Topic

# Curvature

About: Curvature is a research topic. Over the lifetime, 53324 publications have been published within this topic receiving 981776 citations.

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TL;DR: The PSC algorithm as mentioned in this paper approximates the Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws, which can be used also for more general surface motion problems.

Abstract: New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be used also for more general Hamilton-Jacobi-type problems. The algorithms are demonstrated by computing the solution to a variety of surface motion problems.

13,020 citations

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TL;DR: In this paper, a force density proportional to the surface curvature of constant color is defined at each point in the transition region; this force-density is normalized in such a way that the conventional description of surface tension on an interface is recovered when the ratio of local transition-reion thickness to local curvature radius approaches zero.

Abstract: In the novel method presented for modeling the effects of surface tension on fluid motion, the interfaces between fluids with different, color-represented properties are finite-thickness transition regions across which the color varies continuously. A force density proportional to the surface curvature of constant color is defined at each point in the transition region; this force-density is normalized in such a way that the conventional description of surface tension on an interface is recovered when the ratio of local transition-reion thickness to local curvature radius approaches zero. The properties of the method are illustrated by computational results for 2D flows.

7,863 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: In this paper, an efficient line-of-sight method was implemented to calculate the anisotropy and polarization of the cosmic microwave background for scalar and tensor modes in almost Friedmann-Robertson-Walker models with positive spatial curvature.

Abstract: We implement the efficient line-of-sight method to calculate the anisotropy and polarization of the cosmic microwave background for scalar and tensor modes in almost Friedmann-Robertson-Walker models with positive spatial curvature. We present new results for the polarization power spectra in such models.

4,332 citations