Lecture 1: Discrete Space and Time

In this lecture, we explore the simulation of deformable solids with the aim of developing a discrete, computationally solvable problem. The primary goal is to introduce the abstract algebraic concepts inherent in this problem. We approach elasticity simulation using a top-down architectural view, placing mathematical modeling at the forefront.

The study of classical elastic solids physics largely revolves around Partial Differential Equations (PDEs). In continuum mechanics and finite element analysis literature, the norm is to first derive the continuous form of these PDEs, elaborating on each term's origin, before adapting them to discrete programming languages. Often, this adaptation appears in later sections, creating a sense of anticipation for the reader.

This book, however, takes a different route. It weaves continuum mechanics and PDEs into the discussion as needed, evenly distributing these topics to avoid overwhelming the reader. This method links theory to practice incrementally, enhancing understanding.

We introduce the main problem formulation early, offering an overview of its numerical solutions. This gives readers an initial comprehensive view, sparking curiosity and motivating deeper exploration in later chapters. This strategy makes the learning process smoother and more intuitive, helping readers effortlessly connect complex concepts and quickly grasp the subject's core.

Our aim is to provide a well-rounded, thorough, and engaging exploration of deformable solids simulation, valuable for both students and seasoned researchers in the field.