On the solution of a minimum compliance topology optimisation problem by optimality criteria without a priori volume constraint specification

The paper presents a topology optimisation method based on optimality criteria for total potential energy maximisation with a volume constraint. The final volume of the optimal structural configuration has not to be specified a priori and is directly controlled by the stress, displacement or stiffness constraints defined at the design problem layout phase. The proposed method leads to the identification of well defined structures characterised by a small number of discrete elements with intermediate material properties within a limited number of iterations. The results obtained by solving several two dimensional benchmark problems are shown.     

Path following and critical points for contact problems

This work is concerned with the problem of tracing the equilibrium path in large displacement frictionless contact problems. Conditions for the detection of critical points along the equilibrium path are also given. By writing the problem as a system of non-linear B-differentiable functions, the non-differentiability due to the presence of the unilateral contact constraints is overcome. The path-following algorithm is given as a predictor-corrector method, where the corrector part is performed using Newton's method for B-differentiable functions. A new type of displacement constraints are introduced where the constraining displacement node may change during the corrector iterations. Furthermore it is shown that, in addition to the usual bifurcation and limit points, bifurcation is possible or the equilibrium path may have reached an end point even if the stiffness matrix is non-singular.     

On a two‐level Newton‐type procedure applied for solving non‐linear elasticity problems

The present work is devoted to the damped Newton method applied for solving a class of non‐linear elasticity problems. Following the approach suggested in earlier related publications, we consider a two‐level procedure which involves (i) solving the non‐linear problem on a coarse mesh, (ii) interpolating the coarse‐mesh solution to the fine mesh, (iii) performing non‐linear iterations on the fine mesh. Numerical experiments suggest that in the case when one is interested in the minimization of the L₂‐norm of the error rather than in the minimization of the residual norm the coarse‐mesh solution gives sufficiently accurate approximation to the displacement field on the fine mesh, and only a few (or even just one) of the costly non‐linear iterations on the fine mesh are needed to achieve an acceptable accuracy of the solution (the accuracy which is of the same order as the accuracy of the Galerkin solution on the fine mesh). Copyright © 2000 John Wiley & Sons, Ltd.     

A multiple-scales approach to crack-front waves

Perturbation of a steadily propagating crack with a straight edge is solved using the method of matched asymptotic expansions (MAE). This provides a simplified analysis in which the inner and outer solutions are governed by distinct mechanics. The inner solution contains the explicit perturbation and is governed by a quasi-static equation. The outer solution determines the radiation of energy away from the tip, and requires solving dynamic equations in the unperturbed configuration. The outer and inner expansions are matched via the small parameter ϵ = L/l defined by the disparate length scales: the crack perturbation length L and the outer length scale l associated with the loading. The method is illustrated for a scalar crack model and then applied to the elastodynamic mode I problem. The crack-front wave-dispersion relation is found by requiring that the energy release rate is unaltered under perturbation and dispersive properties of the crack-front wave speed are described for the first time. The example problems considered demonstrate the potential of MAE for moving-boundary-value problems with multiple scales.     

Evolution of stresses in a simple class of oxidation problems

Summary This paper derives the stresses that arise during the oxidation, for example, of a metal. The analysis is carried out in the current or deformed configurations of the oxide and metal, without having to explicitly determine their reference configurations. Elasticity problems are normally solved in a reference configuration, often an initial configuration. In contrast, the appropriate reference configurations for the metal and oxide are not known a priori. The differences between this approach and prior analyses based on an eigenstrain are discussed. To illustrate the method of solution, we discuss silicon oxidation, assuming that molecular oxygen diffuses through the oxide, reacts instantaneously at the metal-oxide interface, and does not diffuse into the metal. The specific geometries chosen are oxidation on the surface of a cylinder and in a cylindrical hole, which are perhaps the simplest one-dimensional oxidation problems with non-zero stress distributions.     

Initial design strategies for iterative design

This paper discusses methods to retrieve designs from a design library in order to achieve a better initial starting point for the iterative model of the design process. The motivation for doing this is to reduce the extensive analysis time required for many iterative design problems. Starting a design at a favorable initial point should help reduce the number of iterations. Four initial design methods have been investigated, varying from a simple nonlibrary method to methods that use designs from a library. The iterative design scheme used is conventional hill-climbing. To evaluate the effectiveness of these methods, the initial design methods were tested on four example problems. They are, a cantilever beam, a gear-pair, a V-belt, and an extruder-die. It was found that the number of iterations reduced approximately to the order of 1/n [referred in the rest of the article as O (1/n)], wheren is the number of stored values in the design library.     

Combined interior‐point method and semismooth Newton method for frictionless contact problems

In the present paper, a solution scheme is proposed for frictionless contact problems of linear elastic bodies, which are discretized using the finite element method with lower order elements. An approach combining the interior‐point method and the semismooth Newton method is proposed. In this method, an initial active set for the semismooth Newton method is obtained from the approximate optimal solution by the interior‐point method. The simplest node‐to‐node contact model is considered in the present paper, that is, pairs of matching nodes exist on the contact surfaces. However, the discussions can be easily extended to a node‐to‐segment or segment‐to‐segment contact model. In order to evaluate the proposed method, a number of illustrative examples of the frictionless contact problem are shown. The proposed combined method is compared with the interior‐point method and the semismooth Newton method. Two numerical examples that are difficult to solve using the semismooth Newton method are solved effectively using the proposed combined method. It is shown that the proposed method converges within far fewer iterations than the semismooth Newton methods or the interior‐point method. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite deformation post-buckling analysis involving inelasticity and contact constraints

This paper is concerned with the numerical solution of large deflection structural problems involving finite strains, subject to contact constraints and unilateral boundary conditions, and exhibiting inelastic constitutive response. First, a three-dimensional finite strain beam model is summarized, and its numerical implementation in the two-dimensional case is discussed. Next, a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed. Finally, basic strategies for tracing limit points are reviewed and a modification of the arc-length method is proposed. The good performance of the procedures discussed is illustrated by means of numerical examples.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

Experience with the conjugate gradient method for stress analysis on a data parallel supercomputer

The storage requirements and performance consequences of a few different data parallel implementations of the finite element method for domains discretized by three-dimensional brick elements are reviewed. Letting a processor represent a nodal point per unassembled finite element yields a concurrency that may be one to two orders of magnitude higher for common elements than if a processor represents an unassembled finite element. The former representation also allows for higher order elements with a limited amount of storage per processor. A totally parallel stiffness matrix generation algorithm is presented. The equilibrium equations are solved by a conjugate gradient method with diagonal scaling. The results from several simulations designed to show the dependence of the number of iterations to convergence upon the Poisson ratio, the finite element discretization and the element order are reported. The domain was discretized by three-dimensional Lagrange elements in all cases. The number of iterations to convergence increases with the Poisson ratio. Increasing the number of elements in one special dimension increases the number of iterations to convergence, linearly. Increasing the element order p in one spatial dimension increases the number of iterations to convergence as p^α, where α is 1·4–1·5 for the model problems.     

Solution procedures for nonlinear structural dynamic analysis

Abstract

A review of the solution techniques most widely used in nonlinear structural dynamics is presented. For nonlinear transient responses several explicit and implicit direct integration methods are compared with respect to accuracy, stability and computational efficiency. It is concluded that the choice of a suitable method depends upon the nature of the method, the formulation of finite element models, and the problem itself. In general, the Park method seems to be superior to the others in nonlinear dynamic analysis. If equilibrium iterations are performed at each time step, the Newmark — ß= 1/4 method should be preferred.     

The w-plane finite element method for the solution of scalar field problems in two dimensions

A simple, efficient finite element method has been presented for the solution of a variety of scalar field problems in two dimensions. It is based on the mapping of the physical problem domain into an ‘image’ domain in the w-plane. The governing equation(s) and the boundary conditions in the physical plane are also appropriately transformed into the w-plane. The processes of standard finite element analysis are then implemented to obtain a solution in the w-plane. The method has been explained in detail, with illustrative examples where appropriate; it has several important advantages over the standard finite element method, particularly for the solution of infinite or semi-infinite domain problems. The method has been demonstrated to be simple, efficient, economical and potentially capable of dealing with a large repartoire of two-dimensional problems, including non-homogeneity, nonlinearity, etc.     

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n ? 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi‐infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupling of physical and modal components for analysis of moving non-linear dynamic systems on general beam structures

A general, well-structured and efficient method is advanced for the solution of-a large class of dynamic interaction problems including a non-linear dynamic system running at a prescribed time-dependent speed on a linear track or guideway. The method uses an extended state-space vector approach in conjunction with a complex modal superposition. It allows for the analysis of structures containing both physical and modal components. The physical components studied here are vehicles modelled as linear or non-linear discrete mass–spring–damper systems. The modal component studied is a linear continuous model of a track structure containing beam elements which can be generally damped and which can be embedded in a three-parameter damped Winkler-type foundation. The complex modal parameters of the track structure are solved for. Algebraic equations are established which impose constraints on the transverse forces and accelerations at the interfaces between the moving dynamic systems and the track. An irregularity function modelling a given non-straight profile of the non-loaded track or a non-circular periphery of the wheels is also accounted for. Loss of contact and recovered contact between a vehicle and the track can be treated. The system of coupled first-order differential equations governing the motion of the vehicles and the track and the set of algebraic constraint equations are together compactly expressed in one unified matrix format. A time-variant initial-value problem is thereby formulated such that its solution can be found in a straightforward way by use of standard time-stepping methods implemented in existing subroutine libraries. Examples for verification and application of the proposed method are given. The present study should be of particular value in railway engineering.     

A FRICTIONAL CONTACT ELEMENT FOR STRONGLY CURVED CONTACT PROBLEMS

Based on a mixed formulation approach, a frictional contact element is proposed for the numerical solution of contact problems including strongly curved rigid obstacles. The implementation of the frictional contact element is analogous to that of a finite element. This feature facilitates its implementation in implicit finite element programmes, since the structure of the code need not be modified. For efficient modelling of the forming tool geometries by Computer Aided Geometric Design techniques and in order to achieve a high performance of the contact search, the numerical schemes of the frictional contact element operate directly on parametric polynomial surface patches. Thus, no discretization of curved contact surfaces is performed. Numerical simulations of deep drawing processes demonstrate the performance of the method in the case of large sliding increments upon curved tools and in the case of elasto-plasticity.     

A novel contact interaction formulation for voxel‐based micro‐finite‐element models of bone

Voxel‐based micro‐finite‐element (μFE) models are used extensively in bone mechanics research. A major disadvantage of voxel‐based μFE models is that voxel surface jaggedness causes distortion of contact‐induced stresses. Past efforts in resolving this problem have only been partially successful, ie, mesh smoothing failed to preserve uniformity of the stiffness matrix, resulting in (excessively) larger solution times, whereas reducing contact to a bonded interface introduced spurious tensile stresses at the contact surface. This paper introduces a novel “smooth” contact formulation that defines gap distances based on an artificial smooth surface representation while using the conventional penalty contact framework. Detailed analyses of a sphere under compression demonstrated that the smooth formulation predicts contact‐induced stresses more accurately than the bonded contact formulation. When applied to a realistic bone contact problem, errors in the smooth contact result were under 2%, whereas errors in the bonded contact result were up to 42.2%. We conclude that the novel smooth contact formulation presents a memory‐efficient method for contact problems in voxel‐based μFE models. It presents the first method that allows modeling finite slip in large‐scale voxel meshes common to high‐resolution image‐based models of bone while keeping the benefits of a fast and efficient voxel‐based solution scheme.     

A numerical solution for dynamic contact problems satisfying the velocity and acceleration compatibilities on the contact surface

For the numerical solution of dynamic contact problems, correct contact points and displacements are determined by iteratively reducing the displacement error vector monotonically toward zero. And spurious oscillations are prevented from the solution by enforcing the velocity and acceleration compatibilities of the contact points with the corresponding error vectors. Economic computation is possible because the accelerated iterative schemes are used and because decomposition of large matrix is not required in the iterations for the contact analysis of elastic bodies. Numerical simulations are conducted to demonstrate the accuracy of the solution and the necessity of the velocity and acceleration compatibilities on the contact surface.This research was supported in part by Korea Research Foundation, 1993.     

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J ₂ plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.