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。     

An adaptive finite element scheme for transient problems in CFD

An adaptive finite element scheme for transient problems is presented. The classic h-enrichment / coarsening is employed in conjunction with a triangular finite element discretization in two dimensions. A mesh change is performed every n timesteps, depending on the Courant number employed and the number of ‘protective layers’ added ahead of the refined region. In order to simplify the refinement/ coarsening logic and to be as fast as possible, only one level of refinement/coarsening is allowed per mesh change. A high degree of vectorizability has been achieved on the CRAY XMP 12 at NRL. Several examples involving shock-shock interactions and the impact of shocks on structures demonstrate the performance of the method, indicating that considerable savings in CPU time and storage can be realized even for strongly unsteady flows.     

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.     

Computation of Dendritic Growth with Level Set Model Using a Multi-Mesh Adaptive Finite Element Method

In this paper, we propose an efficient multi-mesh h-adaptive algorithm to solve the level set model of dendritic growth. Since the level set function is used to provide implicitly the location of the phase interface, it is resolved by an h-adaptive mesh with refinement only around the phase interface, while the thermal field is approximated on another h-adaptive mesh. The proposed method not only can enjoy the merits of the level set function to handle complex evolution of the free boundary, but also can achieve the similar accuracy as the front tracking method for the sharp interface model with about the same degrees of freedom. The algorithm is applied to the simulation of the dendritic crystallization in a pure undercooled melt. The accuracy is verified by comparing the computational dendrite tip velocity with solvability theory. Numerical simulations, both in 2D and 3D cases, are presented to demonstrate its capacity and efficiency.     

A fully 3D adaptive finite element simulation and prototype for gas flow in a porous media

This paper presents a three-dimensional (3D) adaptive finite element solution for gas flow in a porous media. The solution obtained is truly 3D and employs a dynamic h-adaptive refinement technique for efficiency. The adaptive procedure uses a new node-based storage mechanism [Wang GH, Tyler JM, Weltman JS, Callahan JD. Advances in Engineering Software including Computing Systems in Engineering 1999;30:31–41]. This node-based structure substantially reduces the memory necessary to store the finite element mesh. A prototype simulator, written in C++, has been implemented for Eugene Island block 305, a multi-well condensate reservoir off the coast of Louisiana to demonstrate the use of this adaptive procedure. Results, presented in this paper, show dramatic agreement with the actual production data. This prototype simulator uses Windows workstations to support a fully dynamic 3D mesh plus mesh generation.     

Automated adaptive 3D forming simulation processes

In this study, an automated adaptive mesh control scheme, based on local mesh modifications, is developed for the finite element simulations of 3D metal-forming processes. Error indicators are used to control the mesh discretization errors, and an h-adaptive procedure is conducted. The mesh size field used in the h-adaptive procedure is processed to control the discretization and geometric approximation errors of the evolving workpiece mesh. Industrial problems are investigated to demonstrate the capabilities of the developed scheme.     

On the adjoint-consistent formulation of interface conditions in goal-oriented error estimation and adaptivity for fluid–structure interaction

The numerical solution of fluid–structure-interaction problems poses a paradox in that most of the computational resources are consumed by the subsystem of least practical interest, viz., the fluid. Goal-oriented adaptive discretization methods provide a paradigm to bypass this paradox. Based on the solution of a dual problem, the contribution of local residuals to the error in a specific goal functional is estimated, and only the regions that yield a dominant contribution are refined. In the present work, we address a fundamental complication in the application of goal-oriented adaptivity to fluid–structure-interaction problems, namely, that the treatment of the interface conditions has nontrivial consequences for the properties of the dual problem. In the context of a linearized model problem, we consider two equivalent discretizations differing only on the formulation of the interface coupling terms. By means of an adjoint consistency analysis, we show that only one of these discretizations is adjoint consistent. Numerical experiments convey that the two discretizations behave very differently for the dual problem, and that the adjoint-consistent discretization yields more reliable error estimates. Based on the adjoint-consistent discretization, we finally present some h- and hp-adaptive results, confirming that tremendous savings in computational cost can be realized through the use of goal-oriented refinement strategies. The numerical experiments illustrate that the goal-oriented approach effectively equilibrates the error contributions of the fluid and structure subsystems, which is imperative for efficiently resolving the coupled fluid–structure-interaction problem, and which cannot be accomplished by uniform or residual-based refinement strategies.     

C1 continuous h-adaptive least-squares spectral element method for phase-field models

To describe the interfacial dynamics between two phases using the phase-field method, the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics. Due to the high gradients of the solution within the interfacial region and consequent high computational cost, the use of the phase-field method has been limited to the small-scale problems whose characteristic length is similar to the interfacial thickness. By using finer mesh at the interface and coarser mesh in the rest of computational domain, the phase-field methods can handle larger scale of problems with realistic interface thicknesses. In this work, a $C^1$ continuous $h$-adaptive mesh refinement technique with the least-squares spectral element method is presented. It is applied to the Navier–Stokes-Cahn–Hilliard (NSCH) system and the isothermal Navier–Stokes–Korteweg (NSK) system. Hermite polynomials are used to give global differentiability in the approximated solution, and a space–time coupled formulation and the element-by-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested, and they are compared in two numerical examples.     

H-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the $h$ -adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using $h$ -adaptive mesh method. In this paper, we propose a so-called double tolerance adaptive strategy to optimize the overall numerical efficiency by reducing the number of mesh adaptions, as well as preserving the quality of the numerical solution. Numerical results are presented to demonstrate the robustness and effectiveness of our $h$ -adaptive algorithm using the double tolerance adaptive strategy.     

Controlled cost of adaptive mesh refinement in practical 3D finite element analysis

In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper we propose the use of alternative mesh refinement with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria allow to obtain the maximum of accuracy for a prescribed cost. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.     

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.     

An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.     

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.     

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     

Adaptive finite-volume solution of complex turbulent flows

This paper describes an automatic local grid adaptation procedure driven by an evaluation of the differential residuals of the RANS equations computed using a higher-order reconstruction operator. A suitable data structure is developed for the local mesh adaptation process to be flexible and low CPU time consuming. The whole procedure is designed in the framework of finite-volume methods on unstructured grids. To avoid the appearance of ill-conditioned near-wall cells in the vicinity of curved surfaces of bodies a global mesh deformation technique is used. The whole procedure is applied to a complex turbulent flow around a high-lift multiple element airfoil in take-off configuration using the Spalart-Allmaras turbulence model. The adaptation is controlled by as many indicators as there are equations involved in the problem. It is demonstrated that the proposed methodology performs rigorous local adaptive mesh refinement and automatically achieves grid independent results. Thus, interesting gains are obtained in terms of CPU time, memory requirement and user effort compared to single mesh computations.     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

Adaptive element-free Galerkin method applied to the limit analysis of plates

The implementation of an h-adaptive element-free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended for use in error estimation and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions.     

An h-hierarchical adaptive scaled boundary finite element method for elastodynamics

A posteriori h-hierarchical adaptive scaled boundary finite element method (ASBFEM) for transient elastodynamic problems is developed. In a time step, the fields of displacement, stress, velocity and acceleration are all semi-analytical and the kinetic energy, strain energy and energy error are all semi-analytically integrated in subdomains. This makes mesh mapping very simple but accurate. Adaptive mesh refinement is also very simple because only subdomain boundaries are discretised. Two 2D examples with stress wave propagation were modelled. It is found that the degrees of freedom needed by the ASBFEM are only 5%–15% as needed by adaptive FEM for the examples.     

Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

The paper presents postprocessing techniques based on locally improved finite element (FE) solutions of the basic field variables. This opens up the possibility to control both “strain energy” terms and “kinetic energy” terms in the governing equations. The proposed postprocessing technique on field variables is essentially a least square fit of the prime variables (displacements) at superconvergent points. Its performance is compared with other well-known techniques, showing a good performance. A h-adaptive FE strategy for acoustic problems is presented where, for adaptive mesh generation and remeshing the commercial software package i-deas has been applied and for the FE analysis the commercial software package sysnoise. The paper also presents an adaptive h-version FE approach to control the discretisation error in free vibration analysis. The postprocessing technique used here is a mix of local and global updating methods. Rapid convergence of the preconditioned conjugate gradient method is enhanced by choosing the initial trial eigenmodes as the superconvergent patch recovery technique for displacements improved FE eigenmodes. Numerical examples show nice properties of the final local and global updated solution as a basis for an error estimator and the error indicator in an adaptive process.     

Adaptive Spectral Element Simulations of Thin Premixed Flame Sheet Deformations

The high order Spectral Element Method is used to solve the reaction-diffusion-advection problem as described by the premixed flame case. An h–p adaptive refinement-coarsening algorithm is developed based on a posteriori error estimators. Simulations of the wrinkling of a premixed flame front are used to illustrate how the mesh is adaptively refined. A similar, idealized heat transfer problem is used to show the adaptive refinement and coarsening of the mesh. Adaptivity efficiently provides high resolution in areas of the domain where large or rapidly varying physical changes exist, while saving unnecessary computation where the solution is smooth or physical phenomena have passed by.