A Variational Neural Network Approach for Glacier Modelling with Nonlinear Rheology

We propose a mesh-free method to solve the full Stokes equation for modeling the glacier dynamics with nonlinear rheology.Inspired by the Deep-Ritz method proposed in[13],we first formulate the solution to the non-Newtonian Stokes equation as the minimizer of a variational problem with boundary constraints.Then,we approximate its solution space by a deep neural network.The loss function for training the neural network is a relaxed version of the variational form,in which penalty terms are used to present soft constraints due to mixed boundary conditions.Instead of introducing mesh grids or basis functions to evaluate the loss function,our method only requires uniform sampling from the physical domain and boundaries.Furthermore,we introduce a re-normalization technique in the neural network to address the significant variation in the scaling of real-world problems.Finally,we illustrate the performance of our method by several numerical experiments,including a 2D model with the analytical solution,the Arolla glacier model with realistic scaling and a 3D model with periodic boundary conditions.Numerical results show that our proposed method is efficient in solving the non-Newtonian mechanics arising from glacier modeling with nonlinear rheology.

Structural and Rheological Properties of PP/EPR/PE Alloys

Linear and non-linear rheological properties were examined for in-reactor alloy samples composed of isotactic polypropylene(iPP),ethylene-propylene rubber(EPR),and polyethylene(PE).The samples were prepared through two-or three-stage sequential polymerization,where iPP,as the majority,was synthesized in the first stage,and either EPR or PE,as the minority,was synthesized in the second or third stage.Above Tm of both PE and iPP,the linear viscoelasticity shows two relaxation processes,where the fast and slow processes are attributable to the relaxation of the iPP component and that of the EPR/PE domains,respectively.Both the damping behavior upon applying the step-strain measurements and the shear-thinning behavior under the shear-rate sweep measurements are strongly related to the different degrees of nonlinearity of the two relaxation processes.