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.

Physically-Based Modeling for Computer Graphics: A Structured Approach addresses the challenge of designing and managing the complexity of physically-based models. This book will be of interest to researchers, computer graphics practitioners, mathematicians, engineers, animators, software developers and those interested in computer implementation and simulation of mathematical models. - Presents a philosophy and terminology for "Structured Modeling" - Includes mathematicl and programming techniques to support and implement the methodology - Covers a library of model components, including rigid-body kinematics, rigid-body dynamics, and force-based constraint methods - Includes illustrations of several ample models created from these components - Foreword by Al Barr

The booming computer games and animated movie industries continue to drive the graphics community's seemingly insatiable search for increased realism, believability, ad speed. To achieve the quality expected by audiences of today's games and movies, programmers need to understand and implement physics-based animation. To provide this understanding, this book is written to teach students and practitioners and theory behind the mathematical models and techniques required for physics-based animation. It does not teach the basic principles of animation, but rather how to transform theoretical techniques into practical skills. It details how the mathematical models are derived from physical and mathematical principles, and explains how these mathematical models are solved in an efficient, robust, and stable manner with a computer. This impressive and comprehensive volume covers all the issues involved in physics-based animation, including collision detection, geometry, mechanics, differential equations, matrices, quaternions, and more. There is excellent coverage of collision detection algorithms and a detailed overview of a physics system. In addition, numerous examples are provided along with detailed pseudo code for most of the algorithms. This book is ideal for students of animation, researchers in the field, and professionals working in the games and movie industries. Topics Covered: * The Kinematics: Articulated Figures, Forward and Inverse Kinematics, Motion Interpolation * Multibody Animation: Particle Systems, Continuum Models with Finite Differences, the Finite Element Method, Computational Fluid Dynamics * Collision Detection: Broad and Narrow Phase Collision Detection, Contact Determination, Bounding Volume Hierarchies, Feature-and Volume-Based Algorithms

"Digital human characters are a mainstay of video games, film, and interactive computer graphics applications. However, animating hands remains a challenging aspect of human character animation: posing the hand involves coordinating many degrees of freedom, synthesizing a plausible grasp requires careful placement of contacts, and realistic rendering must account for intricate colour and texture variations. Traditional solutions to these problems require significant manual effort by skilled artists. It is therefore of great interest to computer animation researchers to develop fast and automatic methods for animating hands. This thesis presents methods for improving the realism of hands in real-time physics-based virtual environments.We begin by presenting a framework for skilled motion synthesis, wherein reinforcement learning and non-linear continuous optimization are used to generate controllers for single-handed re-orientation tasks. A mid-level multiphase approach breaks the problem into three parts, providing an appropriate control strategy for each phase and resulting in cyclic finger motions that accomplish the task. The exact trajectory is never specified, as the task goals are concerned with the final orientation and position of the object. Offline simulations are used to learn controller parameters, but the resulting control policy is suitable for real-time applications.We then describe a method for the simulation of compliant articulated structures using an approximate model that focuses on plausible endpoint behaviour. The approach is suitable for simulating physics-based characters under static proportional derivative control and stiff kinematic structures, like robotic grippers. The computation time of the dynamical simulation is reduced by an order of magnitude, and faster than real-time frame rates are easily achieved. Additionally, the state of internal bodies is computed independently, and in a parallel fashion.We also demonstrate an approach for synthesizing colour variation in fingers due to physical interaction with objects. A data-driven model relates contact information to visible colour changes for the fingernail and surrounding tissue on the back of the fingertip. The model construction uses the space of hemoglobin concentrations, as opposed to an RGB colour space, which permits transferability across different fingers and different people. Principal component analysis (PCA) on the sample images results in a compact model, enabling efficient implementation as a fragment shader program.Finally, we introduce a system for capturing grasping and dexterous interactions with real-world objects. A novel sensor ensemble collects information about joint motion and pressure distributions for the hand, and the data is used to design grasping controllers for a physics-based climbing simulation. Additionally, we speculate on how the interaction data can be used to derive future control strategies in physics-based animation. Combining interaction data with physical models is a promising approach for skilled motion synthesis involving hands." --

In this thesis, we present a geometrically exact beam theory and a corresponding displacement-based finite-element model for modeling, analysis and natural-looking animation of highly flexible beam components of multi-body systems undergoing huge static/dynamic rigid-elastic deformations. By using Jaumann strains, concepts of local displacements and orthogonal virtual rotations, and three Euler angles to exactly describe the coordinate transformation between the undeformed and deformed configurations, the beam theory can fully explain geometric nonlinearities and initial curvatures. In order to demonstrate the accuracy and capability of this nonlinear beam element, we perform nonlinear static and dynamic analysis of two highly flexible beams examples, which include the twisting of a circular ring into three small rings and the spinup of a flexible helicopter rotor blade. We use this method to analyze helicopter rotor blades' spinup dynamics and to simulate helicopter flight starting from takeoff, loiter, and then landing. These two numerical examples proved that the proposed nonlinear beam element is accurate and versatile for modeling, analysis and 3D rendering and animation of multi-body systems that contain highly flexible beam components.

Physics-based animation is commonplace in animated feature films and even special effects for live-action movies. Think about a recent movie and there will be some sort of special effects such as explosions or virtual worlds. Cloth simulation is no different and is ubiquitous because most virtual characters (hopefully!) wear some sort of clothing. The focus of this book is physics-based cloth simulation. We start by providing background information and discuss a range of applications. This book provides explanations of multiple cloth simulation techniques. More specifically, we start with the most simple explicitly integrated mass-spring model and gradually work our way up to more complex and commonly used implicitly integrated continuum techniques in state-of-the-art implementations. We give an intuitive explanation of the techniques and give additional information on how to efficiently implement them on a computer. This book discusses explicit and implicit integration schemes for cloth simulation modeled with mass-spring systems. In addition to this simple model, we explain the more advanced continuum-inspired cloth model introduced in the seminal work of Baraff and Witkin [1998]. This method is commonly used in industry. We also explain recent work by Liu et al. [2013] that provides a technique to obtain fast simulations. In addition to these simulation approaches, we discuss how cloth simulations can be art directed for stylized animations based on the work of Wojtan et al. [2006]. Controllability is an essential component of a feature animation film production pipeline. We conclude by pointing the reader to more advanced techniques.