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The paper describes the derivation and application of a range of numerical algorithms for implementing the Mohr—Coulomb yield criterion in a non‐linear finite element computer program. Emphasis is placed on the difficulties associated with the corners of the yield surface. In contrast to the more conventional forward‐Euler procedures, a backward‐Euler integration technique is adopted. A range of methods, including a ‘consistent approach’ are used to derive the tangent modular matrix. Numerical experiments are presented which involve solution algorithms including the modified and full Newton—Raphson procedures, ‘line‐searches’ and the arc‐length method. It is shown that the introduction of efficient integration and tangency algorithms can lead to very substantial improvements in the convergence characteristics.

The localized strain softening behaviour of concrete has been modelled by two approaches: (i) the stiffness degrading model based on the total stress‐strain constitutive relationship, and (ii) the tangent softening model based on the incremental stress‐strain relationship. The models are implemented using a new softening initiation criterion proposed for application in multi‐dimensional finite element analysis. Parametric analyses on plain concrete beams, tested experimentally by other researchers, have been carried out to investigate the required numerical efforts, the mesh objectivity, and the energy dissipation characteristics of the structures. The stiffness degrading model is very stable even when applied with relatively coarse finite element meshes. However, the computational demand of this model is relatively high. The combination of a total stress‐strain constitutive relationship to compute the element responses, and an incremental relationship to formulate the stiffness matrix, appears to be computationally efficient and stable, provided that adequately refined finite element mesh is used to model the structure.

In the framework of the finite element method, the problem of elasto‐plastic consolidation gives rise to a system of non‐linear, coupled residual equations which satisfy the conditions of balance of momentum and balance of mass. In determining the roots of these equations it is necessary that the coupled equations be linearized. To this end, the concept of ‘consistent linearization’ proposed by Simo and Taylor for a single‐phase system is applied to the two‐phase soil‐water system. The roots of the coupled residual equations are solved iteratively by employing Newton's method. It is shown that in non‐linear consolidation analyses, the use of a tangent coefficient matrix derived consistently from the integrated constitutive equation defining the characteristics of the solid skeletal phase results in an iterative solution scheme which preserves the asymptotic rate of quadratic convergence of Newton's method. Numerical examples involving combined radial and vertical flows through an elasto‐plastic soil medium are presented to demonstrate the computational superiority of the above technique over the method based on standard ‘elasto‐plastic continuum formulations’ adopted in most finite element codes.

The load carrying capacity of steel structures, built of slender members like bridge cross‐sections, depends on coupled yielding and buckling of the stringers and the plate strips as well as on the global buckling. Therefore, the common techniques of modelling the limit load by an elasto‐plastic layer model fail. In order to overcome the difficulty a material law is developed, in which local buckling failure and yielding is considered. This is based on an energy function, which describes the elasto‐plastic intermediate and ultimate state of plates and webs dependent on only a few parameters. The application is shown on large scale examples of stiffened steel bridge decks.

Describes two low‐order shell elements, one (quadrilateral) with 16 degrees‐of‐freedom; twelve translations and four rotations and another (triangular) with 12 degrees‐of‐freedom; nine translations and three rotations. The elements are formulated in a geometrically non‐linear manner and large strains, which may be hyper‐elastic or elasto‐plastic, are also considered. Hills yield criterion with a Lankford constant for the special case of transversely isotropic problem is introduced into the large‐strain formulations. To illustrate its application, the hydrostatic bulging of rectangular diaphragms with different aspect ratios is analysed and the obtained results are compared with the experimental ones. The elements have advantageous nodal configuration that makes them particularly suitable for analysing structures with junctions. Such a problem is an initially square steel box loaded with internal pressure. This problem is analysed and comparisons are made with experimental results.

Of particular interest is the ability of the extended finite element method (XFEM) to capture transient solution and motion of phase boundaries without adaptive remeshing or moving-mesh algorithms for a physically nonlinear phase change problem. The paper aims to discuss this issue.

The XFEM is applied to solve nonlinear transient problems with a phase change. Thermal conductivity and volumetric heat capacity are assumed to be dependent on temperature. The nonlinearities in the governing equations make it necessary to employ an effective iterative approach to solve the problem. The Newton-Raphson method is used and the incremental discrete XFEM equations are derived.

The robustness and utility of the method are demonstrated on several one-dimensional benchmark problems.

The novel procedure based on the XFEM is developed to solve physically nonlinear phase change problems.

The purpose of this paper is to carry out a finite element simulation of a physically non-linear phase change problem in a two-dimensional space without adaptive remeshing or moving-mesh algorithms. The extended finite element method (XFEM) and the level set method (LSM) were used to capture the transient solution and motion of phase boundaries. It was crucial to consider the effects of unequal densities of the solid and liquid phases and the flow in the liquid region.

The XFEM and the LSM are applied to solve non-linear transient problems with a phase change in a two-dimensional space. The model assumes thermo-dependent properties of the material and unequal densities of the phases; it also allows for convection in the liquid phase. A non-linear system of equations is derived and a numerical solution is proposed. The Newton-Raphson method is used to solve the problem and the LSM is applied to track the interface.

The robustness and utility of the method are demonstrated on several two-dimensional benchmark problems.

The novel procedure based on the XFEM and the LSM was developed to solve physically non-linear phase change problems with unequal densities of phases in a two-dimensional space.

A non‐linear shallow thin shell element is described. The element is a curved quadrilateral one with corner nodes only. At each node, six degrees of freedom (i.e. three translations and three rotations) make the element easy to connect to space beams, stiffeners or intersecting shells. The curvature is dealt with by Marguerre's theory. Membrane bending coupling is present at the element level and improves the element behaviour, especially in non‐linear analysis. The element converges to the deep shell solution. The sixth degree of freedom is a true one, which can be assimilated to the in‐plane rotation. The present paper describes how overstiffness due to membrane locking on the one hand and to the sixth degree of freedom on the other hand can be corrected without making use of numerical adjusted factors. The behaviour of this new element is analysed in linear and non‐linear static and dynamic tests.

This paper aims to describe a method for efficient frequency domain model order reduction. The method attempts to combine the desirable attributes of Krylov reduction and proper orthogonal decomposition (POD) and is entitled Krylov enhanced POD (KPOD).

The KPOD method couples Krylov’s moment-matching property with POD’s data generalization ability to construct reduced models capable of maintaining accuracy over wide frequency ranges. The method is based on generating a sequence of state- and frequency-dependent Krylov subspaces and then applying POD to extract a single basis that generalizes the sequence of Krylov bases.

The frequency response of a pre-stressed microelectromechanical system resonator is used as an example to demonstrate KPOD’s ability in frequency domain model reduction, with KPOD exhibiting a 44 per cent efficiency improvement over POD.

The results indicate that KPOD greatly outperforms POD in accuracy and efficiency, making the proposed method a potential asset in the design of frequency-selective applications.

This paper is concerned with rank analysis of rectangular matrix of a homogeneous set of incremental equations regarded as an element of continuation method. The rank analysis is based on a known feature that every rectangular matrix can be transformed into the matrix of echelon form. By inspection of the rank, correct control parameters are chosen and this allows not only for rounding limit and turning points but also for branch‐switching near bifurcation points.