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 point
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 5-Layer Patch 20 Nonintrusive CSA with SGR
AFOSR Program Review 2014
CSA with “Black ( ⨯ #DV for FD ) Boxes” like Finite Difference
Highly parallel, independent of # design variables
Compute FD
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 21 SORCER implementation @WPAFB
AFOSR Program Review 2014
Last Summer
This Summer
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 22 SORCER Flowchart
NASA Automated Structural Structural Analysis Optimization (NASTRAN) System (ASTROS)
Service- Oriented Computing EnviRonment (SORCER)
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 23 CSA nonintrusive analysis Flow Jacobian available Flow Jacobian not available CSA (SU2) CSA (Fluent)
Perform primary flow analysis in SU2, Perform primary flow analysis in Fluent, get steady-state flow solution and exact get steady-state flow solution Euler Jacobian (cell centered) (vertex centered) 푢 푢 퐶 푢퐶 Use SGR to get the spatial gradients of 푉 Extrapolate Fluent data at cell centers to velocities on the airfoil boundary grid points (mesh vertices)
훻푥푢 푢퐶푉 푇 Create boundary residual by applying weak transpiration BC Restart SU2 from the Fluent steady-state solution, only for one pseudo-time step, to output the exact Euler Jacobian 푅퐶푆퐸
Solve the linear system 푇퐶푉 ′ 푻 횫풖 = 푹푪푺푬 to get local derivatives 푢′ SU2
Add convective term to get total Fluent derivatives ′ Matlab 풖 = 풖 + 훁퐱퐮.V
Continuum Sensitivity Analysis Mandar Kulkarni, Virginia Tech 24 Flow over NACA 0012 airfoil
Flow Problem: Calculate 휌, u, v, p in domain.
Euler eqns. 2-D: • Steady, Compressible, Inviscid • M=0.5, 훼 = 1.25∘ • FVM & Implicit solver: SU2* Analytical solution NOT available! Sensitivity Problem: DV, b = Magnitude of Hicks-Henne bump at center of top surface Calculate sensitivity of 휌, u, v, p w.r.t. shape parameter b
Change in shape of airfoil with Hicks- *SU2: Stanford University Unstructured flow solver, 2015 AIAAJ Vol. 53. No. 9 Henne bump design variable
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 25 Flow derivative results on the airfoil
• CSA results match well with the FD results • CSA nonintrusive results (with Fluent as primary analysis) for velocity and pressure match well
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 26 Next Two Years
• Scale up • Many Variables (Hybrid Adjoint) • Complexity (SORCER / ASTROS) • Fidelity (SU2 / Fluent for CFD) • ALE • Transient Hybrid Adjoint
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 27 Journal Publications 2015–16
– Liu, Shaobin; and Canfield, Robert A. (2016) “Two Forms of Continuum Shape Sensitivity Method for Fluid-Structure Interaction Problems,” Journal of Fluids and Structures, Vol. 62, pp. 46–64, April 2016 – Cross, David M.; and Canfield, Robert A. (2016) “Convergence Study for the Local Continuum Sensitivity Method Using Spatial Gradient Reconstruction,” AIAA Journal, Vol. 54, No. 3, March 2016, pp. 1050–1063, doi: 10.2514/1.J053800 – Kulkarni, Mandar D.; Cross, David M.; and Canfield, Robert A. (2016) “Discrete Adjoint Formulation for Continuum Sensitivity Analysis,” AIAA Journal, Vol. 54, No. 2, February 2016, doi: 10.2514/6.2015-0138, – Cross, David M.; and Canfield, Robert A. (2015), “Local Continuum Shape Sensitivity with Spatial Gradient Reconstruction for Nonlinear Analysis,” Structural and Multidisciplinary Optimization, April 2015, Volume 51, Issue 4, pp 849-865, doi: 10.1007/s00158-014-1178-8
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 28 Conference Publications 2015–16
– Kulkarni, M. D., Canfield, R. A., and Patil, M. "Continuum Sensitivity Analysis for Aeroelastic Shape Optimization," 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Jan 2016, doi: 10.2514/6.2016- 1177 – Kulkarni, Mandar D.; Canfield, Robert A.; and Patil, Mayuresh, “Nonintrusive Continuum Sensitivity Analysis for Aerodynamic Shape Optimization,” 16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Dallas TX, Jun 22–26, 2015, doi: 10.2514/6.2015-3237 – Kulkarni, Mandar D.; Canfield, Robert A.; and Patil, Mayuresh, “Discrete Adjoint Formulation for Continuum Sensitivity Analysis, AIAA SciTech 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee FL, Jan 5–9, 2015, ” doi:10.2514/6.2015-0138
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 29 Back-up Charts
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 30 Built-up & Graduated Meshes
Design Derivative of Cp Design Derivative of u Design Derivative of v Beam-stiffened rectangular Zoukowsky20 Airfoil 15 15 plate modeled in Nastran NL Analytic Analytic Analytic Analytic10 SolutionLocal CSA Local CSA Local CSA SOL 400 with mixed 10 10
boundary conditions 0
x
x x
5 -10
/D 5
p -20 Dv/D
Du/D 0 DC -30 0 -5 -40
-50 -5 -10 0 0.5 1 0 0.5 1 0 0.5 1 x/c x/c x/c Rectangular Membrane Mesh Refinement Using Unstructured Grids
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 31 Grid and SGR Convergence
AFOSR Program Review 2014 Displacement, u(x) Grid Convergence of u(x) 0.14 -4 u p = 1, slope = -2 exact
u )
0.12 FE -6 Szabo & Babuska 1991
2 || 0.1 -8 FE p =2
0.08 -10 u(x) 0.06 -12
0.04 Log(||u-u -14
0.02 -16 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 x p =1 Log(N) Local Sensitivity, u'(x) Grid Convergence of u'(x) 0.55 u' 0 exact True BC, N = 10 ) -2 N = 10 Shape Function, N = 10 2
0.5 SGR: L=3, O=2, N = 10 || -4 SGR: L = 4, O=3, N = 10 SGR: L = 6, O=5, N = 10 FE -6 0.45
-8 u'(x) True BC, slope = -2 Convergence rates of -10 0.4 Shape Function, slope = -2 SGR: L=3, O=2, slope = -1 displacement derivative w.r.t. bar -12 SGR: L=4, O=3, slope = -2 Log(||u'-u' length increases with p-order & SGR: L=6, O=5, slope = -2 0.35 -14 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 SGR Taylor series order x Log(N)
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 32 Accomplishments
• FY11–13 Accomplishments 1st Boundary Velocity CSA for Built-Up Structure: Liu, Canfield JFS 2013 Means to handle interface discontinuity with existing FEM solvers Proved Equivalence of NL Domain Velocity CSA / Discrete Liu, Canfield SMO 2013 1st Comparison of Boundary & Domain Velocity Formulation Results for Static &Transient Aeroelasticity Liu, Canfield submitted Boundary Velocity CSA for Built-Up Structures in Black Box Solver (NASTRAN) with Gradient Reconstruction: Cross, Canfield SMO 2014 • Avoids Mesh Sensitivity and Discrete Stiffness Sensitivity • Nonintrusive, Element-agnostic Algorithm for Black Box Solvers • FY13–14 Accomplishments (submitted journal articles) Comprehensive accuracy & convergence rate comparisons among methods 1st Comparison of p-order vs SGR for Local Form CSE • SGR gives controllable accuracy to FEA order, w/o shape functions
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 33 Guidelines for Convergence
Recommendations for SGR: Choose Taylor series order based on… Order of spatial derivatives in local CSE BC’s Convergence rate of the analysis Choose the number of patch layers based on… Ratio of the number of data points in the patch to the number of coefficients in the Taylor series expansion Needs to be >1, but need not be >2 Recommendations for Adaptive Meshing: Where to refine the mesh Along design dependent boundaries Where design velocity is high At points where material design derivative is desired
Convective terms (훻풙풖 ∙ Ѵ) and (훻풙푸 ∙ Ѵ) should serve as a metric for mesh refinement.
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 34 Publications & Award 2013–2014
• Journal Articles Cross, David M. and Canfield, Robert A. (2014), “Local continuum shape sensitivity with spatial gradient reconstruction,” Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-014-1092-0, June 2014
Liu, Shaobin; and Canfield, Robert. A. (2013), “Equivalence of Continuum and Discrete Analytic Sensitivity Methods for Nonlinear Differential Equations,” Structural and Multidisciplinary Optimization, doi: 10.1007/s00158-013-0951-4 December 2013, Volume 48, Issue 6, pp. 1173–1188
Liu, Shaobin; Canfield, Robert. A. (2013), “Boundary Velocity Method for Continuum Shape Sensitivity of Nonlinear Fluid-Structure Interaction Problems,” Journal of Fluids and Structures, doi: 10.1016/j.jfluidstructs.2013.05.003, Vol. 40, pp. 284–301
Liu, Shaobin; and Canfield, Robert A., “Two Forms of Continuum Shape Sensitivity Method for Fluid-Structure Interaction Problems,” submitted to AIAA Journal • AIAA Multidisciplinary Design Optimization (MDO) Award for Technical Excellence 2014 • Questions?
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 35
Technical Approach
• Continuum Sensitivity Equation (CSE) Method – Boundary Velocity Formulation – Requires Derivation of Sensitivity BC’s • Advantages – Avoid Mesh Sensitivity & Fictitious Load in Domain – Sensitivity BC’s may be used with Black Box • Verify with Analytic, Complex Step & Finite Difference • Compare with – Domain Velocity Formulation – Discrete Analytic Gradients
Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 36 Zoukowsky Airfoil
2-D Potential Flow Joukowsky Airfoils 0.4 =-0.13, =0.01 Finite Element Analysis x y 0.3 =-0.10, =0.01 x y 0.2 =-0.13, =0.05 Validate with Analytic Solution x y 0.1
0 y/c
-0.1
-0.2
-0.3
0 0.2 0.4 0.6 0.8 1 x/c
Coefficient of Pressure HorizontalDesign Derivative Velocity, ofu Cp VerticalDesign Velocity,Derivative v of u Design Derivative of v 1 2 20 2 15 15 AnalyticAnalytic AnalyticAnalytic Analytic 10 SGRLocal CSA SGRLocal CSA Local CSA 0 1.5 1.5 10 10
0
x
x x
-1 1 1 5
-10
p
/D v
u 5 p
C -20 Dv/D
-2 0.5 0.5Du/D 0 DC -30 0 -3 0 0 -5 Analytic -40 FEA -4 -0.5 -50 -0.5 -5 -10 0 0.5 1 0 0 0.50.5 1 1 0 0 0.50.5 1 1 0 0.5 1 x/c x/cx/c x/cx/c x/c Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 37 Comparison between Total Form CSE and Discrete Analytic method Same shape EquilibriumThe boundary Eqn.: velocity approach provides a simple and convenient way for computing sensitivity information. Itfunctions… also supplies a physical understanding of design sensitivity information. For engineering applications that use the numerical method, however, an accurate evaluation of the function value on the boundary is critical. If a domain Weakmethod, Form: such as theShape finite element function method , is used to solve a variational equation, it is well known [85] thatspatial the results gradients of a finite element analysis may not be satisfactory at the boundaries for a system with a nonsmooth load or with interface problems. The domain velocityNL approach terms fromcan be solution used effectively with a high enough accuracy rate when the finite element method is used. In this text, the boundary velocity approach is relied on to convenientlyTotal form explain CSE: analytical examples, while the domain velocity approach is used to accurately solve numerical examples. Discrete weak– Choi, Form:Design Kim, Structural Velocity Sensitivity / FE Analysis and Optimization (2005) map spatial gradient Liu, Shaobin; and Canfield, Robert. A. (2013), “Equivalence of Continuum and Discrete Analytic Sensitivity Methods for Nonlinear Differential Equations,” Same coeff StructuralSame and MultidisciplinaryFE Jacobian Optimization, matrix Discretedoi : 10.1007/s00158-013-0951-4, analyticDecember method: 2013, Volume 48, Issue 6, pp. 1173–1188 Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 38 SGR vs. Shape Functions Final Defense Presentation
Low order elements Shape functions likely to yield poor approximations of high-order derivatives With sufficient number of elements, SGR can yield accurate derivative approximations What about high-order elements?
SGR method amenable to “black box” codes Requires primary and secondary variable output Dependent on accuracy of secondary variable output
David M. Cross, Virginia Tech Canfield, AFOSR Grant Continuum #FA9550-16 -Shape1-0125 DEF Sensitivity 39 39 CPU time for solving the FSI sensitivity of static nonlinear Joined beam with potential flow FD CS Total Local CPU 316.34 s 319.04 s 58.26 s 1.51 s time
40 Canfield, AFOSR Grant #FA9550-16-1-0125 DEF 40