M. Pini*, D. Pasquale‡, G. Persico*, A. Ghidoni‡, S. Rebay ‡

* Gruppo di ricerca di Fluidodinamica delle Macchine

Dipartimento di Energia

ASME ORC 2013

2nd International Seminar on ORC Power Systems

October 7th-8th, De Doelen, Rotterdam , The Netherlands

Adjoint Method for Shape Optimization

in Real Gas Flow Applications

‡ Università degli

Studi di Brescia

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Motivation

ORC Turbomachinery Fluid-Dynamic Design

Optimization Methods + Accurate EoS

• Gradient-based Adjoint

• Gradient-free Metamodel-based GA

MDM

H20

space,cost

space,cost

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Outline

• Adjoint method & Optimization Algorithm

• Shape Optimization of an ORC Supersonic Cascade

• Conclusions & future plans

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Adjoint Method

Advantage:

Efficiency (Independent on the number of design variables)

Drawback:

Intrusive method (Flow Solver differentiation AD)

0

v

u

R

u

TTJ

Discrete adjoint equation

Gradient vector

v

X

R

Xα

X

α geogeo

TTTT

J

d

d

d

dJ

Fitness function: Constraint (flow equations):

)(),,(, geogeophyphy αXααuαJJ 0)(),,(, geogeophyphy αXααuαR

Shape Optimization phyαgeoα )(J

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Design Chain for Shape Optimization

Optimization main features

Inviscid flow solver

in-house zFlow

Discrete adjoint method

automatic differentiation (Tapenade)

Surface gradients

state-of-the-art parameterization techniques

Grid deformation effects

novel interpolation method

Thermodynamic Modeling

consistent look-up table approach

Solver Initialization

fast flowfield interpolation method

M. Pini, G. Persico, and. S. Rebay, Shape Optimization for Fluid-Dynamics Problems on Unstructured Grids based on Adjoint

and Automatic Differentiation, submitted to Computers&Fluids

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Numerical Solvers

• Grid Generator (adMesh)

Fully automated 2D mesh generator

Advancing Delaunnay-front technique

• Flow Solver (zFlow)

Hybrid FE/FV Euler/RANS solver

Look-up Tables + direct EoS

• Adjoint Solver (JzFlow)

GMRES + exact Jacobian preconditioning

Look-up Tables + direct EoS

(AD differentiation)

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Geometry Modeling TTTT

d

d

d

d

d

d

b

b

X

X

P

X

P

X

α

X

geo

Grid Deformation

Based on Radial Basis Functions

Suitable for Unstructured Grids

Algorithmically differentiated

Geometry Parameterization

NURBS Curves

Low number of design variables

Algorithmically differentiated

v

X

R

Xα

X

α geogeo

TTTT

J

d

d

d

dJ

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Modeling of Real Gas Flows

Main features of CLUT approach

Thermodynamic Mesh Generation

multiblock methodology

Construction of Fundamental Equations

bicubic polynomials ( e=e(s,v) )

Computation of Thermodynamic Properties

algebraic derivatives

Fluid-Dynamic Simulation

kdtree-based grid decomposition

)(, 33vsfvsee

sv

evsp

,

sv

evvsc

2

2

,

M. Pini et al., Novel Interpolation Method for Real Gas Properties Calculation, ASME-ORC 2013

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Test Case – Supersonic ORC Turbine

outflow

1i

mixi

outflow

n

MM

J

n

P

Objective:

Uniform Flow

Fitness Function

ist ,p

Optimization Algorithm:

1. SD with Gradient Preconditioning + Projection

2. Preliminary Gradient Analysis

Main Data

Working fluid: siloxane MDM

EoS: Span-Wagner (LUT)

Inlet compr. factor: ~0.7

Expansion Ratio: ~8

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Test Case - Results

Outlet Mach Number

Outlet Flow Angle

Outlet Total Pressure

Base Opti

ηis 95.1 % 98.0 %

mflow 1 0.92

Δpt 12.33 % 5.69 %

αangle 74.92° 76.57°

Baseline Configuration

Optimal Configuration

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Influence of Thermodynamic Model

• Lower performances using

poor accurate EoS

• Strong influence of the blade

shape (shocks)

• Multilevel approach for

demanding optimizations

Outlet Mach Number

Rear Profile Shapes

Mach flowfield (LUT) - shape optimization based on different EoS

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Design Validation for Turbulent Flows

Numerical Model

Ansys CFX + Look-up Tables (50 kcells)

k-ω SST turbulence model (y+ ~ 1)

Base Opti

mflow 1 0.94

Δpt 20.32 % 12.24 %

Baseline Configuration

Optimized Configuration

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Off-Design Performances

1. Preliminary ANOVA

analysis on the baseline

geometry

2. Uniform distribution of

the static backpressure (p)

Deterministic Optimization

unsuitable at off-design

Multipoint Optimization

Robust Optimization

Baseline Configuration (higher p)

Optimal Configuration (higher p)

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Conclusions & Future Plans

Achieved so far

1. Adjoint Method for Real Gas Flows

2. Optimization Methodology for Blade Shape Design

3. Large Variety of Turbine Arrangements Handled

Future Perspectives

1. Extension to Turbulent Flows (Algebraic Models)

2. Extension to Multipoint Optimization

3. Constrained Optimization (Lagrangian)

M. Pini et al.: Adjoint Method for Shape Optimization in Real Gas Flow Applications

Thank you for your attention