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

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.

💡 Research Summary

This paper introduces a novel model‑order reduction (MOR) strategy for the computational homogenization of locally periodic hyperelastic media undergoing large, quasi‑static deformations. The authors call the method “Clustering and Sensitivity‑based Approximation” (CSA). The central idea is to partition the space of macroscopic deformation states into a small number of clusters using K‑means clustering. Each cluster is represented by a centroid deformation, and a single representative microstructure (RVE) is solved on‑the‑fly for that centroid. In addition to the homogenized stress and tangent elasticity tensor obtained from the RVE, the method computes the sensitivities of these quantities with respect to the macroscopic deformation gradient (∂S/∂F and ∂A/∂F). These sensitivities are then used to construct a linear mapping that corrects the homogenized quantities for any deformation within the cluster. The correction is expressed in terms of the polar decomposition of the deformation gradient, allowing a clear separation of stretch and rotation components.

The algorithm proceeds as follows: (1) during each macro‑scale Newton iteration the current deformation gradient is evaluated; (2) the nearest centroid is identified; (3) if the centroid’s micro‑problem has not yet been solved, it is solved once, yielding stress, elasticity tensor, and their sensitivities; (4) the difference between the current deformation and the centroid deformation is measured via the stretch tensor; (5) the linear sensitivity‑based correction is applied to obtain approximated homogenized coefficients; (6) these coefficients are inserted into the macroscopic equilibrium equations and the global system is solved; (7) the process repeats until convergence, with optional reclustering when a user‑defined error tolerance is exceeded.

Because the method requires solving a micro‑problem only for each centroid (rather than for every quadrature point as in classical FE² or for a dense offline snapshot set as in POD), the number of expensive microscopic solves is dramatically reduced. Moreover, the approach does not rely on an extensive offline stage; the representative microstructures are generated “on‑the‑fly” as the macro‑deformation evolves, which is especially advantageous when additional parameters such as design variables or internal state variables are introduced (e.g., in functionally graded material design or inelastic modeling).

The theoretical development is presented for a compressible neo‑Hookean material, but the derivations are generic and can be extended to more complex hyperelastic models (Mooney‑Rivlin, Ogden, Yeoh). The paper details the weak form, the updated Lagrangian linearization, the Lie derivative of the Cauchy stress, and the expression of the tangent stiffness tensor in terms of the deformation gradient. The sensitivity analysis is performed using the “design velocity field” concept borrowed from shape optimization, which provides the required derivatives of the homogenized quantities with respect to the macroscopic deformation.

Numerical experiments are carried out on two‑dimensional benchmark problems. The CSA results are compared against full FE² simulations and a proper orthogonal decomposition (POD) reduced model. The CSA achieves relative errors in stress and strain below 2 % while reducing the number of microscopic solves by 70–85 % relative to FE² and by 60–70 % relative to POD. The authors also address the well‑known issue of abrupt changes in material properties when switching clusters by employing a “cluster freezing” technique, which improves the robustness of the Newton iterations and reduces the number of global iterations by roughly 30 % compared with the standard FE² approach.

Implementation is performed within the open‑source finite element framework SfePy, and the source code is made publicly available. The authors discuss possible extensions, including the treatment of functionally graded materials (where the microstructure itself varies with position) and the incorporation of internal variables for inelastic behavior. They acknowledge that the linear sensitivity correction is a first‑order approximation; for problems with very strong nonlinearity or large jumps in deformation, higher‑order sensitivity terms or adaptive clustering may be required.

In summary, the paper presents a practical, computationally efficient MOR technique that combines clustering of macroscopic deformation states with sensitivity‑based linear updates of homogenized coefficients. By eliminating the need for an extensive offline database and by drastically cutting the number of micro‑scale solves, the CSA method offers a promising route for large‑scale, nonlinear multiscale simulations, especially in contexts where design optimization or material grading is involved.