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Raphson Method (raphson + method)
Selected AbstractsCoupled HM analysis using zero-thickness interface elements with double nodes.INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 18 2008Part I: Theoretical model Abstract In recent years, the authors have proposed a new double-node zero-thickness interface element for diffusion analysis via the finite element method (FEM) (Int. J. Numer. Anal. Meth. Geomech. 2004; 28(9): 947,962). In the present paper, that formulation is combined with an existing mechanical formulation in order to obtain a fully coupled hydro-mechanical (or HM) model applicable to fractured/fracturing geomaterials. Each element (continuum or interface) is formulated in terms of the displacements (u) and the fluid pressure (p) at the nodes. After assembly, a particular expression of the traditional ,u,p' system of coupled equations is obtained, which is highly non-linear due to the strong dependence between the permeability and the aperture of discontinuities. The formulation is valid for both pre-existing and developing discontinuities by using the appropriate constitutive model that relates effective stresses to relative displacements in the interface. The system of coupled equations is solved following two different numerical approaches: staggered and fully coupled. In the latter, the Newton,Raphson method is used, and it is shown that the Jacobian matrix becomes non-symmetric due to the dependence of the discontinuity permeability on the aperture. In the part II companion paper (Int. J. Numer. Anal. Meth. Geomech. 2008; DOI: 10.1002/nag.730), the formulation proposed is verified and illustrated with some application examples. Copyright © 2008 John Wiley & Sons, Ltd. [source] An incremental formulation for the prediction of two-dimensional fatigue crack growth with curved pathsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2007Ki-Seok Kim Abstract This paper presents a new incremental formulation for predicting the curved growth paths of two-dimensional fatigue cracks. The displacement and traction boundary integral equations (BIEs) are employed to calculate responses of a linear elastic cracked body. The Paris law and the principle of local symmetry are adopted for defining the growth rate and direction of a fatigue crack, respectively. The three governing equations, i.e. the BIEs, the Paris law and the local symmetry condition, are non-linear with respect to the crack growth path and unknowns on the boundary. Iterative forms of three governing equations are derived to solve problems of the fatigue crack growth by the Newton,Raphson method. The incremental crack path is modelled as a parabola defined by the crack-tip position, and the trapezoidal rule is employed to integrate the Paris law. The validity of the proposed method is demonstrated by two numerical examples of plates with an edge crack. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical modelling of equilibrium charge separation in poled devicesINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 6 2001A. De Francesco Abstract We describe an efficient numerical procedure for the equilibrium solution of the internal electric field distribution resulting from poling of photo-refractive materials. This technique has been developed to model the equilibrium state in poled bulk devices requiring bulk charge neutrality to facilitate the modelling of blocking boundaries for a high externally applied voltage (bias) in the kV range for a small number of points. This technique is an improvement on existing conventional numerical techniques employed for modelling semiconductor devices that are intended for low bias. This method can also accommodate the modelling of planar insulators and organic optical materials. We develop an algorithm incorporating the existing Newton,Raphson method for solving Kukhtarev's equations that enforces conservation of charge within the modelled system. We apply this technique to model one-dimensional charge separation in ultraviolet (UV) excited poling of glass and, report numerical equilibrium electric field distribution for a 2 kV bias. The convergence behaviour of the algorithm is investigated and compared against the Newton,Raphson method. Copyright © 2001 John Wiley & Sons, Ltd. [source] Remarks on the updated Hessian matrix methodsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 6 2003Josep Maria Bofill Abstract Optimizing a function with respect to a set of variables using the quasi-Newton,Raphson method implies updating the Hessian matrix at each iteration. The Broyden,Fletcher,Goldfarb,Shanno update formula is used for minimization and the Murtagh,Sargent,Powell update formula for optimization of first-order saddle points. Two new formulae are proposed to update the Hessian matrix. One of these formulae is derived using exponential weights and should be used to locate first-order saddle points. The second formula is a modification of the TS,Broyden,Fletcher,Goldfarb,Shanno update and could used for both minimum and first-order saddle point optimizations. These two update Hessian matrix formulae present a performance that is the same and in many cases better that the Broyden,Fletcher,Goldfarb,Shanno and Murtagh,Sargent,Powell formulae. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 324,332, 2003 [source] | ||||||||||