Introduction to Automated Modeling with FEniCS

Introduction to Automated Modeling with FEniCS explores

solution of partial differential equations via the finite

element method. It illustrates the use of automated software

generation via the FEniCS Project systems. The book reviews

most common types of partial differential equations arising

in technical simulation. It is ideal for engineers and for

computational and applied mathematicians.

PDEs are used pervasively in science, engineering, and technology

to model phenomena of interest. The most widely used technique

to convert a PDE into a computable form is the finite element method.

This book is primarily about PDEs as they are used in models. Our

emphasis is on the diversity of PDEs that occur in practice, their

features and their foibles. Our intent is to enable exploration

of new models and to show how easily this can be done. However,

this approach is not without caveats. We describe pitfalls in various

aspects of the use of PDE models. We show how to be sure that a PDE

model is well posed in many cases. In particular, we use this theory

to understand appropriate boundary conditions.

Secondarily, the book introduces basic concepts of numerical methods

for approximating the solutions of PDEs. This is done so that the

language used by software from the FEniCS Project can be properly

understood. We limit the discussion of numerical methods as much as

possible, except when it is essential to avoid catastrophes.

A tertiary objective is to present some examples of the modeling

process. One important type of model is derived by specializing a

more general model. An important example of this is the plate model

in structural mechanics. We show how the plate model is derived from

the general elasticity model and indicate some issues that arise

related to it. When relevant, we explain other modeling approaches as

well. Ultimately, FEniCS can support an automated approach to modeling.