Arbitrary Lagrangian–Eulerian methods for analysis of regressing solid domains and interface tracking

We present the details of the formulation and implementation of the arbitrary Lagrangian–Eulerian (ALE) finite element method for three-dimensional problems involving regressing solid domains and moving boundaries. An example of such problems is the simulation of solid-propellant rockets in which the evolution of a fluid–solid interface is governed by a combustion law and the transfer of mass and momentum across it. The ALE method, while providing a means to track the location of the interface, allows the adaptation of the finite element mesh to the constantly changing solid domain. In this study, the mesh adaptation is achieved via a novel smoothing technique in which the shape of finite elements with smaller volumes, which are more susceptible to mesh-entanglement, are better preserved compared to those with larger volumes. An analysis of the stability of the ALE computations, under certain simplifying assumptions, is also performed. The stability limits determined from this analysis can be utilized as constraints for adjusting mesh velocities or time increments in the convective mesh-motion phase of the ALE computations. In addition, a method is provided for generating verification problems with moving interfaces from those with known solutions on stationary material domains. A problem in which the prescribed growth of a cavity in an infinite medium under a time-varying pressure loading is used to verify the implementation and to demonstrate the verification technique.     

A global arbitrary Lagrangian–Eulerian method for stratified Richtmyer–Meshkov instability

Richtmyer–Meshkov (RM) instability arises when a material interface is accelerated impulsively by shock waves. In this work, an arbitrary Lagrangian–Eulerian method, global ALE method, was proposed for the simulation of stratified RM instability. In the global ALE method, an Eulerian diffusion interface model was implemented based on mass fraction function. Thus all the meshes can be remeshed arbitrarily no matter whether they are material interface or not. Some benchmark problems, such as shock tube problem with different specific ratio, RM instability with small initial perturbation, were computed with the global ALE method, and the numerical results agree well with exact solution or theoretical model. Also, we proposed some stratified RM instability model problems with two or more material interfaces in planar, cylindrical and spherical geometries. Then the stratified RM instabilities were simulated with global ALE method. The interface evolution process was studied and compared in different geometry cases based on simulation results. To overcome the spurious mesh distortion, a sub-zonal Riemann solver method was proposed in appendix part of the paper based on the analysis of the error source of 2D Lagrangian computation due to non-uniform multi-dimensional mesh.     

一种物理量重映方法的研究

§1.引 言 求解流体力学问题,依照采用的坐标可分为Lagrange方法和Euler方法两大类[1].用拉氏方法局部图像可以算得比较精细,物质界面清晰,但是由于二维流体运动中可能出现严重的扭曲现象,可能造成拉氏网格相交,以致于计算不能继续下去.欧拉方法当然没有网格相交的问题,但当系统中含有多种介质时,不加特殊处理,会使物质界面逐渐模糊,得不到正确的结果.为了避免拉氏方法和欧氏方法的缺点,Frank.Lazarus(1964)提出了一种混合欧拉拉格朗日方法,Noh(1964)的耦合欧拉拉格朗日方法(CEL方法),则是将求     

Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing

We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.     

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.     

Template-based quadrilateral mesh generation from imaging data

This paper describes a novel template-based meshing approach for generating good quality quadrilateral meshes from 2D digital images. This approach builds upon an existing image-based mesh generation technique called Imeshp, which enables us to create a segmented triangle mesh from an image without the need for an image segmentation step. Our approach generates a quadrilateral mesh using an indirect scheme, which converts the segmented triangle mesh created by the initial steps of the Imesh technique into a quadrilateral one. The triangle-to-quadrilateral conversion makes use of template meshes of triangles. To ensure good element quality, the conversion step is followed by a smoothing step, which is based on a new optimization-based procedure. We show several examples of meshes generated by our approach, and present a thorough experimental evaluation of the quality of the meshes given as examples.     

Modelling and simulation of moving contact line problems with wetting effects

This paper presents a numerical scheme for computing moving contact line flows with wetting effects. The numerical scheme is based on Arbitrary Lagrangian Eulerian (ALE) finite elements on moving meshes. In the computations, the wetting effects are taken into account through a weak enforcement of the prescribed equilibrium contact angle into the model equations. The equilibrium contact angle is included in the variational form of the model by replacing the curvature with Laplace Beltrami operator and integration by parts. This weak implementation allows that the contact angle determined by the numerical scheme differs from the equilibrium value and develops a certain dynamics. The Laplace Beltrami operator technique with an interface/boundary resolved mesh is well-suited for describing the dynamic contact angle observed in experiments. We consider the spreading and the pendant liquid droplets to investigate this implementation of the contact angle. It is shown that the dynamic contact angle tends to the prescribed equilibrium contact angle when time goes to infinity. However, the dynamics of the contact angle is influenced by the slip at the moving contact line. This work has been partially supported by the German Research Foundation (DFG) through the grant To143/9.     

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

Numerical techniques for moving meshes are many and varied. In this paper we present a novel application of a moving grid finite element method applied to biological problems related to pattern formation where the mesh movement is prescribed through a specific definition to mimic the growth that is observed in nature. Through the use of a moving grid finite element technique, we present numerical computational results illustrating how period doubling behaviour occurs as the domain doubles in size.     

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.     

Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.     

Subdomain generation using emergent ant colony optimization

Finite elements mesh decomposition is a well known optimization problem and is used to split a computationally expensive finite elements mesh into smaller subdomains for parallel finite elements analysis.The ant colony optimization is a type of algorithm that seeks to model the emergent behaviour observed in ant colonies and utilize this behaviour to solve combinatorial problems. This technique has been applied to several problems, most of which are graph related because the ant colony metaphor can be most easily applied to such types of problems. This paper examines the application of ant colony optimization algorithm to the partitioning of unstructured adaptive meshes for parallel explicit time-stepping finite elements analysis.The concept of ant colony optimization technique in addition to the notion of swarm intelligence for finding approximate solutions to combinatorial optimization problems is described. This algorithm combines the features of the classical ant colony optimization technique with swarm intelligence to form a model which is an artificial system designed to perform a certain task.The application of the ant colony optimization for partitioning finite elements meshes based on triangular elements using the swarm intelligence concept is described. A recursive greedy algorithm optimization method is also presented as a local optimization technique to improve the quality of the solutions given by the ant colony optimization algorithm. The partitioning is based on the recursive bisection approach.The mesh partitioning is carried out using normal and predictive modes for which the predictive mode uses a trained multi-layered feedforward neural network that estimates the number of triangular elements that will be generated after finite elements mesh generation is carried out.The performance of the proposed hybrid approach for the recursive bisection of finite elements meshes is examined by decomposing two mesh examples and comparing them with a well known finite elements domain decomposer.     

Variable time-step method with coordinate transformation

The variable time-step methods for solving moving boundary problems are presented by transforming the variable space domain. This results in dissociating the mode of advancement of the boundary from the size of the space mesh. That is, a small movement of the moving boundary may be chosen for computing the time interval while the space domain is subdivided into larger space meshes. As a consequence, an enormous amount of saving in computer time may be achieved by using the proposed method. Two sample problems are selected for the illustration of the method.     

ReALE: A Reconnection Arbitrary-Lagrangian–Eulerian method in cylindrical geometry

This paper deals with the extension to the cylindrical geometry of the recently introduced Reconnection algorithm for Arbitrary-Lagrangian–Eulerian (ReALE) framework. The main elements in standard ALE methods are an explicit Lagrangian phase, a rezoning phase, and a remapping phase. Usually the new mesh provided by the rezone phase is obtained by moving grid nodes without changing connectivity of the underlying mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In ReALE we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and permits to follow Lagrangian features much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tesselation machinery. In this work we focus on the extension of each phase of ReALE to cylindrical geometry. The Lagrangian, rezone with reconnection and remap phases are revamped to take into account the cylindrical geometry. We demonstrate the efficiency of our ReALE in cylindrical geometry on series of numerical examples.     

Large eddy simulation and ALE mesh motion in Rayleigh-Taylor instability simulation

Many large eddy simulation (LES) techniques have been developed for stationary computational meshes. This study applies a single equation LES to Arbitrary Lagrangian-Eulerian (ALE) simulations of Rayleigh-Taylor instability and investigates its effects. Behavior of LES is similar for Eulerian and ALE simulations for the test problem studied. However, the motion of the mesh can be tied to the subgrid scale model in the form of a relaxation weight based on subgrid scale energy. This increases mesh resolution in areas of high subgrid scale energy.     

半监督的三维网格模型层次分割

提出一种半监督K均值聚类和带状区域增长的三维网格模型层次分割算法,包括显著性特征点提取、预分割和后分割3个阶段.该算法在多维标度法的基础上进行显著性特征点提取;利用半监督K均值聚类算法来对原始模型进行初步的粗分割,以提高算法的整体效率;根据预分割结果,利用离散高斯曲率逼近,以带状推进的区域增长法进行层次的后分割.与同类算法相比,文中算法得到的分割边界更有意义,具有较高的边缘准确性和分割区域一致性.     

Moving curved mesh adaptation for higher-order finite element simulations

Higher-order finite element method requires valid curved meshes in three-dimensional domains to achieve the solution accuracy. When applying adaptive higher-order finite elements in large-scale simulations, complexities that arise include moving the curved mesh adaptation along with the critical domains to achieve computational efficiency. This paper presents a procedure that combines Bézier mesh curving and size-driven mesh adaptation technologies to address those requirements. A moving mesh size field drives a curved mesh modification procedure to generate valid curved meshes that have been successfully analyzed by SLAC National Accelerator Laboratory researchers to simulate the short-range wakefields in particle accelerators. The analysis results for a 8-cavity cryomodule wakefield demonstrate that valid curvilinear meshes not only make the time-domain simulations more reliable, but also improve the computational efficiency up to 30%. The application of moving curved mesh adaptation to an accelerator cavity coupler shows a tenfold reduction in execution time and memory usage without loss in accuracy as compared to uniformly refined meshes.     

Geometric signal compression

Compression of mesh attributes becomes a challenging problem due to the great need for efficient storage and fast transmission. This paper presents a novel geometric signal compression framework for all mesh attributes, including position coordinates, normal, color, texture, etc. Within this framework, mesh attributes are regarded as geometric signals defined on mesh surfaces. A planar parameterization algorithm is first proposed to map 3D meshes to 2D parametric meshes. Geometric signals are then transformed into 2D signals, which are sampled into 2D regular signals using an adaptive sampling method. The JPEG2000 standard for still image compression is employed to effectively encode these regular signals into compact bit-streams with high rate/distortion ratios. Experimental results demonstrate the great application potentials of this framework.     

A hybrid ant colony optimization approach for finite element mesh decomposition

This paper examines the application of the ant colony optimization algorithm to the partitioning of unstructured adaptive meshes for parallel explicit time-stepping finite element analysis. The concept of the ant colony optimization technique for finding approximate solutions to combinatorial optimization problems is described.The application of ant colony optimization for partitioning finite element meshes based on triangular elements is described.A recursive greedy algorithm optimization method is also presented as a local optimization technique to improve the quality of the solutions given by the ant colony optimization algorithm. The partitioning is based on the recursive bisection approach.The mesh decomposition is carried out using normal and predictive modes for which the predictive mode uses a trained multilayered feed-forward neural network which estimates the number of triangular elements that will be generated after finite elements mesh generation is carried out.The performance of the proposed hybrid approach for the recursive bisection of finite element meshes is examined by decomposing two mesh examples.     

Using surface variability characteristics for segmentation of deformable 3D objects with application to piecewise statistical deformable model

To cope with the small sample size problem in the construction of Statistical Deformable Models (SDM), this paper proposes two novel measures that quantify the similarity of the variability characteristics among deforming 3D meshes. These measures are used as the basis of our proposed technique for partitioning a 3D mesh for the construction of piecewise SDM in a divide-and-conquer strategy. Specifically, the surface variability information is extracted by performing a global principal component analysis on the set of sample meshes. An iterative face clustering algorithm is developed for segmenting a mesh that favors grouping triangular faces having similar variability characteristics into a same mesh component. We apply the proposed mesh segmentation algorithm to the construction of piecewise SDM and evaluate the representational ability of the resulting piecewise SDM through the reconstruction of unseen meshes. Experimental results show that our approach outperforms several state-of-the-art methods in terms of the representational ability of the resulting piecewise SDM as evaluated by the reconstruction accuracy.