Project C7 (finished)
Project C7 – M. Hinze and T. Rung: Optimization of multi-phase flow using model order reduction techniques
Scientific Background and Motivation
The theoretical investigation, the numerical simulation and the optimal control of multi-phase flow is a vivid field of research with a broad spectrum of applications. The mathematical concept is based on the Cahn-Hilliard / Navier-Stokes (CH-NS) system. A possible approach in order to simulate immiscible multi-phase flow is the diffuse interface model, which assumes a smooth phase transition between the different material phases, with a steep gradient. In practice, the numerical realization of the CH-NS system requires adaptive finite element strategies and leads to large-scale systems, long calculation times and high storage demands. In order to tackle the computational troubles, the main focus of this project lies on the application of model order reduction (MOR) utilizing proper orthogonal decomposition (POD) in combination with adaptive finite element meshes.
Aims and Objectives
The key objective of this project is the application of POD-MOR to the simulation of multi-phase flow in combination with adaptive finite element meshes. The inclusion of an adaptivity concept for the spatial discretization means that the mesh resolution may differ at each time level. As a consequence the snapshots can be of different lengths. By means of structured finite elements and a bisection refinement strategy, we aim to produce a finest reference mesh, on which the solution can be evaluated in order to obtain snapshots of the same length at each time level. Since the evaluation of nonlinearities in the context of POD is a costly issue, we intend to investigate possible remedies. As the numerical calculation of singular value decomposition (SVD) of a highly scaled snapshot ensemble can face its limits, we aim to test incremental SVD. Another way out could be an adaptive snapshot selection strategy which determines an optimal time grid for snapshot locations. Finally, we intend to use MOR in multi-level schemes for the optimization of multi-phase flow, where the inclusion of MOR could be performed by computing the optimization steps with a reduced order model and improving the reduced order model if necessary.
PhD student: Carmen Gräßle succesfully defended her PhD thesis on June 25, 2019.
PhD panel: Michael Hinze (advisor), Thomas Rung (co-advisor), Armin Iske (chair)
1) A. Alla, C. Gräßle, M. Hinze: A-posteriori snapshot location for POD in optimal control of linear parabolic equations. accepted for publication in ESAIM:M2AN, 2016.
2) C. Gräßle, M. Gubisch, S. Metzdorf, S. Rogg, S. Volkwein: POD basis updates for nonlinear PDE control. - at-Automatisierungstechnik, Volume 65, Issue 5, 2017, 298-307.
3) A. Alla, C. Gräßle, M. Hinze: A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations. IFAC-PapersOnLine, Volume 49, Issue 8, 2016, Pages 13-18.
4) C. Gräßle, M. Hinze. Combining POD Model Order Reduction with Adaptivity. To appear in ScienceOpenPosters, 2018
5) C. Gräßle, M. Hinze, N. Scharmacher. POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots. Accepted for publication in ENUMATH 2017 proceedings.
6) C. Gräßle, M. Hinze. The combination of POD model reduction with adaptive finite element methods in the context of phase field models. PAMM. Volume 17, Issue 1. 2017, 47-50.
7) C. Gräßle, M. Hinze. POD reduced order modeling for evolution equations utilizing arbitrary finite element discretizations. Submitted to ACOM, 2017.
8) A. Alla, C. Gräßle, M. Hinze. Towards optimal snapshot location for POD model order reduction in optimal control. ScienceOpenPosters, 2016, DOI: 10.14293/P2199-8442.1.SOP-MATH.PDKGEB.v1
9) A. Alla, C. Gräßle, M. Hinze. Snapshot location for POD in control of a linear heat equation. PAMM. Volume 16, Issue 1. 2016, 781-782