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

Abstract

SUMMARYNon-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 nonsymmetric when this non-associated flow rule is applied. Efficient solution of this large-scale nonsymmetric 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 K ep (ILU(0)-K ep ). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)-K ep . To further reduce computational cost, the global stiffness matrix K ep is divided into two parts. The first part is the linear elastic stiffness matrix K e , 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)-K e preconditioner is more time effective than IDR(s = 6) with ILU(0)-K ep 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.

Elastoplastic analysis of pressure‐sensitive materials by an effective three‐dimensional mixed finite element

“…">2.Equations and are solved through a Newton–Raphson algorithm which furnishes also the consistent tangent matrix of the material, see again . 3.For materials like soils the assumption of associative plasticity may lead to an unrealistic dilatancy behaviour of the material during the plastic phase, however, the use of a dilatancy angle different than the friction one destroys the symmetry of the consistent tangent matrix requiring the use of dedicated solvers as shown in .…”