Parametric stress field solutions for heterogeneous materials using proper generalized decomposition

The proper generalized decomposition (PGD) method is developed for parametric solutions of full stress fields in heterogeneous materials. PGD decouples the multi-dimensional problem into a product of low-dimensional expansion with an enrichment process to approximate the field solutions. The configurations of inclusions (size, location, and material properties) and other model parameters can be included as extra-coordinates in PGD formulations to develop parametric field solutions. Numerical examples of 2D linear elastic heterogeneous materials with an inclusion varying in size, location, and stiffness properties are studied. Almost invisible differences on the full stress fields were observed between FEA and PGD approximate solutions, with the mean squared error (MSE) mostly within 0.25 for the whole stress field. The proposed PGD implementation for heterogeneous materials is able to predict the full stress fields including all localized stress concentration patterns with high accuracy. The parametric solutions from the PGD framework enable online computations of full stress fields for heterogeneous materials, providing a viable way for design optimization, uncertainty quantification, and many other real-time tasks.