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Infotag Nichtlineare Optimierung und stochastische Analysen 27. Juni 2003 in Stuttgart Multidisciplinary Optimization using the Multidisciplinary Optimization using the Successive Response Surface Method Successive Response Surface

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Heiner Müllerschön, Nielen Stander hm@dynamore.de, nielen@lstc.com

Introduction

LS-OPT: Application of the Successive Response Surface Method (SRSM) Example: Multidisciplinary Optimization (MDO)

Topics

Multidisciplinary Optimization using the Multidisciplinary Optimization using the Successive Response Surface Method Successive Response Surface Method

Infotag “Nichtlineare Optimierung und stochastische Analysen”

27. Juni 2003 in Stuttgart

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Introduction

What is LS-OPT? LS-OPT is an environment to explore automatically the design space and find an optimum design LS-OPT is a product of LSTC (Livermore Software Technology Corporation) LS-OPT is based on the Successive Response Surface Method (SRSM). Statistical approaches (Robustness Analysis) and genetic algorithms (Discrete Methods) will be implemented in near future LS-OPT provides a graphical user interface (GUI) LS-OPT can be linked to any simulation code, but it is perfect suitable in combination with LS-DYNA

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Why Response Surface Method and not Gradient Based Methods? Highly Nonlinear Problems Local Sensitivities may lead to local optimums Difficulties by the Computation of Numerical Gradients

If the perturbation intervall is too large: loose accuracy
If the perturbation intervall is too small: find spurious gradients

LS-OPT: Application of the SRSM

Error Log interval Safe interval Round-off error Bias error

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

SRSM: How does it work? Design surfaces are fitted through points in the design space to form approximate optimization problem

LS-OPT: Application of the SRSM

The idea is to find surfaces with the best predictive capability

The idea is to find surfaces with the best predictive capability

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

LS-OPT: Application of the SRSM

Design Variable 1 Design Variable 2 Design Space Region of Interest Range 2 Baseline Design Range 1 E Experimental Design Points Design Space, Region of Interest & Experimental Design Points

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

LS-OPT: Application of the SRSM

Feasible Experimental Design Design Variable 1 Design Variable 2

Constraint f Constraint g Design Space Center of Region

f Interest

(Baseline Design) Feasible Region Region of Interest Basis Experiments

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

LS-OPT: Application of the SRSM

Successive Approximation Scheme Design Variable 1 Design Variable 2 Design Space Region of Interest

Optimum Start 2 3

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

The Optimization Prozess

DOE Simulation Build response surfaces Optimization

Solution

No

Trial Design

Approximate solution

Converged?

Sensitivity Analysis Trade-Off

Preprocessing

Error Analysis Convergence Design Formulation Model Approximation Model Region

f Interest

(Move Limits)

LS-OPT: Application of the SRSM

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Graphical User Interface

LS-OPT: Application of the SRSM

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Graphical User Interface

LS-OPT: Application of the SRSM

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Graphical User Interface

LS-OPT: Application of the SRSM

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Advantages of the Method

Global Optimization: Response Surface have a tendency to capture globally optimal regions. Local minima caused by noisy response as well as the step-size dilemma for numerical gradients are avoided Parallel Computation: Successive Response Surface scheme allows parallel (independent) computation of experimental points within one iteration Flexible Design Exploration: Design exploration can be changed within the optimization process. Thus, control of the computational time and the quality of the Response Surface is possible Trade-Off Studies: Since the Response Surface is determined, easy examination of varying constraint bounds is possible (not reliable with linear approximations)

LS-OPT: Application of the SRSM

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Fully Integrated Optimization - Crash and NVH

Example: Multidisciplinary Optimization (MDO)

Systems Level Optimizer

Goal: Minimize Mass Crashworthiness and NVH Constraints

Design x(k) Multidisciplinary Analysis x(k)

CRASH Crash Analysis

x(k)

NVH NVH Analysis

Response Surfaces Response Surfaces f(x(k)

CRASH)

f(x(k)

NVH )

Iteration (k) x(k)

CRASH

x(k)

NVH

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Full Vehicle – Crash Performance (LS-DYNA)

Example: Multidisciplinary Optimization (MDO)

Baseline:

30 000 elements

Displacement = 552mm Stage1Pulse = 14.34 g Stage2Pulse = 17.57 g Stage3Pulse = 20.76 g

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Full Vehicle – Crash Performance (LS-DYNA)

Example: Multidisciplinary Optimization (MDO)

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

BIW-Modell - NVH Performance (LS-DYNA)

Example: Multidisciplinary Optimization (MDO)

Baseline:

18 000 elements

Torsional Mode 1 Frequency = 38.7 Hz

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Design Variables (Thickness)

Example: Multidisciplinary Optimization (MDO)

Left and right Apron (1) Inner and

uter rail (2)

Front cradle cross members (1) Left and right cradle rails (1) Shotgun outer and inner (2)

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Design Formulation – FULLY SHARED VARIABLES

Example: Multidisciplinary Optimization (MDO)

Design Objective: Minimize (Mass of components) Design Constraints: Displacement > 551.8mm 37.77Hz < Torsional mode 1 frequency < 39.77Hz Stage1Pulse > 14.34g Stage2Pulse > 17.57g Stage3Pulse > 20.76g Thickness Design Variables Shared: 7 Rails (inner and outer), Shotgun (inner and outer), Aprons, Cradle rails, cross member

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Mode Tracking

Example: Multidisciplinary Optimization (MDO)

During NVH optimization necessary to track mode as mode switching can occur due to design changes Search for maximum scalar (dot) product between eigenvector of base mode and each solved mode:

( ) ( )

     

j j T j

M M φ φ

2 1 2 1

max

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Optimization History: Mass (Objective) – FULLY SHARED VARIABLES

Example: Multidisciplinary Optimization (MDO)

93.5 94 94.5 95 95.5 96 96.5 97 97.5 98 1 2 3 4 5 6 7 8 9

Iteration Mass [lbs]

Reduction:

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Optimization History: Maximum Displacement – FULLY SHARED VARIABLES

Example: Multidisciplinary Optimization (MDO)

544 546 548 550 552 554 556 1 2 3 4 5 6 7 8 9 Iteration Maximum displacement [mm]

Maximum displacement (Full) Lower bound

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Optimization History: Stage Pulses – FULLY SHARED VARIABLES

Example: Multidisciplinary Optimization (MDO)

14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9

Iteration Acceleration [g]

Stage1Pulse (Full) Stage2Pulse (Full) Stage3Pulse (Full) Lower bound: Stage 1 Lower bound: Stage 2 Lower bound: Stage 3

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Optimization History: Torsional Mode Frequency – FULLY SHARED VARIABLES

Example: Multidisciplinary Optimization (MDO)

37.5 38 38.5 39 39.5 40

1 2 3 4 5 6 7 8 9

Iteration Frequency [Hz]

Frequency (Full) Upper bound (Full) Lower bound (Full)

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Variable Screening

Example: Multidisciplinary Optimization (MDO)

Maxim um displacem ent

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 Iteration 10-scale of significance rail_in ner rail_outer cradle_rail apron s sh

n _in ner sh

n _outer cradle_csm br

Frequency

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 Iteration 10-scale of significance rail_in n er rail_ou ter cradle_rail apron s sh

n _in n er sh

n _ou ter cradle_csm br

Goal: Remove of less significant variables

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Variable Screening

Example: Multidisciplinary Optimization (MDO)

Coefficient: variable j Coefficient: variable j

From regression analysis

90%

Methodology: ANOVA (ANalysis Of VAriance)

depends on the variance of the simulation points

Use a 90% confidence level and determine the lower bound

b ∆

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Variable Screening

Example: Multidisciplinary Optimization (MDO)

Variables are ranked according to lower bound If the lower bound < 0, regression coefficient is

insignificant

In a linear approximation, a variable can be removed if its

coefficient is insignificant Significant Significant Insignificant Insignificant

Value which determines Value which determines significance significance

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Design Formulation – PARTIALLY SHARED VARIABLES

Example: Multidisciplinary Optimization (MDO)

Design Objective: Minimize (Mass of Components) Design Constraints: Displacement > 551.8mm 38.27Hz < Torsional Mode 1 frequency < 39.27Hz Stage1Pulse > 14.34g Stage2Pulse > 17.57g Stage3Pulse > 20.76g Crashworthiness Design Variables: 6 Rails (inner and outer), Shotgun (inner and outer), Aprons, Cradle Rails NVH Design Variables: 4 Shotgun (inner and outer), Cradle Rails, Cross Member

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Example: Multidisciplinary Optimization (MDO)

92 93 94 95 96 97 98 99 1 2 3 4 5 6 7 8 9

Iteration Mass [lbs] Mass (Full) Mass (Partial)

Optimization History: Mass (Objective)

Reduction:

4.7%

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Example: Multidisciplinary Optimization (MDO)

Optimization History: Maximum Displacement

544 546 548 550 552 554 556 1 2 3 4 5 6 7 8 9 Iteration Maximum displacement [mm]

Maximum displacement (Full) Maximum displacement (Partial) Lower bound

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Example: Multidisciplinary Optimization (MDO)

Optimization History: Stage Pulses

14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9

Iteration Acceleration [g]

Stage1Pulse (Full) Stage2Pulse (Full) Stage3Pulse (Full) Stage1Pulse (Partial) Stage2Pulse (Partial) Stage3Pulse (Partial) Lower bound: Stage 1 Lower bound: Stage 2 Lower bound: Stage 3

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

37.5 38 38.5 39 39.5 40

1 2 3 4 5 6 7 8 9

Iteration Frequency [Hz]

Frequency (Full) Frequency (Partial) Upper bound (Full) Lower bound (Full) Upper bound (Partial) Lower bound (Partial)

Example: Multidisciplinary Optimization (MDO)

Optimization History: Torsional Frequency

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Example: Multidisciplinary Optimization (MDO)

Run Statistics

Run Statistics – Fully Shared MDO 13 experimental points per iteration per discipline 7 hours per crash simulation 10 minutes per NVH simulation (700MB memory each) 9 iterations to converge 117 crash simulations and 117 NVH simulations Run Statistics – Partially Shared MDO 11 experimental points per iteration for crash 8 experimental points per iteration for NVH 6 iterations for good compromised solution 66 crash simulations and 48 NVH simulations More flexibility in using resources (processors and memory)

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Example: Multidisciplinary Optimization (MDO)

Starting from Ligthest and Heaviest Design

25 35 45 55 65 75 1 2 3 4 5 6 7 8 9 10

Iteration Mass [kg]

Max Design Min Design

430 450 470 490 510 530 550 570 590 610 630 1 2 3 4 5 6 7 8 9 10

Iteration Maximum displacement [mm]

Maximum displacement (Min Design) Maximum displacement (Max Design) Equality constraint 9 14 19 24 29 1 2 3 4 5 6 7 8 9 10 Iteration Acceleration [g] Stage1Pulse (Max Design) Stage2Pulse (Max Design) Stage3Pulse (Max Design) Stage1Pulse (Min Design) Stage2Pulse (Min Design) Stage3Pulse (Min Design) Lower bound: Stage 1 Lower bound: Stage 2 Lower bound: Stage 3

34 35 36 37 38 39 40 41 42

1 2 3 4 5 6 7 8 9 10

Iteration Frequency [Hz]

Frequency (Max Design) Frequency (Min Design) Lower bound Upper bound

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Conclusions MDO-Example

Conclusions / Outlook / Remarks Multidisciplinary feasible optimization of a full vehicle model considering crashworthiness and NVH design criteria is described Almost 5% mass reduction is achieved while maintaining or improving of the design criteria of the baseline design Variable Screening allows the detection of unsignificant design variables The capability of partially or non shared variables for MDO may reduce the computational effort dramatically

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Optimization using the Successive Response Surface Method Optimization using the Successive Response Surface Method

Conclusions MDO-Example

Conclusions / Outlook / Remarks Optimization with current full vehicle crash models (500000- 1000000 Elements) is still very time consuming and requires huge hardware resources Gradients of the linear implizit discipline (NVH) may be used for the calculation of the according Response Surface approximation Discrete Methodologies for sheet thickness optimization A two-stage approach with stochastic and deterministic methods, may be very efficient for crash