Assessment of temporal integration schemes for the sensitivity analysis of frictional contact/impact response of axisymmetric composite structures

An assessment is made of three temporal integration schemes for the sensitivity analysis of the frictional contact/impact response of axisymmetric composite structures. The structures considered consist of an arbitrary number of perfectly-bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric non-linearity is included. Sensitivity coefficients measure the sensitivity of the response to variations in the different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the spatial discretization. Normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with fundamental unknowns consisting of both the nodal displacements and the Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using either a penalty method or a Lagrange multiplier technique, in conjunction with the classical Coulomb's friction model. The three temporal integration schemes considered are: the implicit Newmark and Houbolt schemes, and the explicit central difference method. In the case of the implicit methods, the Newton-Raphson iterative technique is used for the solution of the resulting non-linear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. Sensitivity coefficients are evaluated by using a direct differentiation approach in conjunction with the incremental equations. Numerical results are presented for the frictional contact of a composite spherical cap impacting a rigid plate, showing the effects of each of the following factors on the accuracy of the predicted response and sensitivity coefficients: (a) incorporating the normal contact conditions, (b) the magnitude of the penalty parameter in the normal direction (for the perturbed Lagrangian method), and (c) the time step size for the response and the sensitivity analyses.     

Sensitivity analysis of frictional contact/ impact response on distributed-memory computers

Multilevel parallel computational strategies are presented for predicting the frictional contact/impact response and evaluating the sensitivity coefficients of axisymmetric composite structures. Both implicit and explicit temporal integration techniques are considered. For implicit techniques, parallelism is exploited in both the spatial and temporal domains. Spatial parallelism is achieved by using a parallel sparse equation solver based on a nested dissection node-ordering scheme. The explicit techniques exploit parallelism using both an element-based domain decomposition strategy, as well as concurrent evaluation of sensitivity coefficients. Implementations of the strategies on distributed-memory computers are described. The strategies are applied to the problem of an axisymmetric composite spherical cap impacting a rigid surface, and the performance characteristics are assessed on the IBM SP2 and the Cray T3D parallel computer systems.     

Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation

Direct differentiation method of sensitivity analysis is developed for frictional contact problems. As a result of the augmented Lagrangian treatment of contact constraints, the direct problem is solved simultaneously for the displacements and Lagrange multipliers using the Newton method. The main purpose of the paper is to show that this formulation of the augmented Lagrangian method is particularly suitable for sensitivity analysis because the direct differentiation method leads to a non-iterative exact sensitivity problem to be solved at each time increment. The approach is applied to a general class of three-dimensional frictional contact problems, and numerical examples are provided involving large deformations, multibody contact interactions, and contact smoothing techniques.     

Nonlinear finite element analysis for composite structures of axisymmetric geometry and loading

A linear strain, axisymmetric, triangular finite element is formulated for the modeling of materials which are generally orthotropic in the plane of the fibers and loaded axisymmetrically. The fibers may be oriented in planes other than the symmetric plane.This necessitates the use of three displacement degrees of freedom at each node.The element is also formulated to include geometric nonlinearity to take into account any stiffening or weakening of the structure due to deformation.The element is programmed into a finite element code, which is validated using a well-known isotropic nonlinear problem. It shows very good agreement with the results of other programs.The unique application of the element is shown in the analysis of a pneumatic tire.     

Bem frictional contact analysis: Load incremental technique

A boundary element formulation for solving structural problems associated with frictional contact is presented; it uses an efficient, iterative and fully load-incremental technique. Problems with any number of two-dimensional bodies in contact can be analysed using this technique; the bodies may be conforming or non-conforming, of similar or dissimilar materials. The interface may be frictionless or frictional and undergo slip or partial slip. Numerical solutions of both normal and tangential traction distributions can be obtained automatically for successive load increments. The technique utilizes automatic updating procedures in order to model the continuously changing boundary conditions occurring inside the contact region. Results obtained show that an accurate account of the non-linear behaviour, caused by the frictional effects, can be obtained only by following the loading history as well as the contact history.     

Collapse load evaluation of structures with frictional contact supports under combined stresses

This paper extends classical limit analysis to structures for which some supports are subjected to “nonstandard” unilateral frictional contact with the ground. A typical and commonly adopted model is nonassociative Coulomb friction. For such cases, the use of the classical bound theorems is not possible. Moreover, simply solving the governing equations as a mixed complementarity problem (MCP) does not guarantee that the best bound has been calculated. We have therefore developed an approach that attempts to compute, in a single step, the critical (least) upper bound solution by formulating and solving an instance of the challenging class of optimization problems, known as a mathematical program with equilibrium constraints (MPEC). Two examples are provided to illustrate application of the proposed scheme, as well as to highlight some key features of such structures.     

Nonlinear harmonic analysis applied to axisymmetric structures

The total Lagrangian formulation for axisymmetric solids, which was introduced in an earlier paper [Comput. Struct.49, 219–230 (1993)] is investigated in more detail with a series of sample analyses. The formulation uses a cylindrical reference system to define the geometry, a Cartesian reference system for the displacement field and Fourier decomposition to describe the displacement field in the circumferential direction. Loading functions are defined in terms of the Cartesian displacement reference system and the Fourier functions. The capabilities of the formulation are further demonstrated in several examples which compare its solutions with analytical solutions, other Fourier decomposition formulations and physical test of a pipe subjected to combined axial and bending loads.     

An iterative procedure for finite-element stress analysis of frictional contact problems

A simple, efficient, versatile and easily adaptable, iterative finite-element technique is described for solving frictional contact problems. The method is based on logical steps to establish the contact geometry and regions of slip and nonslip. Unlike previous techniques, the approach can be extended readily to multiple contact surfaces. The scheme is demonstrated by analyzing a mechanical joint in orthotropic wood. In this case, mixed coordinate systems are used to enhance accuracy of the stresses near the pin contact region. The numerically computed values agree with those reported elsewhere.     

The effect of contact interface on dynamic characteristics of composite structures

In this project, nonlinear characteristics on the rolling interface of a linear guide were studied by the finite element analysis and experimental verification. Contact of the ball/surface rolling interface in the rolling guides was simulated as a three-dimensional membrane element without thickness. By introducing Hertzian contact theory and applying proper normal/shear stiffness to such contact elements in the overall finite element model, dynamic behaviors of linear guides affected by preload were thus investigated. In the finite element procedure, three contact models, 1D point-to-point, 2D point-to-point and 3D surface-to-surface, were sequentially introduced for purpose of verification with experiments. As a validation in this project, vibrational experiments on linear guides with different preloads were conducted and related frequency spectrums were derived. Both the finite element and the experimental results reveal that the natural frequency of a linear guide increases with the increment of the preload. In addition, the dynamic characteristics predicted by finite element analysis agree well with those measured from instrumental experiments. The proposal of current study may provide an alternate and reliable way for understanding of the dynamic characteristic of the rolling contact components in machine design field.     

Sensitivity analysis in shape optimization of continuum structures

This paper describes an efficient method for sensitivity analysis in shape optimum design. One feature is the use of limited number of master nodes to characterize the surfaces of a set of isoparametric finite elements, and the adoption of their coordinates as design variables of the shape optimization. Another is the derivation of analytical formulations of the gradients of both the stiffness terms and the load vectors with respect to the design variables. A finite element analysis code is adapted to the purposes of the method and numerical examples are performed and comparisons made with sensitivity analysis based on forward finite differences.     

Numerical analysis of coupled thermomechanical frictional contact problems. Computational model and applications

Summary In this paper a numerical model for the analysis of coupled thermomechanical multi-body frictional contact problems at finite deformations is presented. The multi-body frictional contact formulation is fully developed on the continuum setting and then a spatial (Galerkin projection) and temporal (time-stepping algorithm) discretization is applied. A contact pressure and temperature dependent thermal contact model has been used. A fractional step method arising from an operator split of the governing equations has been used to solve the coupled nonlinear system of equations, leading to a staggered solution algorithm. The numerical model has been implemented into an enhanced version of the computational finite element program FEAP. Numerical examples and simulation of industrial metal forming processes show the performance of the numerical model in the analysis of coupled thermomechanical frictional contact problems.     

Nonsmooth spatial frictional contact dynamics of multibody systems

Multibody System Dynamics - Computational prediction of 3D crutch-assisted walking patterns is a challenging problem that could be applied to study different biomechanical aspects of crutch walking...     

Limit analysis of masonry block structures with non-associative frictional joints using linear programming

Although limit analysis has been found to be a valuable tool for analysing the stability of masonry gravity structures, modelling non-associative Coulomb sliding friction can be problematic. A simple iterative procedure which involves the successive solution of linear programming sub-problems is presented in the paper. Using the procedure a specially modified Mohr-Coulomb failure surface is adopted at each contact interface, with all failure surfaces updated at each iteration until a converged solution is obtained. The procedure is applied to problems from the literature and also to new, considerably larger, benchmark problems.     

Thermoelastic analysis of layered structures with imperfect layer contact

The behavior of orthotropic layered slabs and cylinders in which the temperature and stress distributions vary in the thickness direction is investigated. A finite element formulation utilizing quadratic layer elements and linear interface elements are used to perform the analyses. The transient heat conduction response is obtained using implicit linear time interpolation including the Crank-Nicolson, Galerkin and Euler backward schemes. It is shown that the modified Crank-Nicolson and modified Galerkin schemes provide computationally economical solutions which are both accurate and free of temporal oscillations. The effects of heat-transfer resistance at layer contact surfaces are illustrated through numerical examples.     

Two aggregate-function-based algorithms for analysis of 3D frictional contact by linear complementarity problem formulation

Three dimensional frictional contact is formulated as linear complementarity problem (LCP) by using the parametric variational principle and quadratic programming method. Two aggregate-function-based algorithms, called respectively as self-adjusting interior point algorithm and aggregate function smoothing algorithm, are proposed for the solution of the LCP derived from the contact problems. A nonlinear finite element code is developed for numerical analysis of 3D multi-body contact problems. Four numerical examples are computed to demonstrate the applicability and computational efficiency of the methods proposed.     

Optimal design of three-dimensional axisymmetric elastic structures

The problem of maximizing the overall stiffness of an elastic body comprised of given materials will be treated. Particular examples include the optimal shape and structure of shells, plates, domes, cantilevers, etc. The axisymmetry allows us to compute mathematically optimal out-of-plane examples. We will use recently developed variational methods of optimizing local composite structures in conjunction with a computational global minimization strategy. The optimal designs could be simplified into suboptimal projects subject to other practical considerations.     

Elasto-plastic analysis of axisymmetric structures subject to arbitrary loads by hybrid-stress finite elements

A semi-analytic finite element method in conjuction with a hybrid-stress functional based on the initial stress approach is presented for elasto-plastic analysis. A three-dimensional solid element based on hybrid-stress model is also implemented for the elasto-plastic analysis. A procedure to compute the non-linear effects in terms of Fourier series in the hybrid-stress model is described. The accuracy and efficiency of the semi-analytic method is evaluated via numerical examples by comparing the solution with a full 3-D solution. The semi-analytic method is observed to be a viable alternative to 3-D analysis in elasto-plastic analysis of axisymmetric structures subject to arbitrary loads.     

On static analysis of tensile structures with sliding cables: the frictional sliding case

In some structural systems, such as cable structures, membranes and tensegrity structures, the use of sliding cables allows to reduce the number of elements required to be controlled during tensioning or activation. However, using sliding cables modifies the structural behavior of tensile structures since it alters the distribution of axial forces in structural members. This has been experienced in structures with continuous cables under the assumption of frictionless sliding. However, sliding-induced friction can further alter the behavior of the system. An enhancement of the static analysis of tensile structures with sliding-induced friction is investigated in this paper. In the proposed formulations, the finite-element analysis method and the dynamic relaxation method are combined with a linear complementary approach. Sliding-induced friction is integrated in the formulations through the consideration of the Euler–Eytelwein equation. The importance of considering sliding-induced friction in the static analysis of tensile structures is demonstrated through a series of examples, where it is shown that friction significantly affects the mechanical behavior of the structures. The examples also reveal that the proposed formulations do not affect the computational time of the static analyses.