Parallel-vector computations for geometrically nonlinear finite element analysis

Existing procedures for nonlinear finite element analysis are reviewed. Common computational steps among existing methods are identified. Parallel-vector solution strategies for the generation and assembly of element matrices, solution of the resulting system of linear equations, calculations of the unbalanced loads, displacements and stresses are all incorporated into the Newton-Raphson (NR), modified Newton-Raphson (mNR), and BFGS methods. Furthermore, a mixed parallel-vector Choleski-Preconditioned Conjugate Gradient (C-PCG) equation solver is also developed and incorporated into the piecewise linear procedure for nonlinear finite element analysis. Numerical results have indicated that the Newton-Raphson method is the most effective nonlinear procedure and the mixed C-PCG equation solver offers substantial computational advantages in a parallel-vector computer environment.     

Modal representation of geometrically nonlinear behavior by the finite element method

A method is described for representing mild geometrically nonlinear static behavior of thin-type structures, within the finite element method, in terms of the solution of a certain eigenvalue problem. This eigenvalue problem, commonly known as the linear or bifurcation buckling problem for a restricted class of so-called “perfect” structural situations, is thus seen to have a broader significance. The applied loading nonlinearly amplifies the contributions of each mode (eigenvector) present in the linear finite dement solution, and the amplification factors are easily computable functions of the eigenvalues. Computational results for braced frames and arches under asymmetric loading are presented.     

A geometrically nonlinear shell finite element for tire vibration analysis

An axisymmetric finite element is developed which includes such features as orthotropic material properties, doubly curved geometry, and both the first and second order nonlinear stiffness terms. This element can be used to predict the equilibrium state of an axisymmetric shell structure with geometrically nonlinear large displacements. Small amplitude vibration analysis can then be performed based on this equilibrium state. The nonlinear path is predicted by using the self-correcting incremental procedure and any point on the path can be checked by using the Newton-Raphson iterative scheme. The present formulation and solution procedure are evaluated by analyzing a series of examples with results compared with alternative known solutions. Examples include: free vibration of an isotropic cylindrical shell, a conical frustum, and an orthotropic cylindrical shell; buckling of a cylindrical shell; large deflection of a clamped disk, a spherical cap, and a steel belted radial tire. The final example is a free vibration analysis of the inflated tire and the natural frequencies obtained compared well with published experimental data.     

Stochastic linearisation of geometrically non-linear finite element models

The equivalent linearisation method is a well established approximate technique for the probabilistic analysis of non-linear structures subjected to random loads. It is shown here that this technique is readily applicable to geometrically non-linear small strain problems in which the equations of motion of the structure are derived using the Finite Element method. A general formulation for problems of this type is presented, following which the method is applied to the non-linear vibrations of an elastic beam. Good agreement with published results is obtained, demonstrating the general feasibility of the approach.     

A new heterosis plate element for geometrically non-linear finite element analysis of laminated plates

In this study, a new nine-node quadrilateral, shear-deformable heterosis element is developed. In order to model this element, Kirchhoff constraints are modified using Reissner-Mindlin theory assumptions. All of the modifications are performed for first-order shear-deformation theory (FSDT). This new heterosis element is developed by modifying 8-node serendipity and twelve-node cubic polynomials. The new heterosis element is used with nine-node Lagrangian elements in finite element analysis of composite plates. A modified element is used in finite element analysis of linear and non-linear analysis considering the advantages of free of ‘shear locking’. Numerical results are presented by comparing Navier's series solution.     

Subdomain generation for non-convex parallel finite element domains

In this paper the Subdomain Generation Method (SGM), originally formulated in Khan & Topping (1993; Khan, A. I. & Topping, B. H. V., Subdomain generation for parallel finite element analysis. Comput. Syst. Engng, 1993, 4(4/6), 473–488) for convex finite element domains, is generalized for arbitrary shaped domains. Modifications to the original SGM are described which allow partitioning of non-convex domains. These modifications have been made to the formulation of the optimization module and the predictive module. The examples presented in Khan & Topping (1993) have been re-worked and two more examples have been added which demonstrate the application of the method to arbitrary shaped domains. It is shown with the aid of the examples that the method provides well-balanced subdomains very efficiently and allows parallel adaptive mesh generation. The method in its present form may be used to partition unstructured graphs in two or three dimensions. Since the computational cost for the mesh partitioning with this method depends solely upon the initial coarse mesh, hence the computational cost does not increase with the increase in the mesh density of the final mesh. The method in its present form is unsuitable for relatively coarse grained parallel computers, however the modifications which would impart a greater degree of scalability to this method are discussed.     

A Nitsche-extended finite element method for earthquake rupture on complex fault systems

The extended finite element method (XFEM) provides a natural way to incorporate strong and weak discontinuities into discretizations. It alleviates the need to mesh discontinuities, allowing simulation meshes to be nearly independent of discontinuity geometry. Currently, both quasistatic deformation and dynamic earthquake rupture simulations under standard FEM are limited to simplified fault networks, as generating meshes that both conform with the faults and have appropriate properties for accurate simulation is a difficult problem. In addition, fault geometry is not well known; robustness of solution to fault geometry must be determined. Remeshing with varying geometry would make such tests computationally unfeasible. The XFEM makes a natural choice for discretization in these crustal deformation simulations on complex fault systems. Here, we develop a method based upon the XFEM using Nitsche’s method to apply boundary conditions, enabling the solution of static deformation and dynamic earthquake models. We compare several approaches to calculating and applying frictional tractions. Finally, we demonstrate the method with two problems: an earthquake community dynamic code verification benchmark and a quasistatic problem on a fault system model of southern California.     

复杂结构井钻压值的有限元数值计算

通常,钻井过程中钻杆自重生成的钻杆轴向力即钻压的计算是通过解析方法计算的.由于接触条件复杂,钻杆与井壁及套管接触产生的摩擦力一般通过简化的经验系数计入.随着水平井等复杂结构井的普及,解析方法得到的钻杆轴向力的结果误差逐渐明显,不能满足实际需要.因此,采用三维有限元法计算钻杆自重生成的钻杆轴向力即钻压值.模型采用管单元模拟套管和钻柱,并在钻杆与井壁之间设置专门模拟管-管相互作用的ITT摩擦接触单元,判断钻柱与套管和井壁的接触状态,可以准确计算钻杆全长沿井轴方向的轴向力、轴向摩擦力及其引起的钻压变化.在钻杆全长的整体力学行为的有限元计算中,还可以考虑钻杆的屈曲问题.结果表明:有限元法是正确设计钻压的有效理论工具.该数值结果为正确设计钻杆尺寸、得到理想的钻压值提供成功的例子.     

A hybrid method for finite element ordering

Nodal ordering for the formation of suitable sparsity patterns for stiffness matrices of finite element meshes are often performed using graph theory and algebraic graph theory. In this paper a hybrid method is presented employing the main features of each theory. In this method, vectors containing certain properties of graphs are taken as Ritz vectors, and using methods for constructing a complementary Laplacian, a reduced eigenproblem is formed. The solution of this problem results in coefficients of the Ritz vectors, indicating the significance of each considered vector.The present method uses the global properties of graphs in ordering, and the local properties are incorporated using algebraic graph theory. The main feature of this method is its capability of transforming a general eigenproblem into an efficient approach incorporating graph theory. Examples are included to illustrate the efficiency of the presented method.     

A finite element method for flow separation

The finite element model has been developed in order to solve separation pattern of the flow past an obstruction in a two-dimensional flow field. The Helmholtz-Poisson form of the Reynolds equations are solved alternately until a stable flow separation in the neighbourhood of the obstruction is obtained. In order to check the results of the finite element model, an experimental separation pattern using Pitot-tube measurements has been conducted. The computed and the experimental flow separation patterns show a good agreement.     

Mixed finite element method for contact problems

In the paper the model of finite elements for elastic contact problems was used. Real structures are modeled by finite elements and rigid finite elements. We calculate stresses in the surface of two substructures using Coulomb model of friction.The method given here is an iterative procedure which is planed to incorporate this technique in the system allowing for incremental elastic solution. The computer program is adapted to solving spatial problems.     

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.     

On finite element methods for elliptic equations on domains with corners

A finite element method for approximating elliptic equations on domains with corners is proposed. The method makes use of the singular functions of the problem in the trial space and the kernel functions of the adjoint problem in the test space. This leads to good approximates of the coefficients of the singular functions. In the numerical computations, the method is compared with the well known Singular Function Method.     

Superconvergence of mixed finite element methods on rectangular domains

Superconvergence by one power of h along Gauss lines for theBDFM (Brezzi-Douglas-Fortin-Marini) mixed finite element approximation by rectangular elements of the vector field associated with a second order elliptic equation on a rectangular domain is established under an appropriate assumption of smoothness on the exact solution.     

Characteristic-based operator-splitting finite element method for Navier-Stokes equations

A new finite element method, which is the characteristic-based operator-splitting (CBOS) algorithm, is developed to solve Navier-Stokes (N-S) equations. In each time step, the equations are split into the diffusive part and the convective part by adopting the operator-splitting algorithm. For the diffusive part, the temporal discretization is performed by the backward difference method which yields an implicit scheme and the spatial discretization is performed by the standard Galerkin method. The convective...     

A Partitioned finite element method for dynamical systems

A new analytical method is described for deriving the equations of motion of dynamical systems. The concept is to consider the displacements of the domain to be composed of rigid and elastic components. In contrast to other reduction methods, the domain modeled by finite number of degrees of freedom is discretized into two distinctive types of subdomains. Rigid and elastic subdomains are generated by consistent lumping of the domain properties under unique kinematic constraint relations. Equations of motion of the disjoint subdomains are derived by Lagrange's equations, in conjunction with the shape function matrix represented in partitioned form. This allows reduced sizes of matrices and avoids their possible singularities. Based on the invariance of energies under a compatible partitioned procedure, a simple analytical method is introduced for building the equations of motion of the whole domain from those of the subdomains. The dynamic analysis of a two-node domain with application to a blade-shaft combination is presented to illustrate the method.     

Convenient representation method for spatial finite element structures

A calculation procedure is presented which allows the visible and invisible lines of spatial structures to be differentiated.     

无条件稳定的显式降维非线性有限元

针对传统降维非线性有限元计算速度与精确度难以兼顾的问题,提出了一种无条件稳定的显式迭代算法。基于泰勒展开式得到速度、加速度的三阶精度差分表达式从而获得新的有限元显式迭代方程,并分析其单自由度系统下的传递矩阵谱半径。改进迭代方程使谱半径始终小于1从而满足无条件稳定的要求。实验表明,改进后的显式迭代算法在等效阻尼比的精度上优于中心差分法和隐式迭代法;在降维非线性有限元模型计算中的计算耗时优于隐式迭代方法,提高了降维非线性有限元的迭代计算速度。模型在降维后维度数值较高时,仍能维持良好的计算耗时和帧率,保证了模型的精确度。     

A finite element method for consolidation of clay

We consider a model for consolidation of clay in the case of an elasto-plastic soil skeleton. We prove existence of a solution and we prove an error estimate for a finite element method for finding approximate solutions of the problem.