A semi-smooth Newton method for elasto-plastic contact problems

In this paper we reformulate the frictional contact problem for elasto-plastic bodies as a set of unconstrained, non-smooth equations. The equations are semi-smooth so that Pang's Newton method for B-differentiable equations can be applied. An algorithm based on this method is described in detail. An example demonstrating the efficiency of the algorithm is presented.     

A Newton type iterative method for heat-conduction inverse problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

A Newton type iterative inverse method for heat-conduction problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

A higher-order boundary element method for three-dimensional potential problems

A method of eliminating the singularities involved in boundary element methods for three-dimensional potential problems is presented and the non-singular expressions of integrals on an element on which the singular point is situated are given for linear and quadratic interpolation functions. Numerical examples are compared with analytical solutions to show that the higher-order interpolations have better precision.     

A three-dimensional fictitious domain method for incompressible fluid flow problems

A new Galerkin finite element method for the solution of the Navier–Stokes equations in enclosures containing internal parts which may be moving is presented. Dubbed the virtual finite element method, it is based upon optimization techniques and belongs to the class of fictitious domain methods. Only one volumetric mesh representing the enclosure without its internal parts needs to be generated. These are rather discretized using control points on which kinematic constraints are enforced and introduced into the mathematical formulation by means of Lagrange multipliers. Consequently, the meshing of the computational domain is much easier than with classical finite element approaches. First, the methodology will be presented in detail. It will then be validated in the case of the two-dimensional Couette cylinder problem for which an analytical solution is available. Finally, the three-dimensional fluid flow inside a mechanically agitated vessel will be investigated. The accuracy of the numerical results will be assessed through a comparison with experimental data and results obtained with a standard finite element method. © 1997 John Wiley & Sons, Ltd.     

三维接触问题的非光滑算法

给出了一种非光滑算法直接用于求解三维摩擦接触问题的不可微非线性互补模型,不再对模型进行光滑化处理,使算法更加简单。文中对非光滑算法的收敛性给出了严格的数学证明,数值实验表明该算法列式简单,但与光滑化算法同样有效。     

三维声学特征频率灵敏度分析的边界元法

声系统特征频率的灵敏度分析为其优化设计提供了基础,具有重要意义。边界元法在声学问题的求解中具有独特优势,但因其系统方程系数矩阵的频率相关性导致的非线性特征值问题给声学特征频率的灵敏度分析带来了很大困难。为此,本文首先对非线性特征值问题进行了线性化处理,利用围道积分投影方法将非线性特征方程转换为小规模广义特征方程,然后对其关于设计变量直接求导,并引入左特征向量和转换矩阵构造了一种适用于内外声场的三维声学单/重特征频率灵敏度分析的边界元法。数值算例验证了该方法的适用性,以及对单/重特征频率灵敏度的计算精度。     

The analytical method for three-dimensional elastic problems in finite deformations

The three-dimensional elastic problems in finite deformations are not known to have been analyzed by the usual stress function and displacement function. By applying Hasegava's presentation and Adkins perturbation method, we propose a new analytical method for three-dimensional elastic problems for compressible materials and incompressible materials, using the displacement function for axisymmetrical elastic problems in finite deformations with surface force or body force. Further, this analytical method is examined by two simple examples.     

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.     

A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

In this paper, a new semi-analytical method is presented for modeling of three-dimensional (3D) elastostatic problems. For this purpose, the domain boundary of the problem is discretized by specific subparametric elements, in which higher-order Chebyshev mapping functions as well as special shape functions are used. For the shape functions, the property of Kronecker Delta is satisfied for displacement function and its derivatives, simultaneously. Furthermore, the first derivatives of shape functions are assigned to zero at any given node. Employing the weighted residual method and implementing Clenshaw–Curtis quadrature, coefficient matrices of equations’ system are converted into diagonal ones, which results in a set of decoupled ordinary differential equations for solving the whole system. In other words, the governing differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. To evaluate the efficiency and accuracy of the proposed method, which is called Decoupled Scaled Boundary Finite Element Method (DSBFEM), four benchmark problems of 3D elastostatics are examined using a few numbers of DOFs. The numerical results of the DSBFEM present very good agreement with the results of available analytical solutions.     

A new nondestructive method for three-dimensional photoelasticity

A new set of equations describing completely the optical phenomena in a model involving continuous rotation of secondary axes and secondary principal-stress differences are obtained. These are solved by Peano-Baker method using experimentally determined characteristic parameters for several wavelengths of light. Experimental verifications are obtained for a rectangular bar subjected to combined torsion and tension.     

Extended finite element method for fretting fatigue crack propagation

In this paper, the extended finite element method (X-FEM) is considered for the analysis of fretting fatigue problems. A two-dimensional implementation of the X-FEM is carried out within the finite element software ABAQUS? by means of user subroutines, and crack propagation in fretting fatigue problems is investigated. On utilizing the non-linear contact capabilities of this code, the numerical technique is applied to a specimen-indenter model. The use of the X-FEM facilitates very accurate stress intensity factor computations on relatively coarse meshes, and furthermore, no remeshing is required for crack growth simulations. The implementation is applied to complete and incomplete contact fretting problems. A study of crack growth is conducted for several bulk loads applied to the specimen, and the influence of the initial crack angle is ascertained. The numerical simulations reveal the merits of applying the X-FEM to fretting fatigue problems.     

A composite three-dimensional photoelastic method

Care must be taken in preparation and testing of three-dimensional composite photoelastic models. Some problems encountered in modeling the prototype and during model testing are: model-material failure, loss of fringe pattern in slicing, inherent shrinkage response in freezing, inadequate bonding between materials, and modular ratio difficulties. The selection of the correct plastic can eliminate the first four problems, but the correct modular ratio between the matrix and the insert has to be obtained. This investigation illustrated the behavior of commercially manufactured plastics as inserts, with a matrix material of Epon 828 epoxy. The effective moduli of elasticity of these plastics are reported for pure tension and for flexure. Since the manufactured plastics produced varying results, the use of Epon 828 epoxy as an insert was investigated. The inserts were cast in tygon tubing and their curing cycle was altered from that used for the matrix material to produce a different effective modulus of elasticity. The Epon 828 inserts gave excellent results in the beams. The use of the same material for matrix and insert eliminates many of the problems associated with composite three-dimensional photoelasticity.     

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.