复杂边界旋转Navier-Stokes方程的几何方法及二度并行算法

本文提出了一种求解复杂边界旋转Navier-Stokes方程的微分几何方法及其二度并行算法.此方法可用于求解透平机械内部叶片间流动和飞行器外部绕流等复杂流动问题.假设流动区域可以用一系列光滑曲面■_k,k=1,2,…,K分割为一系列子区域(称作流层),通过应用微分几何的方法,三维N-S算子可以分解为两类算子之和:建立在曲面■_k切空间上"膜算子"和曲面■_k法线方向的"挠曲算子",将挠曲算子应用欧拉中心差商来逼近,由此得到建立在■_k上的"2D-3C"N-S方程.求解2D-3C N-S方程并且反复迭代直到收敛.我们得到"二度并行算法",它是2D-3C N-S方程并行算法与k方向的同时并行.这个算法的优点在于,(1)可以改进由于复杂边界造成的不规则三维网格引起的逼近解的精度;(2)为克服边界层的数值效应,在边界层内可以构造很密的流层,形成三维多尺度的网格,是一个很好的边界层算法;(3)这个方法不同于经典的区域分解算法,这里的每个子区域只需要求解一个"2D-3C"N-S方程,而经典区域分解方法要在每个子区域上求解三维问题.     

分析土壤半无限域的一种有限层方法(Ⅱ)——几种特殊情形及算例*

本文就几种特殊情形用作者所研究的一种用于横观各向同性体动力学的有限层法作了简化分析,分别讨论了二维问题、轴对称问题以及静力问题,并推广到介质具有粘性性质的情形.对于轴对称情形,本文还给出两个算例,表明作者所研究的有限层法用于分析半无限域层状土壤介质是可行的,因而为研究土壤与结构相互作用问题提供了一条新途径.     

Growth of Self‐Similar Graphs

Locally finite self‐similar graphs with bounded geometry and without bounded geometry as well as non‐locally finite self‐similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor ν and the volume scaling factor μ can be defined similarly to the corresponding parameters of continuous self‐similar sets. There are different notions of growth dimensions of graphs. For a rather general class of self‐similar graphs, it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous self‐similar fractals: . © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 224–239, 2004     

Efficient simulation of wave propagation in heterogeneous materials

In this contribution we explain the core idea of the recently proposed spectral cell method, which combines a fictitious domain approach with finite elements of high order and noticeably relieves the burden of mesh generation. Moreover, it employs mass lumping techniques and significantly reduces the computational expenditure. Our studies show that the spectral cell method leads to similar results yet with less computational effort as compared to standard techniques. These properties turn the method to a viable tool for the wave propagation analysis of structures that obey a complicated geometry. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of Guided Lamb Wave Propagation (GW) in Honeycomb Sandwich Panels

The paper aims to introduce the guided lamb wave propagation (GW) in a honeycomb sandwich panels to be used in the health monitoring applications. Honeycomb sandwich panels are well-known as lightweight structures with a good stiffness behavior and a wide range of applications in different industries. Due to the complex geometry and complicated boundary conditions in such a structure, the development of analytical solutions for describing the wave propagation and the interaction of waves with damages is hardly possible. Therefore dimensional finite element simulations have been used to model GW for different frequency ranges and different sandwich panels with different geometrical properties. The waves, which are highly dispersive, have been excited by thin piezoelectric patches attached to the surface of the structure. In the first step, the honeycomb panel has been simplified as an orthotropic layered continuum medium. The required material data have been calculated by applying a numerical homogenization method for the honeycomb core layer. The wave propagation has been compared in the homogenized model with the real geometry of a honeycomb sandwich panel. Such calculations of high frequency ultrasonic waves are costly, both in creating a proper finite element model as well as in the required calculation time. In this paper the influence of changes in the geometry of the sandwich panel on the wave propagation is presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of silicon reinforcement on load-capacity of porous nanofilters

Square–form microfilters consisting of a two–layer porous membrane with nano–size pores and reinforced by silicon columns have been investigated. The change of layer thicknesses and reinforcement geometry modify the load–capacity of the device through the variation of strength properties. A simple thin plate model was constructed by substituting the effect of silicon columns by an equivalent third layer. Estimates for the load–capacity were made by finite element calculations. These estimates have been verified by experimental results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Holistic Simulation Approach from a Measured Load to Element Stress Using Combined Multi-body Simulation and Finite Element Modelling

The design of vehicle bodies requires the knowledge of the vehicle's structural response to external loads and disturbances. In rigid multi-body simulation the dynamic behaviour of complex systems is calculated with rigid bodies and neglect of body elasticity. On the other hand, in finite element models large degree of freedom numbers are used to represent the elastic properties of a single body. Both simulation methods can be combined, if the finite element model size is reduced to a degree of freedom number feasible to multi-body simulation. The application to practical purposes requires the use and interconnection of several different software tools. In this contribution a holistic method is presented, which starts with the measurement or synthesis of loads and excitations, continues with the integration of a reduced finite element model into a multi-body system, the dynamic response calculation of this combined model, and concludes with the result expansion to the full finite element model for calculating strain and stress values at any point of the finite element mesh. The applied software tools are Simpack, Nastran, and Matlab. An example is given with a railway vehicle simulated on measured track geometry. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Representation of convex geometries by circles on the plane

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an $n$-dimensional space equipped with a convex hull operator, by the result of Kashiwabara et al. (2005). Allowing circles rather than points, as was suggested by Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on the plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.     

Development of a single-layer finite element and a simplified finite element modeling approach for constrained layer damped structures

A new finite element (FE) is formulated based on an extension of previous FE models for studying constrained layer damping (CLD) in beams. Most existing CLD FE models are based on the assumption that the shear deformation in the core layer is the only source of damping in the structure. However, previous research has shown that other types of deformation in the core layer, such as deformations from longitudinal extension, and transverse compression, can also be important. In the finite element formulated here, shear, extension, and compression deformations are all included. As presented, there are 14 degrees of freedom in this element. However, this new element can be extended to cases in which the CLD structure has more than three layers. The numerical study shows that this finite element can be used to predict the dynamic characteristics accurately. However, there is a limitation when the core layer has a high stiffness, as the new element tends to predict loss factors and natural frequencies that are too high. As a result, this element can be accepted as a general computation model to study the CLD mechanism when the core layer is soft. Because the element includes all three types of damping, the computational cost can be very high for large scale models. Based on this consideration, a simplified finite modeling approach is presented. This approach is based on an existing experimental approach for extracting equivalent properties for a CLD structure. Numerical examples show that the use of these extracted properties with commercially available FE models can lead to sufficiently accurate results with a lower computational expense.     

A unified formulation of PVD-based finite cylindrical layer methods for functionally graded material sandwich cylinders

A unified formulation of finite cylindrical layer methods (FCLMs) based on the principle of virtual displacements (PVDs) is developed for the quasi-three-dimensional (3D) bending and free vibration analyses of simply-supported, functionally graded material (FGM) sandwich circular hollow cylinders, in which the material properties of the FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate. In this formulation, the cylinder is divided into a number of cylindrical finite layers, where the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the displacement components of each individual layer, respectively. Because an h-refinement is adopted in this article to yield the convergent solutions, the relative orders used for expansion of the displacement components remain variable, and can be freely chosen as linear, quadratic and cubic ones. The accuracy and convergence rate of a variety of PVD-based FCLMs developed in this article are assessed by comparing their solutions with the available 3D ones.     

Finite element simulation of a layout optimisation technique by photoelastic stress minimisation

Layout optimisation to minimise maximum Tresca stress by photoelastic stress minimisation technique is simulated by finite element method: elements in the design domain that are lowly stressed are slowly removed resulting in a structure having minimum Tresca stress. The FEM simulation consists of analysing-monitoring the Tresca stress of elements in the design domain and “removing” material by declaring the element stiffness matrix of those possessing small stress values as of negligible stiffness in the subsequent step of the optimisation process. The lower bounds and upper bounds of stress limits for the “removal” criterion have to be appropriately chosen and effects of sharp notches introduced by removing finite elements should be properly taken into account for successful optimisation. The FEM simulation can be made fully automatic and can be extended to cases of complex geometry, loading material properties as well as to other objective functions of the optimisation problem.     

Zeta functions of discrete self-similar sets

In this paper we study a class of countable and discrete subsets of a Euclidean space that are “self-similar” with respect to a finite set of (affine) similarities. Any such set can be interpreted as having a fractal structure. We introduce a zeta function for these sets, and derive basic analytic properties of this “fractal” zeta function. Motivating examples that come from combinatorial geometry and arithmetic are given particular attention.     

A high-order enrichment strategy for the finite cell method

Thanks to the application of the immersed boundary approach in the finite cell method, the mesh can be defined independently from the geometry. Although this leads to a significant simplification of the mesh generation, it might cause difficulties in the solution. One of the possible difficulties will occur if the exact solution of the underlying problem exhibits a kink inside an element, for instance at material interfaces. In such a case, the solution turns out less smooth – and the convergence rate is deteriorated if no further measures are taken into account. In this paper, we explore a remedy by considering the partition of unity method. The proposed approach allows to define enrichment functions with the help of a high-order implicit representation of the material interface. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear buckling behavior of 3D-braided composite cylindrical shells subjected to internal pressure loads in thermal environments

The nonlinear buckling behavior of a 3D-braided composite cylindrical shell of finite length subjected to internal pressure in thermal environments is considered. According to a new micromacromechanical model, a 3D-braided composite may be treated as a cell system where the geometry of each cell strongly depends on its position in the cross section of the cylindrical shell. The material properties of the epoxy matrix are expressed as linear functions of temperature. The governing equations are based on Reddy’s higher-order shear deformation theory of shells with a von Karman–Donnell-type kinematic nonlinearity and include thermal effects. The singular perturbation technique is employed to determine the buckling pressure and the postbuckling equilibrium paths of the shell.     

分析土壤半无限域的一种有限层方法(Ⅰ)——一般理论

本文研究了一种用于横观各向同性体动力学的有限层法.将土壤介质半无限域视为一个横观各向同性半空间体,介质的材料函数沿深度变化,将介质沿深度方向分成若干层,在每一层内材料函数用一个指数函数来模拟.这样,使求解问题的方程得到简化,利用Fourier变换我们得到了各层处“节点”力与位移的关系.这种有限层法实际上是一种半解析法.具有一般半解析法所具备的数据前后处理量少及计算量小等优点.本文所研究的这种有限层法为分析土壤与结构相互作用问题提供了一条新途径.     

An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

Vibration and buckling analyses of FGM plates with multiple internal defects using XIGA-PHT and FCM under thermal and mechanical loads

In this paper, the vibration and buckling analyses of the FGM (functionally graded material) plates with multiple internal cracks and cutouts under thermal and mechanical loads are numerically investigated using the combined XIGA-PHT (extended isogeometric analysis based on PHT-splines) and FCM (finite cell method). Material properties are graded only in the thickness direction. The effective material properties are estimated by using either the rule of mixture or the Mori-Tanaka homogenization technique. The plate displacement field is based on the HSDT (higher-order shear deformation plate theory) without any requirement of the SCF (shear correction factor). The HSDT model can exactly represent the shear stress distribution and improve the accuracy of solutions. The PHT-splines can naturally fulfill the C¹-continuous requirement of the HSDT model. The representation of internal defects is mesh-independent. The discontinuous and singular phenomena induced by the cracks are captured using the enrichment pattern in the XIGA, and the influence of cutouts is implemented by the FCM. The geometries of cutouts are captured by means of adaptive quadrature procedure based on a simple unfitted structural mesh, which avoids the need for multiple patches to describe the complex geometry and eliminates the enforcement of C¹-continuity patch-coupling across the patch boundaries. The initial mesh density around the cracks and cutouts can be controlled flexibly utilizing the local refinement property of the PHT-splines. After validating the results of the developed approach with those available in the literature, the effects of material gradient index, side to thickness ratio, boundary conditions, cutout size and crack length on the normalized frequency and the critical buckling parameter are investigated. Numerical results illustrate the effectiveness and accuracy of the present approach.     

Macroscopic mechanical properties of metal foam in dependence on the inhomogenities of the mesoscale

In this contribution, a way of simulating the influence of the mesoscopic irregular structure of metal foams on the macroscale is shown. To this end, mesoscopic periodic volume elements of a foam are derived in order to compute the mechanical properties including the effects of inhomogenities like imperfections, irregular structure and varying cell wall thicknesses. With the help of these volume elements, which are analysed via the finite element method, and their varying mechanical properties, a local varying stiffness can be computed and inserted into the macromechanical model. In this way the propagation of uncertainities from the mesoscale to the macroscale can be assessed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient computation of cellular materials using the finite cell method

The application of cellular materials in lightweight structures is a subject of growing interest. As meshing with surface aligned elements is very time consuming and entails a lot of degrees of freedom, we apply the finite cell method (FCM). The FCM is an extension of the known fictitious domain methods based on using high order polynomials for the Ansatz functions. In order to obtain a realistic representation of the geometry, we derive our discretization directly from the voxel data coming from the computer tomography (CT-scan) of the specimen of interest. Thus, a much coarser Cartesian grid consisting of hexahedral cells can be readily generated in a fully automated fashion. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)