We present a novel method to simulate bending and torsion of elastic rods within the position-based dynamics (PBD) framework. The main challenge is that torsion effects of Cosserat rods are described in terms of material frames which are attached to the centerline of the rod. But frames or orientations do not fit into the classical position-based dynamics formulation. To solve this problem we introduce new types of constraints to couple orientations which are represented by unit quaternions. For constraint projection quaternions are treated in the exact same way as positions. Unit length is enforced with an additional constraint. This allows us to use the strain measures form Cosserat theory directly as constraints in PBD. It leads to very simple algebraic expressions for the correction displacements which only contain quaternion products and additions. Our results show that our method is very robust and accurately produces the complex bending and torsion effects of rods. Due to its simplicity our method is very efficient and more than one order of magnitude faster than existing position-based rod simulation methods. It even achieves the same performance as position-based simulations without torsion effects.
In this paper we propose a novel method to construct hierarchical $hp$-adaptive Signed Distance Fields (SDFs). We discretize the signed distance function of an input mesh using piecewise polynomials on an axis-aligned hexahedral grid. Besides spatial refinement based on octree subdivision to refine the cell size (h), we hierarchically increase each cell's polynomial degree (p) in order to construct a very accurate but memory-efficient representation. Presenting a novel criterion to decide whether to apply h- or p-refinement, we demonstrate that our method is able to construct more accurate SDFs at significantly lower memory consumption than previous approaches. Finally, we demonstrate the usage of our representation as collision detector for geometrically highly complex solid objects in the application area of physically-based simulation.
We introduce deferred warping, a novel approach for real-time deformation of 3D objects attached to an animated or manipulated surface. Our target application is virtual prototyping of garments where 2D pattern modeling is combined with 3D garment simulation which allows an immediate validation of the design. The technique works in two steps: First, the surface deformation of the target object is determined and the resulting transformation field is stored as a matrix texture. Then the matrix texture is used as look-up table to transform a given geometry onto a deformed surface. Splitting the process in two steps yields a large flexibility since different attachment types can be realized by simply defining specific mapping functions. Our technique can directly handle complex topology changes within the surface. We demonstrate a fast implementation in the vertex shading stage allowing the use of highly decorated surfaces with millions of triangles in real-time.
We present a new method for particle based fluid simulation, using a combination of Projective Dynamics and Smoothed Particle Hydrodynamics (SPH). The Projective Dynamics framework allows the fast simulation of a wide range of constraints. It offers great stability through its implicit time integration scheme and is parallelizable in large parts, so that it can make use of modern multi core CPUs. Yet existing work only uses Projective Dynamics to simulate various kinds of soft bodies and cloth. We are the first ones to incorporate fluid simulation into the Projective Dynamics framework. Our proposed fluid constraints are derived from SPH and seamlessly integrate into the existing method. Furthermore, we adapt the solver to handle the constantly changing constraints that appear in fluid simulation. We employ a highly parallel matrix-free conjugate gradient solver, and thus do not require expensive matrix factorizations.
We present a novel algorithm to extract the rotational part of an arbitrary 3x3 matrix. This problem lies at the core of two popular simulation methods in computer graphics, the co-rotational Finite Element Method and Shape Matching techniques. In contrast to the traditional method based on polar decomposition, degenerate configurations and inversions are handled robustly and do not have to be treated in a special way. In addition, our method can be implemented with only a few lines of code without branches which makes it particularly well suited for GPU-based applications. We demonstrate the robustness, coherence and efficiency of our method by comparing it to stabilized polar decomposition in several simulation scenarios.