Catalog Description: Prerequisite: A course on the FEM, or consent of instructor. Advanced topics on classical FEM. Non-classical FEM (discontinuous Galerkin methods, nonconforming/mixed FEM, etc.). Transient problems. Optimization theory applied to mixed FEM; incompressible flows. Generalized Hu-Washizu mixed variational principle. Electromagnetics, heat, fluids, solids. Other advanced topics (e.g., boundary elements, edge elements, meshfree methods, etc.), depending on mutual interest between students and instructor.
Textbook: None required.
Prerequisites by topics: A course on the FEM. Undergraduate linear algebra.
Goals: To provide a solid foundation for the classical FEM, its applications in all areas of engineering, and to treat advanced concepts of non-classical FEM, e.g., discontinuous Galerkin method, mixed FEM, founded on solid mathematical setting. As a complement to the course EGM 6351, this course is designed to sharpen (i) the basic concepts of the FEM, and (ii) the skills to formulate FE solution to solve the PDE's governing physical phenomena. Various types of PDE's are addressed, including hyperbolic PDE's in flow problems. The course prepares engineering students to access the rich mathematics literature on the subject. Further, this course provides an introduction to nonlinear FEM, depending on the mutual interest between students and instructor.
Topics: Classical FE formulation: Poisson equation in R^n, biharmonic equation, elastodynamic equations, Maxwell equations in electromagnetics. Classical FEM as a projection method. FE function spaces: Lagrange elements of various types in R^n, Hermite elements with various types, elements for plate problems (biharmonic equation), etc., mathematical properties. Discontinuous FEM for transient problems: space-time FEM. Problems with differential constraint (e.g., incompressible flow, etc.), mixed FEM. Optimization theory and its application to mixed FEM. Application of mixed FEM to electromagnetics: Edge elements. Navier-Stokes equations, convection-dominated flow problems: Upwind-Petrov Galerkin method, etc. Introductory nonlinear FEM. And more... The selection of the topics to be lectured depends on the mutual interest between the students and the instructor.
Project: To be announced in class. For example, the project for a student could consist of reading any recent paper of interest to this student and relevant to the course. On-campus students will make a presentation on her reading of the selected paper; off-campus students will submit a term paper on the selected paper.
Engineering applications: Electromagnetics, heat, fluid and solid mechanics.
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