Pavel Bochev, Marta D'Elia, Mauro Perego, Denis Ridzal · “Optimization-based mesh correction...
Transcript of Pavel Bochev, Marta D'Elia, Mauro Perego, Denis Ridzal · “Optimization-based mesh correction...
Sandia National Laboratories is a multimission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DEAC0494AL85000.
Supported by
Pavel Bochev, Marta D'Elia, Mauro Perego, Denis Ridzal
Rome, Italy, April 6, 2017
Optimizationbased, property preserving finite element methods
Center for Computing ResearchSandia National Laboratories
Albuquerque, NM, USA
Optimizationbased modeling (OBM)
Use optimization and control ideas to manage externally those objectives that are difficult (or impractical) to handle directly in the discretization process.
Approach:
➪ Coupling of different physics: reformulation into an equivalent optimization problem, constrained by the single physics
➪ Avoid use of limiters: see talk.
➪ Generality with respect to problem discretization: applicable to FE, FV and FD schemes as well as particle methods, on mixed nD grids
➪ Generality with respect to problem type: elliptic, hyperbolic, …
➪ Enabling of efficient reuse of existing codes: solvers, optimization tools,…
Potential payoffs
Challenges:
● Typically, constraints are not preserved automatically under discretization, even with stabilization/regularization
● Automatic preservation of maximum principle, local and global bounds, is required for robust, predictive simulations
physical model:
physical properties:
match target
define target
enforce bounds
Preservation of properties as an optimization problem
“Optimization based remap”, Bochev, Ridzal, Scovazzi, Shashkov JCP, 2011
“Optimization-based transport”, Parts 1-3, Bochev, Peterson, Ridzal, Young, LNCS 2012
“Optimization-based mesh correction with volume and convexity constraints”, D'Elia, Ridzal, Peterson, Bochev,
Shashkov, JCP, 2016
Find the optimal solution that● [objective] minimizes the mismatch with the highorder target solution● [constraints] satisfies the physical bounds/properties
Optimization based approach has been successfully used in several problems including transport remap and mesh correction:
Here we investigate its use in the context of bound preservation for scalar transport equations and compare it with algebraic flux correction schemes.
Transport equation
Accurate Galerkin finite element solution
Low order, bound preserving solution
Flux formulation*:
Local bounds
By construction, fluxes are antisymmetric, which implies global mass conservation.
Spatial DiscretizationFlux formulation
We would like the solution to satisfy local bounds:
*Kuzmin, “Algebraic flux correction I”, in “Fluxcorrected Transport”, Kuzmin, Löhner, Turek Editors, Springer, 2011
weak LED constr.
OptimizationBased Transport (OBT) Flux Target formulation
The optimization problem is solved resorting to a dual formulation featuring simple bound constraints, see “Optimizationbased transport”, Part 2, Young, Ridzal, Bochev, LNCS 2012.
OptimizationBased Transport (OBT) Flux Target formulation
Linearized Algebraic Fluxcorrected Transport (FCT) formulation*
FCT can be viewed as approximation of OBT obtained by replacing the OBT constraints set by this simpler set of box constraints
The FCT solution satisfies the weak LED constraint
The optimization problem is solved resorting to a dual formulation featuring simple bound constraints, see “Optimizationbased transport”, Part 2, Young, Ridzal, Bochev, LNCS 2012.
*Kuzmin, “Algebraic flux correction I”, in “Fluxcorrected Transport”, Kuzmin, Löhner, Turek Editors, Springer, 2011
weak LED constr.
Comparison between OBT and FCT for the combo* problem
*LeVeque, SINUM 33, 1996
Crank Nicholson scheme is used for the time discretization (throughout the talk)
Comparison between OBT and FCT for the combo* problem
*LeVeque, SINUM 33, 1996
Crank Nicholson scheme is used for the time discretization (throughout the talk)
Comparison between OBT with density target and FCT for a different geometryline plot
Smooth Time Deriv. Sharp Time Deriv.
Smooth Time Deriv. Sharp Time Deriv.
Local bounds coputed at previous time step
OBT, Density Target formulation
We consider three different ways to compute the target:
Galerkin finite element solution
SUPG solution
Fluxbased solution, fluxes computed with sharp time deriv. approx.
Advantages of the density target formulation: ● the optimization problem can be solved very efficiently.● Less intrusive. Does not need fluxes, except for the third case.
Global mass conservation
Fast Optimization Algorithm for OBT with density target
Without the equality constraint the KKT conditions are fully separable and can be solved for any fixed value of .λ
Without the equality constraint the QP is fully separable into N onedimensional QPs with simple bounds
Singly linearly constrained QP with simple bounds:
The Lagrangian:
The KarushKuhnTucker (KKT) conditions:
is continuous, piecewise linear and monotonic increasing
- Can solve to machine precision by a simple secant method - is an excellent initial guess:
- solves the QP without the equality constraint, i.e., “almost” a solution
- barely violates the mass conservation constraint
Fast Optimization Algorithm for OBT with density target
Comparison between OBT with density target and FCT for the combo problem
Vanilla DensityTarget OBT, presents large terracing effect. OBTSUPG is more diffusive than the “sharp” FCT, but less diffusive then the “smooth” FCT. Compared to the FluxTarget OBT, DensityTarget OBT is more diffusive.
Terracing
Comparison between OBT with density target and FCT for the combo problemhigher resolution
At higher resolution, the methods perform more similarly, however “vanilla” OBT presents large terracing effect. OBT with SUPG is more diffusive than the other methods.
Comparison between OBT with density target and FCT for a different geometryhigher resolution
Local undershoot
Comparison between OBT with density target and FCT for a different geometryline plot
SmoothTime Deriv.
SmoothTime Deriv.
SharpTime Deriv.
SharpTime Deriv.
OBTSUPG presents smaller local over/undershoots that FCT.Its accuracy is between the two FCT implementations considered
Comparison between OBT with density target and FCT for staircase problemhigher resolution
Local undershoot
Comparison between OBT with density target and FCT for staircase problemline plot
SmoothTime Deriv.
SmoothTime Deriv.
SharpTime Deriv.
SharpTime Deriv.
OBT with SUPG presents smaller local over/undershoots than the sharp FCT.
Conclusions● FluxTarget OBT proved to be the most accurate method, although the costs
associated with the solution of the optimization are too high for most applications
● DensityTarget OBT is generally more diffusive than the FluxTarget counterpart.In its vanilla version, it presents significant terracing.
● DensityTarget OBT, with SUPG target showed good performances in all the problems tested. It is more dissipative than the sharp version of FCT, but less dissipative than the smooth FCT and it presents small violations of monotonicity compared to FCT.
● DensityTarget OBT is very efficient and less intrusive. According to our matlab simulation is the fastest method. It only requires one matrix solve per time step, in contrast with the sharp FCT that requires the two matrix solves.
● Further developments include improvement of the FluxTarget optimization scheme and further understanding of the properties of OBT.
Acknowledgment: S. Mabuza, D. Kuzmin