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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 19 Sep 2024 13:58:17 GMT2024-09-19T13:58:17ZA numerical method based on Taylor series for bifurcation analyses within Föppl-von Karman plate theory
http://hdl.handle.net/10985/17484
A numerical method based on Taylor series for bifurcation analyses within Föppl-von Karman plate theory
TIAN, Haitao; POTIER-FERRY, Michel; ABED-MERAIM, Farid
A new numerical technique for post-buckling analysis is presented by combining the Asymptotic Numerical Method (ANM) and the Taylor Meshless Method (TMM). These two methods are based on Taylor series, with respect to a scalar load parameter for ANM and with respect to the space variables for TMM. The advantage of ANM is an adaptive step length and this is very efficient near bifurcation points. The specificity of TMM is a quasi-exact solution of the PDEs inside the domain, which leads to a strong reduction of the number of degrees of freedom (DOFs).
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/174842018-01-01T00:00:00ZTIAN, HaitaoPOTIER-FERRY, MichelABED-MERAIM, Farid A new numerical technique for post-buckling analysis is presented by combining the Asymptotic Numerical Method (ANM) and the Taylor Meshless Method (TMM). These two methods are based on Taylor series, with respect to a scalar load parameter for ANM and with respect to the space variables for TMM. The advantage of ANM is an adaptive step length and this is very efficient near bifurcation points. The specificity of TMM is a quasi-exact solution of the PDEs inside the domain, which leads to a strong reduction of the number of degrees of freedom (DOFs).Taylor Meshless Method for bending and buckling of thin plates
http://hdl.handle.net/10985/20359
Taylor Meshless Method for bending and buckling of thin plates
TIAN, Haitao; POTIER-FERRY, Michel; ABED-MERAIM, Farid
This paper introduces a new meshless method named Taylor Meshless Method (TMM) using Taylor series to deduce the shape functions. Next the problem is discretized by point-collocation only on the boundary and without integration. The discrete boundary problem is solved by least-squares method. In this talk, this method is applied to bending and buckling of isotropic and anisotropic plates. The shape functions are polynomials that coincide with harmonic polynomials in the case of Laplace equation. These polynomials are computed numerically by solving the PDE approximately in the sense of Taylor series. Of course, this leads to an error that decreases asymptotically with the degree. TMM can be considered as a Trefftz Method, but we search approximated solutions in the sense of Taylor series while Trefftz Method is generally based on exact solutions of the PDE. As a counterpart, one is able to build this complete family of approximated solutions, whatever be the studied equation. A strong reduction of the number of degrees of freedom is the main advantage of this class of discretization techniques. The main drawback is the ill-conditioning of the final matrix, what can limit the size of the solved problem, but it was established that TMM is able to solve large scale problems. In the case of nonlinear PDEs as Föppl-Von Karman plate models, one applies first a linearization technique as Newton iterative technique. Here we apply Asymptotic Numerical Method. Next the resulting linear equations have variable coefficients are they are solved by TMM. Several linear and nonlinear numerical results will be presented for isotropic and anisotropic laminated plates.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/203592017-01-01T00:00:00ZTIAN, HaitaoPOTIER-FERRY, MichelABED-MERAIM, Farid This paper introduces a new meshless method named Taylor Meshless Method (TMM) using Taylor series to deduce the shape functions. Next the problem is discretized by point-collocation only on the boundary and without integration. The discrete boundary problem is solved by least-squares method. In this talk, this method is applied to bending and buckling of isotropic and anisotropic plates. The shape functions are polynomials that coincide with harmonic polynomials in the case of Laplace equation. These polynomials are computed numerically by solving the PDE approximately in the sense of Taylor series. Of course, this leads to an error that decreases asymptotically with the degree. TMM can be considered as a Trefftz Method, but we search approximated solutions in the sense of Taylor series while Trefftz Method is generally based on exact solutions of the PDE. As a counterpart, one is able to build this complete family of approximated solutions, whatever be the studied equation. A strong reduction of the number of degrees of freedom is the main advantage of this class of discretization techniques. The main drawback is the ill-conditioning of the final matrix, what can limit the size of the solved problem, but it was established that TMM is able to solve large scale problems. In the case of nonlinear PDEs as Föppl-Von Karman plate models, one applies first a linearization technique as Newton iterative technique. Here we apply Asymptotic Numerical Method. Next the resulting linear equations have variable coefficients are they are solved by TMM. Several linear and nonlinear numerical results will be presented for isotropic and anisotropic laminated plates.Buckling and wrinkling of thin membranes by using a numerical solver based on multivariate Taylor series
http://hdl.handle.net/10985/20655
Buckling and wrinkling of thin membranes by using a numerical solver based on multivariate Taylor series
TIAN, Haitao; POTIER-FERRY, Michel; ABED-MERAIM, Farid
Buckling and wrinkling of thin structures often lead to very complex response curves that are hard to follow by standard path-following techniques, especially for very thin membranes in a slack or nearly slack state. Many recent papers mention numerical difficulties encountered in the treatment of wrinkling problems, especially with path-following procedures and often these authors switch to pseudo-dynamic algorithms. Moreover, the numerical modeling of many wrinkles leads to very large size problems. In this paper, a new numerical procedure based on a double Taylor series is presented, that combines path-following techniques and discretization by a Trefftz method: Taylor series with respect to a load parameter (Asymptotic Numerical Method) and with respect to space variables (Taylor Meshless Method). The procedure is assessed on buckling benchmarks and on the difficult problem of a sheared rectangular membrane.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10985/206552021-01-01T00:00:00ZTIAN, HaitaoPOTIER-FERRY, MichelABED-MERAIM, Farid Buckling and wrinkling of thin structures often lead to very complex response curves that are hard to follow by standard path-following techniques, especially for very thin membranes in a slack or nearly slack state. Many recent papers mention numerical difficulties encountered in the treatment of wrinkling problems, especially with path-following procedures and often these authors switch to pseudo-dynamic algorithms. Moreover, the numerical modeling of many wrinkles leads to very large size problems. In this paper, a new numerical procedure based on a double Taylor series is presented, that combines path-following techniques and discretization by a Trefftz method: Taylor series with respect to a load parameter (Asymptotic Numerical Method) and with respect to space variables (Taylor Meshless Method). The procedure is assessed on buckling benchmarks and on the difficult problem of a sheared rectangular membrane.