Meshfree analysis of softening elastoplastic solids using variational multiscale method

A meshfree multiscale method is presented for efficient analysis of elastoplastic solids. In the analysis of softening elastoplastic solids, standard finite element methods or meshfree methods typically yield mesh-dependent results. The reason for this well-known effect is the loss of ellipticity of the boundary value problem. In this work, the scale decomposition is carried out based on a variational form of the problem. A coarse scale is designed to represent global behavior and a fine scale to represent local behavior. A fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly. Each scale variable is approximated using a meshfree method. Meshfree approximation is well-suited for adaptivity. As a method of increasing the resolution, a partition of unity based extrinsic enrichment is used. In particular, fine scale approximations are designed to appropriately represent local behavior by using a localization angle. Moreover, the regularization effect through the convexification of non-convex potential is embedded to represent fine scale behavior. Each scale problem is solved iteratively. The proposed method is applied to shear band problems. In the results of analysis about pure shear and compression problems, straight shear bands can be captured and mesh-insensitive results are obtained. Curved shear bands can also be captured without mesh dependency in the analysis of indentation problem.     

Adaptive variational multiscale method for the Stokes equations

An adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd.     

应变局部化分析的嵌入强间断多尺度有限元法

固体材料的应变局部化行为是导致结构破坏失效的重要因素之一,开展相关数值模拟分析对于结构安全性评估具有重要意义.然而由于材料的非均质和多尺度特性,采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构,这将大幅度增加计算规模和成本.针对这一问题,本文提出了一种基于嵌入强间断模型的多尺度有限元方法.该方法从粗细两个尺度离散求解模型,首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性,所附加的跳跃位移自由度则通过凝聚技术进行消除,从而保持细尺度单元刚度阵维度不变.其次,提出了一种增强多节点粗单元技术,其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点,新构造的增强数值基函数可以捕捉细尺度间断特性,完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析;再次,在细尺度解的计算中,将细尺度解分解为降尺度解与单胞局部摄动解,从而消除弹塑性分析时单胞内部的不平衡力.最后,通过两个典型算例分析,并与完全采用细单元的嵌入有限元结果进行对比,验证了所提出算法的正确性与有效性.     

A new hybrid stress element method for the analysis of elastoplastic solids

Based on a modified Hellinger/Reissner variational principle which includes the equivalent stress, equivalent plastic strain and non-conforming displacement increments as independent variables, a quadrilateral isoparametric hybrid stress element for the analysis of elastoplastic problem is proposed. By this formulation, the yield criterion and flow rule are satisfied in an average sense and greater accuracy can be obtained by using non-conforming displacement. A numerical example is presented to show that the present model has high accuracy and computational effectiveness.This project is supported by the Natural Science Foundation of the State Education Commission.     

An efficient Galerkin meshfree formulation for shear deformable beam under finite deformation

This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam. The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection, large deformation as well as finite rotation. The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation. The Lagrangian meshfree shape function is utilized to discretize the variational equation. Subsequently to resolve the shear and membrane locking issues and accelerate the computation, the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation. Numerical results reveal that the present formulation is very effective.     

基于分区径向基函数配点法的大变形分析

无网格法因为不需要划分网格, 可以避免网格畸变问题,使得其广泛应用于大变形和一些复杂问题. 径向基函数配点法是一种典型的强形式无网格法,这种方法具有完全不需要任何网格、求解过程简单、精度高、收敛性好以及易于扩展到高维空间等优点,但是由于其采用全域的形函数, 在求解高梯度问题时 存在精度较低和无法很好地反应局部特性的缺点. 针对这个问题,本文引入分区径向基函数配点法来求解局部存在高梯度的大变形问题. 基于完全拉格朗日格式,采用牛顿迭代法建立了分区径向基函数配点法在大变形分析中的增量求解模式.这种方法将求解域根据其几何特点划分成若干个子域, 在子域内构建径向基函数插值, 在界面上施加所有的界面连续条件,构建分块稀疏矩阵统一求解. 该方法仍然保持超收敛性, 且将原来的满阵转化成了稀疏矩阵, 降低了存储空间,提高了计算效率. 相比较于传统的径向基函数配点法和有限元法, 这种方法能够更好地反应局部特性和求解高梯度问题.数值分析表明该方法能够有效求解局部存在高梯度的大变形问题.      

Multiscale block spectral solution for unsteady flows

Many problems of interest are characterized by 2 distinctive and disparate scales and a huge multiplicity of similar small‐scale elements. The corresponding scale‐dependent solvability manifests itself in the high gradient flow around each element needing a fine mesh locally and the similar flow patterns among all elements globally. In a block spectral approach making use of the scale‐dependent solvability, the global domain is decomposed into a large number of similar small blocks. The mesh‐pointwise block spectra will establish the block‐block variation, for which only a small set of blocks need to be solved with a fine mesh resolution. The solution can then be very efficiently obtained by coupling the local fine mesh solution and the global coarse mesh solution through a block spectral mapping. Previously, the block spectral method has only been developed for steady flows. The present work extends the methodology to unsteady flows of short temporal and spatial scales (eg, those due to self‐excited unsteady vortices and turbulence disturbances). A source term–based approach is adopted to facilitate a two‐way coupling in terms of time‐averaged flow solutions. The global coarse base mesh solution provides an appropriate environment and boundary condition to the local fine mesh blocks, while the local fine mesh solution provides the source terms (propagated through the block spectral mapping) to the global coarse mesh domain. The computational method will be presented with several numerical examples and sensitivity studies. The results consistently demonstrate the validity and potential of the proposed approach.     

Residual‐based turbulence models for moving boundary flows: hierarchical application of variational multiscale method and three‐level scale separation

This paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.     

Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.     

The least-squares meshfree method for rigid-plasticity with frictional contact

The least-squares meshfree method (LSMFM) for rigid-plasticity based on J₂-flow rule and infinitesimal theory is proposed. In the least-squares formulation the squared residuals of the constitutive and equilibrium equations are minimized. Those residuals are represented in a form of first-order differential system using the velocity and stress components as nodal unknowns and thus the proposed formulation is a mixed-type method. Also the penalty scheme for the enforcement of the boundary and frictional contact conditions is devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near the contact interface. The proposed method does not require any structure of extrinsic cells for the construction of shape functions, the treatment of incompressibility, the integration of variational formulation and the reconstruction of approximation. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed.     

Nonlinear Deformation and Stability of Elliptic Cylindrical Shells under Torsion and Bending

The problem of nonlinear deformation and buckling of noncircular cylindrical shells under combined loading is solved by the variational finite-element method in the displacement formulation. A numerical algorithm for solving the problem is proposed. Stability of cylindrical shells with an elliptic cross-sectional contour under a combined action of torsion and bending is analyzed. The effect of cross-sectional ellipticity and nonlinear prebuckling deformation on the critical loads and buckling mode is studied.     

A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism

This paper presents a thermodynamic formulation of a model for finite deformation of materials exhibiting elastoplastic material behaviour with non-linear isotropic and kinematic hardening. Central to this formulation is the notion that the form of the elastic constitutive relation be unaffected by the plastic deformation or transformation in the material, as commonly assumed in particular in the context of crystal plasticity. When generalized to the phenomenological context, this implies that the internal variable representing plastic deformation is an elastic material isomorphism. Among other things, this requirement on the plastic deformation leads directly to the standard elastoplastic multiplicative decomposition of the deformation gradient. In addition, a dependence of the plastic part of the free energy on the plastic deformation itself yields a thermodynamic form for the centre of the elastic range of the material, i.e. the back stress. Finally, we show how this approach can be applied to formulate thermodynamic forms for linear, and non-linear Armstrong-Frederick, kinematic hardening models.     

On the elastoplastic cyclic analysis of plane beam structures using a flexibility-based finite element approach

This paper presents the extension of a flexibility-based large increment method (LIM) for the case of cyclic loading. In the last few years, LIM has been successfully tested for solving a range of non-linear structural problems involving elastoplastic material models under monotonic loading. In these analyses, the force-based LIM algorithm provided robust solutions and significant computational savings compared to the displacement-based finite element approach by using fewer elements and integration points. Although in cyclic analysis a step-by-step solution procedure has to be adopted to account for the plastic history, LIM will still have many advantages over the traditional finite element method. Before going into the basic idea of this extension, a brief discussion regarding LIM governing equations is presented followed by the proposed solution procedure. Next, the formulation is specified for the treatment of the elastic perfectly plastic beam element. The local stage for the beam behavior is discussed in detail and the required improvement for the LIM methodology is described. Illustrative truss and beam examples are presented for different non-linear material models. The results are compared with those obtained from a standard displacement method and again highlight the potential benefits of the proposed flexibility-based approach.     

A theory of large-strain isotropic thermoplasticity based on metric transformation tensors

Summary A formulation of isotropic thermoplasticity for arbitrary large elastic and plastic strains is presented. The underlying concept is the introduction of a metric transformation tensor which maps a locally defined six-dimensional plastic metric onto the metric of the current configuration. This mixed-variant tensor field provides a basis for the definition of a local isotropic hyperelastic stress response in the thermoplastic solid. Following this fundamental assumption, we derive a consistent internal variable formulation of thermoplasticity in a Lagrangian as well as a Eulerian geometric setting. On the numerical side, we discuss in detail an objective integration algorithm for the mixed-variant plastic flow rule. The special feature here is a new representation of the stress return and the algorithmic elastoplastic moduli in the eigenvalue space of the Eulerian plastic metric for plane problems. Furthermore, an algorithm for the solution of the coupled problem is formulated based on an operator split of the global field equations of thermoplasticity. The paper concludes with two representative numerical simulations of thermoplastic deformation processes.     

Elastoplastic Analysis with Limited Plastic Deformations and Displacements*

ABSTRACT

This paper presents solution methods for elastoplastic and shakedown analysis of linearly elastic, perfectly plastic bodies for which the conventional classical formulations of these problems are completed by constraints on overall plastic deformation and elastoplastic displacement. The methods are described in terms of nonlinear mathematical programming and provide solutions when the plastic reserves of the body are not fully exhausted, and the plastic performance and the plastic deformations are controlled. Application of the method is illustrated by an example.     

An inverse elastoplastic problem for plates

We study an inverse elastoplastic problem of determining the residual stresses, the plasticity zone, and the external loads for a plate for known residual deflections which occur after removal of these loads and elastic unloading. Assuming that the deformation theory of plasticity is valid at the active stage of deformation, we prove the theorem of unique solution. An iterative method of solution is proposed and a variational formulation of the problem is given. Some simple examples are considered. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 186–194, July–August, 1999.     

Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment

The problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment M and the angle of twist per unit length of the rod’s length θ is investigated as an inverse problem. The proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the Levenberg-Marquardt iteration method. In the direct problem, these properties are known, and the torsional moment is calculated as a function of the angle of twist from the solution of a non-linear boundary value problem. This non-linear problem results from the Saint-Venant displacement assumption, the Ramberg–Osgood constitutive equation, and the deformation theory of plasticity for the stress–strain relation. To solve the direct problem in each iteration step, the Kansa method is used for the circular cross section of the rod, or the method of fundamental solutions (MFS) and the method of particular solutions (MPS) are used for the prismatic cross section of the rod. The non-linear torsion problem in the plastic region is solved using the Picard iteration.     

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems.