42 research outputs found
Progressive construction of a parametric reduced-order model for PDE-constrained optimization
An adaptive approach to using reduced-order models as surrogates in
PDE-constrained optimization is introduced that breaks the traditional
offline-online framework of model order reduction. A sequence of optimization
problems constrained by a given Reduced-Order Model (ROM) is defined with the
goal of converging to the solution of a given PDE-constrained optimization
problem. For each reduced optimization problem, the constraining ROM is trained
from sampling the High-Dimensional Model (HDM) at the solution of some of the
previous problems in the sequence. The reduced optimization problems are
equipped with a nonlinear trust-region based on a residual error indicator to
keep the optimization trajectory in a region of the parameter space where the
ROM is accurate. A technique for incorporating sensitivities into a
Reduced-Order Basis (ROB) is also presented, along with a methodology for
computing sensitivities of the reduced-order model that minimizes the distance
to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced
optimization framework is applied to subsonic aerodynamic shape optimization
and shown to reduce the number of queries to the HDM by a factor of 4-5,
compared to the optimization problem solved using only the HDM, with errors in
the optimal solution far less than 0.1%
Recent advances in parametric nonlinear model order reduction: treatment of shocks, contact and interfaces, structure-preserving hyper reduction, acceleration of multiscale formulations, and application to design optimization
International audienceParametric, projection-based, Model Order Reduction (MOR) is a mathematical tool for constructing a parametric Reduced-Order Model (ROM) by projecting a given parametric High Dimensional Model (HDM) onto a Reduced-Order Basis (ROB). It is rapidly becoming indispensable for a large number of applications including, among others, computational-based design and optimization, multiscale analysis, statistical analysis, uncertainty quantification, and model predictive control. It is also essential for scenarios where real-time simulation responses are desired. During the last two decades, linear, projection-based, parametric MOR has matured and made a major impact in many fields of engineering including electrical engineering, acoustics, and structural acoustics, to name only a few. By comparison, nonlinear, projection-based, parametric MOR remains somehow in its infancy. Nevertheless, giant strides have been recently achieved in many of its theoretical, algorithmic, and offline/online organizational aspects. The main purpose of this lecture is to highlight some of these advances, discuss their mathematical and computer science underpinnings, and report on their impact for an important class of problems in aerodynamics, fluid mechanics, nonlinear solid mechanics and structural dynamics, failure analysis, multiscale analysis, uncertainty quantification, and design optimization. To this effect, nonlinear, projection-based, parametric MOR will be first interpreted as a constrained semidiscretization on a subset of a compact Stiefel manifold, using a low-dimensional basis of global shape functions constructed a posteriori — that is, after some knowledge about the response of the system of interest has been developed. Usually, such a knowledge is gathered using the given parametric HDM and an offline training procedure where the model parameters are sampled with a greedy strategy based on a cost-effective ROM error indicator. Specifically, a set of problems related to the parametric problem of interest are solved at the sampled parameter points using the given HDM, and the computed solution snapshots are compressed to obtain the desired global ROB. Depending on the mathematical type of the governing equations underlying the given HDM, a dual ROB is also constructed and the sought-after nonlinear parametric ROM is constructed by Galerkin (or Petrov-Galerkin) projection of the HDM onto the global ROB (and its dual counterpart)
Mechanical MNIST: A benchmark dataset for mechanical metamodels
Metamodels, or models of models, map defined model inputs to defined model outputs. Typically, metamodels are constructed by generating a dataset through sampling a direct model and training a machine learning algorithm to predict a limited number of model outputs from varying model inputs. When metamodels are constructed to be computationally cheap, they are an invaluable tool for applications ranging from topology optimization, to uncertainty quantification, to multi-scale simulation. By nature, a given metamodel will be tailored to a specific dataset. However, the most pragmatic metamodel type and structure will often be general to larger classes of problems. At present, the most pragmatic metamodel selection for dealing with mechanical data has not been thoroughly explored. Drawing inspiration from the benchmark datasets available to the computer vision research community, we introduce a benchmark data set (Mechanical MNIST) for constructing metamodels of heterogeneous material undergoing large deformation. We then show examples of how our benchmark dataset can be used, and establish baseline metamodel performance. Because our dataset is readily available, it will enable the direct quantitative comparison between different metamodeling approaches in a pragmatic manner. We anticipate that it will enable the broader community of researchers to develop improved metamodeling techniques for mechanical data that will surpass the baseline performance that we show here.Accepted manuscrip
Model Order Reduction for Rotating Electrical Machines
The simulation of electric rotating machines is both computationally
expensive and memory intensive. To overcome these costs, model order reduction
techniques can be applied. The focus of this contribution is especially on
machines that contain non-symmetric components. These are usually introduced
during the mass production process and are modeled by small perturbations in
the geometry (e.g., eccentricity) or the material parameters. While model order
reduction for symmetric machines is clear and does not need special treatment,
the non-symmetric setting adds additional challenges. An adaptive strategy
based on proper orthogonal decomposition is developed to overcome these
difficulties. Equipped with an a posteriori error estimator the obtained
solution is certified. Numerical examples are presented to demonstrate the
effectiveness of the proposed method
Incremental proper orthogonal decomposition for PDE simulation data: Algorithms and analysis
We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. We introduce an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD). The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. We also modify the algorithm to initialize and incrementally update both the SVDand an error bound during the time stepping in a PDE solver without storing the simulation data. We show the algorithm produces the exact SVD of an approximate data matrix, and the operator norm error between the approximate and exact data matrices is bounded above by the computed error bound. This error bound also allows us to bound the error in the incrementally computed singular values and singular vectors. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers\u27 equation, a 1D FitzHugh-Nagumo PDE system, and a 2D Navier-Stokes problem --Abstract, page iv
A note on incremental POD algorithms for continuous time data
In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6,
1942-1960], we developed an incremental approach to compute the proper
orthogonal decomposition (POD) of PDE simulation data. Specifically, we
developed an incremental algorithm for the SVD with respect to a weighted inner
product for the discrete time POD computations. For continuous time data, we
used an approximate approach to arrive at a discrete time POD problem and then
applied the incremental SVD algorithm. In this note, we analyze the continuous
time case with simulation data that is piecewise constant in time such that
each data snapshot is expanded in a finite collection of basis elements of a
Hilbert space. We first show that the POD is determined by the SVD of two
different data matrices with respect to weighted inner products. Next, we
develop incremental algorithms for approximating the two matrix SVDs with
respect to the different weighted inner products. Finally, we show neither
approximate SVD is more accurate than the other; specifically, we show the
incremental algorithms return equivalent results