Parameter Identification and FE Implementation of a Viscoelastic Constitutive Equation Using Fractional Derivatives

Damping in viscoelastic materials can be described in several ways. In FE codes for transient calculations with direct integration usually Rayleigh‐damping is provided. However, it is known that this model is not qualified to represent the damping properties of viscoelastic material over a broad range of time or frequency. Another approach uses fractional time derivatives of stresses and strains in the constitutive equations. This model requires few parameters, provides good curve fitting properties and is physically proved. In this paper a parameter identification for the fractional 3‐parameter model will be carried out and its implementation into an FE code will be demonstrated.     

An Unified Computational Framework for Parameter Identification of Material Models in Finite Inelasticity

Parameter identification processes concern the determination of parameters in a material model in order to fit experimental data. We provide a distinct, unified algorithmic setting of a generic class of material models and discuss the associated gradient–based optimization problem. Gradient–based optimization algorithms need derivatives of the objective function with respect to the material parameter vector κ . In order to obtain the necessary derivatives, an analytical sensitivity analysis is pointed out for the unified class of algorithmic material models. The quality of the parameter identification is demonstrated for a representative example.     

Experimental Investigations on Polymeric Foams

The knowledge of the material behaviour of polymeric foams and their experimental investigation is the starting point for the structure of the chosen constitutive equations and for the following identification of the material constants therein. Especially for the parameter identification, it is necessary to make an adequate set of experimental data available. In this regard, it is important that the experiments make the different kinds of material behaviour visible like elastic, plastic or viscous material properties. For this reason, the foam is observed under uniaxial tension and compression and under simple shear tests combined with different deformation states in axial direction. Unfortunately, due to different reasons, e.g., the foam must be sticked on the fastener to realize the tests mentioned above, it is very difficult to initialize a homogenous deformation state in the specimen. Therefore, the experiments are recorded with a standard digital camcorder to get local information of the deformation state by tracking single points with algorithms of the digital image processing. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter optimization and uncertainty analysis in a model of oceanic CO2 uptake using a hybrid algorithm and algorithmic differentiation

Methods and results for parameter optimization and uncertainty analysis for a one-dimensional marine biogeochemical model of NPZD type are presented. The model, developed by Schartau and Oschlies, simulates the distribution of nitrogen, phytoplankton, zooplankton and detritus in a water column and is driven by ocean circulation data. Our aim is to identify parameters and fit the model output to given observational data. For this model, it has been shown that a satisfactory fit could not be obtained, and that parameters with comparable fits can vary significantly. Since these results were obtained by evolutionary algorithms (EA), we used a wider range of optimization methods: A special type of EA (called quantum-EA) with coordinate line search and a quasi-Newton SQP method, where exact gradients were generated by Automatic/Algorithmic Differentiation. Both methods are parallelized and can be viewed as instances of a hybrid, mixed evolutionary and deterministic optimization algorithm that we present in detail. This algorithm provides a flexible and robust tool for parameter identification and model validation. We show how the obtained parameters depend on data sparsity and given data error. We present an uncertainty analysis of the optimized parameters w.r.t. Gaussian perturbed data. We show that the model is well suited for parameter identification if the data are attainable. On the other hand, the result that it cannot be fitted to the real observational data without extension or modification, is confirmed.     

基于BP神经网络的动态稳健参数设计

传统的动态稳健参数设计方法(田口方法)虽然在工业生产实践中展现了极大的方便,但是其本身也存在较大的改进空间.当调节变量不存在时,传统的田口方法难以实现;此外,田口方法只能根据所选取的参数水平得到最优参数组合,而这种所谓的最优结果有时并不符合实际的需要.首先构建BP神经网络模型,利用训练后的BP神经网络获得参数设计中质量特性、噪声因子以及各参数间的动态关系;然后,利用超拉丁方抽样,计算信号与特性参数间的斜率,并由此将动态稳健参数设计的寻优问题转化为相应的非线性规划问题;最后,利用次序二次规划(SQP)算法解决并优化动态稳健参数的设计。此外,我们选取了一个简单的数据案例对本文提出的方法的有效性进行了说明.     

A superlinearly convergent method of quasi-strongly sub-feasible directions with active set identifying for constrained optimization

Combining the norm-relaxed sequential quadratic programming (SQP) method and the idea of method of quasi-strongly sub-feasible directions (MQSSFD) with active set identification technique, a new SQP algorithm for solving nonlinear inequality constrained optimization is proposed. Unlike the previous work, at each iteration of the proposed algorithm, the norm-relaxed quadratic programming (QP) subproblem only consists of the constraints corresponding to an active identification set. Moreover, the high-order correction direction (used to avoid the Maratos effect) is yielded by solving a system of linear equations (SLE) which also includes only the constraints and their gradients corresponding to the active identification set, therefore, the scale and the computation cost of the high-order correction directions are further decreased. The arc search in our algorithm can effectively combine the initialization processes with the optimization processes, and the iteration points can get into the feasible set after a finite number of iterations. Furthermore, the arc search conditions are weaker than the previous work, and the computation cost is further reduced. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ). If the strong second-order sufficient conditions are satisfied, then the active constraints are exactly identified by the identification set. Without the strict complementarity, superlinear convergence can be obtained. Finally, some elementary numerical results are reported.     

Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems with General Constraints

In Ref. 1, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints was proposed. At each iteration, this new algorithm only needs to solve four systems of linear equations having the same coefficient matrix, which is much less than the amount of computation required for existing SQP algorithms. Moreover, unlike the quadratic programming subproblems of the SQP algorithms (which may not have a solution), the subproblems of the SSLE algorithm are always solvable. In Ref. 2, it is shown that the new algorithm can also be used to deal with nonlinear optimization problems having both equality and inequality constraints, by solving an auxiliary problem. But the algorithm of Ref. 2 has to perform a pivoting operation to adjust the penalty parameter per iteration. In this paper, we improve the work of Ref. 2 and present a new algorithm of sequential systems of linear equations for general nonlinear optimization problems. This new algorithm preserves the advantages of the SSLE algorithms, while at the same time overcoming the aforementioned shortcomings. Some numerical results are also reported.     

An optimization method to estimate models with store-level data: A case study

The quality of the estimation of a latent segment model when only store-level aggregate data is available seems to be dependent on the computational methods selected and in particular on the optimization methodology used to obtain it. Following the stream of work that emphasizes the estimation of a segmentation structure with aggregate data, this work proposes an optimization method, among the deterministic optimization methods, that can provide estimates for segment characteristics as well as size, brand/product preferences and sensitivity to price and price promotion variation estimates that can be accommodated in dynamic models. It is shown that, among the gradient based optimization methods that were tested, the Sequential Quadratic Programming method (SQP) is the only that, for all scenarios tested for this type of problem, guarantees of reliability, precision and efficiency being robust, i.e., always able to deliver a solution. Therefore, the latent segment models can be estimated using the SQP method when only aggregate market data is available.     

非线性约束优化问题的一个新的全局收敛算法

对于非线性约束的优化问题.最近给出的各种SQP算法均采用罚函数技巧以保证算法的全局收敛性,因而都必须小心地调整惩罚参数。本文给出一个不依赖于惩罚参数、每步迭代的校正矩阵也不需正定且仍具有全局收敛性的SQP方法,而且罚函数形式简单、具有和约束函数同阶的光滑性.     

解约束优化问题的相容SQP滤子方法

提出了解约束优化问题的一类相容SQP滤子算法.利用序列二次规划方法结合信赖域技术计算试探步,而用滤子接受准则选择接受试探步.对二次规划子问题的不相容问题,应用Powell1978年于文[9]提出的方法对其约束引进参数进行了可行化处理.在一般条件下,算法具有全局收敛性.最后,数值试验显示了较好的结果.     

Parameter Identification by Inverse Modelling of Biaxial Tensile Tests for Discontinous Fiber Reinforced Polymers

In the present article we discuss the characterization of the mechanical properties of long fiber reinforced thermoplastics (LFT) and sheet molding compounds (SMC) with biaxial tensile tests and the inverse parameter identification. The full 3D strain field is measured via digital image correlation (DIC). The anisotropic viscoelastic material properties are identified through inverse modelling by comparison of the heterogeneous experimental and simulated strain fields. A Gauss-Newton type algorithm is used to identify the optimal parameter set [1]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter–identification of macroscopic material models based on virtual testing of given material mesostructures

For many heterogeneous materials such as composites and polycrystals, the material modeling for the constituents on a representative mesoscale can be considered as known, including concrete values of their inherent material parameters. Typical examples are isotropic elastic–plastic models for the constituents of composites or anisotropic crystal–plasticity models for the grains of polycrystals. This knowledge can be exploited with regard to the modeling of the homogenized macroscopic response. In particular, parameters in macroscopic models may be identified by virtual experiments provided by a computational deformation–driving of representative mesostructures. This paper outlines the general concept for the parameter–identification of macroscopic materialmodels based on the virtual testing of given material mesostructures. The virtual test data are obtained in the form of multi–dimensional stress–strain paths by applying different deformation gradients to a given mesostructure. After specifying a corresponding macroscopic material model covering the observed effects on the macroscale, the material parameters are identified by a least–square–type optimization procedure that optimizes the macroscopic material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global convergence on an active set SQP for inequality constrained optimization

Sequential quadratic programming (SQP) has been one of the most important methods for solving nonlinearly constrained optimization problems. In this paper, we present and study an active set SQP algorithm for inequality constrained optimization. The active set technique is introduced which results in the size reduction of quadratic programming (QP) subproblems. The algorithm is proved to be globally convergent. Thus, the results show that the global convergence of SQP is still guaranteed by deleting some “redundant” constraints.     

Process flowsheet optimization strategies: recent results and future directions

Optimization strategies based on detailed and rigorous models of process flowsheets have been of interest to process engineers for the past 25 years. Here we review earlier optimization strategies and emphasize more recent strategies based on successive quadratic programming (SQP). Application of SQP strategies to process optimization have reduced the computational effort by more than an order of magnitude over conventional methods and now create the potential to make simulator-based optimization with complex process models a powerful tool. This report describes the process optimization problem with a simple flowsheeting model and briefly discusses the infeasible path approach, which permits simultaneous recycle convergence and optimization with SQP. Several improvements to the infeasible path algorithm are then outlined that include better gradient calculation and scaling strategies, a more efficient line search function and more reliable performance through intermediate recycle convergence. These concepts are applied to a large-scale process problem and resulting improvements are demonstrated. Finally, we present a brief discussion of open research questions and directions for future research.     

最优化两个拓广的SQP和SSLE算法模型及其超线性和二次收敛性

给出一般约束最优化的序列二次规划（SQP）和序列线性方程组（SSLE）算法两个拓广的模型,详细分析和论证两个模型的局部超线性收敛性及二次收敛性条件,其中并不需要严格互补条件,拓广的模型及其收敛速度结果具有更广泛的适用性,为SQP和SSLE算法收敛速度的研究提供了更为完善和便利的理论基础。     

基于并行计算的多方向强次可行模松弛SQP算法

本文研究求解非线性约束优化问题.利用多方向并行方法,提出了一个新的强次可行模松弛序列二次规划(SQP)算法.数值试验表明,迭代次数和计算时间少于只取单一参数的传统算法.     

Parameter estimation of ordinary differential equations

This paper addresses the development of a new algorithm forparameter estimation of ordinary differential equations. Here,we show that (1) the simultaneous approach combined with orthogonalcyclic reduction can be used to reduce the estimation problemto an optimization problem subject to a fixed number of equalityconstraints without the need for structural information to devisea stable embedding in the case of non-trivial dichotomy and(2) the Newton approximation of the Hessian information of theLagrangian function of the estimation problem should be usedin cases where hypothesized models are incorrect or only a limitedamount of sample data is available. A new algorithm is proposedwhich includes the use of the sequential quadratic programming(SQP) Gauss–Newton approximation but also encompassesthe SQP Newton approximation along with tests of when to usethis approximation. This composite approach relaxes the restrictionson the SQP Gauss–Newton approximation that the hypothesizedmodel should be correct and the sample data set large enough.This new algorithm has been tested on two standard problems.     

等式约束优化一个新的SQP算法

提出了一个处理等式约束优化问题新的SQP算法,该算法通过求解一个增广Lagrange函数的拟Newton方法推导出一个等式约束二次规划子问题,从而获得下降方向．罚因子具有自动调节性,并能避免趋于无穷．为克服Maratos效应采用增广Lagrange函数作为效益函数并结合二阶步校正方法．在适当的条件下,证明算法是全局收敛的,并且具有超线性收敛速度．