Numerical material representation using proper orthogonal decomposition and diffuse approximation

From numerical point of view, analysis and optimization in computational material engineering require efficient approaches for microstructure representation. This paper develops an approach to establish an image-based interpolation model in order to efficiently parameterize microstructures of a representative volume element (RVE), based on proper orthogonal decomposition (POD) reduction of density maps (snapshots). When the parameters of the RVE snapshot are known a priori, the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters. Otherwise, a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients. We showcase the approach and its potential applications by considering a set of two-phase composite snapshots. The choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization. The constant increase of computing power coupled with easier-than-ever access to high performance computing platforms enables the computational investigation of materials at the microscopic level: microstructure generation and model-ing [1,2], material property prediction and evaluation [3–5], multi-scale analysis [6–9], and within a stochastic framework to include the effects of the input uncertainties at the material level [10]. At the same time, the progress made in the field of material science allows us to control the material microstructure composition to an unprecedented extent [11,12]. Recently, image-based microstructure modeling and analysis have attracted the interest of more and more researchers. One category of research employs voxel-based finite element models using a mesh that is automatically built by converting each voxel into a finite element [13–15]. Given the high computational cost of a voxel-based approach, another theme of research generates image-based microscopic models by incorporating level-sets and the extended finite element method (XFEM) [16,17]. A comparison of the two approaches has been made in a recent work [18]. Thanks to the proposed analysis approaches, the material constitutive behavior can be predicted by imaged-based numerical models. However, access to microstructural images is economically expensive by experiments or time-consuming by numerical simulations [19–21]. Therefore, there is a great demand for an economical and efficient approach to generate microstructure images.