Two-level Multigrid Research Articles

Variational quadratic shape functions for polygons and polyhedra

Solving partial differential equations (PDEs) on geometric domains is an important component of computer graphics, geometry processing, and many other fields. Typically, the given discrete mesh is the geometric representation and should not be altered for simulation purposes. Hence, accurately solving PDEs on general meshes is a central goal and has been considered for various differential operators over the last years. While it is known that using higher-order basis functions on simplicial meshes can substantially improve accuracy and convergence, extending these benefits to general surface or volume tessellations in an efficient fashion remains an open problem. Our work proposes variationally optimized piecewise quadratic shape functions for polygons and polyhedra, which generalize quadratic P 2 elements, exactly reproduce them on simplices, and inherit their beneficial numerical properties. To mitigate the associated cost of increased computation time, particularly for volumetric meshes, we introduce a custom two-level multigrid solver which significantly improves computational performance.

Aquifer Acceleration in Parallel Implicit Field-Scale Reservoir Simulation

Summary Grid coarsening outside of the areas of interest is a common method to reduce computational cost in reservoir simulation. Aquifer regions are candidates for grid coarsening. In this situation, upscaling is applied to the fine grid to generate coarse-grid flow properties. The efficacy of the approach can be judged easily by comparing the simulation results between the coarse-grid model and the fine-grid model. For many reservoirs in the Middle East bordered by active aquifers, transient water influx is an important recovery mechanism that needs to be modeled correctly. Our experience has shown that the standard grid coarsening and upscaling method do not produce correct results in this situation. Therefore, the objective of this work is to build a method that retains the fine-scale heterogeneities to accurately represent the water movement, but to significantly reduce the computational cost of the aquifer grids in the model. The new method can be viewed as a modified two-level multigrid (MTL-MG) or a specialized adaptation of the multiscale method. It makes use of the vertical-equilibrium (VE) concept in the fine-scale pressure reconstruction in which it is applicable. The method differs from the standard grid coarsening and upscaling method in which the coarse-grid properties are computed a priori. Instead, the fine-scale information is restricted to the coarse grid during Newton's iteration to represent the fine-scale flow behavior. Within the aquifer regions, each column of fine cells is coarsened vertically based on fine-scale z-transmissibility. A coarsened column may consist of a single amalgamated aquifer cell or multiple vertically disconnected aquifer cells separated by flow barriers. The pore volume (PV), compressibility, and lateral flow terms of the coarse cell are restricted from the fine-grid cells. The lateral connectivity within the aquifer regions and the one between the aquifer and the reservoir are honored, inclusive of the fine-scale description of faults, pinchouts, and null cells. Reservoir regions are not coarsened. Two alternatives exist for the fine-scale pressure reconstruction from the coarse-grid solution. The first method uses the VE concept. When VE applies, pressure variation can be analytically computed in the solution update step. Otherwise, the second method is to apply a 1D z-line solve for the fine-scale aquifer pressure from the coarse-grid solution. Simulation results for several examples are included to demonstrate the efficacy and efficiency of the method. We have applied the method to several Saudi Arabian complex full-field simulation models in which the transient aquifer water influx has been identified as a key factor. These models include dual-porosity/dual-permeability (DPDP) models, as well as models with faults and pinchouts in corner-point-geometry grids, for both history match and prediction period. The method is flexible and allows for the optional selection of aquifer regions to be coarsened, either only peripheral aquifers or both the peripheral and bottom aquifers. The new method gives nearly identical results compared with the original runs without coarsening, but with significant reduction in computer time or hardware cost. These results will be detailed in the paper.

Jet noise simulations for realistic jet nozzle geometries

This paper describes a methodology for the direct calculation of noise from realistic nozzle geometries. The focus of the paper is on the numerical approach to this problem to provide noise predictions to engineering accuracy in an efficient manner. In addition, issues related to grid generation are discussed. The methodology uses structured multiblock grids. The block surrounding the jet centerline has a Cartesian form and the surrounding grid blocks have a cylindrical polar form–at least for nearly axisymmetric jet nozzle geometries. Appropriate block interface conditions are used. In the case of military style jet nozzles the nozzles are not smooth in the azimuthal direction but have facets representative of the movable flaps in such variable area nozzles. These features must be included in the grid. To enable efficient calculations, in addition to parallel computation, a dual time-stepping approach is used. The sub-iterations in the fictitious time are accelerated using a two-level multigrid approach. A Detached Eddy Simulation (DES) approach based on the Spalart-Allmaras (S-A) one-equation turbulence model is used. Comparisons are made between flow predictions using the DES with the S-A model everywhere and with the turbulence model turned off in the jet external flow. Noise predictions are made with the permeable surface Ffowcs Williams–Hawkings (FW-H) solution. Noise predictions are presented for both a smooth convergent-divergent nozzle as well as a nozzle representative of a military aircraft engine. Comparisons are made with available experimental data.

Reprint of: Jet Noise Simulations for Realistic Jet Nozzle Geometries

This paper describes a methodology for the direct calculation of noise from realistic nozzle geometries. The focus of the paper is on the numerical approach to this problem to provide noise predictions to engineering accuracy in an efficient manner. In addition, issues related to grid generation are discussed. The methodology uses structured multiblock grids. The block surrounding the jet centerline has a Cartesian form and the surrounding grid blocks have a cylindrical polar form – at least for nearly axisymmetric jet nozzle geometries. Appropriate block interface conditions are used. In the case of military style jet nozzles the nozzles are not smooth in the azimuthal direction but have facets representative of the movable flaps in such variable area nozzles. These features must be included in the grid. To enable efficient calculations, in addition to parallel computation, a dual time-stepping approach is used. The sub-iterations in the fictitious time are accelerated using a two-level multigrid approach. A Detached Eddy Simulation (DES) approach based on the Spalart-Allmaras (S-A) one-equation turbulence model is used. Comparisons are made between flow predictions using the DES with the S-A model everywhere and with the turbulence model turned off in the jet external flow. Noise predictions are made with the permeable surface Ffowcs Williams – Hawkings (FW-H) solution. Noise predictions are presented for both a smooth convergent-divergent nozzle as well as a nozzle representative of a military aircraft engine. Comparisons are made with available experimental data.

Two‐level preconditioner via a rigid body‐based aggregation for the Schur complement system

AbstractMost aggregation multigrid (MG) methods employ rigid body modes in constructing the prolongators and are famous for fast convergence with good efficiency. However, it is nontrivial to extend these methods to the Schur complement (SC) system, which is a partitioned and condensed system, because the rigid body modes utilize the nodal coordinates. In this work, we proposed a prolongator in an explicit form by transforming the SC system into an alternative system. The two‐level preconditioner derived from the proposed prolongator guaranteed the convergence to be the same as a two‐level MG in a single‐domain system. As demonstrated in a typical shell problem, the two‐level preconditioner showed much faster convergence rate, which is estimated by the condition number based on the Lanczos connection, and also improved the dependency of the convergence on the size of elements and subdomains as compared with a single‐level preconditioner. Copyright © 2008 John Wiley & Sons, Ltd.

An efficient semi-coarsening multigrid method for variable diffusion problems in cylindrical coordinates

In this paper, we present an efficient multigrid (MG) algorithm for solving the three-dimensional variable coefficient diffusion equation in cylindrical coordinates. The multigrid V-cycle combines a semi-coarsening in azimuthal direction with the red-black Gauss–Seidel plane (radial-axial plane) relaxation. On each plane relaxation, we further semi-coarsen the axial direction with red-black line relaxation in the radial direction. We also prove the convergence of two-level MG with plane Jacobi relaxation. Numerical results show that the present multigrid method indeed is scalable with the mesh size.

On a two-level multigrid solution method for finite Markov chains

We study a two-level algebraic multigrid scheme for computing the stationary distribution of a homogeneous Markov chain with a finite state space. We reveal its relationship to an iterative aggregation-disaggregation method, provide an error analysis, and prove its local convergence. Moreover, we discuss the stochastic meaning of a block SOR procedure that can be used as iterative smoother of the multigrid scheme.

A hierarchical two-level multigrid solver

In this paper we present a fast, inherently parallel, two-level multigrid (TLMG) algorithm for solving quadratic finite element models containing tetrahedral meshes. The basic idea is to integrate a TLMG algorithm with hierarchical octree structure based on Recursive Spatial Decomposition (RSD). The new hierarchical TLMG (HTLMG) algorithm combines direct substructuring and static condensation with a new algorithm for performing hierarchically the SOR iteration. The hierarchical SOR (HSOR) iteration is designed to operate independently on each subdomain in the octree structure. The time and space complexities of the HSOR and the HTLMG iterations are derived for a regular domain. Two test problems are used to compare the computational cost of the HTLMG algorithm with that of a direct solver.

The Use of Defect Correction to Refine the Eigenelements of Compact Integral Operators

We show how the principle of defect correction can be applied to an integral eigenvalue problem. We present as particular cases of this principle the two important methods which are known as the two-level multigrid and iterative refinement.

Preconditioning and Two-Level Multigrid Methods of Arbitrary Degree of Approximation

Preconditioning and Two-Level Multigrid Methods of Arbitrary Degree of Approximation