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.     

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.     

A Novel Approach to Identify the Thermal Conductivities of a Thin Anisotropic Medium by the Boundary Element Method

A common difficulty arises in characterizing the anisotropic properties of a thin sheet of anisotropic material, especially in the transverse direction. This difficulty is even more phenomenal for measuring its mechanical properties on account of its thickness. As the prelude of such investigation, this paper proposes a novel approach to identify the thermal conductivities of an unknown thin layer of anisotropic material. For this purpose, the unknown layer is sandwiched in isotropic materials with known conductivities. Prescribing proper boundary conditions, one may easily measure temperature data on a few sample boundary points. Therefore, the anisotropic thermal conductivities can be calculated inversely. For the inverse analysis, the boundary element method (BEM) is employed to combine with the conjugate gradient method (CGM). For verifying our analysis, numerical experiments were carried out. The obtained results have shown great computational efficiency and accuracy in identifying the thermal conductivities of the thin anisotropic layer.     

A boundary element-based inverse-problem in estimating transient boundary conditions with conjugate gradient method

A Boundary Element Method (BEM)-based inverse algorithm utilizing the iterative regularization method, i.e. the conjugate gradient method (CGM), is used to solve the Inverse Heat Conduction Problem (IHCP) of estimating the unknown transient boundary temperatures in a multi-dimensional domain with arbitrary geometry. The results obtained by the CGM are compared with that obtained by the standard Regularization Method (RM). The error estimation based on the statistical analysis is derived from the formulation of the RM. A 99 per cent confidence bound is thus obtained. Finally, the effects of the measurement errors to the inverse solutions are discussed. Results show that the advantages of applying the CGM in the inverse calculations lie in that (i) the major difficulties in choosing a suitable quadratic norm, determining a proper regularization order and determining the optimal smoothing (or regularization) coefficient in the RM are avoided and (ii) it is less sensitive to the measurement errors, i.e. more accurate solutions are obtained. © 1998 John Wiley & Sons, Ltd.     

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.     

桥梁移动荷载识别的不适定性及其试验研究

对桥梁移动荷载识别方程不适定问题进行研究，提出采用预处理共轭梯度法（PCGM）求解超定方程组，通过选择不同的预优矩阵，改善和解决超定方程组的欠秩和病态问题。为验证基于PCGM方法的现场实用性，设计制作了车桥试验模型，通过试验采集到的桥梁弯矩响应数据识别桥面移动荷载。比较桥梁模态数、预处理共轭梯度法迭代次数、桥面粗糙度、车辆重量以及测点选择对识别结果精度的影响后，研究结果表明：基于PCGM方法能够很好地识别车辆荷载，收敛较快且能较好改善荷载识别方程的不适定性。     

INTERELEMENT STRESS EVALUATION BY BOUNDARY ELEMENTS

The conventional boundary element method employs piecewise shape functions which lead to stress discontinuity at the interelement boundary. This paper derives formulae for boundary stress and boundary stress gradient based on boundary element solutions, and discusses error propagation in stress evaluation due to errors in the displacement boundary values. The nature of stress discontinuity is investigated. A simple post-processing scheme is presented using continuities of stress and stress gradient along a traction boundary as two extra conditions, so that a high-order shape function can be employed for the evaluation of stress and stress gradient on the interelement boundary. Four numerical examples are used to demonstrate the accuracy of the proposed post-processing scheme. The numerical results show that as compared with the conventional method, the post-processing method can significantly improve the stress and stress gradient on the traction boundaries, especially in the area of stress concentration.     

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.     

Weakly singular stress‐BEM for 2D elastostatics

A weakly singular stress‐BEM is presented in which the linear state regularizing field is extended over the entire surface. The algorithm employs standard conforming C⁰ elements with Lagrangian interpolations and exclusively uses Gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space. The linear state stress‐BIE on which the algorithm is based has no free term so that the BEM treatment of external corners requires no special consideration other than to admit traction discontinuities. The self‐regularizing nature of the Somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary. A key observation is the relation between BIE density functions and successful interpolation orders. Numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations. Additionally, the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions. Copyright © 1999 John Wiley & Sons, Ltd.     

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.     

Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations

 In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem for Helmholtz-type equations is analysed. An efficient regularising stopping criterion for CGM proposed by Nemirovskii [25] is employed. The numerical results obtained confirm that the CGM + BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. Received: 5 November 2002 / Accepted: 5 March 2003 L. Marin would like to acknowledge the financial support received from the EPSRC. The authors would like to thank Professor Dinh Nho Hào and Dr. Thomas Johansson for some useful discussions and suggestions.     

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.     

Parameter estimation in heat conduction using a two-dimensional inverse analysis

This article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.     

Application of the inverse elasticity problem to identify irregular interfacial configurations

An inverse elasticity problem is solved to identify the irregular boundary between the components of a multiple connected domain using displacement measurements obtained from an uniaxial tension test. The boundary elements method (BEM) coupled with the particle swarm optimization (PSO) and conjugate gradient method (CGM) are employed. Due to the ill-posed nature of this inverse elasticity problem, and the need for an initial guess of the unknown interfacial boundary when local optimization methods are implemented, a Meta heuristic procedure based on the PSO algorithm is presented. The CGM is then employed using the best initial guess obtained by the PSO to reach convergence. This procedure is highly effective, since the computational time reduces considerably and accuracy of the results is reasonable. Several example problems are solved and the accuracy of obtained results is discussed. The influence of material properties and the effect of measurement errors on the estimation process are also addressed.     

The computation of the J integral with the traction boundary integral equation

The solutions of the displacement boundary integral equation (BIE) are not uniquely determined in certain types of boundary conditions. Traction boundary integral equations that have unique solutions in these traction and mixed boundary cases are established. For two‐dimensional linear elasticity problems, the divergence‐free property of the traction boundary integral equation is established. By applying Stokes' theorem, unknown tractions or displacements can be reduced to computation of traction integral potential functions at the boundary points. The same is true of the J integral: it is divergence‐free and the evaluation of the J integral can be inverted into the computation of the J integral potential functions at the boundary points of the cracked body. The J integral can be expressed as the linear combination of the tractions and displacements from the traction BIE on the boundary of the cracked body. Numerical integrals are not needed at all. Selected examples are presented to demonstrate the validity of the traction boundary integral and J integral. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary element analysis of thermoelastic contact problems

A boundary element method (BEM) is applied to thermoelastic contact problems where thermal resistance at the contact interface is not negligible. The displacement, traction, temperature and temperature gradient in the contact zone are unknown quantities to be determined numerically. Due to the existence of thermal resistance, temperature and stress fields are mutually coupled. To solve the problem, two kinds of methods are presented. In the first method, the solution is obtained by minimizing a suitably defined objective function. In the second method, discretized equations of each of the bodies in contact are computed alternately until all prescribed boundary conditions are satisfied. The applicability of these methods to practical problems is examined through several numerical examples.     

Numerical solutions of hypersingular integral equation for curved cracks in circular regions

In this paper, numerical solutions of a hypersingular integral equation for curved cracks in circular regions are presented. The boundary of the circular regions is assumed to be traction free or fixed. The suggested complex potential is composed of two parts, the principle part and the complementary part. The principle part can model the property of a curved crack in an infinite plate. For the case of the traction free boundary, the complementary part can compensate the traction on the circular boundary caused by the principle part. Physically, the proposed idea is similar to the image method in electrostatics. By using the crack opening displacement (COD) as the unknown function and traction as right hand term in the equation, a hypersingular integral equation for the curved crack problems in the circular regions is obtained. The equation is solved by using the curve length coordinate method. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

An optimal control problem in estimating the optimal control force for the force vibration system

An optimal control problem for the force vibration system based on the iterative regularization method, i.e. the conjugate gradient method (CGM), is examined to estimate the optimal control force in a damped system having time‐dependent system parameters such that the desire (or design) system displacements can be satisfied. It is assumed that no prior information is available on the functional form of the unknown control function in the present study, thus, it is classified as the function estimation. Numerical simulations are performed to test the validity of the present algorithm by using different types of the desire system displacements. Results show that an excellent estimation on the optimal control force can be obtained with arbitrary initial guesses within a couple of second's CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.     

Hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation

In this paper a two-dimensional hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation is presented. The proposed approach incorporates displacement and the traction boundary integral equations as well as finite deformation stress measures, and general crack problems can be solved with single-region formulations. Efficient regularization techniques are applied to the corresponding singular terms in displacement, displacement derivatives and traction boundary integral equations, according to the degree of singularity of the kernel functions. Within the numerical implementation of the hyper-singular boundary element formulation, crack tip and corners are modelled with discontinuous elements. Fracture measures are evaluated at each load increment, using the J-integral. Several cases studies with different boundary and loading conditions have been analysed. It has been shown that the new singularity removal technique and the non-linear elastoplastic formulation lead to accurate solutions.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.