ROBUST DESIGN OF A MECHANICAL LATCH

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LA-UR-15-21296 Page 1 of 20 ROBUST DESIGN OF A MECHANICAL LATCH François Hemez Technical

of California San Diego (UCSD) hemez@lanl.

1 LA-UR Page 1 of 20 ROBUST DESIGN OF A MECHANICAL LATCH François Hemez Technical Staff Member, XTD-Division, Los Alamos National Laboratory Adjunct Professor, University of California San Diego (UCSD) hemez@lanl.gov This presentation is approved for unlimited, public release. Date: Reference: February LA-UR Number of pages: 20

2 Abstract LA-UR Page 2 of 20 Advances in computational sciences in the past three decades, such as those embodied by the finite element method, have made it possible to perform design and analysis using numerical simulations. While they offer undeniable benefits for rapid prototyping, and can shorten the design-test-optimize cycle, numerical simulations also introduce assumptions and various sources of uncertainty. Examples are modeling assumptions to represent a nonlinear material behavior, energy dissipation mechanisms, and environmental conditions, in addition to numerical effects such as truncation error, mesh adaptation, and artificial dissipation. Given these sources of uncertainty, what is the best way to support a design decision using numerical simulations? We propose that an effective simulation-based design hinges on the ability to establish the robustness of its performance to assumptions and sources of uncertainty. Robustness means that the performance requirement is guaranteed even if reality deviates from assumptions of the simulation model. The theory of info-gap for decision-making under severe uncertainty is applied to assess the robustness of a design. Instead of being theoretical, the discussion is articulated around the steps followed to search for a robust design of mechanical latch for a consumer electronics product. (Approved for public release, LA-UR , Unclassified.)

3 How to guarantee the performance of a design in spite of having to rely on an imperfect simulation model? Numerical predictions are used to guide decision-making. LA-UR Page 3 of 20 Models and simulations are fraught with (somewhat arbitrary) choices and assumptions. Reliability Analysis A robust design is one which guarantees the performance even if these assumptions are erroneous. Confidence comes from establishing that performance is as immune as possible to these (erroneous) assumptions. Robustness Analysis

4 Outline LA-UR Page 4 of 20 Formulation of the latch design problem Immunity of performance to modeling assumptions

This application is the robust design of a latch mechanism for a

The objective is to design the latch to ensure a robust performance.

- Plasticity, damage, and failure mechanics not considered.

5 This application is the robust design of a latch mechanism for a consumer electronics product. The objective is to design the latch to ensure a robust performance. - Contact condition simplified. - Severe impact loads not considered. - Plasticity, damage, and failure mechanics not considered. The simulation is a dynamic analysis of stresses where requirement compliance is defined as σ Max σ Yield. LA-UR Page 5 of 20

0 Applied Force or Displacement L C = 2.0 Abaqus TM Geometry 2.0 0.50 0.55 0.95 L T = 0.

6 0.7 The geometry of the latch is read from Abaqus TM and simplified for dynamic stress analysis. 4.0 (Width) Fixed Boundary Condition LA-UR Page 6 of 20 W C = Applied Force or Displacement L C = 2.0 Abaqus TM Geometry L T = 0.50 L M = 0.50 L B = 0.90 D H = 0.4 Units of length = mm. 0.5 D C = 0.8 Actual Dimensions Geometry Simplification

7 LA-UR Page 7 of 20 Always know the solution before you start working on the problem. H. Rogers, LANL Fellow

What can be learned from (static only) back-of-the-envelope calculations? LA-UR-15-21296 Page 8 of 20 The bending of the latch as it opens/closes is simplified using beam bending theory.

8 What can be learned from (static only) back-of-the-envelope calculations? LA-UR Page 8 of 20 The bending of the latch as it opens/closes is simplified using beam bending theory. F Tip x U(x) A tip displacement of U Tip = 0.20 mm, gives a (static) peak stress of σ Max = MPa. (#) Shear Correction Factor This exceeds the yield stress of σ Yield = 55 MPa. Notation: d = height, w = width (out-of-plane), L = length, E = elasticity modulus, G = shear modulus, v = Poisson s ratio, A = cross-sectional area, I xx = moment of inertia. (#) Assuming generic polycarbonate material properties.

9 The dynamic finite element analysis predicts the deflection and peak stress, σ Max, that results from displacements imposed on the contact surface. LA-UR Page 9 of 20 Direction of Motion U Contact Time U Contact Specify Displacement U Contact on the Contact Surface

10 How much numerical error (due to truncation) is generated by running the calculation at mesh size Δx? Run the calculations with multiple meshes. Verify the convergence rate, p, of the solver. LA-UR Page 10 of 20 Estimate the bounds of truncation error, U(Δx). Δx = 0.40 mm Δx = 0.20 mm Select an appropriate mesh size, Δx. Δx = 0.10 mm Δx = 0.05 mm

11 The size of Δx = 0.20 mm is selected after examining the trade-off between time-to-solution and truncation error. LA-UR Page 11 of 20 Δx = 0.40 mm 3.8 min. Δx = 0.20 mm Δx = 0.1 mm 54.5 min. Extrapolation, σ Max * = MPa 0.64 MPa 1.63 MPa 4.75 MPa Hold-out Run at Δx = 0.05 mm

12 Outline LA-UR Page 12 of 20 Formulation of the latch design problem Immunity of performance to modeling assumptions

Modulus of elasticity (#) 2.00 GPa 2.00-3.00 GPa G Shear modulus (#) 0.73 GPa 0.71-1.11 GPa ν Poisson s ratio (#) 0.37 0.35-0.

13 What are the main sources of variability and modeling assumptions? LA-UR Page 13 of 20 Unknown Dynamic Overshoot Material Property Variations Unknown Theory Variable Description Nominal Range E Modulus of elasticity (#) 2.00 GPa GPa G Shear modulus (#) 0.73 GPa GPa ν Poisson s ratio (#) ρ Mass density (#) 1,200 kg/m 3 1,200-1,250 kg/m 3 U Contact Applied contact displacement 0.40 mm mm F OS Dynamic load overshoot factor to-1.50 (#) Material properties for generic polycarbonate plastics.

14 What is the effect of these sources of uncertainty on the performance? Simultaneously varying (E; G; v; ρ; U Contact ; F OS ) gives several predictions ( and dots) that fail the peak stress requirement of σ Max < 55 MPa. LA-UR Page 14 of 20 Probability E Min Sampling E Max Probabilities do not necessarily yield confidence. Population of Predictions Five-factor, full-factorial design with 576 runs shown. Reliability Analysis

15 A robust design is one that meets the performance requirement while being immune to the modeling assumptions. LA-UR Page 15 of 20 Confidence comes from demonstrating robustness of the design performance. An intuitive, communication tool is the robustness function. Level-of-uncertainty, α Requirement Compliant Failure Domain Is the performance requirement guaranteed even if the modeling assumptions deviate from their nominal settings? Performance, R Reference:

16 LA-UR Page 16 of 20 The robustness function is explored by searching for the worst-case performance over increasingly larger uncertainty spaces. θ 1 Level-ofuncertainty, α θ 1 θ 1 (0) α 3 Safe Domain Failure Domain θ 1 (0) α 1 θ 2 (0) θ 2 α 3 θ 1 (0) θ 1 θ 2 (0) θ 2 α 2 α 1 θ 2 (0) θ 2 Performance, R

17 LA-UR Page 17 of 20 The design fails if the six variables (E; G; v; ρ; U Contact ; F OS ) deviate from their nominal settings by more than 40%. Best σ Max Worst σ Max α = 40% Truncation Error Added Failure σ Max 55 MPa

LA-UR-15-21296 Page 18 of 20 Iterations are performed by hand, to search for a design variant that performs better

Achieving a lower peak stress, σ Max, with the nominal modeling assumptions, is better.

18 LA-UR Page 18 of 20 Iterations are performed by hand, to search for a design variant that performs better initially while offering more robustness. Various combinations of design variables (L C ; W C ) are simulated. Achieving a lower peak stress, σ Max, with the nominal modeling assumptions, is better. Tolerating more deviation of the modeling assumptions, while remaining requirement-compliant (σ Max < 55 MPa), is better mm Nominal Geometry 0.48 mm Variant Geometry (+20%)

19 Performance of the variant geometry is initially better. Preference Reversal! Failure σ Max 55 MPa LA-UR Page 19 of 20 The robustness functions help decide which one of the nominal or variant geometry is a better candidate design. Robustness of the variant geometry deteriorates faster than robustness of the nominal geometry. Preference reversal occurs if modeling assumptions deviate from the nominal settings by more than 60% (α 60%).

20 Concluding remarks LA-UR Page 20 of 20 An application of robust design is presented for a mechanical latch of consumer electronics product. The keystone concept is the robustness function, an intuitive tool to communicate the tradeoffs of decision-making. Prediction Bounds The robustness function and its nemesis, the opportuneness function, define rigorous bounds of prediction uncertainty. Confidence results from showing that the performance requirement is met, even in the presence of potentially erroneous modeling assumptions.

However, reliability analysis is not limited to calculation of the probability of failure.

However, reliability analysis is not limited to calculation of the probability of failure. Probabilistic Analysis probabilistic analysis methods, including the first and second-order reliability methods, Monte Carlo simulation, Importance sampling, Latin Hypercube sampling, and stochastic expansions