Reduced-order computational homogenization for hyperelastic media using gradient based sensitivity analysis of microstructures

Abstract

We propose an algorithm for the computational homogenization of locally periodic hyperelastic structures undergoing large deformations due to external quasi-static loading. The algorithm performs clustering of macroscopic deformations into subsets called "centroids", and, as a new ingredient, approximates the homogenized coefficients using sensitivity analysis of micro-configurations with respect to the macroscopic deformation. The novel "model-order reduction" approach significantly reduces the number of microscopic problems that must be solved in nonlinear simulations, thereby accelerating the overall computational process. The degree of reduction can be controlled by a user-defined error tolerance parameter. The algorithm is implemented in the finite element framework SfePy, and its performance effectiveness is demonstrated using two-dimensional test examples, when compared with solutions obtained by the proper orthogonal decomposition method, and by the full "FE-square" simulations. Extensions beyond the present implementations and the scope of tractable problems are discussed.

Figures (25)

• Figure 1: Inner/outer iterations within one "loading-time" step.
• Figure 2: Updating microscopic configurations using the characteristic responses and $\boldsymbol{\omega}^{ij}$ and the gradient of the macroscopic displacement increments $\delta\boldsymbol{u}^0$.
• Figure 3: Calculation of weights $w^{il}$ for the case of three overlapping centroids.
• Figure 4: Approximations of ${\boldsymbol{\mathcal{A}}}$, ${\boldsymbol{\mathcal{S}}}$ between two centroids $O^1 = O([0, 0, 0], 0.01)$, $O^2 = O([0.015, 0, 0], 0.01)$ with $\rho = 0.01$. Comparison of the direct two-scale simulation (FE$^2$) -- black dotted line, approximation using coefficients sensitivities (CSA) -- blue solid line, and approximation without sensitivity information (CnoSA, $\delta\mathring{{\boldsymbol{\mathcal{A}}}}=\boldsymbol{0}$, $\delta\mathring{{\boldsymbol{\mathcal{S}}}}=\boldsymbol{0}$) -- red dashed curve.
• Figure 5: Approximations of ${\boldsymbol{\mathcal{A}}}$, ${\boldsymbol{\mathcal{S}}}$ between two centroids $O^2 = O([0, 0, 0], 0.001)$, $O^1 = O([0.0015, 0, 0], 0.001)$ with $\rho = 0.001$. Comparison of the direct numerical simulation (FE$^2$) -- black dotted line and approximation using coefficients sensitivities (CSA) -- blue solid line.