Finite Element Method Simulation of 3D Deformable Solids (Synthesis Lectures on Visual Computing: Computer Graphics, Animation, Computational Photography, and Imaging)

This book serves as a practical guide to simulation of 3D deformable solids using the Finite Element Method (FEM). It reviews a number of topics related to the theory and implementation of FEM approaches: measures of deformation, constitutive laws of nonlinear materials, tetrahedral discretizations, and model reduction techniques for real-time simulation. Simulations of deformable solids are important in many applications in computer graphics, including film special effects, computer games, and virtual surgery. The Finite Element Method has become a popular tool in many such applications. Variants of FEM catering to both offline and real-time simulation have had a mature presence in computer graphics literature. This book is designed for readers familiar with numerical simulation in computer graphics, who would like to obtain a cohesive picture of the various FEM simulation methods available, their strengths and weaknesses, and their applicability in various simulation scenarios. The book is also a practical implementation guide for the visual effects developer, offering a lean yet adequate synopsis of the underlying mathematical theory. Chapter 1 introduces the quantitative descriptions used to capture the deformation of elastic solids, the concept of strain energy, and discusses how force and stress result as a response to deformation. Chapter 2 reviews a number of constitutive models, i.e., analytical laws linking deformation to the resulting force that has successfully been used in various graphics-oriented simulation tasks. Chapter 3 summarizes how deformation and force can be computed discretely on a tetrahedral mesh, and how an implicit integrator can be structured around this discretization. Finally, chapter 4 presents the state of the art in model reduction techniques for real-time FEM solid simulation and discusses which techniques are suitable for which applications. Topics discussed in this chapter include linear modal analysis, modal warping, subspace simulation, and domain decomposition.

這本書是一本實用指南，介紹了使用有限元法（FEM）模擬三維可變形固體的方法。它回顧了與FEM方法的理論和實施相關的多個主題：變形度量、非線性材料的本構定律、四面體離散化以及用於實時模擬的模型簡化技術。可變形固體的模擬在電腦圖形學的許多應用中非常重要，包括電影特效、電腦遊戲和虛擬手術。有限元法已成為許多這類應用中的一種流行工具。針對離線和實時模擬的FEM變體在電腦圖形學文獻中已經有了成熟的存在。本書專為熟悉電腦圖形學中數值模擬的讀者設計，他們希望獲得各種FEM模擬方法的一個統一的概述，了解它們的優點和缺點，以及它們在不同模擬場景中的應用性。本書還是視覺效果開發人員的實用實施指南，提供了對底層數學理論的簡明但足夠的概述。第1章介紹了用於捕捉彈性固體變形的定量描述，應變能的概念，並討論了力和應力如何作為對變形的響應而產生。第2章回顧了一些本構模型，即將變形與結果力相關聯的分析法則，這些法則在各種圖形導向的模擬任務中得到了成功應用。第3章總結了如何在四面體網格上離散計算變形和力，以及如何在這種離散化周圍結構化隱式積分器。最後，第4章介紹了實時FEM固體模擬中的模型簡化技術的最新研究成果，並討論了哪些技術適用於哪些應用。本章討論的主題包括線性模態分析、模態扭曲、子空間模擬和域分解。