We introduce a construction of subspaces of the spaces of tangential vector, n-vector, and tensor fields on surfaces. The resulting subspaces can be used as the basis of fast approximation algorithms for design and processing problems that involve tangential fields. Important features of our construction are that it is based on a general principle, from which constructions for different types of tangential fields can be derived, and that it is scalable, making it possible to efficiently compute and store large subspace bases for large meshes. Moreover, the construction is adaptive, which allows for controlling the distribution of the degrees of freedom of the subspaces over the surface. We evaluate our construction in several experiments addressing approximation quality, scalability, adaptivity, computation times and memory requirements. Our design choices are justified by comparing our construction to possible alternatives. Finally, we discuss examples of how subspace methods can be used to build interactive tools for tangential field design and processing tasks.

We introduce the Reduced Immersed Method (RIM) for the real-time simulation of two-way coupled incompressible fluids and elastic solids and the interaction of multiple deformables with (self-)collisions. Our framework is based on a novel discretization of the immersed boundary equations of motion, which model fluid and deformables as a single incompressible medium and their interaction as a unified system on a fixed domain combining Eulerian and Lagrangian terms. One advantage for real-time simulations resulting from this modeling is that two-way coupling phenomena can be faithfully simulated while avoiding costly calculations such as tracking the deforming fluid-solid interfaces and the associated fluid boundary conditions. Our discretization enables the combination of a PIC/FLIP fluid solver with a reduced-order Lagrangian elasticity solver. Crucial for the performance of RIM is the efficient transfer of information between the elasticity and the fluid solver and the synchronization of the Lagrangian and Eulerian settings. We introduce the concept of twin subspaces that enables an efficient reduced-order modeling of the transfer. Our experiments demonstrate that RIM handles complex meshes and highly resolved fluids for large time steps at high framerates on of-the-shelf hardware, even in the presence of high velocities and rapid user interaction. Furthermore, it extends reduced-order elasticity solvers such as Hyper-Reduced Projective Dynamics with natural collision handling.

We present a method for the real-time simulation of deformable objects that combines the robustness, generality, and high performance of Projective Dynamics with the efficiency and scalability offered by model reduction techniques. The method decouples the cost for time integration from the mesh resolution and can simulate large meshes in real-time. The proposed hyper-reduction of Projective Dynamics combines a novel fast approximation method for constraint projections and a scalable construction of sparse subspace bases. The resulting system achieves real-time rates for large subspaces enabling rich dynamics and can resolve general user interactions, collision constraints, external forces and changes to the materials. The construction of the hyper-reduced system does not require user-interaction and refrains from using training data or modal analysis, which results in a fast preprocessing stage.

The spectrum and eigenfunctions of the Laplace-Beltrami operator are at the heart of effective schemes for a variety of problems in geometry processing. A burden attached to these spectral methods is that they need to numerically solve a large-scale eigenvalue problem, which results in costly precomputation. In this paper, we address this problem by proposing a fast approximation algorithm for the lowest part of the spectrum of the Laplace-Beltrami operator. Our experiments indicate that the resulting spectra well-approximate reference spectra, which are computed with state-of-the-art eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.

We introduce a variational approach for modeling n-symmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while providing real-time responses. The approach is based on novel biharmonic and m-harmonic energies for n-fields on surface meshes and the integration of hard constraints to the resulting optimization problems. Real-time computation rates are achieved by a model reduction approach employing a Fourier-like n-vector field decomposition, which associates frequencies and modes to n-vector fields on surfaces. To demonstrate the benefits of the proposed n-field modeling approach, we use it for controlling stroke directions in line-art drawings of surfaces and for the modeling of anisotropic BRDFs, which define the reflection behavior of surfaces.