Topic

# Tangent

About: Tangent is a research topic. Over the lifetime, 5904 publications have been published within this topic receiving 99201 citations. The topic is also known as: tangent line.

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TL;DR: An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented, and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point.

Abstract: An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based density-functional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surfa...

6,825 citations

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TL;DR: In this article, a second order algorithm for finding points on a steepest descent path from the transition state of the reactants and products is presented. But the points are optimized so that the segment of the reaction path between any two adjacent points is given by an arc of a circle, and the gradient at each point is tangent to the path.

Abstract: A new algorithm is presented for obtaining points on a steepest descent path from the transition state of the reactants and products. In mass‐weighted coordinates, this path corresponds to the intrinsic reaction coordinate. Points on the reaction path are found by constrained optimizations involving all internal degrees of freedom of the molecule. The points are optimized so that the segment of the reaction path between any two adjacent points is given by an arc of a circle, and so that the gradient at each point is tangent to the path. Only the transition vector and the energy gradients are needed to construct the path. The resulting path is continuous, differentiable and piecewise quadratic. In the limit of small step size, the present algorithm is shown to take a step with the correct tangent vector and curvature vector; hence, it is a second order algorithm. The method has been tested on the following reactions: HCN→CNH, SiH2+H2→SiH4, CH4+H→CH3+H2, F−+CH3F→FCH3+F−, and C2H5F→C2H4+HF. Reaction paths calculated with a step size of 0.4 a.u. are almost identical to those computed with a step size of 0.1 a.u. or smaller.

5,487 citations

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TL;DR: In this article, the authors extended their previous algorithm for following reaction paths downhill to use mass-weighted internal coordinates, which has the correct tangent vector and curvature vectors in the limit or small step size but requires only the transition vector and the energy gradients.

Abstract: Our previous algorithm for following reaction paths downhill (J. Chem. Phys. 1989, 90, 2154), has been extended to use mass-weighted internal coordinates. Points on the reaction path are round by constrained optimizations involving the internal degrees or freedom or the molecule. The points are optimized so that the segment or the reaction path between any two adjacent points is described by an arc or a circle in mass-weighted internal coordinates, and so that the gradients (in mass-weighted internals) at the end points or the arc are tangent to the path. The algorithm has the correct tangent vector and curvature vectors in the limit or small step size but requires only the transition vector and the energy gradients; the resulting path is continuous, differentiable, and piecewise quadratic

5,291 citations

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TL;DR: The method is presented as a generalization of a recursive bicubic B-spline patch subdivision algorithm, which generates surfaces that approximate points lying-on a mesh of arbitrary topology except at a small number of points, called extraordinary points.

Abstract: This paper describes a method for recursively generating surfaces that approximate points lying-on a mesh of arbitrary topology. The method is presented as a generalization of a recursive bicubic B-spline patch subdivision algorithm. For rectangular control-point meshes, the method generates a standard B-spline surface. For non-rectangular meshes, it generates surfaces that are shown to reduce to a standard B-spline surface except at a small number of points, called extraordinary points. Therefore, everywhere except at these points the surface is continuous in tangent and curvature. At the extraordinary points, the pictures of the surface indicate that the surface is at least continuous in tangent, but no proof of continuity is given. A similar algorithm for biquadratic B-splines is also presented.

2,137 citations

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TL;DR: In this paper, it is shown that consistency between the tangent operator and the integration algorithm employed in the solution of the incremental problem plays crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method.

Abstract: It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative J 2 flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a class of non-associative flow rules are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton's method, whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.

1,702 citations