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.     

Upper-bound analysis of axi-symmetric forward spiral extrusion

An axi-symmetric forward spiral extrusion process, AFSE, has been analyzed using the principle of the virtual work rate. Based on an existing kinematically admissible velocity field, an upper bound solution has been developed for the new forming process. The solution also provides estimation of the die reaction torque during the process. AFSE experiments were conducted with lead specimens in order to verify the upper-bound solution. A reasonable agreement between experimental results and the upper bound solution was observed. The upper bound analysis indicated that both deformation in a small zone next to the velocity discontinuity plane and the frictional power constitute the most significant parts of the consumed power, respectively, during AFSE. As a result, reduction in friction will lead to a dramatic decrease in the total forming force in AFSE while maintaining a circumferentially slippage free flow of material during the process. It was shown that by increasing the die helix angle above a critical value, the rate of growth in the die reaction torque and extrusion load with the helix angle increases dramatically.     

A new discontinuous upper bound limit analysis formulation

A new upper bound formulation of limit analysis of two‐ and three‐dimensional solids is presented. In contrast to most discrete upper bound methods the present one is formulated in terms of stresses rather than velocities and plastic multipliers. However, by means of duality theory it is shown that the formulation does indeed result in rigorous upper bound solutions. Also, kinematically admissible discontinuities, which have previously been shown to be very efficient, are given an interpretation in terms of stresses. This allows for a much simpler implementation and, in contrast to existing formulations, extension to arbitrary yield criteria in two and three dimensions is straightforward. Finally, the capabilities of the new method are demonstrated through a number of examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Limit load analysis of strength undermatched welded T-joints under bending

An increasing amount of laser beam welded T‐joints (e.g. skin‐stringer) of aluminium alloys are now in use in advanced fuselage applications designed as ‘integral structures’ for weight and cost savings. It is known that weld joints generally show lower strength (undermatching) than base metal in both laser beam and friction stir welded joints of 6xxx series Al‐alloys. Damage tolerance considerations in terms of the residual strength of such joints require limit load solutions to be used in engineering fitness‐for‐service (FFS) analysis. The paper, therefore, provides an upper bound limit load solution in closed form for welded T‐joints (idealized) with strength undermatching and subject to a bending moment. In addition to the necessary requirements of the upper bound theorem, the kinematically admissible velocity field chosen leads to a stress field, which satisfies the equilibrium equations and some stress boundary conditions in the plastic zone. This is an advantage of the solution and, therefore, it is expected that the upper bound obtained is close to the exact limit load of such joints.     

一种新的予示三维流动的上限条法(UBST)

本文发展了上限元法(UBET)提出了一种新的予示金属变形过程三维流动的上限条法(upper bound strip technique)。单元内采用平行速度场,单元侧面速度均匀分布且为常数。并将它们作为优化参数,通过总变形功率极小化得出了最佳速度场。用该法予示了矩形截面条形工件的变形力,侧向宽度与纵向延伸,所得结果与文献的实验结果和有限元法计算结果进行了比较,吻合较好。     

An efficient time‐domain perfectly matched layers formulation for elastodynamics on spherical domains

Many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain. One widely used approach for truncating the infinite domain is the so‐called method of perfectly matched layers (PMLs). Most existing PML formulations are developed for finite difference methods based on the first‐order velocity‐stress form of the elasticity equations, and they are not straight‐forward to implement using standard finite element methods (FEMs) on unstructured meshes. Some of the problems with these formulations include the application of boundary conditions in half‐space problems and in the treatment of edges and/or corners for time‐domain problems. Several PML formulations, which do work with FEMs have been proposed, although most of them still have some of these problems and/or they require a large number of auxiliary nodal history/memory variables. In this work, we develop a new PML formulation for time‐domain elastodynamics on a spherical domain, which reduces to a two‐dimensional formulation under the assumption of axisymmetry. Our formulation is well‐suited for implementation using FEMs, where it requires lower memory than existing formulations, and it allows for natural application of boundary conditions. We solve example problems on two‐dimensional and three‐dimensional domains using a high‐order discontinuous Galerkin (DG) discretization on unstructured meshes and explicit time‐stepping. We also study an approach for stabilization of the discrete equations, and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

Limit analysis of plates using the EFG method and second‐order cone programming

The meshless element‐free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least‐squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non‐smooth minimization problem can be transformed into a form suitable for solution using second‐order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper‐bound solutions for benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A novel augmented Lagrangian‐based formulation for upper‐bound limit analysis

This paper describes a novel upper‐bound formulation of limit analysis. This formulation is an innovative variant of an existing two‐field mixed formulation based on the augmented Lagrangian method also developed by the authors. A natural approach is used to describe the deformation of each finite element. Furthermore, and in contrast to the previous formulation, two independent field approximations are now both used to define the velocity field, defined globally and at element level. It is shown that this feature allows a governing system of uncoupled linear equations to be obtained. Some numerical examples in plane strain conditions are presented in order to illustrate the current model performance. In conclusion, the potential and advantages of this new approach are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Higher‐order accurate time‐step‐integration algorithms by post‐integration techniques

In this paper, a framework to construct higher‐order‐accurate time‐step‐integration algorithms based on the post‐integration techniques is presented. The prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced. The algorithmic parameters are chosen such that unconditionally A‐stable higher‐order‐accurate time‐step‐integration algorithms with controllable numerical dissipation can be constructed for linear problems. Besides, it is shown that the order of accuracy for non‐linear problems is maintained through the relationship between the present formulation and the Runge–Kutta method. The second‐order differential equations are also considered. Numerical examples are given to illustrate the validity of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Upper bound shakedown analysis with the nodal natural element method

In this paper, a novel numerical solution procedure is developed for the upper bound shakedown analysis of elastic-perfectly plastic structures. The nodal natural element method (nodal-NEM) combines the advantages of the NEM and the stabilized conforming nodal integration scheme, and is used to discretize the established mathematical programming formulation of upper bound shakedown analysis based on Koiter’s theorem. In this formulation, the displacement field is approximated by using the Sibson interpolation and the difficulty caused by the time integration is solved by König’s technique. Meanwhile, the nonlinear and non-differentiable characteristic of objective function is overcome by distinguishing non-plastic areas from plastic areas and modifying associated constraint conditions and goal function at each iteration step. Finally, the objective function subjected to several equality constraints is linearized and the upper bound shakedown load multiplier is obtained. This direct iterative process can ensure the shakedown load to monotonically converge to the upper bound of true solution. Several typical numerical examples confirm the efficiency and accuracy of the proposed method.     

A monolithic finite-element formulation for magnetohydrodynamics

This work develops a new monolithic strategy for magnetohydrodynamics based on a continuous velocity–pressure formulation. The magnetic field is interpolated in the same way as the velocity field, and the entire formulation is within a nodal finite-element framework. The velocity and pressure interpolations are chosen so that they satisfy the Babuska–Brezzi (BB) conditions. In most of the existing formulations, a stabilized formulation is used that requires a stabilization term, and some associated mesh-dependent parameters that need to be adjusted. In contrast, no such parameters need to be adjusted in the current formulation, making it more user-friendly and robust. Both transient and steady-state formulations are developed for two- and three-dimensional geometries. An exact linearization of the monolithic strategy ensures that rapid (quadratic) convergence is achieved within each time (or load) step, while the stable nature of the interpolations used ensures that no instabilities arise in the solution. An existing analytical solution is corrected. The coarse mesh accuracy is shown to be better compared with other existing strategies in several benchmark problems, showing that the developed formulation is both robust and efficient.     

Application of upper bound method to the powder‐forging of cup‐shaped axisymmetric preforms for estimating punch load

This paper is concerned with the analytic derivation of the punch load required to forge cup‐shaped axisymmetric porous preforms. Splitting the two‐dimensional preform domain into three rectangular regions, we assumed the kinematically admissible velocity fields to apply the upper bound method. According to the resulting formulas, we developed an incremental numerical algorithm for computing time‐discretized step‐wise punch loads and preform volumes. In order to illustrate the validity of the proposed estimate, we compared the numerical results obtained by the present analysis with those by the finite element simulation. From the comparison made through different preform dimensions, we observed that the present analysis expects the reliable upper bounds of process step‐wise punch loads. Copyright © 2001 John Wiley & Sons, Ltd.     

A mixed shell formulation accounting for thickness strains and finite strain 3d material models

A non‐linear quadrilateral shell element for the analysis of thin structures is presented. The Reissner–Mindlin theory with inextensible director vector is used to develop a three‐field variational formulation with independent displacements, stress resultants and shell strains. The interpolation of the independent shell strains consists of two parts. The first part corresponds to the interpolation of the stress resultants. Within the second part independent thickness strains are considered. This allows incorporation of arbitrary non‐linear 3d constitutive equations without further modifications. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison with other element formulations. We present results for finite strain elasticity, inelasticity, bifurcation and post‐buckling problems. Copyright © 2007 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.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 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 finite element formulations for nearly incompressible linear elasticity

In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.