Sequential Optimization and Reliability Assessment Method for Metal Forming Processes Atul Sahai* and Uwe Schramm Altair Engineering, Inc. Troy, Michigan
Thaweepat Buranathiti, Wei Chen and Jian Cao Department of Mechanical Engineering Northwestern University Evanston, Illinois
Cedric Z. Xia Ford Research Laboratory Ford Motor Company Dearborn, Michigan Abstract. Uncertainty is inevitable in any design process. The uncertainty could be due to the variations in geometry of the part, material properties or due to the lack of knowledge about the phenomena being modeled itself. Deterministic design optimization does not take uncertainty into account and worst case scenario assumptions lead to vastly over conservative design. Probabilistic design, such as reliability-based design and robust design, offers tools for making robust and reliable decisions under the presence of uncertainty in the design process. Probabilistic design optimization often involves double-loop procedure for optimization and iterative probabilistic assessment. This results in high computational demand. The high computational demand can be reduced by replacing computationally intensive simulation models with less costly surrogate models and by employing Sequential Optimization and reliability assessment (SORA) method. The SORA method uses a single-loop strategy with a series of cycles of deterministic optimization and reliability assessment. The deterministic optimization and reliability assessment is decoupled in each cycle. This leads to quick improvement of design from one cycle to other and increase in computational efficiency. This paper demonstrates the effectiveness of Sequential Optimization and Reliability Assessment (SORA) method when applied to designing a sheet metal flanging process. Surrogate models are used as less costly approximations to the computationally expensive Finite Element simulations.
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Corresponding Author: 1820 E. Big Beaver Road, Troy, Michigan 48083 Phone: 248-614-2400 ext. 261, Fax: 248-614-2411 email :
[email protected]
CP712, Materials Processing and Design: Modeling, Simulation and Applications, NUMIFORM 2004, edited by S. Ghosh, J. C. Castro, and J. K. Lee © 2004 American Institute of Physics 0-7354-0188-8/04/$22.00
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1. INTRODUCTION The product design and development industry strives to shorten the time from concept to market. Mathematical models and simulation tools are used quite widely to design and compare a range of alternatives rapidly and inexpensively. Not only are these mathematical models being used to predict system behavior accurately but optimization techniques are also being used quite extensively to optimize the system behavior. The use of optimization techniques leads to a cheaper product with optimal system behavior. However, most of the current optimization applications are deterministic in nature i.e. uncertainties in analysis and design are ignored. The exclusion of uncertainties from the design and optimization process could lead to designs that are neither robust nor reliable. The inclusion of uncertainty in the design and optimization cycle leads to better understanding of the impact of uncertainty associated with system input on the system output. This understanding can then be applied for managing such uncertainties. Such designs that incorporate uncertainty are more popularly referred to as probabilistic design. The most challenging issue of a probabilistic design is large computational expense associated with design simulations, evaluation of quality characteristics such as robustness and reliability. Before proceeding any further, we will explain the concept of robustness and reliability. "Robustness" is achieved by making the system performance insensitive to the uncertainties. Simultaneously "optimizing the mean performance" and "minimizing the performance variance" leads to a robust design. "Reliability" is the probability of constraint satisfaction.
does reduce the computational effort, but the accuracy of such techniques is subject to question. Du and Chen recently developed the Sequential Optimization and Reliability Assessment (SORA) method [3], which is a single loop method that converts probabilistic design problem into an equivalent deterministic optimization problem followed by the inverse reliability assessment for checking the constraint feasibility. In this study we demonstrate the efficiency of the SORA method to a metal forming problem.
2. PROBABILISTIC OPTIMIZATION 2.1 Probabilistic Optimization Model Both design variables and design parameters can be subject to uncertainty. In a typical probabilistic optimization problem there are three categories of variables, namely, a vector of deterministic variables (d), a vector of random design variables (x ) and a vector of random parameters (p). A probabilistic design optimization model can be represented as follows: Find
µx, d
(1)
Minimize f(d,x,p) Subject to P{g i (d,x,p) ≥ 0} ≥ Ri , i = 1,2,3,…..,m where Pi is the probability of constraint satisfaction and it should not be less than the reliability level Ri .
2.2 The Most Probable Point
Reliability is the key focus of the study presented in this paper. The use of sampling based methods for probabilistic design can be prohibitively expensive. Reliability assessment often involves probability at the tails of probability distribution functions. Such evaluations can demand a lot of computational effort. There are two major categories of probabilistic optimization techniques, i.e., the double loop method and the single loop method. The double loop method involves the evaluation of reliability analysis within the loop of optimization. Each optimization iteration involves multiple reliability analysis calculations for each probabilistic constraint. This requires a lot of computational effort. The single loop method aims to reduce the computational effort required. The reliability analysis loop is confined within the optimization loop. Though the single loop method
The concept of Most Probable Point (MPP) is at the very core of any reliability based optimization strategy. MPP is used to approximate the reliability, expressed as P{g(d,x,p) ≥ 0 }=P{g(d,u) ≥0}=ϕ(β). The first step in the evaluation of MPP point is the transformation of random variables x and random parameter p into a standardized normal space, called U space. The MPP point is defined in the standardized normal space as the minimum distance point on the limit state function from it to the origin. If the limit state function g, is linear in the U space, its reliability is P{g(d,x,p) ≥ 0 }=p{g(d,u) ≥}=ϕ(β)
(2)
Equation (2) is applicable if the degree of nonlinearity of g is not high. When the g is highly
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non-linear, then higher order approximations such as Second Order Reliability Method (SORM) is applied.
2.3 SORA Methodology Du and Chen [3] recently developed the SORA method to improve the computational efficiency of reliability-based optimization. They have improved on the efficiency by decoupling the reliability assessment from optimization. The SORA method transforms the probabilistic optimization model in equation (1) into a set of equivalent deterministic optimization problems, followed by the confirmation of reliability. The deterministic optimization formulation is updated from cycle to cycle based on MPP information obtained from the previous cycle. The three main ideas that were introduced in the SORA method are as follows: (a) The use of R-percentile formulation to evaluate design feasibility only at the desired reliability level (R). (b) The use of equivalent deterministic optimization and single loop strategy to reduce the number of reliability assessments. (c) The use of efficient MPP search algorithms for reliability assessment. The use of R-percentile formulation is based on the fact that more computational effort is needed if the actual reliability is high. The real need for a probabilistic constraint is not to find the actual reliability of the limit state function but to determine whether it is probabilistic feasible. For more details the readers are referred to Du and Chen [1,3] and Du [2].
process when tooling is retracted, the elastic strain energy stored in the material recovers and achieves a new equilibrium. The elastic recovery causes a geometric distortion. The shape discrepancy between the fully loaded and unloaded configurations is referred to as springback. The material properties, part geometry, die design and other process parameters influence the amount of springback. If springback phenomenon can be efficiently modeled, simulated and controlled, it can reduce product development cycle and cost. The schematic of a springback is shown in Figure 1. The FEA simulations were conducted using ABAQUS. We use solid elements (CPE4R) with reduced integration scheme to simulate the straight flanging process. Softed contact is conducted. MPC is used for changing meshing density of the blank between deformed and clamped parts. The objective of this study is to find a combination of sheet metal and tooling configurations that will result in a desired final springback under the condition that the maximum of the absolute values of strain in flanged sheet metal is less than a specified value. The sheet thickness t and gap g is random design variables because of variations in material and manufacturing properties. Their mean values µx are design variables. The die corner radius r is a deterministic design variable. It is not subject to any uncertainty. The Young's modules E and Yield Stress Y are random parameters because of uncertainty due to material processing. This design problem is also subject to one constraint. The maximum of absolute value of strains of the flanged sheet metal should not exceed a specified value. This constraint is a probabilistic constraint, which must meet the required reliability level. 99.99% reliability level is used in this study. To reduce the computational effort of multiple finite element simulations, we have used a second order polynomial as the surrogate model. The reliabilitybased design problem can be formulated as follows.
3. METAL FORMING PROBLEM
Find
µt , µg and R
A sheet metal flanging process is chosen to demonstrate the effectiveness of the probabilistic optimization approach explained in Section 2.
Minimize
[θ ( x, p, d) − Θ]2
Subject to
P( Max ε ij
3.1 Sheet Metal Flanging Process
where
Sheet metal flanging is a very widely used process in the manufacturing industry. Flanging is used to bend an edge of the part to increase the stiffness of the panel. It is also used to create mating surface for assemblies. The end effect of flanging is a phenomenon called ‘springback’. During the flanging
e
∀e
< ε max ) > PR
µt = mean value of sheet thickness µg = mean value of gap r = die corner radius θ = flange angle in unloaded configuration Θ = desired flange angle
ε max = constraint on maximum strain
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Bounds of design variables 0.50 ≤ µt ≤ 1.20 mm 0.20 ≤ µg ≤ 2.00 cm 0.20 ≤ r ≤ 2.00 cm
though desired reliability is achieved, but convergence is not. To achieve convergence three more optimization cycles are done. This brings the theta value closer to the target of 110° and keeps the constraint on maximum strain within the desired reliability range. The number of function evaluations for each cycle is shown in Table 1.
Probabilistic distribution information σt = 0.08 mm σg = 0.05 cm σE = 3.50 GPa σY = 9.25 MPa
Table 1. Number of function evaluations
Optimization cycle 1 Reliability checking Optimization cycle 2 Reliability checking Optimization cycle 3 Reliability checking Optimization cycle 4 Reliability checking Optimization cycle 5 Reliability checking Total
Optimization Target Value = Θ = 110° Constraint Bound =
ε max = 0.075
g Binder
θO
Punch
No. of function evaluations 22 661 31 201 15 56 21 6 21 6 1040
Table 2. Optimization cycles Optimization cycle 1 Die Corner radius (r) Sheet thickness (t) Gap (g) Theta Maximum strain Optimization cycle 2 Die Corner radius (r) Sheet thickness (t) Gap (g) Theta Maximum strain Optimization cycle 3 Die Corner radius (r) Sheet thickness (t) Gap (g) Theta Maximum strain Optimization cycle 4 Die Corner radius (r) Sheet thickness (t) Gap (g) Theta Maximum strain Optimization cycle 5 Die Corner radius (r) Sheet thickness (t) Gap (g) Theta Maximum strain
Die
θ ∆θ
Figure 1. Schematic of a springback The cycle history of probabilistic optimization is shown in Table 2. The optimization takes five probabilistic design cycles to converge. The constraint on maximum strain in optimization cycle 1 is less than the constraint bound, but the reliability of this constraint does not meet the desired reliability of 99.99%. It must be noted that the springback angle achieves a value that is very close to the target value of 110°. The number of function evaluations for optimization cycle 1 are 22 and the reliability checking are 661. Optimization cycle 2 improves upon the reliability of the constraint and brings it within the desired range. This is achieved at a slight deterioration in the objective value. Please note that the theta value moves further away from the target value of 110°. After Optimization cycle 2, even
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1.089806 0.983463 0.200141 109.9997 0.065211 1.424798 0.500000 0.200141 112.3779 0.005571 1.451011 0.500000 0.200141 111.7442 0.014228 1.451246 0.500000 0.200141 111.7442 0.014292 1.451345 0.500000 0.200141 111.7442 0.014319
Efficient Probabilistic Design”, ASME 2002 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2002/DAC-34127, Montreal, Canada, to appear in ASME Journal of Mechanical Design, March 2004.
4. CONCLUSION The computational requirement associated with reliability based optimization using the finite element method for metal forming problems has inhibited the application of probabilistic optimization to such problems. In this study we effectively applied the SORA method to a sheet metal flanging problem as a new probabilistic optimization strategy. More analysis on comparison of the SORA method to other reliability based design optimization methods is under investigation and will be presented at the conference. The application of SORA method also demonstrates the pitfalls of ignoring uncertainty in the system and only performing deterministic optimization.
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Du, X. and Chen, W., 2001, “A Most Probable Point Based Method for Uncertainty Analysis”, Journal of Design and Manufacturing Automation, 4, pp. 47-66.
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Du, X. and Chen, W., 2002,,”Sequential Optimization and Reliability Assessment Method for
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