Elastoplasticity with linear tetrahedral elements: A variational multiscale method

We present a computational framework for the simulation of J₂‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.     

A variational multiscale method with subscales on the element boundaries for the Helmholtz equation

In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

A variational multiscale method for particle-cloud tracking in turbomachinery flows

We present a computational method for simulation of particle-laden flows in turbomachinery. The method is based on a stabilized finite element fluid mechanics formulation and a finite element particle-cloud tracking method. We focus on induced-draft fans used in process industries to extract exhaust gases in the form of a two-phase fluid with a dispersed solid phase. The particle-laden flow causes material wear on the fan blades, degrading their aerodynamic performance, and therefore accurate simulation of the flow would be essential in reliable computational turbomachinery analysis and design. The turbulent-flow nature of the problem is dealt with a Reynolds-Averaged Navier–Stokes model and Streamline-Upwind/Petrov–Galerkin/Pressure-Stabilizing/Petrov–Galerkin stabilization, the particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence–particle interaction, and one-way dependence is assumed between the flow field and particle dynamics. We propose a closure model utilizing the scale separation feature of the variational multiscale method, and compare that to the closure utilizing the eddy viscosity model. We present computations for axial- and centrifugal-fan configurations, and compare the computed data to those obtained from experiments, analytical approaches, and other computational methods.     

A variational multiscale method to model crack propagation at finite strains

This contribution presents a hierarchical variational multiscale framework to model propagating discontinuities at finite strains. Thereby the deformation map is decomposed into coarse‐scale and fine‐scale displacements, which results in a decoupled system of coarse‐scale and fine‐scale equations. Both are solved numerically by means of the finite element method whereby crack propagation is taken into account at the fine scale. Growing cracks are numerically handled by the introduction of discontinuous elements. A locality assumption on the fine‐scale solution and an adaptive scheme to resize the fine‐scale domain are introduced and demonstrated to increase the efficiency of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

A dynamic variational multiscale method for viscoelasticity using linear tetrahedral elements

In this article, we develop a dynamic version of the variational multiscale (D‐VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement, and velocity are approximated with piecewise linear, continuous finite element functions. To prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations.     

Residual-based variational multiscale simulation of free surface flows

In this work we apply the residual-based variational multiscale method (RB-VMS) to the volume-of-fluid (VOF) formulation of free-surface flows. Using this technique we are able to solve such problems in a Large Eddy Simulation framework. This is a natural extension of our Navier–Stokes solver, which uses the RB-VMS finite element formulation, edge-based data structures, adaptive time step control, inexact Newton solvers and supports several parallel programming paradigms. The VOF interface capturing variable is advected using the computed coarse and fine scales velocity field. Thus, the RB-VMS technique can be readily applied to the free-surface solver with minor modifications on the implementation. We apply this technique to the solution of two problems where available data indicate complex free-surface behavior. Results are compared with numerical and experimental data and show that the present formulation can achieve good accuracy with minor impacts on computational efficiency.     

Fluid structure interaction by means of variational multiscale reduced order models

A reduced order model designed by means of a variational multiscale stabilized formulation has been applied successfully to fluid-structure interaction problems in a strongly coupled partitioned solution scheme. Details of the formulation and the implementation both for the interaction problem and for the reduced models, for both the off-line and on-line phases, are shown. Results are obtained for cases in which both domains are reduced at the same time. Numerical results are presented for a semistationary and a fully transient case.     

The role of continuity in residual-based variational multiscale modeling of turbulence

This paper examines the role of continuity of the basis in the computation of turbulent flows. We compare standard finite elements and non-uniform rational B-splines (NURBS) discretizations that are employed in Isogeometric Analysis (Hughes et al. in Comput Methods Appl Mech Eng, 194:4135–4195, 2005). We make use of quadratic discretizations that are C ⁰-continuous across element boundaries in standard finite elements, and C ¹-continuous in the case of NURBS. The variational multiscale residual-based method (Bazilevs in Isogeometric analysis of turbulence and fluid-structure interaction, PhD thesis, ICES, UT Austin, 2006; Bazilevs et al. in Comput Methods Appl Mech Eng, submitted, 2007; Calo in Residual-based multiscale turbulence modeling: finite volume simulation of bypass transition. PhD thesis, Department of Civil and Environmental Engineering, Stanford University, 2004; Hughes et al. in proceedings of the XXI international congress of theoretical and applied mechanics (IUTAM), Kluwer, 2004; Scovazzi in Multiscale methods in science and engineering, PhD thesis, Department of Mechanical Engineering, Stanford Universty, 2004) is employed as a turbulence modeling technique. We find that C ¹-continuous discretizations outperform their C ⁰-continuous counterparts on a per-degree-of-freedom basis. We also find that the effect of continuity is greater for higher Reynolds number flows.     

复合材料跨尺度失效准则及其损伤演化

提出了一种新的基于物理失效模式的复合材料跨尺度失效准则, 从细观层面分别对纤维和基体的失效模式进行了表征, 将纤维失效分为拉伸失效和压缩失效, 将基体失效分为膨胀失效和扭曲失效。建立了相应的失效准则及损伤演化方法。通过正方形和六边形的代表体积单元(RVE)模型, 计算了宏观应力到细观应力的机械应力放大系数和热应力放大系数。以IM7/5250-4复合材料拉伸试验作为算例对失效模型进行了验证。计算结果与试验结果吻合较好, 表明跨尺度失效准则能够准确预测复合材料层合板的破坏。     

复合材料跨尺度失效准则及其损伤演化

提出了一种新的基于物理失效模式的复合材料跨尺度失效准则,从细观层面分别对纤维和基体的失效模式进行了表征,将纤维失效分为拉伸失效和压缩失效,将基体失效分为膨胀失效和扭曲失效.建立了相应的失效准则及损伤演化方法.通过正方形和六边形的代表体积单元(RVE)模型,计算了宏观应力到细观应力的机械应力放大系数和热应力放大系数.以IM7/5250-4复合材料拉伸试验作为算例对失效模型进行了验证.计算结果与试验结果吻合较好,表明跨尺度失效准则能够准确预测复合材料层合板的破坏.     

Stochastic parametrization of multiscale processes using a dual-grid approach

Some speculative proposals are made for extending current stochastic sub-gridscale parametrization methods using the techniques adopted from the field of computer graphics and flow visualization. The idea is to emulate sub-filter-scale physical process organization and time evolution on a fine grid and couple the implied coarse-grained tendencies with a forecast model. A two-way interaction is envisaged so that fine-grid physics (e.g. deep convective clouds) responds to forecast model fields. The fine-grid model may be as simple as a two-dimensional cellular automaton or as computationally demanding as a cloud-resolving model similar to the coupling strategy envisaged in 'super-parametrization'. Computer codes used in computer games and visualization software illustrate the potential for cheap but realistic simulation where emphasis is placed on algorithmic stability and visual realism rather than pointwise accuracy in a predictive sense. In an ensemble prediction context, a computationally cheap technique would be essential and some possibilities are outlined. An idealized proof-of-concept simulation is described, which highlights technical problems such as the nature of the coupling.     

Galerkin's method and the variational procedure

Galerkin's method and the variational procedure, when applied to most practical problems in electromagnetics, lead to matrix equations of the same form. Variational procedures for self-adjoint and nonself-adjoint operators also result in the same form of matrix equations for a large subclass of problems. However, the three cases may yield different matrix equations in general. This paper examines the subclass of problems for which these methods result in the same matrix equation and provides systematic ways for classification of problems for which two or all three of the cases lead to the same matrix equation. It also describes properties of the coefficient matrix in the matrix equation     

Spline-based variational method with constraints forspectrophotometric data correction

The raw results of spectrophotometric measurements are subject to systematic errors of an instrumental type which may be reduced provided a mathematical model of the instrumental imperfections is identified. It is assumed that this model has the form of an integral, convolution-type equation of the first kind. The correction of the spectrometric data consists in numerically solving this equation on the basis of the raw results of measurements. An algorithm of correction is proposed which is based on the approximation of the solution with a spline function whose parameters are determined using a variational method with the positivity constraint imposed on the set of feasible solutions. The efficiency of the incorporation of this constraint into the algorithm of correction is demonstrated using synthetic data. The possibility of improving resolution of spectrometric data is shown on a set of real spectrophotometric measurements     

Void coalescence processes quantified through atomistic and multiscale simulation

Simulation of ductile fracture at the atomic scale reveals many aspects of the fracture process including specific mechanisms associated with void nucleation and growth as a precursor to fracture and the plastic deformation of the material surrounding the voids and cracks. Recently we have studied void coalescence in ductile metals using large-scale atomistic and continuum simulations. Here we review that work and present some related investigations. The atomistic simulations involve three-dimensional strain-controlled multi-million atom molecular dynamics simulations of copper. The correlated growth of two voids during the coalescence process leading to fracture is investigated, both in terms of its onset and the ensuing dynamical interactions. Void interactions are quantified through the rate of reduction of the distance between the voids, through the correlated directional growth of the voids, and through correlated shape evolution of the voids. The critical inter-void ligament distance marking the onset of coalescence is shown to be approximately one void radius based on the quantification measurements used, independent of the initial separation distance between the voids and the strain-rate of the expansion of the system. No pronounced shear flow is found in the coalescence process. We also discuss a technique for optimizing the calculation of fine-scale information on the fly for use in a coarse-scale simulation, and discuss the specific case of a fine-scale model that calculates void growth explicitly feeding into a coarse-scale mechanics model to study damage localization. The U.S. Government’s right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged.     

A computational library for multiscale modeling of material failure

We present an open-source software framework called PERMIX for multiscale modeling and simulation of fracture in solids. The framework is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH. Fracture on the continuum level is modeled by the extended finite element method (XFEM). Using several novel or state of the art methods, the piece software handles semi-concurrent multiscale methods as well as concurrent multiscale methods for fracture, coupling two continuum domains or atomistic domains to continuum domains, respectively. The efficiency of our open-source software is shown through several simulations including a 3D crack modeling in clay nanocomposites, a semi-concurrent FE-FE coupling, a 3D Arlequin multiscale example and an MD-XFEM coupling for dynamic crack propagation.