Local Multigrid Method Research Articles

Local multigrid method for solving adaptive finite element equations of three-dimensional elasticity problems

Local multigrid method for solving adaptive finite element equations of three-dimensional elasticity problems

Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nedelec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dorfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e. , the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

Parallel-adaptive simulation with the multigrid-based software framework UG

In this paper we present design aspects and concepts of the unstructured grids (UG) software framework that are relevant for parallel-adaptive simulation of time-dependent, nonlinear partial differential equations. The architectural design is discussed on system, subsystem and component level for distributed mesh management and local adaptation capabilities. Parallelization is founded on top of the innovative programming model dynamic distributed data (DDD). Newly introduced modules and extensions of DDD are discussed. Local multigrid methods are introduced as optimal linear solvers in the solution process. The demands of local parallel mesh adaptation are further described: Beside a mesh manipulation module further steps dynamic load balancing and migration have to be introduced. Their realization in the context of local multigrid methods is significantly non-trivial and makes the major contribution to the paper presented here. Parallel I/O provides an efficient mechanism for restart, postprocessing and long-term, large-scale computations. The UG approach is verified through a considerable code-reuse fraction of nearly 90% for simulations of complicated phenomena like porous media flow and transport as well as elastoplasticity. Parallel simulations with up to 108 unknowns are shown for the Couplex benchmark. Therefore a grid convergence study to verify the reliability of the computed results is possible. For an parallel-adaptive elastoplasticity computation the speedup of the multigrid solver, which is the most scalability critical simulation part, exceeds on 512 processor a value of 300. The overhead introduced by the parallel-adaptive scheme turns out to be below 10% of the whole simulation time.

Adaptive local multigrid methods for solving time-harmonic eddy-current problems

The efficient computation of large eddy-current problems with finite elements requires adaptive methods and fast optimal iterative solvers such as multigrid methods. This paper provides an overview of the most important implementation aspects of an adaptive multigrid scheme for time-harmonic eddy currents. Numerical experiments show that the standard multigrid scheme can be modified to yield an O(N) complexity even for general adaptive refinement strategies, where the number of unknowns N can grow slowly from one to the next refinement level. Algorithmic details and numerical examples are given.

Large‐scale density‐driven flow simulations using parallel unstructured Grid adaptation and local multigrid methods

AbstractAdvanced parallel applications based on the message‐passing paradigm are difficult to design and implement, especially when solution adaptive techniques are used and three‐dimensional problems on complex geometries are faced, which yield the use of unstructured Grids. We present the building blocks for a parallel‐adaptive scheme for the solution of time‐dependent and nonlinear partial differential equations. To minimize computational requirements, h‐adaptivity is introduced via parallel, local Grid adaptation. Novel techniques to avoid hanging nodes are introduced, these assure conforming meshes of hybrid element type in three space dimensions. As a core of the adaptive scheme, local multigrid methods are used to solve the arising linear systems rapidly in parallel. Dynamic Grid changes from h‐adaptivity lead to load imbalance during run time, therefore dynamic load balancing and migration is performed to exploit the aggregated performance of large processor sets efficiently. Real‐world calculations arising from density‐driven flow problems in porous media are performed using the presented parallel‐adaptive solution strategy. The computations are analyzed with regard to speedup. Timings of Grid adaptation, dynamic load balancing/migration and numerical solution scheme show that large‐scale runs on 512 processors gain an overall parallel, numerical speedup of up to 278. A further reduction of the element count by h‐adaptivity by a factor of up to 195 shows the enormous capabilities of the presented parallel‐adaptive multigrid based solution scheme. Copyright © 2005 John Wiley & Sons, Ltd.

A multigrid adaptive mesh refinement strategy for 3D aerodynamic design

Presently, improving the accuracy and reducing computational costs are still two major CFD objectives often considered incompatible. This paper proposes to solve this dilemma by developing an adaptive mesh refinement method in order to integrate the 3D Euler and Navier–Stokes equations on structured meshes, where a local multigrid method is used to accelerate convergence for steady compressible flows. The time integration method is a LU-SGS method (AIAA J 1988; 26: 1025–1026) associated with a spatial Jameson-type scheme (Numerical solutions of the Euler equations by finite volume methods using Runge–Kutta time-stepping schemes. AIAA Paper, 81-1259, 1981). Computations of turbulent flows are handled by the standard k–ω model of Wilcox (AIAA J 1994; 32: 247–255). A coarse grid correction, based on composite residuals, has been devised in order to enforce the coupling between the different grid levels and to accelerate the convergence. The efficiency of the method is evaluated on standard 2D and 3D aerodynamic configurations. Copyright © 2004 John Wiley & Sons, Ltd.

Modelling inclusions, holes and fibre reinforced composites using the local multi-grid finite element method

This paper examines the application of the local multi-grid finite element method to model both a single inclusion in an infinite plate and unidirectional laminae consisting of square and hexagonal packed arrays of fibres. Single and arrays of holes are modelled by assigning a negligible inclusion/fibre modulus of elasticity. The paper demonstrates that the local multi-grid method provides a convenient and accurate method for modelling single and arrays of inclusions compared to conventional structured and unstructured finite element models. Convenient in that local inclusion/fibre ‘primitive’ patches greatly assist in generating meshes consisting of arrays of fibres, and accurate in that structured base and patch meshes can be used, even though interpolation is used between levels.

Modelling inclusions, holes and fibre reinforced composites using the local multi-grid finite element method

This paper examines the application of the local multi-grid finite element method to model both a single inclusion in an infinite plate and unidirectional laminae consisting of square and hexagonal packed arrays of fibres. Single and arrays of holes are modelled by assigning a negligible inclusion/fibre modulus of elasticity. The paper demonstrates that the local multi-grid method provides a convenient and accurate method for modelling single and arrays of inclusions compared to conventional structured and unstructured finite element models. Convenient in that local inclusion/fibre ‘primitive’ patches greatly assist in generating meshes consisting of arrays of fibres, and accurate in that structured base and patch meshes can be used, even though interpolation is used between levels.

A One-Cell Local Multigrid Method for Solving Unsteady Incompressible Multiphase Flows

An original local multigrid method for solving incompressible two-phase flow with surface tension is described. The dynamics of the interface are resolved on a hierarchy of structured and uniform grids (orthogonal Cartesian meshes). A new type of composite boundary condition is proposed to solve the dynamics of the multigrid calculation domains. The interface tracking is described by a TVD VOF algorithm and the equations of motion are solved using an augmented Lagrangian method. The surface tension is calculated using a continuous surface force method. The one-cell local multigrid method is compared to relevant analytical scalar advection tests. Several classical two-phase flow problems, including nonlinear drop oscillations, Rayleigh–Taylor instabilities, and the drop impact on liquid film, have also been considered. The local character of the method and the differences between a single-grid and a multigrid solution are discussed. For unsteady problems, such as the Rayleigh–Taylor instability, the memory costs and the computational time have been reduced by up to 50%.

Numerical solving of incompressible Navier-Stokes equations using an original local multigrid refinement method

Numerical simulation of complex flows involving unsteady structures, boundary layers or free surfaces on a unique grid is very greedy in time and CPU because a great number of calculation point is needed to ensure a satisfactory description of the different scales of the problem. An original local multigrid method, able to refine the grid at the cell scale, is proposed to solve the Navier-Stokes and energy equations. The results obtained for the natural convection in a square cavity are compared to the one of the literature for Rayleigh numbers in the range 10 5–10 8.

LOCAL MULTIGRID REFINEMENT METHOD FOR NATURAL CONVECTION IN FISSURED POROUS MEDIA

Abstract The problem of natural convection in a fissured porous layer is studied. A local multigrid method is used in order to take local variations induced by the fissure into account. The method is based on a correction of the fluxes at the interface between the different grids. It is applied to Darcy's law and to the energy balance equation. The principle of the method is first described, and its adaptation to the coupled equations of Darcy and energy is then detailed. We present results in two dimensions in the case of a fissured porous layer with permeable top heated from below. These results are compared to reference solutions.

An adaptive multigrid technique for three‐dimensional elasticity

AbstractA program for finite element analysis of 3D linear elasticity problems is described. The program uses quadratic hexahedral elements. The solution process starts on an initial coarse mesh; here error estimators are determined by the standard Babuška‐Rheinboldt method and local refinement is performed by partitioning of indicated elements, each hexahedron into eight new elements. Then the discrete problem is solved on the second mesh and the refinement process proceeds in the following way‐on the ith mesh only the elements caused by refinement on the (i‐1)th mesh can be refined. The control of refinement is the task of the user because the dimension of the discrete problem grows very rapidly in 3D. The discrete problem is being solved by the frontal solution method on the initial mesh and by a newly developed and very efficient local multigrid method on the refined meshes. The program can be successfully used for solving problems with structural singularities, such as re‐entrant corners and moving boundary conditions. A numerical example shows that such problems are solved with the same efficiency as regular problems.

An algebraic study of a local multigrid method for variational problems

A two-level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed. The smoothing steps consists of Gauss-Seidel relaxations applied only to the local mesh. A direct solver is required on the global coarse mesh. We analyze this scheme by modifying the standard algebraic convergence theory. The estimates of the convergence factor do not require any regularity assumption. The estimates can be evaluated element-by-element. The evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples. The theory also applies to a method with a direct solver on the local fine mesh. In this case we have obtained very sharp (and in fact optimal) estimates of the convergence factors.