The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill–Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss’ and Voigt’s bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE calculations is presented. The performances of the proposed schemes are illustrated via a series of twoand three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks. [DOI: 10.1115/1.4034024]

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Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

A method that utilises atomic trajectories and velocities from molecular dynamics simulations has been suitably adapted and employed for the implicit calculation of the phonon dispersion curves of graphene Classical potentials widely used in the literature were employed Their performance was assessed for each individual phonon branch and the overall phonon dispersion, using available inelastic x-ray scattering data The method is promising for systems with large scale periodicity, accounts for anharmonic effects and non-bonding interactions with a general environment, and it is applicable under finite temperatures The temperature dependence of the phonon dispersion curves has been examined with emphasis on the doubly degenerate Raman active Γ-E2g phonon at the zone centre, where experimental results are available The potentials used show diverse behaviour The Tersoff-2010 potential exhibits the most systematic and physically sound behaviour in this regard, and gives a first-order temperature coefficient of χ = -005 cm(-1)/K for the Γ-E2g shift in agreement with reported experimental values

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Phonon properties of graphene derived from molecular dynamics simulations.

An adaptive multiscale phase field method for brittle fracture

Biodegradable magnesium alloy stents (MAS) are a promising solution for long-term adverse events caused by interactions between vessels and permanent stent platforms of drug eluting stents. However, the existing MAS showed severe lumen loss after a few months: too short degradation time may be the main reason for this drawback. In this study, a new design concept of MAS was proposed and a shape optimization method with finite element analysis was applied on two-dimensional (2D) stent models considering four different magnesium alloys: AZ80, AZ31, ZM21, and WE43. A morphing procedure was utilized to facilitate the optimization. Two experiments were carried out for a preliminary validation of the 2D models with good results. The optimized designs were compared to an existing MAS by means of three-dimensional finite element analysis. The results showed that the final optimized design with alloy WE43, compared to the existing MAS, has an increased strut width by approximately 48%, improved safety properties (decreased the maximum principal stress after recoil with tissue by 29%, and decreased the maximum principal strain during expansion by 14%) and improved scaffolding ability (increased by 24%). Accordingly, the degradation time can be expected to extend. The used methodology provides a convenient and practical way to develop novel MAS designs.

Finite element shape optimization for biodegradable magnesium alloy stents.

Multiobjective design optimisation of coronary stents.

An extended multiscale finite element method is developed for small-deformation elasto-plastic analysis of periodic truss materials. The base functions constructed numerically are employed to establish the relationship between the macroscopic displacement and the microscopic stress and strain. The unbalanced nodal forces in the micro-scale of unit cells are treated as the combined effects of macroscopic equivalent forces and microscopic perturbed forces, in which macroscopic equivalent forces are used to solve the macroscopic displacement field and microscopic perturbed forces are used to obtain the stress and strain in the micro-scale to make sure the correctness of the results obtained by the downscale computation in the elastic-plastic problems. Numerical examples are carried out and the results verify the validity and efficiency of the developed method by comparing it with the conventional finite element method.

Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials

Shape optimization of coronary artery stent based on a parametric model

A new multiscale computational method is developed for the elasto-plastic analysis of heterogeneous continuum materials with both periodic and random microstructures. In the method, the multiscale base functions which can efficiently capture the small-scale features of elements are constructed numerically and employed to establish the relationship between the macroscopic and microscopic variables. Thus, the detailed microscopic stress fields within the elements can be obtained easily. For the construction of the numerical base functions, several different kinds of boundary conditions are introduced and their influences are investigated. In this context, a two-scale computational modeling with successive iteration scheme is proposed. The new method could be implemented conveniently and adopted to the general problems without scale separation and periodicity assumptions. Extensive numerical experiments are carried out and the results are compared with the direct FEM. It is shown that the method developed provides excellent precision of the nonlinear response for the heterogeneous materials. Moreover, the computational cost is reduced dramatically.

H. W. Zhang

Papers

Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials

Shape optimization of coronary artery stent based on a parametric model

A new multiscale computational method for elasto-plastic analysis of heterogeneous materials

Boundary recovery for 3D Delaunay triangulation

Phonon dispersion analysis of carbon nanotubes based on inter-belt model and symplectic solution method