The boundary element-linear complementarity method for the Signorini problem

The boundary element-linear complementarity method for solving the Laplacian Signorini problem is presented in this paper. Both Green's formula and the fundamental solution of the Laplace equation have been used to solve the boundary integral equation. By imposing the Signorini constraints of the potential and its normal derivative on the boundary, the discrete integral equation can be written into a standard linear complementarity problem (LCP). In the LCP, the unique variable to be affected by the Signorini boundary constraints is the boundary potential variable. A projected successive over-relaxation (PSOR) iterative method is employed to solve the LCP, and some numerical results are presented to illustrate the efficiency of this method.     

Frictionless contact-detachment analysis: iterative linear complementarity and quadratic programming approaches

The object of the paper concerns a consistent formulation of the classical Signorini’s theory regarding the frictionless contact problem between two elastic bodies in the hypothesis of small displacements and strains. The employment of the symmetric Galerkin boundary element method, based on boundary discrete quantities, makes it possible to distinguish two different boundary types, one in contact as the zone of potential detachment, called the real boundary, the other detached as the zone of potential contact, called the virtual boundary. The contact-detachment problem is decomposed into two sub-problems: one is purely elastic, the other regards the contact condition. Following this methodology, the contact problem, dealt with using the symmetric boundary element method, is characterized by symmetry and in sign definiteness of the matrix coefficients, thus admitting a unique solution. The solution of the frictionless contact-detachment problem can be obtained: (i) through an iterative analysis by a strategy based on a linear complementarity problem by using boundary nodal quantities as check quantities in the zones of potential contact or detachment; (ii) through a quadratic programming problem, based on a boundary min-max principle for elastic solids, expressed in terms of nodal relative displacements of the virtual boundary and nodal forces of the real one.     

An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity

The alternating iterative algorithm proposed by Kozlov et al. [An iterative method for solving the Cauchy problem for elliptic equations. USSR Comput Math Math Phys 1991;31:45–52] for obtaining approximate solutions to the Cauchy problem in two-dimensional anisotropic elasticity is analysed and numerically implemented using the boundary element method (BEM). The ill-posedness of this inverse boundary value problem is overcome by employing an efficient regularising stopping. The numerical results confirm that the iterative 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.     

A contact algorithm for the Signorini problem using space–time finite elements

The most common approach in the finite‐element modelling of continuum systems over space and time is to employ the finite‐element discretization over the spatial domain to reduce the problem to a system of ordinary differential equations in time. The desired time integration scheme can then be used to step across the so‐called time slabs, mesh configurations in which every element shares the same degree of time refinement. These techniques may become inefficient when the nature of the initial boundary value problem is such that a high degree of time refinement is required only in specific spatial regions of the mesh. Ideally one would be able to increase the time refinement only in those necessary regions. We achieve this flexibility by employing space–time elements with independent interpolation functions in both space and time. Our method is used to examine the classic contact problem of Signorini and allows us to increase the time refinement only in the spatial region adjacent to the contact interface. We also develop an interface‐tracking algorithm that tracks the contact boundary through the space–time mesh and compare our results with those of Hertz contact theory. Copyright 2004 John Wiley & Sons, Ltd.     

An iterative algorithm for singular Cauchy problems for the steady state anisotropic heat conduction equation

In this paper, we propose an alternating iterative algorithm to solve a singular Cauchy problem for the anisotropic heat conduction equation. The numerical algorithm is based on the boundary element method (BEM), modified to take into account the form of the singularity, without substantially increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in the boundary conditions and the second with a singularity caused by a sharp re-entrant corner, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, the iterative BEM is efficient in dealing with the difficulties arising from both the instabilities produced by the boundary condition formulation and the slow rate of convergence of standard numerical methods around the singular point.     

Reconstruction of compact binary images from limited Fourier amplitude data

The problem of reconstructing a binary image from undersampled Fourier amplitude data is considered. This problem maps to an image reconstruction problem in x-ray crystallography. The binary constraint is sufficient to overcome the undersampling and enforce uniqueness, but is insufficient in the case of the additional loss of data that can occur in practice. An iterative projection algorithm is developed that uses binary, connectivity, and compactness constraints to solve the image reconstruction problem. Simulations show the utility of the reconstruction algorithm.     

Use of boundary element method for the determination of tool shape in electrochemical machining

The problem of cathode shape determination for a given anode shape in electrochemical machining is considered. An algorithm based on the boundary integral equation technique and non-linear optimization for this inverse problem is developed. The additional flux condition at the anode is used as the constraint in this ill-posed problem. Through an iterative process, the shape of the cathode is determined by minimizing a functional. The algorithm is tested on two examples. It is shown that the algorithm is consistently superior compared to published numerical techniques based on the embedding method or the method of the lines.     

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.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd.     

Contact Interaction of Crack Faces under Oblique Incidence of a Harmonic Wave

The contact interaction of the opposite faces of a crack under oblique incidence of a harmonic wave is considered. The problem is solved by the method of boundary integral equations using an iterative algorithm. The contact forces and the displacement discontinuity on the crack faces are studied for different values of the friction coefficient.     

凸组合投影算法中的组合因子对算法效率的影响

在求解生产生活中各类实际问题的优化模型的算法研究中,投影梯度算法在解决凸约束最优化问题上一直被学者所重视．本文考虑凸组合投影算法求解凸约束最优化问题,在此凸组合投影算法中,由投影梯度法得到的点与上一步迭代点的凸组合得到新的迭代点．此算法不仅利用投影算法得到的点的信息而且也利用了前一步点的信息．进一步,通过数值实验分析凸组合算法的效率及凸组合因子对算法的影响．数值试验结果表明,这种凸组合算法总体比原来投影梯度法更稳定,而且这种凸组合算法在适当的凸组合因子下较投影梯度法收敛更快．     

An efficient BE iterative-solver-based substructuring algorithm for 3D time-harmonic problems in elastodynamics

This work is concerned with the development of an efficient and general algorithm to solve frequency-domain problems modelled by the boundary element method based on a sub-region technique. A specific feature of the algorithm discussed here is that the global sparse matrix of the coupled system is implicitly considered, i.e. problem quantities are not condensed into interface variables. The proposed algorithm requires that only the block matrices with non-zero complex-valued coefficients be stored and manipulated during the analysis process. In addition, the efficiency of the technique presented is improved by using iterative solvers. The good performance of pre-conditioned iterative solvers for systems of equations having real-valued coefficients, well demonstrated in the literature, is confirmed for the present case where the system matrix coefficients are complex. The efficiency of the algorithm described here is verified by analysing a soil–machine foundation interaction problem. CPU time and accuracy are the parameters used for estimating the computational efficiency.     

Inverse wave scattering in 2-D elastic solids

Summary This paper is concerned with an inverse problem for two-dimensional elastic solids. It seeks to recover the subsurface density profile based on the measurements obtained at the boundary. The method considers a temporal interval for which time dependent measurements are provided. It formulates an optimal estimation problem which seeks to minimize the error difference between the given data and the response from the system. It uses a boundary regularization term to stabilize the inversion. The method leads to an iterative algorithm which, at every iteration, requires the solution to a two-point boundary value problem. Several numerical results are presented which indicate that a close estimate of the unknown density function can be obtained based on the boundary measurements only.     

Retrieval of aerosol extinction coefficient profiles from Raman lidar data by inversion method

We regard the problem of differentiation occurring in the retrieval of aerosol extinction coefficient profiles from inelastic Raman lidar signals by searching for a stable solution of the resulting Volterra integral equation. An algorithm based on a projection method and iterative regularization together with the L-curve method has been performed on synthetic and measured lidar signals. A strategy to choose a suitable range for the integration within the framework of the retrieval of optical properties is proposed here for the first time to our knowledge. The Monte Carlo procedure has been adapted to treat the uncertainty in the retrieval of extinction coefficients.     

基于SCAD的压缩感知阈值迭代算法的收敛性分析

基于SCAD罚函数的压缩感知在有噪声稀疏信号重建中具有优良的理论及应用效果,开展其快速重建算法研究有着重要的意义,阈值迭代算法是解决压缩传感问题最有效的算法之一.本文研究了基于SCAD罚函数的压缩感知阈值迭代算法的收敛性问题,给出了算法收敛到稀疏解的充分条件,并证明了迭代估计值以指数阶速率收敛于最优值.进一步,本文给出了基于AMP改进的SCAD阈值迭代算法的收敛性分析.     

An iterative algorithm for modeling crack closure and sliding

In this paper, a simple and efficient iterative algorithm is adopted to model crack closure and sliding with the displacement discontinuity method (DDM). The problem of a subsurface crack in a half-plane under indentation loading considered by Elvin and Leung (Elvin N, Leung C. A fast iterative boundary element method for solving closed crack problems. Engng Fract Mech 1999;63:631-48) and DeBremaecker and Ferris (DeBremaecker J-Cl, Ferris MC. A comparison of two algorithm for solving closed crack problems. Engng Fract Mech 2000;66:601-05) is reexamined using the iterative DDM. The cases of a partially closed crack with both frictionless and frictional contacts are considered. Benchmark results for the stress intensity factors and the interfacial crack mechanisms for a subsurface crack are presented and compared with the results obtained from the fast iterative algorithm of Elvin and Leung and those from the PATH algorithm of DeBremaecker and Ferris.     

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative 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. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.     

Physics-constrained non-Gaussian probabilistic learning on manifolds

An extension of the probabilistic learning on manifolds (PLoM), recently introduced by the authors, has been presented: In addition to the initial data set given for performing the probabilistic learning, constraints are given, which correspond to statistics of experiments or of physical models. We consider a non-Gaussian random vector whose unknown probability distribution has to satisfy constraints. The method consists in constructing a generator using the PLoM and the classical Kullback-Leibler minimum cross-entropy principle. The resulting optimization problem is reformulated using Lagrange multipliers associated with the constraints. The optimal solution of the Lagrange multipliers is computed using an efficient iterative algorithm. At each iteration, the Markov chain Monte Carlo algorithm developed for the PLoM is used, consisting in solving an Itô stochastic differential equation that is projected on a diffusion-maps basis. The method and the algorithm are efficient and allow the construction of probabilistic models for high-dimensional problems from small initial data sets and for which an arbitrary number of constraints are specified. The first application is sufficiently simple in order to be easily reproduced. The second one is relative to a stochastic elliptic boundary value problem in high dimension.     

Integrating a micromechanical model for multiscale analyses

The Chang‐Hicher micromechanical model based on a static hypothesis, not unlike other models developed separately at around the same epoch, has proved its efficiency in predicting soil behaviour. For solving boundary value problems, the model has now integrated stress‐strain relationships by considering both the micro and macro levels. The first step was to solve the linearized mixed control constraints by the introduction of a predictor–corrector scheme and then to implement the micro–macro relationships through an iterative procedure. Two return mapping schemes, consisting of the closest‐point projection method and the cutting plane algorithm, were subsequently integrated into the interparticle force‐displacement relations. Both algorithms have proved to be efficient in studies devoted to elementary tests and boundary value problems. Closest‐point projection method compared with cutting plane algorithm, however, has the advantage of being more intensive cost efficient and just as accurate in the computational task of integrating the local laws into the micromechanical model. The results obtained demonstrate that the proposed linearized method is capable of performing loadings under stress and strain control. Finally, by applying a finite element analysis with a biaxial test and a square footing, it can be recognized that the Chang‐Hicher micromechanical model performs efficiently in multiscale modelling.