SHAPE OPTIMIZATION WITH ADAPTIVE MESH REFINEMENT: TARGET ERROR SELECTION STRATEGIES

The multi-point approximation method in conjunction with Adaptive Mesh Refinement (AMR) for shape optimization of thin-walled structures is studied. Application of AMR is done in such a manner that in the beginning of an optimization process large discretization errors are accepted, while finite element discretizations become more accurate as the optimization process progresses. In this paper several strategies for selecting the target discretization errors are investigated. Special attention is paid to both the overall computational effort and the convergence of the optimization processes.     

Calibration of a class of non-linear viscoelasticity models with adaptive error control

The calibration of constitutive models is considered as an optimization problem where parameter values are sought to minimize the discrepancy between measured and simulated response. Since a finite element method is used to solve an underlying state equation, discretization errors arise, which induce errors in the calibrated parameter values. In this paper, adaptive mesh refinement based on the pertinent dual solution is used in order to reduce discretization errors in the calibrated material parameters. By a sensitivity assessment, the influence from uncertainties in experimental data is estimated, which serves as a threshold under which there is no need to further reduce the discretization error. The adaptive strategy is employed to calibrate a viscoelasticity model with observed data from uniaxial compression (i.e., homogeneous stress state), where the FE-discretization in time is studied. The a posteriori error estimations show an acceptable quality in terms of effectivity measures.     

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.     

EXTENSION OF THE ZIENKIEWICZ–ZHU ERROR ESTIMATOR TO SHAPE SENSITIVITY ANALYSIS

The application of the Zienkiewicz–Zhu estimator was extended to the estimation of the discretization error arising from shape sensitivity analysis using the finite element method. The sensitivity error was quantified from the sensitivity of the energy norm by using an estimator specially developed for this purpose. Sensitivity analyses were carried out using the discrete analytical approach, which introduced no additional errors other than the discretization error. In this work, direct nodal averaging was used for linear triangular elements and the SPR technique for quadratic elements in order to obtain the smoothed stress and sensitivities fields. Two examples with an exact solution are used to analyse the effectivity of the proposed estimator and its convergence with the h-adaptive refinement. © 1997 by John Wiley & Sons, Ltd.     

直角坐标系下的水下被动目标跟踪自适应卡尔曼滤波算法

针对纯方位被动目标跟踪中，直角坐标系下的扩展卡尔曼滤波器容易发散，导致滤波精度很差的情况，文章中提出了一种直角坐标系下自适应卡尔曼滤波算法，对虚拟噪声进行了估计，动态补偿观测模型的线性化误差，削减系统的观测误差，并对其滤波理论及其算法进行了研究和仿真，结果表明，该算法提高了滤波的稳定性、快速性和精确性，优于一般的扩展卡尔曼滤波算法。     

Probabilistic Modeling of Errors from Structural Optimization Based on Multiple Starting Points

With optimization increasingly used in engineering applications, a series of optimization runs may be required, and it may be too expensive to converge them to very high accuracy. A procedure for estimating average optimization convergence errors from a set of poorly converged optimization runs is developed. A probabilistic model is fitted to the errors in optimal objective function values of poorly converged runs. The Weibull distribution was identified as a reasonable error model both for the Rosenbrock function problem and the structural optimization of a high speed civil transport. Once a statistical model for the error is identified, it can be used to estimate average errors from a set of pairs of runs. In particular, by performing pairs of optimization runs from two starting points, accurate estimates of the mean and standard deviation of the convergence errors can be obtained.     

Numerical quenchback in thermofluid simulations of superconducting magnets

One of the most important thermofluid processes encountered in internally cooled superconducting magnets is that of quenching. Numerical simulation of the quench propagation involves accurately modelling a moving boundary layer at the quench front. Due to the highly non-linear nature of the quench process, slightest numerical errors can rapidly grow to unacceptable limits. The quench propagation in such a non-converged solution exhibits a very rapid propagation velocity which resembles a ‘quenchback’ effect. Hence, the term ‘Numerical Quenchback’ is used to characterize a numerically unstable solution of the governing quench model. This paper presents the underlying physical phenomena that causes a numerical discretization scheme to have error terms that increase exponentially with time, causing the numerical quenchback effect. Specifically, by analytically solving the equivalent differential equation of the numerical scheme, we are able to obtain closed-form relations for the error terms associated with the propagation velocity. This allows us to define error criteria on the space and time steps used in the simulation. The reliability of the error criteria is proven by detailed convergence studies of the quench process. © 1998 John Wiley & Sons, Ltd.     

Local discretization error bounds using interval boundary element method

In this paper, a method to account for the point‐wise discretization error in the solution for boundary element method is developed. Interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented. The developed method does not assume any special properties besides the Laplace equation being a linear elliptic partial differential equation whose Green's function for an isotropic media is known. Numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error. Copyright © 2008 John Wiley & Sons, Ltd.     

CALCULATION OF THE STRESS INTENSITY FACTOR AND ESTIMATION OF ITS ERROR BY A SHAPE SENSITIVITY ANALYSIS

Abstract— The precision with which the stress intensity factor (SIF) can be calculated from a finite element solution depends essentially on the extraction method and on the discretization error. In this paper, the influence of the discretization error in the SIF calculation was studied and a method for estimating the resulting error was developed. The SIF calculation method used is based on a shape design sensitivity analysis; this assures that the resulting error in the extracted SIF depends solely on the global discretization error present in the finite element solution. Moreover, this method allows us to extend the Zienkiewicz-Zhu discretization error estimator to the SIF calculation. The reliability of the proposed method was analysed solving a two-dimensional problem using an h -adaptive process. Also the convergence of the error with the h -adaptive refinement was studied.     

Adaptive analysis of thin shells using facet elements

An adaptive mesh refinement (AMR) procedure is used in static thin shell analysis using triangular facet shell elements. The procedure described herein uses the h-version of adaptive refinement based on an error estimate determined by using the best guess values of bending moments and membrane forces obtained from a previous solution. It includes the use of a relaxation factor to achieve better convergence. Some examples are presented to illustrate this method. The results obtained are compared with those of uniform mesh refinement (UMR).     

Some benchmark problems and basic ideas on the accuracy of boundary element analysis

Taken the linear elasticity problems as examples, some benchmark problems are presented to investigate the influence of calculation error and discretization error on the accuracy of boundary element analysis. For the conventional boundary element analysis based on singular kernel function of fundamental solution and using Gaussian elimination method, the main calculation error comes from the integration of kernel and shape function product on each element. Based on some benchmark problems of “simple problem” without discretization error, it can be observed that sometimes a large number of integration points in Gaussian quadrature should be adopted. To guarantee the integration accuracy reliably, an improved adaptive Gaussian quadrature approach is presented and verified. The main error of boundary element analysis is the discretization error, provided the calculation error has been reduced effectively. Based on some benchmark problems, it can be observed that for the bending problems both the constant and linear element are not efficient, the quadratic element with a reasonable refined mesh is required to guarantee the accuracy of boundary element analysis. An error indicator to guide the mesh refinement in the convergence test towards the converged accurate results based on the distribution of boundary effective stress is presented and verified.     

A general methodology for structural shape optimization problems using automatic adaptive remeshing

The proposed methodology is based on the use of the adaptive mesh refinement (AMR ) techniques in the context of 2D shape optimization problems analysed by the finite element method. A suitable and very general technique for the parametrization of the optimization problem, using B-splines to define the boundary, is first presented. Then mesh generation, using the advancing frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines. finite element mesh, structural behaviour and error estimator) is studied in detail. The sensitivities of the finite element mesh and error estimator permit their projection from one design to the next one leading to an a priori knowledge of the finite element error distribution on the new design without the necessity of any additional structural analysis. With this information the mesh refinement criterion permits one to build up a finite element mesh on the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked by means of several examples.     

谱表示法模拟风场的误差分析

研究了原型谱表示法模拟的非各态历经性多变量风场的统计矩的时域估计值和目标值之间误差的概率描述。基于原型谱表示法的模拟公式，以三变量风场为例，导出了模拟结果的均值、相关函数、功率谱密度函数和根方差等四项统计特征的单样本时域估计表达式，它们是随机变量或随机过程。运用概率论的计算方法，推导出了上述随机变量或过程的前二阶矩的解析表达式，得到了模拟风场的统计特征时域估计的偏度误差和随机误差。将三变量过程的结果加以推广，给出了误差计算的通式。通过算例中统计误差值和理论误差值的对比，验证解析解的正确性。探讨了可能的降低随机误差的方法。求得的误差闭合解将有利于结合误差传播理论进行可靠性分析。     

Thermal investigations of an experimental active magnetic regenerative refrigerator operating near room temperature

In this paper, numerical and experimental investigations on a magnetic refrigeration device based upon the active magnetic regeneration (AMR) cycle operating near room temperature are presented. A numerical 1D model based on the transient energy equations is proposed for modelling the heat exchange between the magnetocaloric material and the carrier fluid in the regenerator bed. The validity of 1D AMR-numerical model is investigated through the recently developed magnetic cooling demonstrator by Clean Cooling Systems SA (CCS) at the University of Applied Sciences of western Switzerland (HES−SO). The obtained results including the temperature span, the coefficient of performance and the cooling power are presented and discussed. In general, good agreements have been noted between the experimental and numerical results.     

Aberrations and convergence characteristics of a concentric-circular focusing grating coupler: analysis

The convergence characteristics of the previously proposed concentric-circular focusing grating coupler (CFGC) are analyzed, and aberration functions for such typical errors as wavelength errors, effective index errors, grating pattern distortion, and CFGC eccentricity are derived for evaluation of their allowances. The analyzed results prove that the Strehl intensity deterioration caused by a wavelength error and an effective index offset can be minimized by optimization of the annular aperture of the CFGC. In the case of Marechal's criterion, a wavelength error of ±8.6% at the annular aperture of NA = 0.440-0.607 and an effective index error of ±20.4% at NA = 0.500-0.652 are permissible, and these values are ～100 times greater than those that are seen in a conventional focusing grating coupler that has a rotationally asymmetrical structure with respect to its optical axis.     

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.     

ERROR ESTIMATION AND ADAPTIVITY IN ELASTOPLASTICITY

In this paper, a method is developed to control the parameters of a finite element computation for time-dependent material models. This method allows the user to obtain a prescribed accuracy with a computational cost as low as possible. To evaluate discretization errors, we use a global error measure in constitutive relation based on Drucker's inequality. This error includes, over the studied time interval, the error of the finite element model and the error of the algorithm being used. In order to master the size of the elements of the mesh and the length of the time increments, an error estimator, which permits estimating the errors due to the time discretization, is proposed. These tools are used to elaborate two procedures of adaptivity. Various examples for monotonous or non-monotonous loadings, for 2-D or axisymmetric problems, show the reliability of these procedures.     

基于互信息PSO-LSSVM的SO2浓度预测

针对火电厂SO2污染物排放问题,提出了一种基于互信息的粒子群寻优(PSO)最小二乘支持向量机(LSSVM)模型预测方法,通过筛选出与SO2实测入口浓度相关性较高的辅助变量,将其作为模型的输入,实现对主导变量SO2浓度的预测.利用互信息筛选出的辅助变量相比于机理分析、皮尔逊相关性筛选出的辅助变量具有更高的相关性.利用互信...     

CFD modelling of pump-around jet mixing tanks: a reliable model for overall mixing time prediction

Over the past two decades or so, computational fluid dynamics (CFD) has been employed to predict overall mixing times inside jet mixing tanks instead of non-universal mixing time correlations obtained by experiments. However, the numerical methods for jet mixing tank simulations were not clearly tested and the discretization errors of the previous CFD models were not assessed. So, in this paper, the suitable turbulence model and numerical methods for pump-around jet mixing tank simulations were investigated. Further, the discretization errors of the present CFD models were estimated with the help of grid convergence index (GCI). The results revealed that the realizable k-epsilon model, SIMPLE, second order upwind, and first order implicit were proper turbulence model and numerical methods for pump-around jet mixing tank simulations. From GCI analyses, the maximum discretization uncertainty in overall mixing time of the present CFD models was about ±0.08 s.