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Article
Publication date: 4 September 2017

Xiaoying Zhao, Yanren Hou and Guangzhi Du

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

Abstract

Purpose

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

Design/methodology/approach

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

Findings

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

Originality/value

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 9
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 2 November 2015

Guangzhi Du and Yanren Hou

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

Abstract

Purpose

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

Design/methodology/approach

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

Findings

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.

Originality/value

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 7 August 2017

Guangzhi Du and Liyun Zuo

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

Abstract

Purpose

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

Design/methodology/approach

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

Findings

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

Originality/value

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 16 March 2020

Mateus Rauen, Roberto Dalledone Machado and Marcos Arndt

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

Abstract

Purpose

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.

Design/methodology/approach

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.

Findings

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

Originality/value

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.

Details

Engineering Computations, vol. 37 no. 7
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 February 2001

Suvranu De and Klaus‐Jürgen Bathe

Computational efficiency and reliability are clearly the most important requirements for the success of a meshless numerical technique. While the basic ideas of meshless…

Abstract

Computational efficiency and reliability are clearly the most important requirements for the success of a meshless numerical technique. While the basic ideas of meshless techniques are simple and well understood, an effective meshless method is very difficult to develop. The efficiency depends on the proper choice of the interpolation scheme, numerical integration procedures and techniques of imposing the boundary conditions. These issues in the context of the method of finite spheres are discussed.

Details

Engineering Computations, vol. 18 no. 1/2
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 December 2003

C.K. Lee, X. Liu and S.C. Fan

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

Abstract

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 partitionofunity 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.

Details

Engineering Computations, vol. 20 no. 8
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 4 July 2016

Marcos Arndt, Roberto Dalledone Machado and Adriano Scremin

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…

Abstract

Purpose

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.

Design/methodology/approach

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.

Findings

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.

Originality/value

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.

Details

Engineering Computations, vol. 33 no. 5
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 17 May 2021

Diego Amadeu Furtado Torres

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…

Abstract

Purpose

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. Herein, the topological enrichment pattern is revisited 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.

Design/methodology/approach

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 used. 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 loading.

Findings

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.

Originality/value

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.

Details

Engineering Computations, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 2 May 2017

Mohammad Malekan, Felício Barros, Roque Luiz da Silva Pitangueira, Phillipe Daniel Alves and Samuel Silva Penna

This paper aims to present a computational framework to generate numeric enrichment functions for two-dimensional problems dealing with single/multiple local…

Abstract

Purpose

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.

Design/methodology/approach

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.

Findings

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.

Originality/value

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.

Details

Engineering Computations, vol. 34 no. 3
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 4 July 2016

Nasrin Jafari and Mojtaba Azhari

The purpose of this paper is to present a simple HP-cloud method as an accurate meshless method for the geometrically nonlinear analysis of thick orthotropic plates of

Abstract

Purpose

The purpose of this paper is to present a simple HP-cloud method as an accurate meshless method for the geometrically nonlinear analysis of thick orthotropic plates of general shape. This method is used to investigate the effects of thickness, geometry of various shapes, boundary conditions and material properties on the large deformation analysis of Mindlin plates.

Design/methodology/approach

Nonlinear analysis of plates based on Mindlin theory is presented. The equations are derived by the Von-Karman assumption and total Lagrangian formulations. Newton-Raphson method is applied to achieve linear equations from nonlinear equations. Simple HP-cloud method is used for the construction of the shape functions based on Kronecker-δ properties, so the essential boundary conditions can be enforced directly. Shepard function is utilized for a partition of unity and complete polynomial is used as an enrichment function.

Findings

The suitability and efficiency of the simple HP-cloud method for the geometrically nonlinear analysis of thin and moderately thick plates is studied for the first time. Large displacement analysis of various shapes of plates, rectangular, skew, trapezoidal, circular, hexagonal and triangular with different boundary conditions subjected to distributed loading are considered.

Originality/value

This paper shows that the simple HP-cloud method is well suited for the large deformation analysis of Mindlin plates with various geometries, because it uses a set of a few arbitrary nodes placed in a plate of general shape. Moreover the convergence rate of the proposed method is high and the cost of solving equations is low.

Details

Engineering Computations, vol. 33 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

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