Space-Time Isogeometric Analysis for linear and non-linear elastodynamics

A simultaneous space‐time discretization approach to the inverse dynamics of geometrically exact strings

“…22 In contrast to the semi-discretization approach commonly applied in structural dynamics (see, e.g., References 23,24), the simultaneous space-time discretization approach proves to be the natural choice for the solution of the inverse dynamics problem at hand. A readable account of previous developments in the field of space-time finite elements can be found in the recent work, 25 see also We further show that the classical method of characteristics (see, e.g., References 30,31) can also be applied to solve the present space-time BVP. In particular, the method of characteristics provides further insight into the present space-time BVP, explaining why the simultaneous space-time discretization is much better suited to solve the inverse problem than the standard semi-discretization approach.…”

“…Example 3. In the case of the linear-elastic bar (Example 2), the semi-discrete equations of motion (25) in conjunction with the servo-constraints (28) boil down to the DAEs…”

“…However, these DAEs have to be distinguished from the DAEs governing the motion of discrete mechanical systems subject to holonomic constraints. If the constraints were holonomic, the term B T 0 f 0 in (25) would correspond to constraint forces with associated Lagrange multiplier f 0 for the enforcement of holonomic constraints ‡ B 0 q(t) = 𝜸(t). Moreover, in the standard case of holonomic systems the DAEs have (differentiation) index 𝜈 d = 3 and are thus amenable to a direct numerical solution (see, e.g., Reference 37).…”

“…In the inverse dynamics problem stated in Section 2.2, the actuating force f 0 (t) plays the role of an unknown which needs be determined by accounting for the partially specified motion of the string at its right boundary. This can be accomplished by appending servo constraints of the form r p+1 (t) = 𝜸(t) to the semi-discrete equations of motion (25). The servo-constraints can be recast in the form…”

“…In contrast to the semi‐discretization approach commonly applied in structural dynamics (see, e.g., References 23,24), the simultaneous space‐time discretization approach proves to be the natural choice for the solution of the inverse dynamics problem at hand. A readable account of previous developments in the field of space‐time finite elements can be found in the recent work, 25 see also References 26‐29.…”