Multiscale temporal integration Bottasso, Carlo L.
Computer methods in applied mechanics and engineering,
04/2002, Letnik:
191, Številka:
25
Journal Article
Recenzirano
We study the application of the variational multiscale method to the problem of temporal integration, with the final goal of designing integration schemes that go beyond the classical notion of ...upwinding.
We develop a formulation based on a mixed hybrid finite element method, where the fine scale mode problems automatically decouple at the element level without the need to resort to a localization assumption. We give general orthogonality conditions for the trial and test spaces that allow to construct hierarchical
p methods.
We test some simple ideas for the modeling of the unresolved scales. The resulting algorithms are analyzed using classical analytical measures.
We propose a novel discontinuous mixed finite element formulation for the solution of second-order elliptic problems. Fully discontinuous piecewise polynomial finite element spaces are used for the ...trial and test functions. The discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions, and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous Galerkin methods. Static condensation is performed at the element level, leading to a solution procedure based on the sole interface unknowns.
The resulting family of discontinuous dual–primal mixed (DPM) finite element methods is presented in the one and two-dimensional cases. In the one-dimensional case, we show the equivalence of the method with implicit Runge–Kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis. Numerical experiments in one and two dimensions demonstrate the order accuracy of the new method, confirming the results of the analysis.
We consider the Discontinuous Petrov–Galerkin method for the advection–diffusion model problem, and we investigate the application of the variational multiscale method to this formulation. We show ...the exact modeling of the fine scale modes at the element level for the linear case, and we discuss the approximate modeling both in the linear and in the non-linear cases. Furthermore, we highlight the existing link between this multiscale formulation and the
p-version of the finite element method. Numerical examples illustrate the behavior of the proposed scheme.
This work presents a novel methodology for the dynamic analysis of general non-linear flexible multibody systems. In Part I we develop the 6-D compact representation of motion for those body models ...which motion may be described by a displacement field plus an independent rotation field. This approach explores the fundamental properties of rigid body motion, and in particular the coupled nature of linear and angular quantities in both kinematics and dynamics, inspiring a novel parameterization technique based on the exponential map. Using the proposed approach, we derive the governing equations for the case of multibody systems composed by rigid bodies and geometrically non-linear beams connected by holonomic constraints. These equations provide the starting point for the derivation of a class of numerical algorithms characterized by non-conventional conservation properties. In Part II of this work we develop the algorithms and illustrate their properties with the aid of some numerical applications.
We show that integration schemes derived from some well known finite elements in time formulations are Runge-Kutta methods, and we discuss the implications of this finding.
We develop two unconditionally stable displacement based time stepping schemes for the non-linear dynamic response of beams. The first algorithm guarantees the exact discrete conservation of energy ...and momentum. The second is associated with an energy decay inequality that achieves control of the unresolved frequencies by means of a numerical dissipation mechanism.
Both schemes emanate from a weak form of the equations of dynamic equilibrium referred to a fixed pole. Space and time discretizations are based on the exponential parameterization of motion. This implies that the beam reference line and the trajectories of the beam nodes are helicoids in space. The exponential mapping approach allows a unified treatment of translations and rotations, greatly simplifying the derivation of the algorithms and their analysis.
The capabilities and performance of the new schemes are demonstrated and discussed with the aid of numerical simulations.
A novel integration scheme for nonlinear dynamics of geometrically exact shells is developed based on the inextensible director assumption. The new algorithm is designed so as to imply the strict ...decay of the system total mechanical energy at each time step, and consequently unconditional stability is achieved in the nonlinear regime. Furthermore, the scheme features tunable high frequency numerical damping and it is therefore stiffly accurate. The method is tested for a finite element spatial formulation of shells based on mixed interpolations of strain tensorial components and on a two-parameter representation of director rotations. The robustness of the scheme is illustrated with the help of numerical examples.
This paper considers the problem of computing optimal trajectories for rotorcraft systems. The vehicle is described through a flight mechanics model, and the optimal control problem is solved by ...discretizing the vehicle governing equations using a finite-element method, followed by optimization of the resulting finite-dimensional problem. It is found that the computed control policies exhibit oscillations and very high—and therefore unrealistic—time rates, especially for aggressive or emergency maneuvers. Highly oscillatory controls can affect the vehicle trajectory by, for example, exciting short period type oscillations. We argue that this behavior of the computed controls is due to the lack of modeling detail of the vehicle actuators, implied by the classical treatment of the system controls as algebraic variables. We propose a simple, low-cost solution that is based on the recovery of the control time rates through a Galerkin projection. This approach is motivated by the desire to avoid direct modeling of the actuator dynamics, which typically requires one to resolve fine temporal scales in the solution. The recovered control rates can then be constrained to remain within physically acceptable bounds during the solution and can also be included in the optimization cost functions. Numerical experiments are shown to demonstrate that smoother control time histories and vehicle trajectories are computed through this approach.
. We develop a methodology for introducing regions of high anisotropy in existing isotropic unstructured grids in complex, curved, three-dimensional domains. The new procedures are here applied to ...the capturing of solution features in the proximity of model boundaries (e.g. boundary layers). Suitable voids are created in an existing grid in the regions of localization using a mesh motion algorithm that solves a fictitious elasticity problem. The voids are then filled with stacks of prisms that are subsequently tetrahedronized to yield a simplicial mesh. The mesh motion algorithm allows us to deal in a simple and effective manner with the problem of self-intersection of elements in concave regions of the model boundaries, and in the case of closely spaced model faces, avoiding the need for cross-over checks and complex grid correction procedures. The capabilities and performance of the proposed methodology are illustrated with the help of practical examples. PUBLICATION ABSTRACT
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