Differential Algebraic Equation Formulation Research Articles

Efficient modeling and simulation of gas separations applying Maxwell-Stefan approach and Ideal Adsorbed Solution Theory

We present an efficient way to incorporate the Maxwell-Stefan approach (Krishna, 1990) for modeling gas separation processes, i.e., adsorbent particles and membranes. Ideal Adsorbed Solution Theory (Myers and Prausnitz, 1965) is applied for the description of competitive equilibria. The proposed calculation method introduces analytical expressions for the partial derivatives of the loadings w.r.t. fluid-phase concentrations, developed by Rubiera Landa et al. (2013), in the calculation of the thermodynamic correction factors, as required by Maxwell-Stefan flux equations. These expressions are solved simultaneously with transient mass balances that describe the gas separation processes in a differential-algebraic equation formulation. Robustness and advantages of the proposed solution approach are demonstrated with several examples adapted from the literature.

Computationally efficient battery model for microgrid applications using the Chebyshev spectral method

Model-based optimization and control are often employed to regulate batteries; however, such modeling tools are required to predict battery status fast and accurately. In this work, we present two modeling frameworks that help significantly decrease computational time while modeling battery physics accurately. These frameworks employ the Chebyshev spectral method for discretization of the governing equations, while integration of the resulting discretized system is implemented using either a segregated or a differential-algebraic equation formulation. The segregated formulation relies on linearized reaction rate kinetics, which enables easy coupling with the governing equations. Our choice of the Chebyshev spectral method is due to its exponential convergence and use of few discretization nodes, compared to conventional methods, such as finite difference method and finite element method. An average of about 98% reduction in computational time for the differential-algebraic equation framework was gained, while for the segregated algorithm, on average, a 51% reduction in computational time relative to the finite element method reference was realized. For all of these frameworks, the Chebyshev spectral method results were on the average within 0.07% - 0.39% of the high-fidelity finite element method reference.

Aircraft attitude reconstruction via novel quaternion-integration procedure

Unit quaternion representation is widely used in flight simulation to overcome the limitations of the standard numerical ordinary-differential-equations (ODEs) based on three-parameters rotation variables (such as Euler angels), as they may impose kinematic singularities during aircraft's attitude reconstruction. However, these benefits do not come without a price, since the classical way of integrating rotational quaternions includes solving of differential-algebraic equations (DAEs) that requires post-integration numerical stabilization of the additional algebraic constraint enforcing the quaternion unitary norm. This can pose a problem in the case of longer flight simulations since improper numerical treatment of the quaternion-normalization constraint may induce numerical drift into the simulation results. As a remedy, the proposed novel algorithm circumvents DAE problem of quaternion integration by shifting update-integration-process from configuration manifold to the local tangential level of the incremental rotations (reducing thus integration to standard three ODEs problem at tangential Lie algebra level). This can be done due to the isomorphism of the Lie algebras of the rotational SO(3) group and the configuration manifold unit quaternion Sp(1) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem, the proposed algorithm also exhibits numerical advantages as it is discussed in the presented example.

An index 0 Differential-Algebraic equation formulation for multibody dynamics: Holonomic constraints

ABSTRACTThe Lagrange multiplier form of index 3 differential-algebraic equations of motion for holonomically constrained multibody systems is transformed using tangent space generalized coordinates to an index 0 form that is equivalent to an ordinary differential equation. The index 0 formulation includes embedded tolerances that assure satisfaction of position, velocity, and acceleration constraints and is solved using established explicit and implicit numerical integration methods. Numerical experiments with two spatial applications show that the formulation accurately satisfies constraints, preserves invariants due to conservation laws, and behaves as if applied to an ordinary differential equation.

High-order ALE schemes for incompressible capillary flows

The spatial discretization of problems with moving boundaries is considered, incorporating the temporal evolution of not just the mechanical variables, but also of the nodal positions of the moving mesh. The outcome is a system of Differential-Algebraic Equations (DAE) of index 2 or, in the case of inertialess flow, just 1. From the DAE formulation it its possible to define strategies to build schemes of arbitrary accuracy. We introduce here several schemes of order 2 and 3 that avoid the solution of a nonlinear system involving simultaneously the mechanical variables and the geometrical ones. This class of schemes has been the one adopted by the majority of practitioners of Computational Fluid Dynamics up to now. The proposed schemes indeed achieve the design accuracy, and further show stability restrictions that are not significantly more severe than those of popular first order schemes. The numerical experimentation is performed on capillary problems, discretized by both div-stable (P2/P1, P1+/P1) and equal-order (P1/P1, stabilized) finite elements, and incorporating surface tension and triple (contact) line effects.

Contributions to the Method of Coordinate Splitting for Solving Differential Algebraic Equations

The Coordinate Splitting method is a numerical technique which was developed to aid in solving constrained multibody dynamic systems modelled as differential algebraic equations (DAEs). This paper presents a discussion about the implementation of this technique, analyzing what happens to the structure of the DAEs and how the quantity called the annihilation matrix is utilized to compute a solution. Also, an explicit formula is described for computing the Jacobian of the DAE formulation of the Coordinate Splitting method when using QR-factorization to determine the annihilation matrix. Finally, an illustrative example is used to demonstrate that the Coordinate Splitting method can be successfully applied to systems involving a singular inertia matrix.

Rigid body dynamics with a scalable body, quaternions and perfect constraints

In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the 7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton–Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrators.

Stabilized index-1 differential-algebraic formulations for sensitivity analysis of multi-body dynamics

Convergence results are shown for the Hiller-Anantharaman stabilized index-1 differential algebraic equation (DAE) formulation of the DAE of motion of a multi-body system. The formulation is extended to the direct differentiation DAE and the adjoint (A) DAE resulting from the design sensitivity analysis of multi-body dynamics. Convergence of backward differential formula methods applied to the Hiller-Anantharaman formulation is proven not only for the DAE of motion but also for the differentiation DAE and the ADAE. The Hiller-Anantharaman formulation is attractive for multi-body dynamics because of its stabilization properties, resulting from a stabilized index reduction of the original index-3 DAE, and because it yields a DAE that includes the position and velocity constraint equations. In addition, the Hiller-Anantharaman index-1 DAE formulation has the advantage that the currently available index-1 DAE integrators are in a more mature phase than higher-index DAE integrators.

Numerical simulation of aerated powder consolidation

When a fine powder is dumped into a silo, the gas trapped by the particles will slowly escape by diffusing through the material. The corresponding uneven gas pressure distribution creates a body force that is taken into account through Darcy's law. By using spatial averaging, the formulation, even though essentially one-dimensional in space, includes effects due the geometry of the container. An efficient and robust numerical scheme based on a differential algebraic equation formulation is proposed and implemented. Various computational results are presented and discussed to establish the validity of the approach.