Exploiting semantics of temporal multi‐scale methods to optimize multi‐level mesh partitioning

Multi‐scale problems are often solved by decomposing the problem domain into multiple subdomains, solving them independently using different levels of spatial and temporal refinement, and coupling the subdomain solutions back to obtain the global solution. Most commonly, finite elements are used for spatial discretization, and finite difference time stepping is used for time integration. Given a finite element mesh for the global problem domain, the number of possible decompositions into subdomains and the possible choices for associated time steps is exponentially large, and the computational costs associated with different decompositions can vary by orders of magnitude. The problem of finding an optimal decomposition and the associated time discretization that minimizes computational costs while maintaining accuracy is nontrivial. Existing mesh partitioning tools, such as METIS, overlook the constraints posed by multi‐scale methods and lead to suboptimal partitions with a high performance penalty. We present a multi‐level mesh partitioning approach that exploits domain‐specific knowledge of multi‐scale methods to produce nearly optimal mesh partitions and associated time steps automatically. Results show that for multi‐scale problems, our approach produces decompositions that outperform those produced by state‐of‐the‐art partitioners like METIS and even those that are manually constructed by domain experts. Copyright © 2017 John Wiley & Sons, Ltd.     

Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.     

Multi‐scale analysis and FE computation for the structure of composite materials with small periodic configuration under condition of coupled thermoelasticity

The two‐scale asymptotic (TSA) expressions of the increment of temperature and the displacement for the structure of composite materials with small periodic configuration under coupled thermoelasticity condition are derived formally in this paper, especially, the two‐scale coupled relation between the increment of temperature and the displacements are set up. Then the approximate solutions and their error estimations are presented, and the multi‐scale finite element algorithm corresponding to TSA is described. Finally, simple numerical results evaluated by multi‐scale FE computation are shown. They demonstrate that the basic configuration and the increment of temperature strongly influence upon local strains and local stresses inside basic cell. Copyright © 2004 John Wiley & Sons, Ltd.     

Multi‐scale modeling of plastic deformations in nano‐scale materials; Transition to plastic limit

A large amount of research in computational mechanics has biased toward atomistic simulations. This trend, on one hand, is due to the increased demand to perform computations in nanoscale and, on the other hand, is due to the rather simple applications of pairwise potentials in modeling the interactions between atoms of a given crystal. The Cauchy–Born (CB) hypothesis has been used effectively to model the behavior of crystals under different loading conditions, in which the comparison with molecular dynamics simulations presents desirable coincidence between the results. A number of research works have been devoted to the validity of CB hypothesis and its application in post‐elastic limit. However, the range of application of CB hypothesis is limited, and it remains questionable whether it is still applicable beyond the validity limit. In this paper, a multi‐scale technique is developed for modeling of plastic deformations in nanoscale materials. The deformation gradient is decomposed into the plastic and elastic parts, i.e., F  =  F ^p F ^e. This decomposition is in contrast to the conventional decomposition, F  =  F ^e F ^p, generally encountered in continuum and crystal plasticity. It is shown that the former decomposition is more appropriate for the problem dealt within this work. Inspired by crystal plasticity, the plastic part is determined from the slip on potential slip systems. Based on the assumption that the CB hypothesis remains valid in the homogeneous deformation, the elastic deformation gradient resulting from the aforementioned decomposition is employed in conjunction with the CB hypothesis to update the state variables for face‐centered cubic crystals. The assumption of homogeneity of elastic deformation gradient is justified by the fact that elastic deformations are considerably smaller than the plastic deformations. The computational algorithms are derived in details, and numerical simulations are presented through several examples to demonstrate the capability of the proposed computational algorithm in the modeling of golden crystals under different loading conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

A computational framework for scale‐bridging in multi‐scale simulations

A computational framework for scale‐bridging in multi‐scale simulations is presented. The framework enables seamless combination of at‐scale models into highly dynamic hierarchies to build a multi‐scale model. Its centerpiece is formulated as a standalone module capable of fully asynchronous operation. We assess its feasibility and performance for a two‐scale model applied to two challenging test problems from impact physics. We find that the computational cost associated with using the framework may, as expected, become substantial. However, the framework has the ability of effortlessly combining at‐scale models to render complex multi‐scale models. The main source of the computational inefficiency of the framework is related to poor load balancing of the lower‐scale model evaluation We demonstrate that the load balancing can be efficiently addressed by recourse to conventional load‐balancing strategies. Copyright © 2016 John Wiley & Sons, Ltd.     

Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling

The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three‐dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short‐term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright © 2006 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effect via the boundary Cauchy–Born method

In this paper, a novel multi‐scale approach is developed for modeling of the surface effect in crystalline nano‐structures. The technique is based on the Cauchy–Born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter‐atomic potentials. The notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures, called as an indicator of material behavior. The information of quadratures is derived by interpolating the data from probable representative atoms in their proximity. The technique is implemented by the definition of reference boundary CB elements, which enable to capture not only the surface but also the edge and corner effects. As the surface effect is important in small‐scale simulation, the relative number of boundary CB elements increases which leads to predomination of boundary effects in the model. In order to implement the equivalent continua in boundary value problems, the updated‐Lagrangian formulation of nonlinear finite element is derived. The numerical simulation of the proposed model together with the direct comparison with fully atomistic model indicates that the technique provides promising results for facile modeling of boundary effects and investigating its effect on the mechanical response of metallic nano‐scale devices. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

Multi‐scale experimental study on fatigue damage behaviour and its effect on structural nonlinear response

Experimental analyses on the structural response caused by local fatigue damage accumulation in welded details are accomplished to perform failure process and nonlinear effect analysis at different structural levels. The experiment is carried out by using welded compact tension (CT) specimens and a scaled truss specimen, and all of them have a notch at the weld toe to facilitate damage initiation. Cyclic loads are applied to those specimens to generate accumulative fatigue damage, respectively. The process of fatigue accumulation including initiation and propagation of fatigue cracks in the welded detail and resultant structural responses of CT specimens and the truss are measured with integration of multiple testing techniques. Multi‐scale experimental results show that microscopic‐/mesoscopic‐concentrated strain and extension of plastic zone in the vicinity of notch tip are both affected significantly by the fatigue damage accumulation and present appreciable nonlinear behaviour; however, the macroscopic response such as the frequency and stiffness parameters of the welded truss specimen are less sensitive to the low‐level fatigue damage. It is concluded that the fatigue failure of the welded truss is a multi‐scale progressive process due to fatigue damage trans‐scale evolving, in which the local meso‐damage firstly affects local strain of plastic zone in the vicinity of the notch tip, and then fatigue damage evolving from meso‐ to macro‐scale affects nonlinear responses of the damaged components; lastly, the fatigue failure could be expected as the results of the propagation of macroscopic fatigue cracks.     

kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains

We propose a novel finite‐element method for polygonal meshes. The resulting scheme is hp‐adaptive, where h and p are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis‐aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices.     

Two‐scale computational modelling of water flow in unsaturated soils containing irregular‐shaped inclusions

The focus of this paper is two‐dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two‐scale Richards’ equation‐based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro‐scale as a two‐dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro‐scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two‐scale model in a completely coupled manner. Numerical simulations are presented for a two‐dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

Problems involving reaction and species diffusion involve field and flux jumps at a moving reaction front. In multi‐scale problems such as carbon fiber composite oxidation, these effects need to be tracked at the microscopic scale of individual carbon fibers. A multi‐scale model is derived in this paper for predicting species distribution in such problems using a fully coupled multi‐scale homogenization approach. The homogenized fluxes from the micro‐scale are derived using Hill's macro‐homogeneity condition accounting for both flux jumps and species density field jumps at the reacting interface in the micro‐scale unit cell. At the macro‐scale, the competition between the transport of reacting species (oxygen) and the reaction product (carbon dioxide) is modeled using homogenized mass conservation equations. The moving reaction front in carbon fibers at the micro‐scale is tracked using level set method and an adaptive meshing strategy. The macroscopic weight loss of the composite when exposed to oxygen is simulated as a function of time using a coupled finite element methodology at various locations in a validated macroscopic model. Copyright © 2010 John Wiley & Sons, Ltd.     

Computational aspects of a new multi‐scale dispersive gradient elasticity model with micro‐inertia

Computational aspects of a recently developed gradient elasticity model are discussed in this paper. This model includes the (Aifantis) strain gradient term along with two higher‐order acceleration terms (micro‐inertia contributions). It has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy. In this paper, the discretisation details of this model are thoroughly investigated, including both discretisation in time and in space. Firstly, the critical time step is derived that is relevant for conditionally stable time integrators. Secondly, recommendations on how to choose the numerical parameters, primarily the element size and time step, are given by comparing the dispersion behaviour of the original higher‐order continuum with that of the discretised medium. In so doing, the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length‐scales. A set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy. These guidelines are then verified numerically by examining the wave propagation in a one‐dimensional bar as well as in a two‐dimensional example. Copyright © 2016 John Wiley & Sons, Ltd.     

A new multi‐scale dispersive gradient elasticity model with micro‐inertia: Formulation and ‐finite element implementation

Motivated by nano‐scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi‐scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro‐inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain‐gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro‐inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth‐order equations are rewritten in two sets of symmetric second‐order equations so that ‐continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi‐scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

A logarithmic‐exponential backward‐Euler‐based split of the flow rule for anisotropic inelastic behaviour at small elastic strain

A basic aspect of modern algorithmic formulations for large‐deformation hyperelastic‐based isotropic inelastic material models is the exponential backward‐Euler form of the algorithmic flow rule in the context of the multiplicative decomposition of the deformation gradient. Advantages of this approach in the isotropic context include the exact algorithmic fulfilment of inelastic incompressibility. The purpose of this short work is to show that such an algorithm can be formulated for anisotropic inelastic models as well under assumption of small elastic strain, i.e. for metals. In particular, the current approach works for both phenomenological anisotropy as well as for crystal plasticity. The major difference between the current and previous approaches lies in the fact that the elastic rotation is reduced algorithmically to a dependent internal variable, resulting in a smaller internal variable system. Copyright © 2006 John Wiley & Sons, Ltd.     

Effects of Multi‐Scale Patterning on the Run‐In Behavior of Steel–Alumina Pairings under Lubricated Conditions

In nature, many examples of multi‐scale surfaces with outstanding tribological properties such as reduced friction and wear under dry friction and lubricated conditions can be found. To determine whether multi‐scale surfaces positively affect the frictional and wear performance, tests are performed on a ball‐on‐disk tribometer under lubricated conditions using an additive‐free poly‐alpha‐olefine oil under a contact pressure of around 1.29 GPa. For this purpose, stainless steel specimens (AISI 304) are modified by micro‐coining (hemispherical structures with a structural depth of either 50 or 95 μm) and subsequently by direct laser interference patterning (cross‐like pattern with 9 μm periodicity) to create a multi‐scale pattern. The comparison of different sample states (polished reference, laser‐patterned, micro‐coined, and multi‐scale) shows a clear influence of the fabrication technique. In terms of the multi‐scale structures, the structural depth of the coarser micro‐coining plays an important role. In case of lower coining depths (50 μm), the multi‐scale specimens show an increased coefficient of friction compared to the purely micro‐coined surfaces, whereas larger coining depths (95 μm) result in stable and lower friction values for the multi‐scale patterns.     

Two‐scale diffusion–deformation coupling model for material deterioration involving micro‐crack propagation

In this paper, we introduce a two‐scale diffusion–deformation coupled model that represents the aging material deterioration of two‐phase materials involving micro‐crack propagations. The mathematical homogenization method is applied to relate the micro‐ and macroscopic field variables, and a weak coupling solution method is employed to solve the two‐way coupling phenomena between the diffusion of scalar fields and the deformation of quasi‐brittle solids. The macroscopic mechanical behavior represented by the derived two‐scale two‐way coupled model reveals material nonlinearity due to micro‐scale cracking induced by the scalar‐field‐induced deformation, which can be simulated by the finite cover method. After verifying the fundamental validity of the proposed model and the analysis method, we perform a simple numerical example to demonstrate their ability to predict aging material deterioration. Copyright © 2010 John Wiley & Sons, Ltd.     

Truncation error in mesh‐free particle methods

A truncation error analysis has been developed for the approximation of spatial derivatives in smoothed particle hydrodynamics (SPH) and related first‐order consistent methods such as the first‐order form of the reproducing kernel particle method. Error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length, Δx/h. For uniformly spaced particles in one dimension, analysis shows that as h is reduced while maintaining constant Δx/h, error decays as h² until a limiting discretization error is reached, which is independent of h. If Δx/h is reduced while maintaining constant h (i.e. if the number of neighbours per particle is increased), error decreases at a rate which depends on the kernel function's smoothness. When particles are distributed non‐uniformly, error can grow as h is reduced with constant Δx/h. First‐order consistent methods are shown to remove this divergent behaviour. Numerical experiments confirm the theoretical analysis for one dimension, and indicate that the main results are also true in three dimensions. This investigation highlights the complexity of error behaviour in SPH, and shows that the roles of both h and Δx/h must be considered when choosing particle distributions and smoothing lengths. Copyright © 2005 John Wiley & Sons, Ltd.