Hybrid-conventional finite element for gradient-dependent plasticity

The hybrid-conventional finite element method is applied to the analysis of quasi-static, gradient-dependent elastoplastic problems in solid mechanics. The stresses within the element domain and the displacements on the boundary are simultaneously and independently approximated using Trefftz constraints, which lead to boundary integrals. The plastic multipliers are conventionally approximated with regard to C₀ continuity of the multiplier field of the gradient-dependent plasticity. The finite element formulation is derived using a Galerkin-weighted residual approach. The plastic boundary conditions are examined and plastic radiations are set to zero on the plastic boundaries. The effectiveness of the present method is demonstrated with three numerical applications.     

轴对称Poisson方程的Trefftz有限元解法

将径向基函数应用到一类轴对称Poisson方程的数值求解中,提出了一种Trefftz有限元计算格式.非0右端项将问题的特解引入Trefftz单元域内场,致使单元刚度方程涉及区域积分.利用径向基函数对特解近似处理,可消除区域积分,从而保持Trefftz有限元法只含边界积分的优势.为获得特解,选取求解域内所有单元的节点和形心作为基本插值点,而在求解域之外构造一个虚拟边界,在其上布置一定数目的虚拟点作为额外插值点.数值算例验证了该方法的有效性和可行性.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

A stabilized mixed formulation for finite strain deformation for low-order tetrahedral solid elements

A stabilized mixed finite element formulation for four-noded tetrahedral elements is introduced for robustly solving small and large deformation problems. The uniqueness of the formulation lies within the fact that it is general in that it can be applied to any type of material model without requiring specialized geometric or material parameters. To overcome the problem of volumetric locking, a mixed element formulation that utilizes linear displacement and pressure fields was implemented. The stabilization is provided by enhancing the rate of deformation tensor with a term derived using a bubble function approach. The element was implemented through a user-programmable element of the commercial finite element software ANSYS. Using the ANSYS platform, the performance of the element was evaluated by comparing the predicted results with those obtained using mixed quadratic tetrahedral elements and hexahedral elements with a B-bar formulation. Based on the quality of the results, the new element formulation shows significant potential for use in simulating complex engineering processes.     

正交各向异性轴对称位势问题的Trefftz有限元分析

Trefftz有限元法（Trefftz finite element method, TFEM）因其独特的优良品质而备受关注.针对正交各向异性轴对称位势问题,提出了一种4节点四边形环状单元．在该单元模型中,首先假设两套独立的位势插值模式:即单元域内场和网线场,然后代入修正变分泛函并利用Gauss散度定理消除区域积分,最后根据驻值原理导得只含边界积分的单元刚度方程．数值算例表明了该单元的准确性、稳定性以及对网格畸变的不敏感性.     

Numerical–analytical model for transient dynamics of elastic-plastic plate under eccentric low-velocity impact

The aim of this study is to present an efficient model for the analysis of complicated nonlinear transient dynamics of an elastic-plastic plate subjected to a transversely eccentric low-velocity impact. A mixed numerical–analytical model is presented to predict the transient dynamic behaviours consisting of either plate impact responses or wave propagations induced by the impact in a plate with an arbitrary shape and support. This hybrid approach has been validated by comparison with results of laboratory tests performed on an elastic-perfectly plastic narrow plate eccentrically struck by an elastic sphere, and results of a three-dimensional finite element (FE) analysis for an elastic-perfectly plastic simply-supported rectangular plate eccentrically struck by an elastic sphere. The advantages of this hybrid approach are in the simplification of local contact force formulation, computational efficiency over the FE model, and convenient application to parametric study for eccentric impact behaviour. The hybrid approach can provide accurate predictions of the plate impact responses and plate wave propagations.     

A Study on Mixed Finite Element Formulations Applied to Diffusion Problems

The objective of this contribution is to study various mixed finite element schemes applied to classical and non-classical diffusion problems. The Fickean diffusion is considered using a three-field formulation where the concentration gradient and the diffusive flux are additionally chosen as independent variables. Furthermore, the non-classical diffusion model given by the Cahn–Hilliard equation is discussed where mixed finite elements are used to cast the fourth-order PDE into two second-order PDEs. A penalized version of a mixed formulation for the underlying second gradient model is given by specific micromorphic models. Another approach is a classical second order splitting method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trefftz method: A general theory

A precise definition of Trefftz method is proposed and, starting with it, a general theory is briefly explained. This leads to formulating numerical methods from a domain decomposition perspective. An important feature of this approach is the systematic use of “fully discontinuous functions” and the treatment of a general boundary value problem with prescribed jumps. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces in which the functions together with their derivatives may have jump discontinuities and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz—Jirousek) and indirect (Trefftz—Herrera) methods. In turn, each one of them can be divided into overlapping and nonoverlapping. The generality of the resulting theory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The article is dedicated to Professor Jiroslav Jirousek, who has been a very important driving force in the modern development of Trefftz method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 561–580, 2000     

Recent advances in the numerical modeling of constitutive relations

In this paper we present an overview of the recent developments in the area of numerical and finite element modeling of nonlinear constitutive relations. The paper discusses elastic, hyperelastic, elastoplastic and anisotropic plastic material models. In the hyperelastic model an emphasis is given to the method by which the incompressibility constraint is applied. A systematic and general procedure for the numerical treatment of hyperelastic model is presented. In the elastoplastic model both infinitesimal and large strain cases are discussed. Various concerns and implications in extending infinitesimal theories into large strain case are pointed out. In the anisotropic elastoplastic case, emphasis is given to the practicality of proposed theories and its feasible and economical use in the finite element environment.     

Geometrically nonlinear shell finite element based on the geometry of a planar curve

An engineering approach for constructing a curved triangular finite element of a thin shell is considered. The approach is based on the assumption that the triangle sides are planar nearly circular curves before and after deformation. A geometrically nonlinear formulation of a triangular finite element of a thin Kirchhoff–Love shell is given. The predictive capabilities of the element are tested using benchmark problems of nonlinear deformation of elastic plates and shells.     

Computation of the Three-Dimensional Stress State in Composite Shell Structures with Mixed Finite Elements

Being able to compute the complete three-dimensional stress state in layered composite shell structures is essential in order to examine complicated interlaminar failure modes such as delamination. We lay out a mixed finite element formulation with independent displacements, rotations, stress resultants and shell strains. A mixed hybrid shell element with 4 nodes and 5 or 6 nodal degrees of freedom is developed, so that the element formulation can also be used for problems with shell intersections. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hybrid boundary element formulation for acoustic wave propagation

The so‐called hybrid stress boundary element method (HSBEM) is introduced in a frequency domain formulation for the computation of acoustic radiation and scattering in closed and in finite domains. Different from other boundary element formulations, the HSBEM is based on an extended Hellinger‐Reissner variational principle and leads to a Hermitian, frequency‐dependent stiffness equation. Due to this, the method is very well suited for treating fluid structure interaction problems since the effort for the coupling the structure, discretized by a finite elements, and the fluid, discretized by the HSBEM is strongly reduced. To arrive at a boundary integral formulation, the field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions weighed by source strength. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. Comparing to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix for exterior domain problems are communicated here.     

Theory of differential equations in discontinuous piecewise‐defined functions

A truly general and systematic theory of finite element methods (FEM) should be formulated using, as trial and test functions, piecewise‐defined functions that can be fully discontinuous across the internal boundary, which separates the elements from each other. Some of the most relevant work addressing such formulations is contained in the literature on discontinuous Galerkin (dG) methods and on Trefftz methods. However, the formulations of partial differential equations in discontinuous functions used in both of those fields are indirect approaches, which are based on the use of Lagrange multipliers and mixed methods, in the case of dG methods, and the frame, in the case of Trefftz method. This article addresses this problem from a different point of view and proposes a theory, formulated in discontinuous piecewise‐defined functions, which is direct and systematic, and furthermore it avoids the use of Lagrange multipliers or a frame, while mixed methods are incorporated as particular cases of more general results implied by the theory. When boundary value problems are formulated in discontinuous functions, well‐posed problems are boundary value problems with prescribed jumps (BVPJ), in which the boundary conditions are complemented by suitable jump conditions to be satisfied across the internal boundary of the domain‐partition. One result that is presented in this article shows that for elliptic equations of order 2m, with m ≥ 1, the problem of establishing conditions for existence of solution for the BVPJ reduces to that of the “standard boundary value problem,” without jumps, which has been extensively studied. Actually, this result is an illustration of a more general one that shows that the same happens for any differential equation, or system of such equations that is linear, independently of its type and with possibly discontinuous coefficients. This generality is achieved by means of an algebraic framework previously developed by the author and his collaborators. A fundamental ingredient of this algebraic formulation is a kind of Green's formulas that simplify many problems (some times referred to as Green‐Herrera formulas). An important practical implication of our approach is worth mentioning: “avoiding the introduction of the Lagrange multipliers, or the ‘frame’ in the case of Trefftz‐methods, significantly reduces the number of degrees of freedom to be dealt with.” © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A new boundary-type scheme for sensitivity analysis using Trefftz formulation

This paper presents a new boundary-type scheme for a sensitivity analysis of the two-dimensional potential problem by using the Trefftz formulation.

Since the Trefftz method is the boundary-type solution procedure, input data generation is easier than the domain-type solution procedure. Moreover, the physical quantities are expressed by the regular equations, their sensitivities, which is derived from the direct differentiation of the original quantaties, are also regular. Therefore, they can be calculated more easily than the ordinary boundary element method using the singular boundary integral equation. The present schemes are applied to simple numerical examples in order to confirm the validity of the present formulation.     

An enhanced finite volume method to model 2D linear elastic structures

This paper details the evaluation and enhancement of the vertex-centred finite volume method for the purpose of modelling linear elastic structures undergoing bending. A matrix-free edge-based finite volume procedure is discussed and compared with the traditional isoparametric finite element method via application to a number of test-cases. It is demonstrated that the standard finite volume approach exhibits similar disadvantages to the linear Q4 finite element formulation when modelling bending. An enhanced finite volume approach is proposed to circumvent this and a rigorous error analysis conducted. It is demonstrated that the developed finite volume method is superior to both standard finite volume and Q4 finite element methods, and provides a practical alternative to the analysis of bending-dominated solid mechanics problems.     

在材料非线性问题中的摄动有限元法

摄动法是解决非线性连续介质力学问题的一种有效方法.这种方法是建立在该问题的线性解析解的基础上的,因此,若得不到一个简单的解析解,应用这种方法去解决一些复杂的非线性问题将遇到困难.有限元法对解非线性问题也是一种十分有用的工具,然而一般来说,它需要相当长的计算时间. 本文介绍摄动有限元法.这种方法吸取上述两种方法的优点,能够解决更复杂的非线性问题,而且也能大量节省计算机的计算时间. 本文讨论了比例加载下的弹塑性力学问题,并提出一个带孔拉板的数值解.     

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations.

A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.     

MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS

In this paper,we consider a modified alternating positive semidefinite splitting precon-ditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model.The eigenvalue distri-bution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology.Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.