Method of fundamental solutions and high order algorithm to solve nonlinear elastic problems

Askour, Omar; Tri, Abdeljalil; Braikat, Bouazza; Zahrouni, Hamid; Potier‐Ferry, Michel

doi:10.1016/j.enganabound.2018.01.007

Cited by 47 publications

(22 citation statements)

References 35 publications

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“…They used the Tikhonov regularization method and the discrepancy principle and obtained very accurate solutions. Askour et al [Askour, Tri, Braikat et al (2018)], by combination of the MFS, the asymptotic numerical method, and the analog equation method, developed a technique for analysis of nonlinear elastic problems. They also used the TSVD and Tikhonov regularization methods for solving the resulting ill-conditioned system of equations.…”

Section: Introductionmentioning

confidence: 99%

Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

Hematiyan¹,

Arezou²,

Dezfouli³

et al. 2019

Computer Modeling in Engineering &Amp; Sciences

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In this paper, some remarks for more efficient analysis of two-dimensional elastostatic problems using the method of fundamental solutions are made. First, the effects of the distance between pseudo and main boundaries on the solution are investigated and by a numerical study a lower bound for the distance of each source point to the main boundary is suggested. In some cases, the resulting system of equations becomes ill-conditioned for which, the truncated singular value decomposition with a criterion based on the accuracy of the imposition of boundary conditions is used. Moreover, a procedure for normalizing the shear modulus is presented that significantly reduces the condition number of the system of equations. By solving two example problems with stress concentration, the effectiveness of the proposed methods is demonstrated.

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Section: Introductionmentioning

confidence: 99%

Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

Hematiyan¹,

Arezou²,

Dezfouli³

et al. 2019

Computer Modeling in Engineering &Amp; Sciences

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“…Obviously, eliminating of the aforementioned drawbacks encountered in the high mesh‐dependent imposes the use of mesh‐free methods. These methods have been successfully introduced firstly into high‐order algorithms for structure mechanics and then for fluid mechanics 32‐39 …”

Section: Introductionmentioning

confidence: 99%

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

Rammane

Mesmoudi

Tri

et al. 2020

Numerical Methods in Fluids

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In this article, we propose to investigate numerically the steady bifurcation points and bifurcated branches in fluid mechanics by employing high‐order mesh‐free geometric progression algorithms. These algorithms are based on the use of the geometric progression (GP) in a high‐order mesh‐free approach. The first proposed algorithm is applied on a strong formulation using the moving least squares (MLS) approximation coupled with GP (HO‐MLS‐GPM). While the second proposed algorithm is applied on a weak formulation using the element‐free Galerkin (EFG) coupled also with GP (HO‐EFG‐GPM). The incompressibility condition is taken by introducing the penalty technique to transform the stationary Navier–Stokes equations verified by the pressure and velocity into ones verified by only the velocity. The high‐order mesh‐free algorithm permits to transform this nonlinear equations into a succession of linear ones. The GP allows to detect with precision the bifurcation points and the Lyapunov–Schmidt reduction is coupled with HO‐MLS‐GPM and HO‐EFG‐GPM as a continuation procedure to follow the many bifurcated branches. The aim of this resolution strategy concerns the treatment of the bifurcation phenomena for a fluid flow through an expansion in several geometries, where the steady flow becomes unstable after a critical Reynolds value. The obtained results are compared with those presented in literature and with those computed using the high‐order finite element algorithm coupled with GP.

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“…This method, combined with iterative methods as Newton–Raphson method and Picard iteration, has been extended to solve some nonlinear problems . Tri et al and Askour et al have associated MFS to ANM for solving nonlinear Poisson problems, nonlinear elasticity problems, and computing bifurcation branches.…”

Section: Introductionmentioning

confidence: 99%

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

Tri

Askour

Braikat

et al. 2019

Numerical Methods Partial

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New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi-harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms. KEYWORDS asymptotic numerical method, bifurcation branch, bifurcation indicator, bi-harmonic problem, method of fundamental solutions 1 Numer Methods Partial Differential Eq. 2019;35:2091-2102. wileyonlinelibrary.com/journal/num

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Method of fundamental solutions and high order algorithm to solve nonlinear elastic problems

Cited by 47 publications

References 35 publications

Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

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