765 research outputs found
Space-time adaptive finite elements for nonlocal parabolic variational inequalities
This article considers the error analysis of finite element discretizations
and adaptive mesh refinement procedures for nonlocal dynamic contact and
friction, both in the domain and on the boundary. For a large class of
parabolic variational inequalities associated to the fractional Laplacian we
obtain a priori and a posteriori error estimates and study the resulting
space-time adaptive mesh-refinement procedures. Particular emphasis is placed
on mixed formulations, which include the contact forces as a Lagrange
multiplier. Corresponding results are presented for elliptic problems. Our
numerical experiments for -dimensional model problems confirm the
theoretical results: They indicate the efficiency of the a posteriori error
estimates and illustrate the convergence properties of space-time adaptive, as
well as uniform and graded discretizations.Comment: 47 pages, 20 figure
Computational methods and software systems for dynamics and control of large space structures
Two key areas of crucial importance to the computer-based simulation of large space structures are discussed. The first area involves multibody dynamics (MBD) of flexible space structures, with applications directed to deployment, construction, and maneuvering. The second area deals with advanced software systems, with emphasis on parallel processing. The latest research thrust in the second area involves massively parallel computers
Inverse problems for linear hyperbolic equations using mixed formulations
We introduce in this document a direct method allowing to solve numerically
inverse type problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave equation posed in - a bounded subset of - from a partial
distributed observation. We employ a least-squares technique and minimize the
-norm of the distance from the observation to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the optimality
conditions are reduced to a mixed formulation involving both the state to
reconstruct and a Lagrange multiplier. Under usual geometric optic conditions,
we show the well-posedness of this mixed formulation (in particular the inf-sup
condition) and then introduce a numerical approximation based on space-time
finite elements discretization. We prove the strong convergence of the
approximation and then discussed several examples for and . The
problem of the reconstruction of both the state and the source term is also
addressed
Variational data assimilation for two interface problems
“Variational data assimilation (VDA) is a process that uses optimization techniques to determine an initial condition of a dynamical system such that its evolution best fits the observed data. In this dissertation, we develop and analyze the variational data assimilation method with finite element discretization for two interface problems, including the Parabolic Interface equation and the Stokes-Darcy equation with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the VDA into an optimization problem, we establish the existence, uniqueness and stability of the optimal solution for each concerned case. Based on weak formulations of the Parabolic Interface equation and Stokes-Darcy equation, the dual method and Lagrange multiplier rule are utilized to derive the first order optimality system (OptS) for both the continuous and discrete VDA problems, where the discrete data assimilations are built on certain finite element discretization in space and the backward Euler scheme in time. By introducing auxiliary equations, rescaling the optimality system, and employing other subtle analysis skills, we present the finite element convergence estimation for each case with special attention paid to recovering the properties missed in between the continuous and discrete OptS. Moreover, to efficiently solve the OptS, we present two classical gradient methods, the steepest descent method and the conjugate gradient method, to reduce the computational cost for well-stabilized and ill-stabilized VDA problems, respectively. Furthermore, we propose the time parallel algorithm and proper orthogonal decomposition method to further optimize the computing efficiency. Finally, numerical results are provided to validate the proposed methods”--Abstract, page iii
Asymptotic Stability of POD based Model Predictive Control for a semilinear parabolic PDE
In this article a stabilizing feedback control is computed for a semilinear
parabolic partial differential equation utilizing a nonlinear model predictive
(NMPC) method. In each level of the NMPC algorithm the finite time horizon open
loop problem is solved by a reduced-order strategy based on proper orthogonal
decomposition (POD). A stability analysis is derived for the combined POD-NMPC
algorithm so that the lengths of the finite time horizons are chosen in order
to ensure the asymptotic stability of the computed feedback controls. The
proposed method is successfully tested by numerical examples
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