Mesh refinement strategies for solving singularly perturbed reaction-diffusion problems

We consider the numerical approximation of a singularly perturbed reaction-diffusion problem over a square. Two different approaches are compared namely: adaptive isotropic mesh refinement and anisotropic mesh refinement. Thus, we compare the h-refinement and the Shishkin mesh approaches numerically with PLTMG software [1]. It is shown how isotropic elements lead to over-refinement and how anisotropic mesh refinement is much more efficient in thin boundary layers.     

栅格法三维六面体网格局部加密算法

有限元数值分析的精度和效率与网格单元的划分质量以及疏密程度密切相关,针对三维六面体网格单元之间疏密过渡必须平缓和协调的要求,提出了一套基于8分法的六面体网格加密模板,并给出了相应的数据结构和模板应用方式.为使所有加密单元都有相对应的加密模板,建立了加密信息场调整规则;对需要进行加密的区域首先补充加密单元,按照节点加密属性调整加密信息场,然后根据单元加密属性对加密单元进行分类,按照全加密单元、面加密单元、边加密单元以及过渡加密单元的顺序依次采用相应的模板进行加密,从而实现三维六面体网格的局部协调加密.实例结果表明,采用该套加密模板的六面体网格局部加密算法能够保障密集网格向稀疏网格的平缓和协调过渡,所生成的网格可满足有限元数值计算的要求.     

HP90: A general and flexible Fortran 90 hp-FE code

A general 2D-hp-adaptive Finite Element (FE) implementation in Fortran 90 is described. The implementation is based on an abstract data structure, which allows to incorporate the full hp-adaptivity of triangular and quadrilateral finite elements. The h-refinement strategies are based on h2-refinement of quadrilaterals and h4-refinement of triangles. For p-refinement we allow the approximation order to vary within any element. The mesh refinement algorithms are restricted to 1-irregular meshes. Anisotropic and geometric refinement of quadrilateral meshes is made possible by additionally allowing double constrained nodes in rectangles. The capabilities of this hp-adaptive FE package are demonstrated on various test problems. Received: 18 December 1997 / Accepted: 17 April 1998     

Development of adaptive mesh refinement scheme for engine spray simulations

Spray modeling is a critical component to engine combustion and emissions simulations. Accurate spray modeling often requires a fine computational mesh for better numerical resolutions. However, computations with a fine mesh will require extensive computer time. This study developed a methodology that uses a locally refined mesh in the spray region. The fine mesh virtually moves with the liquid spray. Such adaptive mesh refinement can enable greater resolution of the liquid-gas interaction while incurring only a small increase in the total number of computational cells. The present study uses an h-refinement adaptive method. A face-based approach is used for the inter-level boundary condition. The prolongation and restriction procedure preserves conservation of properties in performing grid refinement/coarsening. The refinement criterion is based on the total mass of liquid drops and fuel vapor in each cell. The efficiency and accuracy of the present adaptive mesh refinement scheme is described in the paper. Results show that the present scheme can achieve the same level of accuracy in modeling sprays with significantly lower computational cost as compared to a uniformly fine mesh.     

Adaptive multiple-levelh-refinement in automated finite element analyses

This paper discusses an automatic, adaptive finite element modeling system consisting of mesh generation, finite element analysis, and error estimation. The individual components interact with one another and efficiently reduce the finite element error to within an acceptable value and perform only a minimum number of finite element analyses.One of the necessary components in the automated system is a multiple-level local remeshing algorithm. Givenh-refinement information provided by an a posteriori error estimator, and adjacency information available in the mesh data structures, the local remeshing algorithm grades the refinement toward areas requesting refinement. It is shown that the optimal asymptotic convergence rate is achieved, demonstrating the effectiveness of the intelligent multiple-level localh-refinement.     

三维实体仿真建模的网格自动生成方法

有限元网格模型的生成与几何拓扑特征和力学特性有直接关系。建立网格模型时，为了更真实地反映原几何形体的特征，在小特征尺寸或曲率较大等局部区域网格应加密剖分；为提高有限元分析精度和效率，在待分析的开口、裂纹、几何突变、外载、约束等具有应力集中力学特性的局部区域，网格应加密剖分。为此，该文提出了基于几何特征和物理特性相结合的网格自动生成方法。该方法既能有效地描述几何形体，又能实现应力集中区域的网格局部加密及粗细网格的均匀过渡。实例表明本方法实用性强、效果良好。     

The NIRVANA code: Parallel computational MHD with adaptive mesh refinement

I report on a new version of the magnetohydrodynamics code NIRVANA¹ which is targeted at the study of astrophysical problems. The new version allows for distributed-memory simulations supporting adaptive mesh refinement. Numerical algorithms include dissipative terms (viscosity, Ohmic diffusion, thermal heat conduction) in a conservative form. Domain decomposition is preferably block-wise in case of unigrid applications but adopts space-filling curve techniques for adaptive mesh applications with a hierarchical block-structured mesh. The code architecture facilitates workload balancing among processors for arbitrary mesh refinement depths maintaining intra-level data locality via space-filling curve mappings and, at the same time, ensuring inter-level data locality by applying a novel technique called block sharing. This way, it is demonstrated that comparable performance can be achieved for problems with locally highly refined grid. The data transfer between processors extensively utilizes the coarse-granularity concept of parallel computing and makes use of the MPI library. Conservation properties of the numerical method carry over to the parallel framework. In particular, the solenoidality condition for the magnetic field is preserved to roundoff precision applying the constrained transport machinery. This paper has its focus of discussion on implementation details related to the parallelization and on a code performance analysis.     

An optimal constrained approach for divergence-free velocity interpolation and multilevel VOF method

The use of mesh refinement techniques is becoming more and more popular in computational fluid dynamics, from multilevel approaches to adaptive mesh refinement. In this paper we present a new method to interpolate the coarse velocity field which is based on an optimal approach and is characterized by a constrained minimization of an objective functional. The functional contains the sum of the square difference between the velocity components and their target average value subject to a number of divergence-free constraints. In this work we describe this approach in two- and three-dimensional geometries with different discrete velocity field configurations. This technique is applied to a multilevel Volume-of-Fluid (VOF) method where the volume fraction function is used to reconstruct and advect the interface between two immiscible phases. The coarse velocity field is interpolated to a fixed fine grid with the optimal approach over a given number of refinement levels. The results of several kinematic tests are presented, where the mass and geometrical errors are compared with those obtained with refined velocity fields interpolated with a simple midpoint rule.     

Hybrid Discretization Methods with Adaptive Yield Surface Detection for Bingham Pipe Flows

We devise a hybrid low-order method for Bingham pipe flows, where the velocity is discretized by means of one unknown per mesh face and one unknown per mesh cell which can be eliminated locally by static condensation. The main advantages are local conservativity and the possibility to use polygonal/polyhedral meshes. We exploit this feature in the context of adaptive mesh refinement to capture the yield surface by means of local mesh refinement and possible coarsening. We consider the augmented Lagrangian method to solve iteratively the variational inequalities resulting from the discrete Bingham problem, using piecewise constant fields for the auxiliary variable and the associated Lagrange multiplier. Numerical results are presented in pipes with circular and eccentric annulus cross-section for different Bingham numbers.     

Comparison of the meccano method with standard mesh generation techniques

The meccano method is a novel and promising mesh generation technique for simultaneously creating adaptive tetrahedral meshes and volume parameterizations of a complex solid. The method combines several former procedures: a mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. In this paper we present the main advantages of our method against other standard mesh generation techniques. We show that our method constructs meshes that can be locally refined using the Kossaczky bisection rule and maintaining a high mesh quality. Finally, we generate volume T-mesh for isogeometric analysis, based on the volume parameterization obtained by the method.     

Transitive mesh space of a progressive mesh

The paper investigates the set of all selectively refined meshes that can be obtained from a progressive mesh. We call the set the transitive mesh space of a progressive mesh and present a theoretical analysis of the space. We define selective edge collapse and vertex split transformations, which we use to traverse all selectively refined meshes in the transitive mesh space. We propose a complete selective refinement scheme for a progressive mesh based on the transformations and compare the scheme with previous selective refinement schemes in both theoretical and experimental ways. In our comparison, we show that the complete scheme always generates selectively refined meshes with smaller numbers of vertices and faces than previous schemes for a given refinement criterion. The concept of dual pieces of the vertices in the vertex hierarchy plays a central role in the analysis of the transitive mesh space and the design of selective edge collapse and vertex split transformations.     

HexBox: Interactive Box Modeling of Hexahedral Meshes

We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid-based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques.     

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.     

Development of an object-oriented finite element program with adaptive mesh refinement for multi-physics applications

In this paper, an object-oriented framework for numerical analysis of multi-physics applications is presented. The framework is divided into several basic sets of classes that enable the code segments to be built according to the type of problem to be solved. Fortran 2003 was used in the development of this finite element program due to its advantages for scientific and engineering programming and its new object-oriented features. The program was developed with h-type adaptive mesh refinement, and it was tested for several classical cases involving heat transfer, fluid mechanics and structural mechanics. The test cases show that the adaptive mesh is refined only in the localization region where the feature gradient is relatively high. The overall mesh refinement and the h-adaptive mesh refinement were justified with respect to the computational accuracy and the CPU time cost. Both methods can improve the computational accuracy with the refinement of mesh. The overall mesh refinement causes the CPU time cost to greatly increase as the mesh is refined. However, the CPU time cost does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved by up to 90%, especially for the simulated system with a large number of elements and nodes.     

A consistent octree hanging node elimination algorithm for hexahedral mesh generation

A new algorithm for hanging node elimination in octree structures is developed. The proposed algorithm utilizes hanging node elimination by refinement templates and a new mesh conditioning technique based on decoupling templates. Refinement templates insert transition elements to eliminate hanging nodes. Decoupling templates insert circular loops in the dual mesh without introducing or removing hanging nodes. Decoupling templates are introduced to avoid full refinement in the cases that do not match any of the available refinement templates. The proposed algorithm eliminates hanging nodes for concavely refined regions without excessive refinement. Another advantage of the proposed algorithm lies in eliminating narrow gaps of coarse meshes between refined regions. This step has a positive effect on the mesh quality as it avoids introducing non-regular templates with a limited penalty of uniform refinement. The presented algorithm produces good quality meshes and provides a consistent and complete method for producing conformally refined octree structures.     

Massively parallel adaptive mesh refinement and coarsening for dynamic fracture simulations

We use the graphical processing unit (GPU) to perform dynamic fracture simulation using adaptively refined and coarsened finite elements and the inter-element cohesive zone model. Due to the limited memory available on the GPU, we created a specialized data structure for efficient representation of the evolving mesh given. To achieve maximum efficiency, we perform finite element calculation on a nodal basis (i.e., by launching one thread per node and collecting contributions from neighboring elements) rather than by launching threads per element, which requires expensive graph coloring schemes to avoid concurrency issues. These developments made possible the parallel adaptive mesh refinement and coarsening schemes to systematically change the topology of the mesh. We investigate aspects of the parallel implementation through microbranching examples, which has been explored experimentally and numerically in the literature. First, we use a reduced-scale version of the experimental specimen to demonstrate the impact of variation in floating point operations on the final fracture pattern. Interestingly, the parallel approach adds some randomness into the finite element simulation on the structured mesh in a similar way as would be expected from a random mesh. Next, we take advantage of the speedup of the implementation over a similar serial implementation to simulate a specimen whose size matches that of the actual experiment. At this scale, we are able to make more direct comparisons to the original experiment and find excellent agreement with those results.     

A Fast Discontinuous Galerkin Method for a Bond-Based Linear Peridynamic Model Discretized on a Locally Refined Composite Mesh

We develop a family of fast discontinuous Galerkin (DG) finite element methods for a bond-based linear peridynamic (PD) model in one space dimension. More precisely, we develop a preconditioned fast piecewise-constant DG scheme on a geometrically graded locally refined composite mesh which is suited for the scenario in which the jump discontinuity of the displacement field occurs at the one of the nodes in the original uniform partition. We also develop a preconditioned fast piecewise-linear DG scheme on a uniform mesh that has a second-order convergence rate when the jump discontinuity occurs at one of the computational nodes or has a convergence rate of one-half order otherwise. Motivated by these results, we develop a preconditioned fast hybrid DG scheme that is discretized on a locally uniformly refined composite mesh to numerically simulate the PD model where the jump discontinuity of the displacement field occurs inside a computational cell. We use a piecewise-constant DG scheme on a uniform mesh and a piecewise-linear DG scheme on a locally uniformly refined mesh of mesh size \(O(h^2)\), which has an overall convergence rate of O(h). Because of their nonlocal nature, numerical methods for PD models generate dense stiffness matrices which have \(O(N^2)\) memory requirement and \(O(N^3)\) computational complexity, where N is the number of computational nodes. In this paper, we explore the structure of the stiffness matrices to develop three preconditioned fast Krylov subspace iterative solvers for the DG method. Consequently, the methods have significantly reduced computational complexity and memory requirement. Numerical results show the utility of the numerical methods.     

A control approach for pore size distribution in the bone scaffold based on the hexahedral mesh refinement

Tissue engineering is the application of that knowledge to the building or repairing of tissues. Generally, engineered tissue is a combination of living cells and a support structure called scaffolds. Modeling, design and fabrication of tissue scaffold with intricate architecture, porosity and pore size for desired tissue properties presents a challenge in tissue engineering. In this paper, a control approach for pore size distribution in the bone scaffold based on the hexahedral mesh refinement is presented. Firstly, the bone scaffold modeling approach based on the shape function in the finite element method is provided. The resulting various macroporous morphologies can be obtained. Then conformal refinement algorithm for all-hexahedral element mesh is illustrated. Finally, a modeling approach for constructing tissue engineering (TE) bone scaffold with defined pore size distribution is presented. Before the conformal refinement of all-hexahedral element mesh, a 3D mesh with various hexahedral elements must be provided. If all the pores in the bone scaffold need to be reduced, that means that the whole hexahedral mesh needs to be refined. Then the solid entity can be re-divided with altered subdivision parameters. If the pores in the local regions of bone need to be reduced, that means that 3D hexahedral mesh in the local regions needs to be refined. Based on SEM images, the pore size distribution in the normal bone can be obtained. Then, according to the conformal refinement of all-hexahedral element meshes, defined hexahedral size distribution can be gained, which leads to generate defined pore size distribution in the bone scaffold, for the pore morphology and size are controlled by various subdivided hexahedral elements. Compared to other methods such as varying processing parameters in supercritical fluid processing and multi-interior architecture design, the method proposed in this paper enjoys easy-controllability and higher accuracy.