Spatial Discretization
The model equations for elastic bodies developed in Chaps. 3 and 4 are now discretized with respect to the spatial variable. In case of the unconstrained equations of motion, this Galerkin projection is a widespread approach and usually applied with fin
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Spatial Discretization
The model equations for elastic bodies developed in Chaps. 3 and 4 are now discretized with respect to the spatial variable. In case of the unconstrained equations of motion, this Galerkin projection is a widespread approach and usually applied with finite elements or eigenfunctions as spatial approximations. It leads in a natural way to a system of second order ODEs, which is then coupled with other components in the assembly to express the corresponding interactions. As argued in Part I, the most general model equations include the constraints already before discretization, which raises the question how to discretize the transient saddle point models from above. In this chapter, we generalize the Galerkin projection to such models and investigate the properties of the resulting differentialalgebraic system. For this purpose, we employ some of the results that have become standard for the classes of mixed and hybrid finite element methods. Performing first the discretization in space—as opposed to starting with the time discretization—is mainly motivated by the requirements of commercial simulation software where elastic bodies are treated as discrete substructures. Even more, standard interfaces between finite element and multibody codes allow an easy model set-up by the finite element method followed by the export of data such as mass and stiffness matrices to the multibody formalism. This approach, however, requires a careful selection of the ansatz functions for the Galerkin projection, in particular if additional model reduction techniques are applied.
5.1 Finite Element Approximation of Elastic Body This section considers, in analogy to Chap. 3, the equations of motion for a single elastic body under the presence of boundary constraints (3.42a), (3.42b) and introduces the corresponding Galerkin projection. Floating reference frames and additional rigid motion variables will be treated in the subsequent section. To get started, we review briefly the typical spatial discretization for the equations of unconstrained motion (3.31). B. Simeon, Computational Flexible Multibody Dynamics, Differential-Algebraic Equations Forum, DOI 10.1007/978-3-642-35158-7_5, © Springer-Verlag Berlin Heidelberg 2013
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5 Spatial Discretization
5.1.1 Unconstrained Equations The Galerkin projection for the dynamic equations of motion (3.31) stems from a straightforward generalization of the stationary case [Hug87, Chap. 7]. Let uh (x, t) denote the approximation to the displacement field with the subscript h standing for the spatial grid. We decompose the approximation into uh = w h + r,
w h |Γ0 = 0 and r|Γ0 = u0 ,
(5.1)
where the inhomogeneous part r is assumed to be known, cf. Sect. 3.3.1. Next, let V0,h ⊂ V0 be a finite-dimensional subspace of V0 . By restriction onto V0,h , the variational problem (3.31) or (3.55), respectively, is transformed into ¨ h , v + a(wh , v) = ψ, v ρ w
∀v ∈ V0,h .
(5.2)
We are seeking the approximation wh (·, t) ∈ V0,h of the homogeneous solution part for the
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