Target Functional-based Adaptation > Daniel Vollmer > 21.06.2006 Folie 1 Target Functional-based...

Folie 1Target Functional-based Adaptation > Daniel Vollmer > 21.06.2006

Target Functional-based AdaptationDaniel Vollmer

TAU Grid Adaptation Workshop21 / 22 June 2006, DLR Göttingen

Target Functional-based Adaptation > Daniel Vollmer > 21.06.2006

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Outline

The Introduction – or what we are doing now…

Our Aim – or where we would rather be…

A Plan – or how we intend to get there…

Some Examples – or what has already been done…

The Future – or where do we go from here…


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The IntroductionWhat we are doing now

The TAU Adaptation consists of two main parts:

Figuring out where to refine the mesh (using an indicator)

Constructing a new mesh where the regions selected by the indicator are better resolved

The current indicators select regions where the gradient of the flow variables (or values computed from them) is large, e.g.

shocks, discontinuities, …

total pressure loss

Does not necessarily converge to the correct (i.e. globally refined) solution, although it works fine in most non-contrived cases


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Our AimWhere we would rather be

Adapt the mesh until it is “accurate enough”

What is “accurate enough”?

Users are often interested in the aerodynamic coefficients to estimate performance

These coefficients are functionals of the flow solution

They don’t care whether discontinuities are well-resolved, as long as drag / lift / … are accurate

For example, we would like to be able to specify

“adapt until the error in drag is smaller than 0.5 drag counts”

Not a solution for every problem (e.g. BVI), but appropriate for when functionals of the solution are of interest


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A PlanHow we intend to get there

Use the adjoint solution to correlate the local truncation error (due to insufficient mesh resolution) with the error in the target functional

Also known as “Dual-Weighted Residual” approach in the Finite-Element (Rannacher) or Discontinuous-Galerkin (Hartmann, Houston) realm

Has been popularized in the Finite-Volume context by Venditti & Darmofal

Only changes the indicator, the TAU Adaptation constructs the refined meshes exactly as it did before


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Need two additional bits of information

Adjoint solution, computed based on the flow solution and the given target functional

Tells us where the flow solution influences the functional

An estimate of the mesh-induced error (residual)

Tells us where a finer mesh would improve the flow solution

Multiply the two together to get an idea where a finer mesh is needed to improve the accuracy of the target functional

The above is an “intuitive” explanation, but there is real maths behind it


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We can compute the adjoint solution with TAU’s adjoint module

But what about the mesh-induced error?

Difficult problem for Finite Volume methods

FE / DG can simply increase the order of approximation to compute a non-zero residual from a lower order solution

Finite Volume methods can increase the order, too – by halving the mesh spacing h

Globally refine the original mesh

Interpolate the original (converged) flow solution onto it

Evaluate the residual

Interpolate residuals back to the original mesh


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Initial Grid Flow Solution

Adaptation

Adapted Grid


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TAU Adjoint

Adjoint Solution

Globally Refined Grid

TAU Residual

Fine ResidualAdaptation

Residual

AdaptationInitial Grid Flow Solution

Adaptation

Adapted Grid


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Some Examples What has already been done…

Tested a lot of the ideas in the structured FLOWer code

Is less complicated

Adjoint solver was working

Implemented the process in a TAU development version

Relies on available TAU functionality, such as

Continuous Adjoint Solver for Euler problems (M. Widhalm, N. Gauger)

Discrete Adjoint Solver for Navier-Stokes problems (R. Dwight)

(Modified) Adaptation Module for global (de-)refinement

Currently, the indicator is computed with an external Python script that processes TAU restart-files of the adjoint solution and the residual


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X

Z

-0.4 -0.2 0 0.2 0.4

-0.5

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-0.4 -0.2 0 0.2 0.4

-0.5

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0

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Example: NACA0012, Euler, M=1.5, α=1.0°, target functional = pressure at tip


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X

Z

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.8

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NACA0012, M=1.5, =1.0°, Adjoint: Pressure at tip


X

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-0.4 -0.2 0 0.2 0.4

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NACA0012, M=1.5, =1.0°, Adjoint: Pressure at tip

Example: NACA0012, Euler, M=1.5, α=1.0°, target functional = pressure at tip


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Example: NACA0012, Euler, M=0.85, α=2.0°, target functional = lift


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Example: NACA0012, Euler, M=0.85, α=2.0°, target functional = lift

Adapted 4 times


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Example: RAE2822, NS/SAE, M=0.73, α=2.76°, target functional = drag


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The FutureWhere do we go from here…

Currently, a given percentage of elements is added

Want to refine all edges exceeding a certain threshold

Some code for this is already there

Determine a good threshold to use for the edges, so that the global error in the target functional is smaller than some prescribed maximum

More testing on selected test-cases

Convergence to the correct (globally refined) solution?

Comparison to normal adaptation

Accuracy in the target functional


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Seek an alternative to the residual computation on the fine mesh

Global (de-)refinement is expensive, especially on large meshes

Makes the process complicated and cumbersome


TAU Residual


Residual

AdaptationFlow Solution


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Seek an alternative to the residual computation on the fine mesh

Global (de-)refinement is expensive, especially on large meshes

Makes the process complicated and cumbersome


TAU Residual


Residual

AdaptationFlow Solution

???????

Residual

Flow Solution


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Try to streamline / simplify the process

Integrate the changes back into Mainline-TAU

...

Give it to users

Questions?

Target Functional-based Adaptation > Daniel Vollmer > 21.06.2006 Folie 1 Target Functional-based...

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