Computational Design Optimization Using Distributed Grid Resources

Computational Design Optimization Using

Distributed Grid Resources

Petri Kere1 and Juha Lento

CSC – the Finnish IT Center for Science

http://www.csc.fi

1_{Tampere University of Technology}

Insitute of Applied Mechanics and Optimization

Acknowledgement: The work of the first author is funded by the Academy of Finland postdoctoral researcher appropriation

Introduction

Parameter studies, genetic algorithms and Monte Carlo type

calculations are examples of pleasantly parallel computational

tasks.

Pleasantly parallel computational tasks can be effectively calculated

in computer clusters or Grids.

Overview:

• Problem formulation

• Modelling

• Optimization task

Problem Statement

Using a given set of layer orientations optimize the laminate lay-up

for maximum failure margin and structural stability with minimum number of layers and

Numerical Example

AS4/3501-6 t_ply = 0.134 mm E1 = 139.3 GPa E2 = 11.1 GPa G12 = 6.0 GPa ν12 = 0.3 G23 = 3.964 GPa ν23 = 0.4 Xt = 1950 MPa Yt = 48 MPa Xc = 1480 MPa Yc = 200 MPa S12 = 79 MPa ρ = 1580 kg/m3 L = 1.0 m, b = w = 0.1 m Fy = 2.4 kN, Fz = 1.6 kN

Tsai-Hill failure criterion

z y z F y F

Computational Modelling Tool

Elmer is a software package for solving Partial Differential

Equations (PDEs) [1].

It has been developed at CSC in collaboration with Finnish universities, research laboratories, and industry.

Elmer includes physical models of fluid dynamics, structural

mechanics, electromagnetics, vibroacoustics, and heat transfer, for instance. These are described by PDEs which Elmer solves by

FEM.

Recently the software capabilities have been extended to cover laminated structures composed of orthotropic Fiber-Reinforced Polymer (FRP) composite layers.

Composite Modelling

The plate bending problem has been formulated for a thin or moderately thick laminated composite plate.

Reissner-Mindlin-Von K´arm´an type plate model [2] has been

implemented including analysis capabilities as follows.

• Laminate load response including stress-strain analysis of layers

• Laminate failure prediction including First Ply Failure (FPF)

analysis with generalized failure criterion

• Linear stability

Elmer

ElmerFront

Ply mechanical properties: .sif Mesh Mesh:

Editing ElmerGrid, Ansys,

Abaqus, Fidap, FemLab, FieldView (GridGen)

Pre-processing: Netgen

Laminate structure: Objective and ESAComp, MSC.Laminate ElmerSolver constraint function

Modeler values

.ep .dat

ElmerPost Curve plotting: Matlab

Optimizer

Post-processing: ESAComp

Structural Optimization Problem

The structural weight minimization problem is formulated in discrete form

S = {~y | ~y = arg min ~ x∈S¯

N(~x)} (1)

where ~x = (x₁, x₂, . . . , x_N) is the layer orientation identity design

variable vector defining the laminate lay-up configuration and

¯

S the feasible set of lay-up configurations

¯

S = {~x | g˜(~x) = 1 − RF˜ (~x) ≤ 0} (2) A set of four allowable layer orientations is defined as

Optimization for Strength and Stability

The multi-criteria optimization problem is formulated as

max ~ x∈S RF(~x) λ(~x) = max ~ x∈S ~z(~x) (3)

The image of the feasible set in the criterion space is

Λ = {~z ∈ _R2 | ~z = ~z(~x), ~x ∈ S}.

A solution ~x∗ is Pareto optimal for the problem (3) if and only if

there exists no ~x ∈ S such that z_i(~x) ≥ z_i(~x∗) for all i = 1, 2 and

z_i(~x) > z_i(~x∗) for at least one i = 1, 2.

The points ~z ∗ = ~z(~x∗) ∈ Λ in the criterion space are called the

Achievement Function Approach

The achievement problem by Wierzbicki [3] to be solved is

min ~ z∈Λ sz¯(~z) (4) s_z¯(~z) = max i=1,2{ρi(¯zi − zi)} + c 2 X i=1 [w_i(max{1 − z_i, 0})] (5)

where z¯_i ∈ _R, i = 1, 2 are arbitrary reference objectives

characterizing aspiration levels for the given criterion vector.

At the jth cycle ρ_i = w_i/(max z_i(j) − min z_i(j)) and z¯_i = max z_i(j)

with some fixed weighting vector w >~ 0 and c > 0.

Search Space

The algorithm begins with the generation of initial population, i.e., permutations of Symmetric Even (SE) laminate lay-ups.

At each iteration cycle design alternatives are generated as follows.

~ x(j+1) ∈ X(j+1) =                  (~x(j), 3, 4), (~x(j), 4, 3), (3, 4, ~x(j)), (4, 3, ~x(j)), (~x(j), 1, 1), (1, ~x(j), 1), (1, 1, ~x(j)), (~x(j), 1, 2), (1, ~x(j), 2), (1, 2, ~x(j)), (~x(j), 2, 1), (2, ~x(j), 1), (2, 1, ~x(j)), (~x(j), 2, 2), (2, ~x(j), 2), (2, 2, ~x(j))                  (6)

Here ~x(j) denotes half of the selected laminate, e.g., [0/90/ ± θ]_SE

Datastructures and Parallelization

u_{w~} i ? ' & $ % u θ, ~x, ~z ? {s(~z)} u_{θ, ~x, {w~}} j ? ' & $ % u u ? ' & $ % u B B B B B Nu ? ' & $ % u ? u ? ' & $ % u ? u ? ' & $ % }    parents chidren ~x ∈ IN 0th Generation }    parents ~x ∈ IN chidren ~x ∈ IN+2 1st Generation }    parents ~x ∈ IN+2 chidren ~x ∈ IN+4 2nd Generation

Generation’s “Flow Chart”

1. Select parents from previous Generation using s_z¯(~z)

2. Compute (nonlinear) constraint function g˜(~x) and check

convergence

3. Generate children from parent’s design point (θ, ~x)

Computation of Objective Functions

optimizer {θ_i, ~x_i} −→ ←− {~z(θ_i, ~x_i)} filter jobf iles −→ ←− resultf iles executor

• Optimizer implements the search algorithm

• Filter transforms optimizer’s requests to Executor’s job descrip-tion and parses job results

• Executor runs the jobs in Grid (or in a cluster)

• Optimizer and executor are persistent, i.e. save their state and automatically continue after restart

Submitting a Single Job to Grid Manually

• User creates the job specification myjob.xrsl

• User “logs in” to the Grid: grid-proxy-init

• User submits the job to Grid: ngsub -f myjob.xrsl

• User monitors the job status: ngstat JOB NAME or ngcat

JOB NAME

• User downloads the results: ngget JOB NAME

These tasks are to be automated by filter and executor modules for

Job Description Example

&(rsl_substitution = (TEMPLATE "/home/jlento/PYTHON/NGMCO/TEMPLATE" )) (jobName = "elmer-TEMPLATE" )

(executable = "wrapper.sh" )

(inputfiles = (wrapper.sh $(TEMPLATE)/wrapper.sh ) (laminate.opt $(TEMPLATE)/laminate.opt ) (Shell.sif $(TEMPLATE)/Shell.sif ) (unchangebles.tar.gz $(TEMPLATE)/unchangebles.tar.gz ) ) (executables = lib/ld-linux.so.3 lib/elements.def ElmerSolver lib/ld-linux.so.3 lib/libc.so.6 lib/libcxa.so.3 lib/libg2c.so.0 lib/libSolver.so Shell) (outputfiles = (layup.opt "" )) (stdout = stdout.txt ) (stderr = stderr.txt ) (cache = yes ) (disk = 150 ) (cpuTime = 3 ) (|(architecture = i686 ) (architecture = i386 ))

Results

w₁ w₂ Lay-up RF˜ RF λ 0.13 0.87 [90/ ± 55/7(0)]_SE 1.00274 1.41723 0.42676 0.15 0.85 [90/ ∓ 50/7(0)]_SE 1.10132 1.43038 0.40723 0.20 0.80 [90/ ∓ 45/7(0)]_SE 1.14626 1.44884 0.38928 0.25 0.75 [90/ ± 40/7(0)]_SE 1.21515 1.47442 0.37288 c = 9.0 · 10−7 z y x z F y F

Summary

NorduGrid is

• Production class grid facility

• Running since summer 2002

• Open source

• Simple to use

• In active developement

References

[1] ELMER web site at www.csc.fi/elmer.

[2] P. Kere and M. Lyly. Nonlinear Analysis and Design of Laminated

Composites using Reissner-Mindlin-Von K´arm´an Type Plate Model. To appear in Proc. 4th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyv¨askyl¨a, Finland, 24-28 July 2004. P. Neittaanm¨aki et. al., eds.

[3] A. P. Wierzbicki. The Use of Reference Objectives in Multiobjective Optimization. In G. Fandel and T. Gal, eds., Multiple Criteria Decision Making Theory and Applications, Lecture Notes in Economics and

Mathematical Systems 177, 468–486, Berlin: Springer-Verlag, 1980. [4] NorduGrid web site at www.nordugrid.org.