Narrow Results

In this paper, we propose a novel efficient surface reconstruction method from unorganized point cloud data in three-dimensional Euclidean space. The proposed method is based on the Allen–Cahn partial differential equation, with an edge indicating function to restrict the evolution. We applied the explicit Euler’s method to solve the discrete equation, and use the operator splitting technique to split the governing equation. Furthermore, we also modify the double well form to a periodic potential. Then we find that the proposed model can reconstruct the surface well even in the case of insufficient data. After selecting the appropriate parameters, we carried out various numerical experiments to demonstrate the robustness and accuracy of the proposed method. We adopt the proposed method to reconstruct the surfaces on simple, irregular and complex models, respectively, and can obtain smooth three-dimensional surfaces and visual effects. In addition, we also perform comparison tests to show the superiority of the proposed model. Statistic metrics such as the $\sigma$, $d_{max}$, $d_{\mathrm{\text{mean}}}$, CPU time, and vertex numbers are evaluated. Results show that our model performs better than the other methods in statistical metrics even use far less point cloud data, and with the faster CPU computing speed.

We study, in this paper, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional (2D) bounded domain. The model consists of the Navier–Stokes equations (NSEs) for the velocity, coupled with a Allen–Cahn model for the order (phase) parameter. We prove the existence and the uniqueness of a variational solution.

In this paper, we derive a large deviation principle for a stochastic 2D Allen–Cahn–Navier–Stokes system with a multiplicative noise of Lévy type. The model consists of the Navier–Stokes equations for the velocity, coupled with a Allen–Cahn system for the order (phase) parameter. The proof is based on the weak convergence method introduced in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincar Probab. Stat. 47(3) (2011) 725747].