Feature Preserving Parameterization for Quadrilateral Mesh Generation Based on Ricci Flow and Cross Field

We propose a newmethod to generate surface quadrilateralmesh by calculating a globally defined parameterization with feature constraints.In the field of quadrilateral generation with features,the cross field methods are wellknown because of their superior performance in feature preservation.The methods based on metrics are popular due to their sound theoretical basis,especially the Ricci flow algorithm.The cross field methods’major part,the Poisson equation,is challenging to solve in three dimensions directly.When it comes to cases with a large number of elements,the computational costs are expensive while the methods based on metrics are on the contrary.In addition,an appropriate initial value plays a positive role in the solution of the Poisson equation,and this initial value can be obtained from the Ricci flow algorithm.So we combine the methods based on metric with the cross field methods.We use the discrete dynamic Ricci flow algorithm to generate an initial value for the Poisson equation,which speeds up the solution of the equation and ensures the convergence of the computation.Numerical experiments show that our method is effective in generating a quadrilateral mesh for models with features,and the quality of the quadrilateral mesh is reliable.

A Geometric Understanding of Deep Learning

This work introduces an optimal transportation(OT)view of generative adversarial networks(GANs).Natural datasets have intrinsic patterns,which can be summarized as the manifold distribution principle:the distribution of a class of data is close to a low-dimensional manifold.GANs mainly accomplish two tasks:manifold learning and probability distribution transformation.The latter can be carried out using the classical OT method.From the OT perspective,the generator computes the OT map,while the discriminator computes the Wasserstein distance between the generated data distribution and the real data distribution;both can be reduced to a convex geometric optimization process.Furthermore,OT theory discovers the intrinsic collaborative-instead of competitive-relation between the generator and the discriminator,and the fundamental reason for mode collapse.We also propose a novel generative model,which uses an autoencoder(AE)for manifold learning and OT map for probability distribution transformation.This AE–OT model improves the theoretical rigor and transparency,as well as the computational stability and efficiency;in particular,it eliminates the mode collapse.The experimental results validate our hypothesis,and demonstrate the advantages of our proposed model.

Simple Fourth-Degree Cubature Formulae with Few Nodes over General Product Regions

A simple method is proposed for constructing fourth-degree cubature formulae over general product regions with no symmetric assumptions. The cubatureformulae that are constructed contain at most n2 + 7n + 3 nodes and they are likelythe first kind of fourth-degree cubature formulae with roughly n2 nodes for nonsymmetric integrations. Moreover, two special cases are given to reduce the numberof nodes further. A theoretical upper bound for minimal number of cubature nodesis also obtained.

Prismatic mesh generation based on anisotropic volume harmonic field

In this paper,we present an effective prismatic mesh generation method for viscous flow simulations.To address the prismatic mesh collisions in recessed cavities or slit areas,we exploit 3D tensors controlled anisotropic volume harmonic field to generate prismatic meshes.Specially,a well-fitting tetrahedral mesh is first constructed to serve as the discrete computation domain of volume harmonic fields.Then,3D tensors are exploited to control the volume harmonic field that better fits the shape geometry.From the topological perspective,the generation of the prismatic mesh can be treated as a topology-preserved morphing of the viscous wall.Therefore,iso-surfaces in the volume harmonic field should be homeomorphic to the viscous wall while fitting its shapes.Finally,a full prismatic mesh can be induced by estimating the forward directions and visible regions in the volume harmonic field.Moreover,to be compatible with different simulation situations,the thickness of prismatic meshes should be variable.Our approach provides local adjustable thickness of prismatic meshes,which can be achieved by controlling local 3D tensors.The proposed anisotropic volume harmonic field based prismatic meshes are efficient and robust,and a full prismatic mesh can be guaranteed without low quality collisions.Various experiments have shown that our proposed prismatic meshes have obvious advantages in terms of efficiency and effectiveness.