Nonlocal damage model using the phase field method: Theory and applications

A new nonlocal, gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials. The general framework of the phase field model (PFM) is discussed and the order parameter is related to the damage variable in continuum damage mechanics (CDM). The time dependent Ginzburg–Landau equation which is also termed the Allen–Cahn equation is used to describe the damage evolution process. Specific length scale which addresses the interface region in which the process of changing undamaged solid to fully damaged material (microcracks) occurs is defined in order to capture the effect of the damaged localization zone. A new implicit damage variable is proposed through the phase field theory. Details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated.     

Mesh regularization for an ALE code based on the limitation of the Lagrangian mesh velocity

The Lagrangian approach is usually used for the simulation of flow with strong shock waves. Moreover, this approach is particularly well suited to treatment of material interfaces in the case of multimaterial flows.Unfortunately, this formulation leads to very large deformations in the mesh. The arbitrary Lagrangian‐Eulerian method overcomes this drawback by using a mesh regularization that is based on an analysis of cell geometry. The regularization step may be considered as a method used to correct the nonconvex and potentially tangled cells that constitute the mesh. In this paper, we present a new approach to mesh regularization. Instead of using a purely geometric criterion, we propose that the mesh evolution is computed on the basis of the flow vorticity. This approach is called the large Eddy limitation method, and it is aimed here to be used in finite volume direct arbitrary Lagrangian‐Eulerian methods. The large Eddy limitation method is general, which means that it is not restricted to applications in the finite volume framework dedicated to fluid flow simulation; for instance, it could also be naturally applied to the finite element framework.     

A smoothed inverse eigenstrain method for reconstruction of the regularized residual fields

A smoothed inverse eigenstrain method is developed for reconstruction of residual field from limited strain measurements. A framework for appropriate choice of shape functions based on the prior knowledge of expected residual distribution is presented which results in stabilized numerical behavior. The analytical method is successfully applied to three case studies where residual stresses are introduced by inelastic beam bending, laser-forming and shot peening. The well-rehearsed advantage of the proposed eigenstrain-based formulation is that it not only minimizes the deviation of measurements from its approximations but also will result in an inverse solution satisfying a full range of continuum mechanics requirements. The smoothed inverse eigenstrain approach allows suppressing fluctuations that are contrary to the physics of the problem. Furthermore, a comprehensive discussion is performed on regularity of the asymptotic solution in the Tikhonov scheme and the regularization parameter is then exactly determined utilizing Morozov discrepancy principle. Gradient iterative regularization method is also examined and shown to have an excellent convergence to the Tikhonov–Morozov regularization results.     

Stress state of flexible plates with a hole that are subject to bending

This paper proposes and mathematically proves a method to regularize singular iterations used to solve geometrically nonlinear problems of bending of thin plates with a hole. It is established that the new approximations for deflections and stress functions satisfy the boundary conditions exactly and Karman’s equations asymptotically exactly. The regularization method and the approximations for solutions are based on new methods of summing functional series and sequences __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 1, pp. 100–116, January 2007.     

A new semi-analytical method for the non-linear static analysis of an infinite beam on a non-linear elastic foundation: A general approach to a variable beam cross-section

We propose a new non-linear method for the static analysis of an infinite non-uniform beam resting on a non-linear elastic foundation under localized external loads. To this end, an integral operator equation is newly formulated, which is equivalent to the original differential equation of non-uniform beam. By using the integral operator equation, we propose a new functional iterative method for static beam analysis as a general approach to a variable beam cross-section. The method proposed is fairly simple as well as straightforward to apply. An illustrative example is presented to examine the validity of the proposed method. It shows that just a few iterations are required for an accurate solution.     

二阶非定常多宗量热传导反问题的正则解

引入Bregman距离函数及其加权函数作为正则项,应用Tikhonov正则 化方法,对二阶非定常多宗量热传导反问题进行求解. 利用测量信息和计算信息构造最小二 乘函数,将多宗量反演识别问题转化为一个优化问题. 空间上采用8节点等参元进行离散, 时域上采用时域精细算法进行离散,建立了二阶非定常多宗量热传导问题的有限元正/反演数 值模型. 该模型不仅考虑了非均质和参数分布的影响,而且也便于正反演问题的敏度分析, 可对导热系数和边界条件等宗量进行有效的单一和组合识别. 给出了相关的数值验证,对信 息测量误差以及不同正则项的计算效率作了探讨. 数值结果表明,该方法能够对二阶非定常 多宗量热传导反问题进行有效的求解,并具有较高的计算精度.     

Formulation and numerical implementation of a constitutive law for concrete with strain-based damage and plasticity

A triaxial constitutive law for concrete within the framework of isotropic damage combined with plasticity is proposed in this paper. It covers typical characteristics of concrete like non-linear uniaxial compression and tension, bi- and triaxial failure criteria and dilatancy with a unified strain-based approach. Thus, this model is quite simple and especially suitable for strain-driven methods like common finite elements. It is complemented with a regularization method based on the crack band approach. A further issue is discussed with procedures for the model parameter determination for a wide range of concrete grades. The application of the model is demonstrated with typical benchmark tests for plain concrete.     

双相介质波动方程孔隙率反演的同伦方法

从材料响应的理论合成应与实际测量数据相拟合这一出发点，将双相介质波劝方程参数的反演问题转化为非线性算子方程的零点求解问题，从而应用一种大范围收敛的同伦方尘土注来解非线性算子方程，并把这种方法用于Simon(1984)给出的具有解析的一维双相介质模型的数值模拟，最后的数值结果表明，给出的算法是十分有效的。     

Identification of velocity fields for geophysical fluids from a sequence of images

We propose an algorithm to estimate the motion between two images. This algorithm is based on the nonlinear brightness constancy assumption. The number of unknowns is reduced by considering displacement fields that are piecewise linear with respect to each space variable, and the Jacobian matrix of the cost function to be minimized is assembled rapidly using a finite-element method. Different regularization terms are considered, and a multiscale approach provides fast and efficient convergence properties. Several numerical results of this algorithm on simulated and experimental geophysical flows are presented and discussed.     

Inversion of dispersion coefficient in water quality model using optimal perturbation algorithm

As a primary parameter in the water quality model for shallow bays, the dispersion coefficient is traditionally determined with a trial-and-error method, which is time-consuming and requires much experience. In this paper, based on the measured data of chemical oxygen demand （COD）, the dispersion coefficient is calculated using an inversion method. In the process, the regularization method is applied to treat the ill-posedness, and an operator identity perturbation method is used to obtain the solu- tion. Using the model with an inverted dispersion coefficient, the distributions of COD, inorganic nitrogen （IN）, and inorganic phosphorus （IP） in Bohai Bay are predicted and compared with the measured data. The results indicate that the method is feasible and the inverted dispersion coefficient can be used to predict other pollutant distribution. This method may also be further extended to the inversion of other parameters in the water quality model.     

Prediction of Effective Properties of Porous Carbon Electrodes from a Parametric 3D Random Morphological Model

Pore structures have a major impact on the transport and electrical properties of electrochemical devices, such as batteries and electric double-layer capacitors (EDLCs). In this work we are concerned with the prediction of the electrical conductivity, ion diffusivity and volumetric capacitance of EDLC electrodes, manufactured from hierarchically porous carbons. To investigate the dependence of the effective properties on the pore structures, we use a structurally resolved parametric model of a random medium. Our approach starts from 3D FIB-SEM imaging, combined with automatic segmentation. Then, a random set model is fitted to the segmented structures and the effective transport properties are predicted using full field simulations by iterations of FFT on 3D pore space images and calculations based on the geometric properties of the structure model. A parameter study of the model is used to investigate the sensitivity of the effective conductivity and diffusivity to changes in the model parameters. Finally, we investigate the volumetric capacitance of the EDLC electrodes with a geometric model, make a comparison with experimental measurements and do a parameter study to suggest improved microstructures.     

An Extended Natural Variable Formulation for Compositional Simulation Based on Tie-Line Parameterization

A nonlinear formulation based on extension of natural variables set is proposed for modeling compositional two-phase flow in porous media. The focus here is on numerical general-purpose simulation using the fully implicit method. In the formulation, the phase fraction and the saturation change “continuously” in the immiscible region of the compositional space (i.e., sub-critical region). Inside the two-phase region, these variables are identical to the saturation and phase-fraction of the standard approach. In the single-phase regions, however, these saturation-like and phase-fraction-like variables can become negative, or larger that unity. We demonstrate that when this variable set is used, the equation-of-state (EoS)-based thermodynamic equilibrium computations are resolved completely within the global Newton loop. That is, need not to separate things into phase stability and flash computations. Compared to the standard natural variables approach, the number of global Newton iterations grows only slightly, but overall, the new approach leads to more efficient simulations. Moreover, the continuous variation of both the saturation and phase fraction across phase boundaries results in improved behavior of the nonlinear (Newton) solver. Two different strategies are used to deal with the densities. The first scheme honors the nonlinear dependence of the overall density on phase fractions and saturation, and the second employs a linearized relation for the overall density. Both schemes are compared with the standard natural variables formulation using several challenging compositional problems.     

Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint

The relaxation spectrum is estimated from dynamic experiments using Bayesian analysis and a new regularization constraint. In the Bayesian framework, a probability can be calculated for each estimate of the spectrum. This offers several advantages; (1) an optimal estimate of the relaxation spectrum may be calculated as the mean of a large number of estimates, and (2) reliable errors for the optimal estimate can be provided using the deviation of all estimates from the mean. Furthermore, the Bayesian approach (3) gives an estimate of the overall noise level of the experiment, which is usually an important but unknown parameter for the calculation of relaxation spectra from dynamic experiments by indirect methods (determining the regularization parameter), and finally, (4) the information content in a given set of experimental data can be quantified. The validity of the Bayesian approach is demonstrated using simulated data.     

An Analysis of a Hybrid Optimization Method for Variational Data Assimilation

In four-dimensional variational data assimilation (4D-Var) an optimal estimate of the initial state of a dynamical system is obtained by solving a large-scale unconstrained minimization problem. The gradient of the cost functional may be efficiently computed using the adjoint modeling, at the expense equivalent to a few forward model integrations; for most practical applications, the evaluation of the Hessian matrix is not feasible due to the large dimension of the discrete state vector. Hybrid methods aim to provide an improved optimization algorithm by dynamically interlacing inexpensive L-BFGS iterations with fast convergent Hessian-free Newton (HFN) iterations. In this paper, a comparative analysis of the performance of a hybrid method vs. L-BFGS and HFN optimization methods is presented in the 4D-Var context. Numerical results presented for a two-dimensional shallow-water model show that the performance of the hybrid method is sensitive to the selection of the method parameters such as the length of the L-BFGS and HFN cycles and the number of inner conjugate gradient iterations during the HFN cycle. Superior performance may be obtained in the hybrid approach with a proper selection of the method parameters. The applicability of the new hybrid method in the framework of operational 4D-Var in terms of computational cost and performance is also discussed.     

An Uncertainty Estimator for Slitting Method Residual Stress Measurements Including the Influence of Regularization

This paper describes the development of a new uncertainty estimator for slitting method residual stress measurements. The new uncertainty estimator accounts for uncertainty in the regularization-based smoothing included in the residual stress calculation procedure, which is called regularization uncertainty. The work describes a means to quantify regularization uncertainty and then, in the context of a numerical experiment, compares estimated uncertainty to known errors. The paper further compares a first-order uncertainty estimate, established by a repeatability experiment, to the new uncertainty estimator and finds good correlation between the two estimates of precision. Furthermore, the work establishes a procedure for automated determination of the regularization parameter value that minimizes total uncertainty. In summary, the work shows that uncertainty in the regularization parameter is a significant contributor to the total uncertainty in slitting method measurements and that the new uncertainty estimator provides a reasonable estimate of single measurement uncertainty.

     

CHAOS-REGULARIZATION HYBRID ALGORITHM FOR NONLINEAR TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

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Stress-based nonlocal damage model

Progressive microcracking in brittle or quasi-brittle materials, as described by damage models, presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results. Such regularization methods, which describe interactions between points, provide some general properties (including objectivity and the non-alteration of a uniform field) as well as drawbacks (damage initiation, free boundary).A modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution. The orientation and intensity of nonlocal interactions are modified in accordance with the stress state. The fundamental framework of the original nonlocal method has been retained, making it possible to maintain the method’s advantages. The modification is introduced through the weight function, which in this modified version depends not only on the distance between two points (as for the original model) but also on the stress state at the remote point.The efficiency of this novel approach is illustrated using several examples. The proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques. Damage initiation and propagation in mode I as well as shear band formation are analyzed herein.     

应用响应面方法结合空间映射方法的可靠性评价

应用响应面结合空间映射方法,在第1次迭代拟合极限状态函数,其它迭代 应用映射技术在第1迭代响应面基础上映射调整得到新的极限状态函数,并进行可靠性分析. 这样就改变了序列响应面方法评价可靠性时需要反复对模型进行试验设计、分析并拟合极限状 态函数的执行过程,从而大大降低了模型分析的计算量.     

网络RTK参考站间模糊度动态解算中病态方程的一种解算方法

在网络RTK参考站间的模糊度估计中,若误差方程严重病态,将导致模糊度解与其准确值偏差较大或整周模糊度无法固定,因此提出了一种适于网络RTK模糊度动态解算的新方案：1）法方程病态性的判断;2）Tikhonov正则化解算病态方程;3）LAMBDA方法搜索固定整周模糊度。同时,深入研究了Tikhonov正则化矩阵的构造方法和正则化参数的选取准则。最后以实例验证了采用此方案解算病态方程是可行的,通过选取合适的正则化参数可以解得准确的整周模糊度;详细讨论了选择不同的正则化参数对模糊度解算结果的影响。     

Instrumental variables approach to identification of a class of MIMO Wiener systems

A new approach to identification of multi-input multi-output (MIMO) Wiener systems using the instrumental variables method is presented. It is assumed that static nonlinear elements are invertible and their inverse characteristics can be expressed or approximated by polynomials of known orders. It is also assumed that the linear part of the Wiener system can be represented by a matrix polynomial form. Based on these assumptions, the Wiener system is transformed introducing a new parameterization and its parameters are estimated using a linear-in-parameters model. To solve the problem of non-consistency of least squares parameter estimates, an instrumental variables method is employed. A numerical example is included to show the effectiveness and the practical feasibility of the presented approach.