Element-by-Element Parallel Computation of Incompressible Navier--Stokes Equations in Three Dimensions

Abstract

Abstract. Development of a stable finite element model for solving steady incompressible viscous fluid flows in three dimensions is the main theme of the present study. For stability reasons, weighting functions are designed in favor of field variables on the upstream side. For accuracy reasons, it is required that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed. In the steady-state analysis of Navier-Stokes equations, we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the Ladyzhenskaya-Babǔska-Brezzi (LBB) stability condition. To resolve difficulties arising from asymmetry and indefiniteness in the resulting large-size matrix equations, we abandon the elimination-like solution solver because the storage demand exceeds the ability of our hardware to solve for three-dimensional problems. A modern iteration solver, known as the biconjugate gradiant stabilized (BICGSTAB) solution solver, is thus implemented in an element-by-element fashion to effectively alleviate the problem. For performance reasons, the finite element code developed here should be implemented in a hardware environment which is suited to the use of an iterative solver. To this end, our analysis is implemented in shared memory parallel architectures, CRAY C-90 and J-90. We benchmark the parallel computing performance through a lid-driven cavity flow problem and a problem amenable to analytic solution.

Parallel numerical methods for incompressible flows

“…已有的并行数值方法大致可分为两大类: 并行离散方法 [6][7][8][9][10][11] 和离散所得代数方 程组的并行求解方法 [12][13][14][15][16] . 并行策略包括基于单元级的 EBE (Element-by-element) [17] , EDE(Edgeby-edge) [18] , NBN(Node-by-node) [19] 方法、 重叠型区域分解 [20][21][22][23] 与非重叠型区域分解算法 [24][25][26][27] 和 并行两重或多重网格方法 [28][29][30][31][32][33] 以及并行自适应方法 [34][35][36] 等. 对于一个并行数值算法而言, 除了应具有通常意义下一个数值算法应具有的数值特征 (稳定性、 收敛性和准确性等) 和良好的并行性外, 较少的通信需求和较容易的编程实现更是应考虑的重要因素.…”

Numerical exploration of flow topology and vortex stability in a curved duct

“…Re ) is derived to obtain the nodally exact solution in the one-dimensional quadratic element [17], where is expressed in terms of (≡ V y i h y i /2 ):…”

Global Matrix-Free Finite-Element Scheme for Natural Convection in a Square Cavity with Step Blockage

“…Though the frontal solver has been proved to be a versatile method, its performance depends mainly on the frontal width, which has to be made optimum by proper renumbering of nodes. As far as the element-by-element solution scheme [2,3] is concerned, its implementation is not straightforward for flow problems, though some attempts [4,5] have been made to take advantage of the element-by-element solution scheme at the time of solving the final equations using the conjugate gradient iterative solver. Another efficient method is the compact vector storage scheme [6], in which only the nonzero entries of the matrices are stored in a vector and the global positions of these entries are stored in a separate vector of the same size.…”