Variational multiscale enrichment method for dynamic response of hyperelastic materials at finite deformation

TL;DR

The paper addresses dynamic wave propagation in architected hyperelastic materials where short wavelengths defy scale separation. It introduces a variational multiscale enrichment (VME) with a two-scale additive decomposition ${\\mathbf{u}} = {\\mathbf{u}}^{\\mathrm{c}} + \\sum_{\\alpha}{\\cal H}(\\Omega_{\\alpha}) {\\mathbf{u}}^{\\mathrm{f},\\alpha}$ to capture micro-inertia and nonlinear effects, and derives coupled coarse- and fine-scale PDEs solved via an operator-split time integration. Three time-stepping schemes are developed: EE-CDM, EE-SSM, and EI-SSM, with stability analyses and a two-scale interpolation framework for unit-cell discretization within coarse-scale patches. Numerical tests in 1D Neo-Hookean settings show the VME framework accurately reproduces dispersion, attenuation, and wave steepening, while dissipative schemes suppress spurious high-frequency oscillations and coarse-scale refinement improves accuracy. The approach provides a scalable foundation for predicting dynamic responses in architected materials and offers a path toward higher-dimensional implementations and data-driven reduced-order surrogates.

Abstract

In this manuscript, we extend the variational multiscale enrichment (VME) method to model the dynamic response of hyperelastic materials undergoing large deformations. This approach enables the simulation of wave propagation under scale-inseparable conditions, including short-wavelength regimes, while accounting for material and geometric nonlinearities that lead to wave steepening or flattening. By employing an additive decomposition of the displacement field, we derive multiscale governing equations for the coarse- and fine-scale problems, which naturally incorporate micro-inertial effects. The framework allows the discretization of each unit cell with a patch of coarse-scale elements, which is essential to accurately capture wave propagation in short-wavelength regimes. An operator-split procedure is used to iteratively solve the semi-discrete equations at both scales until convergence is achieved. The coarse-scale problem is integrated explicitly, while the fine-scale problem is solved using either explicit or implicit time integration schemes, including both dissipative and non-dissipative methods. Numerical examples demonstrate that multiscale dissipative schemes effectively suppress spurious oscillations. The multiscale framework was applied to investigate how material and geometric nonlinearities, along with elastic stiffness contrast in heterogeneous microstructures, influence key wave characteristics such as dispersion, attenuation, and steepening. This multiscale computational framework provides a foundation for studying the dynamic response of architected materials.

Figures (7)

• Figure 1: Evolution of displacement with time predicted using the explicit central difference methods for the homogeneous material case with (a) DNS and (b) VME methods.
• Figure 2: Averaged stretch over the element for the homogeneous material case in DNS.
• Figure 3: Evolution of displacement profile with time for homogeneous microstructures using (a) EE-SSM, (b) EI-SSM in VME simulations. In both figures, the colored solid lines correspond to VME results, while the dotted lines in (a) correspond to the EE-SSM-based VME with coarse-grid approximation alone, and in (b) correspond to DNS, at the same non-dimensional times as the colored solid lines.
• Figure 4: Evolution of displacement profile with time using EE-SSM for VME and explicit sub-step method for DNS simulations of heterogeneous microstructures (a) $C=2$, (b) $C=0.2$. In both figures, the colored solid lines correspond to VME results, and the black dashed lines correspond to the DNS results.
• Figure 5: Evolution of displacement profile with time using EE-SSM for VME simulations for a heterogeneous microstructure with $C=0.01$ and $\mathrm{CFL} = 0.1$. The dashed black lines show DNS results at the same non-dimensional times using the explicit sub-step method.