Local Rectangular Refinement in Three Dimensions (LRR3D): Development of a Solution-Adaptive Gridding Technique with Application to Convection-Diffusion Problems

Abstract

The local rectangular refinement (LRR) solution-adaptive gridding method, developed a decade ago to solve coupled nonlinear elliptic partial differential equations in two dimensions, has been extended to three dimensions. Like LRR2D, LRR3D automatically generates orthogonal unstructured adaptive grids, discretizes governing equations using multiple-scale finite differences, and solves the discretized system at all points simultaneously using Newton's method. The computational=programming challenges overcome in developing LRR3D are described. Next, a three-dimensional convection-diffusionreaction problem with a known solution is used to demonstrate the accuracy and efficiency of LRR3D versus a Newton solver on a structured grid. Finally, natural convection within a tilted differentially heated cubic cavity with perfectly conducting nonisothermal walls is examined for a range of Rayleigh numbers (10 3 to 4 Â 10 4 ) and for two inclinations (45 and 90 ) of the isothermal walls. Excellent agreement with published computational and experimental results is observed.

A parallel solution – adaptive method for three-dimensional turbulent non-premixed combusting flows

“…An alternative strategy for adaptive mesh refinement relies on equidistribution of solution error based on local estimates of the gradient and curvature of the solution [88,89,23,[27][28][29][30][31]. Although not implemented herein, this methodology has been applied to steady [24,90,91,25,26,88,89,23,[27][28][29][30][31] and unsteady [46,28] combustion simulations. Note however, in most applications involving nonlinear partial-differential equations, selecting the error indicators is not straightforward and sometimes, the error indicators lack theoretical justifications, as noted in [92].…”

“…Computational grids are automatically adapted to the solution of the governing equations and this can be very effective in treating problems with multiple scales, providing the required spatial resolution while minimizing memory and storage requirements. AMR approaches have since been developed for a wide variety of engineering problems [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and combustion simulations [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Large massively-parallel distributed-memory computers provide another approach by enabling a many fold increase in processing power and memory resources beyond those of conventional single-processor computers.…”

Hybrid mesh adaptation applied to industrial numerical combustion

“…When the metric is the identity matrix, one finds the Euclidean notion of distance. A Riemannian metric allows the local properties of an element to be described through its decomposition shown in Equation (18). The lengths of the elements in each direction are obtained through the eigenvalues such that:…”

“…Currently, the preferred approach in the simulation of reacting flow seems to be mesh refinement through local element subdivision, see [6,[16][17][18][19][20][21][22][23]. All of these methods use isotropic refinement and coarsening operations on quadrilateral meshes in 2D and hexahedral meshes in 3D.…”

MC-Smooth: A mass-conserving, smooth vorticity–velocity formulation for multi-dimensional flows

“…Specifically, since the Poisson approach does not involve any integrodifferential or hyperbolic equations in its set of governing equations, its computational cost is moderate, and its BCs are straightforward to implement. In addition, the Poisson approach does not require the employment of staggered grids, so it can effectively alleviate difficulties associated with complex geometries, non-orthogonal curvilinear coordinates, and the employment of advanced numerical techniques such as adaptive grid refinement [28][29][30][31] or multigrid methods [32]. Being able to maintain the second-order ellipticity, and thus advantageous convergence properties, is another merit of the Poisson approach because, when solved by a fully implicit solver, the second-order elliptic nature of the governing equations will enable an implicit solution to be achieved within a comparatively short CPU time [33,34].…”