SciTech 2015 Student Paper Competition Discrete Adjoint Formulation for Continuum Sensitivity Analysis * Mandar D. Kulkarni, † Robert A. Canfield and ‡ Mayuresh J. Patil Virginia Tech, Blacksburg, VA 24060 Continuum Sensitivity Analysis (CSA) is an approach for calculating analytic derivatives. A direct CSA formulation is advantageous for computing derivatives of many state variables or performance functions. An adjoint formulation of CSA is beneficial for computing derivatives with respect to many design variables, although adjoint CSA boundary conditions are often problematic. For the proposed continuum-discrete hybrid adjoint approach, the adjoint variable is introduced after discretization which simplifies boundary conditions. The sensitivity boundary conditions for the hybrid CSA are posed in terms of the continuum state variables. Thus, the hybrid adjoint formulation of CSA results in design derivatives that are as accurate as those obtained from direct CSA, in addition to making the analysis efficient for the case of large number of design variables. Two test cases, first of an axial bar and second of a cantilever beam modeled with solid elements, illustrate how the hybrid adjoint formulation inherits the benefits of the direct and adjoint CSA formulations. This is also the first application of CSA to obtain design derivatives nonintrusively using three dimensional (3-D) Spatial Gradient Reconstruction (SGR) method for 3-D solid elements. Nomenclature A L Linear differential operator A NL (u) Nonlinear differential operator B Boundary algebraic or differential operator b Shape design variable f External body force applied on the domain g Essential or natural force applied on the boundary t Temporal variable u State variable x Spatial variable V Vector of continuous design velocity components ∇ x Spatial gradient operator {z} Virtual load vector [K] Stiffness matrix N Number of degrees of freedom in the finite element model n ψ Number of performance measures n b Number of design variables ˆ ı Unit vector in the horizontal direction of the Cartesian coordinate system T Mapping function A Cross sectional area of the bar E Elastic modulus of the bar L Length of the bar or length of the cantilever beam * PhD Candidate, Department of Aerospace and Ocean Engineering; Student Member, AIAA; kmandar@vt.edu † Professor and Assistant Department Head for Academic Affairs, Department of Aerospace and Ocean Engineering; Associate Fellow, AIAA; bob.canfield@vt.edu ‡ Associate Professor, Department of Aerospace and Ocean Engineering; Associate Fellow, AIAA; mpatil@vt.edu 1 of 21 American Institute of Aeronautics and Astronautics Downloaded by VIRGINIA TECHNICAL UNIVERSITY on January 22, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0138 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 5-9 January 2015, Kissimmee, Florida AIAA 2015-0138 Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. AIAA SciTech