Monitoring a PGD solver for parametric power flow problems with goal‐oriented error assessment

The parametric analysis of electric grids requires carrying out a large number of power flow computations. The different parameters describe loading conditions and grid properties. In this framework, the proper generalized decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to (1) iterating algebraic solver, (2) number of terms in the separable greedy expansion, and (3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal‐oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end‐user. The paper discusses how to compute the goal‐oriented error estimates. This requires linearizing the error equation and the quantity of interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Parametric uncertainty quantification using proper generalized decomposition applied to neutron diffusion

In this paper, a proper generalized decomposition (PGD) approach is employed for uncertainty quantification purposes. The neutron diffusion equation with external sources, a diffusion-reaction problem, is used as the parametric model. The uncertainty parameters include the zone-wise constant material diffusion and reaction coefficients as well as the source strengths, yielding a large uncertain space in highly heterogeneous geometries. The PGD solution, parameterized in all uncertain variables, can then be used to compute mean, variance, and more generally probability distributions of various quantities of interest. In addition to parameterized properties, parameterized geometrical variations of three-dimensional models are also considered in this paper. To achieve and analyze a parametric PGD solution, algorithms are developed to decompose the model's parametric space and semianalytically integrate solutions for evaluating statistical moments. Varying dimensional problems are evaluated to showcase PGD's ability to solve high-dimensional problems and analyze its convergence.     

Iterative LCP solvers for non‐local loading–unloading conditions

This paper deals with the finite element analysis of a certain class of non‐local dissipative constitutive models, where the canonical pointwise backward‐Euler scheme cannot be employed for satisfying the loading–unloading conditions. In the presence of a non‐local dissipation, the admissibility conditions in a point depend on the inelastic strain increment of the surrounding points and can be cast as a linear complementarity problem (LCP) involving all Gauss points of the process zone. In order to actually solve the LCP, the use of iterative algorithms that can be easily embodied into existing FE codes is discussed. The performance of the proposed algorithms is tested in 1D and 2D examples for both elastoplastic and damaging materials. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Parametric constitutive model of plain-weave fabric reinforced composite ply

A computational model that allows to explicitly determine orthotropic elastic constants of plain-weave fabric-reinforced composite ply as functions of microstructure parameters has been developed in this study. These relationships are not given in the form of analytical formulae (as it is in the case of approximate analytical models) but in the form of an extensive database of numerically evaluated results for different microstructure instances and a numerical scheme that interpolates the results. To build the database, a standard finite-element-based homogenization technique of a periodic representative volume element is employed. As a result, a numerical algorithm is provided that may be easily employed in FE codes as a part of a regular constitutive subroutine. Sensitivity of the composite elastic constants with respect to the microstructure parameters is also directly available from the model.     

Rapid evaluation of notch stress intensity factors using the peak stress method: Comparison of commercial finite element codes for a range of mesh patterns

The peak stress method (PSM) is an engineering, finite element (FE)‐oriented method to rapidly estimate the notch stress intensity factors by using the singular linear elastic peak stresses calculated from coarse FE analyses. The average element size adopted to generate the mesh pattern can be chosen arbitrarily within a given range. Originally, the PSM has been calibrated under pure mode I and pure mode II loadings by means of Ansys FE software. In the present contribution, a round robin between 10 Italian universities has been carried out to calibrate the PSM with 7 different commercial FE codes. To this aim, several two‐dimensional mode I and mode II problems have been analysed independently by the participants. The obtained results have been used to calibrate the PSM for given stress analysis conditions in (i) FE software, (ii) element type and element formulation, (iii) mesh pattern, and (iv) criteria for stress extrapolation and principal stress analysis at FE nodes.     

Real‐time direct integration of reduced solid dynamics equations

A new method is developed here for the real‐time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black‐box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so‐called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.     

ITERATIVE SOLUTION OF COUPLED FE/BE DISCRETIZATIONS FOR PLATE–FOUNDATION INTERACTION PROBLEMS

In the present paper, a scheme is developed for the coupled FE/BE analysis of a plate–foundation interaction problem, in which the boundary element equations of the foundation are not explicitly assembled with the finite element equations of the plate, but instead an iterative procedure is proposed to obtain the final coupled solution. This iterative scheme preserves the nature of the BE and FE approaches and the coupled procedure can be easily implemented within an integrated FEM/BEM software environment. The scheme also reduces the computer storage requirement and avoids the error introduced by symmetrization of the BE equations. In addition, some important issues related to the scheme, such as convergence conditions and parameter selection, are discussed. A numerical example is provided to illustrate pthe benefits of the scheme. It is noted, however, that the overall performance of the proposed scheme when compared with the conventional direct solution of the unsymmetric equations arising from the explicit coupling of the FE and BE equations, depends on the choice of a free parameter and a matrix contained in the scheme.     

Three-dimensional Finite Element Buckling Analysis of Honeycomb Sandwich Composite Shells with Cutouts

This paper investigates the buckling response of honeycomb sandwich composite shells with cutouts under axial compression. The Wilson's incompatible solid Finite Element (FE) is used around cutouts to obtain the detail stress distribution there. While to reduce the computational expense, a special multilayered relative degrees-of-freedom (DOF) shell FE is used to model the regions far from the cutouts. The efficiency and accuracy of this modeling scheme are illustrated by two benchmarks. Then parametric studies are carried out to reveal how the buckling response is influenced by the area, the shape and the orientation of cutouts.     

PAM-CRASH碰撞模拟中主要控制参数影响的分析

在汽车耐撞性研究中有限元模拟计算已成为主要的研究手段。本文基于非线性显式有限元软件PAM－CRASH对某车车架前部碰撞过程的模拟计算，分析了有限元模型的主要控制参数对计算结果和效率的影响，其中包括网格尺寸、沿单元厚度的积分点个数、单元类型和沙漏（Hourglass)控制。比较了采用不同控制参数对计算结果的载荷－时间曲线、时间和沙漏能量等的影响。确定该车架的模拟计算控制参数选择范围。     

复合材料正交加筋层合圆柱壳结构阶梯式挖补修理的参数化研究

利用三维参数化有限元分析模型,对复合材料正交加筋层合圆柱壳结构的阶梯式挖补修理进行了参数分析.主要考虑了载荷情况、阶梯数、铺层顺序以及补片铺层错误等修理参数对阶梯式挖补修理结构胶层剥离应力和剪切应力分布情况的影响规律,并利用绝对值的平均值和均方差来描述胶层剪切应力剥离应力的分布情况,以确定最优的阶梯式挖补修理参数.基于该文建立的三维有限元模型可以深入地了解复合材料正交加筋层合圆柱壳结构阶梯式挖补修理区域胶层应力分布的详细情况,而且参数化分析结论也可为复合材料正交加筋层合板圆柱壳结构的阶梯式挖补修理方案的设计和分析提供参考.     

Thermal-stress analysis of RC beams reinforced with GFRP bars

This paper aims to develop a 3D nonlinear finite element (FE) model that is capable of accurately predicting the performance of reinforced concrete (RC) beams reinforced with internal Glass Fiber-Reinforced Polymer (GFRP) bars when exposed to fire loading. The developed FE model is based on tested experimental data collected from the open literature. The model accounts for the variation in the thermal and mechanical constituent materials with temperature associated with the RC beam. To study the heat transfer mechanism and mechanical behavior of the RC beam, transient thermal-stress finite element analysis is performed using the ANSYS. It was shown that the FE predicted temperature and mid-span deflection results are in a good agreement with that of the measured experimental data. The validated FE model is used to conduct a parametric study to investigate the effect of the different parameters on the flexural performance of the reinforced beam specimens. The parametric study consisted of varying the concrete cover thickness as well as exposing the FE model to different fire curves. It is concluded that successful FE modeling of this structure would provide an economical and alternative solution to expensive and time consuming experimental testing. Other observations and recommendations are also discussed.     

An adaptive finite element framework for fatigue crack propagation

A new finite element (FE) framework for fatigue crack propagation (FCP) analysis is proposed. This framework combines the simplicity of standard industrial FCP analysis with the generality and accuracy of a full FE analysis and can be implemented on a small computer by combining standard existing computational tools. In this way it constitutes an attractive alternative to existing approaches. The framework is based on linear elastic fracture mechanics and on FE mesh adaptation. Some novel features are introduced in several of its steps in order to make it efficient and at the same time reasonably accurate. Various computational aspects of the scheme are discussed. A few two‐dimensional numerical examples involving FCP in thin sheets under plane‐stress conditions are presented to demonstrate the performance of the framework. Some of the numerical results are compared to those of laboratory experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

Parametric topology optimization with multiresolution finite element models

We present a methodical procedure for topology optimization under uncertainty with multiresolution finite element (FE) models. We use our framework in a bifidelity setting where a coarse and a fine mesh corresponding to low- and high-resolution models are available. The inexpensive low-resolution model is used to explore the parameter space and approximate the parameterized high-resolution model and its sensitivity, where parameters are considered in both structural load and stiffness. We provide error bounds for bifidelity FE approximations and their sensitivities and conduct numerical studies to verify these theoretical estimates. We demonstrate our approach on benchmark compliance minimization problems, where we show significant reduction in computational cost for expensive problems such as topology optimization under manufacturing variability, reliability-based topology optimization, and three-dimensional topology optimization while generating almost identical designs to those obtained with a single-resolution mesh. We also compute the parametric von Mises stress for the generated designs via our bifidelity FE approximation and compare them with standard Monte Carlo simulations. The implementation of our algorithm, which extends the well-known 88-line topology optimization code in MATLAB, is provided.     

A meshless method for axi‐symmetric poroelastic simulations: numerical study

A meshless model, based on the meshless local Petrov–Galerkin (MLPG) approach, is developed and implemented for the solution of axi‐symmetric poroelastic problems. The solution accuracy and the code performance are investigated on a realistic application concerning the prediction of land subsidence above a deep compacting reservoir. The analysis addresses several numerical issues, including the parametric selection of the optimal size of the local sub‐domains for the weak form and the nodal supports, the appropriate integration rule, and the linear system solver. The results show that MLPG can be more accurate than the standard finite element (FE) method on coarse discretizations, with its superiority decreasing as the nodal resolution increases. This is due to both a slower convergence rate and a progressively higher computational cost compared to FE. These drawbacks can be partially mitigated by improving the efficiency of the numerical integration and the system solver with the aid of projection techniques based on Krylov subspace methods. The outcome of the present analysis supports the development of coupled methods where a limited number of MLPG nodes are used to locally improve a FE solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Solution of viscoelastic scattering problems in linear acoustics using hp Boundary/Finite Element Method

The interaction of acoustic waves with submerged structures remains one of the most difficult and challenging problems in underwater acoustics. Many techniques such as coupled Boundary Element (BE)/Finite Element (FE) or coupled Infinite Element (IE)/Finite Element approximations have evolved. In the present work, we focus on the steady‐state formulation only, and study a general coupled hp‐adaptive BE/FE method. A particular emphasis is placed on an a posteriori error estimation for the viscoelastic scattering problems. The highlights of the proposed methodology are as follows: (1) The exterior Helmholtz equation and the Sommerfeld radiation condition are replaced with an equivalent Burton–Miller (BM) boundary integral equation on the surface of the scatterer. (2) The BM equation is coupled to the steady‐state form of viscoelasticity equations modelling the behaviour of the structure. (3) The viscoelasticity equations are approximated using hp‐adaptive FE isoparametric discretizations with order of approximation p⩾5 in order to avoid the ‘locking’ phenomenon. (4) A compatible hp superparametric discretization is used to approximate the BM integral equation. (5) Both the FE and BE approximations are based on a weak form of the equations, and the Galerkin method, allowing for a full convergence analysis. (6) An a posteriori error estimate for the coupled problem of a residual type is derived, allowing for estimating the error in pressure on the wet surface of the scatterer. (7) An adaptive scheme, an extension of the Texas Three Step Adaptive Strategy is used to manipulate the mesh size h and the order of approximation p so as to approximately minimize the number of degrees of freedom required to produce a solution with a specified accuracy. The use of this hp‐scheme may exhibit exponential convergence rates. Several numerical experiments illustrate the methodology. These include detailed convergence studies for the problem of scattering of a plane acoustic wave on a viscoelastic sphere, and adaptive solutions of viscoelastic scattering problems for a series of MOCK0 models. Copyright © 1999 John Wiley & Sons, Ltd.     

不锈钢压弯构件整体稳定性能有限元研究与承载力计算方法

该文采用有限元软件ANSYS建立了焊接不锈钢压弯构件有限元模型,并基于试验结果验证了该模型的有效性。采用经验证的有限元模型分析了残余应力和几何初始缺陷对构件整体稳定性能的影响,并在广泛的截面等级、荷载偏心和长细比组合下进行了参数分析以产生足够的数据点。最后,利用有限元分析结果对我国现行《不锈钢结构技术规程》（CECS 410―2015）中的计算方法进行评价,并给出了提高计算精度的改进方法。结果表明：CECS 410―2015中的计算方法对于不锈钢焊接工字形和箱形截面压弯构件整体稳定承载力的预测均偏于保守,改进的计算方法能够更精确地预测不锈钢焊接工字形和箱形截面压弯构件的整体稳定承载力。     

Finite element buckling analysis of rotationally periodic laminated composite shells

A finite element (FE) buckling analysis of rotationally periodic laminated composite shells is performed in this paper. Because the buckling mode of such structures is characterized as rotationally periodic, a corresponding FE buckling analysis scheme is proposed to reduce the computational expenses. Moreover, a new kind of relative degrees‐of‐freedom element is developed, which can be connected to other solid elements with ease and can yield satisfactory results with a relatively coarse FE mesh. Numerical results of two laminated cylindrical shells subjected to lateral pressure are compared with theoretical ones. The good agreement of them shows the validity of this new computational strategy. Finally, a practical structure is analysed to demonstrate the advantage of this method. Copyright © 2001 John Wiley & Sons, Ltd.     

A note on finite element implementation of sandwich shell homogenization

A finite element (FE) implementation for sandwich shell through‐thickness homogenization is presented. The homogenization is performed within the analysis constitutive procedure and is suitable for the FE analysis of sandwich shells using explicit time‐integration scheme. Copyright © 2000 John Wiley & Sons, Ltd.