A Note on the Effect of the Choice of Weak Form on GMRES Convergence for Incompressible Nonlinear Elasticity Problems

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
Pras Pathmanathan ◽  
Jonathan P. Whiteley ◽  
S. Jonathan Chapman ◽  
David J. Gavaghan
Keyword(s):  
Linear Systems ◽  
Nonlinear Elasticity ◽  
Stokes Flow ◽  
Symmetric Matrix ◽  
Incorrect Choice ◽  
Incompressible Nonlinear Elasticity ◽  
The Matrix ◽  
Elasticity Problems ◽  
Incompressibility Constraint ◽  
Symmetric Block

The generalized minimal residual (GMRES) method is a common choice for solving the large nonsymmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes’ flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of the Navier–Stokes flow (where the matrix is block-symmetric/block-skew-symmetric); however, there has been very little investigation into the GMRES performance in incompressible nonlinear elasticity problems, where the nonlinearity of the incompressibility constraint means the matrix is not block-symmetric/block-skew-symmetric. In this short paper, we identify one feature of the problem formulation, which has a huge impact on unpreconditioned GMRES convergence. We explain that it is important to ensure that the matrices are perturbations of a block-skew-symmetric matrix rather than a perturbation of a block-symmetric matrix. This relates to the choice of sign before the incompressibility constraint integral in the weak formulation (with both choices being mathematically equivalent). The incorrect choice is shown to have a hugely detrimental effect on the total computation time.

On Stability of Linear Systems with Impulsive Action at the Matrix

2018 ◽  
Vol 230 (5) ◽  
pp. 673-676 ◽  
Author(s):  
N. I. Zhelonkina ◽  
A. N. Sesekin
Keyword(s):  
Linear Systems ◽  
Impulsive Action ◽  
The Matrix

Richardson's Iteration With Dynamic Parameters and the SIP Incomplete Factorization for the Solution of Linear Systems of Equations

1981 ◽  
Vol 21 (06) ◽  
pp. 699-708
Author(s):  
Paul E. Saylor
Keyword(s):  
Linear Systems ◽  
Field Data ◽  
Symmetric Matrix ◽  
Simple Problem ◽  
Test Problems ◽  
Physical Parameters ◽  
Incomplete Factorization ◽  
The Real ◽  
Hydrocarbon Reservoir ◽  
Linear Algebraic Equations

Abstract Reservoir simulation yields a system of linear algebraic equations, Ap=q, that may be solved by Richardson's iterative method, p(k+1)=p(k)+tkr(k), where r(k)=q-Ap(k) is the residual and t0, . . . tk are acceleration parameters. The incomplete factorization, Ka, of the strongly implicit procedure (SIP) yields an improvement of Richardson's method, p(k+1)=p(k)+tkKa−1r(k). Parameter a originates from SIP. The product of the L and U factors produced by SIP gives Ka=LU. The best values of the tk acceleration parameters may be computed dynamically by an efficient algorithm; the best value of a must be found by trial and error, which is not hard for only one value. The advantages of the method are (1) it always converges, (2) with the exception of the a parameter, parameters are computed dynamically, and (3) convergence is efficient for test problems characterized by heterogeneities and transmissibilities varying over 10 orders of magnitude. The test problems originate from field data and were suggested by industry personnel as particularly difficult. Dynamic computation of parameters is also a feature of the conjugate gradient method, but the iteration described here does not require A to be symmetric. Matrix Ka−1 A must be such that the real part of each eigenvalue is nonnegative, or the real part of each is nonpositive, but not both positive and negative. It is in this sense that the method always converges. This condition is satisfied by many simulator-generated matrices. The method also may be applied to matrices arising from the simulation of other processes, such as chemical flooding. Introduction The solution of a linear algebraic system, Ap=q, is a basic, costly step in the numerical simulation of a hydrocarbon reservoir. Many current solution methods are impractical for large linear systems arising from three-dimensional simulations or from reservoirs characterized by widely varying and discontinuous physical parameters. An iterative solution is described with these two main advantages:it is efficient for difficult problems andthe selection of iteration parameters is straightforward. The method is Richardson's method applied to a preconditioned linear system. Matrix A may be symmetric or nonsymmetric. In the simulation of multiphase flow, it is usually nonsymmetric. Convergence behavior is shown for four examples. Two of these, Examples 3 and 4, were provided by an industry laboratory (Exxon Production Research Co.), and were suggested by personnel as especially difficult to solve; SIP failed to converge and only the diagonal method1 was effective. Convergence of Richardson's method is compared with the diagonal method using data from a laboratory run. The other two examples are: Example 1, a matrix not difficult to solve, generated from field data, and Example 2, a variant of a difficult matrix described by Stone.2 The easy matrix of Example 1 is included to show the performance of Richardson's method (with preconditioning) on a simple problem.

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

Pell Numbers, Pell–Lucas Numbers and Modular Group

2007 ◽  
Vol 14 (01) ◽  
pp. 97-102 ◽  
Author(s):  
Q. Mushtaq ◽  
U. Hayat
Keyword(s):  
Modular Group ◽  
Symmetric Matrix ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Pell Numbers ◽  
The Matrix

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.

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