The course will present a brief deduction of the partial differential equations (PDEs) that govern continuum mechanics: These are the conservation laws of mass, momentum and energy. The introduction will give a unified frame of reference to solve any kind of continuum mechanics problem (solid, fluid, thermal, transport, etc). The concepts of Lagrangian and Eulerian frame of references, and of material and spatial time derivatives will also be clarified. The numerical solution of the different PDEs via Finite Elements (FE), Finite Differences (FD) and/or Finite Volumes (FV) will be explained. The three main numerical problems (instabilities) that are commonly encountered will be outlined.
The first instability (spurious oscillations) is due to the incompressibility constraint, which appears in problems of incompressible solid mechanics, incompressible fluid mechanics (Stokes problem), plate or shell problems, plasticity, etc.
The second is due to the presence of convective terms, and appears for heat transfer problems, flows at high speeds (Navier Stokes problem), problems of transport of species, free-surface problems, etc.
The last one is due to the presence of high reactive terms. These are found for problems of transport of species with reactive terms, turbulence problems, flows in a rotating frame of reference, etc.
Among the methods used to treat such an instabilities, the following will be presented: Div-stable elements, Galerkin Least Square type methods (GLS), Orthogonal Subgrid Stabilization (OSS), fractional steps (FS), Streamline Upwind (SD), Characteristic Galerking, Taylor Galerkin, Shock capturing techniques, and Riemann and FCT solvers with limitors. Most of the methods will be treated in detail (convergence and stability analisys will be presented), and the main equivalences and differences among them will be outlined (Riemann, FCT solvers, and FS methods will be briefly mentioned. A more detailed treatment of them can be found in the courses CFD I and II).
A unified methodology to implement in a computer code the different methods using finite elements of any order and numerical integration in any spatial dimension (1D, 2D or 3D) will be given (chunks of pseudo-code will be presented and explained).
Finally, if time allows, the treatment of non-linear terms using fixed point methods (Picard), and Newton type methods (Newton-Raphson, quasi-Newton) will be presented. Such non-linearities may be due to non-linear material properties, non-linear convective terms (Navier-Stokes problems), geometric non-linearities (problems with high deformations and/or high displacements), etc. Some concepts for an efficient implementation using low order elements, edge based data structures and iterative solvers with sparse matrix storage will also be introduced.
As in any class, you are allowed to study with other students. However, tests and homework assignments (unless otherwise specified) must be completed on your own. SPECIFICALLY - YOU MAY NOT COPY ANY TEXT OR MATERIAL AND REPRESENT IT AS YOUR OWN WORK. For both papers and for code, you may reference or link to other peoples work (if it is consistent with the assignment), but you MUST cite the source it came from. Failure to follow these guidelines will be considered a violation of GMU's academic honor code and will be treated as such.