Segmenting mechanically heterogeneous domains via unsupervised learning

Abstract

From biological organs to soft robotics, highly deformable materials are essential components of natural and engineered systems. These highly deformable materials can have heterogeneous material properties, and can experience heterogeneous deformations with or without underlying material heterogeneity. Many recent works have established that computational modeling approaches are well suited for understanding and predicting the consequences of material heterogeneity and for interpreting observed heterogeneous strain fields. In particular, there has been significant work toward developing inverse analysis approaches that can convert observed kinematic quantities (e.g., displacement, strain) to material properties and mechanical state. Despite the success of these approaches, they are not necessarily generalizable and often rely on tight control and knowledge of boundary conditions. Here, we will build on the recent advances (and ubiquity) of machine learning approaches to explore alternative approaches to detect patterns in heterogeneous material properties and mechanical behavior. Specifically, we will explore unsupervised learning approaches to clustering and ensemble clustering to identify heterogeneous regions. Overall, we find that these approaches are effective, yet limited in their abilities. Through this initial exploration (where all data and code are published alongside this manuscript), we set the stage for future studies that more specifically adapt these methods to mechanical data.

Appendices

Appendix A : K-means clustering result for circle inclusion with boundary conditions varying from equibiaxial to biaxial to uniaxial extension

In Fig. 6, 7, and 8, we show the results of applying our clustering pipeline to samples with equibiaxial, uniaxial in x, uniaxial in y, shear, and confined compression boundary conditions. In this Appendix, we provide an additional supplementary result where we vary the x and y displacements such that the boundary conditions fall between uniaxial and equibiaxial extension. In Fig. 10, we show the results of applying k-means clustering to a circle inclusion with these varying boundary conditions. In brief, this result shows that k-means clustering works well for a wide range of different biaxial extensions in the experimental setting. Overall, we observe that equibiaxial extension and near-equibiaxial extension boundary conditions provide the best clustering results (i.e., ARI $\ge 0.95$), while boundary conditions closer to uniaxial extension provide worse clustering results (i.e., ARI $< 0.90$). Despite the poorer performance in uniaxial extension cases, k-means still identifies the circle inclusion quite well with the lowest $ARI=0.88$. Finally, as expected, k-means fails to provide any reasonable results when the sample experiences no deformation.

Appendix B: K-means clustering result for circle inclusion with a linearly varying stiffness inclusion with boundary conditions varying from equibiaxial to biaxial to uniaxial extension

In this Appendix, we provide the clustering results from using k-means on a circle inclusion with a linearly varying stiffness. Specifically, the Young’s modulus of the circle inclusion has a value of 1.1 at the edge of the circle, and linearly increases to 10.0 at the center. And, the background of the sample has a Young’s modulus of 1.0. In Fig. 11, we show the results of applying k-means clustering to a circle inclusion with different biaxial extension boundary conditions. These results demonstrate that k-means clustering still performs well even when the inclusion has varying stiffness. However, the clustering results for the varying stiffness inclusion example (i.e., best performing $\text {ARI} = 0.92$) are slightly worse than that of single Young’s modulus inclusion (i.e., best performing $\text {ARI} = 0.96$, Appendix 1). Despite the slightly worse performance, k-means still identifies the inclusions with good results (i.e., the lowest $ARI=0.87$ excluding the no deformation case).