are considered to be unknown and can be determined using a proper strategy so that the accuracy of the solution is optimized. This strategy is based on the minimization of an error function. The results obtained from the proposed method are compared with those obtained from the conventional IGA.     

An adaptive quadrature-based factorization method for inverse acoustic scattering problems

We propose a novel adaptive algorithm, which generates nonuniform sampling points that automatically concentrate near the boundary of an unknown scatterer, to dramatically speed up Kirsch’s factorization method for inverse acoustic scattering problems. Built upon the widely used adaptive Simpson quadrature method, our proposed adaptive algorithm approximates the integral of an indicator function over the search domain and yields reliable and accurate reconstructions significantly faster than the standard factorization method. Numerical experiments are performed to validate the effectiveness of our proposed algorithm and make comparisons with the established multilevel linear sampling method.     

Basis mapping methods for forward and inverse problems

This paper describes a novel method for mapping between basis representation of a field variable over a domain in the context of numerical modelling and inverse problems. In the numerical solution of inverse problems, a continuous scalar or vector field over a domain may be represented in different finite‐dimensional basis approximations, such as an unstructured mesh basis for the numerical solution of the forward problem, and a regular grid basis for the representation of the solution of the inverse problem. Mapping between the basis representations is generally lossy, and the objective of the mapping procedure is to minimise the errors incurred. We present in this paper a novel mapping mechanism that is based on a minimisation of the L² or H¹ norm of the difference between the two basis representations. We provide examples of mapping in 2D and 3D problems, between an unstructured mesh basis representative of an FEM approximation, and different types of structured basis including piecewise constant and linear pixel basis, and blob basis as a representation of the inverse basis. A comparison with results from a simple sampling‐based mapping algorithm shows the superior performance of the method proposed here. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

First-order and second-order adjoint methods for parameter identification problems with an application to the elasticity imaging inverse problem

We devise a fast and reliable computational framework for the elasticity imaging inverse problem of detecting cancerous tumors in the human body using an output least-squares (OLS) approach. From a mathematical standpoint, this inverse problem requires identifying a parameter in a mixed variational problem. We develop, in a continuous setting, a first-order adjoint method and two second-order adjoint methods. The continuous formulae are then used to devise a scheme for an efficient computation of the gradient and the Hessian of the OLS objective. We give detailed numerical examples.     

A hybrid regularization method for inverse heat conduction problems

This paper presents a hybrid regularization method for solving inverse heat conduction problems. The method uses future temperatures and past fluxes to reduce the sensitivity to temperature noise. A straightforward comparison technique is suggested to find the optimal number of the future temperatures. Also, an eigenvalue reduction technique is used to further improve the accuracy of the inverse solution. The method provides a physical insight into the inverse problems under study. The insight indicates that the inverse algorithm is a general purpose algorithm and applicable to various numerical methods (although our development was based on FEM), and that the inverse solutions can be obtained by directly extending Stolz's equation in the least‐squares error (LSE) sense. Direct extension of the present method to the inverse internal heat generation problems is made. Four numerical examples are given to validate the method. The effects of the future temperatures, the past fluxes, the eigenvalue reduction, the varying number of future temperatures and local iterations for non‐linear problems are studied. Copyright © 2005 John Wiley & Sons, Ltd.     

求解病态问题的一种新的正则化子与正则化算法

根据紧算子的奇异系统理论,提出了一种新的正则化子,进而建立了一类新的求解病态问题的正则化方法。证明了正则解的收敛性并得到了其最优的渐近收敛阶,数值算例说明文中建立的正则化算法是可行而有效的。     

Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems

We present a model reduction approach to the solution of large‐scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non‐linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non‐linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient‐function approximation. The resulting model reduction methodology is applied to a highly non‐linear combustion problem governed by an advection–diffusion‐reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non‐linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three‐dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full‐order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright © 2009 John Wiley & Sons, Ltd.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

Computational framework for the BIE solution to inverse scattering problems in elastodynamics

The focus of this paper is a computational platform for the non-intrusive, active seismic imaging of subterranean openings by means of an elastodynamic boundary integral equation (BIE) method. On simulating the ground response to steady-state seismic excitation as that of a uniform, semi-infinite elastic solid, solution to the 3D inverse scattering problem is contrived as a task of minimizing the misfit between experimental observations and BIE predictions of the surface ground motion. The forward elastodynamic solution revolves around the use of the half-space Greens functions, which analytically incorporate the traction-free boundary condition at the ground surface and thus allow the discretization and imaging effort to be focused on the surface of a hidden cavity. For a rigorous approach to the gradient-based minimization employed to resolve the cavity, sensitivities of the trial boundary element model with respect to (geometric) void parameters are evaluated using an adjoint field approach. Details of the computational treatment, including the regularized (i.e. Cauchy principal value-free) boundary integral equations for the primary and adjoint problem, the necessary evaluation of surface displacement gradients and their implementation into a parallel code, are highlighted. Through a suite of numerical examples involving the identification of an ellipsoidal cavity, a parametric study is presented which illustrates the importance of several key parameters on the imaging procedure including the prior information, measurement noise, and the amount of experimental input. The support provided by the National Science Foundation through CAREER Award No.CMS-9875495 to B. Guzina and the University of Minnesota Supercomputing Institute during the course of this investigation is gratefully acknowledged. Special thanks are due to MTS Systems Corporation for providing the opportunity for M. Bonnet to visit the University of Minnesota through the MTS Visiting Professorship of Geomechanics.     

Analysis of ultrasonic wave scattering for characterization of defects in solids: I. Spherical inclusions and reciprocity

This paper, as part of a series on elastic wave scattering, presents results of measurements and calculations on scattering of ultrasonic waves by a solid spherical inclusion (tungsten carbide) embedded in titanium alloy by the diffusion bonding process. Both direct scattering and mode-converted scattering angular distributions are reported for shear and compressional incident waves. The consequences upon the signals when transmitter and receiver were interchanged are explored in a reciprocity rule.     

Iteratively-improved Robin boundary conditions for the finite element solution of scattering problems in unbounded domains

An iterative procedure is described for the finite-element solution of scalar scattering problems in unbounded domains. The scattering objects may have multiple connectivity, may be of different materials or with different boundary conditions. A fictitious boundary enclosing all the objects involved is introduced. An appropriate Robin (mixed) condition is initially guessed on this boundary and is iteratively improved making use of Green's formula. It will be seen that the best choice for the Robin boundary condition is an absorbing-like one. A theorem about the theoretical convergence of the procedure is demonstrated. An analytical study of the special case of a circular cylindrical scatterer is made. Comparisons are made with other methods. Some numerical examples are provided in order to illustrate and validate the procedure and to show its applicability whatever the frequency of the incident wave. Although particular emphasis is laid in the paper on electromagnetic problems, the procedure is fully applicable to other kinds of physical phenomena such as acoustic ones. © 1998 John Wiley & Sons, Ltd.     

A numerical reconstruction method in inverse elastic scattering

In this paper a new numerical method for the shape reconstruction of obstacles in elastic scattering is proposed. Initially, the direct scattering problem for a rigid body and the mathematical setting for the corresponding inverse one are presented. Inverse uniqueness issues for the general case of mixed boundary conditions on the boundary of our obstacle, which are valid for a rigid body as well are established. The inversion algorithm based on the factorization method is presented into a suitable form and a new numerical scheme for the reconstruction of the shape of the scatterer, using far-field measurements, is given. In particular, an efficient Tikhonov parameter choice technique, called Improved Maximum Product Criterion (IMPC) and its linchpin within the framework of the factorization method is exploited. Our regularization parameter is computed via a fast iterative algorithm which requires no a priori knowledge of the noise level in the far-field data. Finally, the effectiveness of IMPC is illustrated with various numerical examples involving a kite, an acorn, and a peanut-shaped object.     

An inverse problem technique for detecting irregular cavities in circular cylinders by infrared scanning

An inverse problem technique has been developed for detecting irregular cavities in circular cylinders. In this method, the cavity is considered a part of the unknown geometry of the investigated system, and the evaluated temperature is used to locate this geometry. An auxiliary problem is introduced in the solution of this problem; and in the solution, the cavity wall is located by forcing the temperature to satisfy the condition imposed at the cavity. The new methodology is validated by an experiment presented in this paper, and the test results indicate that this method is highly successful in locating cavities. The accuracy of the method is closely related to the accuracy of the temperature that can be measured at the surface. A small error in the surface temperature results in a slight cavity error for deep cavities, while a shallow cavity is not severely affected by a surface temperature error. This method is particularly attractive in detecting shallow cavities in nondestructive evaluation.     

Hermite spectral method for solving inverse heat source problems in multiple dimensions

In this paper, we in multiple dimensions. We present a stability estimate for determining the source term in the multiple dimensional heat equation in an unbounded domain, and the regularization parameter is chosen by a discrepancy principle. Error estimate between the exact solution and its regularization solution is given. Numerical experiments for the one-dimension and two-dimension cases show the effectiveness of the method.     

Infrared scanning thermography for a quantitative detection of cavities in a plane slab and a rectangular prism

An approach for treating nondestructive testing as the solution of inverse problems in mathematical physics has been used for the detection of cavities. The approach is developed based on the use of an additional boundary condition of scanned temperature on the surface to solve for the cavity geometry. For the present study, the condition at the cavity side is taken to be that of a specified temperature, and the experiment is carried out to meet this condition. Two specimens are tested in this paper, a plane slab and a rectangular prism. In both bodies the cavity is rectangular in shape. For the testing of the plane slab, the method is able to detect the cavity wall with high accuracy, whereas the cavity depth error is larger (6%). The detection of the cavity position in the rectangular prism has an error ranging from –9.7 to 7.7%. Errors in the experiment are attributed to the uncertainties in the measurements of temperature and the Biot number. The former is read off from the analog data output of the infrared scanner. The latter is not measured separately, but is computed from the scanned data and thus becomes a portion of the total nondestructive testing output. A final note is also made in this paper to relate how the presented method can be used in actual practice.     

Mathematical analysis and solution methodology for an inverse spectral problem arising in the design of optical waveguides

We analyse mathematically the problem of determining refractive index profiles from some desired/measured guided waves propagating in optical fibres. We establish the uniqueness of the solution of this inverse spectral problem assuming that only one guided mode is known. We then propose an iterative computational procedure for solving numerically the considered inverse spectral problem. Numerical results are presented to illustrate the potential of the proposed regularized Newton algorithm to efficiently and accurately retrieve the refractive index profiles even when the guided mode measurements are highly noisy.