On the Importance of Adaptivity for Higher-order Discretizations in - PowerPoint PPT Presentation

On the Importance of Adaptivity for Higher-order Discretizations in Aerospace Applications Joshua Krakos, Eric Liu, Huafei Sun, Masayuki Yano David Darmofal, Laslo Diosady, Bob Haimes Aerospace Computational Design Laboratory Massachusetts Institute of Technology August 6, 2012

Motivation Mach number distribution CFD solutions for complex problems have been made possible by increases in computational and algorithmic power The Launch Abort Vehicle simulation above took 30 minutes to complete on 16 CPUs (Nemec et al, 2008, CART3D group) AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Mesh “convergence” comparison (Chaffin, 2009, DPW4 Presentation) Same CFD code (NSU3D) run on two “best practice” meshes of about 40 million nodes CESSNA mesh NASA mesh C D =262 counts C D =277 counts Which solution is most realistic? How would this level of uncertainty be detected in practice? AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Output-based adaptation • Problem • Flow solution Calculate Estimate • Output Error met or • Outputs flow and error in ? • Max output error time over • Estimated errors outputs outputs • Max time Adapt grid to control error Objective: Increase reliability of CFD by estimating and autonomously controlling error in outputs (e.g. drag or lift) AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Higher-order Discontinuous Galerkin Finite Element Method (DGFEM) • Approximations are degree p polynomials within elements but discontinuous between elements u h,p ( x, y ) u(x,y) y x T H DGFEM approximation: Find u h,p ∈ V h,p such that R h,p ( u h,p , v h,p ) = 0 , ∀ v h,p ∈ V h,p AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Why higher order for CFD? • Higher-order methods known to be more efficient than lower-order for problems with smooth flows • Aerospace flows typically have limited smoothness • Can higher-order methods be beneficial in aerospace applications? • Adaptation key to realizing benefits of higher- order discretization on practical problems AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Outputs & adjoints • J h,p ( u h,p ) is an output such as lift, drag, ef- Transonic RANS example ficiency, etc. Primal (Mach) • Consider a (infinitessimal) perturbation to the residual such that, R h,p ( u h,p + δ u h,p , v h,p ) + ( δ r, v h,p ) = 0 • The adjoint ψ h,p ∈ V h,p is the sensitivity of the output to a residual perturbation, Drag adjoint (mass) δ J h,p ≡ ( δ r, ψ h,p ) • Interpretation: adjoint is transfer function between δ r and J h,p . • In the infinite-dimensional case, the adjoint satisfies a linear PDE. AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Outputs & adjoints • J h,p ( u h,p ) is an output such as lift, drag, ef- Transonic RANS example ficiency, etc. Primal (Mach) Drag error indicator, η κ • Consider a (infinitessimal) perturbation to the residual such that, R h,p ( u h,p + δ u h,p , v h,p ) + ( δ r, v h,p ) = 0 • The adjoint ψ h,p ∈ V h,p is the sensitivity of the output to a residual perturbation, Drag adjoint (mass) δ J h,p ≡ ( δ r, ψ h,p ) Transonic RANS example • Interpretation: adjoint is transfer function between δ r and J h,p . • In the infinite-dimensional case, the adjoint satisfies a linear PDE. AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Continuous Optimization: Mesh-metric Duality Metric field, M ( x ) Mesh, T h 0.1 0.1 Mesh 0.09 0.09 generation 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 Implied 0.02 0.02 metric 0.01 0.01 0 0 • (Intractable) discrete optimization problem T ∗ h = arg inf E ( T h ) s.t. C ( T h ) = Cost T h • Continuous relaxation (Loiselle, 2009) M ∗ = arg inf E ( M ) s.t. C ( M ) = Cost M AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Local sampling         • For each configuration, solve local problems keeping states outside of κ 0 fixed • Determine error estimate η κ i = R h,p ( u κ i h,p , ψ h,p +1 | κ 0 ) • Produces a set of pairs, {M κ i , η κ i } AEROSPACE COMPUTATIONAL DESIGN LABORATORY Yano & Darmofal, 2012

MOESS Algorithm ( M esh O ptimization via E rror S ampling & S ynthesis) • Solve flow and adjoint on current grid • Determine error-metric gradients via local sampling • Utilize steepest descents algorithm to improve metric • Remesh using improved metric AEROSPACE COMPUTATIONAL DESIGN LABORATORY Yano & Darmofal, 2012

Impact of Adaptation on Higher-order Efficiency Subsonic Euler NACA 0012, M = 0 . 5, α = 2 � − 3 10 • Adaptive refinement is per- formed at 2 , 500 and 5 , 000 − 4 10 DOFs generating “optimal” meshes c d error estimate − 5 10 − 1.44 • Uniform refinement (each el- ement subdivided into four) − 0.69 − 6 10 is performed • Uniform refinement compared − 7 p=1 (uniform) 10 to adaptive refinement at 10 , 000 p=3 (uniform) − 3.28 p=1 (adapt) and 20 , 000 DOFs p=3 (adapt) − 8 10 2.5k 5k 10k 20k degrees of freedom When singularities are present, adaptive refinement critical to realize benefits of higher order AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Impact of Adaptation on Higher-order Efficiency Subsonic Euler NACA 0012, M = 0 . 5, α = 2 � p = 3: Distribution of (log) elemental error (log 10 η κ ) 20K DOF mesh Optimized 20K DOF mesh uniformally refined from 5K DOF mesh Adaptive mesh resolves trailing edge singularity more effectively AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Adaptation, higher-order, and RANS Subsonic RANS RAE2822, RANS, M = 0 . 3, Re = 6 . 5 × 10 6 , α = 2 . 31 � Error indicator for160K DOF, p=3 1 10 c d error estimate (counts) 0 10 − 0.98 − 1 Uniform refinement of 40K DOF to 160K DOF 10 − 1.71 − 2 10 p=1 (uniform) − 3.28 p=3 (uniform) p=1 (adapt) − 3 10 p=3 (adapt) 20k 40k 80k 160k degrees of freedom Adaptive 160K DOF To see full benefit of higher-order approximations, solution irregularities must be controlled: adaptation is critical AEROSPACE COMPUTATIONAL DESIGN LABORATORY Yano, Modisette, Darmofal 2011

MDA-3 RANS M = 0 . 2, Re = 9 × 10 6 , Lift adaptation, 11 α values from 0 to 24 . 5 � boundary conforming 5 p = 2 results 4.5 4 α =0.0 c l 3.5 3 2.5 α =8.1 2 0 5 10 15 20 25 angle of attack 5 4.5 α =16.21 4 3.5 c l 3 α =23.28 2.5 2 AEROSPACE COMPUTATIONAL DESIGN LABORATORY 0.02 0.04 0.06 0.08 0.1 0.12 c d

MDA-3 RANS M = 0 . 2, Re = 9 × 10 6 , Lift adaptation, 11 α values from 0 to 24 . 5 � boundary conforming • An appropriate mesh is critical (and subtle) • Consider α sweep using α = 8 . 1 � optimized grid 2 5 10 fixed mesh fixed mesh adaptive adaptive 4.5 1 10 4 c l error estimate 0 10 3.5 c l − 1 10 3 − 2 10 2.5 − 3 2 10 0 5 10 15 20 25 0 5 10 15 20 25 angle of attack angle of attack c l error estimate c l AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Multi-element Airfoil RANS Case: Higher-order Workshop (2012) c d convergence with WU MIT MIT UMich UMich Bergano Bergano Wyoming Wyoming p = 1 p = 2 0 0 10 10 • UMich and Bergamo results use uniform mesh refinement − 1 − 1 10 10 • Wyoming is an hp adaptive result |drag coefficient error| |drag coefficient error| − 2 − 2 10 10 ~10% error • MIT (h-adaptive) results 10-100 x − 3 − 3 more efficient 10 ~1% error 10 • MIT work includes adaptive time − 4 − 4 10 10 1000 work units from a coarse initial mesh − 5 − 5 10 10 • With adaptivity, p=2 error drops rapidly with small additional work − 6 − 6 10 10 2 3 4 5 2 3 4 5 10 10 10 10 10 10 10 10 Work units Work units AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Cut-cell vs. boundary conforming Transonic RANS RAE2822, RANS, M = 0 . 729, Re = 6 . 5 × 10 6 , α = 2 . 31 � Initial grid Final grid, p=3: 40K DOF − 2 10 − 3 10 c d error estimate − 4 10 − 1.17 − 5 10 − 2.08 p=1 (cutcell) p=2 (cutcell) − 6 p=3 (cutcell) 10 p=1 (conforming) Mach − 2.53 p=2 (conforming) p=3 (conforming) − 7 10 20k 40k 80k 160k AEROSPACE COMPUTATIONAL DESIGN LABORATORY degrees of freedom Yano, Modisette, Darmofal 2011

Laminar delta wing example Higher Order Workshop Test Case 2012: M = 0 . 3, Re = 4000, α = 12 . 5 � − 2 10 1% error − 3 10 ref | |C D − C D 0.1% error − 4 10 HOW mesh (p=1) HOW mesh (p=2) L&H (p=1, hexa) MOESS (p=1) − 5 10 MOESS (p=2) 4 5 6 10 10 10 dof Adaptation critical for achieving higher-order convergence (apriori meshes are uniform refinements) AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Key take-away Adaptation is critical to realize the performance benefits of higher-order discretizations on aerospace applications AEROSPACE COMPUTATIONAL DESIGN LABORATORY

Adaptive higher-order methods: Current Status & Challenges • Higher-order 2D & 3D RANS (including shocks) demonstrated on a priori meshes (Bassi; Darmofal; Fidkowski; Hartmann; Mavriplis; Peraire, etc.) • Robust anisotropic adaptation demonstrated for 2D steady RANS (Darmofal; Fidkowski; Hartmann) • Proof of concept demonstrations for adaptive 3D RANS (Darmofal; Fidkowski; Hartmann) • Challenges: higher-order adaptive meshing; robustness for under-resolved RANS; unsteadiness AEROSPACE COMPUTATIONAL DESIGN LABORATORY