An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function $ϕ(x,y)$. Our adaptive mesh refinement is done within a small tube of $|ϕ(x,y)|≤δ$ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.     

An Adaptive Moving Mesh Method for the Five-Equation Model

The five-equation model of multi-component flows has been attracting muchattention among researchers during the past twenty years for its potential in the studyof the multi-component flows. In this paper, we employ a second order finite volume method with minmod limiter in spatial discretization, which preserves local extrema of certain physical quantities and is thus capable of simulating challenging testproblems without introducing non-physical oscillations. Moreover, to improve thenumerical resolution of the solutions, the adaptive moving mesh strategy proposedin [Huazhong Tang, Tao Tang, Adaptive mesh methods for one- and two-dimensionalhyperbolic conservation laws, SINUM, 41: 487-515, 2003] is applied. Furthermore, theproposed method can be proved to be capable of preserving the velocity and pressure when they are initially constant, which is essential in material interface capturing.Finally, several classical numerical examples demonstrate the effectiveness and robustness of the proposed method.     

An Adaptive Moving Mesh Method for Two-Dimensional Relativistic Hydrodynamics

This paper extends the adaptive moving mesh method developed by Tang and Tang [36] to two-dimensional (2D) relativistic hydrodynamic (RHD) equations. The algorithm consists of two "independent" parts: the time evolution of the RHD equations and the (static) mesh iteration redistribution. In the first part, the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations. The second part is an iterative procedure. In each iteration, the mesh points are first redistributed, and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.     

An Adaptive Moving Mesh Method for Two-Dimensional Incompressible Viscous Flows

In this paper, we present an adaptive moving mesh technique for solving the incompressible viscous flows using the vorticity stream-function formulation. The moving mesh strategy is based on the approach proposed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562–588] to separate the mesh-moving and evolving PDE at each time step. The Navier-Stokes equations are solved in the vorticity stream-function form by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation in conjunction with a more sophisticated monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. This paper discusses the implementation of the periodic boundary conditions, where the physical domain is allowed to deform with time while the computational domain remains fixed and regular throughout. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.     

Moving Mesh Kinetic Simulation for Sheared Rodlike Polymers with High Potential Intensities

The Doi-Hess equation that describes the evolution of an orientational distribution function is capable of predicting several rheological features of nematic polymers. Since the orientational distribution function becomes sharply peaked as potential intensity increases, powerful numerical methods become necessary in the relevant numerical simulations. In this paper, a numerical scheme based on the moving grid techniques will be designed to solve the orientational distribution functions with high potential intensities. Numerical experiments are carried out to demonstrate the effectiveness and robustness of the proposed scheme.     

An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in scienceand engineering. It is an integro-differential equation involving time, frequency, spaceand angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needsto handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions.Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinatemethod for angular discretization. The former employs a dynamic mesh adaptationstrategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency aredemonstrated in a selection of one- and two-dimensional numerical examples.     

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.     

A Moving Mesh Method for the Euler Flow Calculations Using a Directional Monitor Function

This paper is concerned with the adaptive grid method for computations of the Euler equations in fluid dynamics. The new feature of the present moving mesh algorithm is the use of a dimensional-splitting type monitor function, which is to increase grid concentration in regions containing shock waves and contact discontinuities or their interactions. Several two–dimensional flow problems are computed to demonstrate the effectiveness of the present adaptive grid algorithm.     

Moving Mesh Method with Conservative Interpolation Based on L^2-Projection

We develop an efficient one-dimensional moving mesh algorithm for solving partial differential equations. The main contribution of this paper is to design an effective interpolation scheme based on L²-projection for the moving mesh method. The proposed method preserves not only the mass-conservation but also the first order momentum of the underlying numerical solution at each mesh redistribution step. Numerical examples are presented to demonstrate the effectiveness of the new interpolation technique.     

Mesh Sensitivity for Numerical Solutions of Phase-Field Equations Using r-Adaptive Finite Element Methods

There have been several recent papers on developing moving mesh methods for solving phase-field equations. However, it is observed that some of these moving mesh solutions are essentially different from the solutions on very fine fixed meshes. One of the purposes of this paper is to understand the reason for the differences. We carried out numerical sensitivity studies systematically in this paper and it can be concluded that for the phase-field equations, the numerical solutions are very sensitive to the starting mesh and the monitor function. As a separate issue, an efficient alternating Crank-Nicolson time discretization scheme is developed for solving the nonlinear system resulting from a finite element approximation to the phase-field equations.     

A General Moving Mesh Framework in 3D and Its Application for Simulating the Mixture of Multi-Phase Flows

In this paper, we present an adaptive moving mesh algorithm for meshes of unstructured polyhedra in three space dimensions. The algorithm automatically adjusts the size of the elements with time and position in the physical domain to resolve the relevant scales in multiscale physical systems while minimizing computational costs. The algorithm is a generalization of the moving mesh methods based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible, the key is to develop an efficient mesh redistribution procedure so that this part will cost as little as possible comparing with the solution evolution part. Since the mesh redistribution procedure normally requires to solve large size matrix equations, we will describe a procedure to decouple the matrix equation to a much simpler block-tridiagonal type which can be efficiently solved by a particularly designed multi-grid method. To demonstrate the performance of the proposed 3D moving mesh strategy, the algorithm is implemented in finite element simulations of fluid-fluid interface interactions in multiphase flows. To demonstrate the main ideas, we consider the formation of drops by using an energetic variational phase field model which describes the motion of mixtures of two incompressible fluids. Numerical results on two- and three-dimensional simulations will be presented.     

直肠阴道疝Y型补片修补与悬吊的初步研究

目的探讨应用倒“Y”字形补片对直肠阴道疝进行修补,并行骶前阴道悬吊术的临床效果。方法从门诊盆底功能不全的患者中,选择具有具有直肠阴道疝的妇科患者共8例作为研究对象。常规缝合直肠阴道疝的疝囊,应用到“Y”型补片加强阴道后壁,并将阴道顶端悬吊于第2～3骶骨的前筋膜处。结果直肠阴道疝修补均成功,手术时间（110．0±16．9）min,出血（91．6±52．0）ml,排气时间（33．5±5．8）h,住院时间（6．8±0．7）d。所有患者术后1年,POP—Q评估均为。度。1例患者术后出现补片排斥反应,出现阴道腐蚀。结论直肠阴道疝应用倒Y型补片修补并行骶前阴道悬吊术效果确切,消除了临床症状,比单纯的修补效果佳,不易复发,加强了盆底功能,对防止复发及盆腔脏器的脱垂具有重要意义。．     

Large-Pore PDS Mesh Compared to Small-Pore PG Mesh

ABSTRACT

Background: Currently, absorbable meshes are used as temporary closure in case of laparostoma. Unfortunately the multifilament polyglycolic acid (PG) meshes with small pores reveal little elasticity acting rather as a fluid barrier than permitting drainage of intra-abdominal fluids. Therefore, a new mesh was constructed of absorbable polydioxanon monofilaments (PDS) with increased porosity and longer degradation time. Material and Methods: For evaluation of the tissue response the new PDS mesh was implanted as abdominal wall replacement in each five rats for 7, 21, or 90 days, respectively, and compared to a PG mesh. Histological analysis included HE staining with measurement of the size of the granuloma and immunoshistochemistry for TUNEL, Ki67, TNF-R2, MMP-2, YB1, FVIII, gas6, AXL. Parameters for neovascularization and nerve ingrowth were analyzed. Results: The inflammatory and fibrotic tissue reaction is attenuated with PDS in comparison to PG, e.g., the size of the granuloma was smaller with less cell turnover, and less remodeling as represented by, e.g., reduction of apoptosis, expression of MMP-2, or TNF-R2. The number of ingrowing nerves and vessels explored via AXL, gas6, and factor VIII was increased in the PDS mesh. Conclusion: The results from the present investigation showed that a mesh can be constructed of monofilament PDS that induce significant less inflammatory and fibrotic reaction, however permits fluid drainage and preserves elasticity.