In this paper, we present a novel pairwise-force smoothed particle hydrodynamics (PF-SPH) model to allow modeling of
various interactions at interfaces in real time. Realistic capture of interactions at interfaces is a challenging problem for SPH-based simulations, especially for scenarios involving multiple interactions at different interfaces. Our PF-SPH model can readily handle multiple kinds of interactions simultaneously in a single simulation; its basis is to use a larger support radius than that used in standard SPH. We adopt a novel anisotropic filtering term to further improve the performance of interaction forces. The proposed model is stable; furthermore, it avoids the particle clustering problem which commonly occurs at the free surface. We show how our model can be used to capture various interactions. We also consider the close connection between droplets and bubbles, and show how to animate bubbles rising in liquid as well as bubbles in air. Our method is versatile, physically plausible and easy-to-implement. Examples are provided to demonstrate the capabilities and effectiveness of our approach.
Surface flow phenomena, such as rain water flowing down a tree trunk and progressive water front in a shower room, are common in real life. However, compared with the 3D spatial fluid flow, these surface flow problems have been much less studied in the graphics community. To tackle this research gap, we present an efficient, robust and high-fidelity simulation approach based on the shallow-water equations. Specifically, the standard shallow-water flow model is extended to general triangle meshes with a feature-based bottom friction model, and a series of coherent mathematical formulations are derived to represent the full range of physical effects that are important for real-world surface flow phenomena. In addition, by achieving compatibility with existing 3D fluid simulators and by supporting physically realistic interactions with multiple fluids and solid surfaces, the new model is flexible and readily extensible for coupled phenomena. A wide range of simulation examples are presented to demonstrate the performance of the new approach.
We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the previously determined analytic continuation of the fundamental solution of the QDE of the Hilbert scheme. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves.
The equivariant orbifold Gromov-Witten theory of the symmetric product of the plane is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture.
In this paper, we formulate the deep residual network (ResNet) as a control problem of transport equation. In ResNet, the transport equation is solved along the characteristics. Based on this observation, deep neural network is closely related to the control problem of PDEs on manifold. We propose several models based on transport equation, Hamilton-Jacobi equation and Fokker-Planck equation. The discretization of these PDEs on point cloud is also discussed.
This talk will give a very brief outline how a portion of geometric analysis was carried out in relation to geometry, nonlinear partial differential equations and mathematical physics. My personal experience is that this is a beautiful subject with great depth and it touches almost all branch of mathematics.
We present a real-time approach for acquiring 3D objects with high fidelity using hand-held consumer-level RGB-D scanning
devices. Existing real-time reconstruction methods typically do not take the point of interest into account, and thus might fail
to produce clean reconstruction results of desired objects due to distracting objects or backgrounds. In addition, any changes
in background during scanning, which can often occur in real scenarios, can easily break up the whole reconstruction process. To address these issues, we incorporate visual saliency into a traditional real-time volumetric fusion pipeline. Salient regions detected from RGB-D frames suggest user-intended objects, and by understanding user intentions our approach can put more emphasis on important targets, and meanwhile, eliminate disturbance of non-important objects. Experimental results on real world scans demonstrate that our system is capable of effectively acquiring geometric information of salient objects in cluttered real-world scenes, even if the backgrounds are changing.
In this paper, we generalize the point integral method to solve the general elliptic PDEs with variable coefficients and corresponding eigenvalue problems with Neumann, Robin and Dirichlet boundary conditions on point cloud. The main idea is using integral equations to approximate the original PDEs. The integral equations are easy to discretize on the point cloud. The truncation error of the integral approximation is analyzed. Numerical examples are presented to demonstrate that PIM is an effective method to solve the elliptic PDEs with smooth coefficients on point cloud.