Summation rules for a fully nonlocal energy-based quasicontinuum method

Abstract

a b s t r a c tThe quasicontinuum (QC) method coarse-grains crystalline atomic ensembles in order to bridge the scales from individual atoms to the micro-and mesoscales. A crucial cornerstone of all QC techniques, summation or quadrature rules efficiently approximate the thermodynamic quantities of interest. Here, we investigate summation rules for a fully nonlocal, energy-based QC method to approximate the total Hamiltonian of a crystalline atomic ensemble by a weighted sum over a small subset of all atoms in the crystal lattice. Our formulation does not conceptually differentiate between atomistic and coarsegrained regions and thus allows for seamless bridging without domain-coupling interfaces. We review traditional summation rules and discuss their strengths and weaknesses with a focus on energy approximation errors and spurious force artifacts. Moreover, we introduce summation rules which produce no residual or spurious force artifacts in centrosymmetric crystals in the large-element limit under arbitrary affine deformations in two dimensions (and marginal force artifacts in three dimensions), while allowing us to seamlessly bridge to full atomistics. Through a comprehensive suite of examples with spatially non-uniform QC discretizations in two and three dimensions, we compare the accuracy of the new scheme to various previous ones. Our results confirm that the new summation rules exhibit significantly smaller force artifacts and energy approximation errors. Our numerical benchmark examples include the calculation of elastic constants from completely random QC meshes and the inhomogeneous deformation of aggressively coarse-grained crystals containing nano-voids. In the elastic regime, we directly compare QC results to those of full atomistics to assess global and local errors in complex QC simulations. Going beyond elasticity, we illustrate the performance of the energy-based QC method with the new second-order summation rule by the help of nanoindentation examples with automatic mesh adaptation. Overall, our findings provide guidelines for the selection of summation rules for the fully nonlocal energy-based QC method.

The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids

“…Since A 0 (x) is typically not known for heterogeneous materials, the ansatz of FE-HMM is to approximate the virtual work expression at point x K l in the semidiscrete form (10) by another bilinear form using the known microheterogeneous elasticity tensor A , see (11). According to this approximation, the solution u h K δ is obtained on microsampling domains K δ = x K l + δ [−1/2, +1/2] d , δ ≥ , which are each centered at the quadrature points x K l of K, l = 1, .…”

Fatigue crack growth characteristics of Fe and Ni under cyclic loading using a quasi-continuum method

“…Dow and Baskes have proposed the EAM potential, which is a multibody potential [20–21]. In this study, the temperature used in the simulations was 0 K for all cases.…”

The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE2 method

“…Since A 0 (x) is typically not known for heterogeneous materials, the ansatz of FE-HMM is to approximate the virtual work expression at point x K l in the semidiscrete form (10) by another bilinear form using the known microheterogeneous elasticity tensor A , see (11). According to this approximation, the solution u h K δ is obtained on microsampling domains…”