8 research outputs found
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
Model Order Reduction by Proper Orthogonal Decomposition
We provide an introduction to POD-MOR with focus on (nonlinear) parametric
PDEs and (nonlinear) time-dependent PDEs, and PDE constrained optimization with
POD surrogate models as application. We cover the relation of POD and SVD, POD
from the infinite-dimensional perspective, reduction of nonlinearities,
certification with a priori and a posteriori error estimates, spatial and
temporal adaptivity, input dependency of the POD surrogate model, POD basis
update strategies in optimal control with surrogate models, and sketch related
algorithmic frameworks. The perspective of the method is demonstrated with
several numerical examples.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0505
Time adaptivity in model predictive control
The core of the Model Predictive Control (MPC) method in every step of the
algorithm consists in solving a time-dependent optimization problem on the
prediction horizon of the MPC algorithm, and then to apply a portion of the
optimal control over the application horizon to obtain the new state. To solve
this problem efficiently, we propose a time-adaptive residual a-posteriori
error control concept based on the optimality system of this optimal control
problem. This approach not only delivers a tailored time discretization of the
the prediction horizon, but also suggests a tailored length of the application
horizon for the current MPC step. We apply this concept for systems governed by
linear parabolic PDEs and present several numerical examples which demonstrate
the performance and the robustness of our adaptive MPC control concept
Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities
We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property
Damage Identification in Fiber Metal Laminates using Bayesian Analysis with Model Order Reduction
Fiber metal laminates (FML) are composite structures consisting of metals and
fiber reinforced plastics (FRP) which have experienced an increasing interest
as the choice of materials in aerospace and automobile industries. Due to a
sophisticated built up of the material, not only the design and production of
such structures is challenging but also its damage detection. This research
work focuses on damage identification in FML with guided ultrasonic waves (GUW)
through an inverse approach based on the Bayesian paradigm. As the Bayesian
inference approach involves multiple queries of the underlying system, a
parameterized reduced-order model (ROM) is used to closely approximate the
solution with considerably less computational cost. The signals measured by the
embedded sensors and the ROM forecasts are employed for the localization and
characterization of damage in FML. In this paper, a Markov Chain Monte-Carlo
(MCMC) based Metropolis-Hastings (MH) algorithm and an Ensemble Kalman
filtering (EnKF) technique are deployed to identify the damage. Numerical tests
illustrate the approaches and the results are compared in regard to accuracy
and efficiency. It is found that both methods are successful in multivariate
characterization of the damage with a high accuracy and were also able to
quantify their associated uncertainties. The EnKF distinguishes itself with the
MCMC-MH algorithm in the matter of computational efficiency. In this
application of identifying the damage, the EnKF is approximately thrice faster
than the MCMC-MH
POD based inexact SQP methods for optimal control problems governed by a semilinear heat equation
This diploma thesis is focused on the application of a POD based inexact SQP method to an optimal control problem governed by a semilinear heat equation. The theoretical foundation for the solution theory of the optimal control problem is laid by discussing the unique solvability of the state equation, investigating the existence of an optimal solution and deriving necessary optimality conditions utilizing the Lagrange technique. Due to the nonlinearity, the discussion of second order sufficient optimality criteria is needed. The numerical solution of the optimal control problem is realized by an inexact SQP method. To illustrate the presented SQP strategy, numerical test examples are carried out and discussed in detail. A POD based model reduction is applied and persues the aim to decrease computational complexity of the high-dimensional FE systems while providing solutions of good accuracy
Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities
We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property