Nonsingular Terminal Sliding Mode Guidance with Impact Angle Constraints

Guidance laws based on a conventional sliding mode ensures only asymptotic convergence. However, convergence to the desired impact angle within a finite time is important in most practical guidance applications. These finite time convergent guidance laws suffer from singularity leading to control saturation. In this paper, guidance laws to intercept targets at a desired impact angle, from any initial heading angle, without exhibiting any singularity, are presented. The desired impact angle, which is defined in terms of a desired line-of-sight angle, is achieved in finite time by selecting the interceptor's lateral acceleration to enforce nonsingular terminal sliding mode on a switching surface designed using nonlinear engagement dynamics. Numerical simulation results are presented to validate the proposed guidance laws for different initial engagement geometries and impact angles. Although the guidance laws are designed for constant speed interceptors, its robustness against the time-varying speed of interceptors is also evaluated through extensive simulation results.

Generalized scaling of misorientation angle distributions at meso-scale in deformed materials

Scaling behaviour has been observed at mesoscopic level irrespective of crystal structure, type of boundary and operative micro-mechanisms like slip and twinning. The presence of scaling at the meso-scale accompanied with that at the nano-scale clearly demonstrates the intrinsic spanning for different deformation processes and a true universal nature of scaling. The origin of a 1/2 power law in deformation of crystalline materials in terms of misorientation proportional to square root of strain is attributed to importance of interfaces in deformation processes. It is proposed that materials existing in three dimensional Euclidean spaces accommodate plastic deformation by one dimensional dislocations and their interaction with two dimensional interfaces at different length scales. This gives rise to a 1/2 power law scaling in materials. This intrinsic relationship can be incorporated in crystal plasticity models that aim to span different length and time scales to predict the deformation response of crystalline materials accurately.

Modal tailoring and closed-form solutions for rotating non-uniform Euler-Bernoulli beams

In this paper, the free vibration of a rotating Euler-Bernoulli beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a rotating beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and fifth order stiffness variations of the beam which are typical of helicopter blades. Thus, it is found that an infinite number of such beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the first, second and third modes of a rotating cantilever beam and the first and second elastic modes of a rotating pinned-free beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these cases. The derived results can be used as benchmark solutions for the validation of rotating beam numerical methods and may also guide nodal tailoring. (C) 2014 Elsevier Ltd. All rights reserved.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

Three Dimensional Impact Angle Constrained Guidance Law using Sliding Mode Control

In this paper, three dimensional impact angle control guidance laws are proposed for stationary targets. Unlike the usual approach of decoupling the engagement dynamics into two mutually orthogonal 2-dimensional planes, the guidance laws are derived using the coupled dynamics. These guidance laws are designed using principles of conventional as well as nonsingular terminal sliding mode control theory. The guidance law based on nonsingular terminal sliding mode guarantees finite time convergence of interceptor to the desired impact angle. In order to derive the guidance laws, multi-dimension switching surfaces are used. The stability of the system, with selected switching surfaces, is demonstrated using Lyapunov stability theory. Numerical simulation results are presented to validate the proposed guidance law.

Nonlinear Differential Games-Based Impact-Angle-Constrained Guidance Law

The problem of intercepting a maneuvering target at a prespecified impact angle is posed in nonlinear zero-sum differential games framework. A feedback form solution is proposed by extending state-dependent Riccati equation method to nonlinear zero-sum differential games. An analytic solution is obtained for the state-dependent Riccati equation corresponding to the impact-angle-constrained guidance problem. The impact-angle-constrained guidance law is derived using the states line-of-sight rate and projected terminal impact angle error. Local asymptotic stability conditions for the closed-loop system corresponding to these states are studied. Time-to-go estimation is not explicitly required to derive and implement the proposed guidance law. Performance of the proposed guidance law is validated using two-dimensional simulation of the relative nonlinear kinematics as well as a thrust-driven realistic interceptor model.

Implicit scheme for meshless compressible Euler solver

In this paper, an implicit scheme is presented for a meshless compressible Euler solver based on the Least Square Kinetic Upwind Method (LSKUM). The Jameson and Yoon's split flux Jacobians formulation is very popular in finite volume methodology, which leads to a scalar diagonal dominant matrix for an efficient implicit procedure (Jameson & Yoon, 1987). However, this approach leads to a block diagonal matrix when applied to the LSKUM meshless method. The above split flux Jacobian formulation, along with a matrix-free approach, has been adopted to obtain a diagonally dominant, robust and cheap implicit time integration scheme. The efficacy of the scheme is demonstrated by computing 2D flow past a NACA 0012 airfoil under subsonic, transonic and supersonic flow conditions. The results obtained are compared with available experiments and other reliable computational fluid dynamics (CFD) results. The present implicit formulation shows good convergence acceleration over the RK4 explicit procedure. Further, the accuracy and robustness of the scheme in 3D is demonstrated by computing the flow past an ONERA M6 wing and a clipped delta wing with aileron deflection. The computed results show good agreement with wind tunnel experiments and other CFD computations.

Simulations of impinging droplets with surfactant-dependent dynamic contact angle

An arbitrary Lagrangian-Eulerian (ALE) finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface is presented. The numerical scheme solves the time-dependent Navier-Stokes equations for the fluid flow, scalar convection-diffusion equation for the surfactant transport in the bulk phase, and simultaneously, surface evolution equations for the surfactants on the free surface and on the liquid-solid interface. The effects of surfactants on the flow dynamics are included into the model through the surface tension and surfactant-dependent dynamic contact angle. In particular, the dynamic contact angle (theta(d)) of the droplet is defined as a function of the surfactant concentration at the contact line and the equilibrium contact angle (theta(0)(e)) of the clean surface using the nonlinear equation of state for surface tension. Further, the surface forces are included into the model as surface divergence of the surface stress tensor that allows to incorporate the Marangoni effects without calculating the surface gradient of the surfactant concentration on the free surface. In addition to a mesh convergence study and validation of the numerical results with experiments, the effects of adsorption and desorption surfactant coefficients on the flow dynamics in wetting, partially wetting and non-wetting droplets are studied in detail. It is observed that the effects of surfactants are more in wetting droplets than in the non-wetting droplets. Further, the presence of surfactants at the contact line reduces the equilibrium contact angle further when theta(0)(e) is less than 90 degrees, and increases it further when theta(0)(e) is greater than 90 degrees. Nevertheless, the presence of surfactants has no effect on the contact angle when theta(0)(e) = 90 degrees. The numerical study clearly demonstrates that the surfactant-dependent contact angle has to be considered, in addition to the Marangoni effect, in order to study the flow dynamics and the equilibrium states of surfactant droplet impingement accurately. The proposed numerical scheme guarantees the conservation of fluid mass and of the surfactant mass accurately. (C) 2015 Elsevier Inc. All rights reserved.

Meshless Local Petrov-Galerkin Method for Rotating Euler-Bernoulli Beam

Free vibration problem of a rotating Euler-Bernoulli beam is solved with a truly meshless local Petrov-Galerkin method. Radial basis function and summation of two radial basis functions are used for interpolation. Radial basis function satisfies the Kronecker delta property and makes it simpler to apply the essential boundary conditions. Interpolation with summation of two radial basis functions increases the node carrying capacity within the sub-domain of the trial function and higher natural frequencies can be computed by selecting the complete domain as a sub-domain of the trial function. The mass and stiffness matrices are derived and numerical results for frequencies are obtained for a fixed-free beam and hinged-free beam simulating hingeless and articulated helicopter blades. Stiffness and mass distribution suitable for wind turbine blades are also considered. Results show an accurate match with existing literature.

Gauge invariance in the theoretical description of time-resolved angle-resolved pump/probe photoemission spectroscopy

Nonequilibrium calculations in the presence of an electric field are usually performed in a gauge, and need to be transformed to reveal the gauge-invariant observables. In this work, we discuss the issue of gauge invariance in the context of time-resolved angle-resolved pump/probe photoemission. If the probe is applied while the pump is still on, one must ensure that the calculations of the observed photocurrent are gauge invariant. We also discuss the requirement of the photoemission signal to be positive and the relationship of this constraint to gauge invariance. We end by discussing some technical details related to the perturbative derivation of the photoemission spectra, which involve processes where the pump pulse photoemits electrons due to nonequilibrium effects.

Composite Guidance for Impact Angle Control Against Higher Speed Targets

In the literature, the impact angle control problem has been addressed mostly against lower speed or stationary targets. However, in the current defense scenario, targets of much higher speeds than interceptors are a reality. Moreover, approaching a higher speed target from a specified angle is important for effective seeker acquisition and enhanced warhead effectiveness. This paper proposes a composite proportional navigation guidance law using a combination of the standard proportional navigation and the recently proposed retroproportional navigation guidance laws for intercepting higher speed nonmaneuvering targets at specified impact angles in three-dimensional engagements. An analysis of the set of achievable impact angles by the composite proportional navigation guidance law is presented. It is shown that there exists an impulse bias that, when added to the composite proportional navigation guidance command, expands this set further by reversing the direction of the line-of-sight angular rotation vector. A bound on the magnitude of the bias is also derived. Finally, an implementation of this impulse bias, in the form of a series of pulses, is proposed and analyzed. Simulation results are also presented to support the analysis.

Composite Guidance for Impact Angle Control Against Higher Speed Targets

In the literature, the impact angle control problem has been addressed mostly against lower speed or stationary targets. However, in the current defense scenario, targets of much higher speeds than interceptors are a reality. Moreover, approaching a higher speed target from a specified angle is important for effective seeker acquisition and enhanced warhead effectiveness. This paper proposes a composite proportional navigation guidance law using a combination of the standard proportional navigation and the recently proposed retroproportional navigation guidance laws for intercepting higher speed nonmaneuvering targets at specified impact angles in three-dimensional engagements. An analysis of the set of achievable impact angles by the composite proportional navigation guidance law is presented. It is shown that there exists an impulse bias that, when added to the composite proportional navigation guidance command, expands this set further by reversing the direction of the line-of-sight angular rotation vector. A bound on the magnitude of the bias is also derived. Finally, an implementation of this impulse bias, in the form of a series of pulses, is proposed and analyzed. Simulation results are also presented to support the analysis.

求解Euler方程的区域分解方法与并行算法

将复杂形状区域划分成多块子区域，研究发展了一种多块区域之间迎风守恒型的内边界耦合方法，实现相邻子区域解的光滑过渡，使多区耦合得到总体流场的数值解。对二维翼型跨音速流动和圆板形隆起物超音速流动等进行了分区数值计算，并将计算结果与单区计算结果和实验结果作了比较。并行分区计算引入“先进先出”的同步控制等待机制，实现了高效率并行计算，还分析了影响并行效率的主要因素。