Implicit state space solution

We derive now an expression for the implicit state space solution of a linear multibody system. To simplify notation, we will restrict ourselves to BDF methods. The results can easily be extended to the general case. 

An immediate consequence is that by (5.3.17) the conditions given in Def. 5.3.1 hold in X. Thus, this iteration gives an implicit state space solution also in the nonlinear case. 

The implicit state space form is motivated by a transformation to state space form, discretization of the state space form equations and back transformation. Integrating constrained mechanical systems by first transforming them to state space form is used frequently, [HY90, Yen93]. This leads to approaches known as coordinate splitting methods. The method we discuss here does not carry out the transformation to state space form explicitly. It embeds the choice of state variables in the iterative solution of the nonlinear system (5.3.2), while the integration method always treats the full set of coordinates. 

We pointed out earlier that for equations of motion of constrained mechanical systems written in index-1 form the position and velocity constraints form integral invariants, see (5.1.16). Thus the coordinate projection and the implicit state space method introduced in the previous section can be viewed as numerical methods for ensuring that the numerical solution satisfies these invariants. 

The terms j/kj contained in the solution for the variables zjj (Equation (B.14)) become indeterminate as j —> 0, and are implicitly determined by the additional constraints obtained in the jth time scale. Using the solution for zm in Equation (B.14), a state-space realization of the DAE system in Equation (B.13) is obtained as... 

Discretization of the state space form by an implicit BDF method. The resulting system is - in contrast to the discretized form of (5.3.7) - a square linear system with a well defined solution. 

This high-level description obviously hides many details. For example, the differential equation may involve either explicitly or implicitly various constraints on the system over time. The state of the system may be discontinuous (usually not in space, but velocity discontinuities are often fundamental to treatments of rigid bodies in contact). If the number of variables describing the system state is high, we may have to deal with very large systems of coupled equations. In some cases even the inherent computational complexity of the simulation itself may be questionable for example, some treatments of rigid-body simulation give rise to individual steps for which the solution is NP-... 

Half-Cells. The concept of a supported electrolyte has proven quite valuable in solution electrochemistry by allowing great theoretical simplification at (usually) only a small cost in accuracy. The several (often implicit) assumptions made in treating the electrolyte in a given cell as supported, however, deserve careful attention as they generally do not apply in the case of solid state electrochemical systems. It should also be noted that it is usually possible in solution electrochemistry to use a large, essentially kineticaUy reversible counterelectrode so that aU but a negligible fraction of the applied potential difference falls across the electrode-electrolyte interface of interest. In its simplest form, the supported approach assumes that all the potential difference in the system falls across the compact double layer—approximately one solvent molecule diameter in thickness— at this electrode, and the approach of the electroactive species to the boundary of the compact layer, the outer Helmholtz plane, occurs purely by diffusion. Corrections for the buildup of space... 

Stirred tanks typically contain one or more impellers mounted on a shaft, and optionally, baffles and other internals. Although it is a straightforward matter to build a 3D mesh to contour to the space between these elements, the mesh must be built so that the solution of the flow field incorporates the motion of the impeller. This can be done in two ways. First, the impeller geometry can be modeled directly, or explicitly, and the grid and solution method chosen so as to incorporate the motion of the impeller using either a steady-state or time-dependent techniqne. This approach is discussed in detail in Section 5-5. Second, the motion of the impeller can be modeled implicitly, using time-averaged experimental velocity data to represent the impeller motion. The second approach is the subject of this section. 

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