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The initial stiffness method has been extensively adopted for elasto‐plastic finite element analysis. The main problem associated with the initial stiffness method, however, is its slow convergence, even when it is used in conjunction with acceleration techniques. The Newton‐Raphson method has a rapid convergence rate, but its implementation resorts to non‐symmetric linear solvers, and hence the memory requirement may be high. The purpose of this paper is to develop more advanced solution techniques which may overcome the above problems associated with the initial stiffness method and the Newton‐Raphson method.

In this work, the accelerated symmetric stiffness matrix methods, which cover the accelerated initial stiffness methods as special cases, are proposed for non‐associated plasticity. Within the computational framework for the accelerated symmetric stiffness matrix techniques, some symmetric stiffness matrix candidates are investigated and evaluated.

Numerical results indicate that for the accelerated symmetric stiffness methods, the elasto‐plastic constitutive matrix, which is constructed by mapping the yield surface of the equivalent material to the plastic potential surface, appears to be appealing. Even when combined with the Krylov iterative solver using a loose convergence criterion, they may still provide good nonlinear convergence rates.

Compared to the work by Sloan et al., the novelty of this study is that a symmetric stiffness matrix is proposed to be used in conjunction with acceleration schemes and it is shown to be more appealing; it is assembled from the elasto‐plastic constitutive matrix by mapping the yield surface of the equivalent material to the plastic potential surface. The advantage of combining the proposed accelerated symmetric stiffness techniques with the Krylov subspace iterative methods for large‐scale applications is also emphasized.

This paper aims to provide a series of new methods for projecting a three-dimensional (3D) object onto a free-form surface. The projection algorithms presented can be divided into three types, namely, orthogonal, perspective and parallel projection.

For parametric surfaces, the computing strategy of the algorithm is to obtain an approximate solution by using a geometric algorithm, then improve the accuracy of the approximate solution using the Newton–Raphson iteration. For perspective projection and parallel projection on an implicit surface, the strategy replaces Newton–Raphson iteration by multi-segment tracing. The implementation takes two mesh objects as an example of calculating an image projected onto parametric and implicit surfaces. Moreover, a comparison is made for orthogonal projections with Hu’s and Liu’s methods.

The results show that the new method can solve the 3D objects projection problem in an effective manner. For orthogonal projection, the time taken by the new method is substantially less than that required for Hu’s method. The new method is also more accurate and faster than Liu’s approach, particularly when the 3D object has a large number of points.

The algorithms presented in this paper can be applied in many industrial applications such as computer aided design, computer graphics and computer vision.

This study aims to develop an adaptive finite element method for structural eigenproblems of cracked Euler–Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems).

The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler–Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, using the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton–Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems.

Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler–Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

The proposed combination of methodologies described in the paper leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

The present paper is directed towards elasto‐plastic large deformation analysis of thin shells based on the concept of degenerated solids. The main aspect of the paper is the derivation of an efficient computational strategy placing emphasis on consistent elasto‐plastic tangent moduli and stress integration with the radial return method under the restriction of ‘zero normal stress condition’ in thickness direction. The advantageous performance of the standard Newton iteration using a consistent tangent stiffness matrix is compared to the classical scheme with an iteration matrix based on the infinitesimal elasto‐plastic constitutive tensor. Several numerical examples also demonstrate the effectiveness of the standard Newton iteration with respect to modified and quasi‐Newton methods like BFGS and others.

This paper proposes an improved algorithm to compute selective harmonics elimination pulse width modulation (SHEPWM) angles, based on the Newton‐Raphson (NR) iteration for cascaded multilevel inverter (CMI).

Newton Raphson (NR) is a very popular numerical method for transcendental equations that lack analytical solutions. It has been successfully used to compute the angles for selective harmonics elimination pulse width modulation (SHEPWM) schemes. Despite its effectiveness, NR has not been used for SHEPWM with cascaded multilevel inverter (CMI) structure with equal and non‐equal DC voltage sources. It is known that for CMI, inappropriate selection of initial angles causes long‐iteration time and possibly non‐convergence takes place. The computational difficulty is compounded by the fact that the SHEPWM switching angles need to be correctly sequenced, i.e. each angle must be assigned to the correct output voltage level of the CMI. In this work, an attempt is made to reduce the iteration time and to resolve the non‐convergence problem. The main idea is to relax the switching angle constraint by placing the switching angle sequencing outside the main loop of NR iteration. This allows for the program to run more freely and able to generate more possible solutions for the switching angles. Then these angles are selected to fulfill the requirements of multilevel sequencing. The performance of the proposed technique will be compared with the standard NR for CMI with equal and non‐equal DC sources. The latter case is quite common for CMI with renewable energy applications because the sources normally have different voltage levels.

Using MATLAB simulation, it will be shown that using this scheme, accurate SHEPWM angles can be achieved for a wide range of fundamental components. Furthermore, significant reduction in iteration time to compute the SHEPWM switching angles is achieved.

This paper proposes an improved algorithm to compute SHEPWM angles based on NR iteration.

A computational procedure is presented for the efficient non‐linear dynamic analysis of quasi‐symmetric structures. The procedure is based on approximating the unsymmetric response vectors, at each time step, by a linear combination of symmetric and antisymmetric vectors, each obtained using approximately half the degrees of freedom of the original model. A mixed formulation is used with the fundamental unknowns consisting of the internal forces (stress resultants), generalized displacements and velocity components. The spatial discretization is done by using the finite element method, and the governing semi‐discrete finite element equations are cast in the form of first‐order non‐linear ordinary differential equations. The temporal integration is performed by using implicit multistep integration operators. The resulting non‐linear algebraic equations, at each time step, are solved by using iterative techniques. The three key elements of the proposed procedure are: (a) use of mixed finite element models with independent shape functions for the stress resultants, generalized displacements, and velocity components and with the stress resultants allowed to be discontinuous at interelement boundaries; (b) operator splitting, or restructuring of the governing discrete equations of the structure to delineate the contributions to the symmetric and antisymmetric vectors constituting the response; and (c) use of a two‐level iterative process (with nested iteration loops) to generate the symmetric and antisymmetric components of the response vectors at each time step. The top‐ and bottom‐level iterations (outer and inner iterative loops) are performed by using the Newton—Raphson and the preconditioned conjugate gradient (PCG) techniques, respectively. The effectiveness of the proposed strategy is demonstrated by means of a numerical example and the potential of the strategy for solving more complex non‐linear problems is discussed.

This paper aims to present an improved finite element method used for achieving faster convergence in simulations of incompressible fluid flows. For stable computations of incompressible fluid flows, it is important to ensure that the flow field satisfies the equation of continuity in each element of a generally distorted mesh. The study aims to develop a numerical approach that satisfies this requirement based on the highly simplified marker-and-cell (HSMAC) method and increases computational speed by introducing a new algorithm into the simultaneous relaxation of velocity and pressure.

First, the paper shows that the classical HSMAC method is equivalent to a Jacobi-type method in terms of the simultaneous relaxation of velocity and pressure. Then, a Gauss–Seidel or successive over-relaxation (SOR)-type method is introduced in the Newton–Raphson iterations to take into account all the derivative terms in the first-order Taylor series expansion of a nodal-averaged error explicitly. Here, the nine-node quadrilateral (Q2–Q1) elements are used.

The new finite element approach based on the improved HSMAC algorithm is tested on fluid flow problems including the lid-driven square cavity flow and the flow past a circular cylinder. The results show significant improvement of the convergence property with the accuracy of the numerical solutions kept unchanged even on a highly distorted mesh.

To the best of the author’s knowledge, the idea of using the Gauss–Seidel or SOR method in the simultaneous relaxation procedure of the HSMAC method has not been proposed elsewhere.

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.

Some frictional contact problems are characterized by significant variations in the location and size of the contact area occurring in the process of deformation. When this feature is combined with strongly non‐linear, path‐dependent material behaviour, difficulties with convergence of the typically used iterative processes can be encountered. Demonstrates this by analysis of press‐fit connection, a typical problem in which both of those characteristics can be present. Offers an explanation as to the possible source of those difficulties. Suggests in support of this explanation, two simple modifications of the usual iterative schemes. In spite of their simplicity, they are found to be more robust than those usual schemes which are normally used in numerical analysis of similar problems.

This work addresses the computational aspects of a model for elastoplastic damage at finite strains. The model is a modification of a previously established model for large strain elastoplasticity described by Perić et al. which is here extended to include isotropic damage and kinematic hardening. Within the computational scheme, the constitutive equations are numerically integrated by an algorithm based on operator split methodology (elastic predictor—plastic corrector). The Newton—Raphson method is used to solve the discretized evolution equations in the plastic corrector stage. A numerical assessment of accuracy and stability of the integration algorithm is carried out based on iso‐error maps. To improve the stability of the local N—R scheme, the standard elastic predictor is replaced by improvedinitial estimates ensuring convergence for large increments. Several possibilities are explored and their effect on the stability of the N—R scheme is investigated. The finite element method is used in the approximation of the incremental equilibrium problem and the resulting equations are solved by the standard Newton—Raphson procedure. Two numerical examples are presented. The results are compared with those obtained by the original elastoplastic model.