Algebraic Constraints Research Articles

A robust explicit integration of elasto-plastic constitutive models, based on simple subincrement size estimation

PurposeThe purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit integration algorithm for rate-independent elasto-plastic constitutive models.Design/methodology/approachThe proposed automatic substepping algorithm is founded on an explicit integration scheme. The estimation of the maximal subincrement size is based on the stability analysis.FindingsIn contrast to other explicit substepping schemes, the algorithm is self-correcting by definition and generates no cumulative drift. Although the integration proceeds with maximal possible subincrements, high level of accuracy is attained. Algorithmic tangent stiffness is calculated in explicit form and optionally no analytical second-order derivatives are needed.Research limitations/implicationsThe algorithm is convenient for elasto-plastic constitutive models, described with an algebraic constraint and a set of differential equations. This covers a large family of materials in the field of metal plasticity, damage mechanics, etc. However, it cannot be directly used for a general material model, because the presented algorithm is convenient for solving a set of equations of a particular type.Practical implicationsThe estimation of the maximal stable subincrement size is computationally cheap. All expressions in the algorithm are in explicit form, thus the implementation is simple and straightforward. The overall performance of the approach (i.e. accuracy, time consumption) is fully comparable with a default (built-in) ABAQUS/Standard algorithm.Originality/valueThe estimated maximal subincrement size enables the algorithm to be stable by definition. Subincrements are much larger than those in conventional substepping algorithms. No error control, error correction or local iterations are required even in the case of large increments.

Space-time CFTs from the Riemann sphere

We consider two-dimensional chiral, first-order conformal field theories governing maps from the Riemann sphere to the projective light cone inside Minkowski space — the natural setting for describing conformal field theories in two fewer dimensions. These theories have a SL(2) algebra of local bosonic constraints which can be supplemented by additional fermionic constraints depending on the matter content of the theory. By computing the BRST charge associated with gauge fixing these constraints, we find anomalies which vanish for specific target space dimensions. These critical dimensions coincide precisely with those for which (biadjoint) cubic scalar theory, gauge theory and gravity are classically conformally invariant. Furthermore, the BRST cohomology of each theory contains vertex operators for the full conformal multiplets of single field insertions in each of these space-time CFTs. We give a prescription for the computation of three-point functions, and compare our formalism with the scattering equations approach to on-shell amplitudes.

TLISMNI/Adams algorithm for the solution of the differential/algebraic equations of constrained dynamical systems

This paper examines the performance of the 3rd and 4th order implicit Adams methods in the framework of the two-loop implicit sparse matrix numerical integration method in solving the differential/algebraic equations of heavily constrained dynamical systems. The variable-step size two-loop implicit sparse matrix numerical integration/Adams method proposed in this investigation avoids numerical force differentiation, ensures satisfying the nonlinear algebraic constraint equations at the position, velocity, and acceleration levels, and allows using sparse matrix techniques for efficiently solving the dynamical equations. The iterative outer loop of the two-loop implicit sparse matrix numerical integration/Adams method is aimed at achieving the convergence of the implicit integration formulae used to solve the independent differential equations of motion, while the inner loop is used to ensure the convergence of the iterative procedure used to satisfy the algebraic constraint equations. To solve the independent differential equations, two different implicit Adams integration formulae are examined in this investigation; a 3rd order implicit Adams-Moulton formula with a 2nd order explicit predictor Adams Bashforth formula, and a 4th order implicit Adams-Moulton formula with a 3rd order explicit predictor Adams Bashforth formula. A standard Newton–Raphson algorithm is used to satisfy the nonlinear algebraic constraint equations at the position level. The constraint equations at the velocity and acceleration levels are linear, and therefore, there is no need for an iterative procedure to solve for the dependent velocities and accelerations. The algorithm used for the error check and step-size change is described. The performance of the two-loop implicit sparse matrix numerical integration/Adams algorithm developed in this investigation is evaluated by comparison with the explicit predictor-corrector Adams method which has a variable-order and variable-step size. Simple and heavily constrained dynamical systems are used to evaluate the accuracy, robustness, damping characteristics, and effect of the outer-loop iterations of the proposed implicit schemes. The results obtained in this investigation show that the two-loop implicit sparse matrix numerical integration methods proposed in this study can be more efficient for stiff systems because of their ability to damp out high-frequency oscillations. Explicit integration methods, on the other hand, can be more efficient in the case of non-stiff systems.

Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

In program analysis, the synthesis of models of logical theories representing the program semantics is often useful to prove program properties. We use order-sorted first-order logic as an appropriate framework to describe the semantics and properties of programs as given theories. Then we investigate the automatic synthesis of models for such theories. We use convex polytopic domains as a flexible approach to associate different domains to different sorts. We introduce a framework for the piecewise definition of functions and predicates. We develop its use with linear expressions (in a wide sense, including linear transformations represented as matrices) and inequalities to specify functions and predicates. In this way, algorithms and tools from linear algebra and arithmetic constraint solving (e.g., SMT) can be used as a backend for an efficient implementation.

On the use of two-dimensional Euler parameters for the dynamic simulation of planar rigid multibody systems

This paper introduces a new coordinate formulation for the kinematic and dynamic analysis of planar multibody systems composed of rigid bodies. The methodology presented in this work is called planar reference point coordinate formulation (RPCF) with Euler parameters. In the planar RPCF with Euler parameters, the rotational coordinates used for describing the body orientation are the redundant components of a two-dimensional unit quaternion that identify a planar set of Euler parameters. It is shown in the paper that the planar RPCF with Euler parameters allows for obtaining consistent kinematic and dynamic descriptions of two-dimensional rigid bodies. In the numerical solution of the equations of motion, the well-known generalized coordinate partitioning method can be effectively utilized to stabilize the violation of the algebraic constraints at the position and velocity levels leading to physically correct and numerically stable dynamic simulations. Furthermore, a standard numerical integration procedure can be employed for calculating an approximate solution of the equations of motion resulting from the planar RPCF with Euler parameters. In the paper, the computer implementation of the proposed formulation approach is demonstrated considering four rigid multibody systems which serve as simple benchmark problems.

Spherically symmetric sector of self-dual Ashtekar gravity coupled to matter: Anomaly-free algebra of constraints with holonomy corrections

Using self dual Ashtekar variables, we investigate (at the effective level) the spherically symmetry reduced model of loop quantum gravity, both in vacuum and when coupled to a scalar field. Within the real Ashtekar-Barbero formulation, the system scalar field coupled to spherically symmetric gravity is known to possess a non closed (quantum) algebra of constraints once the holonomy corrections are introduced, which forbids the loop quantization of the model. Moreover, the vacuum case, while not anomalous, introduces modifications which are usually interpreted as an effective signature change of the metric in the deep quantum region. We show in this paper that both those complications disappear when working with self dual Ashtekar variables, both in the vacuum case and in the case of gravity minimally coupled to a scalar field. In this framework, the algebra of the holonomy corrected constraints is anomaly free and reproduces the classical hypersurface deformation algebra without any deformations. A possible path towards quantization of this model is briefly discussed.

H∞ dynamic observer design for linear discrete-time systems

The objective of this paper is to propose a new form of H∞ dynamic observer for linear discrete-time systems. The widely used proportional observer (PO) and proportional integral observer (PIO) can be considered as particular cases of the proposed observer. The observer design is derived from the solution of linear matrix inequalities (LMIs), based on the solutions of algebraic constraints obtained from the unbiasedness conditions of estimation error. A numerical example is provided to show the performance of the proposed observer, compared with PO and PIO.

Nonlinear State Estimation for Trajectory Tracking of a Flexible Parallel Manipulator

Lightweight robots can be advantageous when considering the lower energy consumption, the possibility to use smaller motors and the lower material cost. Nevertheless, in such lightweight structures non-negligible flexibilities are inherent which can lead to significant oscillations making the control more difficult. Within this research end-effector trajectory tracking of a parallel manipulator with a highly flexible link is considered. As centerpiece, a new approach to estimate the state of the system of differential algebraic equations is discussed to obtain the end-effector position and velocity. For the estimator, the kinematic loop is treated by a projection tangential to the constraint manifold which is based on a QR decomposition. Subsequently, an Unscented Kalman Filter is applied and the dependent coordinates are obtained by satisfying the algebraic constraint equations with the Newton-Raphson method. The utilized signals for the estimator are on the one hand position measurements of the actuators which are realized as direct drives and on the other hand measurements of strain gauges attached to the long and highly flexible link. As application, a basic end-effector output controller based on an equivalent rigid model and the transfer function of the dominant first bending eigenmode is utilized to enhance the tracking results. The concepts are applied to an experimental rig for validation purposes and to show first results for end-effector estimation with feedback control.

Robust and reliable estimation via recursive nonlinear dynamic data reconciliation based on cubature Kalman filter

Since measurements of process variables are subject to measurements errors as well as process variability, data reconciliation is the procedure of optimally adjusting measured date so that the adjusted values obey the conservation laws and constraints. Thus, data reconciliation for dynamic systems is fundamental and important for control, fault detection, and system optimization. Attempts to successfully implement estimators are often hindered by serve process nonlinearities, complicated state constraints, and un-measurable perturbations. As a constrained minimization problem, the dynamic data reconciliation is dynamically carried out to product smoothed estimates with variances from the original data. Many algorithms are proposed to solve such state estimation such as the extended Kalman filter (EKF), the unscented Kalman filter, and the cubature Kalman filter (CKF). In this paper, we investigate the use of CKF algorithm in comparative with the EKF to solve the nonlinear dynamic data reconciliation problem. First we give a broad overview of the recursive nonlinear data dynamic reconciliation (RNDDR) scheme, then present an extension to the CKF algorithm, and finally address the issue of how to solve the constraints in the CKF approach. The CCRNDDR method is proposed by applying the RNDDR in the CKF algorithm to handle nonlinearity and algebraic constraints and bounds. As the sampling idea is incorporated into the RNDDR framework, more accurate estimates can obtained via the recursive nature of the estimation procedure. The performance of the CKF approach is compared with EKF and RNDDR on nonlinear process systems with constraints. The conclusion is that with an error optimization solution of the correction step, the reformulated CKF shows high performance on the selection of nonlinear constrained process systems. Simulation results show the CCRNDDR is an efficient, accurate and stable method for real-time state estimation for nonlinear dynamic processes.

Precise algebraic-based swept volumes for arbitrary free-form shaped tools towards multi-axis CNC machining verification

This paper presents an algebraic based approach and a computational framework for the simulation of multi-axis CNC machining of general freeform tools. The boundary of the swept volume of the tool is precisely modeled by a system of algebraic constraints, using B-spline basis functions. Subdivision-based solvers are then employed to solve these equations, resulting in a topologically guaranteed construction of the swept volume. The presented algebraic-based method readily generalizes to accept tools of arbitrary free-form shape as input, and at the same time, delivers high degree of precision.Being a common representation in CNC simulations, the computed swept volume can be reduced to a dexels’ representation. Several multi-axis test cases are exhibited using an implementation of our algorithm, demonstrating the robustness and efficacy of our approach.

Modeling and Stability Analysis of Tethered Kites at High Altitudes

A systematic analysis of the role played by several physical mechanisms in the longitudinal stability of a tethered kite is presented. A simple model, which artificially constrains the pitch motion of the kite and approximates the tether by a massless and rigid bar, is improved progressively to include the kite pitch motion as well as the tether inertia, flexibility, wind load, and elasticity. The models are presented as compact sets of ordinary differential equations without algebraic constraints, which are explicitly eliminated by making an extensive use of Lagrangian mechanics. The contributions of each physical mechanism on kite stability are investigated separately, and a tradeoff between the complexity and computational costs of the models against their accuracy and reliability is carried out. The wind load on the tether is identified as a key effect stabilizing the steady state of the kites. The optimal bridle design and tether length selections to compute the kite ceiling are discussed.

Enforcing an Algebraic Constraint in Extended Kalman Filter Design

A new method is presented for optimally enforcing an algebraic constraint in Kalman filtering. The method is illustrated in the setting of inertial navigation using extended Kalman filtering. It is shown that to first order in the estimation error the constrained optimal update of the quaternion portion of the state vector is a projection of the unconstrained update onto the constraint space, and the update increment for the quaternion is perpendicular to the a priori estimate. The procedure is easy to implement and appears to provide a nearly optimal solution to the problem of quaternion normalization in the setting of inertial navigation. More importantly, it provides a correction for the covariance update that significantly enhances filter stability. Examples involving initial alignment and inertial navigation illustrate the efficacy of the approach, and the method is compared to several other methods in the literature.

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

The tt * equations define a flat connection on the moduli spaces of $${2d, \mathcal{N}=2}$$ quantum field theories. For conformal theories with c = 3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the tt * equations, restricted to the conformal directions, in the cases d = 1, 2, 3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with the generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of tt * and provides these with a finer and algebraic meaning. For sigma models into lattice polarized K3 manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for d = 1 and 3. The generators of the differential rings of special functions are given by quasi-modular forms for d = 1 and their generalizations in d = 2, 3. Some explicit examples are worked out including the case of the mirror of the quartic in $${\mathbbm{P}^3}$$ , where due to further algebraic constraints, the differential ring coincides with quasi modular forms.

Output Feedback Admissible Control for Singular Systems: Delta Operator (Discretised) Approach

AbstractSingular systems simultaneously capture the dynamics and algebraic constraints in many practical applications. Output feedback admissible control for singular systems through a delta operator method is considered in this article. Two novel admissibility conditions, derived for the singular delta operator system (SDOS) from a singular continuous system through sampling, can not only produce unified admissibility for both continuous and discrete singular systems but also practical procedures. To solve the problem of output feedback admissible control for the SDOS, an existence condition and design procedure is given for the determination of a physically realisable observer for the state estimation, and then a suitable state-feedback-like admissible controller design based on the observer is developed. All of the conditions presented are necessary and sufficient, involving strict linear matrix inequalities (LMI) with feasible solutions obtained with low computational costs. Numerical examples illustrate our approach.

Gauging the Relativistic Particle Model on the Noncommutative Plane

We construct a new model for relativistic particle on the noncommutative plane, using the symplectic formalism of constrained systems. We suggest a shortcut approach to construct the gauged Lagrangian, using the Poisson algebra of constraints, without calculating the whole procedure of symplectic formalism. We also propose an approach for the systems, in which the symplectic formalism is not applicable, due to the truncation of production of secondary constraints at the first level. After gauging the model, we obtain the corresponding generators of gauge transformations of the physical system. Finally, by extracting the Poisson structure of all constraints, we show the effect of gauging on the canonical structure of the phase spaces of both primary and gauged models.

On the solution of differential-algebraic equations through gradient flow embedding

In this paper Gradient Flow methods are used to solve systems of differential-algebraic equations via a novel reformulation strategy, focusing on the solution of index-1 differential-algebraic equation systems. A reformulation is first effected on semi-explicit index-1 differential-algebraic equation systems, which casts them as pure ordinary differential equation systems subject to an embedded pointwise least-squares problem. This is then formulated as a gradient flow optimization problem. Rigorous proofs for this novel scheme are provided for asymptotic and epsilon convergence. The computational results validate the predictions of the effectiveness of the proposed approach, with efficient and accurate solutions obtained for the case studies considered. Beyond the theoretical and practical value for the solution of DAE systems as pure ODE ones, the methodology is expected to have an impact in similar cases where an ODE system is subjected to algebraic constraints, such as the Hamiltonian necessary conditions of optimality in optimal control problems.

Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

On Some Properties of Calibrated Trifocal Tensors

In two-view geometry, the essential matrix describes the relative position and orientation of two calibrated images. In three views, a similar role is assigned to the calibrated trifocal tensor. It is a particular case of the (uncalibrated) trifocal tensor and thus it inherits all its properties but, due to the smaller degrees of freedom, satisfies a number of additional algebraic constraints. Some of them are described in this paper. More specifically, we define a new notion --- the trifocal essential matrix. On the one hand, it is a generalization of the ordinary (bifocal) essential matrix, and, on the other hand, it is closely related to the calibrated trifocal tensor. We prove the two necessary and sufficient conditions that characterize the set of trifocal essential matrices. Based on these characterizations, we propose three necessary conditions on a calibrated trifocal tensor. They have a form of 15 quartic and 99 quintic polynomial equations. We show that in the practically significant real case the 15 quartic constraints are also sufficient.

Data reconciliation for chemical reaction systems using vessel extents and shape constraints

Abstract Concentrations measurements are typically corrupted by noise. Data reconciliation techniques improve the accuracy of measurements by using redundancies in the material and energy balances expressed as relationships between measurements. Since in the absence of kinetic models these relationships cannot integrate information regarding past measurements, they are expressed in the form of algebraic constraints. This paper shows that, even in the absence of a kinetic model, one can use shape constraints to relate measurements at different time instants, thereby improving the accuracy of reconciled estimates. The construction of shape constraints depends on the operating mode of the reactor. Moreover, it is shown that the representation of the reaction system in terms of vessel extents helps identify additional shape constraints. A procedure for deriving shape constraints from measurements is also described. Data reconciliation using both numbers of moles and extents is illustrated via a simulated case study.

Deformed symmetries in noncommutative and multifractional spacetimes

We clarify the relation between noncommutative spacetimes and multifractional geometries, two quantum-gravity-related approaches where the fundamental description of spacetime is not given by a classical smooth geometry. Despite their different conceptual premises and mathematical formalisms, both research programs allow for the spacetime dimension to vary with the probed scale. This feature and other similarities led to ask whether there is a duality between these two independent proposals. In the absence of curvature and comparing the symmetries of both position and momentum space, we show that $\kappa$-Minkowski spacetime and the commutative multifractional theory with $q$-derivatives are physically inequivalent but they admit several contact points that allow one to describe certain aspects of $\kappa$-Minkowski noncommutative geometry as a multifractional theory and vice versa. Contrary to previous literature, this result holds without assuming any specific measure for $\kappa$-Minkowski. More generally, no well-defined $\star$-product can be constructed from the $q$-theory, although the latter does admit a natural noncommutative extension with a given deformed Poincar\'e algebra. A similar no-go theorem may be valid for all multiscale theories with factorizable measures. Turning gravity on, we write the algebras of gravitational first-class constraints in the multifractional theories with $q$- and weighted derivatives and discuss their differences with respect to the deformed algebras of $\kappa$-Minkowski spacetime and of loop quantum gravity.