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CSA with SGR Air Force Office of Scientific Research Dr. Jean-Luc Cambier Computational Mathematics Program Officer Continuum Shape Sensitivity Analysis with Spatial Gradient Reconstruction for Fluid-Structure Interaction Bob Canfield Computational Math Program Review Arlington, Virginia 9 Aug 2016 Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 1 Continuum Shape Sensitivity for Nonlinear Aeroelastic Gust Response Boeing • Grant Awarded Dec 2015 Joined-Wing – Motivated by HALE SensorCraft Aeroelastic Gust Response – Supports David Sandler • Related AFRL/RQ MSTC Grant – Mandar Kulkarni – David Cross • AFOSR FY09–FY14 – Shaobin Liu • AFIT PhD FY08–09 – Maj Doug Wickert Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 2 Research Motivation Novel aircraft configurations that exhibit geometric nonlinearity during an aeroelastic gust response require a high fidelity fluid- structure interaction (FSI) modeling capability. “The board determined that the mishap resulted from the inability to predict, using available analysis methods, the aircraft's increased sensitivity to atmospheric disturbances such as turbulence, following vehicle configuration changes required for the long- NASA’s Helios (NASA, 2004) duration flight demonstration.” Boeing Sensorcraft (Johnson, 2001) [NASA 2004] Gradient-based design optimization requires FSI sensitivity analysis Fast evaluation of nearby-flows, e.g., aircraft stability/control derivatives DARPA Tactical Technology Office Vulture program concept http://www.darpa.mil/Our_Work/TTO/Programs/Vulture.aspx More accurate, efficient, and robust SA method Boundary Velocity that is amenable to black box analysis tools. CSA with SGR Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 3 Outline • Motivation for Continuum Sensitivity Analysis (CSA) • FY11–14 Accomplishments Built-Up Structures (Local CSA) Higher-order p-elements vs SGR Equivalence of Domain Velocity (Total) CSE to Discrete Transient Aeroelastic Gust Sensitivity • CSA with Spatial Gradient Reconstruction FY16–18 Arbitrary Lagrangian-Eulerian Hybrid Adjoint Nonintrusive “Black Box Algorithm” (using 3D SGR) Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 4 Continuum Sensitivity Analysis (CSA) Literature “The domain and the boundary ‘75 ‘85 ‘95 methods‘05 are analytically ‘15 Structures Fluids Fluid-Structure Interaction equivalent… However, when FEM is used for analysis of built-up • Haug & Aurora (1978): Introduced CSA for structural applicationsstructures, (intrusive) accuracy of numerical • Choi & Kim (2005): CSA text book for structural applications (intrusive) results… on interface boundaries • Liu, Cross & Canfield (2014, 2016): CSA for linear/nonlinear, 1D/ 2D applications (nonintrusive) may not be satisfactory. [Babuska, o (MISSING): Nonintrusive CSA for many design variables & 3D applications Aziz 1972]” -- Choi, Kim (2005) • Jameson (1988) Fluid Flow Continuous Adjoint Resolved: Liu, Canfield JFS (2013), • Borggaard, Burns, Stanley, Stewart (1994, 1997, 2002): CSA for Euler/ N-CrosS, FE/s, FD Canfield based CFD SMO (intrusive) (2014) • Godfrey, Cliff (2001): CSA with FV for structured meshes (FD spatial gradients and possible nointrusive) • Duvigneau & Pelletier (2006): CSA with l-patch (~SGR) for Euler/ N-S, FE based CFD (intrusive) • Gobal & Grandhi (2015): CSA with immersed bndy.,FV (FD spatial gradients and possible nointrusive) o (MISSING) Nonintrusive CSA for CFD with FV discretization and unstructured meshes • Etienne, Hay, Garon, Pelletier (2005): CSA for coupled incompressible flow and elastic solid • Wickert, Liu, Cross, Canfield (2008-2015): CSA for aeroelastic applications (typical section/pot. flow) o (MISSING) Nonintrusive CSA for aeroelastic applications with Euler solution aerodynamics Continuum Sensitivity Analysis Mandar Kulkarni, Virginia Tech 5 (Shape) Sensitivity Analysis Methods * Liu, Canfield, SMO 2013 Differentiate, then Discretize (No mesh sensitivity!) Discretize, then Structures Differentiate (Lagrangian) Fluids (Eulerian) Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 6 Local (Partial) Derivative Shape Sensitivity of Flow • Local or Total Sensitivity may be desired Point of Interest Local Derivative 휕푢 휕푏 • Boundary Velocity Formulation yields local (partial) derivative Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 7 Material (Total) Derivative Shape Sensitivity of Structural Response • Total Sensitivity typically desired Material Derivative Point of Interest Material Derivative Local Derivative Du(x,t;b) ¶u ¶u dx = + Db ¶b ¶x db FSI Solution Geometric d x 푢 = 푢′ + 푢 Ѵ V ,푥 Spatial Sensitivity, db Gradient (Design Velocity) Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 8 CSA—Two Formulations 퐷푢 휕푢 휕푢 푑푥 = + = 푢′ + 훻 푢 ∙ 푉 where b is the design parameter 퐷푏 휕푏 휕푥 푑푏 푥 Du u u = is total sensitivity variable, u' is local sensitivity variable Db b 퐷¶ 푅 푢 = 퐴푢 − 푓 = 0 Subject to B.C. 퐵푢 = 푔 퐷푏¶b Domain Velocity (Total) Form Boundary Velocity (Local) Form Take Material Derivative Take Partial Derivative Not 0 0 (often) 퐴 Total 푢 = 푓Total 푢, 푥, 푡 on Ω 퐴 Local 푢′ = 푓′ 푥, 푡 − 퐴′ 푢 on Ω 퐵 Total 푢 = 푔Total 푢, 푥, 푡 on Γ 퐵Local 푢′ = 푔 푥, 푡 − 훻푥퐵 푢 ∙ 푽 on Γ 0 (often) Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 9 Boundary Velocity CSA Local and total differentiation of domain and boundary equations, respectively: 0 푨푏풖′ = 풇′ 풙, 푡; 풃 − 푨푏′풖 푩푏풖 = 품 풙, 푡; 풃 − 푩 푏풖 Express Material derivative of BC .i.t.o. local derivative 0 ′ ′ 푩푏 풖 + 훻풙풖 ∙ Ѵ = 품 풙, 푡; 풃 − 푩푏 + 훻풙푩푏 ∙ Ѵ 풖 Convective Terms Requiring SGR: Essential BC: 푩푏 푢′; 푏 = 푢′ Γ푒 = 푔 푥; 푏 − 훻풙푢 ∙ Ѵ Γ푒 Γ푒 Natural BC: 푩푏 푢′; 푏 = 푄′ Γ푛 = 푔 푥; 푏 − 훻풙푄 ∙ Ѵ Γ푛 Γ푛 Why use it? Expand, recover high-order Cheaper (no mesh sensitivity) primary-variable derivatives More accurate w/ SGR from FEM shape functions Nonintrusive (element agnostic) or recover forces from SGR Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 10 CSA for ALE ALE Reference Frame Eqs Added to FSI Eqs. ALE Reference Frame Velocity Coupling occurs in Boundary Conditions Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 11 Szabo and Babuska Axial Bar AFOSR Program Review 2016 휋푥 푥, 푢(푥) 푇(푥) = sin 퐸퐴 = 1 퐿 푃 = 1/휋 푘 = 10 퐿 = 1 푢,푥푥 + sin 휋푥 = 0 푢,푥 0 = 1/휋 and 푢,푥 1 = −10 푢(1) 1 1 푢 푥 = sin 휋푥 + 휋2 10휋 Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 12 Boundary Vel. CSA for Axial Bar AFOSR Program Review 2014 휵풙 푩풃풖 0 1 Advantages: Partial differentiation commutes Linear sensitivity equations Disadvantages: High-order derivatives Discontinuities at structural interfaces Can be difficult to implement with general purpose codes Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 13 3D cantilever beam • Cantilever beam modeled with 8 noded brick finite elements (NASTRAN) • Parabolic shear applied on tip face • Displacement prescribed at the root • Obtain derivative of transverse displacement with respect to L • Des. Var. 1: Points move to the right • Des. Var. 2: Points move to the left • Grid convergence done using 4 meshes (#elem=80, 640, 5120, 40960) Solid element cantilever beam model (e=40960) Colors indicate shear stress in the beam Design variable 1: L Design variable 2: L Parameterization: points move to the right Parameterization: points move to the left Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 14 14 3D Cantilever beam Finite difference results (for Des. Var. L) Step size study: 50 step sizes Total derivatives of transverse 51 NASTRAN runs for forward finite difference 100 NASTRAN runs for central finite difference displacement w.r.t. L Continuum Sensitivity Analysis Mandar Kulkarni, Virginia Tech 15 3D Cantilever beam CSA results (for Length Des. Var.) Performance measures 흍ퟏ, 흍ퟐ, 흍ퟑ, 흍ퟒ, 흍ퟓ = Displacements at five locations (x = 0.2, 1.2, 2.2, 3.2, 4.2) Local Total derivatives derivatives 휓 휓′3,2 3,2 휓 3,1 휓′3,1 • Material derivatives obtained from hybrid CSA match FD solution • Direct CSA required 2 linear solutions (for 2 design variables, and so on) • Hybrid Adjoint CSA required only 1 linear solution (for any # of design variables) Continuum Sensitivity Analysis Mandar Kulkarni, Virginia Tech 16 Continuum-Discrete Hybrid Adjoint AFOSR Program Review 2014 Performance Metric of Discrete System Response 휓 = {푧}푇{푢} SGR Material Design Derivative D휓 ′ 푇 푇 = 휓 = 휓 + {푧} [훻푥푢 {풱} D푏푖 Add adjoint-weighted Boundary Velocity CSE ′ 휕휓 푇 ′ 푇 푇 휓 = = 푧 푢 + 휆 퐹퐿표푐푎푙 − 퐊 푢′ = 휆 퐹퐿표푐푎푙 휕푏푖 Adjoint equation (same as Discrete Adjoint) continuous adjoint {z}, {F} may differ 휆 = 퐾 −푇 푧 Mesh Contrast to Discrete Derivative with Adjoint Sensitivity D휓 풅[퐊] = 휓 = 휆 푇 퐅 − {푢} Semi-analytic D푏푖 풅푏 causes notorious inaccuracies Other shape design variables Des. Var. 1 Des. Var. 3 Des. Var. 5 Const. height (bottom) Linear taper (bottom) Cubic taper (bottom) Des. Var. 2 Des. Var. 4 Des. Var. 6 Const. height (top) Linear taper (top) Cubic taper (top) Nonzero terms in CSA traction loads: Continuum Sensitivity Analysis Mandar Kulkarni, Virginia Tech 18 Derivatives of stresses at the root wrt Height Derivatives of root normal stress Derivatives of root shear stress Super-convergent stress derivatives Continuum Sensitivity Analysis Mandar Kulkarni, Virginia Tech 19 Spatial Gradient Reconstruction AFOSR Program Review 2014 Zienkiewicz, Zhu 1996: Patch recovery of stresses & error estimates Duvigneau, Pelletier 2006: Patch recovery for boundary conditions Cross, Canfield 2014: SGR for essential & natural boundary conditions Spatial derivatives of a response are approximated locally via Taylor series expansion about the node of interest, matching nearby response data in a least-squares sense by fitting the Taylor series derivatives. Taylor series expansion
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