一种有效的边界元/有限元混合法迭代算法及其在回转体自由扭振分析中的应用*

本文给出一种新的边界元/有限元混合法迭代算法,基本做法是将近似的固有频率值代入自由振动问题的基本解,按一般混合法列式,通过迭代逐步修正近似解的值.这种算法避开了一般边界元法需要求解非代数特征值问题的困难,同时数值结果的精度基本上不依赖于区域内单元网格的疏密程度,这都给实际计算带来很多方便.应用于回转体自由扭振问题的分析,得到令人满意的数值结果.     

二次Lagrangian有限元方程的几何多重网格法

为了构造快速求解二次Lagrangian有限元方程的几何多重网格法,在选择二次Lagrangian有限元空间和一系列线性Lagrangian有限元空间分别作为最细网格层和其余粗网格层以及构造一种新限制算子的基础上,提出了一种新的几何多重网格法,并对它的计算量进行了估计.数值实验结果,与通常的几何多重网格法和AMG01法相比,表明了新算法计算量少且稳健性强.     

混合有限元非嵌套网格型预处理法

1 引言 众所周知,二阶椭圆型问题混合有限元离散以后的矩阵是不定的,所以对混合法很难形成一种有效的区域分解法,在文[9]、[10]、[11]中提出了一些混合有限元方法的区域分解法,但在实际计算中有很多局限性。最近Chen对混合有限元法提出一种全新的解释并把它应用到多重网格法中,他的基本思想是混合有限元离散的代数系统实际上等价于某个非协调有限元离散的代数系统,这样可把一个不定问题转化为一个正定问题,本文将基于这种思想考虑混合有限元的区域分解法。 若按传统的Dryia-widlund两水平加性Schwarz方法,要求两层网格间具有嵌套关系,这样在应用中将带来很大的不便。本文将不要求粗网格嵌入细网格中,减少两层网格间的     

离散系统运动方程的Galerkin有限元EEP法自适应求解

对于结构动力分析中的离散系统运动方程,现有算法的计算精度和效率均依赖于时间步长的选取,这是时间域问题求解的难点.基于EEP(element energy projection)超收敛计算的自适应有限元法,以EEP超收敛解代替未知真解,估计常规有限元解的误差,并自动细分网格,目前已对诸类以空间坐标为自变量的边值问题取得成功.对离散系统运动方程建立弱型Galerkin有限元解,引入基于EEP法的自适应求解策略,在时间域上自动划分网格,最终得到所求时域内任一时刻均满足给定误差限的动位移解,进而建立了一种时间域上的新型自适应求解算法.     

任意区域的网格自动生成和任意高阶有限元方法

本文首先简要介绍非拟合网格有限元方法求解复杂区域上椭圆问题的发展现状.然后结合最近本文作者发展的非拟合网格有限元方法,针对二阶椭圆方程提出一种任意光滑区域上的任意高阶协调有限元方法.本文在带悬点的Cartesian网格上自动生成诱导网格,在诱导网格上构造协调的高阶有限元空间,采用Nitsche技术处理Dirichlet边界条件,并证明方法的适定性和hp先验误差估计.数值算例验证了本文的理论结果.     

有限元网格的超收敛测度及其应用

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.     

径向基无网格法在蒸发作用下一维排水问题中的应用

无网格方法是一种只需要节点信息而不需要划分网格的数值计算方法.利用径向基无网格法求解一维排水问题,导出了在地下水蒸发强度与埋深成线性关系时,一维不稳定流方程的无网格算法,通过迭代求其数值解.与常用的地下水位近似理论解、有限元解以及实测数据进行比较,均表明该算法误差小、收敛性好、实用性强,可以很好地应用于农田排水问题的计算.     

抛物方程初边值问题连续有限元的超收敛性

研究了一类一维抛物方程初边值问题的连续有限元方法.在空间上进行任意m次有限元半离散,在时间方向上进行二次连续有限元后,获得了一个稳定的全离散计算格式.利用单元分析法校正技术的新思想进行理论分析,连续有限元解在剖分网格节点上具有超收敛性.     

定常Navier-Stokes方程的一种高效稳定有限元方法

提出了二维定常Navier-Stokes(N-S)方程的一种两层稳定有限元方法.该方法基于局部高斯积分技术,通过不满足inf-sup条件的低次等阶有限元对N-S方程进行有限元求解.该方法在粗网格上解定常N-S方程,在细网格上只需解一个Stokes方程.误差分析和数值试验都表明:两层稳定有限元方法与直接在细网格上采用的传统有限元方法得到的解具有同阶的收敛性,但两层稳定有限元方法节省了大量的工作时间.     

加罚Navier—Stokes方程的最佳非线性Galerkin算法

该文提出了求解二维加罚Navier-Stokes方程的最佳非线性Galerkin算法．这个算法在于在粗网格有限元空间上求解一非线性子问题，在细网格增量有限元空间Wh上求解一线性子问题．如果线性有限元被使用及，则该算法具有和有限元Galerkin算法同阶的收敛速度．然而该文提出的算法可以节省可观的计算时间．     

Stoßwellenausbreitung in Rohrverzweigungen

Summary The interaction of a shock wave with the junction of a straight duct having a side branch is considered. The paper checks the validity of three hypotheses. The first and second one calculate the shock waves after the junctions only by a geometrical consideration. A comparison with experimental values shows only an unsatisfactory result for weak shock waves. The third theory assumes that the pressure drop across the junction, in the quasi-steady flow which takes place after the incident shock wave, is the same as in steady flow tests. The pressure drops were measured for different junctions and the results of the calculation were compared with experimental values from a shock tube, both for circular and rectangular channels. A good agreement was obtained.     

Morse Theory for Geodesics in Conical Manifolds

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided.     

A New Conical Regularization for Some Optimization and Optimal Control Problems: Convergence Analysis and Finite Element Discretization

In this article, we study an abstract constrained optimization problem that appears commonly in the optimal control of linear partial differential equations. The main emphasis of the present study is on the case when the ordering cone for the optimization problem has an empty interior. To circumvent this major difficulty, we propose a new conical regularization approach in which the main idea is to replace the ordering cone by a family of dilating cones. We devise a general regularization approach and use it to give a detailed convergence analysis for the conical regularization as well as a related regularization approach. We showed that the conical regularization approach leads to a family of optimization problems that admit regular multipliers. The approach remains valid in the setting of general Hilbert spaces and it does not require any sort of compactness or positivity condition on the operators involved. One of the main advantages of the approach is that it is amenable for numerical computations. We consider four different examples, two of them elliptic control problems with state constraints, and present numerical results that completely support our theoretical results and confirm the numerical feasibility of our approach. The motivation for the conical regularization is to overcome the difficulties associated with the lack of Slater's type constraint qualification, which is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, vector optimization, set-valued optimization, sensitivity analysis, variational inequalities, among others.     

Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.     

Frequency analysis of rotating truncated conical shells using the Haar wavelet method

In this study, we present an analysis of the frequency characteristics of rotating truncated conical shells using the Haar wavelet method. Based on the Love first-approximation theory, the governing equations are formulated by considering the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotation. The displacement field is expressed as the Haar wavelet series in the axial direction and trigonometric functions in the circumferential direction. By considering the boundary conditions, the eigenvalue equation is obtained to determine the vibration behaviors of rotating conical shells. To validate the current analysis, the results obtained by the proposed method are compared with those reported previously, where the agreement is very good. Finally, we investigate the effects of the geometrical parameters, rotation speed, and boundary conditions on the vibration characteristics of rotating conical shells and the results are presented.     

Optimal design of a sewer line using Linear Programming

Wastewater collection systems greatly contribute to the cost of the overall municipal sewerage system; a cost-effective design of the collection system will provide significant savings towards the cost of wastewater services. It is impossible to evaluate the full impact that each pipe size and slope would have on the overall cost of the collection system with intuitive designs. However, these solutions generally satisfy the design objectives within the given constraints. A survey of the literature indicates that various optimisation techniques are being applied for least-cost solutions. In general these approaches provide continuous pipe sizes, which are converted to closest commercial sizes for adoption, which would heavily dilute the optimal outcome. Search methods are also adopted to obtain cost-effective design solutions using directly commercial pipe sizes, which are computationally expensive. In the design of a sewerage system, a sewer line is a basic unit occurring repeatedly in the design-process and finally the combinations of these basic units formulate the complete sewer system. However, the branch sewer lines, main sewers, trunk sewers, pumping stations, treatment plant and outfall sewers are in general the main components of an urban wastewater collection, treatment and disposal systems. A method has been developed to optimise this basic unit using Linear Programming technique without transforming nonlinear objective function or constraint equations into linear functions and incorporating commercially available pipe sizes directly in the problem formulation. The current research area of optimal sewer system design is focusing equally on economic considerations and hydraulic feasibility and moving away from conventional design guidelines based on only self cleaning velocity concepts for node to node sewer link hydraulic design. This paper is a step forward in developing optimal design approaches of sewer systems.     

Modeling and numerical analysis of the thread rolling process

The paper presents the physical and mathematical models of deformations (displacements and strains) and the stress in the cold process of the thread rolling. The process is considered as a geometrical and physical nonlinear, initial as well as a boundary value problem. The phenomena of a typical incremental step were described using a step-by-step incremental procedure, in the updated Lagrangian formulation. The state of strains was described by Green-Lagrange's tensor, while the state of stress was described by the second symmetrical Pioli-Kirchhoff's tensor. The object was treated as an elastic (in the reversible zone) and visco-plastic body (in the non-reversible zone) with mixed hardening. The variational equation of the motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. In a numerical analysis, boundary condition for a displacement increment, was determined in the model investigation. The results of a numerical analysis were compared and verified in an experimental investigation. Examples of calculations of the influence of a friction coefficient on the state of the deformation and stress, and an example application for this method of thread rolling were presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of time-dependent material properties on the crack behavior in the interface of two polymeric materials

Crack penetration through the bimaterial interface of two polymers is investigated numerically. Due to the practical importance of the problem, a crack in a three-layer pipe consisting of a main and two, inner and outer, protective layers is analyzed in this paper. The prime aim is to formulate the conditions under which the crack stays arrested at the interface between the protective layer and the main pipe or penetrates into the interface and causes failure of the main pipe and consequently of the entire pipe system. The crack tip stress field is described by using a generalized stress intensity factor for cases where the crack touches the interface and the stress singularity exponent differs from 1/2. In the case of short-term applications, the stress state on the interface is given simply by a combination of the elastic properties of materials of the main pipe and the protective layers. In long-term applications, the time-dependent properties of the materials can significantly influence the stress state of the interface and can lead to considerable changes in failure conditions. The results presented here may contribute to a more accurate estimation of the residual lifetime of multilayer pipes.     

改进的非协调广义混合单元及性能分析

非协调广义混合单元最突出的特点是避免了传统混合单元中系数矩阵主对角线上存在零元素的问题，因此位移和应力结果的收敛是稳定的.以最小势能原理和H R变分原理为基础，联合增强假设应变理论建立了新的8结点非协调广义混合单元.一方面，该单元保持了已有非协调广义混合单元的全部优点；另一方面，该单元简化了积分计算.数值实例表明，改进的非协调广义混合单元的数值结果精度高，计算速度快并且对单元的几何扭曲敏感度低.