A Sequential, Multi-Complexity Topology Optimization Process for Aeroelastic Wing Structure Design

Transcription

1 A Sequential, Multi-Complexity Topology Optimization Process for Aeroelastic Wing Structure Design Bill Crossley, Significant content from graduate student Mark Guiles and input from colleague Terry Weisshaar Research Consortium for Multidisciplinary System Design Workshop 27 July 2011, Ann Arbor, MI

2 Overview Develop a process that can be used as a design tool to guide the selection and sizing of the internal layout of aeroelastic wing structures. (Based on some informal discussions with participants at previous MDO Consortium workshops) Approach: Use commercially available sizing optimization software to guide removal of non-essential elements from an initially over-populated structure Define model in terms of common structural members (i.e. ribs and spars) to simplify interpretation and application of results to wing design problems Results: Reduction of weight and structural complexity are both important for the optimization process Presence of aileron divides structure into two distinct regions Directionality associated with results suggest need for composites 2

3 Outline Purpose Statement and General Strategy Related Work Process and Problem Statement Overview Description of the Process Process Demonstration Discussion Future Work Conclusions 4

4 Purpose Statement and General Strategy Develop a process that can be used as a design tool to guide the selection of the internal layout of wing structures. Strategy: Formulate a process to optimize topologies of wings using wellestablished software for structural sizing optimization Include ability to study aeroelastic and dynamic effects (here, ASTROS used) Define model in terms of preliminary geometric data to allow for use in preliminary design stage (i.e. span, chord, sweep, airfoil section defined) Define model in terms of similar to common wing structural elements (i.e. ribs and spars) to simplify comparison with and application to wing design problems 5

5 Some Related Work Yurkovich (1994): Goal: Study the effect of external geometry and internal structural parameters on minimum wing weight. Approach: Compared minimum weights of several traditional wing structures and determined main variables by Taguchi design of experiments techniques. Results: Minimum wing weight is mainly affected by external geometry, and is independent of number of ribs and spars. Limitation: Could not include skin buckling constraint. Taylor (2000): Goal: Study the use of evolutionary finite element modeling as a guide for preliminary wing design. Approach: Optimize continuous distribution of material to obtain load paths to guide the topology selection process. Results: Minimum weight is independent of number of spars. Secondary measure of merit is needed to drive topology selection. Both studies examined traditional designs. Several other studies have obtained weight savings by allowing unconventional designs. 6

6 Process Overview Begin by discretizing design space into overpopulated ground structure Initial finite element model contains more elements than expected for the final model Provides a large number of potential load paths Structural weight is minimized through optimization of element thicknesses Filter process removes thin, lightly-stressed elements: Based on optimal thickness and stress distributions Creates a new structural layout New topologies are optimized and then filtered so that the structure approaches an optimal layout Variable-complexity approach: Computational cost is regulated through a gradual increase in the number of optimization variables Weight minimization of initial structure is simplified through extensive use of variable linking Number of linked groups is gradually reduced to produce a refined model 7

7 Process Objective Function Goal of the process is twofold: Minimize structural weight Reduce number of elements (reflect manufacturing cost?) Product of weight and complexity factor Favors reduction in number of elements Allows small weight increase if offset by larger complexity decrease Minimize: nw nw 0 0 where: W/W 0 = normalized weight n/n 0 = complexity factor (normalized element count) 8

8 Constraints for Aeroelastic Wing 0-1 i i i i i i y cr 0 1, 1.5 i cr p 2 All stresses must be below maximum allowable values. Skin loads must be below critical buckling levels. Roll rate must be positive for a steady roll load case. 9

9 Objective is reduced through two distinct steps: Filter removes non-essential elements Sizing optimization reduces overall weight by determining element thicknesses Design constraints are enforced directly through the sizing optimization Objective function is preserved as a constraint in the filter process Supported by variable delinking scheme (multi-complexity) School of Aeronautics and Astronautics Two-step Approach 10

10 Optimization Process Overview Implementing two-step approach and variable-complexity modeling requires a larger process Optimization procedure follows seven steps: Definition of the initial model Optimization of initial model thicknesses Management of variable linking for next weight minimization Weight minimization of current topology Checks of improvement and termination criteria Element deletion filter Final result processing Optimization Process Initial Model Initial Weight Minimization Variable Delinking and Weight Minimization Loop Improvement / Termination Checks Filter Process Final Result 11

11 Definition of Initial Overpopulated Model Internal wing structure modeled as a skeletal structure: A grid of node points is distributed across the design space Find the optimal set of connections between grid points Ideally, initial model includes all possible connections, but this is impractical for large, three-dimensional structures For this study, nodes were only connected to closest neighbors Initial model is defined in terms of linked groups: Elements of the same type that are part of a single member form one group Each group is initially defined by one thickness variable; i.e. rod elements along the top of a single spar share the same thickness Linked groups lack the desired resolution and can be delinked as the process progresses An initial set of thicknesses is needed at the start of the process 12

12 Weight Minimization of Intermediate Topologies ASTROS optimization finds minimum weight for each configuration produced by the filter Design constraints are enforced directly in this step of the process This study included three types of constraints: Element stresses within allowable range Skin loads below critical levels for buckling Aileron reversal at or above a target dynamic pressure (using roll rate) ASTROS optimization outputs a new set of thicknesses and stresses, and the minimum weight of the current topology The initial structure may be infeasible, and the initial minimization may cause a weight increase The weight of the first feasible design is W 0 in the process objective function 13

13 Filter Process for Element Removal Motivated by procedure from: Performance-based Optimization of Structures by Qing Quan Liang Filter here removes non-essential elements based on load magnitude i th element is deleted if: s i R s max R: filter threshold (R 0 R R max < 1) s i : pseudo-force (s i = t i σ i ) t i : thickness or area σ i : maximum element principal stress across all load cases Elements are compared to the maximum valued element of the same type, but skin elements are not filtered Filter is part of a process designed to delete a small number of elements during each iteration Filter Process Optimized Design from ASTROS Filter Threshold Initialization Element Deletion via Filter Rule Any Elements Filtered? No Increase R if R < R max (R + ΔR) Yes To Delinking with New Structural Layout 14

14 Model Complexity Control by Variable Delinking Variable delinking gradually increases the number of unique variables in a filtered design Within each element type, i th group delinked if: t i - t min /t min D or t i - t max /t max D t: element thickness or area D: delink tolerance (D 0 < D < 1) Prepares lowest-valued variables for deletion in the next iteration by allowing clear identification of non-essential elements Keeps highest valued groups from dominating the design Delinking and Weight Minimization Filtered Design Delink Tolerance Initialization Delink (if any groups remain) ASTROS Optimization Successful Optimization? No Increase Tolerance Value (D + ΔD) Yes Improvement / Termination Checks 15

15 Variable Management and Improvement Checks Variable management controls input to filter, delinking, and ASTROS Accepted design data is saved Lightly loaded elements filtered Lowest and highest groups delinked New design / topology undergoes ASTROS optimization Process objective value is updated and checked against old value: If the objective decreases, the design is accepted. If the objective increases, the design is rejected. Structure reverts to previous configuration, and the delinking tolerance is increased. Process continues until a design is accepted or the termination criteria are met Accepted Design: Save design vector and objective value. Increment Delink Tolerance Reject Design: Revert to saved design. No Improvement Check: Objective decrease? Yes Accept Design: Save new design vector and objective value. Filtering Routine: x* x filtered New n Delinking Routine: x 0 x x filtered delinked ASTROS Optimization: x 0 x * New W Update Objective Value: n f n W 0 W 0 Next Iteration

16 Termination Criteria The process is terminated if all variable groups have been delinked and if one of the following is true: No further elements can be filtered for any R R max The last sizing optimization produced an objective increase The last sizing optimization could not find a feasible solution Process at the layout with the lowest objective value, but not necessarily lightest structure Identified a need to determine a way to reduce the number of unnecessary iterations by terminating the process soon after the least-weight iteration 17

17 Summary of Investigated Problems Plane truss for validation Recovered near Michell truss Wing structures for process demonstration Applied to wings from three types of aircraft Low aspect ratio, supersonic capable Medium aspect ratio, transonic capable High aspect ratio, subsonic capable Model features Multiple element types (shear elements, rod elements, beam elements) Three load cases Steady-state roll at or above maximum dynamic pressure. Low-speed (corner velocity) pull up at maximum load factor. High-speed pull up at maximum load factor. This presentation, only low aspect ratio, supersonic capable application shown 18

18 Wing Modeling: Unit Cell Structure Wing structure is defined by hexahedral unit cells: Allows development of structure in chord-wise, span-wise and diagonal directions Simplifies structure and reduces memory requirement by connecting only closest neighboring nodes Cells are scaled, tapered and skewed to evenly fill the wing planform, with regions excluded for compatibility with flaps Node point heights are adjusted to match airfoil shape (symmetric NACA 4-digit for simplicity) All elements were modeled with the properties of aluminum Wing Structure Unit Cell (Exploded View) Grid Points: 5 On Each Skin Surface Diagonal Spar: Spar Caps Shear Web Posts: Located at each intersection Upper Skin: 4 Triangular Plates Rib Segment: Shear Web Spar Segment: Spar Caps Shear Web Lower Skin: 4 Triangular Plates 19

19 Wing Modeling: Geometric Data Three test cases loosely based on existing aircraft: Low AR wing (Based on T-38 data) Moderate AR wing (Based on DC-9 data) High AR wing (Based on Global Hawk data) Low AR Moderate AR High AR Aspect Ratio Sweep Wingspan (ft) Root Chord (ft) Taper Ratio Max. Root Thickness (ft) Thickness Taper Ratio Aileron Position (% Half Span) Allowable Design Region (% Chord) 50% - 70% 50% % 50% - 75% 20% - 65% 15% - 80% 5% - 85% 20

20 Wing Modeling: Load Case Data Each wing topology undergoes sizing optimization to minimize weight Subject to three load cases: Steady-state roll at or above maximum dynamic pressure Low-speed (corner velocity) pull up at maximum load factor High-speed pull up at maximum load factor Steady state roll Low speed pull-up High-speed pull-up Aircraft Weight 12,000 lb Mach Number 1.55 Dynamic Pressure 2016 psf Load Factor 1 Mach Number 0.85 Dynamic Pressure psf Load Factor 6 Mach Number 1.44 Dynamic Pressure 1700 psf Load Factor 6 21

21 Wing Modeling and Process Parameters Weight minimizations were constrained by: Maximum allowable principle stress Skin loads below critical levels for buckling Non-negative roll rate for steady-state roll condition Variable linking divided structure into large-scale members: Spar and rib elements of the same type were initially linked along their lengths (i.e. upper spar cap, lower spar cap and shear web) All skin elements grouped in rectangular patches between two neighboring ribs and on a common surface Process parameters were set to reduce memory cost through slow delinking and fast filtering: R 0 = 10%, ΔR = 10% D 0 = 0.5%, ΔD = 0.5% 22

22 Definition of Skin Buckling Constraints Required extra steps ASTROS plate buckling constraint has two major limitations: Eigenvalues are calculated from an approximate rectangular panel with user-defined dimensions Multiple elements forming a single panel are not grouped together for buckling analysis Loads are taken from a single element and applied to the rectangular panel Extra steps define the constraint by: Identifying skin panels and their constituent elements Finding dimensions and orientation of the approximate panel Defining a constraint for each skin element Approximation gives good estimate for nearly rectangular panels and a weaker estimate for irregular panels. Buckling Constraint Definition From Filter Search for Grouped Skin Elements Calculate Panel Size and Orientation Define Skin Elements and Buckling Constraints To Delinking and Weight Optimization 23

23 Iteration History: Low Aspect Ratio Wing Initial design: Rods: 1 in 2 Shear webs: 0.25 in Skin plates: 0.25 in Optimal design: Weight: lb Complexity: 53.0% Minimum weight design occurred for iteration 36 Weight increases were accepted for iterations 24, 31 and 34 Designs were rejected for iterations 29 and 30 Complexity: n/n 0 Weight: W/W 0 Objective: W/W 0 n/n 0 24

24 Low Aspect Ratio Wing Example 1 Structure undergoes a transition across the aileron position Inboard structure: Forward swept spars Stringers in all three directions Large shear-bearing structure near the trailing edge Thick stringers near trailing edge support aileron loads Outboard structure: Span-wise stringers Single-cell torsion box Transition zone ahead of the aileron Bending supported mainly by spanwise stringers Shear Webs and Posts Spar Caps/Stringers Skins

25 Low Aspect Ratio Wing Example 2 Different initial design: Rods: 0.5 in 2 Shear Webs: 1 in Skin plates: 1 in Optimal design: Weight: lb Complexity: 67.4% Minimum weight design occurred for iteration 24 Structure is lighter but more complex than the first example Iterations do not improve the design weight Need better stopping criteria Complexity Weight Objective ASTROS Optimization Number

26 Second Example: Low Aspect Ratio Wing Structure undergoes a transition across the aileron position. Inboard structure: Greater number of forward swept spars Network of stringers ends at aileron Larger shear-bearing structure near the trailing edge Outboard structure: Stringers connecting tip to leading and trailing edges Larger torsion box Bending supported by skin, with stringer support for aileron loads Spar Caps/Stringers Shear Webs and Posts Skins

27 Discussion Transition across aileron position is present in all cases: May be caused by a jump in stress levels produced by aileron deflection Inboard structure follows shortest path to the root Outboard structure follows shortest path to major inboard structure. Inboard structure has a higher priority; load paths individually optimal Outboard structure has lower priority; add least weight by attaching to inboard features Outboard Region First Example (Low AR) Shear webs and Posts Aileron Location (Transition) Inboard Region 28

28 Discussion Development of areas with composite-like directional stiffnesses: Present in moderate and high aspect ratio cases Inboard sections resist torsion and delay aileron reversal Outboard sections produce greater washout for upward bending May alleviate reduce root stress and decrease total wing weight Moderate AR wing example Spar Caps and Stringers 29

29 Discussion Second low aspect ratio case produced a lighter design: First low AR example: lb Second low AR example: lb Key features of lighter design: Favored support of bending loads by skin Greater number of spars near the root Use of stringers to support aileron loads Some or all of these features present in the moderate and high aspect ratio results Process probably has local minima or path dependence 30

30 Possible Stopping Criterion Minimum weight design followed by rejected designs in three of four cases Termination might be based on the number of rejected designs Avoid premature stopping, but allow for minimum weight design If the routine were stopped after three rejected designs, all examples would find the same minimum weight designs found for the original criterion 2 Rejected Designs Rejected Designs Weight Complexity Objective Minimum Weight Minimum Weight First Example (Low AR) ASTROS Optimization Number Third Example (Moderate AR) 31

31 Future Work Potential to improve process through future refinement Improved filter process Ability to treat each element type with its own threshold Ability to partially accept designs resulting in a weight increase Investigate impact of optimization parameters (R 0, ΔR, D 0, ΔD) Selection of initial structure Examples suggest dependence on initial design Finer initial structure might yield better results, but at computational time and memory cost Improved treatment of skin elements Greater accuracy in calculation of critical buckling loads for panels modeled from multiple elements Direct incorporation of composite skins 32

32 Conclusion Demonstrated ability to find optimal result for plane truss (not shown here for time) Structural complexity and weight both important for optimal topologies Demonstrated application to wing layout combining sizing and topology considerations Final topology appears sensitive to initial design (may be analog to process local minima) Directional stiffness may suggest need for composite wing skins Inclusion of aileron Divided topology into direct and indirect load path zones Suggests areas of high and low priority structure 33