An adaptive FE–MD model coupling approach

This contribution introduces an adaptively coupled finite element (FE)–molecular dynamics (MD) model based on the Quasicontinuum (QC) method. The idea for obtaining constitutive laws from the underlying lattice structure (local QC model) will be discussed in detail. The outline of the formulation for the quasi-static MD model (nonlocal QC model) will also be derived in the same mathematical structure. A new type of element is proposed to solve the boundary problems and to couple the FE and MD models. The interpolation techniques for the atomic stress and strain fields are introduced. A two-step adaptive mechanism is applied to the multiscale model, including the mesh refinement step for the FE model and the FE–MD conversion step. A 3D nanoindentation example is used for demonstrating accuracy and the efficiency of the coupled FE–MD model at the end.     

A quasicontinuum methodology for multiscale analyses of discrete microstructural models

Many studies in different research fields use lattice models to investigate the mechanical behavior of materials. Full lattice calculations are often performed to determine the influence of localized microscale phenomena on large‐scale responses but they are usually computationally expensive. In this study the quasicontinuum (QC) method (Phil. Mag. A 1996; 73 :1529–1563) is extended towards lattice models that employ discrete elements, such as trusses and beams. The QC method is a multiscale approach that uses a triangulation to interpolate the lattice model in regions with small fluctuations in the deformation field, while in regions of high interest the exact lattice model is obtained by refining the triangulation to the internal spacing of the lattice. Interpolation ensures that the number of unknowns is reduced while summation ensures that only a selective part of the underlying lattice model must be visited to construct the governing equations. As the QC method has so far only been applied to atomic lattice models, the existing summation procedures have been revisited for structural lattice models containing discrete elements. This has led to a new QC method that makes use of the characteristic structure of the considered truss network. The proposed QC method is, to the best of the authors' knowledge, the only QC method that does not need any correction at the interface between the interpolated and the fully resolved region and at the same time gives exact results unlike the cluster QC methods. In its present formulation, the proposed QC method can only be used for lattice models containing nearest neighbor interactions, but with some minor adaptations it can also be used for lattices with next‐nearest neighbor interactions such as atomic lattices. Copyright © 2011 John Wiley & Sons, Ltd.     

The Quasicontinuum Method: Overview, applications and current directions

The Quasicontinuum (QC) Method, originally conceived and developed by Tadmor, Ortiz and Phillips [1] in 1996, has since seen a great deal of development and application by a number of researchers. The idea of the method is a relatively simple one. With the goal of modeling an atomistic system without explicitly treating every atom in the problem, the QC provides a framework whereby degrees of freedom are judiciously eliminated and force/energy calculations are expedited. This is combined with adaptive model refinement to ensure that full atomistic detail is retained in regions of the problem where it is required while continuum assumptions reduce the computational demand elsewhere. This article provides a review of the method, from its original motivations and formulation to recent improvements and developments. A summary of the important mechanics of materials results that have been obtained using the QC approach is presented. Finally, several related modeling techniques from the literature are briefly discussed. As an accompaniment to this paper, a website designed to serve as a clearinghouse for information on the QC method has been established at www.qcmethod.com. The site includes information on QC research, links to researchers, downloadable QC code and documentation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Insights on slip transmission at grain boundaries from atomistic simulations

To fully understand the plastic deformation of metallic polycrystalline materials, the physical mechanisms by which a dislocation interacts with a grain boundary must be identified. Recent atomistic simulations have focused on the discrete atomic scale motions that lead to either dislocation obstruction, dislocation absorption into the grain boundary with subsequent emission at a different site along the grain boundary, or direct dislocation transmission through the grain boundary into the opposing lattice. These atomistic simulations, coupled with foundational experiments performed to study dislocation pile-ups and slip transfer through a grain boundary, have facilitated the development and refinement of a set of criteria for predicting if dislocation transmission will occur and which slip systems will be activated in the adjacent grain by the stress concentration resulting from the dislocation pile-up. This article provides a concise review of both experimental and atomistic simulation efforts focused on the details of slip transmission at grain boundaries in metallic materials and provides a discussion of outstanding challenges for atomistic simulations to advance this field.     

Adaptive atomistic‐to‐continuum modeling of propagating defects

An adaptive atomistic‐to‐continuum method is presented for modeling the propagation of material defects. This method extends the bridging domain method to allow the atomic domain to dynamically conform to the evolving defect regions during a simulation, without introducing spurious oscillations and without requiring mesh refinement. The atomic domain expands as defects approach the bridging domain method coupling domain by fine graining nearby finite elements into equivalent atomistic subdomains. Additional algorithms coarse grain portions of the atomic domain to the continuum scale, reducing the degrees of freedom, when the atomic displacements in a subdomain can be approximated by FEM or extended FEM elements to within a certain homogeneity tolerance. The extended FEM approximations are created by fitting the broken inter‐atomic bonds of fractured surfaces and dislocation slip planes. Because atomic degrees of freedom are maintained only where needed for each timestep, the solution retains the advantages of multiscale modeling, with a reduced computational cost compared with other multiscale methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive superposition of finite element meshes in elastodynamic problems

An adaptive finite element procedure is developed for modelling transient phenomena in elastic solids, including both wave propagation and structural dynamics. Although both temporal and spatial adaptivity are addressed, the novel feature of the formulation is the use of mesh superposition to produce spatial refinement (referred to as s‐adaptivity) in transient problems. Spatial error estimation is based on superconvergent patch recovery of higher‐order accurate stresses and is used to guide mesh adaptivity, while the temporal error estimation is based on the assumption of linearly varying third‐order time derivatives of the displacement field and is used to adjust the time step size for the HHT‐α variant of the Newmark direct numerical integration method. Spatial adaptivity of the mesh is performed using a hierarchical h‐refinement scheme that is efficiently implemented using a structured version of finite element mesh superposition. This particular spatial adaptivity scheme is extremely fast and consequently makes it feasible to repeatedly update both the mesh and the time increment as required in an adaptive transient analysis. This work represents the initial effort in applying this type of spatial adaptivity to transient problems. Three example problems are given to demonstrate the performance characteristics of the s‐adaptive procedure. Copyright © 2005 John Wiley & Sons, Ltd.     

An embedded statistical method for coupling molecular dynamics and finite element analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with finite element (FE) nodes at their common interface, necessarily requiring that the FE mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modelling to two‐dimensional material domains due to difficulties in simulating full three‐dimensional material processes. In the present work, a new approach to MD–FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and FE nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. Thus, the method lends itself for use with any FEM or MD code. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three‐dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied. Published in 2009 by John Wiley & Sons, Ltd.     

Ultrasonic wave characteristics of a monolayer graphene on silicon substrate

The present work deals with an ultrasonic type of wave propagation characteristics of monolayer graphene on silicon (Si) substrate. An atomistic model of a hybrid lattice involving a hexagonal lattice of graphene and surface atoms of diamond lattice of Si is developed to identify the carbon-silicon bond stiffness. Properties of this hybrid lattice model is then mapped into a nonlocal continuum framework. Equivalent force constant due to Si substrate is obtained by minimizing the total potential energy of the system. For this equilibrium configuration, the nonlocal governing equations are derived to analyze the ultrasonic wave dispersion based on spectral analysis. From the present analysis we show that the silicon substrate affects only the flexural wave mode. The frequency band gap of flexural mode is also significantly affected by this substrate. The results also show that, the silicon substrate adds cushioning effect to the graphene and it makes the graphene more stable. The analysis also show that the frequency bang gap relations of in-plane (longitudinal and lateral) and out-of-plane (flexural) wave modes depends not only on the y-direction wavenumber but also on nonlocal scaling parameter. In the nonlocal analysis, at higher values of the y-directional wavenumber, a decrease in the frequency band gap is observed for all the three fundamental wave modes in the graphene–silicon system. The atoms movement in the graphene due to the wave propagation are also captured for all the tree fundamental wave modes. The results presented in this work are qualitatively different from those obtained based on the local analysis and thus, are important for the development of graphene based nanodevices such as strain sensor, mass and pressure sensors, atomic dust detectors and enhancer of surface image resolution that make use of the ultrasonic wave dispersion properties of graphene.     

Numerical simulations of strain localization in inelastic solids using mesh‐free methods

In this paper, a comprehensive account on using mesh‐free methods to simulate strain localization in inelastic solids is presented. Using an explicit displacement‐based formulation in mesh‐free computations, high‐resolution shear‐band formations are obtained in both two‐dimensional (2‐D) and three‐dimensional (3‐D) simulations without recourse to any mixed formulation, discontinuous/incompatible element or special mesh design. The numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi‐uniform, which, to a large extent, relieves the mesh alignment sensitivity that finite element methods suffer. Moreover, a simple h‐adaptivity procedure is implemented in the explicit calculation, and by utilizing a mesh‐free hierarchical partition of unity a spectral (wavelet) adaptivity procedure is developed to seek high‐resolution shear‐band formations. Moreover, the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine‐scale finite element simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

Elastic crack propagation model for crystalline solids using a self-consistent coupled atomistic–continuum framework

Deformation and failure processes of crystalline materials are governed by complex phenomena at multiple scales. It is necessary to couple these scales for physics-based modeling of these phenomena, while overcoming limitations of modeling at individual scales. To address this issue, this paper develops self-consistent elastic constitutive and crack propagation relations of crystalline materials containing atomic scale cracks, from observations made in a concurrent multi-scale simulation system coupling atomistic and continuum domain models. The concurrent multi-scale model incorporates a finite temperature atomistic region containing the crack, a continuum region represented by a self-consistent crystal elasticity constitutive model, and a handshaking interphase region. Atomistic modeling is done by the molecular dynamics code LAMMPS, while continuum modeling is conducted by the finite element method. For single crystal nickel a nonlinear and nonlocal crystal elasticity constitutive relation is derived, consistent with the atomic potential function. An efficient, staggered solution scheme with parallel implementation is designed for the coupled problem. The atomistic–continuum coupling is achieved by enforcing geometric compatibility and force equilibrium in the interphase region. Quantitative analyses of the crack propagation process focuses on size dependence, strain energy release rate, crack propagation rate and degradation of the local stiffness. The self-consistent constitutive and crack propagation relations, derived from the concurrent model simulation results are validated by comparing results from the concurrent and full FE models. Excellent accuracy and enhanced efficiency are observed in comparison with pure MD and concurrent model results.     

Goal-oriented error estimation and mesh adaptivity in 3d elastoplasticity problems

We propose a 3D adaptive method for plasticity problems based on goal-oriented error estimation, which computes the error with respect to a prescribed quantity of interest. It is a dual-based scheme which requires an adjoint problem. The computed element-wise errors at each load/displacement increment are utilized for the mesh adaptivity purpose. Mesh adaptivity procedure is performed based on refinement and coarsening by introducing hanging nodes in quadrilateral and hexahedral elements in 2d and 3d, respectively. Several numerical simulations are investigated and the results are compared with available analytical solutions, existing experimental data and results of mesh adaptivity based on other conventional error estimation methods.     

A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

We describe how wavelets constructed out of finite element interpolation functions provide a simple and convenient mechanism for both goal‐oriented error estimation and adaptivity in finite element analysis. This is done by posing an adaptive refinement problem as one of compactly representing a signal (the solution to the governing partial differential equation) in a multiresolution basis. To compress the solution in an efficient manner, we first approximately compute the details to be added to the solution on a coarse mesh in order to obtain the solution on a finer mesh (the estimation step) and then compute exactly the coefficients corresponding to only those basis functions contributing significantly to a functional of interest (the adaptation step). In this sense, therefore, the proposed approach is unified, since unlike many contemporary error estimation and adaptive refinement methods, the basis functions used for error estimation are the same as those used for adaptive refinement. We illustrate the application of the proposed technique for goal‐oriented error estimation and adaptivity for second and fourth‐order linear, elliptic PDEs and demonstrate its advantages over existing methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models

Nonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality, they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy, mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by $c\delta^{−2}$ (where $c$ is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.     

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

In this paper, we present an adaptive level set method for motion of high codimensional objects (e.g., curves in three dimensions). This method uses only two (or a few fixed) levels of meshes. A uniform coarse mesh is defined over the whole computational domain. Any coarse mesh cell that contains the moving object is further divided into a uniform fine mesh. The coarse‐to‐fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. Refinement and coarsening (removing the fine mesh within a coarse grid cell) are performed dynamically during the evolution. In this adaptive method, the computation is localized mostly near the moving objects; thus, the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high‐order numerical methods. This method is examined by numerical examples of moving curves and applications to dislocation dynamics simulations. This two‐level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57 :891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.     

A cohesive finite element for quasi-continua

In this paper, a cohesive finite element method (FEM) is proposed for a quasi-continuum (QC), i.e. a continuum model that utilizes the information of underlying atomistic microstructures. Most cohesive laws used in conventional cohesive FEMs are based on either empirical or idealized constitutive models that do not accurately reflect the actual lattice structures. The cohesive quasi-continuum finite element method, or cohesive QC-FEM in short, is a step forward in the sense that: (1) the cohesive relation between interface traction and displacement opening is now obtained based on atomistic potentials along the interface, rather than empirical assumptions; (2) it allows the local QC method to simulate certain inhomogeneous deformation patterns. To this end, we introduce an interface or discontinuous Cauchy–Born rule so the interfacial cohesive laws are consistent with the surface separation kinematics as well as the atomistically enriched hyperelasticity of the solid. Therefore, one can simulate inhomogeneous or discontinuous displacement fields by using a simple local QC model. A numerical example of a screw dislocation propagation has been carried out to demonstrate the validity, efficiency, and versatility of the method. An erratum to this article can be found at     

Stabilized nonlocal model for dispersive wave propagation in heterogeneous media

In this paper, a general-purpose computational model for dispersive wave propagation in heterogeneous media is developed. The model is based on the higher-order homogenization with multiple spatial and temporal scales and the C⁰-continuous mixed finite element approximation of the resulting nonlocal equations of motion. The proposed nonlocal Hamilton principle leads to the stable discrete system of equations independent of the mesh size, unit cell domain and the excitation frequency. The method has been validated for plane harmonic analysis and for transient wave motion insemi-infinite domain with various microstructures.This work was supported by the Sandia National Laboratories under Contract DE-AL04–94AL8500, the Office of Naval Research through grant number N00014–97–1-0687, and the Japan Society for the Promotion of Science under contract number Heisei11-nendo 06542.     

Adaptive vs. non‐adaptive strategies for the computation of optical flow

Confident adaptive algorithms are described, evaluated, and compared with other algorithms that implement the estimation of motion. A Galerkin finite element adaptive approach is described for computing optical flow, which uses an adaptive triangular mesh in which the resolution increases where motion is found to occur. The mesh facilitates a reduction in computational effort by enabling processing to focus on particular objects of interest in a scene. Compared with other state‐of‐the‐art methods in the literature our adaptive methods show only motion where main movement is known to occur, indicating a methodological improvement. The mesh refinement, based on detected motion, gives an alternative to methods reported in the literature, where the adaptation is usually based on a gradient intensity measure. A confidence is calculated for the detected motion and if this measure passes the threshold then the motion is used in the adaptive mesh refinement process. The idea of using the reliability hypothesis test is straightforward. The incorporation of the confidence serves the purpose of increasing the optical flow determination reliability. Generally, the confident flow seems most consistent, accurate and efficient, and focuses on the main moving objects within the image. © 2006 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 16, 35–50, 2006     

rp-adaptation for compressible flows

We present a novel rp-adaptation strategy for high-fidelity simulations of compressible inviscid flows with shocks. The mesh resolution in regions of flow discontinuities is increased by using a variational optimizer to r-adapt the mesh and cluster degrees of freedom there. In regions of smooth flow, we locally increase or decrease the local resolution through increasing or decreasing the polynomial order of the elements, respectively. This dual approach allows us to take advantage of the strengths of both methods for best computational performance, thereby reducing the overall cost of the simulation. The adaptation workflow uses a sensor for both discontinuities and smooth regions that is cheap to calculate, but the framework is general and could be used in conjunction with other feature-based sensors or error estimators. We demonstrate this proof-of-concept using two geometries in transonic and supersonic flow regimes. The method has been implemented in the open-source spectral/hp element framework Nektar++, and adaptivity is performed by its dedicated high-order mesh generation tool NekMesh. The results show that the proposed rp-adaptation methodology is a reasonably cost-effective way of improving simulation accuracy.     

Determining material constants in nonlocal micromorphic theory through phonon dispersion relations

This paper aims to determine the material constants in nonlocal micromorphic theory through phonon dispersion relations. The nonlocal constitutive relations for isotropic elastic solids in micromorphic theory are derived. The numerical algorithm to determine the material constants is presented. The material constants for single crystal silicon and diamond in local and nonlocal micromorphic theory are determined by fitting the phonon dispersion relations obtained from atomistic calculations or experimental measurements. The solutions of the local micromorphic theory can only fit those dispersion relations well in half Brillouin zone, while it is found that the nonlocal solutions are in very good agreement with the lattice dynamic results throughout the entire Brillouin zone. The applicability of nonlocal micromorphic theory is discussed.