Second-order Formulation Research Articles

Delta-V-optimal centralized guidance strategy for under-actuated N-satellite formations

This paper addresses the computation of Delta-V-optimal, safe, relative orbit reconfigurations for satellite formations in a centralized fashion. The formations under consideration comprise an uncontrolled chief spacecraft flying with an arbitrary number, N, of deputy satellites, where each deputy is equipped with a single electric thruster. Indeed, this represents a technological solution that is becoming widely employed by the producers of small-satellite platforms. While adopting a single electric thruster does reduce the required power, weight, and size of the orbit control system, it comes at the cost of rendering the satellite under-actuated. In this setting, the satellite can provide a desired thrust vector only after an attitude maneuver is carried out to redirect the thruster nozzle opposite to the desired thrust direction. In order to further extend the applicability range of such under-actuated platforms, guidance strategies are developed to support different reconfiguration scenarios for N-satellite formations. This paper starts from a classical non-convex quadratically constrained trajectory optimization formulation, which passes through multiple simplifications and approximations to arrive to two novel convex formulations, namely a second-order cone programming formulation, and a linear programming one. Out of five guidance formulations proposed in this article, the most promising three were compared through an extensive benchmark analysis that is applied to fifteen of the most widely-used solvers. This benchmark experiment provides information about the key distinctions between the different problem formulations, and under which conditions each one of them can be recommended.

Flexibility-Constrained Energy Storage System Placement for Flexible Interconnected Distribution Networks

Configuring energy storage systems (ESSs) in distribution networks is an effective way to alleviate issues induced by intermittent distributed generation such as transformer overloading and line congestion. However, flexibility has not been fully taken into account when placing ESSs. This paper proposes a novel ESS placement method for flexible interconnected distribution networks considering flexibility constraints. An ESS siting and sizing model is formulated aiming to minimize the life-cycle cost of ESSs along with the annual network loss cost, electricity purchasing cost from the upper-level power grid, photovoltaic (PV) curtailment cost, and ESS scheduling cost while fulfilling various security constraints. Flexible ramp-up/-down constraints of the system are added to improve the ability to adapt to random changes in both power supply and demand sides, while a fluctuation rate of net load constraints is also added for each bus to reduce the net load fluctuation. The nonconvex model is then converted into a second-order cone programming formulation, which can be solved in an efficient manner. The proposed method is evaluated on a modified 33-bus flexible distribution network. The simulation results show that better flexibility can be achieved with slightly increased ESS investment costs. However, a large ESS capacity is needed to reduce the net load fluctuation to low levels, especially when the PV capacity is large.

Mixed variational formulation and finite-element implementation of second-order poro-elasticity

We present a variational formulation of second-order poro-elasticity that can be readily implemented into finite-element codes by using standard Lagrangian interpolation functions. Point of departure is a two-field minimization principle in terms of the displacement and the fluid flux as independent variables. That principle is taken as a basis for the derivation of continuous and incremental saddle-point formulations in terms of an extended set of independent variables. By static condensation this formulation is then reduced to a minimization principle in terms of the displacement and fluid flux as well as associated higher-order fields. Once implemented into a finite-element code, the resulting formulation can be applied to the numerical simulation of porous media in consideration of second-order effects. Here, we analyze the model response by means of several example problems including two standard tests in poro-elasticity, namely the consolidation problems of Terzaghi and Mandel, and compare the results with those of a corresponding first-order model. As becomes clear, the second-order formulation can unleash its full potential when applied to the study of porous media having spatial dimensions comparable to the size of their microstructure. In particular, it is capable to regularize steep field gradients at external as well as internal surfaces and to describe material dilatation effects known from experiments.

Carroll fermions

Using carefully chosen projections, we consider different Carroll limits of relativistic Dirac fermions in any spacetime dimensions. These limits define Carroll fermions of two types: electric and magnetic. The latter type transforms as a reducible but indecomposable representation of the Carroll group. We also build action principles for all Carroll fermions we introduce; in particular, in even dimensions we provide an action principle for a minimal magnetic Carroll fermion, having the same number of components as a Dirac spinor. We then explore the coupling of these fermions to magnetic Carroll gravity in both its first-order and second-order formulations.

A novel immersed boundary approach for irregular topography with acoustic wave equations

Irregular terrain has a pronounced effect on the propagation of seismic and acoustic wavefields but is not straightforwardly reconciled with structured finite-difference (FD) methods used to model such phenomena. Accurate wavefield simulation is paramount in subsurface imaging applications such as reverse time migration and full-waveform inversion, requiring suitable topography handling. Methods currently detailed in the literature generally are limited in scope application-wise or nontrivial to apply to real-world geometries. A general immersed boundary treatment capable of imposing a range of boundary conditions in a relatively equation-agnostic manner has been developed, alongside a framework implementing this approach to complement emerging code-generation paradigms. The approach is distinguished by the use of N-dimensional Taylor-series extrapolants constrained by boundary conditions imposed at some suitably distributed set of surface points. The extrapolation process is encapsulated in modified derivative stencils applied in the vicinity of the boundary, using hyperspherical support regions. This method ensures boundary representation is consistent with the FD discretization. Furthermore, high-dimensional and vector boundary conditions can be applied without approximation prior to discretization. A consistent methodology can thus be applied across free and rigid surfaces with first- and second-order acoustic wave equation formulations. Application to both equations is demonstrated, and numerical examples based on analytic and real-world topography implementing free and rigid surfaces in 2D and 3D are presented. Numerical examples and convergence tests demonstrate the accuracy of boundary treatments devised by the prescribed approach, their suitability to practical applications, and the feasibility of automatically generating treatments to suit each case.

Rocket landing guidance based on second-order Picard-Chebyshev-Newton type algorithm

This paper proposes a rocket substage vertical landing guidance method based on the second-order Picard-Chebyshev-Newton type algorithm. Firstly, the continuous-time dynamic equation is discretized based on the natural second-order Picard iteration formulation and the Chebyshev polynomial. Secondly, the landing problem that considers terminal constraints is transformed into a nonlinear least-squares problem with respect to the terminal constraint function and solved with the Gauss-Newton method. In addition, the projection method is introduced to the iteration process of the Gauss-Newton method to realize the inequality constraints of the thrust. Finally, the closed-loop strategy for rocket substage vertical landing guidance is proposed and the numerical simulations of the rocket vertical landing stage are carried out. The simulation results demonstrate that compared with the sequential convex optimization algorithm, the proposed algorithm has higher computational efficiency.

Optimal Dispatch of DERs and Battery-Based ESS in Distribution Grids While Considering Reactive Power Capabilities and Uncertainties: A Second-Order Cone Programming Formulation

Optimal Dispatch of DERs and Battery-Based ESS in Distribution Grids While Considering Reactive Power Capabilities and Uncertainties: A Second-Order Cone Programming Formulation

A robust unstructured finite volume scheme for urban flash flood simulation: Duhok city as a case study

ABSTRACT A robust well-balanced finite volume scheme is proposed for simulating the flash floods due to rainfall-runoff process. The solver’s implementation on unstructured triangular meshes is based on a Godunov-type second-order upwind finite volume formulation, and Roe’s average scheme is used to determine the fluxes. The bed slope, friction and rainfall source terms are discretized using an upwind approach. The suggested algorithm uses the slope limiter and Hancock’s predictor-corrector method in order to secure numerical stability. The proposed model overwhelms the drawbacks of many commercial codes that use the finite difference, finite elements and finite volume in their scheme, especially in dealing with rapid change in topography. The prepared scheme solves the two-dimensional shallow water equations over a real topography of Hishkaro basin located in Duhok city in Kurdistan of Iraq where the variation of bed elevation reaches 717 m. The scheme is validated against real observed data in Duhok city provided by the Sewage directory. The data of five observation points over the study area were used to validate the code. The results showed that the max water level reaches 3.67, 2.12, 1.20, 0.67 and 0.3 m at observed points 3, 2, 4, 1 and 5, respectively.

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

Distributionally robust joint chance-constrained programming: Wasserstein metric and second-order moment constraints

In this paper, we propose a new approximate linear reformulation for distributionally robust joint chance programming with Wasserstein ambiguity sets and an efficient solution approach based on Benders decomposition. To provide a convex approximation to the distributionally robust chance constraint, we use the worst-case conditional value-at-risk constrained program. In addition, we derive an approach for distributionally robust joint chance programming with a hybrid ambiguity set that combines a Wasserstein ball with second-order moment constraints. This approach, which allows injecting domain knowledge into a Wasserstein ambiguity set and thus allows for less conservative solutions, has not been considered before. We propose two formulations of this problem, namely a semidefinite programming and a computationally favorable second-order cone programming formulation. The models and algorithms proposed in this paper are evaluated through computational experiments demonstrating their computational efficiency. In particular, the Benders decomposition algorithm is shown to be more than an order of magnitude faster than a standard solver allowing for the solution of large instances in a relatively short time.

A second-order resolvent formulation for the analysis of turbulent flow structures

A second-order resolvent formulation for the analysis of turbulent flow structures

Time-explicit hybrid high-order method for the nonlinear acoustic wave equation

We devise a fully explicit scheme for the nonlinear acoustic wave equation in its second-order formulation in time, using the HHO method for space discretization and the leapfrog scheme for time integration. The key idea for the explicitation is an iterative procedure to approximate at each time step the static nonlinear coupling between the cell and face unknowns. This procedure is based on a splitting of the HHO stabilization operator and, in the linear case, is proved to converge for a large enough weight of the stabilization uniformly in the mesh size. Increasing the stabilization weight turns out to have a moderate impact on the CFL condition. The numerical experiments demonstrate the computational efficiency of the splitting procedure compared to a semi-implicit scheme for the static coupling between cell and face unknowns.

Second-order multi-scale modelling of natural and architected materials in the presence of voids: Formulation and numerical implementation

This contribution proposes a second-order computational homogenisation formulation for natural and architected materials in the presence of voids. The macro-scale is described by a second gradient continuum theory in the finite strain regime, and the micro-scale is modelled by the concept of representative volume element (RVE) within the classical first-order continuum mechanics. The Method of Multi-scale Virtual Power is employed to link the two scales, ensuring a variationally consistent scale transition. Unlike the multi-scale model proposed by Rodrigues Lopes and Andrade Pires (2022), the developed formulation allows the consideration of voids in the RVE domain and boundary, which is critical for modelling cellular materials, lattice structures, and metamaterials, among others. This is achieved by defining the kinematical quantities only in the solid domain of the RVE and postulating a new homogenisation relation for the second-order gradient. The kinematic constraints are imposed on the RVE by the Lagrange multiplier method and particularised for minimal (lower bound), periodic and direct (upper bound) conditions. It is demonstrated that the homogenised macroscopic stress tensors can be expressed in terms of the Lagrange multipliers. The finite element method is adopted for the numerical solution of the micro and macro equilibrium. The Newton–Raphson scheme is employed to solve the non-linear systems of equations at both scales and the consistent macroscopic tangents required for the FE2 framework are derived. Several numerical examples of porous solids, lattice structures and metamaterials illustrate the consistency and applicability of the formulation for two and three-dimensional problems.

Mean flow modeling in high-order nonlinear Schrödinger equations

The evaluation and consideration of the mean flow in wave evolution equations are necessary for the accurate prediction of fluid particle trajectories under wave groups, with relevant implications in several domains, from the transport of pollutants in the ocean to the estimation of energy and momentum exchanges between the waves at small scales and the ocean circulation at large scale. We derive an expression of the mean flow at a finite water depth, which, in contrast to other approximations in the literature, accurately accords with the deep-water limit at third order in steepness and is equivalent to second-order formulations in intermediate water. We also provide envelope evolution equations at fourth order in steepness for the propagation of unidirectional wave groups either in time or space that include the respective mean flow term. The latter, in particular, is required for accurately modeling experiments in water wave flumes in arbitrary depths.

Dirichlet and Neumann boundary conditions for a lattice Boltzmann scheme for linear elastic solids on arbitrary domains

We propose novel, second-order accurate boundary formulations of Dirichlet and Neumann boundary conditions for arbitrary curved boundaries, within the scope of our recently introduced lattice Boltzmann method for linear elasticity. The proposed methodology systematically constructs and analyzes the boundary formulations on the basis of the asymptotic expansion technique; thus, it is expected to be of general applicability to boundary conditions for lattice Boltzmann formulations, well beyond the focus of this paper. For Dirichlet boundary conditions, a modified version of the bounce-back method is developed, whereas the Neumann boundary conditions are based on a novel generalized ansatz proposed in this work. The Neumann boundary formulation requires information from one additional neighbor node, for which a finite difference algorithm is introduced that enables the handling of very general boundary geometries. The theoretical derivations are verified by convergence studies using manufactured solutions, as well as by comparison with the analytical solution of the classical plate with hole problem.

Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in generalised harmonic gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background—the inversion (conformal) representation of the Minkowski spacetime—and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

Finite element method for radiative transfer in multi-dimensional graded index media based on second-order form formulation

Gradient distribution of refractive index appears commonly in gas of non-uniform temperature/pressure and water of non-uniform salinity. Efficient and accurate solution of radiative transfer in graded index media is essential for related applications. Finite element method (FEM) is a very effective tool to solve various problems governed with partial differential equations. However, FEM may suffer stability problem for solving radiative transfer in semitransparent media due to the convection-dominated property of the transfer equation. The existence of two angular derivative terms complicates the solution of radiative transfer in graded index media as compared to uniform index media. In this paper, a new stabilized FEM is proposed to solve radiative transfer in multi-dimensional graded index media, based on the second-order form transformation approach. Using the approach, a second order diffusion term is naturally introduced in the transfer equation, which ensures numerical stability, and avoids the false smearing effects as compared to the common schemes based on artificial diffusion. The performance of the presented FEM is verified with one- and two-dimensional test cases. It is demonstrated that the new FEM owns good numerical stability and accuracy, and could potentially be an effective tool to solve radiative transfer in graded index media.

Modeling material length-scale effect using the second-order peridynamic material correspondence model

In this paper, a study on modeling material length-scale effect using peridynamics via the second-order material correspondence formulation is presented. Connections of the second-order model with all three types of Mindlin first gradient theory are established. An implicit solution scheme based on the automatic differentiation technique for the construction of the stiffness matrix for linear elastic problems is developed. Numerical examples are used to examine the material stability of the model and verify its capability in modeling material length-scale effect for different length-scale parameters. From the wave dispersion analysis, it is found that the second-order model is stable as long as the material has nonzero length-scale parameter. In the linear elastic deformation study, great agreements between the model predictions and the analytical solutions for nonzero length-scale parameters are observed.

Implicit inverse force identification method for vibroacoustic finite element model

The two-field vibroacoustic finite-element (FE) model requires a relatively large number of degrees of freedom compared to the monophysics model, and the conventional force identification method for structural vibration can be adjusted for multiphysics problems. In this study, an effective inverse force identification method for an FE vibroacoustic interaction model of an interior fluid–structure system was proposed. The method consists of: (1) implicit inverse force identification based on the Newmark-β time integration algorithm for stability and efficiency, (2) second-order ordinary differential formulation by avoiding the state-space form causing large degrees of freedom, (3) projection-based multiphysics reduced-order modeling for further reduction of degrees of freedom, and (4) Tikhonov regularization to alleviate the measurement noise. The proposed method can accurately identify the unmeasured applied forces on the in situ application and concurrently reconstruct the response fields. The accuracy, stability, and computational efficiency of the proposed method were evaluated using numerical models and an experimental testbed. A comparative study with the augmented Kalman filter method was performed to evaluate its relative performance.

Real-Time Response Estimation of Structural Vibration with Inverse Force Identification

Virtual sensing is an effective method to identify the inaccessible state of the structural systems by compensating the limitations of the conventional physical sensing techniques. Recently, it becomes popular in structural vibration field and their relevant physical domains such as civil, mechanical, aerospace engineering, thermal dynamics, and acoustics. This study aimed to develop a virtual sensing algorithm of structural vibration for the real-time identification of unmeasured information. First, certain local point vibration responses (such as displacement and acceleration) are measured using physical sensors, and the data sets are extended using a numerical model to cover the unmeasured quantities through the entire spatial domain in the real-time computation process. A modified time integrator is then proposed to synchronize the physical sensors and the numerical model using inverse dynamics. In particular, an efficient inverse force identification method is derived using implicit time integration. The second-order ordinary differential formulation and its projection-based reduced-order modeling are used to avoid two times larger degrees of freedom within the state-space form. Then, the Tikhonov regularization noise-filtering algorithm is employed instead of Kalman filtering. The performance of the proposed method is investigated on both numerical and experimental testbeds under sinusoidal and random excitation loading conditions. In the numerical test, the system could identify the status of the motor housing structure in the speed of 16,181 S a m p l e s / s . The FDE and RMSE values are bounded under 0.1816 and 1503.2. In the case of the experimental test, the algorithm is implemented to the beam structure using a single-board computer, including inverse force identification and unmeasured response prediction. Even in the limited computational environment, the system could identify the applied forces in real time in the speed of 2,173 S a m p l e s / s . Through all experimental cases, FDE and RMSE values are bounded under 0.2925 and 0.1660. The results show that the virtual sensing algorithm can accurately identify unmeasured information, forces, and displacements throughout the vibration model in real time in a very limited computing environment.