A simple alternative to the conjugate gradient (CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy, i.e. it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the A-orthogonality in n-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.
Exact arithmetic as a tool for convergence assessment of the IRM-CG method This is an author produced version of a paper submitted to a peer-review Exact arithmetic as a tool for convergence assessment of the IRM-CG method
AbstractUsing exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by rationalisation of numerically given input data. In the latter case there is an initial roundoff error, but this does not propagate during the solution process. If this system is first exactly solved, then by the floating-point arithmetic, convergence of the numerical method is easily followed. As one example, IRM-CG, a special case of the more general Iterated Ritz method and interesting replacement for a standard or preconditioned CG, is verified. Further, because the computer demands and execution time grow enourmously with the number of unknowns using this strategy, the possibilities for larger systems are also provided.
Original scientific paper The distribution of stresses and strains during the portal section excavation for the right tube of the Sleme Tunnel is analysed by the Finite Element Method (FEM). Results of analyses obtained using 2D and 3D models have been compared to in-situ measurements of tunnel convergence and ground surface settlements. Multistage excavations with pipe roof support of the working face were modelled. The numerical models verified that for this particular case a sufficiently safe and cost-efficient construction technology was applied.Keywords: Finite Element Method; pipe roof; stress and strain analysis; tunnel excavation
Analiza stanja naprezanja i deformacija tijekom iskopa tunela SlemeIzvorni znanstveni članak Analizirano je stanja naprezanja i deformacija tijekom iskopa portalne dionice desne cijevi tunela "Sleme" primjenom metode konačnih elemenata. Napravljena je usporedba rezultata proračuna ravninskog (2D) i prostornog (3D) modela s rezultatima terenskih mjerenja konvergencija u tunelu i slijeganja površine terena. Modelirani su višefazni iskopi s ojačanjem čela cijevnim krovom. Proračunski model je pokazao da je u navedenom slučaju primijenjena dovoljno sigurna i ekonomična tehnologija iskopa.Ključne riječi: analiza stanja naprezanja i deformacija; cijevni krov; iskop tunela; metoda konačnih elemenata
In this paper, basic structural stability phenomena are described. After some general comments about stability in the fi eld of civil engineering, four elementary sources of nonlinearity are mentioned: of equilibrium equations, strain (geometry) relations, material (stress-strain) law, force and displacement boundary conditions. Four fundamental stability models are analysed, both ideal (perfect) and with geometric imperfection. Besides geometrically exact theory, initial post-buckling behaviour and linearization are briefl y sketched. This paper is concluded with comments about the infl uence of plasticity.
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