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The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent Stokes problems.

This paper solved the time-dependent Stokes equations using the finite element method and the partition of unity method.

The proposed method in this paper obtained the same accuracy as the standard Galerkin method, but it, in general, saves time.

Based on a combination of the partition of unity method and the finite element method, the authors, in this paper, propose a new parallel partition of unity method to solve the unsteady Stokes equations. The idea is that, at each time step, one need to only solve a series of independent local sub-problems in parallel instead of one global problem. Numerical tests show that the proposed method not only reaches the same convergence orders as the fully discrete standard Galerkin method but also saves ample computing time.

The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent convection-diffusion equations.

This paper opted for the time-dependent convection-diffusion equations using the finite element method and the partition of unity method.

This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method (SGM) but saves a lot of computational time.

In this paper, a parallel partition of unity method is proposed for the time-dependent convection-diffusion equations. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

The purpose of this paper is to propose a parallel partition of unity method (PPUM) to solve the nonstationary Navier-Stokes equations.

This paper opted for the nonstationary Navier-Stokes equations by using the finite element method and the partition of unity method.

This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method but saves a lot of computational time.

In this paper, a PPUM is proposed for nonstationary Navier-Stokes. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

This study aims to present a new hybrid formulation based on non-uniform rational b-splines functions and enrichment strategies applied to free and forced vibration of straight bars and trusses.

Based on the idea of enrichment from generalized finite element method (GFEM)/extended finite element method (XFEM), an extended isogeometric formulation (partition of unity isogeometric analysis [PUIGA]) is conceived. By numerical examples the methods are tested and compared with isogeometric analysis, finite element method and GFEM in terms of convergence, error spectrum, conditioning and adaptivity capacity.

The results show a high convergence rate and accuracy for PUIGA and the advantage of input enrichment functions and material parameters on parametric space.

The enrichment strategies demonstrated considerable improvements in numerical solutions. The applications of computer-aided design mapped enrichments applied to structural dynamics are not known in the literature.

The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames.

The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented.

The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode.

The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.

It has been well recognized that interface problems often contain strong singularities which make conventional numerical approaches such as uniform h‐ or p‐version of finite element methods (FEMs) inefficient. In this paper, the partition‐of‐unity finite element method (PUFEM) is applied to obtain solution for interface problems with severe singularities. In the present approach, asymptotical expansions of the analytical solutions near the interface singularities are employed to enhance the accuracy of the solution. Three different enrichment schemes for interface problems are presented, and their performances are studied. Compared to other numerical approaches such as h‐p version of FEM, the main advantages of the present method include: easy and simple formulation; highly flexible enrichment configurations; no special treatment needed for numerical integration and boundary conditions; and highly effective in terms of computational efficiency. Numerical examples are included to illustrate the robustness and performance of the three schemes in conjunction with uniform h‐ or p‐refinements. It shows that the present PUFEM formulations can significantly improve the accuracy of solution. Very often, improved convergence rate is obtained through enrichment in conjunction with p‐refinement.

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 check the efficiency of isogeometric analysis (IGA) by comparing its results with classical finite element method (FEM), generalized finite element method (GFEM) and other enriched versions of FEM through numerical examples of free vibration problems.

Since its conception, IGA was widely applied in several problems. In this paper, IGA is applied for free vibration of elastic rods, beams and trusses. The results are compared with FEM, GFEM and the enriched methods, concerning frequency spectra and convergence rates.

The results show advantages of IGA over FEM and GFEM in the frequency error spectra, mostly in the higher frequencies.

Isogeometric analysis shows a feasible tool in structural analysis, with emphasis for problems that requires a high amount of vibration modes.

The purpose of this paper is to propose a new three-node triangular element in the framework of the numerical manifold method (NMM), which is designated by Trig3-MLScns.

The formulation uses the improved parametric shape functions of classical triangular elements (Trig3-0) to construct the partition of unity (PU) and the moving least square (MLS) interpolation method to construct the local approximation function.

Compared with the classical three-node element (Trig3-0), the Trig3-MLScns element has a higher order of approximations, much better accuracy and continuous nodal stress. Moreover, the linear dependence problem associated with many PU-based methods with high-order approximations is eliminated in the present element. A number of numerical examples indicate the high accuracy and robustness of the Trig3-MLScns element.

The proposed element inherits the individual merits of the NMM and the MLS.

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.