Abstract: We propose an example-based approach for simulating complex elastic material behavior. Supplied with a few poses that characterize a given object, our system starts by constructing a space of prefered deformations by means of interpolation. During simulation, this example manifold then acts as an additional elastic attractor that guides the object towards its space of prefered shapes. Added on top of existing solid simulation codes, this example potential effectively allows us to implement inhomogeneous and anisotropic materials in a direct and intuitive way. Due to its example-based interface, our method promotes an art-directed approach to solid simulation, which we exemplify on a set of practical examples.

Abstract: We develop an accurate, unified treatment of elastica. Following the method of resultant-based formulation to its logical extreme, we derive a higher-order integration rule, or elaston, measuring stretching, shearing, bending, and twisting along any axis. The theory and accompanying implementation do not distinguish between forms of different dimension (solids, shells, rods), nor between manifold regions and non-manifold junctions. Consequently, a single code accurately models a diverse range of elastoplastic behaviors, including buckling, writhing, cutting and merging. Emphasis on convergence to the continuum sets us apart from early unification efforts.

Abstract: We present a novel approach for animating elastically deformable solids in a shape-preserving manner. Standard approaches to animate this kind of objects are based on classic FEM discretizations of the elasticity theory, combined with embedding techniques to deform highly-detailed object geometries. However, these approaches are usually not able to preserve fine geometric features at sub-element scales, showing visually disturbing deformations. We propose to use Green Coordinates (GC) for the representation of the deformation field to get shape preservation by construction and describe how to discretize the elastic energy using these cage-based coordinates. By linearizing the deformation field we arrive at a simple approach which leads to just a few additional terms compared to classic FEM discretizations.

Abstract: This technical report describes an implementation of the discontinuous Galerkin (DG) finite element method for thin shells presented by [Noels and Radovitzky 2008]. After a short summary of the Kirchhoff-Love shell theory, the DG weak form is reviewed and the assembly of the stiffness matrix is described in detail. We also present a co-rotational extension to the method which allows us to simulate large rotational deformations without the typical linearization artifacts of a linear shell model. The proposed model has been successfully applied to the simulation of cutting and fracturing of thin shells by means of harmonic enrichments [Kaufmann et al. 2009].

Abstract: We present a method for simulating highly detailed cutting and fracturing of thin shells using low-resolution simulation meshes. Instead of refining or remeshing the underlying simulation domain to resolve complex cut paths, we adapt the extended finite element method (XFEM) and enrich our approximation by customdesigned basis functions, while keeping the simulation mesh unchanged. The enrichment functions are stored in enrichment textures, which allows for fracture and cutting discontinuities at a resolution much finer than the underlying mesh, similar to image textures for increased visual resolution. Furthermore, we propose harmonic enrichment functions to handle multiple, intersecting, arbitrarily shaped, progressive cuts per element in a simple and unified framework. Our underlying shell simulation is based on discontinuous Galerkin (DG) FEM, which relaxes the restrictive requirement of C1 continuous basis functions and thus allows for simpler, C0 continuous XFEM enrichment functions.

Abstract: We propose a simulation techniques for elastically deformable objects based on the discontinuous Galerkin finite element method (DG FEM). In contrast to traditional FEM, it overcomes the restrictions of conforming basis functions by allowing for discontinuous elements with weakly enforced continuity constraints. This added exibility enables the simulation of arbitrarily shaped, convex and non-convex polyhedral elements, while still using simple polynomial basis functions. For the accurate strain integration over these elements we propose an analytic technique based on the divergence theorem. Being able to handle arbitrary elements eventually allows us to derive simple and efficient techniques for volumetric mesh generation, adaptive mesh renement, and robust cutting. Furthermore, we show DG FEM not to suer from locking artifacts even for nearly incompressible materials, a problem that in standard FEM requires special handling.

Abstract: Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.

Abstract: We propose a simulation technique for elastically deformable objects based on the discontinuous Galerkin finite element method (DG FEM). In contrast to traditional FEM, it overcomes the restrictions of conforming basis functions by allowing for discontinuous elements with weakly enforced continuity constraints. This added flexibility enables the simulation of arbitrarily shaped, convex and non-convex polyhedral elements, while still using simple polynomial basis functions. For the accurate strain integration over these elements we propose an analytic technique based on the divergence theorem. Being able to handle arbitrary elements eventually allows us to derive simple and efficient techniques for volumetric mesh generation, adaptive mesh refinement, and robust cutting.

Abstract: In this master thesis the application of field lines from electrostatic fields to the task of surface parameterization is examined. Especially mappings of genus-0 surfaces to their natural domain, the sphere, are considered. Further, the suitability of such mapping for a subsequent optimization procedure, ensuring special properties of the mapping, is investigated.
A related, second method is also examined, motivated by incompressible fluid flows, which directly finds parameterizations being area-preserving.
The Finite Difference Method and the Finite Element Method for the solution of the involved partial differential equation are compared and their strengths and weaknesses are discussed. Further, the feasibility and scalability of the methods are examined and the properties of their application to surfaces with higher-order genus is presented.