Keywords: coupled problems, finite element method, shield tunnelling, mixed formulation, block-preconditioning, preconditioned Krylov solver.

This paper features a block preconditioning strategy for the solution of coupled problem of partially saturated soils in the numerical simulation of shield tunnelling. It is based upon a similar strategy for two-phase materials proposed by White & Borja in [1] where the block preconditioning strategy is used for a fully coupled flow problem. Although these two problems differ a lot in their mathematical and mechanical formulation, the finite element formulation of both problems leads to a well-known block structure of the stiffness matrix which can be separated into phase blocks. The block preconditioning strategy employs an algebraic LU decomposition such that a lower triangular block matrix is obtained. The preconditioning problem can be simplified to finding preconditioner for the lower triangular block which is easier to compute and flexible to construct.

The first part of the paper introduces the investigated coupled problem by providing an introduction to the tunnelling process to be modelled and to derive the underlying coupled problem. Here, the previous works in this field performed by Nagel & Meschke [2] and Nagel, Stascheit & Meschke [3] are recapitulated. In the second and third part, the effect of the finite element formulations for partially saturated soil with inelastic material behaviour of the soil skeleton on the properties of the system matrix is discussed. In particular, the block structure of the matrix as a result of the coupled formulation and the numerical properties of the matrix due to the choice of primary variables is addressed. In the fourth part, the block-preconditioning operator in case of one or multiple domains is introduced. The preconditioning technique is verified by means of a numerical example: the problem of dewatering of a sand column is described and analysed in terms of convergence properties to investigate the advantages and disadvantages of the proposed method.

1

J. White, R. Borja, "Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics", Computational Geoscience, 4, 2011. DOI 10.1007/s10596-011-9233-7

2

F. Nagel, G. Meschke, "An elasto-plastic three phase model for partially saturated soil for the finite element simulation of compressed air support in tunnelling", International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 34, 605-625, 2010.

3

F. Nagel, J. Stascheit, G. Meschke, "Process-oriented numerical simulation of shield supported tunnelling in soft soils", Geomechanics and Tunnelling, Vol. 3, p. 268-282, 2010. DOI: 10.1002/geot.201000024

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