Mesh Morphing and the Adjoint Solver in ANSYS R14 · An adjoint solver computes sensitivity data...

© 2011 ANSYS, Inc. September 26, 2011

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Mesh Morphing and the Adjoint Solver in

ANSYS R14.0

Genong Li

Chi-Yang Cheng


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• Fluent Morphing-Optimization Feature

• The Adjoint Solver for Optimzation

• Summary

• A Morpher Demo

Outline


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FLUENT Morpher-Optimization feature


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FLUENT Morpher-Optimization feature

•Allows users to optimize product design based on shape deformation to achieve design objective

•Based on Free-Form Deformation tool coupled with various optimization methods


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Region Defined

Examples

Some Basic examples…

Optimization based morphing…

Baseline Modified


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Process • Morpher coupled with

Optimizer 1. Setup the CFD problem

2. Invoke Mesh Morpher Tool

3. Define Objective Function

4. Define deformation region and assign deformation of control points through “Optimizer”

5. Perform solution to get the optimized design

OR What if? Optimizer

Setup Case

Run

Setup Morph

Evaluate

Choose “best” design

Regions

Parameters

Deformation

Setup Case

Run

Setup Optimizer

Optimize

Optimal Solution

Morph

Optimizer

Auto


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Morpher/Optimizer vs the WB Approach

Similarity:

The design tools have the similar idea: modify the mesh, design a design-of-experiments (DOE) space and conduct multiple CFD runs to obtain the optimal design

Differences:

• FLUENT Morpher/Optimizer does everything within FLUENT solver

• FLUENT morpher: usually handles small deformation

– Mesh count and mesh connectivity do not change, grid points are moved locally according to some smoothing algorithms

– Does not preserve the shape of the original geometry


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Example – Simple Sedan

Sequential Tabs

• Define Control Region(s)


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Deformation Definition

• Define constraint(s) (if any)

• Select control points and prescribe the relative ranges of motion


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Optimizer Algorithms; Compass, Powell, Rosenbrock, Simplex, Torczon

Auto

• Optimize!


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Results

Incompressible turbulent flow

Objective Function; Minimize Drag

Baseline Design Optimized Design

• Questions?

• Please contact ANSYS Tech support for help in applying this technology


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The Adjoint Solver in Fluent


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Gradient-based versus the Adjoint Methods in Design Optimization

• In traditional gradient-based design optimization methods, the cost of finding the sensitivities of the objective function to each design variable is proportional to the number of design variables

• In contrast, the control-theory approach, called the “adjoint method,” computes the gradients (dir. and mag.) directly by solving a set of adjoint equations.* The adjoint method therefore is extremely efficient: its cost is independent of the number of design variables

• The gradients obtained from the adjoint solver are then combined with existing design optimization methods

=================================================

* The adjoint equation is derived by using the method of Lagrange multipliers in calculus in minimizing the objective function, with the CFD governing equations as the constraint equations


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Fundamentals

FLOW SOLVER

Inputs

• Boundary mesh

• Interior mesh

• Material properties

• Boundary condition 1

• Flow angle

• Inlet velocity

• …

• …

Outputs

• Field data

• Contour plots

• Vector plots

• xy-plots

• Scalar values

• Lift

• Drag

• Total pressure drop

High-level “system” view of a conventional flow solver

HOW ARE CHANGES TO KEY OUTPUTS DEPENDENT ON CHANGES TO THE INPUTS?


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Workflow

Workflow

• Solve the flow equations and post-process the results as usual.

• Pick an observation that is of engineering interest.

Lift, drag, total pressure drop?

• Set up and solve the adjoint problem for this observation

Define solution advancement controls

Set convergence criteria

Initialize

Iterate to convergence

• Post-process the adjoint solution to get

Shape sensitivity

Sensitivity to boundary condition settings

Contour & vector plots

• Modify the mesh using a mesh deformation approach

Mesh morphing


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Shape Sensitivity

Shape sensitivity: Sensitivity of the observed value with respect to (boundary) grid node locations

mesh

nnxwDrag .)(

Shape sensitivity coefficients:

Vector field defined on mesh nodes

Node displacement

Visualization of shape sensitivity

• Uses vector field visualization.

• Identifies regions of high and low

sensitivity.

• These are the places where changes

to the shape can have a big impact on

the quantity of interest.

• The guidance is specific to the

quantity of interest, and the current flow

state.

Drag sensitivity for NACA0012


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Full discrete adjoint for shape sensitivity

Frozen turbulence

Reduce total pressure drop, DP, through system

Internal flow – Simple 2D Example

DP = -232.8 Expect change 10.0

Baseline 1

Actual change 9.0 DP = -223.8 Expect change 8.9


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Actual change 3.1 DP = -213.8 Total improvement of 8%

3


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180° Elbow optimization (pressure drop)

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0 10 20 30

Dp

tot [P

a]

Run [-]

Thanks to Hauke Reese ANSYS Germany


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180° Elbow: Optimization Loop

Base configuration End design


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Goal is to reduce the total pressure drop through the system

Set up and solve the adjoint system with a total pressure drop objective function

Total Pressure Drop in a Duct

Flow residuals

Adjoint residuals

Flow

Outflow


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Aggressive adjustment results in a 17% reduction in loss in just one design iteration

Total Pressure Drop in a Duct Total Pressure Drop (Pa)

Geometry Predicted Result

Original --- -22.0

Modified -14.8 -18.3


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Summary

• FLUENT continues to offer enhancements and improvements in efficiency and accuracy in the core solver

• FLUENT offers a number of practical and high-end design/optimization tools to accelerate the use of FLUENT as a design tool


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Demo Problem

Inlet

Exit 1

Exit 2

The goal is to modify the shape of the straight duct section so that flow rates from two exits are the same


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Examples


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Full discrete adjoint for shape sensitivity

Frozen turbulence

Reduce total pressure drop, DP, through system

Total Pressure Drop in a Bend

DP = -232.8 Expect change 10.0

Baseline 1



2

Actual change 3.1 DP = -213.8 Total improvement of 8%

3


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The adjoint solver will be released with R14

An adjoint solver computes sensitivity data that can be used to aid with design decisions in 4 main ways:

1. Qualitative identification of critical parts of the system of interest.

2. Quantitative predictions of the optimal choice for a design change and a prediction of the effect of that change.

3. Aiding in the numerical analysis of the flow solution to improve solution quality.

4. Gradient-based optimization.

Supporting technologies such as mesh morphing, and the application of design constraints, are seen as important.

The adjoint solver for the present release is limited to steady incompressible flows, with other restrictions on models.

Summary


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Objective Function

Baseline Design Optimized Design

• Objective Function: Equal flow rate


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Mesh Morphing

Applies a geometric design change directly to the mesh in the solver

Uses a Bernstein polynomial-based morphing scheme

• Freeform mesh deformation defined on a matrix of control points leads to a smooth deformation

• Works on all mesh types (Tet/Prism, CutCell, HexaCore, Polyhedral)

User prescribes the scale and direction of deformations to control points distributed evenly through the rectilinear region.


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RBF-Morph


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RBF-Morph


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How RBF-Morph Works? • Once displacements are defined by the user at the source points, Radial Basis Function interpolation is used to derive the displacement at any location in the space, so it is also available at every grid node. • The RBF problem definition is mesh independent, same set up can be applied to different meshes


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External flow example


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RBF-Morph main features

• Fully integrated within FLUENT and Workbench

• Easy to use

• Parallel calculation allows to morph large size models (many millions of cells) in a short time

• Mesh independent solution works with all element types (tetrahedral, hexahedral, polyhedral, etc.)

• Superposition of multiple RBF-solutions makes the FLUENT case truly parametric (only 1 mesh is stored)

– RBF-solution can also be applied on the CAD

• Precision: exact nodal movement and exact feature preservation.


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Ship hull: Series 60, CB=0.6

external hydrodynamics

multiphase flow (air & water)

ship advancing steadly in calm water

trim and sinkage fixed

displaced volume as constraint

resistance prediction

Objective: Optimization of the hull shape with no displacement reduction

Reduction of the resistance

Test case description

Conducted by Pranzitelli & Caridi


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CAD

Mesh ICEM-CFD

Baseline sim. Fluent

Workbench and RBF-morph setup

DOE RUNS

Optimization

Final solution

op

era

tor

wo

rkb

ench

Process

grid cells

Coarse 331,652

Medium 692,984

Fine 1,274,742 CT

ΔCT

Coarse 5.81x10-3 -2.52%

Medium 5.94x10-3 -0.34%

Fine 5.96x10-3 0%

Exp.* 5.96x10-3 -


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CAD

Mesh ICEM-CFD



DOE RUNS

Optimization

Final solution

op

era

tor

wo

rkb

ench

Process

Symmetry plane fixed

Morphing domain defined Eight cross sections specified

Section deformation applied


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CAD

Mesh ICEM-CFD



DOE RUNS

Optimization

Final solution

op

era

tor

wo

rkb

ench

Process

DX builds a DOE and drives Fluent and RBF Morph

Parameters are defined and transferred to the parameter set bar for use with ANSYS DesignXplorer


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• Design of Experiments

• 45 Design Points

• Solved in Batch

• Input parameters

• Output parameters

• DOE Settings

ANSYS DesignXplorer • Results

• Sensitivity analysis

• Response Surface


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baseline

optimized

baseline

optimized

• Optimize

Optimize with ANSYS DesignXplorer

Baseline Optimized

Fx 6.83N 6.29N

• 7.9% resistance reduction

• No volume reduction


41 *one Intel® i7 quad-core processor, 2.8GHz

Performance with RBF-Morph in Workbench: • Mesh generation: 6 man-hours • Fluent case setup: 1 man-hours • Baseline simulation (coarse grid): 4 CPU*-hours • Workbench and RBF-Morph setup:1 man-hours • DOE (45 simulations): 45 CPU*-hours • Optimization: Minutes

8 man hrs

2 CPU days

Without Workbench & RBF-Morph....? • Mesh generation (first mesh): 6 man-hours • Geometry (CAD) and mesh modification for each case (considering mesh automation in ICEM-CFD): 1x45 = 45 man-hours • Cases management (Fluent): 1x46 = 46 man-hours • Cases execution: 4+45 = 49 CPU*-hours • use of other optimization tools: ??

~100 man hrs

2 CPU days (optimistically)


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Preface

• The release of the adjoint solver in ANSYS Fluent 14 is the culmination of several years of R&D effort.

• This project was risky, but the rewards are great for ANSYS clients.

• There were a number of false starts and dead-ends.

• Writing an adjoint solver that meets the needs of the engineering community is not a trivial task.

• We are pleased to have come so far, and look forward to going much further.


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An adjoint solver allows specific information about a fluid system to be computed that is very difficult to gather otherwise.

The adjoint solution itself is a set of derivatives.

• They are not particularly useful in their raw form and must be post-processed appropriately.

• The derivative of an engineering quantity with respect to all of the inputs for the system can be computed in a single calculation.

– Example: Sensitivity of the drag on an airfoil to its shape.

There are 4 main ways in which these derivatives can be used:

1. Qualitative guidance on what can influence the performance of a system strongly.

2. Quantitative guidance on the anticipated effect of specific design changes.

3. Guidance on important factors in solver numerics.

4. Gradient-based design optimization.

What is an adjoint solution and how do we use those results?


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GOAL: Identify features of a system design that are most influential in the performance of the system.

EXAMPLE: – Sensitivity of the Drag on a NACA 0012 airfoil to changes in the

shape of the airfoil.

– The shape sensitivity field is extracted from the adjoint solution in a post-processing step.

How to use the results - Qualitative

High sensitivity – changes to shape have a big effect on drag

Low sensitivity – changes to shape have a small effect on drag


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GOAL: Identify specific system design changes that benefit the performance and quantify the improvement in performance that is anticipated.

EXAMPLE: – Design modifications to turning vanes in a 90 degree elbow to

reduce the total pressure drop.

– The optimal adjustment that is made to the shape is defined by the shape sensitivity field (steepest descent algorithm).

– Effect of each change can be computed in advance based on linear extrapolation.

How to use the results - Quantitative

Original DP = -232.8 Pa Expected change computed using the adjoint and linear extrapolation = 10.0 Pa Make the change and recompute the solution. Actual change = 9.0 Pa

Baseline Modified


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GOAL: Identify aspects of the solver numerics and computational mesh that have a strong influence on quantities that are being computed that are of engineering interest.

EXAMPLE: – Use the adjoint solution to identify parts of the mesh where mesh

adaption will benefit the computed drag by reducing the influence of discretization errors.

How to use the results – Solver Numerics

Baseline Mesh Adapted Mesh

Adapted Mesh Detail


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GOAL: Perform a sequence of automated design modifications to improve a specific performance measure for a system

EXAMPLE: – Gradient-based optimization of the total pressure drop in a pipe.

– Flow solution is recomputed and the adjoint recomputed at each design iteration.

How to use the results – Optimization

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0 10 20 30D

pto

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Initial design

Final design

30% reduction in total pressure drop after 30 design iterations


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Standard CFD Workflow elements

– Define a flow problem.

– Create a geometric representation of the problem and create a computational mesh.

– Setup and solve the flow problem.

– Post-process the results.

If the design is not meeting performance requirements

– Use insight, experience and intuition to decide how to select design changes that will improve the performance of the system

or

Adjoint workflow elements

– Use the results to improve the design systematically using one of the 4 strategies outlined

– Pick an observation that is of engineering interest.

• Lift, drag, total pressure drop?

– Set up and solve the adjoint problem for this observation for the specific computed flow field

• Define adjoint solution advancement controls

• Set adjoint convergence criteria

• Initialize the adjoint solution field

• Iterate to convergence

• Post process

How does an adjoint analysis fit into the familiar CFD workflow?


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– Mesh morphing

– Mesh morphing & Adjoint Data

– Mesh Morphing, Adjoint Data & Constraints

Supporting Technologies


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Once a desired change to the geometry of the system has been selected, how is that change to be made?

• Mesh morphing provides a convenient and powerful means of changing the geometry and the computational mesh.

– Use Bernstein polynomial-based morphing scheme discussed earlier

Mesh Morphing


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• Example: Sensitivity of lift to surface shape

• Select portions of the geometry to be modified

• Adjoint to deformation operation

• Surface shape sensitivity becomes control point sensitivity (chain rule for differentiation)

• Benefit of this approach is two-fold

• Smooths the surface sensitivity field

• Provides a smooth interior and boundary mesh deformation

Mesh Morphing & Adjoint Data

Flow


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The adjoint solution is determined based on the specific flow physics of the problem in hand.

The effect of other practical engineering constraints must be reconciled with the adjoint data to decide on an allowable design change.

Example:

– Some walls within the control volume may be constrained not to move.

– A minimal adjustment is made to the control-point sensitivity field so that deformation of the fixed walls is eliminated.

Mesh Morphing, Adjoint Data & Constraints

Fixed wall

Fixed wall

Moveable walls


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Current Functionality

ANSYS-Fluent flow solver has very broad scope

Adjoint is configured to compute solutions based on some assumptions

– Steady, incompressible, laminar flow.

– Steady, incompressible, turbulent flow with standard wall functions.

– First-order discretization in space.

– Frozen turbulence.

The primary flow solution does NOT need to be run with these restrictions

– Strong evidence that these assumptions do not undermine the utility of the adjoint solution data for engineering purposes.

Fully parallelized

Gradient algorithm for shape modification

– Mesh morphing using control points.

Adjoint-based solution adaption


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The adjoint solver is an addon that will be part of the Fluent 14 distribution.

Documentation is available

– Theory

– Usage

– Tutorial

– Case study

Training is available.

Functionality is activated by loading the adjoint solver addon module.

A new menu item is added at the top level.

Limitations include unsupported models (porous media, MRF etc.), convergence can be challenging for large cases (5-10M+ cells) and cases that exhibit unsteady flow or strong shear flows

– Stabilized solution advancement algorithm is in place

Current Functionality


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GUI

• Follow as closely as possible the same design layout as Fluent solver – Specify observable

– Adjoint solution advancement controls

– Residual monitors

– Initialization and iteration

– Post-processing: contours, vectors.

– Results reporting

– Mesh-morphing with pre-calculation of expected change in observable.

TUI

User-Interface

/adjoint>

controls morphing/ reporting/

monitors/ observable/ run/

Mesh Morphing and the Adjoint Solver in ANSYS R14 · An adjoint solver computes sensitivity data...

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