The boundary element method for nonlinear problems

Katsikadelis, John T.; Nerantzaki, M. S.

doi:10.1016/s0955-7997(98)00093-9

Cited by 62 publications

(43 citation statements)

References 1 publication

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“…The method is based on the concept of the analog equation, according to which the three coupled nonlinear differential equations governing the equilibrium of the membrane are replaced by three Poisson's equations with ®ctitious domain source under the same boundary conditions. The ®ctitious sources are established using a procedure based on BEM as developed by Katsikadelis and Nerantzaki (1999) and the three displacements components as well as their derivatives are computed from their integral representations, which are used as mathematical formulas. The stress resultants at any interior point and the reactions at any boundary point are also evaluated.…”

Section: Introductionmentioning

confidence: 99%

The analog equation method for large deflection analysis of membranes. A boundary-only solution

Katsikadelis

Nerantzaki

Tsiatas

2001

Computational Mechanics

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In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse de¯ections inuence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with ®ctitious sources under the same boundary conditions. Subsequently, the ®ctitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its ef®ciency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary.

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Section: Introductionmentioning

confidence: 99%

The analog equation method for large deflection analysis of membranes. A boundary-only solution

Katsikadelis

Nerantzaki

Tsiatas

2001

Computational Mechanics

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“…Equation (2) is the analog equation of Eq. (1a) [4][5][6]. In this context, the fictitious source is approximated by MQ-RBFs series.…”

Section: The Quintessence Of the Methodsmentioning

confidence: 99%

The meshless analog equation method: I. Solution of elliptic partial differential equations

Katsikadelis

2009

Arch Appl Mech

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A new purely meshless method for solving elliptic partial differential equations (PDEs) is presented. The method is based on the principle of the analog equation of Katsikadelis, hence its name meshless analog equation method (MAEM), which converts the original equation into a simple solvable substitute one of the same order under a fictitious source. The fictitious source is represented by multiquadric radial basis functions (MQ-RBFs). The integration of the analog equation yields new RBFs, which are used to approximate the sought solution. Then inserting the approximate solution into the PDE and boundary conditions (BCs) and collocating at the mesh-free nodal points results in a system of linear equations, which permit the evaluation of the expansion coefficients of the RBFs series. The method exhibits key advantages over other RBF collocation methods as it is highly accurate and the coefficient matrix of the resulting linear equations is always invertible. The accuracy is achieved using optimal values for the shape parameters and the centers of the multiquadrics as well as of the integration constants of the analog equation, which are obtained by minimizing the functional that produces the PDE. Without restricting its generality, the method is illustrated by applying it to the general second order 2D and 3D elliptic PDEs. The studied examples demonstrate the efficiency and high accuracy of the developed method.

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“…Hartmann and Zomantel [25], further developed the method, and implemented it for plate with relatively complex geometries and loadings. Following the above work, others (Tanaka and Miyazaki [62], Fernandes and Venturini [68], Paiva and Aliabadi [42], Albuquerque [2] among others [31,32,55,3,9,41,58,60,23,29,34,64,72]) have further extended the formulation for the classical plate.…”

Section: Introductionmentioning

confidence: 99%

Boundary element analysis of stiffened panels with repair patches

Pisa¹,

Aliabadi²

2015

Engineering Analysis with Boundary Elements

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In this paper a boundary element method for analysis of fractured stiffened panels repaired with riveted or adhesively bonded patches is presented. In order to achieve such a formulation, several boundary element formulations involving the membrane and bending of displacement, and, stress resultants are coupled together to analyse the model. The multi-region formulation is used to simulate the stiffeners, which are modelled as an assembly. They are then connected to the sheets to form the stiffened panels by means of a rivet formulation. The dual boundary element method is used to simulate the presence of cracks, and the patches, treated as independent plates, are joined to the panel either with rivets or adhesive. A boundary element formulation for adhesive bonding is implemented to model the adhesively bonded patches. The Crack Opening Displacements (COD) method and the Jintegral are implemented to evalute the required fracture parameters. Examples presented include a wing box with a three spar section, with fully stiffened skin, and, the skin is considered to have a crack and a repair patch is used on top of the crack to stop its growth.Response to Reviewers: Thank you for highlighting the missing references and figures numbers as well as certain typo's. We have corrected all the highlighted items.Thank you for highlighting the missing references and figures numbers as well as certain typo's. We have corrected all the highlighted items. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Detailed Response to Reviewers AbstractIn this paper a boundary element method for analysis of fractured stiffened panels repaired with riveted or adhesively bonded patches is presented. In order to achieve such a formulation, several boundary element formulations involving the membrane and bending of displacement, and, stress resultants are coupled together to analyse the model. The multi-region formulation is used to simulate the stiffeners, which are modelled as an assembly. They are then connected to the sheets to form the stiffened panels by means of a rivet formulation. The dual boundary element method is used to simulate the presence of cracks, and the patches, treated as independent plates, are joined to the panel either with rivets or adhesive. A boundary element formulation for adhesive bonding is implemented to model the adhesively bonded patches. The Crack Opening Displacements (COD) method and the J-integral are implemented to evalute the required fracture parameters. Examples presented include a wing box with a three spar section, with fully stiffened skin, and, the skin is considered to have a crack and a repair patch is used on top of the crack to stop its growth.

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The boundary element method for nonlinear problems

Cited by 62 publications

References 1 publication

The analog equation method for large deflection analysis of membranes. A boundary-only solution

The analog equation method for large deflection analysis of membranes. A boundary-only solution

The meshless analog equation method: I. Solution of elliptic partial differential equations

Boundary element analysis of stiffened panels with repair patches

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