The most widely used method for simulating the non-linear behaviour of concrete and masonry structures is the Newton–Raphson method with arc-length control (N-R method). However, this method may fail to produce converged results because of softening, negative tangent stiffness, bifurcations or snap-back. Sometimes, convergence can be obtained by controlling degrees of freedom in the failure process zone or by applying sequentially linear analysis (SLA). However, the location of the failure is often not known a priori and geometrical non-linearity needs to be included. Recently, incremental sequentially linear analysis (ISLA) has been proposed, which is based on a combination of the N-R method and SLA. The solution search path follows damage cycles sequentially with secant stiffness corresponding to local damage increments, which traces both damage history (explicit) and displacement history (implicit). The objective of this paper is to demonstrate that ISLA can be applied to problems that behave geometrically nonlinear in addition to physically nonlinear. In this paper, we introduce a method that combines ISLA with indirect displacement control. This method stabilises localised damage process areas and avoids the global unloading caused by geometrical and physical non-linearity. The method uses one or more control points, which are positioned independently of the failure process zones. Two masonry walls were tested and analysed. The load was perpendicular to their planes and evenly distributed. The walls were supported on two or four edges. Stable post-peak results were computed for large geometrical non-linear displacements, and localised crack propagation was computed robustly and correctly.

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It is difficult to accurately predict the strength of masonry and concrete structures. The most widely used method for simulating their behaviour is finite element analysis with the Newton-Raphson method and arch length control. However, the Newton-Raphson method can diverge and not produce a result, for example in bifurcations or during snap-back. In order to enhance the robustness of solving non-linear problems, a new method – called incremental sequentially linear analysis (ISLA) – is proposed. The method is based on a combination of the Newton-Raphson method and a total approach called sequentially linear analysis. In ISLA, local damage is induced by reducing the material secant stiffness of the element that fails a unity check. The load is applied in force increments or displacement increments, which are adjusted to trace the complete structural response. It has been showed that ISLA can handle non-proportional loading, geometrically non-linear analysis and transient analysis. The robustness of ISLA has been demonstrated in four examples: a concrete beam with both prestress and vertical load; out-of-plane bending of a masonry wall with overburden; a differential settlement test on a pre-loaded masonry façade and a 3D pushover analysis of a masonry house. @en

Incremental Sequentially Linear Analysis (ISLA) is a new algorithm for non-linear finite element analysis. It is an extension of Sequentially Linear Analysis (SLA) which has been applied since 2001 as an alternative to the Newton-Raphson method when bifurcation, snap-back or divergence problems arise. ISLA is an incremental procedure with an implicit scheme, which starts and ends with an equilibrium state. The solution search path fol-lows damage steps sequentially with secant stiffness. In each iteration only one element is selected for damaging in the next iteration, which is a similar procedure as used in SLA. In this paper, ISLA is explained and demonstrated for a notched beam test. Because of the incremental procedure, ISLA can be extended to non-proportional loading, geometrically non-linear analysis and transient analysis. The searching path of ISLA is based on physical parameters (damage and history) rather than guided by numerical parameters. In addition, the method keeps the same incremental format throughout the entire analysis, circumventing the need to switch intermittently from incremental to total approaches or vice versa.@en

Quasi brittle materials, such as un-reinforced masonry or concrete are difficult to analyse because often the traditional Newton–Raphson (N-R) procedure fails to converge. Many solutions have been proposed such as Sequentially Linear Analysis (SLA), but these may fail in case of non-proportional loading with a large prestress. In this paper a new method is proposed that is based on a combination of the Newton–Raphson method and Sequentially Linear Analysis. The method is incremental; each increment starts and ends with an equilibrium state. The solution search path follows damage cycles sequentially with secant stiffness. The proposed method is demonstrated to be robust and accurate. It has been tested on prestressed concrete beams. It can be naturally extended to other types of analyses (e.g. geometrically non-linear analysis and transient analysis) due to the incremental procedure. In addition, it is shown that high prestress values can transform the behaviour of a concrete beam from softening to hardening.@en

Sequentially linear analysis (SLA) is an alternative to the Newton-Raphson method for analyzing the nonlinear behavior of reinforced concrete and masonry structures. In this paper SLA is extended to load cases that are applied one after the other, for example first dead load and then wind load. It is shown that every nonlinear analysis step can be made in just two linear elastic analysis steps. The proposed algorithm is extremely robust, which is demonstrated in a prestressed concrete beam analysis. A comparison is made between results of SLA and Newton-Raphson with arch length control.@en

The identification of structured state-space model has been intensively studied for a long time but still has not been adequately addressed. The main challenge is that the involved estimation problem is a non-convex (or bilinear) optimization problem. This paper is devoted to developing an identification
method which aims to find the global optimal solution under mild computational burden. Key to the developed identification algorithm is to transform a bilinear estimation to a rank constrained optimization problem and further a difference of convex programming (DCP) problem. The initial condition
for the DCP problem is obtained by solving its convex part of the optimization problem which happens to be a nuclear norm regularized optimization problem. Since the nuclear norm regularized optimization is the closest convex form of the low-rank constrained estimation problem, the obtained initial
condition is always of high quality which provides the DCP problem a good starting point. The DCP problem is then solved by the sequential convex programming method. Finally, numerical examples are included to show the effectiveness of the developed identification algorithm.@en