An isogeometric finite element formulation for phase transitions on deforming surfaces

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$\alpha$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to the text, added supplementary movie file

Fast and deep deformation approximations

Character rigs are procedural systems that compute the shape of an animated character for a given pose. They can be highly complex and must account for bulges, wrinkles, and other aspects of a character's appearance. When comparing film-quality character rigs with those designed for real-time applications, there is typically a substantial and readily apparent difference in the quality of the mesh deformations. Real-time rigs are limited by a computational budget and often trade realism for performance. Rigs for film do not have this same limitation, and character riggers can make the rig as complicated as necessary to achieve realistic deformations. However, increasing the rig complexity slows rig evaluation, and the animators working with it can become less efficient and may experience frustration. In this paper, we present a method to reduce the time required to compute mesh deformations for film-quality rigs, allowing better interactivity during animation authoring and use in real-time games and applications. Our approach learns the deformations from an existing rig by splitting the mesh deformation into linear and nonlinear portions. The linear deformations are computed directly from the transformations of the rig's underlying skeleton. We use deep learning methods to approximate the remaining nonlinear portion. In the examples we show from production rigs used to animate lead characters, our approach reduces the computational time spent on evaluating deformations by a factor of 5×-10×. This significant savings allows us to run the complex, film-quality rigs in real-time even when using a CPU-only implementation on a mobile device

Beyond developable: computational design and fabrication with auxetic materials

We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications

Generating 3D faces using Convolutional Mesh Autoencoders

Learned 3D representations of human faces are useful for computer vision problems such as 3D face tracking and reconstruction from images, as well as graphics applications such as character generation and animation. Traditional models learn a latent representation of a face using linear subspaces or higher-order tensor generalizations. Due to this linearity, they can not capture extreme deformations and non-linear expressions. To address this, we introduce a versatile model that learns a non-linear representation of a face using spectral convolutions on a mesh surface. We introduce mesh sampling operations that enable a hierarchical mesh representation that captures non-linear variations in shape and expression at multiple scales within the model. In a variational setting, our model samples diverse realistic 3D faces from a multivariate Gaussian distribution. Our training data consists of 20,466 meshes of extreme expressions captured over 12 different subjects. Despite limited training data, our trained model outperforms state-of-the-art face models with 50% lower reconstruction error, while using 75% fewer parameters. We also show that, replacing the expression space of an existing state-of-the-art face model with our autoencoder, achieves a lower reconstruction error. Our data, model and code are available at http://github.com/anuragranj/com