An adaptive finite-element strategy for the three-dimensional time-dependent Navier-Stokes equations (original) (raw)
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A simple algoritlinl for adaptive and automatic h refinement of t,hree dimensional tetrahedral finite element meshes is presented. The algorithm is based on an aposteriori error indicator that is calculated by the finite element solver. The elrment subdivision algorithm is robust and rccilrsive. Smooth transition between large and small c~l(~lnents is achieved without significant degradation of the aspect rat,io of t,he elements in the n~e s h. An cxaniplr is presented to illustrate the utility of the approach.
31st Aerospace Sciences Meeting, 1993
This paper describes recent developments of high resolution finite element schemes for the solution of the unsteady compressible Euler and Navier-Stokes equations on unstructured meshes. These finite element algorithms use an edge-based data structure, as opposed to a more traditional element-based data structure. The advantage of using such an edge-based data structure is that it provides a unified approach in which the relation between centered and upwind schemes becomes apparent, improves the efficiency of the algorithms, and reduces the storage requirements. A variety of numerical schemes using such edgebased data structure, ranging from Godunov schemes to centered schemes with blended dissipation, is presented and discussed. Adaptive mesh refinement is then added to these solvers to enhance the solution accuracy and efficiency. Various numerical results for a wide range of flow conditions, from subsonic to hyperaonic in both 2D and 30, are presented to demonstrate the performance and versatility of the proposed schemes.-'
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We discuss the development of parallel algorithms for adaptive mesh refinement (AMR) and their application to large-scale problems in simulating fluid dynamics. Our approach to AMR can be described as using a forest of octrees (or quadtrees in 2D) that are adaptively refined. The storage of elements is distributed in parallel, and fast and scalable algorithms exist for dynamic refinement/coarsening and other important tasks, such as partitioning and the extraction of one layer of off-process (ghost) neighbours. Our contributions are twofold: (a) We use the p4est software as a basis to create numerical applications to simulate the flow of gas in the atmosphere (advection equations), variably saturated subsurface flow (Richards), and the free flow of liquid (Navier-Stokes). (b) In addition to using quadrilateral/cubic elements, we are developing space filling curves and high-level AMR algorithms for triangles and tetrahedra. We include scalability results and simulation snapshots obtained on the JUQUEEN supercomputer.
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This paper presents a numerical study of 3D Navier-Stokes benchmark problems defined within the DFG high-priority research program in 1996. Specifically, we investigate the accuracy of an equal-order finite element method based on piecewise quadratic shape functions with local projections stabilization on locally refined meshes for stationary laminar flows around an obstacle with circular and square cross-section. It turns out that on globally refined meshes the new stabilization method is comparable to Q 2 /P disc 1 element which was the best one in recent investigations of John [1]. Furthermore, on locally refined meshes we are able to produce reference values for the geometry with singularities (square cross section) which were still unknown up to now.
Parallel adaptive mesh refinement for incompressible flow problems
Computers & Fluids, 2013
The present article describes a simple element-driven strategy for the conforming refinement of simplicial finite element meshes in a distributed environment. The proposed algorithm is effective both for local adaptive refinement and for the division of all the elements within an existing mesh. We aim to provide sufficient detail to allow the practical implementation of the algorithm, which can be coded with minimal effort provided that a distributed linear algebra library is available. The proposed refinement strategy is composed of three basic components: a global splitting strategy, an elemental splitting procedure and an error estimation technique, which are combined so to guarantee obtaining a conformant refined mesh. A number of benchmark examples show the capabilities of the proposed method. Error is estimated for the incompressible fluid-flow benchmarks using a novel indicator based on the computation of the sub-scale velocity.
12th Computational Fluid Dynamics Conference, 1995
This paper presents a finite volume cell-centered technique for computing steady state solutions of the full Euler and Navier-Stokes equations on unstructured meshes. We aim t o design a scheme which is the most possible insensitive t o grid distortions, while however remaining of practical interest in terms of CPU time, storage and convergence. For that purpose, we use an original quadratic reconstruction with a fixed stencil and a high order flux integration by the Gauss quadrature rule t o compute the advective term of the equations. Time evolution is presently performed with an explicit multi-step Runge-Kutta scheme. A very general adaptation procedure based on h-refinement and coarsening is employed t o improve the resolution of complex flow features. The accuracy of the method is demonstrated for a linear equation and for inviscid and viscous flow computations. The inviscid flow over the NACA0012 airfoil is computed at various Mach numbers. These calculations illustrate the effectiveness of the adaptation procedure. We investigate the supersonic flow over a compression ramp t o validate the Navier-Stokes solver by using a hybrid grid.
Adaptive refinement strategies in three dimensions
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3-D Adaptive mesh refinement seems to be the answer towards full automation in the analysis of electromagnetic devices. The computational burden of 3-D problems can be reduced since with an efficient error estimator, optimal tetrahedral meshes are produced and the computational cost is minimized Further, the process of mesh generation, a headache for a finite element analyst, is completely automatic, requiring no user intervention. So self-adaptive mesh generation is proved to be an indispensable tool in 3-dimensional finite element analysis. In this paper 3 different criteria for error estimation are presented that are able to estimate the error present in the approximation model and the strategy followed in their implementation is analyzed Application to a test problem shows the effectiveness of the self-adaptive mesh generation algorithm
Controlled cost of adaptive mesh refinement in practical 3D finite element analysis
Advances in Engineering Software, 2007
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.
A convergent finite element method with adaptive√ 3 refinement
We develop an adaptive finite element method (AFEM) using piecewise linears on a sequence of triangulations obtained by adaptive √ 3 refinement. The motivation to consider √ 3 refinement stems from the fact that it is a slower topological refinement than the usual red or red-green refinement, and that it alternates the orientation of the refined triangles, such that certain features or singularities that are not aligned with the initial triangulation might be detected more quickly. On the other hand, the use of √ 3 refinement introduces the additional difficulty that the corresponding finite element spaces are nonnested. This makes the setting nonconforming. First we derive a BPX-type preconditioner for piecewise linears on the adaptively refined triangulations and we show that it gives rise to uniformly bounded condition numbers, so that we can solve the linear systems arising from the AFEM in an efficient way. Then we introduce the AFEM of Morin, Nochetto, and Siebert adapted to our special case for solving the Poisson equation. We prove that this adaptive strategy converges to the solution within any prescribed error tolerance in a finite number of steps. Finally we present some numerical experiments that show the optimality of both the BPX preconditioner and the AFEM.
Adaptive mesh refinement for high-resolution finite element schemes
International Journal for Numerical Methods in Fluids, 2006
New a posteriori error indicators based on edgewise slope-limiting are presented. The L2-norm is employed to measure the error of the solution gradient in both global and element sense. A second-order Newton–Cotes formula is utilized in order to decompose the local gradient error from a 1 finite element solution into a sum of edge contributions. The slope values at edge midpoints are interpolated from the two adjacent vertices. Traditional techniques to recover (superconvergent) nodal gradient values from consistent finite element slopes are reviewed. The deficiencies of standard smoothing procedures—L2-projection and the Zienkiewicz–Zhu patch recovery—as applied to nonsmooth solutions are illustrated for simple academic configurations. The recovered gradient values are corrected by applying a slope limiter edge-by-edge so as to satisfy geometric constraints. The direct computation of slopes at edge midpoints by means of limited averaging of adjacent gradient values is proposed as an inexpensive alternative. Numerical tests for various solution profiles in one and two space dimensions are presented to demonstrate the potential of this postprocessing procedure as an error indicator. Finally, it is used to perform adaptive mesh refinement for compressible inviscid flow simulations. Copyright © 2006 John Wiley & Sons, Ltd.