基于Bézier提取的三维等几何分析

等几何分析(IGA)将非均匀有理B样条(NURBS)函数作为有限元形函数,具有几何精确、高阶连续和精度高等优点。与常规有限元法C0连续的形函数不同,高阶IGA基函数不是定义在一个单元上,而是跨越由几个单元组成的参数空间,因而编程复杂且无法嵌入现有的有限元法计算框架及相应算法。本文建立了基于Bézier提取的三维IGA,将NURBS函数分解成伯恩斯坦多项式的线性组合,从而实现把NURBS单元分解为C0连续Bézier单元,这些单元与Lagrange单元相似,使IGA的实现和常规有限元一样,以便将IGA分析嵌入现有的有限元软件中。两个三维算例结果表明,基于Bézier提取的IGA和传统IGA的收敛性和精度相同。     

C 1 natural element method for strain gradient linear elasticity and its application to microstructures

C 1 natural element method (C 1 NEM) is applied to strain gradient linear elasticity, and size effects on microstructures are analyzed. The shape functions in C 1 NEM are built upon the natural neighbor interpolation (NNI), with interpolation realized to nodal function and nodal gradient values, so that the essential boundary conditions (EBCs) can be imposed directly in a Galerkin scheme for partial differential equations (PDEs). In the present paper, C 1 NEM for strain gradient linear elasticity is constructed, and several typical examples which have analytical solutions are presented to illustrate the effectiveness of the constructed method. In its application to microstructures, the size effects of bending stiffness and stress concentration factor (SCF) are studied for microspeciem and microgripper, respectively. It is observed that the size effects become rather strong when the width of spring for microgripper, the radius of circular perforation and the long axis of elliptical perforation for microspeciem come close to the material characteristic length scales. For the U-shaped notch, the size effects decline obviously with increasing notch radius, and decline mildly with increasing length of notch.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Hermitian space-time finite elements for estuarine mass transport

Space-time finite element solutions of the convection–dispersion equation using higher-order nodal continuity and Hermitian polynomial shape functions are described. Five separate elements ranging from a complete linear element with C^0,0 nodal continuity to a complete first-order Hermitian element with C^1,1 nodal continuity are subjected to detailed analysis. Wave deformation analyses identify the source of leading or trailing edge oscillations, trailing edge oscillations being the major source of difficulty. These observations are confirmed by numerical experiments which further demonstrate the potential of higher-order nodal continuity. The performance of the complete first-order Hermitian element is quite satisfactory and measurably superior to the linear element.     

An advanced higher-order theory for laminated composite plates with general lamination angles

This paper proposes a higher-order shear deformation theory to predict the bending response of the laminated composite and sandwich plates with general lamination configurations.The proposed theory a priori satisfies the continuity conditions of transverse shear stresses at interfaces.Moreover,the number of unknown variables is independent of the number of layers.The first derivatives of transverse displacements have been taken out from the inplane displacement fields,so that the C 0 shape functions are only required during its finite element implementation.Due to C 0 continuity requirements,the proposed model can be conveniently extended for implementation in commercial finite element codes.To verify the proposed theory,the fournode C 0 quadrilateral element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate.Numerical results show that following the proposed theory,simple C 0 finite elements could accurately predict the interlaminar stresses of laminated composite and sandwich plates directly from a constitutive equation,which has caused difficulty for the other global higher order theories.     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Configuration Design Sensitivity Analysis of Nonlinear Structural Systems with Elastic Material*

ABSTRACT

A continuum-based design sensitivity analysis (DSA) method is presented for configuration design of nonlinear structural systems using Mindlin plate and Tim-oshenko beam theories. Both displacement and critical load performance measures are considered. Configuration design variables are characterized by shape and orientation changes of structural components. The material derivative that is used to develop the continuum-based shape DSA method is extended to account for effects of configuration design variation. The piecewise linear design velocity field, i.e., C⁰-regular, is used to support configuration design changes for a broad class of built-up structures with beams and plates. To allow use of the C⁰-design velocity field, mathematical models of beam and plate bending must be second-order partial differential equations, so that only first-order derivatives appear in the integrand of the energy equation and, thus, in the integrand of the configuration design sensitivity expression. Since the Mindlin plate and Timoshenko beam theories use displacement and rotation to describe structural response, mathematical models of beam and plate bending are reduced to second-order partial differential equations. The isoparametric finite element formulations are used for numerical evaluation of continuum design sensitivity expressions.     

Free vibration of composite laminated plate with complicated cutout

Abstract

This paper presents the free vibration analysis of a composite laminated square plate with complicated cutout. The problem formulation is based on the higher order shear deformation plate theory HDST C⁰ coupled with a curved quadrilateral p-element. The elements of the stiffness and mass matrices are calculated analytically. The curved edges are accurately represented using the blending function method. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as the cutout shape, plate thickness, fiber orientation angle, and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New results for the fundamentals frequencies of a composite laminated plate with complicated cutout are presented.     

A new general-purpose least-squares finite element model for steady incompressible low-viscosity laminar flow using isoparametric C1-continuous Hermite elements

This paper deals with a critical evaluation of various finite element models for low-viscosity laminar incompressible flow in geometrically complex domains. These models use Galerkin weighted residuals UVP, continuous penalty, discrete penalty and least-squares procedures. The model evaluations are based on the use of appropriate tensor product Lagrange and simplex quadratic triangular elements and a newly developed isoparametric Hermite element. All of the described models produce very accurate results for horizontal flows. In vertical flow domains, however, two different cases can be recognized. Downward flows, i.e. when the gravitational force is in the direction of the flow, usually do not present any special problem. In contrast, laminar flow of low-viscosity Newtonian fluids where the gravitational force is acting in the direction opposite to the flow presents a difficult case. We show that only by using the least-squares method in conjunction with C¹-continuous Hermite elements can this type of laminar flow be modelled accurately. The problem of smooth isoparametric mapping of C¹ Hermite elements, which is necessary in dealing with geometrically complicated domains, is tackled by means of an auxiliary optimization procedure. We conclude that the least-squares method in combination with isoparmetric Hermite elements offers a new general-purpose modelling technique which can accurately simulate all types of low-viscosity incompressible laminar flow in complex domains.     

Long-time Stability in Systems of Conservation Laws,Using Relative Entropy/Energy

We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L ²∩ L ^∞. We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.     

On Finite Shear

If a pair of material line elements, passing through a typical particle P in a body, subtend an angle Θ before deformation, and Θ+γ after deformation, the pair of material elements is said to be sheared by the amount γ. Here all pairs of material elements at P are considered for arbitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line elements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane π(S) with normal S passing through P are considered. Provided π(S) is not a plane of central circular section of the C-ellipsoid at P (where C is the right Cauchy-Green strain tensor), it is seen that corresponding to any material element in π(S) there is, in general, one companion material element in π(S) such that the element and its companion are unsheared. There are, however, two elements in π(S) which have no companions. We call their corresponding directions \textit{limiting directions.} Equally inclined to the direction of least stretch in the plane π(S), the limiting directions play a central role. It is seen that, in a given plane π(S), the pair of material line elements which suffer the maximum shear lie along the limiting directions in π(S). If Θ _L is the acute angle subtended by the limitig directions in π(S) before deformation, then this angle is sheared into its supplement π−Θ _L so that the maximum shear γ^*;(S) is γ^*=π− 2 Θ _L . If S is given and C is known, then Θ _L may be determined immediately. Its calculation does not involve knowing the eigenvectors or eigenvalues of C. When all possible planes through P are considered, it is seen that the global maximum shear γ^* _G occurs for material elements lying along the limiting directions in the plane spanned by the eigenvectors of C corresponding to the greatest principal stretch λ₃ and the least λ₁. The limiting directions in this principal plane of C subtend the angle and . Generally the maximum shear does not occur for a pair of material elements which are originally orthogonal. For a given material element along the unit vector N, there is, in general, in each plane π(S passing through N at P, a companion vector M such that material elements along N and M are unsheared. A formula, originally due to Joly (1905), is presented for M in terms of N and S. Given an unsheared pair π(S), the limiting directions in π(S) are seen to be easily determined, either analytically or geometrically. Planar shear, the change in the angle between the normals of a pair of material planar elements at X, is also considered. The theory of planar shear runs parallel to the theory of shear of material line elements. Corresponding results are presented. Finally, another concept of shear used in the geology literature, and apparently due to Jaeger, is considered. The connection is shown between Cauchy shear, the change in the angle of a pair of material elements, and the Jaeger shear, the change in the angle between the normal N to a planar element and a material element along the normal N. Although Jaeger's shear is described in terms of one direction N, it is seen to implicitly include a second material line element orthogonal to N. Accepted: May 25, 1999     

A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation

A new finite element method is presented to solve one‐dimensional depth‐integrated equations for fully non‐linear and weakly dispersive waves. For spatial integration, the Petrov–Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C²‐continuity. For the time integration an implicit predictor–corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth‐order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd.     

Shape optimization of compliant pressure actuated cellular structures

Biologically inspired pressure actuated cellular structures can alter their shape through pressure variations. Previous work introduced a computational framework for pressure actuated cellular structures that is limited to two cell rows and central cell corner hinges. This article rigorously extends these results by taking into account an arbitrary number of cell rows, a more complicated cell kinematic that includes hinge eccentricities and varying side lengths as well as rotational and axial cell side springs. The nonlinear effects of arbitrary cell deformations are fully considered. Furthermore, the optimization is considerably improved by using a second-order approach. The presented framework enables the design of compliant pressure actuated cellular structures that can change their form from one shape to another within a set of one-dimensional C¹ continuous functions. Several examples are used to demonstrate the performance of the proposed framework.     

A novel ellipsoidal acoustic infinite element

A novel ellipsoidal acoustic infinite element is proposed. It is based a new pressure representation, which can describe and solve the eUipsoidal acoustic field more exactly. The shape functions of this novel acoustic infinite element are similar .to the Burnett‘s method, while the weight functions are defined as the product of the complex conjugates of the shaped functions and an additional weighting factor. The code of this method is cheap to generate as for 1-D element because only 1-D integral needs to be numerical. Coupling with the standard finite element, this method provides a capability for very efficiently modeling acoustic fields surrounding structures of virtually any practical shape. This novel method was deduced in brief and the conclusion was kept in detail. To test the feasibility of this novel method efficiently, in the examples the infinite elements were considered, excluding the finite elements relative. This novel eUipsoidal acoustic infinite element can deduce the analytic solution of an oscillating sphere. The example of a prolate spheroid shows that the novel infinite element is superior to the boundary element and other acoustic infinite elements. Analytical and numerical results of these examples show that this novel method is feasible.     

Solution of three-dimensional problems for thick elastic shells by the method of reference surfaces

A new method for solving the elasticity problem for thick and thin shells is proposed. The method is based on the concept of reference surfaces inside the shell. According to this method, N reference surfaces are introduced in the body of the shell so that they are parallel to the midsurface and located at the Chebyshev polynomial nodes, which permits taking the displacement vectors u ¹, u ², …, u ^N of these surfaces for the desired functions. This choice of the desired functions allows one to represent the resolving equations of the proposed theory of higher-order shells in a sufficiently concise form and obtain deformation relations which permit describing the shell displacements as motions of a rigid body.     

C~0型高阶锯齿理论及复合材料厚梁热应力分析

基于已有锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。针对已有锯齿理论存在的问题,本文首先发展了C0型锯齿理论。通过虚位移原理推导出在热载荷作用下复合材料梁的平衡方程,并给出了简支复合材料层合梁解析解。基于发展的锯齿理论分析了复合材料夹层梁和层合梁热膨胀问题,并与其他理论结果对比。数值结果表明,发展的C0型锯齿理论能克服已有锯齿理论的难题。     

The application of compatible stress iterative method in dynamic finite element analysis of high velocity impact

There is a common difficulty in elastic-plastic impact codes such as EPIC ^[2,3].NONSAP ^[4],etc. Most of these codes use the simple linear functions usually taken from static problem to represent the displacement components. In such finite element formulation, the stress components are constant in each element and they are discontinuous in any two neighboring elements. Therefore, the bases of using the virtual work principle in such elements are unreliable. In this paper, we introduce a new method, namely, the compatible stress iterative method, to eliminate the above-said difficulty. The calculated examples show that the calculation using the new method in dynamic finite element analysis of high velocity impact is valid and stable, and the element stiffness can be somewhat reduced.This is a part of the author's Ph. D dissertation under the supervision of Professor Chien Wei-zang.     

复合材料层合厚板锯齿理论及有限元分析

对于较厚复合材料弯曲问题,已有锯齿型厚板理论最大误差超过35%。为了合理地分析较厚复合材料弯曲问题,发展了准确高效的锯齿型厚板理论。此理论位移变量个数独立于层合板层数,其面内位移不含有横向位移一阶导数,构造有限元时仅需C0插值函数,故称此理论为C0型锯齿厚板理论。基于发展的锯齿理论,构造了六节点三角形单元并推导了复合材料层合/夹层板弯曲问题有限元列式。为验证C0型锯齿厚板理论性能,分析了复合材料层合/夹层厚板弯曲问题,并与已有C1型锯齿理论对比。结果表明,本文的C0型锯齿厚板理论最大误差15%,比已有锯齿型厚板理论准确高效。     

Perturbation approach to elastic constitutive relations of polycrystals

Herein we obtain a formula for the effective elastic stiffness tensor C^eff of an orthorhombic aggregate of cubic crystallites by the perturbation method. The effective elastic stiffness tensor of the polycrystal gives the relationship between volume average stress and volume average strain. Under Voigt's model, Reuss’ model and Man's theory, the elastic constitutive relation accounts for the effect of the orientation distribution function (ODF) up to terms linear in the texture coefficients. However, the formula derived in this paper delineates the effect of crystallographic texture on elastic response and shows quadratic texture dependence. The formula is very simple. We also consider the influence of grain shape to elastic constitutive relations of polycrystals. Some examples are given to compare computational results of the formula with those given by Voigt's model, Reuss's model, the finite element method, and the self-consistent method. In Section 3, we also present an expression of the perturbation displacement field, in which Green's function for an orthorhombic aggregate of cubic crystallites is included.     

On the Green's Functions and Boundary Integral Formulation of Elastic Planes with Cutouts

ABSTRACT

The problem of an infinite elastic plane that contains a hole of arbitrary shape and is subjected to a concentrated unit load is considered. The Green's function (influence function) for the problem is formulated by means of two complex potential functions. This is accomplished by mapping the region that is exterior to the hole onto a unit circle. A class of closed contour hole shapes is analyzed. Green's functions for an elliptical hole and a class of triangular holes are determined. Green's functions for a class of rectangular holes are also discussed. In order to determine stress and displacement fields for the finite plane problem, Green's function is employed and an indirect boundary integral equation is formulated, with the integrand of the integral equation incorporating the effect of the hole. The contour of the hole is no longer considered a part of the boundary and only the contour of the region that is exterior to the hole is subdivided into boundary elements. Examples for elliptical and triangular holes are solved.