eartius hmge algorithm applied to optimization tasks with 10,000 design variables

©Copyright eArtius Inc 2012 All Rights Reserved October 22, 2012

eArtius HMGE Algorithm Applied to Optimization Tasks with 10,000

Design VariablesVladimir Sevastyanov

eArtius, Inc., Irvine, CA 92614, [email protected]

Boosting Optimization Standards


A number of algebraic models are designed at eArtius with the following properties:

2 objective functions

10,000 design variables

Only a few hundred design variables are significant; all others are not significant

Design space has a predetermined number of sub-areas (“craters”) where objective functions really depend on design variables;

Each such a sub-area has own list of significant design variables

The models are designed to test scalability of eArtius optimization algorithms


Hybrid Multi-Gradient Explorer (HMGE) Optimization Method


How HMGE algorithm works:

Basically, HMGE works as a Genetic Algorithm;

Periodically HMGE evaluates gradients, and performs gradient-based steps, which improves convergence dramatically

How gradient based steps are performed:

Evaluate the model on 5-7 points generated in a small sub- region around current point;

Recognize the most significant design variables, and build an approximation for each output variable;

Estimate gradients based on the approximations;

Determine a direction of simultaneous improvement for all objective functions;

Perform a step in the direction


Hybrid Multi-Gradient Explorer (HMGE) Optimization Algorithm

Synergy of the features brings HMGE on unparalleled level of efficiency and scalability

HMGE is believed to be the first global multi-objective optimization algorithm which provides:

- Efficiency in finding the global Pareto frontier

- High convergence typical for gradient- based methods

- Scalability: Equal efficiency optimizing models with dozens, hundreds, and even thousands of design variables

Genetic Algorithm Framework

Random Mutation Gradient Mutation

DDRSM – Super Fast Gradient Estimation


DDRSM Benefits:

Equally efficient and accurate for any task dimension

Requires just 0-7 model evaluations regardless of task dimension

Fast— it builds a local approximation in 10-30 milliseconds

Automatic and hidden from users

Eliminates necessity in global response surface methods

Eliminates necessity in a sensitivity analysis

DDRSM evaluates gradients necessary for any gradient based optimization algorithms.

How DDRSM operates:Start iteration:

Determines the most significant design variables

for each responsevariable separately

Start iteration:Determines the most

significant design variablesfor each responsevariable separately

Builds local approximations for each response based

only on the most significant design variables

Builds local approximations for each response based

only on the most significant design variables

Analytically estimates gradients based on local

approximations

Analytically estimates gradients based on local

approximations

Performs a gradient based step

Performs a gradient based step

Dynamically Dimensioned Response Surface Method (DDRSM) for Gradient Estimation


Optimization Results for Tasks with 10,000 design Variables


eArtius optimization technology is not sensitive to the model dimension because it performs optimization in a sub-space of the design space related to the most significant design variables.

All non-significant design variables are dynamically recognized in runtime, and simply ignored.

Thus, eArtius algorithms are equally efficient for low-dimensional and high- dimensional tasks.


Algebraic Model Description eArtius has developed an algorithm

which allows to generate algebraic models with a predetermined number of design variables, and a predetermined type of topology.

The functions are similar to Gaussian, but high-dimensional, with 10,000

design variables, and with a given number of “craters”. We tried to

optimize functions with 7, 8, and 10 “craters”.

The following fragments of the algebraic models give an idea about thefunctions (both functions require 425KB memory in a text format):

F1 = (-3.309 + (-2300 * exp(-0.03176 * ((X0 + X1 + X2) / 3 + 5))) + (2.655 + (-4200 * exp(-0.03589 * ((X3 + X4 + X5 + X6 + X7) / 5 + 5))) + … F2 = (3.02 + (-2100 * exp(-0.02191 * ((X0 + X1 + X2) / 3 + 3))) + (-4.109 + (-3700 * exp(-0.03553 * ((X3 + X4 + X5) / 3 + -5))) + …


Optimization Results for 7 “Craters”

– 2 objectives to be minimized– 10,000 design variables with the range [-10, 10]– 1000 model evaluations– 53 Pareto optimal solutions



– 2 objectives to be minimized– 10,000 design variables with the range [-10, 10]– 1000 model evaluations– 97 Pareto Optimal Points



– 2 objectives to be minimized– 10,000 design variables with the range [-10, 10]– 909 model evaluations– 54 Pareto Optimal Points


An Engineering Example of an Optimization Task Solved by HMGE

Optimization Method in ANSYS Workbench Environment


Multi-Physics Steady State Thermoelectric Simulation coupled with

Solid Works Shape Optimization and Transient Radiative Heat Transfer for

Substrate Heat-up

Solid WorksDesign Modeler (imports geometry parameters from Solid Works, modifies model adding symmetry

Workbench+eArtius

substrateleft

right


Complex Multi Physics Problem

Optimization Log output

Optimization Messages updates (# of data points computed)

Optimization Method selection (MGE)

WorkBench status Bar (stop button)

Steady State Thermo- Electric with Surface to Surface Radiation

Transient Radiation with Surface to Surface

Design Modeler Parametric Geometry interface with Solid Works

Project folders


Optimization Parameters

Heater Geometry dimension 2, from Solid Works

Heater Geometry dimension 1, from Solid Works

P23=P24 (heating on left side=heating on right)

Electrical current runs through 3 separate heating elements creating temperature distribution. Electrical power in each heater equals I*V and to minimize P18 we need to find optimal ratio of power between center and left/Right elements.

Substrate Temperature after short term transie exposure to heater

F1= Tmax-350 F2=Tmin-350350 =>desired process temperature we want to reach


Optimization Results

Heater Geometry dimension 1

Heater Geometry dimension 2

dT,Deg. C

dT,Deg. C

dT,Deg. C

dT,Deg. C

(Tmax-350), Deg. C

(Tmin-350), Deg. C

Want to pick bestValues for geometry dimensions 1 and 2


Most Essential Result

These are demo results of overnight –run, so study is not complete. However, we instantly see relationship between key conflicting variables (P18- maximum temperature difference in substrate) vs F1 – deviation from desired maximum temperature. The larger F1 the lower maximum temperature during heat-up, that means lower thermal ramp (gradient), lower power and thus lower temperature difference P18.

It is easy to have low temperature difference if you heat less, it means you loose less heat as well and thermal uniformity is better. In this problem we need to heat more, thus we are interested in Pareto frontier distribution looking for multiple trade- offs.

dT,Deg. C

(Tmax-350), Deg. C

3135Optimal Range of

interest found


Summary Result

Global computational optimizationof heating module and element designs to minimize temperature difference on substrate surface (DeltaT).

Optimization uses state-of-the art hybrid genetic-multi-gradient optimization methodology.

Optimal power ratio ~2Increase in power reducesuniformity

Plotting by EXCEL using CSV export from eArtius

Dimension 1Dimension 2

eartius hmge algorithm applied to optimization tasks with 10,000 design variables

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