A selective strategy for shakedown analysis of engineering structures

Determining the load‐bearing capacity of engineering structures is essential for their design. In the case of varying thermo‐mechanical loading beyond the elastic limit, the statical shakedown analysis constitutes a particularly suitable tool for this. The application of the statical shakedown theorem, however, leads to a nonlinear convex optimization problem, which is typically characterized by large numbers of variables and constraints. In the present work, this optimization problem is solved by a primal–dual interior‐point algorithm, which shows a remarkable performance due to its problem‐tailored formulation. Nevertheless, the solution procedure remains still a demanding task from computational point of view. Thus, the aim of this paper is to tackle the task of solving large‐scale problems by use of a new so‐called selective algorithm. This algorithm detects automatically the plastically most affected zones within the structure, which have the highest influence on the solution. The entire system is then reduced to a substructure consisting of these zones, based upon which a new optimization problem can be set up, which is solved with significantly less effort. Consequently, the running time decreases drastically, as is shown by application to numerical examples from the field of power plant engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis

A mathematical programming formulation of strain‐driven path‐following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc‐length strain‐driven elastoplastic analyses, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other non‐linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain‐driven algorithms become more effective and less non‐linear with respect to a self‐equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal–dual interior point method suitable to solve the problems at hand. Copyright © 2011 John Wiley & Sons, Ltd.     

A non‐linear programming approach to kinematic shakedown analysis of composite materials

Using a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non‐linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement‐based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non‐linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non‐linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non‐linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials. Copyright © 2005 John Wiley & Sons, Ltd.     

Kinematical shakedown analysis with temperature‐dependent yield stress

This paper describes a direct shakedown analysis of structures subjected to variable thermal and mechanical loading. The classical kinematical shakedown theorem is modified to be implemented with any displacement‐based finite elements. The plastic incompressibility condition is imposed by the penalty function method. The shakedown limit is found via a non‐linear mathematical programming procedure. Two numerical shakedown methods are developed and implemented to provide alternative numerical means. The temperature‐dependent material model is included in theoretical and numerical calculation in a simple way. Its effect on shakedown limit is investigated. The numerical examination for some pressure vessel structures subjected to thermal and mechanical loading shows a satisfying precision and efficiency of the methods presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive multigrid method using duality in plane elasticity

This work presents an adaptive multigrid method for the mixed formulation of plane elasticity problems. First, a mixed‐hybrid formulation is introduced where the continuity of the normal components of the stress tensor is indirectly imposed using a Lagrange multiplier. Two different numerical approximations, naturally associated with the primal problem and the dual problem, are then proposed. The Complementary Energy Principle provides an a posteriori error estimate. For the effective solving of both systems of equations, a non‐standard multigrid algorithm has been designed that allows us to solve the two problems, dual and primal, with reasonable cost and in an integrated way. Finally, a significant numerical application is presented to check the efficiency of the error estimator and the good performance of the algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

An interior‐point algorithm for elastoplasticity

The problem of small‐deformation, rate‐independent elastoplasticity is treated using convex programming theory and algorithms. A finite‐step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the Lagrangian associated with a convex mathematical program. Next, an algorithm, based on the classical primal–dual interior point method, is developed. Several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem. The resulting method is compared to state‐of‐the‐art elastoplastic procedures for which both similarities and differences are found. Finally, a number of examples are solved, demonstrating the capabilities of the algorithm when applied to standard perfect plasticity, hardening multisurface plasticity, and problems involving softening. Copyright © 2006 John Wiley & Sons, Ltd.     

A stabilized discrete shear gap finite element for adaptive limit analysis of Mindlin–Reissner plates

This paper presents a numerical formulation for computation of collapse load of Mindlin–Reissner plates that uses stabilized discrete shear gap finite elements and second‐order cone programming. Displacement fields are approximated using the discrete shear gap in combination with a stabilized strain smoothing technique, ensuring that shear‐locking problem can be avoided and that accurate solutions can be obtained. The underlying optimization problem is formulated in the form of a standard second‐order cone programming, so that it can be solved using highly efficient primal‐dual interior‐point algorithm. An error indicator based on plastic dissipation will be used in the adaptive refinement scheme. Various plates with arbitrary geometries and boundary conditions are examined to illustrate the performance of the proposed procedure. Copyright © 2013 John Wiley & Sons, Ltd.     

On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

A dual mortar approach for 3D finite deformation contact with consistent linearization

In this paper, an approach for three‐dimensional frictionless contact based on a dual mortar formulation and using a primal–dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two‐dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi‐smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

A finite deformation mortar contact formulation using a primal–dual active set strategy

In recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal–dual active set strategy for contact constraint enforcement. We consider linear and higher‐order (quadratic) interpolations throughout this work. So‐called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non‐penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi‐smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results. Copyright © 2009 John Wiley & Sons, Ltd.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites

A three‐dimensional microstructure‐based finite element framework is presented for modeling the mechanical response of rubber composites in the microscopic level. This framework introduces a novel finite element formulation, the meshfree‐enriched FEM, to overcome the volumetric locking and pressure oscillation problems that normally arise in the numerical simulation of rubber composites using conventional displacement‐based FEM. The three‐dimensional meshfree‐enriched FEM is composed of five‐noded tetrahedral elements with a volume‐weighted smoothing of deformation gradient between neighboring elements. The L₂‐orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubeke functional to be degenerated to an assumed strain method, which leads to a displacement‐based formulation that is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numerical examples are analyzed to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.     

An implicit finite wear contact formulation based on dual mortar methods

Finite deformation contact problems with frictional effects and finite shape changes due to wear are investigated. To capture the finite shape changes, a third configuration besides the well‐known reference and spatial configurations is introduced, which represents the time‐dependent worn state. Consistent interconnections between these states are realized by an arbitrary Lagrangean–Eulerian formulation. The newly developed partitioned and fully implicit algorithm is based on a Lagrangean step and a shape evolution step. Within the Lagrangean step, contact constraints as well as the wear equations are weakly enforced following the well‐established mortar framework. Additional unknowns due to the employed Lagrange multiplier method for contact constraint enforcement and due to wear itself are eliminated by condensation procedures based on the concept of biorthogonality and the so‐called dual shape functions. Several numerical examples in both 2D and 3D are provided to demonstrate the performance and accuracy of the proposed numerical algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

The first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element shakedown analysis of two‐dimensional structures

In the present paper, the formulation proposed by Casciaro and Garcea (Comput. Meth. Appl. Mech. Eng., 2002; 191 :5761–5792) and applied to the shakedown analysis of plane frames, is extended to the analysis of two‐dimensional flat structures in both the cases of plane‐stress and plane‐strain. The discrete formulation is obtained using a mixed finite element in which both stress and displacement fields are interpolated. The material is assumed to be elasto‐plastic and a linearization of the elastic domain is performed. The result is a versatile iterative scheme well suited to implementation in general purpose FEM codes. An extensive series of numerical tests is presented showing the reliability of the proposed formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel scaled boundary finite element formulation with stabilization and its application to image‐based elastoplastic analysis

Digital images are increasingly being used as input data for computational analyses. This study presents an efficient numerical technique to perform image‐based elastoplastic analysis of materials and structures. The quadtree decomposition algorithm is employed for image‐based mesh generation, which is fully automatic and highly efficient. The quadtree cells are modeled by scaled boundary polytope elements, which eliminate the issue of hanging nodes faced by standard finite elements. A novel, simple, and efficient scaled boundary elastoplastic formulation with stablisation is developed. In this formulation, the return‐mapping calculation is only required to be performed at a single point in a polytope element, which facilitates the computational efficiency of the elastoplastic analysis and simplicity of implementation. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed technique for performing the elastoplastic analysis of high‐resolution images.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

Mesh adaptive computation of upper and lower bounds in limit analysis

An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound. The second step consists of solving the resulting discrete non‐linear optimization problems. These are reformulated as second‐order cone programs, which allows for the use of primal–dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized, a novel method for adaptive meshing is introduced. The method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions. The efficiency of the methodology is illustrated with applications in plane stress and plane strain problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Algorithmic formulation and numerical implementation of coupled electromagnetic‐inelastic continuum models for electromagnetic metal forming

The purpose of this work is the algorithmic formulation and implementation of a recent coupled electromagnetic‐inelastic continuum field model (Continuum Mech. Thermodyn. 2005; 17 :1–16) for a class of engineering materials, which can be dynamically formed using strong magnetic fields. Although in general relevant, temperature effects are for the applications of interest here minimal and are neglected for simplicity. In this case, the coupling is due, on the one hand, to the Lorentz force acting as an additional body force in the material. On the other hand, the spatio‐temporal development of the magnetic field is very sensitive to changes in the shape of the workpiece, resulting in additional coupling. The algorithmic formulation and numerical implementation of this coupled model is based on mixed‐element discretization of the deformation and electromagnetic fields combined with an implicit, staggered numerical solution scheme on two meshes. In particular, the mechanical degrees of freedom are solved on a Lagrangian mesh and the electromagnetic ones on an Eulerian one. The issues of the convergence behaviour of the staggered algorithm and the influence of data transfer between the meshes on the solution is discussed in detail. Finally, the numerical implementation of the model is applied to the modelling and simulation of electromagnetic sheet forming. Copyright © 2006 John Wiley & Sons, Ltd.