High‐performance multilevel iterative aggregation solver for large finite‐element structural analysis problems

Bulgakov, V. E.; Kühn, G.

doi:10.1002/nme.1620382010

Cited by 39 publications

(31 citation statements)

References 12 publications

(3 reference statements)

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“…227 (80) 221 (78) 337 (128 (5) ) 660 (308 (6) ) * (6) The incomplete factorization applies to the midside node domain only; a complete factorization of the vertex node domain was used in all cases. The number of factoring failures due to negative pivots, if any, is given in parentheses beside the number of iterations An interesting feature of the graphs is that they each have a hump in the vicinity of = 10 −2 , resulting from an excessive number of factoring failures occurring with this drop tolerance.…”

Section: Results For the Larger Problemsmentioning

confidence: 99%

Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems

Graham

1999

Int. J. Numer. Meth. Engng.

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Finite element models of linear elasticity arise in many application areas of structural analysis. Solving the resulting system of equations accounts for a large portion of the total cost for large, three-dimensional models, for which direct methods can be prohibitively expensive. Preconditioned Conjugate Gradient (PCG) methods are used to solve di cult problems with small (1) average element aspect ratios. Incomplete Cholesky (ILL T ) factorizations based on a drop tolerance parameter are used to form the preconditioning matrices. Various new techniques known as reduction techniques are examined. Combinations of these reduction techniques result in highly e ective preconditioners for problems with very poor aspect ratios. Standard and hierarchical triquadratic basis functions are used on hexahedral elements, and test problems comprising a variety of geometries with up to 50 000 degrees of freedom are considered. Manteu el's method of perturbing the sti ness matrix to ensure positive pivots occur during factorization is used, and its e ects on the convergence of the preconditioned system are discussed.

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Section: Results For the Larger Problemsmentioning

confidence: 99%

Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems

Graham

1999

Int. J. Numer. Meth. Engng.

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“…The fine grid and the coarse grid in the aggregate are determined in Equation (4). CG with the approximated SC (singlelevel preconditioner) for the SC system is utilized in comparison with Bi-CG with the proposed two-level preconditioner.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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

Eom

2008

Numer Methods Biomed Eng

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SUMMARYMost 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.

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“…For the Poisson operators consider here this is simply the constant vector (eg, a vector of all ones). Many plane aggregation methods have been developed [9,14,15]. Aggregation methods, as the name implies, aggregate nodes and then inject the kernel vectors onto these nodes sets resulting in piecewise constant coarse grid space functions.…”

Section: Smoothed Aggregation Multigridmentioning

confidence: 99%

Unstructured finite element solvers in gyrokinetic turbulence simulations of burning plasmas

Adams¹,

Nishimura²,

Henson³

2005

J. Phys.: Conf. Ser.

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Abstract. Many plasma physics simulation applications utilize an unstructured finite element method (FEM) discretization of an elliptic operator, of the form −∇ 2 u + αu = f ; when α is equal to zero a "pure" Laplacian or Poisson equation results and when α is greater than zero a Helmholtz equation is produced. Discretized equations of this form, often 2D, occur in many tokamak fusion plasma, or burning plasma, applications -from MHD to Gyrokinetic codes. This report investigates the performance characteristics of basic classes of linear solvers (ie, direct, one-level iterative, and multilevel iterative methods) on 2D unstructured FEM problems of the form −∇ 2 u + αu = f , with both α = 0 and α = 0. The purpose of this work is to provide computational physicists guidelines as to appropriate linear solvers for their problems via detailed performance analysis, in terms of both the scalability and the constants in the solution times. We show, as expected, almost perfect scalability of multilevel methods and quantify the solution costs on a common computational platform -the IBM SP Power3.

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High‐performance multilevel iterative aggregation solver for large finite‐element structural analysis problems

Cited by 39 publications

References 12 publications

Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems

Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems

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

Unstructured finite element solvers in gyrokinetic turbulence simulations of burning plasmas

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