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Research Papers

Reliability Based Design Optimization Using a Single Constraint Approximation Point

[+] Author and Article Information
Tomas Dersjö

Department of Solid Mechanics, Royal Institute of Technology (KTH), Stockholm SE-100 44, Sweden; Dynamics and Strength Analysis, Truck Chassis Development, Scania CV AB, Södertälje SE-151 87, Swedentdersjo@kth.se

Mårten Olsson

Department of Solid Mechanics, Royal Institute of Technology (KTH), Stockholm SE-100 44, Swedenmart@kth.se

J. Mech. Des 133(3), 031006 (Mar 01, 2011) (9 pages) doi:10.1115/1.4003410 History: Received January 11, 2010; Revised December 23, 2010; Published March 01, 2011; Online March 01, 2011

The computational effort for reliability based design optimization (RBDO) is no longer prohibitive even for detailed studies of mechanical integrity. The sequential approximation RBDO formulation and the use of surrogate models have greatly reduced the amount of computations necessary. In RBDO, the surrogate models need to be most accurate in the proximity of the most probable point. Thus, for multiply constrained problems, such as fatigue design problems, where each finite element (FE)-model node constitutes a constraint, the computational effort may still be considerable if separate experiments are used to fit each constraint surrogate model. This paper presents an RBDO algorithm that uses a single constraint approximation point (CAP) as a starting point for the experiments utilized to establish all surrogate models, thus reducing the computational effort to that of a single constraint problem. Examples of different complexities from solid mechanics applications are used to present the accuracy and versatility of the proposed method. In the studied examples, the ratio of the computational effort (in terms of FE-solver calls) between a conventional method and the single CAP algorithm was approximately equal to the number of constraints and the introduced error was small. Furthermore, the CAP-based RBDO is shown to be capable of handling over 10,000 constraints and even an intermittent remeshing. Also, the benefit of considering other objectives than volume (mass) is shown through a cost optimization of a truck component. In the optimization, fatigue-specific procedures, such as shot peening and machining to reduce surface roughness, are included in the cost as well as in the constraints.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Flowchart for the RBDO algorithm employed in this work

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Figure 2

Illustration of constraints in u-space. The active constraints are shown in solid lines and the regarded constraints are the solid and dashed lines. The dotted constraint is not regarded in the CAP computation. The single CAP, u, found when employing method A is indicated with a dot.

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Figure 3

Tapered cantilever beam with cross-sectional heights Xi,i=1,2,3 and width B loaded at the end with a point force P. The dots symbolize nodes, i.e. points, where constraints are evaluated (j=1–5).

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Figure 4

Truss structure loaded by point force P. The 13 bar radii constitute the design variables Xi,i=1,2,…,13 and the stress in each bar is used to construct the 13 constraints Gj,j=1,2,…,13.

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Figure 5

Original design of the drag link arm. (a) Drag link arm cross sections are indicated. The six numbered cross section geometries can be varied, whereas the outer two cross sections (red) define the boundaries of the optimization domain. (b) Drawing that show the definition of the geometric design variables in a cross section.

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Figure 6

(a) Original design with fatigue crack indicated by white paint. (b) Simulated damage for original design and converged CAP load and materials. (c) Simulated damage for optimized design without shot peening using the converged CAP. (d) Simulated damage for optimized design with shot peening using the converged CAP. The white areas in simulated results are neither included in the fatigue damage computations nor in the optimization. The contour level scale is logarithmic and the red interval covers D=0.1 to D=1. Black signifies D>1.

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Figure 7

The probability of failure pf for the cantilever beam example computed with (a) 105 Monte Carlo samples and (b) FORM (and Nx+1 additional “FE”-calls per constraint) using the converged solutions μi(NK,NL(NK)) obtained with the use of each respective MPP for constraint approximation (legend: γ=1, (NX MPPs)) or a single CAP (legend: γ=1–2, (CAP))

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Figure 8

The probability of failure pf for the truss example computed with (a) 105 Monte Carlo samples and (b) FORM (and NX+1 additional FE-calls per constraint) using the converged solutions μi(NK,NL(NK)) obtained with the use of each respective MPP for constraint approximation (legend: γ=1, (NX MPPs)) or a single CAP (legend: γ=1–2, (CAP))

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Figure 9

The probability of failure pf for the truss example computed with (a) 105 Monte Carlo samples and (b) FORM (and NX+1 additional FE-calls per constraint) using the converged solutions μi(NK,NL(NK)) obtained with the use of each respective MPP for constraint approximation (legend: γ=1, (NX MPPs)) or a single CAP (legend: γ=1–2, (CAP)). The set of regarded constraints varies with design variable.

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Figure 10

(a) Damage distribution for the shot peened converged CAP with positions of active constraints indicated. (b) The probability of failure pf for the optimized shot peened drag link arm computed with FORM (and NX+1 additional ABAQUS and FEMFAT calls per active constraint) using the converged solutions μi(NK,NL(NK)) obtained with a single CAP (γ=1, (CAP)).

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