On the adaptive finite element analysis of the Kohn–Sham equations: methods,algorithms, and implementation

In this paper, details of an implementation of a numerical code for computing the Kohn–Sham equations are presented and discussed. A fully self‐consistent method of solving the quantum many‐body problem within the context of density functional theory using a real‐space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co‐aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon‐monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley & Sons, Ltd.     

相场法模拟动力学各向异性对过冷熔体中晶体生长的影响

采用相场模型定量模拟了动力学各向异性作用下过冷熔体中的晶体生长过程.模拟结果表明,仅存在动力学各向异性时,各向异性系数大小对生长方式选择起着决定性作用.当各向异性较低时,固相以分形方式生长,在生长过程中不存在占优势的生长方向,同时也不存在稳态生长状态;而当各向异性系数大于0.02时,固相以枝晶方式沿〈110〉方向生长.进一步研究表明,枝晶生长稳定性系数随各向异性值的增加而增加,而与动力学系数取值无关.     

Adaptive finite element computation of dielectric and mechanical intensity factors in piezoelectrics with impermeable cracks

The paper deals with the application of an adaptive, hierarchic‐iterative finite element technique to solve two‐dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.     

二维自由振动问题的自适应有限元分析初探

自由振动反映结构动力特性,是抗震分析和结构设计的重要基础。近年来,基于单元能量投影(EEP)法的自适应有限元分析已在一系列线弹性及非线性问题中取得成功,而有限元线法(FEMOL)自适应分析在二维自由振动问题中的应用也被证实是有效的。在此基础上,该文进一步提出二维自由振动问题的自适应有限元分析方法。通过将特征值问题线性化,合理引入二维线性问题的 EEP 超收敛计算和自适应求解技术,该法可得到满足精度要求的自振频率和按最大模度量满足用户给定误差限的振型。该文以弹性薄膜为例,介绍了这一进展,并给出数值算例以表明该方法的有效性和可靠性。     

基于自适应有限元相场模型模拟强制流动条件下的枝晶生长(英文)

应用投影算法与相场法相结合的数学模型,采用基于非均匀网格的自适应有限元法求解该模型,并对强制流动作用下镍过冷熔体中枝晶生长行为进行模拟。模拟结果表明,强迫对流的引入导致枝晶生长的不对称性。当流速小于临界值时,流动对枝晶的不对称生长影响较小;当流速达到或超过临界值时,枝晶生长的控制因素逐渐从热扩散过渡到对流。随着流速的增大,流动法向的一次枝晶臂朝逆流方向倾斜角度增大。而枝晶生长对熔体流动具有明显的影响。随着枝晶尺寸的增大,在顺流区域产生涡流效应,涡流区逐渐扩大并在枝晶尖端出现重熔现象。此外,非均匀网格的自适应有限元方法的CPU耗费时间比均匀网格方法降低一个数量级,并且加速比与计算域尺寸成正比。     

三维弹性自适应有限元及其在拱坝分析中的应用

本文将自适应有限元用于拱坝,应用ZZ后验误差估计方法以及h-型自适应策略,对典型的悬臂梁和拱坝进行三维线弹性自适应有限元计算。计算结果表明,本方法可有效提高计算精度和计算效率,特别是对于具有应力集中的结构,具有显著的效益。     

Development of interior permanent magnet motors reducing harmonic iron losses under field weakening control

In this paper, we present the development of interior magnet motors reducing iron loss at high rotational speed under the flux weakening control. The rotor core and magnet shapes are determined by the automatic numerical calculation using combination of the optimization method and the adaptive finite element method. The optimized motor is manufactured to prove the effectiveness by the measurement of the iron loss. Both results of the calculation and the measurement indicate that the iron loss of the proposed motor at the high rotational speed under the flux weakening control is reduced by half compared with the initial rotor shape while the torque is nearly constant. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 168(3): 59–66, 2009; Published online in Wiley InterScience ( www.interscience. wiley.com ). DOI 10.1002/eej.20827     

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

An s‐adaptive finite element procedure is developed for the transient analysis of 2‐D solid mechanics problems with material non‐linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user‐specified tolerances. The spatial error is quantified by the Zienkiewicz–Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third‐order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s‐adaptive procedure is the use of finite element mesh superposition (s‐refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non‐linear transient problems since it is faster, simpler and more efficient than traditional h‐refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s‐adaptive method for quasi‐static and transient problems with material non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive Finite Element Methods for Microstructures? Numerical Experiments for a 2-Well Benchmark

Macroscopic simulations of non-convex minimisation problems with enforced microstructures encounter oscillations on finest length scales – too fine to be fully resolved. The numerical analysis must rely on an essentially equivalent relaxed mathematical model. The paper addresses a prototype example, the scalar 2-well minimisation problem and its convexification and introduces a benchmark problem with a known (generalised) solution. For this benchmark, the stress error is studied empirically to asses the performance of adaptive finite element methods for the relaxed and the original minimisation problem. Despite the theoretical reliability-efficiency gap for the relaxed problem, numerical evidence supports that adaptive mesh-refining algorithms generate efficient triangulations and improve the experimental convergence rates optimally. Moreover, the averaging error estimators perform surprisingly accurate.     

Adaptive finite element method for shape modification in optimization of electric machines

A method for the modification of finite element meshes based on adaptive meshing for optimization of electric machines is proposed. In the optimization of the machine, the proposed method generates the mesh for the new shape from the previous mesh with minor modification. The locations of the nodes for the new shape are decided by solving a Laplacian equation whose unknowns are the displacements of the nodes. The advantages are illustrated by application to the shape optimization of an IPM motor. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 168(1): 21–28, 2009; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20773