Application of Genetic Algorithms to CFD Cameron McCartney
Introduction ●
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define and describe genetic algorithms (GAs) and genetic programming (GP) propose possible applications of GA/GP to CFD
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Genetic Algorithms (GAs) ●
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a highly parallel mathematical algorithm that uses evolution operations to transform a population of data objects, each with individual fitness values, into a new population with higher average fitness values [Koza] an optimization method analogous to biological evolution and similar to simulated annealing “GAs provide a tool in areas that do not yield readily to standard approaches” [Holland]
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Genetic Algorithms (GAs) ●
generally applied to optimization problems where the system dynamics are known but the optimal set of input variables is not
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Genetic Algorithms (GAs) ●
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goal is to determine values for the set of input variables which, when processed by the system dynamics, most closely match the output data GA automatically determines the optimal values of the input variables with minimal initial direction from the user
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Genetic Algorithms: Process 1.Generate a random initial population of potential solutions as generation 0 (G0) 2. Evaulate the fitness of each member of G0 using the known system dynamics and a fitness function 3.Apply genetic operations to members of G0, selecting based on their fitness values, to produce a new set of potential solutions (G1) 4.Continue to evaulate and apply genetic operations to each generation until an optimization criteria has been satisifed
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Genetic Algorithms: Process
i F0 ∑ F 0=
N0
i F1 ∑ F 1=
N1
i F2 ∑ F 2=
N2
F 0F 1F 2
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Population Members ●
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population: a set of similar data objects, each describing a complete combination of the input variables each population member represents a potential solution to the optimization problem members typically express a combination of the input variable values using a binary representaion
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Fitness: Defintion ●
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a quantitative measure of how well a specific member satisfies the optimization problem a fitness function usually compares the output data from a potential solution to the desired set of output data (i.e. error, residual) examples of fitness functions: – – –
least squares error for curve fitting, other optimization applications minimum path length for topographical applications lowest residual energy for protein synthesis applications Application of Genetic Algorithms to CFD
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Fitness: Importance to GA ●
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fitness is the key driving force behind GA/GP; population members with higher fitness are more likely to be 'copied' to the next generation. any factors not explicited included in the fitness measure will not be optimized GA generally increases the average fitness of the population; successive iterations will continue to improve the fitness of the population.
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Genetic Operations ●
two main genetic operations are performed on population members in order to progressively improve their fitness values: crossover – mutation analagous to biological evolutionary processes –
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Genetic Operations
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Genetic Operations ●
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crossover combines parts of two population members in hopes of creating a new member with better fitness that either contributing member population members are selected for crossover based on fitness values i.e. better members are copied into the next generation more frequently crossover is the main cause of improved average fitness from generation to generation; mutation only serves to restore diversity into populations which are too homogeneous (i.e. local solution) Application of Genetic Algorithms to CFD
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GA vs. Simulated Annealing ●
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simulated annealing (SA) is a multi-dimenaionsl optimization technique bsaed on following local gradients to a minima/maximum main difference: SA requires the user to choose some optimization parameters (e.g. transition probability function, transition step size); GA requires less explict input by the user proposed: GA may be less efficient than SA for wellbehaved optimization problems, but may be better suited to complex parameter spaces
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GA: General Optimization ●
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GA has been shown to be a general optimization technique with applications in engineering design GA is well suited parallelization on computing clusters: evaluation of each member's fitness is mutually independant; only combined process is the application of the genetic operations the ability of GA to optimize multiple schemata (i.e. combinations of input variables) at the same time leads to inherent parallelism
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Genetic Programming (GP) ●
a specific application of GA techniques used to evolve structured algorithms or computer programs rather than static combinations of input variables
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GA vs. GP ●
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GA determines the set of input data which, when transformed by a known system dynamics algorithm, most closely matches a set of known output data GP determines the system dynamics algorithm which most closely transforms a set of known input data into a set of known output data Application of Genetic Algorithms to CFD
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GP: Program Structure ●
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GP algorithms are usually represented as linked trees tree elements are either program flow control elements (IF, FOR, etc.), input variable values (x1, x2, etc.) or randomly generated numerical constants.
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GP: Genetic Operations ●
GP crossover exchange branches of two trees Generation i
Generation i+1
Source: Koza Application of Genetic Algorithms to CFD
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GP: Genetic Operations ●
GP mutation changes a single tree element: – –
IF(x1 < x2) → IF(x1 > x2) (x1 + x2 – x3) → (x1 + 2.324 – x3)
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GP: Applications ●
GP is well-suited to problems involving unknown system dynamics: – – –
financial forecasting electronic filter design curve-fitting (symbolic regression)
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GA/GP Research ●
GAs initially developed by Holland (1975)
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GP developed by Koza (~1992)
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GA/GP applied to problems in many different disciplines: – – –
engineering design (e.g. airfoil optimization, control system tuning, artificial vision systems) financial forecasting biological system modeling
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GA/GP Application to CFD ●
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GA/GP may be applied to CFD simulations in two different ways: – act as an optimization technique driving CFD software (external) – allow more efficient solution of CFD problems (internal)
GA/GP may also facilitate research into CFD turbulence modeling Application of Genetic Algorithms to CFD
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GA/GP→CFD: Optimization Driver ●
CFD is often used as an evaluation method for optimization designs: heat sinks – airfoils – etc. current optimization methods driving the CFD software include: –
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Markov chains Bayesian inference simulated annealing Application of Genetic Algorithms to CFD
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GA/GP→CFD: Source Determination ●
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one CFD application is the determination of particulate source 2D locations and strengths based on limited measurements involves searching a large parameter space (x, y, S(t)) with moderate nonlinearity GA may represent a feasible alternative to Bayesian inference
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GA/GP→CFD: Airfoil Design ●
Zhang, Chen and Khalid at the National Research Council Canada have successfully used GA to optimize airfoil geometries
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GA/GP→CFD: Zone Model Solution ●
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zone models are simple two-zone CFD approximations used for modeling heat and smoke transport through buildings solving zone model simulations of a few compartments is trivial; as the number of compartments reaches 50, 100, 1000, the underlying set of ODEs become difficult to solve due to their stiffness GA may be used to evolve a set of functions for each compartment which satisfies the ODEs and constraints
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GA/GP→CFD: Matrix Solvers ●
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can GA be used as a more efficient matrix solution technique for CFD? instead of backsolving a large set of linearized equations, guess and interatively improve a solution field using GA. if forward calculation of error requires substantially less computational power than current matrix solution techniques, improvements in computational efficiency may result
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GA/GP→CFD: Turbulence Modeling ●
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many current tuirbulence models have functional forms and parameter values derived via correlation with experimental or DNS data e.g. (k,ε) = ƒ(Cμ,σk,σε,C1ε,C2ε) GA may be used to tune the parameter values to produce a better correlation GP may be used to derive new functional forms which are of higher complexity i.e. highly non-linear derivation of new functionl forms for tubulence models may lead to new understanding of complex parameter relationships Application of Genetic Algorithms to CFD
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Research Directions ● ●
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review literature for exisiting application of GA/GP to CFD review literature for uses of GA/GP in related applications (e.g. solution of sets of ODEs, symbolic regression) determine suitability of various research topics using simple benchmark problems (e.g. 1-D Fourier equation, Euler's equation)
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References ●
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Holland, J. H. Adaptation in natural and artificial systems (2nd ed.). Massachusettes Institute of Technology Press. 1992. ISBN 0-262-08213-6. Koza, J. R. Genetic programming: on the programming of computers by means of natural selection. Massachusettes Institute of Technology Press. 1992. ISBN 0-262-11170-5. Zhang, F., Chen, S., Khalid, M. Inverse problem study of isolated wing by genetic algorithm. Proc. of the 12th Annual Conference of the CFD Soc. of Canada. 2004.
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Contact ●
[email protected]
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