Analysis Seminar: Analysis of a two membrane thermoelastic problem

Speaker: Nat Fink-Humes, GMU

Title:  Analysis of a two membrane thermoelastic problem

Abstract:   In the field of precision plastic manufacturing, mould construction is largely heuristic. The manufacturing process involves heating a plastic sheet, or membrane, which is then mechanically forced onto the mould. A complicated thermal exchange occurs over the contact region of the membrane and mould during the "cooling" phase, over which both mould and membrane experience thermal deformations prior to setting of the final shape. In 2022 Alphonse, Rautenberg, and Rodrigues analyzed both a stationary and evolutionary model of this system under the assumptions requiring the mould to be both colder and significantly more rigid than the membrane. In this talk we examine a further generalized stationary model which removes both assumptions on the mould. To capture the complex multiphysics involved our model couples two stationary heat equations connected over a contact region with an interdependent pair of contact problems. These problems govern the mould’s and product membrane’s shapes. The system presents difficult to resolve interdependencies as fixing any single equation (temperature, mould, or membrane) behaves like an Ouroboros, eventually leading to dependence upon itself. This presents theoretical and numerical difficulties unresolved in general settings. Mathematically, we identify weak solutions to the model when presented as a system of interdependent, quasi-variational inequalities and elliptic equations using a connection to the N-membrane problem, standard regularization, and fixed-point methods. Further, we demonstrate sufficient conditions for regular solutions via a novel hybridization of formulations from minimization problems and proper regularization.

Date/Time: Friday, February 20, 11:30am

Location: Exploratory Hall, Room 4106 or Zoom