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.     

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.     

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.     

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.     

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.     

An inverse problem in simultaneously estimating boundary moisture fluxes in a porous annular cylinder

Summary Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a porous annular cylinder. While knowing the moisture history at the measuring positions, the unknown time-dependent inner-and-outer boundary moisture fluxes can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown moisture fluxes. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent boundary moisture fluxes can be obtained with any arbitrary initial guesses. Moreover, the methodology presented in this paper can also be used to calculate the cutting forces in nanomachining by atomic force microscopy (AFM), and to determine the heat sources in an X-ray lithographic process.     

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.     

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.     

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.     

Flaw identification using the boundary element method

In this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two-dimensional structures. An introduction to inverse analysis is given, with special reference to methods of flaw identification, along with a brief review of the optimization methods employed. Both the standard boundary element and the dual boundary element method are presented, with the dual boundary element method proposed as the basis for the new formulation. The flaw identification method is presented, along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries. Examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location.     

A Lie-Group Adaptive Method for Imaging a Space-Dependent Rigidity Coefficient in an Inverse Scattering Problem of Wave Propagation

We are concerned with the reconstruction of an unknown space-dependent rigidity coefficient in a wave equation. This problem is known as one of the inverse scattering problems. Based on a two-point Lie-group equation we develop a Lie-group adaptive method (LGAM) to solve this inverse scattering problem through iterations, which possesses a special character that by using onlytwo boundary conditions and two initial conditions, as those used in the direct problem, we can effectively reconstruct the unknown rigidity function by aself-adaption between the local in time differential governing equation and the global in time algebraic Lie-group equation. The accuracy and efficiency of the present LGAM are assessed by comparing the imaged results with some postulated exact solutions. By means of LGAM, it is quite versatile to handle the wave inverse scattering problem for the image of the rigidity coefficient without needing any extra information from the wave motion.     

Inverse problem of simultaneously estimating the thermal conductivity and boundary shape

An inverse analysis is used to simultaneously estimate the thermal conductivity and the boundary shape in steady-state heat conduction problems. The numerical scheme consists of a body-fitted grid generation technique to mesh the heat conducting body and solve the heat conduction equation – a novel, efficient, and easy to implement sensitivity analysis scheme to compute the sensitivity coefficients, and the conjugate gradient method as an optimization method to minimize the mismatch between the computed temperature distribution on some part of the body boundary and the measured temperatures. Using the proposed scheme, all sensitivity coefficients can be obtained in one solution of the direct heat conduction problem, irrespective of the large number of unknown parameters for the boundary shape. The obtained results reveal the accuracy, efficiency, and robustness of the proposed algorithm.     

The boundary element method applied to a moving free boundary problem

In this paper a boundary problem is considered for which the boundary is to be determined as part of the solution. A time‐dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed. The fundamental solution is introduced and Green’s Theorem is used to yield a non‐linear system of integral equations for the unknown solution and the location of the boundary. The boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem. Graphical results for a two‐dimensional problem are presented. Published in 1999 by John Wiley & Sons, Ltd.     

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.     

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.     

S32750双相不锈钢相界与晶界特征对其力学性能和耐蚀性能的影响EI北大核心CSCD

通过固溶处理获得不同初始组织状态的S32750双相不锈钢样品,然后进行厚度压下量80%的冷轧变形和1050℃的退火处理,采用SEM-EBSD和XRD技术研究合金相界与晶界特征以及相组成分布情况,并利用拉伸实验、纳米压痕和双环电化学动电位再活化法(DL-EPR)分析不同初始状态样品的组织对力学性能与耐晶间腐蚀性能的影响规律。结果表明:高温固溶处理的合金样品经冷轧退火后晶粒细小均匀,两相分布接近1∶1,且相界占内界面(晶界+相界)比例较高,同相晶粒团簇程度最低,表现出优异的综合力学性能。合金样品经敏化处理后,σ相易沿α相晶界析出,高温固溶并经轧制退火后的样品中,由于α晶界比例较少且满足K-S取向关系的相界比例较高则又表现出良好的晶间腐蚀抗力。因此,通过适当的工艺来调控合金的相界与晶界分布可以实现材料强度和晶间腐蚀抗力的同步改善。     

A hybrid boundary node method

A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.     

A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.     

A homotopy method for bioluminescence tomography

Bioluminescence tomography (BLT) aims at the determination of the distribution of a bioluminescent source quantitatively. The mathematical problem involved is an inverse source problem and is ill-posed. With the Tikhonov regularization, an optimization problem is formed for the light source reconstruction and it is usually solved by gradient-type methods. However, such iterative methods are often locally convergent and thus the solution accuracy depends largely on initial guesses. In this paper, we reformulate the reduced regularized optimal problem as a nonlinear equation and apply a homotopy method, which is a powerful tool for solving nonlinear problem due to its globally convergent property, to it. Numerical experiments show that the application of the homotopy technique is feasible and can produce satisfactory approximate solutions for a very large range of initial guesses.