Preconditioned IDR(s) iterative solver for non-symmetric linear system associated with FEM analysis of shallow foundation

• Published in: International journal for numerical and analytical methods in geomechanics 2013-12, Vol.37 (17), p.2972-2986
• Main Authors: Tran, H.H.T., Toh, K.C., Phoon, K.K.
• Format: Article
• Language: English
• Subjects: Applied sciences; Buildings. Public works; Computation methods. Tables. Charts; Earthwork. Foundations. Retaining walls; Exact sciences and technology; Finite element method; Geotechnics; induced dimension reduction; Iterative methods; Linear systems; Mathematical analysis; Mathematical models; non-associated; non-symmetric linear system; Nonlinearity; plasticity; preconditioning; Solvers; Stiffness matrix; Structural analysis. Stresses
• ISSN: 0363-9061; 1096-9853
• DOI: 10.1002/nag.2171

SUMMARY Non‐associated flow rule is essential when the popular Mohr–Coulomb model is used to model nonlinear behavior of soil. The global tangent stiffness matrix in nonlinear finite element analysis becomes non‐symmetric when this non‐associated flow rule is applied. Efficient solution of this large‐scale non‐symmetric linear system is of practical importance. The standard Krylov solver for a non‐symmetric solver is Bi‐CGSTAB. The Induced Dimension Reduction [IDR(s)] solver was proposed in the scientific computing literature relatively recently. Numerical studies of a drained strip footing problem on homogenous soil layer show that IDR(s = 6) is more efficient than Bi‐CGSTAB when the preconditioner is the incomplete factorization with zero fill‐in of global stiffness matrix Kep (ILU(0)‐Kep). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)‐Kep. To further reduce computational cost, the global stiffness matrix Kep is divided into two parts. The first part is the linear elastic stiffness matrix Ke, which is formed only once at the beginning of solution step. The second part is a low‐rank matrix Δ, which is re‐formed at each Newton–Raphson iteration. Numerical studies show that IDR(s = 6) with this ILU(0)‐Ke preconditioner is more time effective than IDR(s = 6) with ILU(0)‐Kep when the percentage of yielded Gauss points in the mesh is less than 15%. The total computation time is reduced by 60% when all the recommended optimizing methods are used. Copyright © 2013 John Wiley & Sons, Ltd.