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.     

Combining topological sensitivity and genetic algorithms for identification inverse problems in anisotropic materials

The identification inverse problem is solved here for flaw detection in anisotropic materials by means of an innovative approach: the combination of Genetic Algorithm and the Topological Sensitivity in anisotropic elasticity. The Topological Sensitivity provides a measure of the susceptibility of a defect being at a given location. This is based on a linearized topological expansion, applying Boundary Integral Equations and using solely information of the non-damaged state. It is proved that the Topological Sensitivity provides an accurate tool for estimating the location and size of defects. First, it is shown that the minimum of the residual (cost function) topological sensitivity pinpoints the location and size of the actual flaws, and secondly, the minimization of the residual topological sensitivity is carried out using Genetic Algorithm. When the Genetic Algorithm is applied to the residual Topological Sensitivity instead of to the full residual, the applicability of this method is enhanced since the computational effort, which is the major drawback of this type of search methods, is drastically reduced. In this paper, the formulation for linearly anisotropic elastic media is composed for the case of circular flaws, although the procedure is extensible to other kinds of defects like elliptical cavities, elastic or rigid inclusions or cracks.     

Solution of inverse coefficient problems by the regularization method using spline functions

The problem of determining the unknown coefficient in an equation of conservation of matter is discussed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 2, pp. 332–337, February, 1978.     

Crack and flaw identification in elastodynamics using Kalman filter techniques

Filter-driven optimization based on the extended Kalman filter concept is used here for the numerical solution of crack and flaw identification problems in elastodynamics. The mechanical modeling of the studied two-dimensional problem, which includes the effect of unilateral contact along the sides of the crack, is done with the help of the boundary element method. The effect of various dynamical test loads and the applicability of this method for crack and defect identification in disks are investigated. The work has been supported by the German Research Foundation (DFG). Partial support has been provided by a Greek-German Research Cooperation Grant (IKYDA2001). This support is greatfully acknowledged.     

Solution of multi-parameter inverse problems of heat conduction

We develop methods and present results for the solution of inverse problems of heat conduction in connection with the simultaneous determination of thermophysical characteristics and heat exchange parameters.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 60, No. 1, pp. 136–144, January, 1991.     

Damage identification using inverse methods

This paper gives an overview of the use of inverse methods in damage detection and location, using measured vibration data. Inverse problems require the use of a model and the identification of uncertain parameters of this model. Damage is often local in nature and although the effect of the loss of stiffness may require only a small number of parameters, the lack of knowledge of the location means that a large number of candidate parameters must be included. This paper discusses a number of problems that exist with this approach to health monitoring, including modelling error, environmental effects, damage localization and regularization.     

Solution of linear boundary inverse problems of heat conduction

New methods of solving linear one-dimensional boundary inverse problems of heat conduction are proposed; the methods are convenient for practical realization.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 3, pp. 529–535, March, 1978.     

Solution of inverse heat-conduction problems on specialized analog computers

Recommendations on the application of specialized analog computers for the solution of inverse problems of heat conduction are given. The presence of a zone of sensitivity delimiting the possible location of a primary information source is established.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 6, pp. 1123–1130, December, 1977.     

Solution of inverse problems of the mechanics of reactive media

The general inverse problem of heat and mass transfer in a porous reactive material is reduced to a set of particular problems by using splitting of the problem according to chemical and physical processes. A brief exposition is given of the methods for solving these particular inverse problems.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 3, pp. 459–464, March, 1989.     

Solution of integral equations of inverse problems of heat conduction

A solution is given of integral equations of inverse problems of heat conduction by the method of successive approximations and also by means of expansions in orthogonal systems of functions. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 29, No. 1, pp. 120–123, July, 1975.     

Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

This article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional J is expanded in powers of the characteristic size a of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading O(a²) approximation of J, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the O(a⁴) approximation of J. Simpler and more explicit versions of this formulation are obtained for a centrally symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the O(a⁴) expansion of J over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the O(a⁴) approximation of J and the subsequent numerical experiments.     

A numerical solution of axisymmetric problems in elastodynamics

The equations governing the axisymmetric dynamic deformation of an elastic solid are considered as a symmetric hyperbolic system of linear first-order partial differential equations. The characteristic properties of the system are determined and a numerical method for obtaining the solution of mixed initial and boundary value problems in elastodynamics is presented. Two examples are considered.     

Solution of inverse heat transfer problems by a two-model method

The problem of selecting exact and approximate models of heat transfer is analyzed for the solution of inverse problems.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 3, pp. 500–505, September, 1989.     

Multi-domain BEM for two-dimensional problems of elastodynamics

An advanced implementation of the boundary element technique for the periodic and transient dynamic analyses of two-dimensional elastic or visco elastic solids of arbitrary shape and connectivity is presented. For transient dynamic analysis the problem is first solved in the Laplace transform space and then the time domain solutions are obtained by numerical inversion of transformed domain solutions. The present analysis is capable of treating very large, multi-domain problems by substructuring and satisfying the equilibrium and compatibilities at the interfaces. With the help of this substructuring capability, problems related to the layered media and soil–structure interaction can all be analysed. This paper also introduces a new type of element called ‘Enclosing Element’, which has been developed and used to model the infinitely extending boundaries of a half-space or a layered medium. A number of numerical examples are presented, and through comparisons with available analytical and numerical results, the accuracy, stability and efficiency of the present analysis are established.     

Analysis of elastostatic problems using a semi-analytical method with diagonal coefficient matrices

A new semi-analytical method is proposed for solving boundary value problems of two-dimensional elastic solids, in this paper. To this end, the boundary of the problem domain is discretized by specific non-isoparametric elements that are proposed for the first time in this research. These new elements employ higher-order Chebyshev mapping functions and new special shape functions. For these shape functions, Kronecker Delta property is satisfied for displacement function and its derivatives. Furthermore, the first derivatives of shape functions are assigned to zero at any given control point. Eventually, implementing a weak form of weighted residual method and using Clenshaw–Curtis quadrature, coefficient matrices of equations system become diagonal, which results in a set of decoupled governing equations to be used for solving the whole system. In other words, the governing equation for each degree of freedom (DOF) is independent from other DOFs of the problem. Validity and accuracy of the present method are fully demonstrated through four benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.     

Nonsingular traction BIEs for crack problems in elastodynamics

The nonsingular traction BIEs are derived for the Laplace transforms in elastodynamic crack problems. Two different forms of the final nonsingular traction BIEs are received with respect to the leading singularity of the integral kernels involved. In the first one, the traction BIE is derived from the integral representation of stresses which involves hypersingular kernels. In the second way, the partially regularized integral representation of stresses with strongly singular kernels is used as a starting point. The singular integrals are transformed to nonsingular ones by making use of the Stokes theorem.     

Dual boundary element formulation for inverse potential problems in crack identification

In this paper a new boundary element formulation is presented for the identification of the location and size of internal cracks in two dimensional structures. The method is presented, as a supplement to the experimental non-destructive testing (NDT) methods, for more accuracy in the identification procedure. The identification method is presented, proposing the dual boundary element method (DBEM) as the basis for design sensitivity computation. Examples are presented to demonstrate the performance of the crack identification method for various cracks.     

Solution of inverse problems with an unknown model of the process

A method is proposed for identifying systems with distributed parameters in the case of no a priori information about the form of an adequate model of the process being studied.