A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems

In this paper, we develop an a posteriori error analysis for operator decomposition iteration methods applied to systems of coupled semilinear elliptic problems. The goal is to compute accurate error estimates that account for the combined effects arising from numerical approximation (discretization) and operator decomposition iteration. In an earlier paper, we considered ‘triangular’ systems that can be solved without iteration. In contrast, operator decomposition iterative methods for fully coupled systems involve an iterative solution technique. We construct an error estimate for the numerical approximation error that specifically addresses the propagation of error between iterates and provide a computable estimate for the iteration error arising because of the decomposition of the operator. Finally, we develop an adaptive discretization strategy to systematically reduce the discretization error.Copyright © 2013 John Wiley & Sons, Ltd.     

Implementation of an exact finite reduction scheme for steady‐state reaction‐diffusion equations

In this paper we propose the numerical solution of a steady‐state reaction‐diffusion problem by means of application of a non‐local Lyapunov–Schmidt type reduction originally devised for field theory. A numerical algorithm is developed on the basis of the discretization of the differential operator by means of simple finite differences. The eigendecomposition of the resulting matrix is used to implement a discrete version of the reduction process. By the new algorithm the problem is decomposed into two coupled subproblems of different dimensions. A large subproblem is solved by means of a fixed point iteration completely controlled by the features of the original equation, and a second problem, with dimensions that can be made much smaller than the former, which inherits most of the non‐linear difficulties of the original system. The advantage of this approach is that sophisticated linearization strategies can be used to solve this small non‐linear system, at the expense of a partial eigendecomposition of the discretized linear differential operator. The proposed scheme is used for the solution of a simple non‐linear one‐dimensional problem. The applicability of the procedure is tested and experimental convergence estimates are consolidated. Numerical results are used to show the performance of the new algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

A BENCHMARK COMPUTATIONAL STUDY OF FINITE ELEMENT ERROR ESTIMATION

Benchmark solutions are presented for a simple linear elastic boundary value problem, as analysed using a range of finite element mesh configurations. For each configuration, various estimates of local (i.e. element) and global discretization error have been computed. These show that the optimal mesh corresponds not only to minimization of global energy (or L₂) norms of the error, but also to equalization of element errors as well. Hence, this demonstrates why element error equalization proves successful as a criterion for guiding the process of mesh refinement in mesh adaptivity. The results also demonstrate the effectiveness of the stress projection method for smoothing discontinuous stress fields which, for this investigation, are more extreme as a consequence of the assumption of nearly incompressible material behaviour. In this case, lower order smoothing produces a continuous stress field which is in close agreement with the exact solution.     

Isoparametric finite point method in computational mechanics

In this paper, a new meshless method, the isoparametric finite point method (IFPM) in computational mechanics is presented. The present IFPM is a truly meshless method and developed based on the concepts of meshless discretization and local isoparametric interpolation. In IFPM, the unknown functions, their derivatives, and the sub-domain and its boundaries of an arbitrary point are described by the same shape functions. Two kinds of shape functions that satisfy the Kronecker-Delta property are developed for the scattered points in the domain and on the boundaries, respectively. Conventional point collocation method is employed for the discretization of the governing equation and the boundary conditions. The essential (Dirichlet) and natural (Neumann) boundary conditions can be directly enforced at the boundary points. Several numerical examples are presented together with the results obtained by the exact solution and the finite element method. The numerical results show that the present IFPM is a simple and efficient method in computational mechanics.     

Rosenau-Burgers 方程的修正局部 Crank-Nicolson 格式

本文给出了 Rosenau-Burgers 方程的两种修正局部 Crank-Nicolson 格式．首先,求解原有的偏微分方程对空间方向进行有限差分离散而得到的常微分方程．其次,利用矩阵分裂技术对这个方程的指数系数矩阵分别按行和元素进行逼近．最后,利用修正局部 Crank-Nicolson 方法得到了两种格式．讨论了格式的稳定性、收敛性和先验误差估计．数值实验结果表明了理论证明的正确性及格式的有效性．该格式具有结构简单、精度高的优点．     

<Emphasis Type="Italic">p</Emphasis>-Version error estimation for linear elasticity

An error estimator, formulated earlier for h-adaptive strategies, is extended for use in the p-version finite element analysis. The estimation of error is based on solving a series of local problems, based on patches consisting of elements surrounding each node, with prescribed homogeneous essential boundary conditions. Unlike the original approach in which a patch was constructed based on one element, each patch in the present scheme is automatically formed based on a number of elements surrounding a corresponding node. The present scheme, based on enhancing the degree of interpolation, provides a better estimate than the original h-scheme while still preserving the original lower bound property. The capability of the new scheme is investigated in some numerical examples in terms of its global and local performance.     

High‐order global–regional model interaction: Extension of Carpenter's scheme

A two‐dimensional global–regional model interaction problem for linear time‐dependent waves is considered. The setup, which is sometimes called ‘one‐way nesting,’ arises in numerical weather prediction as well as in other fields concerning waves in very large domains. It involves the interaction of a coarse global model and a fine limited‐area (regional) model through an ‘open boundary.’ The multiscale nature of this general problem is described. The Carpenter scheme, originally proposed in a note by K. M. Carpenter in 1982 for this type of problem, is then revisited, in the context of the linear scalar wave equation. The original Carpenter scheme is based on the Sommerfeld radiation operator and thus is associated with low‐order accuracy. By replacing the Sommerfeld operator with the high‐order Hagstrom–Warburton absorbing operator, a modified Carpenter open‐boundary condition emerges, which possesses high‐order accuracy. This boundary condition is incorporated in a computational scheme, which uses finite element discretization in space and Newmark time‐stepping. Error analysis and numerical tests for wave guides demonstrate the performance of the modified scheme for combinations of incoming and outgoing waves. Copyright © 2008 John Wiley & Sons, Ltd.     

Modeling of uncertainties in long fiber reinforced thermoplastics

The present study is concerned with a numerical scheme for the prediction of the uncertainty of the effective elastic properties of long fiber reinforced composites with thermoplastic matrix (LFT) produced by standard injection or press molding technologies based on the uncertainty of the microstructural geometry and topology. The scheme is based on a simple analysis of the single-fiber problem using the rules of mixture. The transition to the multi-fiber problem with different fiber orientations is made by the formulation of an ensemble average with defined probability distributions for the fiber angles. In the result, the standard deviations of the local fiber angles together with the local fiber content are treated as stochastic variables. The corresponding probability distributions for the effective elastic constants are determined in a numerically efficient manner by a discretization of the space of the random variables and the analysis of predefined cases within this space.     

An augmented Lagrangian quasi-Newton solver for constrained nonlinear finite element applications

A solution scheme is presented for constrained non-linear equations of evolution that result, for example, from the finite element discretization of mechanical contact problems. The algorithm discussed utilizes a quasi-Newton non-linear equation solving strategy, with constraints enforced by an augmented Lagrangian iteration procedure. Through presentation of a simple model problem and its generalization, it is shown that the iterations associated with both the quasi-Newton algorithm and the augmentation procedure can be interwoven to produce a highly efficient and robust solution strategy.     

A fast weakly intrusive multiscale method in explicit dynamics

This paper presents new developments on a weakly intrusive approach for the simplified implementation of space and time multiscale methods within an explicit dynamics software. The ‘substitution’ method proposed in previous works allows to take advantage of a global coarse model, typically used in an industrial context, running separate, refined in space and in time, local analyses only where needed. The proposed technique is iterative, but the explicit character of the method allows to perform the global computation only once per global time step, while a repeated solution is required for the small local problems only. Nevertheless, a desirable goal is to reach convergence with a reduced number of iterations. To this purpose, we propose here a new iterative algorithm based on an improved interface inertia operator. The new operator exploits a combined property of velocity Hermite time interpolation on the interface and of the central difference integration scheme, allowing the consistent upscaling of interface inertia contributions from the lower scale. This property is exploited to construct an improved mass matrix operator for the interface coupling, allowing to significantly enhance the convergence rate. The efficiency and robustness of the procedure are demonstrated through several examples of growing complexity. Copyright © 2014 John Wiley & Sons, Ltd.     

Error estimation for crack simulations using the XFEM

The extended finite element method (XFEM) is by now well‐established for crack calculations in linear elastic fracture mechanics. An advantage of this method is its discretization independence for crack simulations. Nevertheless, discretization errors occur when using the XFEM. In this paper, a simple recovery based error estimator for the discretization error in XFEM‐calculations for cracks is presented. The method is based on the Zienkiewicz and Zhu error estimator. Enhanced smoothed stresses incorporating the discontinuities and singularities because of the cracks are recovered to enable the error estimation for arbitrary distributed cracks. This approach also allows the consideration of materials with generally inelastic behaviour. The enhanced stresses are computed by means of a least square fit problem. To assess the quality of the error estimator, global norms and the effectivity index for the global energy norm for examples with known analytical solutions are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses

In the context of global/goal-oriented error estimation applied to computational mechanics, the need to obtain reliable and guaranteed bounds on the discretization error has motivated the use of residual error estimators. These estimators require the construction of admissible stress fields verifying the equilibrium exactly. This article focuses on a recent method, based on a flux-equilibration procedure and called the element equilibration + star-patch technique (EESPT), that provides for such stress fields. The standard version relies on a strong prolongation condition in order to calculate equilibrated tractions along finite element boundaries. Here, we propose an enhanced version, which is based on a weak prolongation condition resulting in a local minimization of the complementary energy and leads to optimal tractions in selected regions. Geometric and error estimate criteria are introduced to select the relevant zones for optimizing the tractions. We demonstrate how this optimization procedure is important and relevant to produce sharper estimators at affordable computational cost, especially when the error estimate criterion is used. Two- and three-dimensional numerical experiments demonstrate the efficiency of the improved technique.     

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models.     

Dual adaptive finite element refinement for multiple local quantities in linear elastostatics

In this paper, we summarize how dual analysis techniques can be used to determine upper bounds of the discretization error, both in terms of global and local outputs. We present formulas for the bounds of the error in local outputs, based on the approach proposed by Greenberg in 1948 and we show that the resulting intervals are the same as those previously presented, based on the approach proposed by Washizu in 1953. We then explain how the elemental contributions to these bounds can be used to define an adaptive strategy that considers multiple quantities and we present its application to a simple plane stress problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Fourier‐based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

A modified Green operator is proposed as an improvement of Fourier‐based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite. Furthermore, it is shown that the method produces much more accurate local fields inside highly conducting and quasi‐insulating phases, as well as in the vicinity of phase boundaries. These good properties stem from the discretization of Green's function, which is consistent with the pixel grid while retaining the local nature of the operator that acts on the polarization field. Finally, a fast implementation of the ‘direct scheme’ of Moulinec et al. (1994) that allows for parsimonious memory use is proposed. Copyright © 2014 John Wiley & Sons, Ltd.     

A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress

A new methodology for recovering equilibrated stress fields is presented, which is based on traction‐free subdomains' computations. It allows a rather simple implementation in a standard finite element code compared with the standard technique for recovering equilibrated tractions. These equilibrated stresses are used to compute a constitutive relation error estimator for a finite element model in 2D linear elasticity. A lower bound and an upper bound for the discretization error are derived from the error in the constitutive relation. These bounds in the discretization error are used to build lower and upper bounds for local quantities of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

LATIN: A new view and an extension to wave propagation in nonlinear media

The LATIN (acronym of LArge Time INcrement) method was originally devised as a non‐incremental procedure for the solution of quasi‐static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time‐dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.     

圆环形截面压电纤维复合材料有效电弹性性能研究

研究含周期分布压电纤维的压电复合材料的有效电弹性性能。通过在材料代表性体积单元边界上施加位移和电势周期边界条件,利用有限元法求得了代表性体积单元内的电弹性场。由平均电弹性场和压电复合材料有效电弹性性能定义,预测了圆环形截面压电纤维复合材料的有效电弹性系数。通过算例,比较了相同压电材料体积分数下圆环形截面压电纤维复合材料与圆截面压电纤维复合材料有效电弹性性能的差异,讨论了圆环形截面压电纤维内部非压电填充物的力学性质对有效压电系数的影响。该文结论可为高灵敏度压电复合材料设计提供 参考。     

From Ordered to Disordered: The Effect of Microstructure on Composite Mechanical Performance

The microstructural variation in fiber-reinforced composites has a direct relationship with its local and global mechanical performance. When micromechanical modeling techniques for unidirectional composites assume a uniform and periodic arrangement of fibers, the bounds and validity of this assumption must be quantified. The goal of this research is to quantify the influence of microstructural randomness on effective homogeneous response and local inelastic behavior. The results indicate that microstructural progression from ordered to disordered decreases the tensile modulus by 5%, increases the shear modulus by 10%, and substantially increases the magnitude of local inelastic fields. The experimental and numerical analyses presented in this paper show the importance of microstructural variability when small length scale phenomena drive global response.     

非等距网格上一维对流扩散方程的保正格式

对流扩散方程广泛存在于很多领域,为适应一些实际问题模型的求解,对离散格式,不仅要求满足一些基本性质,如稳定性和解的存在唯一性等,还要求离散格式的保正性．采用有限体积格式求解对流扩散方程的工作较少,但在保正性方面所做的工作不多．本文构造了任意非等距网格上一维对流扩散方程的非线性保正有限体积格式．其中,扩散通量的离散,在等距网格上,当扩散系数为标量时可退化为标准的二阶中心差分格式．而对流通量的离散,为避免数值振荡而使其保持迎风特性,提出一种新的方法使格式精度提高到二阶．该方法在上游单元中心处作泰勒级数展开,通过相关辅助未知量来完成梯度的重构,并对出负情形作正性校正,使得格式满足保正性要求．新格式只含有区间单元中心未知量,并满足区间端点处通量的局部守恒性．数值结果表明,本文所提格式是有效的,对于处理扩散占优、对流占优问题,扩散系数连续和间断情形均具有良好的适应性,并且保持二阶精度．另外,新格式适用于扩散系数间断问题的求解．