Tuesday September 28 at 3:00 PM to 4:00 PM in Zoom
Numerical Analysis Seminar
Longfei Li, University of Louisiana at Lafayette, Numerical methods for fourth-order PDEs on overlapping grids with application to Kirchhoff-Love plates
We propose novel numerical methods for solving a class of high-order hyperbolic PDEs on general geometries, which involve 2nd-order derivatives in time and up-to 4th-order derivatives in space. These PDEs are widely used in modeling thin-walled elastic structures such as beams, plates and shells, etc. High-order spatial derivatives together with general geometries bring a number of challenges for any numerical methods. In this work, we resolve these challenges by discretizing the spatial derivatives in domains with general geometries using the composite overlapping grid method. The discretization on overlapping grids requires numerical interpolations to couple solutions on different component grids. However, the presence of interpolation equations causes weak numerical instability. To address this, a fourth-order hyper-dissipation term is included for stabilization. Investigation of incorporating the hyper-dissipation term into several time-stepping schemes for solving the semi-discrete system leads to the development of a series of algorithms. Carefully designed numerical experiments and two benchmark problems concerning the Kirchhoff-Love plate model are presented to demonstrate the accuracy and efficiency of our approaches. This work shows that finite difference methods on overlapping grids are well-suited for solving high-order PDEs in complex domains for realistic applications.
Wednesday September 29 at 2:00 PM to 3:00 PM in Zoom
Geometry and Topology Seminar
Irina Kogan, NC State, Group Actions, Invariants, and Applications
I will overview some important milestones in the development of the Invariant Theory from its classical times to modern days, leading into a discussion of the current progress in theory, computation, and applications. The highlights include Hilbert's basis theorem, geometric invariant theory, differential algebra of invariants, the moving frame approach, Lie's work on symmetries of differential equations, and Tresse's theorem. I will try to emphasize the relationships between the algebraic and differential theories.
This talk is a teaser to the special topic class I will offer in Spring 2022. Students are encouraged to join.
Zoom invitation is sent to the geometry and topology seminar list. If you are not on the list, please, contact Peter McGrath host to get the link.
Wednesday September 29 at 3:00 PM to 4:00 PM in Zoom
Differential Equations Seminar
Riccardo Sacco, Politecnico di Milano, Italy, A Nonlinear Heterogeneous Transmission Model for Organic Polymer Retinal Prostheses
In this talk we propose a model for the simulation of retinal prostheses based on the use of organic polymer nanoparticles (NP). The model consists of a nonlinearly coupled system of elliptic partial differential equations accounting for:
(1) light photoconversion into free charged carriers in the NP bulk;
(2) charge transport in the NP bulk due to drift and diffusion forces;
(3) net charge recombination in the NP bulk due to the balance between light
absorption and particle-particle recombination;
(4) electron-driven molecular oxygen reduction and capacitive coupling at the NP-solution interface;
(5) ion electrodiffusion in the solution bulk;
(6) capacitive and conductive coupling across the neuronal membrane.
Model dependent variables are represented by the electric potential, the number densities of photogenerated charge carriers in the NP bulk and the molar densities of moving ions in the aqueous solution surrounding the NP. The physical mechanisms (4) and (6) are mathematically expressed by nonlinear transmission conditions across interfaces which in turn give rise to the nonlinear coupling among the dependent variables throughout the whole computational domain. The proposed model is solved in stationary conditions and in one spatial dimension by resorting to a solution map which is a modification of the Gummel Decoupled algorithm conventionally used in inorganic semiconductor simulation. System discretization is conducted using the Finite Element Method, with stabilization terms to prevent spurious unphysical oscillations in the electric potential and ensure positivity of carrier and ion concentrations. Model predictions suggest that the combined effect of NP polarization and resistivity of the NP-neuron interface results in neuron depolarization and supports the efficacy of organic NPs in the design and development of retinal prostheses.
Zoom link: https://ncsu.zoom.us/j/8027642791?pwd=d1lNaWZyUW4zeUFvaTA5VmlsTWtjdz09