In order to understand, predict, and eventually control a complex natural or manufactured physical system, one often models it using partial differential equations. Nevertheless, in practically all models arising from complex modern applications, obtaining an exact solution in the form of basic mathematical functions is not a possibility. Thus, a practitioner must resort to what is called a numerical method for computing an approximate solution to the partial differential equations defining the model. Numerical simulations thus play a key role in modern science and technology. They also allow significant reduction in manufacturing costs by designing and testing various models merely on computers before actually building a physical model. Successful computation of approximate solutions to practical problems of interest in part depends on advances in computer technology. However, more importantly, it hinges upon the design, analysis, and implementation of efficient, reliable, accurate, and robust numerical methods.
One of the most widely used family of numerical methods is the finite element method, which has become an indispensable tool for simulation of a wide variety of phenomena arising in science and engineering such as the design of aircrafts, automobiles, bridges, oil platforms, and more recently of nano-materials, to name a few. Discontinuous Galerkin (DG) methods constitute a special subfamily of finite element methods which are known for their stability, robustness, versatility, and high-order accuracy. In this project, the PI will develop and analyze hybridizable DG (HDG) methods for problems arising in structural mechanics. Particular emphasis will be on devising such methods for problems dealing with thin domains such as beams, plates, and shells, since they pose challenges which have attracted much interest in the scientific computing community. The hybridization procedure allows the elimination of many of the globally coupled degrees of freedom rendering the linear system significantly smaller than that of its classical DG counterparts. The resulting HDG methods enjoy desirable properties of DG methods such as stability, high-order convergence, and robustness, and in certain cases they exhibit even better properties. Mathematical analysis of such phenomena is also a part of the proposed project. This project consists of several parts: HDG methods for Naghdi arches; biharmonic problems; Reissner-Mindlin plates; and fourth-order time-dependent problems. Notwithstanding each of these steps is worthy of interest in its own right, one of the ultimate goals of the PI is to devise efficient numerical methods for shell models, and each one of the above steps constitute a stepping stone towards this goal.