Multidisciplinary Optimization and Reliability Analysis of Undersea Weapons

1. Multidisciplinary Optimization and Reliability Analysis of Undersea Weapons Computational Design and Optimization Center Wright State University

2. Acoustic Response Shape Optimization Reliability- based MDO of undersea weapons System Reliability UNDEX Response Composite Design Project Overview

3. Supercavitating Torpedo Design Cavity sizing optimization Torpedo body optimization Uncertainty-based design RBDO of the cavitator Lightweight Torpedo Design Multi-Disciplinary System Reliability Quantification Composite lightweight torpedo modeling RBDO of a composite lightweight torpedo MDO for Underwater Explosion Survivability Modeling and fluid-structure interaction MDO of lightweight torpedo subject to UNDEX MDO for Acoustic Signature Minimization Noise source modeling MDO of lightweight torpedo for improved acoustic behavior Accomplished Tasks

4. P Cavitation Number V d Pcavity dm Drag Properties Cavity Size Lc Optimal Cavity Sizing Problem Cavity Shape • Maximize cavity volume • Minimize cavitator drag • Nominal values set for s = 0.01 and d = 5 cm • Constrained to fit in torpedo tube • 5.8 m long x 54 cm max diameter • Constrained to operate in “stable” cavity conditions • Re-entrant jet closing regime

5. Torpedo Shape Solutions: Cavity Sizing Problem • Maximize internal torpedo volume • Minimize structural weight • Use reduced order models to explore pareto space: for W=0.01, 1, 1.5, 3 solution is the same Cavitator Diameter Drag Cavity Length Maximum volume solution also minimizes the structural mass

6. Torpedo Shape in Cavity Variable Skin Thickness Design • Material – Titanium/Aluminum alloy • Von Mises Stress < 900 MPa • Thrust Buckling Factor > 1.25 • Collapse Buckling Factor > 1.25 • Avoid natural frequencies of 30 Hz (control actuator frequency) • Loading Conditions • 8000 N thrust/drag force • 600 m depth pressure force • 600 m ~ max depth of Russian K-141 Kursk submarine • Must fit inside torpedo tube and cavity Russian Squall

7. Represent correlated random variables with KL expansion l Create scree plot to determine critical eigenvalues and reduced KL expansion i Use full KL expansion for input into deterministic solver (FEA) Create polynomial chaos expansion Output Accurate PCE of random input and output Efficient Probabilistic Reliability Analysis • Large scale problems have a high computational cost • Efficient methods for cost reduction are developed

8. Application to Supercavitating Torpedo • Given a covariance function of how modulus of elasticity varies throughout the torpedo body construct Karhunen-Loeverepresentation Realization of Random Field Spectral Decomposition Covariance Function Analysis of Covariance Function Covariance function eigenvalue eigenfunction Karhunen-Loeve Expansion Standard normal distributed variable

9. Reliability Results Simulation Reduction from 10,000 to 700 over Monte Carlo Simulation PDF Scree Plot CDF Maximum von Mises Stress (4th PCE) mean = 32303.34 psi, stdv = 56.87 psi 640 Random Variables  5 Random Variables (98.67%)

10. Cavitator Mount Cavitator Cavitator Structural Design • Last year we presented cavitator shape optimization • Now we have added the structure into the MDO problem • Such that: • Only realistic shapes are considered • Maximum stress constraints are met • The structure does not buckle Design Variables: Shape and Variable Skin Thickness

11. Deterministic Design of the Cavitator • Evidence Theory – Handles Epistemic and Aleatory Uncertainty • Characterized By • Belief • Plausibility • Where m(A) is the Basic Belief Assignment (BBA) for the occurrence of event A Deterministic optimal points are associated with high failure probability

12. Reliable Design of the Cavitator • Probabilistic gradients are difficult to determine for Evidence Theory because Belief and Plausibility are not continuous • Finite difference will not work! • Solutions • Introduce new continuous measure between belief and plausibility (Dr. Bae 2004) • Estimate gradient based on limit state variation from failure

13. Reliability Analysis Reliability-Based MDO of Lightweight Torpedo MDO of Lightweight Torpedo UNDEX Response RBDO of Composite Model Acoustic Analysis

14. RBDO of Composite Torpedo • Last time: Deterministic Optimization with Reliability estimate at deterministic optimum • This year RBDO has been completed using Fast Fourier Transform technique to improve efficiency of failure probability integration • This capability allows the efficient handling of multiple limit states such as: stress maximums, buckling limitations, frequency constraints, etc …

15. Joint Failure Region System Reliability Estimation • Accurate estimation of system reliability by modeling the joint failure surface • Reduction in computational cost by using high fidelity surrogate models Stress Frequency … Buckling Using TANA2 Functions to construct Approximations Sample Points on Failure Region Combine the sub domains to obtain Pfof the system Construct RSM(s) R2 >= 0.99 yes ... no Divide Design Space Integrate using high fidelity models and Fast Fourier Transforms

16. Reliability-Based Optimization Problem • Objective • Minimize the weight of the composite torpedo shell • Design Variables • Thicknesses of the laminates and the honeycomb core • Uncertain Variables • Thicknesses of the laminates and the honeycomb core • Material properties of the laminates • Constraint on System Reliability 

17. UNDEX Response Reliability-Based MDO of Lightweight Torpedo MDO of Lightweight Torpedo RBDO of Composite Model UNDEX Response Acoustic Analysis

18. Metallic Lightweight Torpedo Optimal Design for UNDEX Response Composite Lightweight Torpedo UNDEX response for stiffened torpedo Main Goal Composite modeling & UNDEX response Stiffener configuration relative to the explosion MDO of Lightweight Torpedo Subject to UNDEX Critical distance for composite model Sensitivity to explosive locations Sensitivity to stiffener dimensions

19. Charge: HBX-1 explosive Charge Weight: 70 Kg Standoff distance: 35 m (Critical distance) Investigate effect of ring and longitudinal stiffeners on the UNDEX response of the torpedo model Minimize critical stress for different combinations of stiffeners LH RH RW Radial LW Shell thickness Longitudinal Stiffened Lightweight Torpedo Maximum von Mises stress w.r.t. number of longitudinal and ring stiffeners

20. 2 1 3 Stand off Point Variability of HBX-1 Explosive • Three different positions of the explosive (source point) are considered • As source point changes, the standoff point changes Position 2: Shock wave hitting nose first from side-on Position 1: Shock wave hitting center first from side-on Position 3: Shock wave hitting tail first from side-on

21. Case 3 Case 2 90o 45o Case 1 Sensitivity to Stiffener Configuration and Torpedo Orientation to Explosion • Change in response due to change in cross-section sizes of stiffeners • Width and height of the stiffeners are considered with 3 % variation Pressure waves Explosion Failed combinations

22. Objective Minimize torpedo weight Design Constraints Max. von Mises stress <= 413 MPa, yield stress of aluminum Fundamental natural frequency >= 22 Hz Design Variables Shell thickness Width and breadth of ring and longitudinal stiffeners Constant Parameters Standoff distance Source point Stiffener position Optimization Results Lt. Wt. Torpedo Optimization for UNDEX

23. Objective Minimize torpedo weight Design Constraints Failure criterion <= 0.9 Fundamental natural frequency >= 22 Hz Design Variables (5) Thickness of each layer Composite Torpedo Optimization for UNDEX

24. Acoustic Analysis Reliability-Based MDO of Lightweight Torpedo MDO of Lightweight Torpedo UNDEX Response RBDO of Composite Model Acoustic Analysis

25. Fluid Experimental Data FEA Modeling OptimizationBased Source Modeling Structure Modal Analysis Noise Source Model Acoustic Analysis Acoustic Optimization Mass Minimize: Mass Sound Modified Structure Subject to : Constraints Frequency Sound NO Optimal Design ? YES Optimum Structure Acoustic Signature Reduction • Acoustic Analysis involves Fluid-Structure Interaction • Package-NASTRAN Pareto Optimization Curve

26. 116 dB 113 dB 110 dB Fans, Converters, Pumps etc. Exhaust-wake interaction 114 dB 109 dB Propulsor Gear Noise 106 dB 110 dB 110 dB 103 dB 110 dB Engine assembly Noise Boundary Layer Guidance & Control Acoustic Sources and Experimental Data Important Noise Sources for Lightweight Torpedo Experimental Noise levels for MK-40 torpedo* • Gear Noise – Expt. data available • Optimization based noise source modeling approach • Sphericalmeasurementconcept *From Naval Ordnance Report # 6569

27. Computational Matching of Experimental Data • Minimize, Squared Error = Ai= Noise value from FE simulation at location i Bi= Noise value from experiment at location I • Design Variables: Source Strengths Source Frequency • 4 % deviation in Experimental and NASTRAN noise values • Need two sources with strengths 0.9 watts and 0.15 watts and frequency 77.85 Hz

28. Acoustic Optimization Formulation and Results Source • Optimization Formulation: • Objective: • Minimize : Structural Mass • Constraints: • Sound < = 74 dB • Frequency >= 23 Hz Design Variable: • Width of ring stiffeners • Thickness of ring stiffeners • Width of long. stiffeners • Thickness of long. stiffeners • Shell thickness Transmission Warhead / Guidance and Control Source Location Pareto Optimization Trade-off Curve

29. Summary • Optimal torpedo shape in a cavity • Reliability analysis of a Supercavitating torpedo • System reliability of a lightweight composite torpedo • Optimal torpedo design for UNDEX • Acoustic signature reduction

30. Current and Future Work • Topology optimization for cavitator shape, compliant mechanism design for variable shape cavitator • Supercavity ventilation system design: shape, location, number • Joint analysis and redesign in order to incorporate metallic to composite transition • Requires identifying optimum location for the joint based on the shell thickness distribution

31. Current and Future Work Continued • Composite lay up design based on manufacturing requirements for higher torpedo performance

32. Thank You Computational Design and Optimization Center Wright State University

33. Optimized Cavitator Shape Cavitator animation – Topology animation

34. Extra slides for UNDEX

35. Contribution of a slave node to the coupling term in acoustic equation Contribution of a slave node to the coupling term in structural equation Fluid mastersurface Solid slave surface nodes Fluid-Structure interaction in ABAQUS Surface based fluid-structure interaction, master and slave surface concept is used Fluid as master surface and structure as slave surface Shape functions : first order, 4 node linear tetrahedron element, g,h,r – local coordinates

36. Deriving discretized finite element equations Equations 1 & 2 define variational problem for coupled fields and p Interpolation functions in structure N is no. of displacement DOF N,M displacement DOF Interpolation functions in fluid P is no. of pressure nodes P,Q pressure DOF Substituting interpolation functions in eqn. 1 &2 (a) (b) (Coupled fluid-structure equations)

37. Finite element equations Contd…. • Eqn’s (a) & (b) couple total pressure in the fluid to the displacements in the structure • Matrix is defined over all the interacting fluid and solid surfaces , in eqn (a) & (b) Known from incident Pressure wave equations Unknown calculated from above eqn The above 2 eqns are solved together with as unknown variable

38. Density of fluid Bulk modulus of fluid Radiation Boundary Condition • Radiation Boundary Condition • Pressure release boundary condition, p=0 The boundary traction term is given by, Where, and

39. (for spherical waves) (for spherical waves) = standoff point = source point = spatial point on structure = incident pressure = pressure due to spatial variation = wave speed in fluid Pressure distribution on structure Incident pressure wave eqns.

40. Longitudinal Radial Torpedo parameters. Finite Element Modeling of Fluid and Structure Fluid 2.42 m Stiffeners bars 4 m Mass elements Torpedo • Structure is modeled with plate and shell elements • Solid elements are used for fluid with fluid properties Ring stiffeners

41. Metallic Lightweight Torpedo Design for UNDEX Response Composite Lightweight Torpedo UNDEX response for stiffened torpedo Main Goal Stiffener configuration relative to the explosion Composite modeling & UNDEX response MDO of Lightweight Torpedo Subject to UNDEX Critical distance for composite model Sensitivity to explosive locations Sensitivity to stiffener dimensions

42. Fluid-Structure Interaction Response highly non-linear for low number of stiffeners Stress Response due to Proximity Charge for Different Torpedo Designs Maximum von Mises stress w.r.t. number of longitudinal and ring stiffeners

43. Objective Minimize torpedo weight Design Constraints Failure criterion <= 0.9 Fundamental natural frequency >= 22 Hz Design Variables (5) Thickness of each layer Constant Parameters Standoff distance (20 m) Source point Orientation angle Ply sequence Composite Torpedo Optimization for UNDEX Optimization Results

44. Topologically Optimized Cavitator Shape Future work animation topology optimization