Two new least-squares mixed finite element procedures for convection-dominated Sobolev equations

Two new convection-dominated are derived under the approximate solutions least-squares mixed finite element procedures are formulated for solving Sobolev equations. Optimal H（div;Ω）×H1（Ω） norms error estimates standard mixed finite spaces. Moreover, these two schemes provide the with first-order and second-order accuracy in time increment, respectively.

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR A CLASS OF STOKES EQUATION

A least-squares mixed finite element method was formulated for a class of Stokes equations in two dimensional domains. The steady state and the time-dependent Stokes' equations were considered. For the stationary equation, optimal H-t and L-2-error estimates are derived under the standard regularity assumption on the finite element partition ( the LBB-condition is not required). Far the evolutionary equation, optimal L-2 estimates are derived under the conventional Raviart-Thomas spaces.

An Adaptive Least-Squares Mixed Finite Element Method for Fourth Order Parabolic Problems

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.

Least-Squares Finite Element Method for the Steady Upper-Convected Maxwell Fluid

In this paper, a least-squares finite element method for the upper-convected Maxell (UCM) fluid is proposed. We first linearize the constitutive and momentum equations and then apply a least-squares method to the linearized version of the viscoelastic UCM model. The L2 least-squares functional involves the residuals of each equation multiplied by proper weights. The corresponding homogeneous functional is equivalent to a natural norm. The error estimates of the finite element solution are analyzed when the conforming piecewise polynomial elements are used for the unknowns.

A Unified Dimensionless Parameter for Finite Element Mesh for Beams Resting on Elastic Foundation

Discretising a structure into elements is a key step in finite element(FE)analysis.The discretised geometry used to formulate an FE model can greatly affect accuracy and validity.This paper presents a unified dimensionless parameter to generate a mesh of cubic FEs for the analysis of very long beams resting on an elastic foundation.A uniform beam resting on elastic foundation with various values of flexural stiffness and elastic supporting coefficients subject to static load and moving load is used to illustrate the application of the proposed parameter.The numerical results show that(a)Even if the values of the flexural stiffness of the beam and elastic supporting coefficient of the elastic foundation are different,the same proposed parameter“s”can ensure the same accuracy of the FE solution,but the accuracy may differ for use of the same element length;(b)The proposed dimensionless parameter“s”can indeed be used as a unified index to generate the mesh for a beam resting on elastic foundation,whereas the use of the same element length as a criterion may be misleading;(c)The errors between the FE and analytical solutions for the maximum vertical displacement,shear force and bending moment of the beam increase with the dimensionless parameter“s”;and(d)For the given allowable errors for the vertical displacement,shear force and bending moment of the beam under static load and moving load,the corresponding values of the proposed parameter are provided to guide the mesh generation.

Pullout of soil nail with circular discs: A three-dimensional finite element analysis

An internal failure mode for a soil-nailed system consists of failure at nail heads,slope facing,nail strength,along groutesoil interface and pullout failure.A better understanding of pullout of soil nail thus becomes important to assess the stability of a soil-nailed system.In the present study,an investigation into the pullout behaviour of soil nail with circular discs along the shaft has been carried out by a threedimensional finite element analysis using Abaqus/Explicit routine.A total of 67 simulations have been performed to accurately predict the pullout behaviour of soil nail.The soil nail under study has circular discs along its shaft varying in numbers from 1 to 4.The pullout of this soil nail in a pullout test box has been simulated with a constant overburden pressure of 20 kPa acting on the nail.The pullout load edisplacement characteristics,stresses around soil nail and failure mechanism during pullout are studied.Variations of dimensionless factors such as normalised pullout load factor and bearing capacity factor have been obtained with different combinations of parameters in terms of relative disc spacing ratio,anchorage length ratio,embedment ratio,diameter ratio and displacement ratio.From the results of analyses,it is found that nail with more circular discs requires higher pullout load.There are critical relative disc spacing ratio and diameter ratio which significantly affect the pullout behaviour of nail.

NEGATIVE NORM LEAST-SQUARES METHODS FOR THE INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS

The purpose of this article is to develop and analyze least-squares approximations for the incompressible magnetohydrodynamic equations. The major advantage of the least-squares finite element method is that it is not subjected to the so-called Ladyzhenskaya-Babuska-Brezzi （LBB） condition. The authors employ least-squares functionals which involve a discrete inner product which is related to the inner product in H^-1（Ω）.

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR SADDLE-POINT PROBLEM

In this paper, a least-squares mixed finite element method for the solution of the primal saddle-point problem is developed. It is proved that the approximate problem is consistent ellipticity in the conforming finite element spaces with only the discrete BB-condition needed for a smaller auxiliary problem. The abstract error estimate is derived. [ABSTRACT FROM AUTHOR]

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENT FOR SECOND-ORDER ELLIPTIC PROBLEMS

In this paper the least-squares mixed finite element is considered for solving second-order elliptic problems in two dimensional domains. The primary solution u and the flux σ are approximated using finite element spaces consisting of piecewise polynomials of degree k and r respectively. Based on interpolation operators and an auxiliary projection, superconvergent H1-error estimates of both the primary solution approximation uh and the flux approximation σh are obtained under the standard quasi-uniform assumption on finite element partition. The superconvergence indicates an accuracy of O(hr+2) for the least-squares mixed finite element approximation if Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order r are employed with optimal error estimate of O(hr+1).

A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS

In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.

无量纲最小二乘有限元法GPU实现及其在变压器绕组流场仿真中的应用研究

为了提高变压器流体场的计算效率,利用统一计算设备架构(compute unified device architecture,CUDA)实现流场的并行计算。首先基于C语言实现无量纲最小二乘有限元法的串行计算方法。然后在Visual Studio 2019+CUDA 10.2的环境下实现并行运算,对于串行程序中最耗时的线性方程组求解部分调用了CUDA自带的函数库进行优化;在大规模模型的数值计算中使用了十字链表格式存储整体刚度阵形成时的非零元素,以解决满阵存储时的内存不足问题。为验证所提方法的有效性,针对方腔模型,分析了不同剖分网格数量下的加速比,分析结果表明,数据规模越大并行效果越好,验证了无量纲最小二乘有限元并行程序的准确性和高效性。最后将该方法应用到大型变压器绕组的流体场分析中,取得了18.3倍的加速效果,为产品级变压器的流体场仿真奠定了基础。

LEAST-SQUARES MIXED FINITE ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

Focuses on the formulation of a number of least-squares mixed finite element schemes to solve the initial boundary value problem of a nonlinear parabolic partial differential equation. Convergence analysis; Information on the least-squares mixed element schemes for nonlinear parabolic problem.

基于GPU并行加速变压器二维瞬态流场问题的研究及应用

针对采用无量纲最小二乘有限元法计算变压器二维瞬态流体场问题时计算时间长,效率低的问题,拟采用GPU对瞬态流体场程序进行并行加速。将瞬态流体场计算程序中计算量最大的两部分,即单元刚度阵的形成和稀疏线性方程组的求解,移植到GPU上运算,从而大幅减少计算时间。同时采用十字链表法和CSR稀疏存储结构存储方程组稀疏矩阵中的非零元素,以降低内存消耗。使用方腔驱动流模型验证了GPU并行程序的有效性,并且并行程序加速比随方腔模型规模的增大而增大。将GPU并行程序应用于变压器绕组模型的瞬态流体场仿真分析,分析结果表明,相较于串行程序,GPU并行程序加速比达到16倍左右。论文所实现的基于GPU并行计算方法为产品级变压器瞬态流体场仿真奠定了基础。

激光冲击强化TC4钛合金强化层弹塑性本构参数反演分析

目的 获取TC4钛合金激光冲击强化层的弹塑性本构模型参数,结合纳米压痕试验和有限元模拟技术,进行激光冲击强化TC4钛合金的材料参数反演计算。方法 首先,在TC4钛合金试样侧面沿强化层深度方向进行纳米压痕测试,获得距表面不同距离处的载荷-压入深度曲线。进而,基于幂律应变硬化模型,通过无量纲方程和有限元模拟反演得到激光冲击强化TC4钛合金梯度强化层的弹塑性参数。最后,将反演获得的弹塑性本构模型材料参数用于有限元模拟,将模拟结果与试验结果进行对比,验证参数反演结果的合理性。结果 强化层表面的弹性模量和纳米硬度较母材分别提高了11%和30%,强化层内的应变硬化指数和屈服强度沿深度方向分别递增和递减。模拟的载荷-压入深度曲线与试验曲线吻合较好,最大压入载荷、弹性模量和纳米硬度的模拟误差分别小于1%、7%和3%,证实了参数反演结果的合理性。结论 通过无量纲方程反演算法得到的强化层本构参数有较强的可信度。激光冲击强化可有效提升TC4钛合金的表面力学性能,强化层的本构参数呈梯度分布,表面的抗塑性变形能力大幅提升。

球形压痕法评价材料屈服强度和应变硬化指数的有限元分析

采用量纲分析法,建立了直径1 mm球形压头的无量纲函数.通过此无量纲函数和有限元计算,可推算金属材料的屈服强度和应变硬化指数.通过模拟,得到了屈服应变从0.00769到0.04范围的材料常数的拟合函数,并利用此拟合函数得到了屈服强度和应变硬化指数.通过模拟验证,此方法提高了计算精度并扩大了材料的计算范围.所获得的屈服强度平均误差是1.6%,应变硬化指数的平均误差12.6%.

基于混合有限元法和降阶技术的油浸式变压器绕组2维瞬态流-热耦合场分析

为准确快速计算油浸式变压器绕组的瞬态温度场,提出了一种基于混合有限元法和降阶技术的2维瞬态流-热耦合计算方法。采用无量纲最小二乘有限元法计算绕组瞬态流场分布,在奇异值分解的基础上构建瞬态流场的降阶模型。采用迎风有限元法计算瞬态温度场,计算过程中考虑变压器油的物性参数受温度影响和绕组损耗的温度效应。采用顺序迭代法求解变压器绕组的瞬态流–热耦合问题。以产品级油浸式变压器局部绕组实验模型为例,采用所提的混合算法和Fluent软件对此实验模型的瞬态流场和温度场进行仿真计算,并与实验测量数据进行对比,所提混合算法和Fluent软件计算出的热点温度最大误差分别为1.02 K和2.5 K,加入降阶技术后,计算速度提升了6.43倍,验证了此混合算法的准确性和高效性。

岩石断裂韧度试样CCNBD临界应力强度因子的全新数值标定

国际岩石力学学会(ISRM)提出的用人字型切槽巴西圆盘(Cracked Chevron Notched Brazilian Disc—CCNBD)测试岩石I型断裂韧度所必需的量纲为一的临界应力强度因子Y*min的精度需要进一步改进。本文对CCNBD试样的Y*min进行了系统的重新标定,CCNBD的几何参数有效范围为0.44≤αB≤1.04、0≤α0≤0.69、0.4≤α1≤0.8。采用有限单元法对CCNBD做三维数值分析,得到了435种CCNBD试样的Y*min值标定结果;在此基础上全面修正了CCNBD试样Y*min计算公式中u和v的取值表;给出了不用查表直接确定CCNBD试样Y*min值的近似表达式,该表达式计算结果与标定结果的相对误差绝对值在1.87%以内。对于ISRM建议的CCNBD标准试样的Y*min值:ISRM标定值0.84比本文结果0.957小12.2%;分片合成法标定值0.947比本文结果小1.0%;子模型法标定值0.943比本文结果小1.5%。本文特别强调了任何CCNBD试样Y*min的取值必须在它对应的上限和下限所限定的范围内,这一要求对判断Y*min标定值是否合理是很重要的。

基于有限元极限分析的双孔隧道稳定性分析

为进一步探讨在考虑土体内摩擦角情况下,双孔隧道在支护反力及地面超载共同作用下的相互作用规律及稳定性分析,首先,基于有限元极限分析法,采用MATLAB平台编制了相关计算程序;此外,为减少各参数之间的交叉影响,引入无量纲分析方法;然后,将本文模型的计算结果与已有成果进行对比,验证了本文分析方法的合理性;最后,在此基础上求得各影响因素下的隧道稳定性参数(Ns),并绘制了无量纲分析图表。结果表明:(1)Ns随φ的增大逐渐增大,且当φ较大时,在Ns未达到稳定值之前,对其产生显著影响;(2)Ns随S/B的增大而增大并趋于稳定;(3)φ的存在与否对Ns随着γB/c的变化产生相反的规律,且随φ的增大,γB/c的变化对临界间距比(S/B)的值影响减弱;(4)临界间距比随H/B的增大不断增大;(5)S/B和φ是影响隧道破坏模式的关键因素。

含中心圆孔有限板动应力集中问题的有限元分析

基于有限元软件ANSYS的计算,对含有中心圆孔的有限板在板边界受到上阶跃波的作用时孔边动应力集中的响应进行分析。得到不同无量纲几何尺寸以及无量纲波数下的动应力响应曲线,并由此得到一些关于动应力响应规律性的结论。另外,对动应力集中系数、最大动应力与静态特征进行比较,并对比较结果进行讨论。

无量纲分析在采空区稳定性分析中的应用

采空区稳定性分析不仅是矿山安全生产的保障,也是确保矿区周边居民及生态环境良好的重要工作。采用有限元分析采空区稳定性时,岩体物理力学参数的准确性对分析结果有着重大的影响。当不能采取直接手段确定岩体物理力学参数时,可借助π定理的无量纲分析,对岩体物理力学参数之间的函数关系进行整理,得到目标参数与相关参数之间的控制方程,利用已知的相关参数来确定目标参数。以岩体弹性模量E为例,结合垴窖铁矿的数据资料解释说明。分析结果表明,该方法在确定岩体物理力学参数时具备一定的可取性。