Description
Computational Modeling Initiative LLC Introduction to Automated Modeling with FEniCS by L Ridgway Scott
Introduction to Automated Modeling with FEniCS explores_x000D_solution of partial differential equations via the finite_x000D_element method. It illustrates the use of automated software_x000D_generation via the FEniCS Project systems. The book reviews_x000D_most common types of partial differential equations arising_x000D_in technical simulation. It is ideal for engineers and for_x000D_computational and applied mathematicians._x000D__x000D_PDEs are used pervasively in science, engineering, and technology_x000D_to model phenomena of interest. The most widely used technique_x000D_to convert a PDE into a computable form is the finite element method._x000D_This book is primarily about PDEs as they are used in models. Our_x000D_emphasis is on the diversity of PDEs that occur in practice, their_x000D_features and their foibles. Our intent is to enable exploration_x000D_of new models and to show how easily this can be done. However, _x000D_this approach is not without caveats. We describe pitfalls in various_x000D_aspects of the use of PDE models. We show how to be sure that a PDE_x000D_model is well posed in many cases. In particular, we use this theory_x000D_to understand appropriate boundary conditions._x000D__x000D_Secondarily, the book introduces basic concepts of numerical methods_x000D_for approximating the solutions of PDEs. This is done so that the_x000D_language used by software from the FEniCS Project can be properly_x000D_understood. We limit the discussion of numerical methods as much as_x000D_possible, except when it is essential to avoid catastrophes._x000D__x000D_A tertiary objective is to present some examples of the modeling_x000D_process. One important type of model is derived by specializing a_x000D_more general model. An important example of this is the plate model_x000D_in structural mechanics. We show how the plate model is derived from_x000D_the general elasticity model and indicate some issues that arise_x000D_related to it. When relevant, we explain other modeling approaches as_x000D_well. Ultimately, FEniCS can support an automated approach to modeling._x000D_ Table of contents :- _x000D_
Part I Groebner bases: Monomial Ideals.- A short introduction to Groebner bases.- Monomial orders and weights.- Generic initial ideals.- The exterior algebra.- Part II: Hilbert functions and resolutions.- Hilbert functions and the theorems of Macaulay and Kruskal-Katona.- Resolutions of monomial ideals and the Eliahou-Kervaire formula.- Alexander duality and resolutions.- Part III Combinatorics: Alexander duality and finite graphs.- Powers of monomial ideals.- Shifting theory.- Discrete Polymatroids.- Some homological algebra.- Geometry_x000D_