Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.     

Wrinkling prediction of rectangular shell-membrane under transverse in-plane displacement

A novel shell-membrane concept is presented in this paper to define the wrinkling analytical object. A stress field model is established and applied to determine the different regions (taut, wrinkled and slack regions) of the shell-membrane. An analytical model based on the bifurcation theory of thin-plate is introduced to predict the wrinkling wavelength and amplitude. Numerical simulation incorporating nonlinear post-wrinkling analysis is employed to simulate the detailed nonlinear wrinkling behavior. To stabilize the finite element model, the initial prestress is introduced to the model at first. The out-of-plane disturbing forces are then applied to the model to initiate the wrinkling and complete the post-wrinkling analysis. Results from numerical simulation show good agreement with the analytical prediction.     

A study on wrinkling characteristics and dynamic mechanical behavior of membrane

An eigenvalue method considering the membrane vibration of wrinkling out-of-plane deformation is introduced, and the stress distributing rule in membrane wrinkled area is analyzed. A dynamic analytical model of rectangular shear wrinkled membrane and its numerical analysis approach are also developed. Results indicate that the stress in wrinkled area is not uniform, i.e. it is larger in wrinkling wave peaks along wrinkles and two ends of wrinkle in vertical direction. Vibration modes of wrinkled membrane are strongly correlated with the wrinkling configurations. The rigidity is larger due to the heavier stress in the part of wrinkling wave peaks. Therefore, wave peaks are always located at the node lines of vibration mode. The vibration frequency obviously increases with the vibration of wave peaks.     

空间充气薄膜结构的褶皱分析

引入一个表征褶皱在纹理方向收缩特性的参数``收缩系数'来替代泊松比参与计算褶皱形变,并联合褶皱产生时的应力条件和褶皱特性,实现了对空间充气薄膜结构的褶皱分析,得到了褶皱存在时的充气薄膜结构的应力场和变形. 分析中重点讨论了``收缩系数'与泊松比的关系,并讨论了引入``收缩系数'约束变形后的变形协调条件. 将两维的褶皱分析方法拓展应用到三维充气结构的褶皱分析中,分析了充气管的弯皱变形,得到了临界皱曲和极限弯矩随结构及力学参数的变化规律. 通过简单的褶皱试验,定性地分析了褶皱的形变特性并与分析结果进行了比较验证,结果一致.      

Experiment and evaluation of wrinkling strain in a corner tensioned square membrane

Prediction of wrinkling characteristics is strongly correlated with the strain perpendicular to wrinkling direc- tion. In this paper, the strain field of wrinkled membrane is tested by VIC-3D system based on the digital image correlation technique. Experimental results are validated by the tension wrinkling simulation. The experimental strain perpendicular to wrinkling direction is analyzed in depth. The wrinkling strain of a square wrinkled membrane under corner tension is extracted from experimental strain perpendicular to wrinkling direction. A quantitative characterization format of the experimental wrinkling strain is proposed. A modified prediction method of wrinkling amplitude is presented based on the experimental wrinkling strain. The re- sults show that the precision of modified prediction model has improved 13.2% compared with the classical prediction model. The results reveal that the modified model can give an accurate prediction of the wrinkling amplitude.     

Boundary conformed co-ordinate systems for selected two-dimensional fluid flow problems. Part I: Generation of BFGs

In this paper the generation of general curvilinear co-ordinate systems for use in selected two-dimensional fluid flow problems is presented. The curvilinear co-ordinate systems are obtained from the numerical solution of a system of Poisson equations. The computational grids obtained by this technique allow for curved grid lines such that the boundary of the solution domain coincides with a grid line. Hence, these meshes are called boundary fitted grids (BFG). The physical solution area is mapped onto a set of connected rectangles in the transformed (computational) plane which form a composite mesh. All numerical calculations are performed in the transformed plane. Since the computational domain is a rectangle and a uniform grid with mesh spacings Δξ = Δη = 1 (in two-dimensions) is used, the computer programming is substantially facilitated. By means of control functions, which form the r.h.s. of the Poisson equations, the clustering of grid lines or grid points is governed. This allows a very fine resolution at certain specified locations and includes adaptive grid generation. The first two sections outline the general features of BFGs, and in section 3 the general transformation rules along with the necessary concepts of differential geometry are given. In section 4 the transformed grid generation equations are derived and control functions are specified. Expressions for grid adaptation arc also presented. Section 5 briefly discusses the numerical solution of the transformed grid generation equations using sucessive overrelaxation and shows a sample calculation where the FAS (full approximation scheme) multigrid technique was employed. In the companion paper (Part II), the application of the BFG method to selected fluid flow problems is addressed.     

New computational approaches for wrinkled and slack membranes

The static response of thin, wrinkled membranes is studied using both a tension field approximation based on plane stress conditions and a 3D nonlinear elasticity formulation, discretized through 8-noded Cosserat point elements. While the tension field approach only obtains the wrinkled/slack regions and at best a measure of the extent of wrinkliness, the 3D elasticity solution provides, in principle, the deformed shape of a wrinkled/slack membrane. However, since membranes barely resist compression, the discretized and linearized system equations via both the approaches are ill-conditioned and solutions could thus be sensitive to discretizations errors as well as other sources of noises/imperfections. We propose a regularized, pseudo-dynamical recursion scheme that provides a sequence of updates, which are almost insensitive to the regularizing term as well as the time step size used for integrating the pseudo-dynamical form. This is borne out through several numerical examples wherein the relative performance of the proposed recursion scheme vis-à-vis a regularized Newton strategy is compared. The pseudo-time marching strategy, when implemented using 3D Cosserat point elements, also provides a computationally cheaper, numerically accurate and simpler alternative to that using geometrically exact shell theories for computing large deformations of membranes in the presence of wrinkles.     

On the incremental equations in non-linear elasticity — I. Membrane theory

A new formulation of the equations of membrane theory in non-linear elasticity is described. It is based on the consistent use of certain conjugate variables averaged through the (undeformed) thickness of the thin shell which the membrane approximates. The deformation gradient is taken as the basic measure of deformation, and its average value as the membrane measure of deformation. It is shown that the average elastic strain energy can be regarded as a function of the average deformation gradient to within an error which is of the second order in a certain small parameter. Moreover, to the same order, the average strain energy is a potential function for the average nominal stress. This means that the averages of the conjugate variables (nominal stress and deformation gradient) are also conjugate.In terms of the average conjugate variables, the membrane equilibrium equations are obtained by averaging from the equilibrium equations of the full three-dimensional theory. Discussion of the order of magnitude of the errors involved in the membrane approximation is a feature of the analysis.The corresponding incremental equations are also derived as a prelude to their application in certain bifurcation problems. One such problem is examined in the companion paper (Part II) in which results for thick shells and membranes are compared.     

Wrinkled membrane element based on the wrinkling potential

For a membrane structure, wrinkles have an important effect on its mechanical behaviors. Wrinkling level characterizes the development of wrinkles and reflects the performance of a membrane in its service, and it plays a very significant role in the wrinkling analysis. The shell elements combined with the stability theory would be an ideal solution to the wrinkling problem. However, this approach requires a dense mesh and the computation is very time consuming. Also the wrinkling parameters are very sensitive to the size of shell elements. Existing wrinkling models based on membrane elements are derived from the Tension Field Theory which are incapable of describing fully the wrinkling behaviors.A new wrinkling model adopting the wrinkling strain as a measure of the wrinkling level is proposed in this paper to address these issues. According to the analogy between the wrinkling strain and the elasto-plastic strain, a wrinkling potential surface is assumed to exist and its normal direction defines the direction of the wrinkling strain tensor by virtue of the flow rule. Based on the consistent condition of the wrinkling potential surface, a modified constitutive tensor is obtained. To avoid the switching of the wrinkling state in the numerical solution, a new wrinkling criterion is proposed, in which the predominant influence of the previous state is included. Besides, a new approach to determine the wrinkling orientation is given to improve the efficiency of convergence in the slack region. The objectivity of the wrinkling coordinate frame is also demonstrated as an accompanying set of results. Finally, two benchmark problems are analyzed with the proposed wrinkling model, and their results are compared with those in the literatures. Results indicate that the proposed wrinkling model is valid and accurate to characterize the wrinkling level of a membrane and it exhibits efficient convergence even in the slack region.     

Computational analysis of liquid crystalline elastomer membranes: Changing Gaussian curvature without stretch energy

Liquid crystalline elastomers (LCEs) can undergo extremely large reversible shape changes when exposed to external stimuli, such as mechanical deformations, heating or illumination. The deformation of LCEs result from a combination of directional reorientation of the nematic director and entropic elasticity. In this paper, we study the energetics of initially flat, thin LCE membranes by stress driven reorientation of the nematic director. The energy functional used in the variational formulation includes contributions depending on the deformation gradient and the second gradient of the deformation. The deformation gradient models the in-plane stretching of the membrane. The second gradient regularises the non-convex membrane energy functional so that infinitely fine in-plane microstructures and infinitely fine out-of-plane membrane wrinkling are penalised. For a specific example, our computational results show that a non-developable surface can be generated from an initially flat sheet at cost of only energy terms resulting from the second gradients. That is, Gaussian curvature can be generated in LCE membranes without the cost of stretch energy in contrast to conventional materials.     

Multipatch discontinuous Galerkin isogeometric analysis of composite laminates

This paper is concerned with the isogeometric analysis (IGA) of composite laminates under cylindrical bending. Non-uniform rational B-splines (NURBS) are employed as basis functions for both geometric and computational implementations. In order to account for multiple domains, each lamina is modeled as a single NURBS patch. This multipatch representation corresponds to decomposition of the computational domain (composite laminate) into non-overlapping subdomains. As NURBS patches are discontinuous across their boundaries, a standard FEA-like procedure does not work for multipatch IGA; an additional numerical technique is required for coupling NURBS patches. Therefore, in this paper, one of the discontinuous Galerkin (DG) methods, namely symmetric interior penalty Galerkin formulation, is employed to allow for discontinuities. For numerical calculations, a composite laminate with stacking sequences $$0^{\circ }{/}90^{\circ }$$ and $$0^{\circ }{/}90^{\circ }{/}0^{\circ }$$, respectively, is adopted. The stresses are calculated along the thickness of the composite laminate, subjected to a sinusoidal load, and they are compared with the analytical solutions. It is shown that DG–IGA gives a better approximation in comparison with the standard IGA.     

弹性薄板的广义多维微分求积分析方法

多维微分求积DC(differential cubature)法是近年迅速发展起来的一种处理多维微分方程的高精度数值方法。本文围绕多元函数插值适定性这一基本问题，探讨了实现DC法时插值结点组和插值空间应满足的条件，研究了离散结点组选择及其构造适定试函数的方法。提出了一种沿各坐标方向可以布置不同数量结点的广义交错网格DC方法，并给出了确定相应的插值空间及其选择试函数的具体方案。通过弹性薄板变形分析研究了广义交错网格DC法解决实际结构力学问题的可靠性及其潜力。研究结果表明，广义交错网格DC法较传统的交错网格DC格式具有更强的适用性和灵活性，采用数量少得多的离散结点即可达到传统交错网格DC法相同的数值精度，表现出精度高和计算工作量小的优点。     

Constitutive analysis of thin biological membranes with application to radial stretching of a hollow circular membrane

The constitutive analysis of the mechanical response of thin elastic membranes under inplane deformation is presented by using the multiplicative decomposition of the deformation gradient into its areal and distortional parts. Specific results are obtained for the Evans-Skalak form of the strain energy function. The solution to the problem of radial stretching of a hollow circular membrane obeying this constitutive model is then derived. The stress concentration factor is determined as a function of the relative hole size and the magnitude of the applied tension. The tension boundary is identified above which no compressive stress appears in the membrane. The limit boundary is introduced below which the membrane cannot support the applied loading without unstable wrinkling. For the loading between the tension and the limit boundary, nonuniformly distributed infinitesimal wrinkles appear within the inner portion of the membrane, carrying radial tension but no circumferential stress (tension field). The specific form of the strain energy function is used to describe this behavior, and to calculate the amount of the membrane area absorbed by infinitesimal wrinkles. The wrinkled portion is surrounded by the outer portion of the membrane carrying both radial and circumferential stresses. The limit boundary is reached when wrinkles spread throughout the membrane. It is shown that for a sufficiently large tension at the outer boundary, the wrinkling does not spread throughout the membrane no matter how large the applied tension at the inner boundary of the membrane is, provided that no rupture takes place. The limiting extent of the tension field in such cases is calculated. The linearized version of the analysis is characterized by a closed form solution.     

等几何分析中采用Nitsche法施加位移边界条件

等几何分析使用NURBS基函数统一表示几何和分析模型,消除了传统有限元的网格离散误差,容易构造高阶连续的协调单元.对于结构分析,选择合适的几何参数可以得到光滑的应力解,避免了后置处理的应力磨平.但是由于NURBS基函数不具备插值性,难以直接施加位移边界条件.针对这一问题,提出一种基于Nitsche变分原理的边界位移条件"弱"处理方法,它具有一致稳定的弱形式,不增加自由度,方程组对称正定和不会产生病态矩阵等优点.同时给出方法的稳定性条件,并通过求解广义特征值问题计算稳定性系数.最后,数值算例表明Nitsche方法在h细化策略下能获得最优收敛率,其结果要明显优于在控制顶点处直接施加位移约束.     

Wrinkling of Orthotropic Membranes: An Analysis by the Polar Method

In the present paper, a simple membrane model based on the wrinkle strain approach is revisited with the aim of examining how the material elastic constants affect the static response of anisotropic membranes when wrinkling is taken into account. Employing the polar method, we analyze the role played by the polar moduli, which enable expressing the elasticity matrix components of an anisotropic material in terms of its invariant quantities. With reference to orthotropic materials, we first address the issue of membrane susceptibility to wrinkling by investigating the influence of the three polar parameters characterizing the anisotropic part of the constitutive law. The stress and strain states at any given point in a wrinkled membrane are analyzed by searching for explicit expressions for the principal values of stress and wrinkle strain. Finally, a comparison between our results and those obtained by a numerical solution available in the literature is made in the basic case of a membrane subjected to a pure shear strain state.     

IMPOSING DISPLACEMENT BOUNDARY CONDITIONS WITH NITSCHE＇S METHOD IN ISOGEOMETRIC ANALYSIS

等几何分析使用 NURBS 基函数统一表示几何和分析模型, 消除了传统有限元的网格离散误差, 容易构造高阶连续的协调单元. 对于结构分析, 选择合适的几何参数可以得到光滑的应力解, 避免了后置处理的应力磨平. 但是由于 NURBS 基函数不具备插值性, 难以直接施加位移边界条件. 针对这一问题, 提出一种基于 Nitsche 变分原理的边界位移条件“弱”处理方法, 它具有一致稳定的弱形式, 不增加自由度, 方程组对称正定和不会产生病态矩阵等优点. 同时给出方法的稳定性条件, 并通过求解广义特征值问题计算稳定性系数. 最后, 数值算例表明 Nitsche 方法在h细化策略下能获得最优收敛率, 其结果要明显优于在控制顶点处直接施加位移约束.}     

空间薄膜结构褶皱的数值模拟最新研究进展

褶皱问题是高精度空间薄膜结构最关心的问题之一, 它是影响薄膜结构形面精度的关键因素. 由于薄膜结构褶皱机理的复杂性, 该问题一直没能得到很好的解决, 人们对褶皱的认识, 目前主要是通过试验观察和数值模拟等方法. 本文综述了空间薄膜结构褶皱数值研究的发展和现状, 并对其中重要的和被广泛采纳的方法进行了重点介绍和评述. 最后对褶皱数值分析方法所面临的问题及其发展趋势作了初步展望.      

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites. Part I: Closed form expression for the effective higher-order constitutive tensor

It is shown that second-order homogenization of a Cauchy-elastic dilute suspension of randomly distributed inclusions yields an equivalent second gradient (Mindlin) elastic material. This result is valid for both plane and three-dimensional problems and extends earlier findings by Bigoni and Drugan [Bigoni, D., Drugan, W.J., 2007. Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753] from several points of view: (i) the result holds for anisotropic phases with spherical or circular ellipsoid of inertia; (ii) the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material; (iii) a perfect energy match is found between heterogeneous and equivalent materials (instead of an optimal bound). The constitutive higher-order tensor defining the equivalent Mindlin solid is given in a surprisingly simple formula. Applications, treatment of material symmetries and positive definiteness of the effective higher-order constitutive tensor are deferred to Part II of the present article.     

空间薄膜结构的褶皱研究进展

大型柔性薄膜结构是目前国际上十分关注的可满足多项空间任务需要的新型空间结构. 由于薄膜材料柔性且抵抗压缩应力作用的能力十分有限, 因此这些空间结构在轨时多处于带有褶皱的工作状态. 褶皱的存在会改变结构的力学性能且会影响结构表面形状, 因此开展薄膜褶皱研究对柔性结构的设计和力学性能评价等具有重要意义. 本文综述了空间薄膜结构褶皱研究的发展和现状, 重点介绍了薄膜褶皱的理论研究概况, 并讨论了其中存在的问题.根据褶皱研究的最新进展对其未来的发展趋势进行了展望.