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It has been usual to prefer an enrichment pattern independent of the mesh when applying singular functions in the Generalized/eXtended finite element method (G/XFEM). This choice, when modeling crack tip singularities through extrinsic enrichment, has been understood as the only way to surpass the typical poor convergence rate obtained with the finite element method (FEM), on uniform or quasi-uniform meshes conforming to the crack. Then, the purpose of this study is to revisit the topological enrichment strategy in the light of a higher-order continuity obtained with a smooth partition of unity (PoU). Aiming to verify the smoothness' impacts on the blending phenomenon, a series of numerical experiments is conceived to compare the two GFEM versions: the conventional one, based on piecewise continuous PoU's, and another which considers PoU's with high-regularity.

The stress approximations right at the crack tip vicinity are qualified by focusing on crack severity parameters. For this purpose, the material forces method originated from the configurational mechanics is employed. Some attempts to improve solution using different polynomial enrichment schemes, besides the singular one, are discussed aiming to verify the transition/blending effects. A classical two-dimensional problem of the linear elastic fracture mechanics (LEFM) is solved, considering the pure mode I and the mixed-mode loadings.

The results reveal that, in the presence of smooth PoU's, the topological enrichment can still be considered as a suitable strategy for extrinsic enrichment. First, because such an enrichment pattern still can treat the crack independently of the mesh and deliver some advantage in terms of convergence rates, under certain conditions, when compared to the conventional FEM. Second, because the topological pattern demands fewer degrees of freedom and impacts conditioning less than the geometrical strategy.

Several outputs are presented, considering estimations for the J–integral and the angle of probable crack advance, this last computed from two different strategies to monitoring blending/transition effects, besides some comments about conditioning. Both h- and p-behaviors are displayed to allow a discussion from different points of view concerning the topological enrichment in smooth GFEM.

This paper aims to present a computational framework to generate numeric enrichment functions for two-dimensional problems dealing with single/multiple local phenomenon/phenomena. The two-scale generalized/extended finite element method (G/XFEM) approach used here is based on the solution decomposition, having global- and local-scale components. This strategy allows the use of a coarse mesh even when the problem produces complex local phenomena. For this purpose, local problems can be defined where these local phenomena are observed and are solved separately by using fine meshes. The results of the local problems are used to enrich the global one improving the approximate solution.

The implementation of the two-scale G/XFEM formulation follows the object-oriented approach presented by the authors in a previous work, where it is possible to combine different kinds of elements and analyses models with the partition of unity enrichment scheme. Beside the extension of the G/XFEM implementation to enclose the global–local strategy, the imposition of different boundary conditions is also generalized.

The generalization done for boundary conditions is very important, as the global–local approach relies on the boundary information transferring process between the two scales of the analysis. The flexibility for the numerical analysis of the proposed framework is illustrated by several examples. Different analysis models, element formulations and enrichment functions are used, and the accuracy, robustness and computational efficiency are demonstrated.

This work shows a generalize imposition of different boundary conditions for global–local G/XFEM analysis through an object-oriented implementation. This generalization is very important, as the global–local approach relies on the boundary information transferring process between the two scales of the analysis. Also, solving multiple local problems simultaneously and solving plate problems using global–local G/XFEM are other contributions of this work.

The purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered.

Numerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results.

Known numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities.

This study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies.

The aim of this paper is to present a novel data transfer technique to simulate, by G/XFEM, a cohesive crack propagation coupled with a smeared damage model. The efficiency of this technique is evaluated in terms of processing time, number of Newton–Raphson iterations and accuracy of structural response.

The cohesive crack is represented by the G/XFEM enrichment strategy. The elements crossed by the crack are divided into triangular cells. The smeared crack model is used to describe the material behavior. In the nonlinear solution of the problem, state variables associated with the original numerical integration points need to be transferred to new points created with the triangular subdivision. A nonlocal strategy is tailored to transfer the scalar and tensor variables of the constitutive model. The performance of this technique is numerically evaluated.

When compared with standard Gauss quadrature integration scheme, the proposed strategy may deliver a slightly superior computational efficiency in terms of processing time. The weighting function parameter used in the nonlocal transfer strategy plays an important role. The equilibrium state in the interactive-incremental solution process is not severely penalized and is readily recovered. The advantages of such proposed technique tend to be even more pronounced in more complex and finer meshes.

This work presents a novel data transfer technique based on the ideas of the nonlocal formulation of the state variables and specially tailored to the simulation of cohesive crack propagation in materials governed by the smeared crack constitutive model.

In this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.

An adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.

The results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.

The proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

The purpose of this paper is to evaluate the capabilities of the generalized finite element method (GFEM) under the context of the geometrically nonlinear analysis. The effect of large displacements and deformations, typical of such analysis, induces a significant distortion of the element mesh, penalizing the quality of the standard finite element method approximation. The main concern here is to identify how the enrichment strategy from GFEM, that usually makes this method less susceptible to the mesh distortion, may be used under the total and updated Lagrangian formulations.

An existing computational environment that allows linear and nonlinear analysis, has been used to implement the analysis with geometric nonlinearity by GFEM, using different polynomial enrichments.

The geometrically nonlinear analysis using total and updated Lagrangian formulations are considered in GFEM. Classical problems are numerically simulated and the accuracy and robustness of the GFEM are highlighted.

This study shows a novel study about GFEM analysis using a complete polynomial space to enrich the approximation of the geometrically nonlinear analysis adopting the total and updated Lagrangian formulations. This strategy guarantees the good precision of the analysis for higher level of mesh distortion in the case of the total Lagrangian formulation. On the other hand, in the updated Lagrangian approach, the need of updating the degrees of freedom during the incremental and iterative solution are for the first time identified and discussed here.

Understanding the fluid flow through rock masses, which commonly consist of rock matrix and fractures, is a fundamental issue in many application areas of rock engineering. As the equivalent porous medium approach is the dominant approach for engineering applications, it is of great significance to estimate the equivalent permeability tensor of rock masses. This study aims to develop a novel numerical approach to estimate the equivalent permeability tensor for fractured porous rock masses.

The radial point interpolation method (RPIM) and finite element method (FEM) are coupled to simulate the seepage flow in fractured porous rock masses. The rock matrix is modeled by the RPIM, and the fractures are modeled explicitly by the FEM. A procedure for numerical experiments is then designed to determinate the equivalent permeability tensor directly on the basis of Darcy’s law.

The coupled RPIM-FEM method is a reliable numerical method to analyze the seepage flow in fractured porous rock masses, which can consider simultaneously the influences of fractures and rock matrix. As the meshes of rock matrix and fracture network are generated separately without considering the topology relationship between them, the mesh generation process can be greatly facilitated. Using the proposed procedure for numerical experiments, which is designed directly on the basis of Darcy’s law, the representative elementary volume and equivalent permeability tensor of fractured porous rock masses can be identified conveniently.

A novel numerical approach to estimate the equivalent permeability tensor for fractured porous rock masses is proposed. In the approach, the RPIM and FEM are coupled to simulate the seepage flow in fractured porous rock masses, and then a numerical experiment procedure directly based on Darcy’s law is introduced to estimate the equivalent permeability tensor.

The purpose of this paper is to propose novel parallel computational techniques for the parallelization of explicit finite element generalized approximate inverse methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems.

The authors' main motive for the derivation of the new Parallel Generalized Approximate Inverse Finite Element Matrix algorithmic techniques is that they can be efficiently used in conjunction with explicit preconditioned conjugate gradient‐type schemes on multicore systems. The proposed parallelization technique of the Optimized Banded Generalized Approximate Inverse Finite Element Matrix (OBGAIFEM) algorithm is achieved based on the concept of the “fish bone” approach with the use of a thread pool pattern. Theoretical estimates on the computational complexity of the parallel generalized approximate inverse finite element matrix algorithmic techniques are also derived.

Application of the proposed method on a two‐dimensional boundary value problem is discussed and numerical results are given on a multicore system using POSIX threads. These results tend to become optimum and are favorably compared to corresponding results from multiprocessor systems, as presented in recent work by Gravvanis et al.

The proposed parallel explicit finite element generalized approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

To propose novel parallel/distributed normalized explicit finite element (FE) approximate inverse preconditioning for solving sparse FE linear systems.

The design of suitable methods was the main objective for which several families of the normalized approximate inverse, based on sparse normalized approximate factorization, are produced. The main motive for the derivation of the new normalized approximate inverse FE matrix algorithmic techniques is that they can be efficiently used in conjunction with normalized explicit preconditioned conjugate gradient (NEPCG) – type schemes on parallel and distributed systems. Theoretical estimates on the rate of convergence and computational complexity of the NEPCG method are also derived.

Application of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results for uniprocessor systems along with speed‐ups and efficiency for multicomputer systems are given. These results tend to become optimum, which are in qualitative agreement with the theoretical results presented for uniprocessor and distributed memory systems, using message passing interface (MPI) communication library.

Further parallel algorithmic techniques will be investigated in order to improve the speed‐ups and the computational complexity of the parallel normalized explicit approximate inverse preconditioning.

The proposed parallel/distributed normalized explicit approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

The purpose of this paper is to propose novel factored approximate sparse inverse schemes and multi-level methods for the solution of large sparse linear systems.

The main motive for the derivation of the various generic preconditioning schemes lies to the efficiency and effectiveness of factored preconditioning schemes in conjunction with Krylov subspace iterative methods as well as multi-level techniques for solving various model problems. Factored approximate inverses, namely, Generic Factored Approximate Sparse Inverse, require less fill-in and are computed faster due to the reduced number of nonzero elements. A modified column wise approach, namely, Modified Generic Factored Approximate Sparse Inverse, is also proposed to further enhance performance. The multi-level approximate inverse scheme, namely, Multi-level Algebraic Recursive Generic Approximate Inverse Solver, utilizes a multi-level hierarchy formed using Block Independent Set reordering scheme and an approximation of the Schur complement that results in the solution of reduced order linear systems thus enhancing performance and convergence behavior. Moreover, a theoretical estimate for the quality of the multi-level approximate inverse is also provided.

Application of the proposed schemes to various model problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than results by other researchers for some of the model problems.

Further enhancements are investigated for the proposed factored approximate inverse schemes as well as the multi-level techniques to improve quality of the schemes. Furthermore, the proposed schemes rely on the definition of multiple parameters that for some problems require thorough testing, thus adaptive techniques to define the values of the various parameters are currently under research. Moreover, parallel schemes will be investigated.

The proposed approximate inverse preconditioning schemes as well as multi-level schemes are efficient computational methods that are valuable for computer scientists and for scientists and engineers in engineering computations.