Development of a genetic algorithm-based solution algorithm for 1-D convection-diffusion analyses

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Proceedings of the 15th Conference of the Computational Fluid Dynamics Society of Canada, 2007-05-27

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Development of a genetic algorithm-based solution algorithm for 1-D convection-diffusion analyses

McCartney, C. J.; Lien, F. S.

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Development of a Genetic Algorithm-Based

Solution Algorithm for 1-D Convection-Diffusion

Analyses

C. J. McCartney1, F. S. Lien1

1

Mechanical & Mechatronics Engineering Department, University of Waterloo, Waterloo, ON

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Overview

● genetic algorithms (GAs): an introduction

● applying GAs to CFD

● development of a GA-based solution algorithm for 1-D

diffusion analyses

● extension of the GA-based solution algorithm to 1-D

convection-diffusion analyses

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Introduction

● CFD solution algorithms must be:

– _accurate

– _robust

– _efficient

● most existing algorithms are iterative in nature, apply different

(5)

Objectives

● can genetic algorithms be used as CFD solution algorithms?

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Genetic Algorithms (GAs)

● multivariate optimization technique first proposed by

Holland [4]

● a complex solution space is iteratively searched to find

optimal values for the input variables based on a proscribed evaluation or fitness function

● main differences from existing matrix solution algorithms:

– _{multiple potential solutions during each iteration}

– _{solutions improved via semi-random evolution operators}

● members are 'copied' more often to the next generation if they

are more accurate i.e. GAs evolve progressively better solutions

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Genetic Algorithms: Process

Generate initial population Evaluate members Crossover Selection Mutation

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Genetic Algorithms: Evolution Operators

● two main classes of operators are used to improve solutions:

– _{Crossover: simulates biological reproduction}

– _{Mutation: random changes to limited number of members}

● wide variety of crossover and mutation operators exist, some

(9)

Genetic Algorithms: Crossover

φ φ φ x_i x x_i Crossover point Parent A Child Crossover

● concatenate pieces of two parent

members to produce a child member

● purpose: combine two 'good' solutions

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Genetic Algorithms: Mutation

φ x_i Child Mutation φ x_i Child

● randomly alter single elements of a member e.g. a single

temperature value in the solution field

● purpose: prevent GA convergence to a sub-optimal solution

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Genetic Algorithms: Parameters

● main GA parameters include:

– _{population size, N (i.e. the number of potential solutions}

per generation)

– _{mutation rate, p}_mut

– _{number of points used in the crossover operator (1, 2, etc.)}

● parameter values affect GA convergence efficiency,

robustness

● impact of these parameters on the performance of a GA-based

solution algorithm for 1-D diffusion analyses is described later

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Overview

● genetic algorithms (GAs): an introduction

● applying GAs to CFD

diffusion analyses

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Literature Review: 'Traditional' GA

● GAs have been applied as an optimization method in many

diverse areas of engineering design e.g. work path optimization, electronic circuit design [e.g. 7]

● GAs have been used in conjunction with CFD simulations to

optimize design parameters for thermofluid systems:

– _{turbomachinery design}

– _{airfoil geometry optimization based on CFD predictions of}

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Literature Review: GAs as CFD Solvers

● GA-based CFD solution algorithms: relatively recent topic,

investigated by a limited number of researchers

● Bourisli and Kaminski [1 3] demonstrate that GAs are able to ‑

solve simple CFD analyses:

– _{potential, backstep and sudden contraction flows}

– _{introduce the application of both domain decomposition}

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GA-Based Solution Algorithms for CFD

● each member defines one or more property fields (e.g.

temperatures, velocities, concentrations); represents a potential solution to the underlying conservation equation(s)

Member A: [T], [u], [φ]

● need a fitness function to evaluate each member; simplest

function calculates how accurately the solution represented by each member satisfies the discretized form of the underlying conservation equations i.e. error:

● members from each generation are 'copied' to the next

generation via crossover based on fitness

F _raw=

∑

i

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GA-Based Solution Algorithms for CFD

● evolving a solution using GA may offer a number of

advantages over traditional matrix solution techniques:

– _{since no special form for the coefficient matrix [A] is}

assumed (i.e. block banded) GA-based solution algorithms can be directly applied to unstructured grids

– _{since the fitness function does not rely on a certain form}

for the computational cell, higher-order discretization schemes may be more easily incorporated

– _{fitness is calculated on a row-by-row basis with regards to}

[A]; lends itself to parallelization

● these and other potential benefits are discussed later in

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Overview

● genetic algorithms (GAs): an introduction ● applying GAs to CFD

diffusion analyses

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● solution of the 1-D diffusion equation is used as a first step in

developing a GA-based solution algorithm for more general thermofluids analyses

1-D Diffusion

q_rad,q_conv T_B T_A x 0 k, A q x_L

Figure 1. Diffusion of heat through a 1-D material.

k d 2

T

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1-D Diffusion: Discretization

● discretize domain into 20 isometric control volumes, each

with a centrally-located node

● two additional nodes are located at the flow domain

boundaries to allow boundary conditions to be specified

● temperature gradients at control volume faces are expressed in

terms of nodal temperatures by applying second-order CDS

● resulting set of discretized equations:

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1-D Diffusion: Implementation

● simple evolution operators:

– _{1- and 2-point crossover}

– _{mutation with a constant mutation distance, ΔT}_mut

● solution with highest fitness in each generation is copied to

the next generation without crossover or mutation applied (i.e. 'elitist selection'); speeds up convergence for simple analyses

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1-D Diffusion: Test Case

● preliminary test case:

– _{Dirichlet boundary conditions: T}_A_{= 300 K, T}_B_= 700 K

– _{k = 1 W/m-K}

– _x_L_= 1 m

– _A = 1 m2

– _{internal heat generation; q = 2 000 W/m}3

● theoretical solution defined by [8]:

● GA parameters: population size N = 100, mutation rate

T  x =[ T B−T A

x_L 

q

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1-D Diffusion: Results

0 250 500 750 1000 0.00 0.25 0.50 0.75 1.00 T [ K ] x [m] Sim ulation C-010 Theoretical 0 250 500 750 1000 0.00 0.25 0.50 0.75 1.00 T [ K ] x [m] Simulation C-010 Theoretical a) G = 1 000 a) G = 10 000

Figure 2. Predicted temperature field for 1-D diffusion analysis vs. theoretical solution (Simulation C-010).

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1-D Diffusion: Analysis

● high-frequency error preferentially dampened out

● 'final' solution for Generation 10 000 still requires further

convergence

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1-D Diffusion: Sensitivity Study

● choosing GA parameter values affects convergence efficiency,

robustness

● identification of appropriate ranges for each of the main

parameters is critical for development of an efficient GA-based CFD algorithm

● a sensitivity study is conducted based on the 1-D diffusion test

case described above; vary three GA parameters:

– _{population size (N): 10, 50, 100}

– _{mutation rate (p}_mut_{): 0.01, 0.1, 0.2}

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1-D Diffusion: Sensitivity Study

● raw fitness used as a measure of convergence; analogous with

error

● due to randomness in the roulette wheel selection process,

performance of a single simulation is not a reliable indicator; fitness trends are based on the average of thirty simulations

● total number of solutions evaluated in each simulation was

106, regardless of population size N; simulate a constant

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1-D Diffusion: Sensitivity Study Results

100 1000 0 250000 500000 750000 1000000 A v e ra g e m in im u m r a w f it n e s s Total members C (N = 100, p_mut = 0.1, 1-point) A (N = 10, p_mut = 0.1, 1-point) F (N = 100, p_mut = 0.01, 1-point) I (N = 100, p_mut = 0.2, 1-point) L (N = 100, p_mut = 0.1, 2-point)

Figure 3. Comparison of selected convergence trends for 1-D diffusion simulations.

● optimal convergence

trend: simulation set C

(N = 100, p_mut = 0.1,

1-point crossover)

● other trends indicate

sub-optimal convergence behaviour:

– _{rapid convergence but}

eventual stalling (A, I)

– _{more robust but}

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1-D Diffusion: Conclusions

● in general, the choice of values for GA parameters balance

two conflicting goals: rapid convergence (higher mutation rates, smaller populations and more aggressive evolution operators) and robustness

● identification of an optimal set of GA parameters for general

CFD applications is difficult, likely application-specific

● efficiency of GA-based CFD solution algorithms varies

inversely with number of nodes

– _{Bourisli and Kaminski [1] demonstrate a grid refinement}

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Overview

diffusion analyses

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1-D Convection-Diffusion

● extend algorithm to a more complex flow analysis: 1-D

convection-diffusion q_rad,q_conv T_B T_A x 0 k, A q x_L u d  ρ u  dx =0 d dx  ρ u T = d dx k dT dx 

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1-D C/D: GA Implementation

● GA-based solution algorithm for 1-D diffusion is modified to

include convection:

– _{nodal velocities u}_i_{added to each member}

– _{continuity equation discretized using fourth-order CDS}

– _{GA fitness function modified to include both the accuracy}

of the temperature field (via heat equation) and velocity field (via continuity equation):

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1-D Convection-Diffusion: Test Case

● simulations are performed using the following values:

T_A = 0.9 K, T_B = 0.1 K, k = 1 W/m-K, x_L = 1 m, A = 1 m2,

u = 0.5 m/s

● following GA parameter values are selected based broadly on

the results of the 1-D diffusion sensitivity study:

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1-D Convection-Diffusion: Results

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 T [ K ] x [m ] Sim ulation Theoretical

Figure 5. Typical temperature field solution for 1-D convection diffusion analysis (G = 106).

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1-D Convection-Diffusion: Results

Figure 6. Typical velocity field solution for 1-D convection diffusion analysis

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 u [ m /s ] x [m] Simulation Theoretical

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1-D Convection-Diffusion: Analysis

● temperature field requires further convergence

● velocity field exhibits chequerboard oscillations; replace CDS

with UDS, hybrid

● further research on improving GA for 1-D

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Overview

diffusion analyses

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Proposed Research: Parallelization

● GA-based solution algorithms do not perform any matrix

inversion or decomposition on coefficient matrix [A]

● fitness is calculated on a row-by-row basis with no interaction

between rows; lends itself to parallelization

F_raw=

∑

i

∣

[ A]i[]i−b_i

∣

Row 1 → CPU 1 Row 2 → CPU 2 Row 3 → CPU 3 [A],[φ]

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Proposed Research: Multigrid

● Bourisli and Kaminski [1] demonstrate how grid refinement

can significantly improve GA convergence behaviour

● GAs appear to be good smoothers

● a more formal multigrid (MG) approach to GA-based solution

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Conclusions

● a GA-based algorithm for solving 1-D diffusion analyses has

been developed and has been shown to predict solutions with acceptable accuracies

● extension of this algorithm to 1 D convection-diffusion ‑

analyses is in progress

● GA convergence efficiency shown to be sensitive to GA

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Acknowledgements

● financial support from Natural Sciences and Engineering

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GA-Based Solution Algorithms for CFD

● main objective of most CFD analyses: solution of large matrix

equation based on discretized conservation equations to determine a field of values for one or more quantities

● existing solution algorithms for analyses of practical interest

are mainly iterative in nature (e.g. SIP, MG), rely on specific properties of these matrices in order to achieve improved convergence efficiency and/or robustness

● although highly efficient for a wide class of analyses, existing

algorithms still limited for certain 'tough' cases e.g. high localized gradients, non-linear processes such as radiation

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1-D Diffusion: Governing Equation

● temperature field in the flow domain governed by the heat

equation [8]:

(2)

● thermal conductivity, k, and cross-sectional area of the

domain, A, are assumed constant

● source term q represents internal heat generation

● application of Dirichlet and/or von Neumann boundary

conditions at either end of the flow domain yield a variety of theoretical solutions [cf.8]

● convective and/or radiative heat transfer from the flow domain

surface may be included through appropriate modifications to

k d 2

T

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1-D Diffusion: Implementation

● a GA-based algorithm to solve 1-D diffusion analyses is

implemented in C

● 'member' structure:

– _{floating point array representing nodal temperatures, T}_i

– _{floating point array representing nodal x co-ordinates, x}_i

– _{raw (F) and normalized (F}_norm_{) fitness values for each}

member

● 'population' structure:

– _{array of member structures}

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1-D Diffusion: Sensitivity Study Parameters

GA Parameter Set Crossover Type

A 10 0.1 1-point 62.9 B 50 0.1 1-point 48.6 C 100 0.1 1-point 47.3 D 10 0.01 1-point 91.5 E 50 0.01 1-point 86.6 F 100 0.01 1-point 86.3 G 10 0.2 1-point 68.4 H 50 0.2 1-point 76.4 I 100 0.2 1-point 66.2 J 10 0.1 2-point 55.4 K 50 0.1 2-point 53.4 L 100 0.1 2-point 48.1 M 10 0.01 2-point 90.5 N 50 0.01 2-point 78.8 O 100 0.01 2-point 85.0 P 10 0.2 2-point 62.9 Q 50 0.2 2-point 68.5 R 100 0.2 2-point 64.1 Population Size N Mutation Rate p_mut Final Fitness Fraw,final

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1-D Diffusion: Sensitivity Study Analysis

● 1-point crossover: final solution accuracy improves with

increasing population size; clear trend not evident for 2-point

– _{Bourisli and Kaminski [1] propose that small population}

sizes are better suited to solution of simple CFD analyses; however, more complex flows may require larger population sizes to avoid convergence to local maxima

● mutation rate of p

mut = 0.1 performs better than 0.01 and 0.2

– _{Bourisli and Kaminski [1] recommend the use of relatively}

high mutation rates for simple CFD analyses to improve convergence efficiency; risk of reduced robustness

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1-D C/D: Governing Equations

● conservation equations describing the temperature and

velocity distribution in a 1-D domain due to convection and diffusion [8]:

(5) (6)

● the inclusion of convection introduces a second conversation

equation as well as a second field quantity (i.e. velocity, u)

● temperature field for the simplified case of constant ρ, u, A

and k is given by [8]: (7) d  ρ u dx =0 T  x =T _AT _B−T _A exp  ρ u x / k −1 exp ρ u x_L/ k −1 d dx  ρ u T = d dx k dT dx 

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Proposed Research: GA-based Solvers

● above examples and results of Bourisli and Kaminski [1-3]

demonstrate that GA-based algorithms are able to solve simple CFD analyses

● GA-based solution algorithms have substantially different

behaviours than traditional matrix solution algorithms

● may offer advantages for 'tough' CFD analyses where

traditional algorithms do not perform well:

– _{Stiff matrices: multiple timescales}

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Proposed Research: Genetic Programming

● perhaps the most interesting new research area related to

GA-based CFD algorithms involves similar algorithms developed using genetic programming (GP)

● GP: optimization technique closely related to GAs but which

evolves solutions in a functional form (i.e. φ = f(x,y,z,t)) rather than a nodal form

● functional solutions require less memory; may improve the

efficiency of parallel implementations since less information must be communicated to each cluster node

● lack of a predefined computational grid in a GP-based CFD

algorithm may also allow for some interesting applications with grid refinement and/or unstructured grid techniques

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Future Research

● demonstrate the benefit of GA-based solution algorithms for

'tough' CFD analyses which cannot be solved efficiently using existing solution algorithms

● implement a multigrid version of a GA-based solution

algorithms

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3. Bourisli, R. I., Kaminski, D. A. Solution of the incompressible Navier-Stokes equations via real-valued evolutionary algorithms. WIT Transactions on Engineering Sciences - Advances in Fluid Mechanics VI v.52:13-20. 2006.

4. Holland, J. H. Adaptation in natural and artificial systems : an introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press. 1975.

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