Adaptive Multigrid Method Research Articles

Multigrid methods for open boundary problems in geophysics

The proposed numerical procedure for static open boundary problems combines fast adaptive multigrid methods with spatial mappings of the infinite region to a finite computational domain. A rigorous analysis of the finite element solution error for ballooning and for spatial mappings is presented. The method is applied to geophysical problems of delogging.

Adaptive multigrid method using duality in plane elasticity

This work presents an adaptive multigrid method for the mixed formulation of plane elasticity problems. First, a mixed-hybrid formulation is introduced where the continuity of the normal components of the stress tensor is indirectly imposed using a Lagrange multiplier. Two different numerical approximations, naturally associated with the primal problem and the dual problem, are then proposed. The Complementary Energy Principle provides an a posteriori error estimate. For the effective solving of both systems of equations, a non-standard multigrid algorithm has been designed that allows us to solve the two problems, dual and primal, with reasonable cost and in an integrated way. Finally, a significant numerical application is presented to check the efficiency of the error estimator and the good performance of the algorithm. Copyright © 2001 John Wiley & Sons, Ltd.

Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves

Partial differential equations can be solved efficiently by adaptive multigrid methods on a parallel computer. We report on the concept of hash-table storage techniques to set up such a program. The code requires substantial less amount of memory than implementations based on tree type data structures and is easier to program in the sequential case. The parallelization takes place by a space-filling curve domain decomposition intimately connected to the hash table. The new data structure simplifies the parallelization of the code substantially and introduces a cheap way to solve the load balancing and mapping problem. We report on the main features of the method and give the results of numerical experiments with the new parallel solver on a cluster of 64 Pentium II/400MHz connected by a Myrinet in a fat tree topology.

An adaptive multigrid technique for evaluating long-range forces in biomolecular simulations

We propose an adaptive multigrid method for evaluating the nonlocal, long-range interactions in biomolecular models. The method provides O( h 3) accuracy for the forces, where h is the mesh size of the finest cubic grid, even though the stencil size is minimal. Two new algorithmic components are developed: an adaptive technique for spreading a three-dimensional charge distribution onto a space mesh, and a procedure for performing coarse-to-fine grid interpolation of the potential, force components, and their derivatives (grid-to-particle interpolation at the finest level) in a consistent fashion. The adaptive mesh used in the spreading procedure contains two types of nodes: regular, positioned at nodes cubic grid, and flexible, one per cubic cell. Thus, the charges located in each grid cell are spread to these nine nodes. The method conserves charge, dipole, and quadrupole moments of cells. The spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils. Implementation is also straightforward. Our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique.

A NON-LINEAR ADAPTIVE FULL TRI-TREE MULTIGRID METHOD FOR THE MIXED FINITE ELEMENT FORMULATION OF THE NAVIER-STOKES EQUATIONS

The full adaptive multigrid method is based on the tri-tree grid generator. The solution of the Navier–Stokes equations is first found for a low Reynolds number. The velocity boundary conditions are then increased and the grid is adapted to the scaled solution. The scaled solution is then used as a start vector for the multigrid iterations. During the multigrid iterations the grid is first recoarsed a specified number of grid levels. The solution of the Navier–Stokes equations with the multigrid residual as right-hand side is smoothed in a fixed number of Newton iterations. The linear equation system in the Newton algorithm is solved iteratively by CGSTAB preconditioned by ILU factorization with coupled node fill-in. The full adaptive multigrid algorithm is demonstr ated for cavity flow. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1037ndash;1047, 1997.

Parallel adaptive multigrid methods in plane linear elasticity problems

Parallel adaptive multigrid methods in plane linear elasticity problems

Parallel adaptive multigrid methods in plane linear elasticity problems

In this paper we discuss the implementation of parallel multigrid methods on unstructured locally refined meshes for 2D linear elasticity calculations. The problem of rebalancing the workload of the processors during execution of the program is discussed in detail and a load balancing algorithm suited for hierarchical meshes is proposed. For large problems the efficiency per multigrid iteration ranges from 65% on 64 processors in the locally refined case to 85% on 256 processors in the uniformly refined case. All calculations were carried out on a CRAY T3D machine. © 1997 John Wiley & Sons, Ltd.

A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent

A fast multigrid boundary element (MBE) method for solving the Poisson equation for macromolecular electrostatic calculations in a solvent is developed. To convert the integral equation of the BE method into a numerical linear equation of low dimensions, the MBE method uses an adaptive tesselation of the molecular surface by BEs with nonregular size. The size of the BEs increases in three successive levels as the uniformity of the electrostatic field on the molecular surface increases. The MBE method provides a high degree of consistency, good accuracy, and stability when the sizes of the BEs are varied. The computational complexity of the unrestricted MBE method scales as O(Nat), where Nat is the number of atoms in the macromolecule. The MBE method is ideally suited for parallel computations and for an integrated algorithm for calculations of solvation free energy and free energy of ionization, which are coupled with the conformation of a solute molecule. The current version of the 3-level MBE method is used to calculate the free energy of transfer from a vacuum to an aqueous solution and the free energy of the equilibrium state of ionization of a 17-residue peptide in a given conformation at a given pH in ∼ 400 s of CPU time on one node of the IBM SP2 supercomputer. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 569–583, 1997

The prolonged adaptive multigrid method for finite element Navier-Stokes equations

A Tri-Tree Method for generating finite element grids in two and three dimensions is developed. The method is based on a new tree search structure. The search tree is built upon triangles in two dimensions and tetrahedra in three dimensions. The density of elements can be varied throughout the computational domain. A Coupled Node Fill-In ILU preconditioner for implicit and itererative solution of the Navier-Stokes equations has been designed. The properties of preconditioner, which is based on new fill-in rules, permit fill-in and preconditioning of the pressure in the equation system. The Tri-Tree grid generation algorithm includes local refinements. The spatial location of refinements are based on the solution calculated for the previous grid. The element Reynolds number is calculated for each Tri-Tree element and Tri-Tree elements with Reynolds number above a predefined limit are recursively refined. Elements with Reynolds number below this limit are recoarsed. A Prolonged Adaptive Multigrid Method, which is based on the Tri-Tree grid generation method and the Coupled Node Fill-In preconditioner is developed. The Prolonged Adaptive Multigrid Method reveals advantages in both robustness and efficiency, especially for nonlinear and nonpositive definite equation systems. In order to reduce the overall work, the element matrices are integrated analytically.

THE PARALLEL BLOCK ADAPTIVE MULTIGRID METHOD FOR THE IMPLICIT SOLUTION OF THE EULER EQUATIONS

A method capable of solving very fast and robust complex non-linear systems of equations is presented. The block adaptive multigrid (BAM) method combines mesh adaptive techniques with multigrid and domain decomposition methods. The overall method is based on the FAS multigrid, but instead of using global grids, locally enriched subgrids are also employed in regions where excessive solution errors are encountered. The final mesh is a composite grid with uniform rectangular subgrids of various mesh densities. The regions where finer grid resolution is necessary are detected using an estimation of the solution error by comparing solutions between grid levels. Furthermore, an alternative domain decomposition strategy has been developed to take advantage of parallel computing machines. The proposed method has been applied to an implicit upwind Euler code (EuFlex) for the solution of complex transonic flows around aerofoils. The efficiency and robustness of the BAM method are demonstrated for two popular inviscid test cases. Up to 19-fold acceleration with respect to the single-grid solution has been achieved, but a further twofold speed-up is possible on four-processor parallel computers.

An efficient multiparticle diffusion simulation by an adaptive multigrid method

An adaptive multigrid method for efficient multiparticle diffusion simulation is presented. This ensures avoidance of the problem of ‘stiffness’ due to large discrepancies of the particles' characteristic lengths by using a local grid decomposition algorithmic strategy in which each particle of the diffusion system has its own adaptive multilevel grid structure. This numerical approach is implemented in the two-dimensional process simulator MUSIC, whose capabilities are illustrated by the results of coupled point defect and impurity diffusion simulation under local wet oxidation conditions, as well as by the results of complete process flow simulation of the low-voltage power VDMOSFET.

The efficient simulation of point diffusion by an adaptive multigrid method

The central goal of this paper is to demonstrate the efficiency of adaptive multilevel numerical method based on full multigrid algorithm for the simulation of point defects diffusion. This numerical approach is implemented in process simulator MUSIC [1] and it is applied for the simulation of interstitials and vacancies diffusion during the local oxidation process.

Adaptive multi-grid methods for finite element systems with BI- and unilateral constraints

Adaptive multi-grid methods for finite element systems with BI- and unilateral constraints