[Paper Review] Computational homogenization of heterogeneous media under dynamic loading

Chenchen Liu, Celia Reina|arXiv (Cornell University)|Oct 8, 2015

Composite Material Mechanics49 references3 citations

TL;DR

This paper presents a variational coarse-graining framework for computational homogenization of heterogeneous materials under dynamic loading, extending the FE² method to arbitrary loading conditions including inertia and body forces. It enables a quasi-explicit multiscale solver (QEMS) that solves macroscopic and microscopic degrees of freedom simultaneously with a single Newton-Raphson iteration per time step, accurately capturing dispersion and convergence behavior in dynamic regimes.

ABSTRACT

A variational coarse-graining framework for heterogeneous media is developed that allows for a seamless transition from the traditional static scenario to a arbitrary loading conditions, including inertia effects and body forces. The strategy is formulated in the spirit of computational homogenization methods (FE$^2$) and is based on the discrete version of Hill's averaging results recently derived by the authors. In particular, the traditional static multiscale scheme is proved here to be equivalent to a direct homogenization of the principle of minimum potential energy and to hold exactly under a finite element discretization. This perspective provides a unifying variational framework for the FE$^2$ method, in the static setting, with Dirichlet or Neumann boundary conditions on the representative volume element; and it directly manifests the approximate duality of the effective strain energy density obtained with these two types of boundary conditions in the sense of Legendre transformation. Its generalization to arbitrary loading conditions and material constitutive relations is then immediate through the incremental minimum formulation of the dynamic problem à la Radovitzky and Ortiz (1999), which, in the discrete setting, is in full analogy to the static problem. These theoretical developments are then translated into an efficient multiscale FE$^2$ computational strategy for the homogenization of a microscopic explicit dynamics scheme, with two noteworthy properties. Firstly, each time incremental problem can be solved exactly with a single Newton-Raphson iteration with a constant Hessian, regardless of the specific non-linearities or history-dependence of the micro-constituents' behavior. Secondly, the scheme concurrently solves for the microscopic and macroscopic degrees of freedom, in contrast to standard approaches based on sequential or nested minimizations.

Motivation & Objective

• Address the limitation of traditional FE² methods, which are restricted to static conditions, by extending them to dynamic loading with inertia and body forces.
• Develop a unified variational formulation that seamlessly transitions from static to dynamic scenarios using incremental minimum potential energy principles.
• Enable simultaneous solution of macroscopic and microscopic degrees of freedom, avoiding sequential or nested minimization strategies.
• Establish a direct link between the homogenized stress in the dynamic setting and the Virial stress in atomistic simulations, enhancing cross-scale consistency.
• Provide a computationally efficient multiscale solver (QEMS) that maintains accuracy across spatial and temporal discretizations

Proposed method

• Formulate the dynamic problem using an incremental minimum potential energy principle à la Radovitzky and Ortiz (1999), adapted to discrete finite element settings.
• Derive the discrete version of Hill’s averaging theorems to ensure consistency between microscopic equilibrium and macroscopic homogenization.
• Introduce a variational coarse-graining strategy that directly homogenizes the incremental potential energy, preserving the duality between Dirichlet and Neumann boundary conditions via Legendre transformation.
• Implement a quasi-explicit multiscale solver (QEMS) where each time step is solved exactly with a single Newton-Raphson iteration due to a constant Hessian matrix.
• Apply periodic boundary conditions on the representative volume element (RVE) to ensure scale separation and enforce symmetry in micro-displacement variations.
• Derive effective stress and strain energy density by taking variations of the averaged incremental potential energy with respect to macroscopic fields, yielding consistent macroscopic driving forces.

Experimental results

Research questions

• RQ1 How can the FE² method be generalized from static to dynamic loading conditions while preserving variational consistency and scale separation?
• RQ2 What is the role of inertia and body forces in the homogenization of heterogeneous media, and how can they be consistently incorporated into the multiscale framework?
• RQ3 Can a single Newton-Raphson iteration solve the coupled micro-macro problem exactly in dynamic simulations, and under what conditions does this hold?
• RQ4 How does the proposed framework relate to the Virial stress in atomistic simulations, and what does this imply for cross-scale consistency?
• RQ5 To what extent does the QEMS method accurately predict dispersion relations and convergence behavior in dynamic wave propagation problems?

Key findings

• The proposed framework generalizes the FE² method to dynamic loading by extending the variational principle of minimum potential energy to time-dependent problems with inertia and body forces.
• The method achieves exact solution of each incremental time step with a single Newton-Raphson iteration due to a constant Hessian matrix, enabling a quasi-explicit multiscale solver (QEMS).
• The effective strain energy density derived via the variational framework exhibits exact duality between Dirichlet and Neumann boundary conditions in the sense of Legendre transformation.
• The homogenized stress in the dynamic setting is mathematically equivalent to the Virial stress tensor used in atomistic simulations, revealing a fundamental analogy between continuum and atomistic homogenization.
• Numerical results for a 1D layered composite show accurate prediction of dispersion properties and convergence to the exact solution with respect to both spatial and temporal discretizations.
• The method successfully captures dynamic behavior in the frequency range permitted by the separation of length scales, validating its applicability to wave propagation in heterogeneous and metamaterials.