Title: An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems

Abstract

We report the fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high-order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary Lagrangian–Eulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit Runge–Kutta scheme is used for the temporal discretization. Quantities of interest that take the form of space–time integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. These quantities of interest and their gradients are used in the context of gradient-based PDE-constrained optimization. The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible Navier–Stokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradientsmore » computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration.« less

Authors:

Zahr, M. J.  ^[1];

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• Search DOE PAGES for ORCID "0000-0003-4066-981X"

Persson, P. -O. ^[2]

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1. Stanford University, CA (United States)
2. University of California, Berkeley, CA (United States)

Publication Date:

Mon Sep 12 00:00:00 EDT 2016

Research Org.:

Krell Institute, Ames, IA (United States); University of California, Berkeley, CA (United States)

Sponsoring Org.:

USDOE Office of Science (SC); USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:

1533951

Alternate Identifier(s):

OSTI ID: 1359311

Grant/Contract Number:  

FG02-97ER25308; AC02-05CH11231

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Volume: 326; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; fully discrete adjoint method; discontinuous Galerkin methods; deforming domain conservation law; optimal control; high-order methods; PDE-constrained optimization