Ballistic learning control： formulation, analysis and convergence

In this paper, we formulate and explore the characteristics of iterative learning in ballistic control problems. The iterative learning control (ILC) theory provides a suitable framework for derivations and analysis of ballistic control under learning process. To overcome the obstacles caused by uncertain gradient and redundant control input, we incorporate extra trials into iterative learning. With the help of trial results, proper control and updating direction can be determined. Then, iterative learning can be applied to ballistic control problem. Several initial state learning algorithms are studied for initial speed control, force control, as well as combined speed and angle control. In the end, shooting angle learning in the basketball shot process is simulated to verify the effectiveness of iterative learning methods in ballistic control problems.     

Efficient formulation of the Gibbs–Appell equations for constrained multibody systems

Multibody System Dynamics - In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell...     

Is my model too complex? Evaluating model formulation using model reduction

While mechanistic models tend to be detailed, they are less detailed than the real systems they seek to describe, so judgements are being made about the appropriate level of detail within the process of model development. These judgements are difficult to test, consequently it is easy for models to become over-parameterised, potentially increasing uncertainty in predictions. The work we describe is a step towards addressing these difficulties. We propose and implement a method which explores a family of simpler models obtained by replacing model variables with constants (model reduction by variable replacement). The procedure iteratively searches the simpler model formulations and compares models in terms of their ability to predict observed data, evaluated within a Bayesian framework. The results can be summarised as posterior model probabilities and replacement probabilities for individual variables which lend themselves to mechanistic interpretation. This provides powerful diagnostic information to support model development, and can identify areas of model over-parameterisation with implications for interpretation of model results. We present the application of the method to 3 example models. In each case reduced models are identified which outperform the original full model in terms of comparisons to observations, suggesting some over-parameterisation has occurred during model development. We argue that the proposed approach is relevant to anyone involved in the development or use of process based mathematical models, especially those where understanding is encoded via empirically based relationships.     

An efficient recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam

Multibody System Dynamics - A recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam is developed, where large displacements, deformations, and rotations are...     

Experimental validation of rigid-flexible coupling dynamic formulation for hub–beam system

The aim of this paper is to compare the accuracy of the absolute nodal coordinate formulation and the floating frame of reference formulation for the rigid-flexible coupling dynamics of a three-dimensional Euler–Bernoulli beam by numerical and experimental validation. In the absolute nodal coordinate formulation, based on geometrically exact beam theory and considering the torsion effect, the material curvature of the beam is derived, and then variational equations of motion of a three-dimensional beam are obtained, which consist of three position coordinates, two slope coordinates, and one rotational coordinate. In the floating frame of reference formulation, the displacement of an arbitrary point on the beam is described by the rigid-body motion and a small superimposed deformation displacement. Based on linear elastic theory, the quadratic terms of the axial strain are neglected, and the curvatures are simplified to the first order. Considering both the linear damping and the quadratic air resistance damping, the equations of motion of the multibody system composed of air-bearing test bed and a cantilevered three-dimensional beam are derived based on the principle of virtual work. In order to verify the results of the computer simulation, two experiments are carried out: an experiment of hub–beam system with large deformation and a dynamic stiffening experiment. The comparison of the simulation and experiment results shows that in case of large deformation, the frequency result obtained by the floating frame of reference formulation is lower than that obtained by the experiment. On the contrary, the result obtained by the absolute nodal coordinate formulation agrees well with that obtained by the experiment. It is also shown that the floating frame of reference formulation based on linear elastic theory cannot reveal the dynamic stiffening effect. Finally, the applicability of the floating frame of reference formulation is clarified.     

Estimating stable delay intervals with a discretized Lyapunov–Krasovskii functional formulation

In general, a system with time delay may have multiple stable delay intervals. Especially, a stable delay interval does not always contain zero. Asymptotically accurate stability conditions such as discretized Lyapunov–Krasovskii functional (DLF) method and sum-of-square (SOS) method are especially effective for such systems. In this article, a DLF-based method is proposed to estimate the maximal stable delay interval accurately without using bisection when one point in this interval is given. The method is formulated as a generalized eigenvalue problem (GEVP) of linear matrix inequalities (LMIs), and an accurate estimate may be reached by iteration either in a finite number of steps or asymptotically. The coupled differential–difference equation formulation is used to illustrate the method. However, the idea can be easily adapted to the traditional differential–difference equation setting.     

Fluid–structure interaction for non-conforming interfaces based on a dual mortar formulation

In the present work the problem of fluid–structure interaction (FSI) with independently space discretized fluid and structure fields is addressed in the context of finite elements. To be able to deal with non-conforming meshes at the fluid–structure interface, we propose the integration of a dual mortar method into the general FSI framework. This method has lately been used successfully to impose interface constraints in other contexts such as finite deformation contact. The main focus is set on monolithic coupling algorithms for FSI here. In these cases the dual mortar approach allows for the elimination of the additional Lagrange multiplier degrees of freedom from the global system by condensation. The resulting system matrices have the same block structure as their counterparts for the conforming case and permit the same numerical treatment. Partitioned Dirichlet–Neumann coupling is also considered briefly and it is shown that the dual mortar approach permits a numerically efficient mapping between fluid and structure quantities at the interface.Numerical examples demonstrate the efficiency and robustness of the proposed method. We present results for a variety of different element formulations for the fluid and the structure field, indicating that the proposed method is not limited to any specific formulation. Furthermore, the applicability of state-of-the-art iterative solvers is considered and the convergence behavior is shown to be comparable to standard simulations with conforming discretizations at the interface.     

A Lie group formulation of Kane’s equations for multibody systems

We propose a Lie group approach to formulate the Kane’s equations of motion for multibody systems. This approach regards the set of rigid body transformations as the special Euclidean group SE(3). By expressing rigid body displacements as exponential maps generated from the Lie algebra se(3), it subsequently manipulates rigid body kinematics as convenient matrix operations. With this approach, all the individual quantities involved in Kane’s equations can be computed explicitly in an intrinsic manner, and the motion equations can be obtained systematically and efficiently. An example is presented to illustrate its use and effectiveness.     

AILU preconditioning for the finite element formulation of the incompressible Navier–Stokes equations

In this paper, the effect of a variable reordering method on the performance of “adapted incomplete LU (AILU)” preconditioners applied to the P2P1 mixed finite element discretization of the three-dimensional unsteady incompressible Navier–Stokes equations has been studied through numerical experiments, where eigenvalue distribution and convergence histories are examined. It has been revealed that the performance of an AILU preconditioner is improved by adopting a variable reordering method which minimizes the bandwidth of a globally assembled saddle-point type matrix. Furthermore, variants of the existing AILU(1) preconditioner have been suggested and tested for some three-dimensional flow problems. It is observed that the AILU(2) outperforms the existing AILU(1) with a little extra computing time and memory.     

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fully-mixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68–95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.     

A mixed computational algorithm for plane elastic contact problems—I. formulation

A mixed variational statement and corresponding finite element model are developed for an arbitrary plane body undergoing large deformations (i.e. large displacements, large rotations and small strains) under external loads using the updated Lagrangian formulation. The mixed finite element formulation allows the nodal displacements and stresses to be approximated independently. Two different contact algorithms are presented for the separate cases of a thin plate in contact with a rigid pin and a flexible pin, and the algorithms account for the computational difficulties that arise from the unknown contact area and the presence of friction between the pin and the plate.     

WEC-C: a distributed,deterministic catchment model — theory,formulation and testing

WEC-C is a distributed, deterministic, catchment-scale water flow and solute transport model containing a number of innovations not present in comparable models. For example, it allows for the imposition of catchment topological changes resulting from land uses such as surface mining. In addition, it features preferential vertical flow in the vadose zone modelled using a dual continuum approach. Numerically, it differs from many existing models in that it employs a direct linkage between the vertical and lateral solvers of its split-solver scheme and, due to its use of explicit solvers, is stable regardless of the form of the soil moisture characteristic curves. The WEC-C model framework is a rectangular grid of uniform cell size in the lateral plane combined with a system of soil layers, of variable thickness, in the vertical direction. Within this structure, the governing equations for flow and transport are discretised and solved. All parameters are defined locally in each computational cell so that all available data on catchment variability can be incorporated directly into the model. This paper describes the formulation of WEC-C, which is based on operator splitting with first-order accurate solvers for both the vertical and lateral flow and transport models. WEC-C was subjected to four stringent, synthetic tests designed to evaluate its accuracy by comparison with available analytical and numerical solutions. These showed that there were scale issues associated with the model, and that induced numeric dispersion of solutes could be significant. Nonetheless, it is suggested that WEC-C is still useful as a distributed catchment model.     

A new finite element formulation for computational fluid dynamics: V. Circumventing the babuška-brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations

A new Petrov-Galerkin formulation of the Stokes problem is proposed. The new formulation possesses better stability properties than the classical Galerkin/variational method. An error analysis is performed for the case in which both the velocity and pressure are approximated by C₀ interpolations. Combinations of C₀ interpolations which are unstable according to the Babuška-Brezzi condition (e.g., equal-order interpolations) are shown to be stable and convergent within the present framework. Calculations exhibiting the good behavior of the methodology are presented.     

An efficient steering control formulation for the articulated body mobile robot “KR-II”

This paper introduces an efficient steering control method for the articulated body mobile robot Koryu-II (KR-II). KR-II is a real robot, composed of six cylindrical segments linked in series and has a long snake-like appearance. The main issue on KR-II's steering control is, given from a remote human operator the velocity and orientation commands for the foremost segment, to automatically generate joint commands for all the following segments, such that they follow the foremost segment's trajectory. The derived method is based on a trajectory planning scheme in the inertial reference frame, and is feasible for real time computation. It also presents good energy efficiency and trajectory tracking performance characteristics, and can be extended for KR-II's W-Shaped Configuration steering control, which augments the lateral stability of the robot, essential for locomotion over uneven terrain. The validity of these methods are verified by experiments on the mechanical model KR-II.     

A geometrically exact beam element based on the absolute nodal coordinate formulation

In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.     

Structures,hypergraphs, and knowledge representation in chemical engineering science. I — Problem formulation

It is shown how algebraic structure theory — a tool developed in the last two decades to describe chemical engineering systems — fits into the conceptual framework of hypergraph theory. A new type of structure — actually a special hereditary hypergraph — is introduced in order to lay the foundations for knowledge representation in expert systems designed for chemical engineering problems. Sketchy algorithms are proposed to solve the problems. An existing expert system called FOREST which is applied in the Hungarian chemical industry is mentioned and it is shown how it has been built upon the theoretical basis outlined here. Several very simple mathematical examples are shown, and a real-life example including chemical reactions, mass and heat transport between not more than two phases are treated in detail in the Appendix. A large set of open problems in modelling, mathematics and computation are presented and possible further directions are outlined.     

Construction of Lyapunov–Krasovskii functionals for networks of iISS retarded systems in small-gain formulation

This paper tackles a problem of verifying stability of retarded dynamical networks in a dissipative formulation. Subsystems are assumed to be integral input-to-state stable (iISS). Time-delays are allowed to reside in both subsystems and interconnection channels, and may be both discrete and distributed. No assumption is made on the interconnection topology. A small-gain methodology is developed for constructing a Lyapunov–Krasovskii functional to establish iISS of such a network.     

The χ-schemes: a new consistent high-resolution formulation based on the normalized variable methodology

This paper deals with the formulation and testing of a new class of consistent high-resolution schemes, denoted as the χ-schemes. These schemes, combine consistency, accuracy and boundedness across systems of equations and are suitable for use in the simulation of multi-phase and multi-component flows. The consistency feature refers to the capability of these schemes to implicitly satisfy the additional algebraic constraint representing a global conservation relation governing certain sets of equations (e.g., species mass fraction, volume fraction, etc.). Four χ-schemes are implemented within an unstructured grid finite-volume framework, tested by solving four multi-component pure-advection test problems, and shown to be consistent.