Modeling Approximations of Microstructure Solutions Using Weak Form-Based Collocation Method
Cymantha Blackmon
Co-Presenters: Individual Presentation
College: The Dorothy and George Hennings College of Science, Mathematics and Technology
Major: Computational Science & Engineering - STEM 5 Year B.S./M.S.
Faculty Research Mentor: Ensela Mema
Abstract:
PURPOSEThis research investigates weak-form mesh-free collocation methods for solutions to variational models of microstructures through a kernel-based approach within the Gaussian process (GP) framework. Within material science, microstructures are structural features ranging between atomic and macroscopic scales, often playing critical roles in the performance of engineering materials. Modeling the behavior of and formulation of microstructures is inherently complex, relying on high-fidelity physical models, such as partial differential equations (PDEs) or variational problems, one that aims to minimize a certain functional quantity satisfying boundary conditions and other external constraints.METHODSThis study focuses on a Gaussian process (GP) combined with deep kernel learning capable of approximating the intended solutions for common one-dimensional problems. GPs offer a non-parametric, assumption-free approach to approximating complex functions while reducing the computational cost. The proposed method discretizes the variational domain into collocation points, including those within the domain and along the boundary. The solution is approximated using a weighted sum of radial basis functions (RBFs). The Gauss-Legendre quadrature rule is considered for numerical integration within the sub-intervals described by the collocation points to compute the variational energy. This approach represents the variational form as an infinite series whose definite integral can be defined as a sum of the integrand evaluated at specific points, weighted by corresponding quadrature weights. The stability and efficiency of the method were examined through computation of its convergence properties and L2-norm errors and validated against the exact solution.SIGNIFICANCE / CONCLUSIONFindings within this research demonstrate the potential to continue to achieve accuracy and address challenges associated with the computational implementation of higher-dimensional variational problems in material sciences.