Material characterization and interfacial boundary identification by solving an inverse elasto statics boundary elements problem

An inverse geometry problem of identifying simultaneously two irregular interfacial boundaries along with the mechanical properties of the interface domain located between the components of multiple (three) connected regions is investigated. A discrete number of displacement measurements obtained from a uniaxial tension test are used as extra information to solve this inverse problem. A unique combination of global and local optimization method is used, that is, the imperialist competitive algorithm (ICA) to find the best initial guesses of the unknown parameters to be used by the local optimization methods, that is, the conjugate gradient method (CGM) and the simplex method (SM). The CGM and SM are used in series. The performance of these local optimization methods is dependents on the initial guesses of the unknown boundaries and the mechanical properties, that is, Poisson’s ratio and Young’s modulus, so ICA provides the best initial guesses. The boundary elements method is employed to solve the direct two-dimensional (2D) elastostatics problem. A fitness function, which is the summation of squared differences between measured and computed displacements at identical locations on the exterior boundary, is minimized. Several example problems are solved and the accuracy of the obtained results is discussed. The influence of the value of the material properties of the subregions and the effect of measurement errors on the estimation process are also addressed.     

An iterative regularization method for inverse phonon radiative transport problem in nanoscale thin films

An inverse phonon radiative transport problem with an alternative form of adjoint equation is solved in this study by using conjugate gradient method (CGM) to estimate the unknown boundary temperature distributions, based on the 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 temperature (or phonon intensity) distributions are to be determined. Finally, it is shown that accurate boundary temperatures can always be obtained with CGM. Copyright © 2006 John Wiley & Sons, Ltd.     

Hybrid FE–IDQ–CG method for dynamic parameters estimation of multilayered half-space

As a first endeavor, a hybrid finite element (FE)–incremental differential quadrature (IDQ) method together with the discrepancy principle and the conjugate gradient method (CGM) is used to develop an inverse algorithm for the parameters estimation of the axisymmetric multilayered half-spaces. The approach is based on the measurement of the dynamic transverse displacement at some boundary points of the half-space to estimate the unknown parameters of its layers. Using the accuracy and unconditional stability of the hybrid FE–IDQ method, the direct problem is solved to get the dynamic transverse displacements. After adding some random errors to the obtained results, they are considered as the measured responses by sensors. Then, the conjugate gradient method as a general and robustness optimization technique is employed to minimize the error between the measured and calculated dynamic surface responses at sensor locations. The sensitivity analysis of the displacement field is performed using a semi-analytical method. The applicability and correctness of the proposed hybrid algorithm is demonstrated through different examples by considering the influence of the layers arrangement, the measurement errors and sensor numbers.     

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.     

Estimation of heat flux and mechanical loads on laminated functionally graded plate

The spatially and temporally varying heat flux and mechanical load on the top (bottom) surface of laminated plates with functionally graded layers are estimated using an inverse algorithm. The temperature and strains at a number of points on the bottom (top) surface of the plate are the only measured input data. The solution of corresponding direct problem is used to simulate the measured temperatures and strains, which are obtained based on a three-dimensional layerwise thermoelastic analysis. The conjugate gradient method as a powerful technique for optimization in conjunction with the discrepancy principle is employed to develop the inverse solution procedure. A semi-analytical approach composed of the layerwise-differential quadrature method and series solution is adopted to discretize the governing differential equations subjected to the related boundary and initial conditions. The influence of measurement errors on the accuracy of the estimated heat flux and mechanical load is investigated. The good accuracy of the results validates the presented inverse approach.     

Solution of identification inverse problems in elastodynamics using semi-analytical sensitivity computation

The purpose of this work is to study a class of inverse problems that arises in solid mechanics areas such as quantitative non-destructive testing (QNDT) or shape optimization. The technique is based on the boundary integral equations (BIEs) used in the classical boundary element method (BEM), which are differentiated semi-analytically with respect to variations of the boundary geometry and used in an iterative search algorithm. The extension of this strategy is presented here for the case of elasticity in dynamics using the displacement or singular BIE, which allows to apply this strategy to QNDT problems based on vibrations or ultrasonics.The central point is the evaluation of the capability of solving numerically a QNDT problem such as the location and characterization of cavity and inclusion-type defects by measuring the dynamic response at an accessible boundary of the specimen. To test this capability, comprehensive convergence tests are made for the badness of the initial guess, the amount of supplied measurements, and simulated errors on measurements, geometry, elastic constants and frequency.     

Solving initial and boundary value problems using learning automata particle swarm optimization

In this article, the particle swarm optimization (PSO) algorithm is modified to use the learning automata (LA) technique for solving initial and boundary value problems. A constrained problem is converted into an unconstrained problem using a penalty method to define an appropriate fitness function, which is optimized using the LA-PSO method. This method analyses a large number of candidate solutions of the unconstrained problem with the LA-PSO algorithm to minimize an error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. This approach is very capable of solving linear and nonlinear ODEs, systems of ordinary differential equations, and linear and nonlinear PDEs. The computational efficiency and accuracy of the PSO algorithm combined with the LA technique for solving initial and boundary value problems were improved. Numerical results demonstrate the high accuracy and efficiency of the proposed method.     

An inverse method for determining elastic material properties and a material interface

A numerical procedure which integrates optimization, finite element analysis and automatic finite element mesh generation is developed for solving a two-dimensional inverse/parameter estimation problem in solid mechanics. The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix. Traction and displacement boundary conditions sufficient for solving a direct problem are applied to the boundary of the domain. In addition, displacements are measured at discrete points on the part of the boundary where the tractions are prescribed. The inverse problem is solved using a modified Levenberg-Marquardt method to match the measured displacements to a finite element model solution which depends on the unknown parameters. Numerical experiments are presented to show how different factors in the problem and the solution procedure influence the accuracy of the estimated parameters.     

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.     

An iterative regularization method in estimating the unknown energy source by laser pulses with a dual-phase-lag model

In the present study an inverse problem for hyperbolic heat conduction with a dual-phase-lag model is solved by the conjugate gradient method (CGM) in estimating the unknown heat generation, due to the ultra-short duration laser heating, based on the interior temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where two different heat source distributions are to be estimated. Results show that the inverse solutions can always be obtained when choosing the initial guesses of the heat sources equal to zero. Finally, it is concluded that accurate heat sources can be estimated in this study. Copyright © 2008 John Wiley & Sons, Ltd.     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.     

Stochastic optimization using a sparse grid collocation scheme

In computational sciences, optimization problems are frequently encountered in solving inverse problems for computing system parameters based on data measurements at specific sensor locations, or to perform design of system parameters. This task becomes increasingly complicated in the presence of uncertainties in boundary conditions or material properties. The task of computing the optimal probability density function (PDF) of parameters based on measurements of physical fields of interest in the form of a PDF, is posed as a stochastic optimization problem. This stochastic optimization problem is solved by dividing it into two problems—an auxiliary optimization problem to construct stochastic space representations from the PDF of measurement data, and a stochastic optimization problem to compute the PDF of problem parameters. The auxiliary optimization problem is solved using a downhill simplex method, whilst a gradient based approach is employed for solving the stochastic optimization problem. The gradients required for stochastic optimization are defined, using appropriate stochastic sensitivity problems. A computationally efficient sparse grid collocation scheme is utilized to compute the solution of these stochastic sensitivity problems. The implementation discussed, requires minimum intrusion into existing deterministic solvers, and it is thus applicable to a variety of problems. Numerical examples involving stochastic inverse heat conduction problems, contamination source identification problems and large deformation robust design problems are discussed.     

Identification of boundary displacements in plane elasticity by BEM

The purpose of this study is to present a possible application of BEM for numerical identification of the boundary conditions for Navier equations in plane elasticity with internal measurements, based on insufficient and noisy information for unique identification. The inverse problem is re-formulated as a minimization problem by the direct variational method. The minimization problem is then recast using the gradient method into successive primary and adjoint boundary value problems in the corresponding plane elasticity problem. For numerical solution of the elasticity problems, the conventional direct boundary element method is employed. From the simple numerical examples considered, it is concluded that our identification scheme is stable and the approximate solutions are convergent to the minimum.     

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.     

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

For an isotropic linear elastic body, only displacement or traction boundary conditions are given on a part of its boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. The inverse problem is different from the Cauchy problems. All the unknown boundary conditions on the whole boundary must be determined with some interior points' information. The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions, and the PCGM is compared with the conjugate gradient method (CGM). Morozov's discrepancy principle is employed to select the iteration step. The analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical solutions of the boundary conditions are not sensitive to the locations of the interior points if these points are distributed along the entire boundary of the considered domain. The numerical results confirm that the PCGM and CGM produce convergent and stable numerical solutions with respect to increasing the number of interior points and decreasing the amount of noise added into the input data.     

A non‐linear inverse vibration problem of estimating the time‐dependent stiffness coefficients by conjugate gradient method

An iterative regularization method, i.e. the conjugate gradient method (CGM) is applied to an inverse non‐linear force vibration problem to estimate the unknown time‐dependent stiffness coefficients (or spring constants) in a damped system by using the measured system displacement. It is assumed that no prior information is available on the functional form of the unknown stiffness coefficients in the present study, thus, it is classified as the function estimation in inverse calculation. The accuracy of the inverse analysis is examined by using the simulated exact and inexact displacement measurements. The numerical simulations are performed to test the validity of the present algorithm by using different types of stiffness coefficients and measurement errors. Results show that an excellent estimation on the time‐dependent spring constants can be obtained with any arbitrary initial guesses within a couple of seconds of CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

Automated algorithm for the nondestructive detection of subsurface cavities by the IR-CAT method

An algorithm is presented to automate the detection of irregular-shaped subsurface cavities within irregular shaped bodies by the IR-CAT method. The algorithm is based on the solution of an inverse geometric steady state heat conduction problem. Cauchy boundary conditions are prescribed at the exposed surface. An inverse heat conduction problem is formulated by specifying the thermal boundary condition along the inner cavities whose unknown geometries are to be determined. An initial guess is made for the location of the inner cavities. The domain boundaries are discretized, and an Anchored Grid Pattern (AGP) is established. The nodes of the inner cavities are constrained to move along the AGP at each iterative step. The location of inner cavities is determined by using the Newton Raphson method with a Broyden update to drive the error between the imposed boundary conditions and computed boundary conditions to zero. During the iterative procedure, the movement of the inner cavity walls is restricted to physically realistic intermediate solutions. A dynamic relocation of the AGP is introduced in the Traveling Hole Method to adaptively refine the detection of inner cavities. The proposed algorithm is general and can be used to detect multiple cavities. Results are presented for the detection of single and multiple irregular shaped cavities. Convergence under grid refinement is demonstrated.     

Electromagnetic transverse electric-wave inverse scattering of a conductor by the genetic algorithm

The genetic algorithm is used to reconstruct the shape of a perfectly conducting cylinder illuminated by transverse electric (TE) waves. A cylinder of unknown shape scatters the incident TE wave in a free space and the scattered field is recorded outside. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is then employed to find out the global extreme solution of the cost function. Numerical results demonstrated that the genetic algorithm can tackle the inverse problem of a larger scatterer. Even when the electrical dimension of the scatterer exceeds one wavelength and the initial guess is far from the exact one, good reconstruction was obtained. In such a case, gradient-based methods often get stuck in a local extreme. In addition, the effect of Gaussian noise on the reconstruction results is investigated. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 388–394, 1998     

An inverse problem in estimating interfacial cracks in bimaterials by boundary element technique

An inverse elasticity problem by utilizing both the regularization method (RM) and the conjugate gradient method (CGM) is presented for estimating the interfacial cracks (including location and shape) of a bimaterial from the measurement of displacements at discrete locations internal to the domain and parallel to the interface. The present algorithm in determining the interfacial cracks is totally different from the conventional one. The comparisons of using the conjugate gradient method and commonly used regularization method are discussed systematically, moreover, the advantages and disadvantages in applying the large matrix (LM) and small matrix (SM) formulations are also examined. To the author's knowledge the present work is the first of its kind. Finally, the effects of the measurement errors on the inverse solutions are discussed. Results show that the present inverse algorithms are not sensitive to measurement errors. The CGM is recommended because it is straightforward, LM formulation is better than SM formulation without the consideration of computer time. Copyright © 1999 John Wiley & Sons, Ltd.