Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming

The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second‐order cone programming (SOCP), for which efficient primal‐dual interior‐point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr–Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker–Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd.     

Rigid plastic model of incremental sheet deformation using second‐order cone programming

This paper describes a method for numerically modelling the incremental plastic deformation of shells and applies the method to incremental sheet forming (ISF). An upper bound finite element shell model is developed based on sequential limit analysis under the rigid plastic assumption, which is solved by manipulating the problem into the form of a second‐order cone program (SOCP). Initially, the static upper bound plate problem is investigated and the results are compared with the existing literature. The approach is then extended to a shell formulation using a linearized form of the Ilyushin yield condition and two methods for treating the Ilyushin condition are presented. The model is solved efficiently using SOCP software. The resulting model shows good geometric agreement when validated against an elasto‐plastic model produced using existing commercial software and with measurements from a real product produced using ISF. Copyright © 2008 John Wiley & Sons, Ltd.     

Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion

The incremental problem for quasistatic elastoplastic analysis with the von?Mises yield criterion is discussed within the framework of the second-order cone programming (SOCP). We show that the associated flow rule under the von?Mises yield criterion with the linear isotropic/kinematic hardening is equivalently rewritten as a second-order cone complementarity problem. The minimization problems of the potential energy and the complementary energy for incremental analysis are then formulated as the primal-dual pair of SOCP problems, which can be solved with a primal-dual interior-point method. To enhance numerical performance of tracing an equilibrium path, we propose a warm-start strategy for a primal-dual interior-point method based on the primal-dual penalty method. In this warm-start strategy, we solve a penalized SOCP problem to find the equilibrium solution at the current loading step. An advanced initial point for solving this penalized SOCP problem is defined by using information of the solution at the previous loading step.     

Lower‐bound limit analysis by using the EFG method and non‐linear programming

Intended to avoid the complicated computations of elasto‐plastic incremental analysis, limit analysis is an appealing direct method for determining the load‐carrying capacity of structures. On the basis of the static limit analysis theorem, a solution procedure for lower‐bound limit analysis is presented firstly, making use of the element‐free Galerkin (EFG) method rather than traditional numerical methods such as the finite element method and boundary element method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced‐basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self‐equilibrium stress subspaces with very low dimensions. The self‐equilibrium stress field is expressed by a linear combination of several self‐equilibrium stress basis vectors with parameters to be determined. These self‐equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto‐plastic incremental analysis. The Complex method is used to solve these non‐linear programming sub‐problems and determine the maximal load amplifier. Numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the EFG method and non‐linear programming. Copyright © 2007 John Wiley & Sons, Ltd.     

SGBEM with Lagrange multipliers applied to elastic domain decomposition problems with curved interfaces using non‐matching meshes

An original approach to the solution of linear elastic domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach is based on searching for the saddle‐point of a new potential energy functional with Lagrange multipliers. The interfaces can be either straight or curved, open or closed. The two coupling conditions, equilibrium and compatibility, along an interface are fulfilled in a weak sense by means of Lagrange multipliers (interface displacements and tractions), which enables non‐matching meshes to be used at both sides of interfaces between subdomains. The accuracy and robustness of the method is tested by several numerical examples, where the numerical results are compared with the analytical solution of the solved problems, and the convergence rates of two error norms are evaluated for h‐refinements of matching and non‐matching boundary element meshes. Copyright © 2010 John Wiley & Sons, Ltd.     

Improvement of semi‐analytical design sensitivities of non‐linear structures using equilibrium relations

The accuracy problem of the semi‐analytical method for shape design sensitivity analysis has been reported for linear and non‐linear structures. The source of error is the numerical differentiation of the element internal force vector, which is inherent to the semi‐analytical approach. Such errors occur for structures whose displacement field is characterized by large rigid body rotations of individual elements. This paper presents a method for the improvement of semi‐analytical sensitivities. The method is based on the element free body equilibrium conditions, and on the exact differentiation of the rigid body modes. The method is efficient, simple to code, and can be applied to linear and non‐linear structures. The numerical examples show that this approach eliminates the abnormal errors that occur in the conventional semi‐analytical method. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐linear analysis of thin‐walled members of open cross‐section

The paper presents a means of determining the non‐linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin‐walled open section finite element that lead to non‐linear stiffness equations. These non‐linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non‐linear matrices is stated herein. It is shown that the method of solution of the non‐linear stiffness matrices is problem dependent. The finite element procedure is used to study non‐linear torsion that illustrates torsional hardening, and the Newton–Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post‐buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non‐linear finite element method of analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimization‐based stability analysis of structures under unilateral constraints

This paper discusses an optimization‐based technique for determining the stability of a given equilibrium point of the unilaterally constrained structural system, which is subjected to the static load. We deal with the three problems in mechanics sharing the common mathematical properties: (i) structures containing no‐compression cables; (ii) frictionless contacts; and (iii) elastic–plastic trusses with non‐negative hardening. It is shown that the stability of a given equilibrium point of these structures can be determined by solving a maximization problem of a convex function over a convex set. On the basis of the difference of convex functions optimization, we propose an algorithm to solve the stability determination problem, at each iteration of which a second‐order cone programming problem is to be solved. The problems presented are solved for various structures to determine the stability of given equilibrium points. Copyright © 2008 John Wiley & Sons, Ltd.     

Active macro‐zone approach for incremental elastoplastic‐contact analysis

The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact‐detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem‐elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic‐contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro‐zones, and uses the well‐known concept of self‐equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self‐stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. Copyright © 2013 John Wiley & Sons, Ltd.     

A Koiter‐Newton approach for nonlinear structural analysis

A new approach termed the Koiter‐Newton is presented for the numerical solution of a class of elastic nonlinear structural response problems. It is a combination of a reduction method inspired by Koiter's post‐buckling analysis and Newton arc‐length method so that it is accurate over the entire equilibrium path and also computationally efficient in the presence of buckling. Finite element implementation based on element independent co‐rotational formulation is used. Various numerical examples of buckling sensitive structures are presented to evaluate the performance of the method. The examples demonstrate that the method is robust and completely automatic and that it outperforms traditional path‐following techniques. This improved efficiency will open the door for the direct use of detailed nonlinear finite element models in the design optimization of next generation flight and launch vehicles. Copyright © 2013 John Wiley & Sons, Ltd.     

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

This paper presents a bubble‐enhanced smoothed finite element formulation for the analysis of volume‐constrained problems in two‐dimensional linear elasticity. The new formulation is derived based on the variational multi‐scale approach in which unequal order displacement‐pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine‐scale displacement approximation. An area‐weighted averaging scheme is employed for the two‐scale smoothed strain calculation under the framework of edge‐based smoothed FEM. The smoothed fine‐scale solution is shown to naturally contain the stress field jump of the smoothed coarse‐scale solution across the boundary of edge‐based smoothing domain and thus provides the possibility to stabilize the global solution for volume‐constrained problems. A global monolithic solution strategy is employed, and the fine‐scale solution is solved without the consideration of approximating the strong form of the fine‐scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

Least‐squares collocation meshless method

A finite point method, least‐squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least‐squares sense. The moving least‐squares interpolant is used to construct the trial functions. The computational effort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution significantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational effort. Copyright © 2001 John Wiley & Sons, Ltd.     

Stress singularity analysis for orthotropic V‐notches in the generalised plane strain state

The singularity for the V‐notch under the generalised plane deformation is investigated by the combination of the asymptotic analysis with the interpolating matrix method developed by part of the authors before. The displacement asymptotic expansions at the vicinity of the V‐notch vertex are introduced into the equilibrium equations, which are transformed into a set of characteristic ordinary differential equations with respect to the notch singularity orders. The boundary conditions and interfacial compatibility conditions are also represented by the combination of the singularity orders and characteristic angular functions. The determination of the singularity orders and characteristic angular functions are transformed into solving the ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method. The present method is suitable for the singularity analysis for isotropic and orthotropic V‐notches. It is versatile for analysing the stress singularity of single material V‐notches, bi‐material V‐notches, interface edges and cracks. The correctness of the results by the proposed method is ensured by the comparison with the published ones.     

A topological optimization approach for structural design of a high‐speed low‐load mechanism using the equivalent static loads method

In high‐speed low‐load mechanisms, the principal loads are the inertial forces caused by the high accelerations and velocities. Hence, mechanical design should consider lightweight structures to minimize such loads. In this paper, a topological optimization method is presented on the basis of the equivalent static loads method. Finite element (FE) models of the mechanism in different positions are constructed, and the equivalent loads are obtained using flexible multibody dynamics simulation. Kinetic DOFs are used to simulate the motion joints, and a quasi‐static analysis is performed to obtain the structural responses. The element sensitivity is calculated according to the static‐load‐equivalent equilibrium, in such a way that the influence on the inertial force is considered. A dimensionless component sensitivity factor (strain energy caused by unit load divided by kinetic energy from unit velocity) is used, which quantifies the significance of each element. Finally, the topological optimization approach is presented on the basis of the evolutionary structural optimization method, where the objective is to find the maximum ratio of strain energy to kinetic energy. In order to show the efficiency of the presented method, we presented two numerical cases. The results of these analyses show that the presented method is more efficient and can be easily implemented in commercial FE analysis software. Copyright © 2011 John Wiley & Sons, Ltd.     

Solution of FE‐BE coupled eigenvalue problems for the prediction of the vibro‐acoustic behavior of ship‐like structures

To predict the vibro‐acoustic behavior of structures, both a structural problem and an acoustic problem have to be solved. For thin structures immersed in water, a strong interaction between the structural domain and fluid domain occurs. This significantly alters the resonance frequencies. In this work, the structure is modeled by the finite element method. The exterior acoustic problem is solved by a fast boundary element method employing hierarchical matrices. An FE‐BE formulation is presented, which allows the solution of the coupled eigenvalue problem and thus the prediction of the coupled eigenfrequencies and mode shapes. It is based on a Schur complement formulation of the FE‐BE system yielding a generalized eigenvalue problem. A Krylov–Schur solver is applied for its efficient solution. Hereby, the compressibility of the fluid is neglected. The coupled eigensolution is then used for a model reduction strategy allowing fast frequency sweep calculations. The efficiency of the proposed formulations is investigated with respect to memory consumption, accuracy, and computation time. Copyright © 2011 John Wiley & Sons, Ltd.     

Mesh‐independent p‐orthotropic enrichment using the generalized finite element method

This paper is aimed at presenting a simple yet effective procedure to implement a mesh‐independent p‐orthotropic enrichment in the generalized finite element method. The procedure is based on the observation that shape functions used in the GFEM can be constructed from polynomials defined in any co‐ordinate system regardless of the underlying mesh or type of element used. Numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p‐orthotropic enrichment. Copyright © 2002 John Wiley & Sons, Ltd.     

Singularity analysis near the vertex of the V‐notch in Reissner's plate by coupling the asymptotic expansion technique with the interpolating matrix method

A new numerical method for calculating the singularity orders of V‐notches in Reissner's plate is proposed in this paper. By introducing the asymptotic expansion of the generalised displacement field at the notch tip into the equilibrium equations of a plate, a set of characteristic ordinary differential equations with respect to the singularity order are established. In addition, by adopting the variable substitution technique, the obtained non‐linear characteristic equations are transformed into linear ones, which are solved by the interpolating matrix method. The singularity orders of moments and shear forces can be obtained simultaneously and can be distinguished from the corresponding characteristic angular functions conveniently. Four types of boundary conditions are proposed to investigate the influence of boundary conditions on the singularity order values. The effect of the Poisson's ratio on the singularity orders of the V‐notch in Reissner's plate is discussed. The present method is versatile for the singularity analysis of single material V‐notches and bi‐material V‐notches, and can be easily extended to multi‐material V‐notches.     

Large strain phase‐field‐based multi‐material topology optimization

A multi‐material topology optimization scheme is presented. The formulation includes an arbitrary number of phases with different mechanical properties. To ensure that the sum of the volume fractions is unity and in order to avoid negative phase fractions, an obstacle potential function, which introduces infinity penalty for negative densities, is utilized. The problem is formulated for nonlinear deformations, and the objective of the optimization is the end displacement. The boundary value problems associated with the optimization problem and the equilibrium equation are solved using the finite element method. To illustrate the possibilities of the method, it is applied to a simple boundary value problem where optimal designs using multiple phases are considered. Copyright © 2015 John Wiley & Sons, Ltd.