Non-singular Formulation Research Articles

Reinforcement learning-based attitude control for a barbell electric sail

The electric solar wind sail (E-sail) is a new propellant-free propulsion concept. The under-actuated and highly nonlinear features of E-sail systems pose a great challenge to their attitude controller design. Conventional control schemes may not be capable of dealing with this tough problem. To this end, a reinforcement learning (RL)-based control scheme, which can explore and obtain optimal policies in the absence of training datasets, is proposed for the attitude control of a barbell E-sail system. The barbell E-sail comprises two end satellites linked to an insulated confluence point through long and conductive tethers. The voltages of the two tethers can be individually modulated for attitude control. The system attitude dynamics is described using a nonsingular formulation. The control scheme has a two-stage design. In the first stage, an RL controller based on the Proximal Policy Optimization (PPO) algorithm is used to obtain an RL control strategy, which is emulated and updated by neural networks. In the second stage, the attitude feedback control is accomplished with low computation and energy consumption and fast convergence speed by performing a real-time mapping from the system state to the control output using the updated control strategy. Finally, the simulation results demonstrate that the proposed RL-based control scheme can effectively adjust the E-sail to the design attitude by regulating the tether voltage difference. The comparisons with the NMPC scheme also indicate that the developed control scheme can significantly reduce the computation time with control accuracy maintained.

Model predictive control for spin-up maneuver of an electrodynamic tether system

This paper focuses on the nonlinear dynamics and control for the spin-up maneuver of an electrodynamic tethered satellite system in an inclined elliptical orbit. A nonsingular formulation is established by using a unit vector along the tether to define the tether orientation and avoid the Euler angle singularity, and is adopted to describe the attitude dynamics of the system and to facilitate the controller design. The spin-up problem of concern is to accelerate the electrodynamic tether system into the prescribed spinning motions and maintain the expected angular velocity. A nonlinear model predictive control law accounting for nonlinear dynamics and system constraints is proposed to achieve this control goal by actively regulating the changing rate of the bounded tether current. Finally, numerical simulations are performed to validate the effectiveness of the proposed control law for the spin-up maneuver and evaluate the control performance. Meanwhile, comparative studies are presented to demonstrate the advantages of the proposed control law in stabilizing the out-of-plane libration motions of the system in inclined orbits.

Nonlinear model predictive control for attitude maneuver of a barbell electric sail through voltage regulation

The paper focuses on the attitude dynamics and control of a barbell electric sail (E-sail). The barbell E-sail is composed of two tip satellites connected via long conductive tethers to a central insulated confluence point. The two tethers are insulated from each other at the confluence point such that the voltages of the two tethers can be independently controlled. A nonsingular formulation based on the dumbbell assumption is proposed to describe the attitude dynamics of the barbell E-sail, and exploited to develop a nonlinear model predictive controller for attitude adjustment and maintenance of the barbell E-sail by regulating the tether voltages. The nonlinear model predictive control law explicitly accounts for the nonlinear attitude dynamics and mission-related constraints of the barbell E-sail. Finally, the efficiency and performance of the proposed control law are demonstrated via numerical case studies.

Energy of periodic discrete dislocation networks

We present an approach to compute the stored energy associated with arbitrary discrete dislocation networks subjected to periodic boundary conditions. To circumvent the issue of conditional convergence while keeping the computational cost tractable, we develop a regularization procedure that involves two equivalent measures of the dislocation network energy. Taking advantage of the non-singular formulation, the energy is first evaluated by explicitly calculating the conditionally convergent sum of all interactions between dislocation segments. Regularization constants are then determined by volume integral of the smooth elastic stress fields produced for large values of the dislocation core radius. The approach is employed to investigate the stored energy of a series of idealized dislocation configurations and large-scale networks generated by discrete dislocation dynamics (DDD) simulations. It is found that (1) the stored energies predicted by DDD simulations are in good agreement with experimental measurements of about 5% of the work done, and (2) Taylor lattice configurations provide surprisingly good energetics models for complex DDD networks, thereby confirming the screening of long-range stresses in work-hardened dislocation structures.

A method for deriving a non-singular state-space formulation based on Rubin's model for Pogo analysis

A method for deriving a non-singular state-space formulation based on Rubin's model for Pogo analysis is presented in this study. Because of the non-singularity, the state-space model can be directly used in frequency-domain analyses and time-domain simulations. To describe the assembly method concisely, the dynamic equations of the nine types of independent elements are described in a standard manner. The nine types of elements are divided into two classes according to characteristics of the dynamic equations. The mapping relationship between the local and global numbers of elements and nodes is obtained by numbering all of the elements and nodes. By integrating the element stiffness matrixes to obtain the total stiffness matrix used in the finite element method, the coefficient matrixes of the improved Rubin's model are assembled from the coefficient matrixes of all of the elements according to the mapping relationship. Based on the non-singular model, the time-varying simulation with the nonlinear property of the accumulator can be done conveniently by revising the state-space model. The successful application of this method to a Pogo analysis of a certain type of CZ rocket used in China verifies the correctness and efficiency of the method of this study.

Computation of virtual X-ray diffraction patterns from discrete dislocation structures

A computational method to generate virtual micro-Laue X-ray diffraction patterns from discrete dislocation representations is presented. In order to capture elastic strain and rotation heterogeneities in space, the method takes a two-step approach, whereby the deformation gradient field is first evaluated at a set of sampling points throughout the volume, after which the diffracting vectors at each sampling point are ray-traced and their intensities are integrated to produce the resulting diffraction patterns. A non-singular formulation for the deformation gradient is developed to avoid numerical artifacts arising when sampling near dislocation cores. As an example, the method is applied to compute virtual diffraction peaks on a large scale strain-hardened microstructure in single crystal copper obtained from a dislocation dynamics simulation.

A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid

This study investigates the problem of a semi-infinite hydraulic fracture that propagates steadily in a permeable formation. The fracturing fluid rheology is assumed to follow a power-law behaviour, while the leak-off is modelled by Carter’s model. A non-singular formulation is employed to effectively analyse the problem and to construct a numerical solution. The problem under consideration features three limiting analytic solutions that are associated with dominance of either toughness, leak-off or viscosity. Transitions between all the limiting cases are analysed and the boundaries of applicability of all these limiting solutions are quantified. These bounds allow us to determine the regions in the parametric space, in which these limiting solutions can be used. The problem of a semi-infinite fracture, which is considered in this study, provides the solution for the tip region of a hydraulic fracture and can be used in hydraulic fracturing simulators to facilitate solving the moving fracture boundary problem. To cater for such applications, for which rapid evaluation of the solution is necessary, the last part of this paper constructs an approximate closed form solution for the problem and evaluates its accuracy against the numerical solution inside the parametric space.

Nonsingular boundary element flexural analysis of stiffened plates

This paper concerns a nonsingular formulation to deal with thin plates reinforced by beams using a BEM-FEM combination. The nonsingular formulation developed for the Kirchhoff׳s plate bending problem is extended to include the effect of the stiffener beams. In the proposed model, the plate is modelled by BEM using quadratic elements with transverse deflection and rotations in two orthogonal directions as the degrees of freedom (w, w,x and w,y). The beam is modelled by FEM using Timoshenko beam theory with quadratic elements and taking into account the transverse shear and bending and twisting moments (three degrees of freedom per node). As the plate and beam are modelled using C0 quadratic interpolation with the same set of three nodal values, these can be easily coupled. The nonsingular BEM formulation for thin plate bending problems is presented and the coupling of the BE equations of the plate and the FE equations of the beam is done by using the compatibility and equilibrium equations. A computer program has been developed using C++ with object oriented programming concepts. Several numerical examples are presented to illustrate the accuracy and efficiency of the present formulation.

A robust and non-singular formulation of the boundary integral method for the potential problem

A non-singular formulation of the boundary integral method (BIM) is presented for the Laplace equation whereby the well-known singularities that arise from the fundamental solution are eliminated analytically. A key advantage of this approach is that numerical errors that arise due to the proximity of nodes located on osculating boundaries are suppressed. This is particularly relevant in multi-scale problems where high accuracy is required without undue increase in computational cost when the spacing between boundaries become much smaller than their characteristic dimensions. The elimination of the singularities means that standard quadrature can be used to evaluate the surface integrals and this results in about 60% savings in coding effort. The new formulation also affords a numerically robust way to calculate the potential close to the boundaries. Detailed implementations of this approach are illustrated with problems involving osculating boundaries, 2D domains with corners and a wave drag problem in a 3D semi-infinite domain. The explicit formulation of problems with axial symmetry is also given.

Nonsingular isogeometric boundary element method for Stokes flows in 3D

Isogeometric analysis (IGA) is emerging as a technology bridging computer aided geometric design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis. In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that describe the geometry define also the approximation spaces. In the FE-IGA approach, the surfaces generated by the CAGD tools need to be extended to volumetric descriptions, a major open problem in 3D. This additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of boundary integral equations, leading to boundary element isogeometric analysis (IGA-BEM). The main advantages of such an approach are given by the dimensionality reduction of the problem (from volumetric-based to surface-based), by the fact that the interface with CAGD tools is direct, and by the possibility to treat exterior problems, where the computational domain is infinite. By contrast, these methods produce system matrices which are full, and require the integration of singular kernels. In this paper we address the second point and propose a nonsingular formulation of IGA-BEM for 3D Stokes flows, whose convergence is carefully tested numerically. Standard Gaussian quadrature rules suffice to integrate the boundary integral equations, and carefully chosen known exact solutions of the interior Stokes problem are used to correct the resulting matrices, extending the work by Klaseboer et al. (2012) [27] to IGA-BEM.

Mass conservative finite volume discretization of the continuity equations in multi-component mixtures

The Stefan–Maxwell equations for multi-component diffusion result in a system of coupled continuity equations for all species in the mixture. We use a generalization of the exponential scheme to discretize this system of continuity equations with the finite volume method. The system of continuity equations in this work is obtained from a non-singular formulation of the Stefan–Maxwell equations, where the mass constraint is not applied explicitly. Instead, all mass fractions are treated as independent unknowns and the constraint is a result of the continuity equations, the boundary conditions, the diffusion algorithm and the discretization scheme. We prove that with the generalized exponential scheme, the mass constraint can be satisfied exactly, although it is not explicitly applied. A test model from the literature is used to verify the correct behavior of the scheme.

Application of the fundamental solutions by Ganowicz in a static analysis of Reissner’s plates by the boundary element method

In this paper the direct non-singular formulation of the boundary element method using the fundamental solutions given by Ganowicz (1966) [6] and its application to a static analysis of plates with intermediate thickness is presented. A more exact calculation of almost singular integrals is possible, thanks to the applied modification of the Gauss integration procedure with the inversed distribution of integration points. The non-singular method is based on an offset of collocation points from a plate boundary. The accuracy of results depends on this offset, so the analysis of a relation between the distance of the collocation point from the plate and the conditioning of the matrix of integral equations is carried out. The optimal distance equal to 0.01 of the boundary element length was determined. The presented approach allows to carry out the static analysis of plates with arbitrary shapes including plates with holes. Solution of thin plates is also possible.

Inclusion of Higher Order Harmonics in the Modeling of Optimal Low-Thrust Orbit Transfer

The higher fidelity modeling of minimum-time transfers using continuous constant acceleration low-thrust is depicted by including the higher zonal harmonics J 3 and J 4for the Earth gravity model. The inclusion of these higher order harmonics is of great benefit in carrying out accurate transfer simulations, especially for long duration flights dwelling in low altitudes where the effects of these zonals are greatest. The analysis presented here can also be coded in the flight guidance computer of spacecraft for autonomous operations and on ground computers for solution uploads and resetting during low-thrust transfers. Equinoctial elements are used to avoid singularities when orbits are circular or equatorial and the applicability of the theory is of a general nature regardless of the size, shape and spatial orientation of the orbits provided they are not of the parabolic or hyperbolic types. To this end, two sets of dynamic and adjoint differential equations in terms of nonsingular orbital elements are derived by further considering a more accurate perturbation model in the form of the higher order Earth zonal harmonics J 3and J 4. Previous analyses involved only the first-order J 2 term in order to model optimal lowthrust transfers between any two given circular or elliptic orbits. The first formulation uses the eccentric longitude as the sixth element of an equinoctial set of elements while describing the thrust as well as the zonal accelerations in the so called direct equinoctial frame. The second formulation makes use of the true longitude as the sixth element instead while resolving the thrust and the zonal accelerations in the rotating Euler-Hill frame simplifying considerably the algebraic derivations leading to the generation of the nonsingular differential equations that are also free of any singularity for the important zero eccentricity and zero inclination cases often encountered in Earth orbit transfer problems. The derivations of both nonsingular formulations are mutually validated by generating an optimal transfer example that achieves the same target conditions regardless of which formulation is used.

Optimal Nonlinear Spacecraft Attitude Control through Hamilton-Jacobi Formulation

Nonlinear optimal feedback control of the complete, three-axis, asymmetric spacecraft attitude dynamics and kinematics is presented. The kinematics description is via minimal modified Rodrigues parameters, which allow nonsingular formulation for eigenaxis rotations greater than 180 degrees. The optimality condition is obtained through Hamilton-Jacobi formulation for the minimization of a cost function in terms of rotational velocities, kinematic parameters, and control torques. A positive-definite Lyapunov function is derived analytically for asymptotic stability. Numerical simulation results for rigid spacecraft undergoing large rotational maneuvers with the present nonlinear controller are compared to a semi-optimal controller and a terminal-time weighted nonlinear optimal controller. With appropriate design parameters, the present nonlinear optimal controller is seen to be globally stabilizing for arbitrarily large initial conditions, whereas the semi-optimal controllers are rendered non-stabilizing. Although being based on an infinite-interval formulation, the present controller is seen to produce a comparable performance with that of the terminal-time weighted optimal controller. In terms of onboard computational resource requirements, the present controller lies between the semi-optimal and the terminal-time weighted optimal controllers.

Non‐singular boundary integral formulations for plane interior potential problems

AbstractIn this article, a non‐singular formulation of the boundary integral equation is developed to solve smooth and non‐smooth interior potential problems in two dimensions. The subtracting and adding‐back technique is used to regularize the singularity of Green's function and to simplify the calculation of the normal derivative of Green's function. After that, a global numerical integration is directly applied at the boundary, and those integration points are also taken as collocation points to simplify the algorithm of computation. The result indicates that this simple method gives the convergence speed of order N −3 in the smooth boundary cases for both Dirichlet and mix‐type problems. For the non‐smooth cases, the convergence speed drops at O(N −1/2) for the Dirichlet problems. Copyright © 2001 John Wiley & Sons, Ltd.

A numerical method for the evaluation of hydrodynamic forces of translating bodies under a free surface

We present a numerical method to evaluate the hydrodynamic forces of translating bodies under a free surface. Both steady and unsteady problems are considered. Analytical and numerical studies are carried out based on the Havelock wave-source function and the integral equation method. Two main problems arising inherently in the proposed solution method are overcome in order to facilitate the numerical implementation. The first lies in evaluating the Havelock function, which involves integrals with highly oscillatory kernels. Particular integration contours leading to non-oscillatory integrands are derived a priori so that the integrals can be evaluated efficiently. The second problem lies in evaluating singular kernels in the boundary integral equation. The corresponding non-singular formulation is derived using some theorems of potential theory, including the Gauss flux theorem and the property related to the equipotential body. The subsequent formulation is amenable to the solution by directly using the standard quadrature formulas without taking another special treatment

Sensor proximity error in the conformal near-field acoustic holography and its solution by using the nonsingular boundary integral equation

In the conformal near-field acoustic holography (NAH) using the boundary element method (BEM), the transfer matrix relating the vibro-acoustic properties of source and field depends solely on the geometrical condition of the problem. This kind of NAH is known to be very powerful in dealing with the sources having irregular boundaries. When the vibro-acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components. The conventional acoustic BEM based on the Kirchhoff–Helmholtz integral equation has the singularity problem in the very-near field of the source surface: computation error increases very rapidly as approaching from the far field. This problem originated from the singular kernel of the fundamental solution of the BIE and can influence the reconstruction accuracy. In this paper, the holographic BIE is reformulated to remove the singularity by introducing an additional propagating plane wave. Reconstructed results by the conventional and the nonsingular BIE are compared for a simple radiator model. It is observed that the nonsingular formulation can yield accurate vibro-acoustic transfer matrix and thus improve the resolution of the reconstructed source field.

A boundary contour formulation for design sensitivity analysis in two-dimensional linear elasticity

A formulation for computing first-order shape design sensitivities in two-dimensional (2-D) linear elastostatics by the boundary contour method (BCM), along with a numerical implementation using quadratic boundary elements, is presented in this paper. Here, the direct differentiation approach is analytically applied to the appropriate boundary contour equations in order to derive the sensitivities of all the physical quantities (displacements, tractions and stresses) on the boundary as well as those for displacements and stresses inside the body under consideration. The nonsingular formulation of the BCM is used for computing the boundary displacements, and boundary stresses at “off contour” regular points. A regular boundary point is a point on the boundary where it is locally smooth; an off contour point lies inside a boundary element. Their corresponding sensitivities are obtained in a straightforward manner from the resulting regular sensitivity formulation. Also, the stress sensitivities at the boundary nodes can be recovered easily from the global displacement shape functions described in a Cartesian coordinate system. Finally, through three numerical examples for which analytical solutions exist, it is shown that the BCM can provide remarkably accurate numerical results for shape sensitivities.

Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard

A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a triangular domain. Our main result is a reformulation of this problem in terms of coupled non-singular integral equations. A non-singular formulation, via Fredholm's theory, guarantees uniqueness and provides a mathematically firm foundation for both numerical and analytic studies. We compare and contrast our reformulation, based on the exact solution for the wedge, with the standard singular integral equations using numerical discretisation techniques. We consider in detail the (integrable) equilateral triangle and the Pythagorean 3–4–5 triangle. Our non-singular formulation produces results which are well behaved mathematically. In contrast, while resolving the eigenvalues very well, the standard approach displays various behaviours demonstrating the need for some sort of “renormalisation.” The non-singular formulation provides a mathematically firm basis for the generation and analysis of periodic orbit expansions. We discuss their convergence, placing particular emphasis on the computational effort required in comparision with Einstein–Brillouin–Keller quantisation and the standard discretisation, which is analogous to the method of Bogomolny. We also discuss the generalisation of our technique to smooth, chaotic billiards.

Optimal low-thrust rendezvous using equinoctial orbit elements

The problem of optimal low-thrust rendezvous using continuous constant acceleration is presented based on the use of the non-singular equinoctial orbit elements. State and adjoint equations are integrated numerically by applying the thrust vector along a direction that maximizes the variational Hamiltonian at each instant of time. The two-point-boundary-value problem is solved via a Newton-Raphson scheme after guessing the initial values of the Lagrange multipliers as well as the total flight time. The transversality condition for the Hamiltonian at the final time is also used to carry out the 7 × 7 search. Equinoctial elements have previously been used to carry out minimum-time orbit transfers by considering only the five slowly varying elements. The inclusion of the sixth element representing the fast variable has allowed us to extend the applicability of the non-singular formulation to problems of orbital rendezvous.