WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:math> Boundary Condition

Koubaiti, Ouadie; Elkhalfi, Ahmed; El-Mekkaoui, Jaouad

doi:10.1155/2020/4879723

Cited by 4 publications

(3 citation statements)

References 25 publications

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“…The work in Sadd (2005); Timoshenko and Goodier (1985); Koubaiti et al (2020a); Ouadie et al (2018) offers more information on elasticity problems.…”

Section: Web-spline Finite Element Methodsmentioning

confidence: 99%

“…Web-spline process B-splines are fundamental in almost all geometric modelling applications. Non-uniform rational B-spline representations (Nurbs) (Bazilevs and Hughes., 2008;Koubaiti et al, 2020a), have become standard in CAD and CAM (Shah and Mäntylä, 1995;Ouadie et al, 2018), work, and by applying the Web method, B-splines have also been found to provide highly efficient finite element approximations. The B-splines located near the edge of a domain offer very little relevant support and thus cause instabilities in the numerical solution; however, they cannot be omitted without affecting the order of approximation and producing a large number of conditions within the resulting linear system.…”

Section: Weak Problem Of Navier-lamé With New Boundary Condition C a Bmentioning

confidence: 99%

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The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

Koubaiti

Fakkoussi

El-Mekkaoui

et al. 2021

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Purpose This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution. Design/methodology/approach In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory. Findings The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity. Originality/value The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

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“…The work in Sadd (2005); Timoshenko and Goodier (1985); Koubaiti et al (2020a); Ouadie et al (2018) offers more information on elasticity problems.…”

Section: Web-spline Finite Element Methodsmentioning

confidence: 99%

Section: Weak Problem Of Navier-lamé With New Boundary Condition C a Bmentioning

confidence: 99%

The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

Koubaiti

Fakkoussi

El-Mekkaoui

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this paper, we suggest a reanalysis method, which reduces the computation time through using data obtained either from an experiment or from a theoretical analysis based on a finite element model of the structure [14,15,16,17]. The originality lies in the fact that the strategy takes into account, in a very significant manner, the contribution of the unknown modes.…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Solve Perturbed Symmetric Eigenvalue Problems

2020

International Journal of Circuits, Systems and Signal Processing

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The objective of this study is to ef-ciently resolve a perturbed symmetric eigen-value problem, without resolving a completelynew eigenvalue problem. When the size of aninitial eigenvalue problem is large, its multipletimes solving for each set of perturbations can becomputationally expensive and undesired. Thistype of problems is frequently encountered inthe dynamic analysis of mechanical structures.This study deals with a perturbed symmetriceigenvalue problem. It propose to develop atechnique that transforms the perturbed sym-metric eigenvalue problem, of a large size, toa symmetric polynomial eigenvalue problem ofa much reduced size. To accomplish this, weonly need the introduced perturbations, the sym-metric positive-de nite matrices representing theunperturbed system and its rst eigensolutions.The originality lies in the structure of the ob-tained formulation, where the contribution of theunknown eignsolutions of the unperturbed sys-tem is included. The e ectiveness of the pro-posed method is illustrated with numerical tests.High quality results, compared to other existingmethods that use exact reanalysis, can be ob-tained in a reduced calculation time, even if theintroduced perturbations are very signi cant.

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Corrigendum to “WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition”

Koubaiti¹,

Elkhalfi²,

El-Mekkaoui

2021

Abstract and Applied Analysis

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WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Cited by 4 publications

References 25 publications

The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

A New Approach to Solve Perturbed Symmetric Eigenvalue Problems

Corrigendum to “WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition”

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