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.

Models of plate tectonics with the Lattice Boltzmann Method

Modern geodynamics is based on the study of a large set of models,with the variation of many parameters,whose analysis in the future will require Machine Learning to be analyzed.We introduce here for the first time how a formulation of the Lattice Boltzmann Method capable of modeling plate tectonics,with the introduction of plastic non-linear rheology,is able to reproduce the breaking of the upper boundary layer of the convecting mantle in plates.Numerical simulation of the earth’s mantle and lithospheric plates is a challenging task for traditional methods of numerical solution to partial differential equations(PDE’s)due to the need to model sharp and large viscosity contrasts,temperature dependent viscosity and highly nonlinear rheologies.Nonlinear rheologies such as plastic or dislocation creep are important in giving mantle convection a past history.We present a thermal Lattice Boltzmann Method(LBM)as an alternative to PDE-based solutions for simulating time-dependent mantle dynamics,and demonstrate that the LBM is capable of modeling an extremely nonlinear plastic rheology.This nonlinear rheology leads to the emergence plate tectonic like behavior and history from a two layer viscosity model.These results demonstrate that the LBM offers a means to study the effect of highly nonlinear rheologies on earth and exoplanet dynamics and evolution.

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.