We present novel analysis methods for vector fields and an intrinsic representation of 2-tensor fields on meshes, and show the benefits they bring to discrete calculus, geometry processing, texture synthesis and fluid simulation. For instance, such vector fields and tensor fields in flat 2D space are necessary for example-based texture synthesis. However, many existing methods cannot ensure the continuity automatically or control the singularities accurately. Moreover, extending such analyses to curved surfaces involves several known challenges. First, vectors at different surface points are defined in different tangent planes, so their comparison necessarily involves a concept calledconnection to transport vectors from one tangent plane to another in a parallel way. The few existing approaches for discrete connections offer neither a globally optimal principled definition nor a consistent disretization of differential operators. Second, symmetric 2-tensors, which play a crucial role in geometry processing, are often discretized as components stored in the predefined local frames. There is no convenient way to perform coordinate-independent computations with arbitrary 2-tensor fields on triangulated surface meshes. Finally, the persistent pursue for efficiency in the processing of vector fields in applications such as incompressible fluid simulation often results in undesired artifacts such as numerical viscosity, which prevents a predictive preview for the fine-resolution simulation at coarse spatial and temporal resolutions.We offer solutions to address these issues using our novel representation and analysis tools.First, we present a framework for example-based texture synthesis with feature alignment to vector fields with two way rotational symmetry, also known as orientation fields. Our contribution is twofold: a design tool for orientation fields with a natural boundary condition and singularity control, and a parallel texture synthesis adapted specifically for such fields in feature alignment.Second, we define discrete connection on triangle meshes, which involves closed-form expressions within edges and triangles and finite rotations between pairs of incident vertices, edges, or triangles. The finite set of parameters of this connection can be optimally computed by minimizing a quadratic measure of the deviation from the connection induced by the embedding of the input triangle mesh. Local integrals of other first-order derivatives as well as the L2-based energies can also be computed.Third, we offer a coordinate-free representation of arbitrary 2-tensor fields on triangle meshes, where we leverage a decomposition of continuous 2-tensors in the plane to construct a finite-dimensional encoding of tensor fields through scalar values on oriented pieces of a manifold triangulation. We also provide closed-form expressions of common operators for tensor fields, including pairing, inner product, and trace for this discrete representation, and formulate a discrete covariant derivative induced by the 2-tensors instead of the metric of the surface. Other operators, such as discrete Lie bracket, can be constructed based on these operators. This approach extends computational tools for tensor fields and offers a numerical framework for discrete tensor calculus on triangulations.Finally, a spectral vector field calculus on embeded irregular shape is introduced to build a model-reduced variational Eulerian integrator for incompressible fluid. The resulting simulation combines the efficiency gains of dimension reduction, the qualitative robustness to coarse spatial and temporal resolutions of geometric integrators, and the simplicity of sub-grid accurate boundary conditions on regular grids to deal with arbitrarily-shaped domains. A functional map approach to fluid simulation is also proposed, where scalar-valued and vector-valued eigenfunctions of the Laplacian operator can be easily used as reduced bases. Using a variational integrator in time topreserve liveliness and a simple, yet accurate embedding of the fluid domain onto a Cartesian grid, our model-reduced fluid simulator can achieve realistic animations in significantly less computation time than full-scale non-dissipative methods but without the numerical viscosity from which current reduced methods suffer.
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