A programming approach to the numerical analysis of elastoplastic continua

The extended kinematic minimum principle is used to formulate a quadratic programming solution to the rate problem in elastoplasticity for plane continua. A simple algorithm is presented which involves the minimisation of a continuous functional subject to inequality constraints. The algorithm has been implemented within an incremental finite element formulation which includes equilibrium iterations and predictor-corrector procedures for calculating plastic strain increments. The potential of this algorithm in situations where the plastic zone at failure is small is discussed.     

Application of the Cosserat continua to numerical studies on the properties of the materials

The finite element models of Cosserat continuum in two- and three-dimensions are presented. The size effects of a cantilever beam and a micro-rod, the well-posedness, the mesh-independent solutions of the boundary value problems with non-associated elastoplastic and strain softening constitutive behavior, and the progressive failure of the two- and three-dimensional vertical excavations are studied. Numerical results illustrate that the proposed Cosserat continuum models are capable of reflecting the size effects of micro-structures, preserving the well-posedness of the boundary value problem characterized by the strain localization, ensuring mesh-independent solutions, and simulating the entire progressive failure process occurring in engineering structures.     

3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations

This paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12: 277–296) mesh objective 2D and 3D finite element formulations are developed using linear and quadratic 2D elements as well as 8‐noded 3D elements. In the formulation of the finite‐element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non‐linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of fixed discontinuities. It is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions. The applicability of the proposed SDA finite‐element model as well as its numerical performance is investigated by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force. Copyright © 2003 John Wiley & Sons, Ltd.     

On error estimator and adaptivity in the meshless Galerkin boundary node method

In this article, a simple a posteriori error estimator and an effective adaptive refinement process for the meshless Galerkin boundary node method (GBNM) are presented. The error estimator is formulated by the difference between the GBNM solution itself and its L ²-orthogonal projection. With the help of a localization technique, the error is estimated by easily computable local error indicators and hence an adaptive algorithm for h-adaptivity is formulated. The convergence of this adaptive algorithm is verified theoretically in Sobolev spaces. Numerical examples involving potential and elasticity problems are also provided to illustrate the performance and usefulness of this adaptive meshless method.     

On error estimator and p‐adaptivity in the generalized finite element method

This paper addresses the issue of a p‐adaptive version of the generalized finite element method (GFEM). The technique adopted here is the equilibrated element residual method, but presented under the GFEM approach, i.e., by taking into account the typical nodal enrichment scheme of the method. Such scheme consists of multiplying the partition of unity functions by a set of enrichment functions. These functions, in the case of the element residual method are monomials, and can be used to build the polynomial space, one degree higher than the one of the solution, in which the error functions is approximated. Global and local measures are defined and used as error estimator and indicators, respectively. The error indicators, calculated on the element patches that surrounds each node, are used to control a refinement procedure. Numerical examples in plane elasticity are presented, outlining in particular the effectivity index of the error estimator proposed. Finally, the ‐adaptive procedure is described and its good performance is illustrated by the last numerical example. Copyright © 2004 John Wiley & Sons, Ltd.     

On the numerical evaluation of Eshelby's tensor and its application to elastoplastic fibrous composites

A numerical evaluation of Eshelby's S tensor for an ellipsoidal inclusion imbedded in a general anisotropic matrix material is performed. The numerical scheme is valid for any degree of matrix anisotropy and for any aspect ratio of the ellipsoid, including the extreme cases of cracks and cylindrical inclusions. The influence of matrix anisotropy on the evaluation of S is tested extensively for cylindrical inclusions by considering plasticity induced anisotropy in the instantaneous properties of an elastic-plastic matrix material. The Mori-Tanaka averaging method is used to study the influence of the evaluation of S on the prediction of instantaneous effective properties of fibrous composites with elastic fibres and elastic-plastic matrix.     

RVE computations with error control and adaptivity: the power of duality

The goal of computational homogenization is to obtain the macro-scale response, normally in terms of macro-scale stress for given macro-scale deformation, via RVE-computations. In this paper we investigate, in a systematic manner, the effects of Dirichlet and Neumann boundary conditions on the RVE. Adaptive computations are carried out with respect to, in particular, control of the error in the macro-scale stress tensor. This requires the corresponding dual solutions. As a new result, it is shown how the same dual solutions can be conveniently used in computing the algorithmic tangent stiffness tensor, thereby demonstrating the “power of duality”.     

A review of error estimation and adaptivity in the boundary element method

When using the boundary element method, the accuracy of the numerical solution depends critically on the discretization of the boundary into elements (panels). The distribution of the panels is one of the most important decisions taken when analyzing a problem, but still the vast majority of users employ empirical guidelines to distribute the panels. This paper reviews the various adaptive schemes that have been proposed for boundary elements. Numerical results are obtained for infinite fluid flow problems and free surface problems and are used to assess the reliability and effectiveness of each method.     

On a variational formulation of problems in the classical continuum mechanics of solids

The basic equations of linear elasticity, of mildly non-linear elasticity and of plastic flow are reduced to a generalization of Hamilton's canonical formalism. Complementary variational principles are deduced. Numerical applications of these principles are demonstrated in some simple cases.     

Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error

This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch‐by‐patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field. Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the ??₂ norm of the error or relating it with the energy norm. The guaranteed upper bounds are obtained by using a pessimistic bound of the error ??₂ norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error ??₂ norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests. Copyright © 2006 John Wiley & Sons, Ltd.     

Applications of inherent strain and interface element to simulation of welding deformation in thin plate structures

Welding-induced distortion not only reduces largely manufacturing accuracy but also decreases significantly productivity due to correction works. If welding distortion can be predicted through a simple and practical method beforehand, the predictions will be helpful for taking active as well as appropriate measures to control the dimension accuracy. Based on inherent strain theory and interface element formulation, we developed a practical prediction system to compute the accumulated distortion during the welding assembly process in the current study. Using the developed prediction method, we calculated the welding distortion in a thin plate structure with considering both the shrinkage due to heat input and the gap/misalignment generated during assembly process. Meanwhile, we investigated the influences of assembly sequence and gap correction on the final distortion.     

Dynamic behavior of concrete at high strain rates and pressures: II. numerical simulation

The response of concrete and mortar under high-strain-rate impact loading are analyzed using fully dynamic finite element simulations. The analyses concern the load-carrying capacity, energy absorbency and the effect of the microstructure. The simulations focus on the plate impact configuration used in the experimental part of this research, allowing for direct comparison of model predictions with experimental measurements. A micromechanical model is formulated and used, accounting for the two-phase composite microstructure of concrete. Arbitrary microstructural phase morphologies of actual concrete used in impact experiments are digitized and explicitly considered in the numerical models. The behavior of the two constituent phases in the concrete are characterized by an extended Drucker–Prager model that accounts for pressure-dependence, rate-sensitivity, and strain hardening/softening. Model parameters are determined by independent impact experiments on mortar and through a parametric study in which the prediction of numerical simulations is matched with measurements from experiments on concrete and mortar. Calculations show that significant inelastic deformations occur in the mortar matrix under the impact conditions analyzed and relatively smaller inelastic strains are seen in the aggregates. The influence of aggregate volume fraction on the dynamic load-carrying capacity of concrete is explored. The strength increases with aggregate volume fraction and an enhancement of approximately 30% over that of mortar is found for an aggregate volume fraction of 42%. Numerical simulations also show increasing energy absorbency with increasing aggregate volume fraction and provide a time-resolved characterization for the history of work dissipation as the deformation progresses.     

Multiscale simulations: application to the heat transfer simulation of sliding solids

Molecular dynamics is a powerful tool allowing the simulation of matter behaviour at the atomic scale. Due to computation time, it is clearly not possible to use molecular dynamics to simulate a forming process. However, atomistic simulations can be used to study and understand the physical phenomena that occur during matter deformation. As an example, heat transfer between the contacting solids in forming processes is one of the important physics phenomena that have to be taken into account in order to do realistic simulations. A multiscale analysis of heat transfer is presented. This analysis leads to two kinds of models: a macroscopic model which can be used for the simulation of the process itself and a microscopic model that is used to determine the parameters of the macroscopic model. In this microscopic model, the friction heat generation phenomena has to be described quite accurately. Friction heat is mainly due to plastic and elastic deformation and adhesion. Thus, to understand the underlying friction heat generation phenomena, atomistic simulations using molecular dynamics are carried out. It is shown that friction heat is the transformation of mechanical work given to the system at the macroscopic scale into potential energy during elastic deformation. This potential energy which is stored in the system is finally transformed into atomic kinetic energy (friction heat) during plastic transformation.     

On strain localization in the high rate extension of nylon fibers

A simple analytical model for the adiabatic high strain rate extension of synthetic textile fibers is presented. The model suggests that, for fibers with particular thermo-mechanical and constitutive properties, initial nominally uniform strain distributions along the fiber will tend to become non-uniform, with localization of axial strain into a thermally softened region. To assess the usefulness of the model in predicting and interpreting fiber behavior, a commercial nylon filament is investigated experimentally. Nylon filaments are extended to break at a low, isothermal strain rate (0.0015 s^–1) and at a high, adiabatic strain rate (70 s^–1). A dimensionless strain localization parameter (SLP), used to characterize the nylon filament in the framework of the model, predicts strain localization to occur during extension at the 70 s^–1 strain rate. Experimental load-extension curves exhibit a sharply reduced elongation-to-break at the high strain rate, consistent with the predicted occurrence of localized, versus uniform, straining. In addition, the transition from homogeneous to localized straining appears to occur at elongations that correspond with the SLP attaining a critical value for onset of localization.     

On the description of ductile fracture in metals by the strain localization theory

Numerical simulations based on the bifurcation and imperfection versions of the strain localization theory are used in this paper to predict the failure loci of metals and applied to an advanced high strength steel subjected to proportional loading paths. The results are evaluated against the 3D unit cell analyses of Dunand and Mohr (J Mech Phys Solids 66(1):133–153, 2014. doi: 10.1016/j.jmps.2014.01.008) available in the literature. The Gurson porous plasticity model (Gurson in J Eng Mater Technol 99(1):2–15, 1977. doi: 10.1115/1.344340) is used to induce strain softening and drive the localization process. The effects of the void growth, void nucleation and void softening in shear are investigated over a large range of stress triaxialities and Lode parameters. A correlation between the imperfection and bifurcation results is established.     

室内指向性声源的声场数值模拟

为了利用计算机对室内指向性声源的声场进行模拟,文中将声线跟踪法,虚源法和Monte Carlo法相结合,编制了一种场场数值模拟仿真程序,该程序能模拟有（或无）指向性声源在室内给定点的声压级等参数值,还能计算出该点的脉冲响应函数,作为例子,分别利用该法和声压叠加法研究了点源和偶极子声源的指向性。     

Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. More precisely, in the context of linear elliptic diffusion problems in randomly packed particulate composites, we develop an approach to strictly bound the error in the expectation and second moment of quantities of interest, without ever solving the fine‐scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine‐scale features. Our computations are purely macroscopic, deterministic and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager–Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss–Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation. Copyright © 2016 John Wiley & Sons, Ltd.