Development and Application of a Flexibility-based Design Method for

Development and Application of a Flexibility-Based Method for Multi-Scale Design Sergio Carlos, 1 Kaarthic Madhavan,1 Gaurav Gupta, 1 Darren Keese, 1 Uma Maheshwaraa, 1 and Carolyn Conner Seepersad. 2 Mechanical Engineering Department, The University of Texas at Austin, Austin, TX 78712

A flexibility-based approach is presented for the solution of multiscale engineering design problems. The methodology is aimed at enhancing distribution of design activities and reducing the number of costly iterations between multiple engineering teams operating on different scales. This goal is achieved by exchanging flexible families of solutions rather than single point solutions, thereby reducing the need for iteration between scales. The effectiveness of the approach is illustrated by a two-level problem involving the system-level design of a gas turbine engine and the mesoscale design of cellular material for the combustor liner. A multi-objective design problem formulation is used to obtain families of prismatic cellular materials that embody a range of tradeoffs between conflicting thermal and structural performance objectives. The results are communicated to the system level and a solution is chosen to meet system-level goals as closely as possible. The effectiveness of the method is evaluated by comparison with a benchmark integrated design method. The approach facilitates identification of satisfactory or nearly optimal solutions quickly and with minimal iterations between scales.

Nomenclature cp CPR Di Do Fthrust γ hPR LHV Lmin m m mfuel ηD ηC

= = = = = = = = = = = = = =

Air specific heat (J/kgK) Compressor pressure ratio Inner combustor liner diameter (m) Outer combustor liner diameter (m) Engine net thrust (N) Air specific heat ratio Enthalpy of reaction (J/kgK) Lower heating value of fuel (J/kg) Minimum combustor length Total mass Air mass flow rate (kg/s) Fuel mass flow rate (kg/s) Diffuser efficiency Compressor efficiency

MULTISCALE

ηT ηN ηth ηp ηo Pi Ti Tlo Ta υ Vin Vliner W C

= = = = = = = = = = = = =

Turbine efficiency Nozzle efficiency Thermal efficiency Propulsive efficiency Overall efficiency Pressure at state i (Pa) Temperature at state i (Pa) Liner exit temperature (K) Ambient air temperature (K) Total volume Air speed (m/s) Volume of material in liner (m3) Power in shaft (W)

I. Introduction

design is a field of simulation-based design in which computational models and simulations, coupled with systems-based design methods, are used to solve design problems that incorporate a number of scales, from material microstructures to overall system configuration. Materials design—the simultaneous design of

1

Graduate Research Assistant, Mechanical Engineering Department. Assistant Professor, Mechanical Engineering Department, AIAA Member. Email: [email protected]. Phone: 512-471-1985. 1 American Institute of Aeronautics and Astronautics

2

a product and the materials from which it is made—is a primary example of multiscale design and an increasingly active area of research.1-5 As illustrated in Figure 1, a multiscale design problem is typically decomposed vertically according to divisions of perspective that occur simultaneously over a range of disciplines and a range of hierarchical scales. In multiscale design, the focus is on coordinating solutions to subproblems so that they can be combined into compatible multiscale solutions that meet system-level objectives as closely as possible. Prominent optimizationbased approaches for solving these types of multidisciplinary and hierarchical system design problems include analytical target cascading,6 simultaneous analysis and design,7 concurrent sub-space optimization,8,9 collaborative optimization,10 and BLISS.11 Multiscale design is a difficult challenge, however, and optimization-based approaches are not necessarily well-suited for the task. Fundamentally, optimization-based approaches are disadvantaged by several signature features: (1) many of the approaches, with the exception of analytical target cascading, have been developed and tested for two hierarchical levels, and extension to multiple levels is not a straightforward task; (2) only a single point solution is alive in each subproblem at any one time, and the burden of coordinating subproblems resides at the system-level, resulting in frequent system-wide iterations; and (3) the goal is to identify only the “best” optimal solution, which is no longer optimal as soon as conditions change anywhere in the system, inducing another round of extensive, system-wide iterations. Although these system-wide iterations are expensive and optimal solutions are illusive for many classes of problems, they are particularly problematic for multiscale problems. These problems are not well-suited to fully automated analysis and synthesis and require active involvement of expert designers. The roles of these expert designers include: (1) formulating simulation models that are often much less mature than those for human-scale systems and validating and interpreting the results, and (2) identifying preliminary design alternatives in highly discontinuous and non-parametric design spaces (e.g., by adding an alloying element or changing the layout of material). Motivated by these challenges, we are developing a flexibility-based approach for multiscale design. The approach involves larger-scale (systems) designers communicating with smaller-scale designers via performance targets and constraints and receiving feedback from them through sets of solutions, all in terms of coupled parameters. The method preserves flexibility in the design process by foregoing conventional, single point solutions in favor of sets of solutions. Collaborating, upper-scale designers have the freedom to select solutions that balance system-level objectives without extensive iterations between designers, in keeping with set-based philosophies advocated in the automotive industry.12 The method has several signature features of its own: (1) multiple concepts are kept alive in each subproblem to preserve enough flexibility to accommodate changes in other parts of the multiscale system; (2) the burden of coordination is distributed throughout the design space rather than concentrated at the system-level; and (3) the goal is to identify satisfactory solutions quickly and with minimal system-wide iterations. The approach benefits from increased coverage of the design space, relative to single point solutions System Specification System Subsystems Components Parts Materials Material Specification Meso Continuum Molecular Quantum

Figure 1. A hierarchy of length scales in complex systems and materials.13 2 American Institute of Aeronautics and Astronautics

associated with optimization-based approaches and intervals associated with robust design approaches.14-16 The process enables flexibility in the mode of communication and keeps subsystem designers consistently in the loop, utilizing their expertise without requiring automated computer simulations. It also avoids centralized, system-level optimization that creates system-level bottlenecks and computational intractability. Finally, it avoids the difficulties associated with metamodeling approaches to complex systems design 17-21 and approaches such as negotiation methods,22 game theoretic techniques,23-27 and analytical target cascading6 that rely heavily on metamodels; namely, the problem of size associated with inefficient or inaccurate metamodels for large numbers of design variables and the difficulty of treating highly nonlinear or discontinuous design spaces. Chanron and coauthors 28 propose a similar approach but focus on multidisciplinary rather than multiscale problems. In the following section, a detailed description of the approach is provided, followed by an application to a system-materials design problem for a gas turbine engine.

II. Description of the Flexibility-Based Method Since multiscale design is a hierarchical activity, it entails multiple levels of interdependent design problems. Each level represents a different spatiotemporal scale and involves a domain-specific design problem, referred to as a local problem. Each local problem is dependent on both coupled parameters that are shared between two or more local problems and uncoupled parameters that are relevant only to a specific local problem. In Figure 2, a generic two-level multiscale design problem is formulated in which the coupled parameters are represented by R. Each local problem is formulated as a multiobjective decision in the form of a compromise Decision Support Problem (DSP).29 For each local problem, the aim is to find the values of design variables (xs and R for the system level and xss for the subsystem level) that satisfy a set of constraints (gs and hs for the system level and gss and hss for the subsystem level) and achieve a set of conflicting goals as closely as possible, as measured by an objective function (Zs for the system level and Zss for the subsystem level). Each local problem is solved by domain experts using specialized analysis models for system- and subsystem-level constraints, goals, and coupled parameters. One of the primary challenges is that the upper level design problem requires the coupled parameters, R, as input, but these coupled parameters can be obtained only as outputs or responses in the lower level problem. To address this challenge, we propose a flexibility-based approach in which the lower level designer generates a flexible family of solutions that represent a variety of coupled parameter values. The family of solutions is generated in lieu of repeated iteration between the levels and offers the upper-level designer the flexibility to select a solution (in terms of coupled parameters) that satisfies her constraints and most closely achieves her goals. The proposed approach is illustrated in Figure 3 and involves the following steps: 1. Partition the overall system design problem, formulate local design problems, and identify coupled parameters. The system-wide problem is partitioned into local problems according to different scales, disciplines, and areas of expertise. Each local problem is formulated as shown in Figure 2, and coupled parameters are identified. 2. Develop analysis tools for local problems. After the problem is properly partitioned and coupled parameters are identified, the “independent” design teams develop the analysis tools needed to solve their local design problems. It is not necessary to create automated analysis centers or approximate models for each local problem. Designers may be kept in the loop for updating models and generating, interpreting, and validating solutions. 3. Generate targets or approximate models for the coupled parameters. Targets for coupled parameters are created by larger-scale designers, providing smaller-scale designers an insight into the large-scale design problem. If the target for a coupled parameter must be specified as a function of other coupled parameters, approximate models are created; however, these approximate models are created only in terms of coupled parameters and do not need to represent the entire design space. 4. Use targets, approximate models, and analysis tools to create a family of feasible solutions. Smaller-scale designers use targets and approximate models from larger-scale designers as guidance for constructing a set of feasible solutions. While solving the local smaller-scale problem, designers minimize deviations from performance targets for coupled parameters, R, and any local goals (yss for the subsystem in Figure 2) without violating constraints and bounds. Since the targets specified by larger-scale designers are almost always ambitious, conflicts usually arise which lead to the creation of tradeoffs among the multiple objectives. Accordingly, Pareto sets of solutions are generated that embody a range of tradeoffs between the coupled parameters.

3 American Institute of Aeronautics and Astronautics

System Design

System Level

Given Analysis models Find

xs, R

Satisfy Constraints

gs ( xs , R) ≤ 0 h s ( xs , R ) = 0

Bounds

xs , lower ≤ xs ≤ xs , upper

Goals

yi ( xs , R) + di− − di+ = yi − target

Minimize

i = 1, …, # system goals

Zs = f (d − , d + )

R Subsystem Design Subsystem Level

Given Analysis models Find

xss Satisfy Constraints

gss ( xs ) ≤ 0 hss ( xss ) = 0

Bounds

xss , lower ≤ xss ≤ xss , upper

Goals

yi , ss ( xs ) + di− − d i+ = yi , ss − target i = 1, …, # subsystem goals (m)

R j , ss ( xss ) + d −j − d +j = R j ,ss − target j = m, …, m+ # coupled parameters Minimize

Zss = f (d − , d + )

Figure 2. General formulation of a multiscale design problem.

Figure 3. Schematic representation of flexibility-based approach to multiscale design. 4 American Institute of Aeronautics and Astronautics

5.

6.

Evaluate and select solutions. Larger-scale designers evaluate the Pareto sets of solutions from smallerscale designers and identify solutions that achieve system performance goals as closely as possible. The Pareto sets of solutions provide flexibility to system designers by allowing them to select the most promising solution and thereby reducing the number of costly iterations in the multiscale design process. Additional iterations are required only if a larger-scale designer requests additional sets of solutions in a specific region of the design space. Propagate design information throughout the system. Global solutions are communicated throughout the system in terms of the final values of coupled parameters.

III. Flexibility-Based Method Applied to a Gas Turbine Engine Example The flexibility-based approach is illustrated through the design of a next-generation gas turbine engine with a combustion chamber lined with prismatic cellular (honeycomb) materials, as illustrated in Figure 4. The cellular materials are actively cooled by forced convection within the cells. Accordingly, the need for combustion side cooling—as required by conventional metallic combustor liners—is reduced or eliminated, increasing engine efficiency while potentially reducing the emission of harmful nitrogen-oxide pollutants into the atmosphere.30,31 Warmed Air Combustor Compressor

Hot Combustion Gases Cooling Air

Inlet

Turbine

Combustor Liner

Compressed Air and Fuel Cellular Material

Combustion Chamber (High Temp And Pressure)

Figure 4. Schematic of a cellular combustor liner. 13 Two teams are introduced in the design of the engine: (1) a system-level team that focuses on overall system performance of the gas turbine engine, and (2) a materials team that focuses on designing the combustor liner’s cellular material for satisfactory structural and heat transfer requirements. The system-level designer is concerned with maximizing the thrust and minimizing the weight of the system. The cellular materials designer influences each of these objectives. Thrust typically increases with the temperature and pressure of the combustion process. These factors place greater load-bearing and cooling demands on the cellular combustor liner, which leads to larger cells, thicker walls, and a heavier combustor liner. The system-level team provides targets for influential coupled parameters, such as the temperature of the combustion process and the mass of the combustor liner and constants for fixed or less influential parameters. In return, the materials designer generates Pareto sets of solutions that represent tradeoffs between the coupled parameters (combustion temperature and combustor liner mass). The materials level problem is formulated as a multi-objective decision. Solutions are obtained using iSIGHT design automation software,32 coupled with ANSYS for modeling the thermal and structural performance of the liner. A combination of local search techniques and exploratory methods are used to identify the set of design specifications for the liner. The final solution is chosen to meet system-level goals as closely as possible. A step-by-step implementation of the method is described in the remainder of this section.


System Design Given Thermodynamic model for system (Sect 3.2.1) with T4, Tlo, Di as inputs Mass model for system with Vliner as input Values of fixed parameters (Do, P3, P4, T3, m ) Find Do, Tlo, Vliner, T4 Satisfy Constraints and Bounds Do, Tlo, Vliner, T4 compatible with materials design Goals

thrust / weight

( thrust / weight )target

+ d1− − d1+ = 1

Minimize

Zs = f (d1− )

R=

Variables : T 4, Tlo, Di, Vliner Fixed Parameters : Do , p3 , p4 , m

Materials Design Given ANSYS thermostructural model of the cellular combustor liner Find

L , h1 , h2 , h3 , h4 , Offset Satisfy Constraints and Bounds

σ ≤ σ y (T )

0.02 m ≤ DISTi ≤ 0.06 m

0.5 m ≤ Di ≤ 0.68 m

DIST = h1 − 0.5 Di

0.005 m ≤ Offset ≤ 0.01 m

DIST1,2,3 = hi +1 − hi

0.6 m ≤ L ≤ 0.85 m

DIST4 = 0.5 Do − h4

L ≥ Lmin

Do , p3 , p4 , m fixed by system designer

Goals

Vliner + d1− − d1+ = 1 Vliner − target

T4 + d 2− − d 2+ = 1 T4− target

Minimize

Zss = f (d1+ , d 2− ) Figure 5. Multiscale design problem formulation for gas turbine engine example. A. STEP 1: Partition the Overall System Design Problem, Formulate Local Design Problems, and Identify Coupled Parameters The first step is to partition the overall design problem. The system designers are responsible for the overall performance of the engine while the subsystem or materials designers are responsible for the geometry of the combustor liner. The local design problems are formulated, and coupled parameters are identified as shown in Figure 5. The system-level design team has is responsible for maximizing the thrust of the entire gas turbine engine while minimizing its total weight. In order to calculate thrust for a particular cellular material solution, the system design team must know the highest temperature, T4, that the combustor liner can withstand, the inner diameter of the liner (Di), and the temperature of the cooling air when it exits the liner (Tlo). Thrust also depends on the outer diameter of the combustor liner, Do, which is fixed by the system designer to satisfy geometric constrains imposed by the rest of the system. None of the other variables that appear in the system thermodynamics model are affected by the choice of liner solution; so T4, Di, and Tlo are the only three coupled parameters related to thrust. The total weight of the 6 American Institute of Aeronautics and Astronautics

system is also a coupled parameter that depends on the volume of material in the combustor liner, Vliner. At the subsystem level, the design problem is to find the values of cell wall dimensions that maximize the combustor exit temperature, T4, and minimize the volume of material in the combustor liner, Vliner. The cellular arrangement is assigned a fixed topology, as illustrated in Figure 6b, for a 1/32 fraction of the axially symmetric combustor liner. The design variables are L, h1, h2, h3, and h4 which represent the length of the combustor liner and the radial distance from the center of the combustor liner to certain points in the topology, as illustrated in Figure 6b. The Offset variable measures the thickness of the radial cell walls. The DISTi variables are dependent on the values of hi and indicate the thickness of the circumferential cell walls. Constraints on the cellular materials design problem include limits on the cell wall dimensions, a stress constraint to avoid temperature-dependent yielding in the cell walls, and a lower bound on the length of the combustor liner that is dependent on T4 and Di and ensures complete combustion.

Do Di L

Figure 6a. Combustor liner.

Figure 6b. Combustor liner cross-section.

To summarize, the coupled parameters are T4, Di, Tlo, and Vliner. Since these variables are shared between the two designers, the system-level designer is limited to selecting values for the coupled design variables that are compatible with the choices of the materials level designer. The characteristics of other engine components are fixed by the system-level designer because this example focuses on combustor design. For the combustor itself, this means that Do, p3, p4, T3, and m are fixed and shared with the materials designer. B. STEP 2: Develop Analysis Tools for Local Problems In the second step of the method, analysis models are constructed for each local design problem. System-Level Model. The system-level model involves a complex thermodynamic analysis that relates input characteristics such as initial air temperature/pressure, incoming airspeed, and compressor pressure ratio to output performance characteristics such as thrust, jet velocity and exiting pressure and temperatures. The entire model is based on recommendations from a standard reference.33 This sequential model is composed of five main components (see Figure 7): 1) The diffuser that accepts and directs the airflow, 2) The compressor that raises both pressure and temperature in preparation for combustion, 3) The combustor where compressed air and fuel are mixed and burned, 4) The turbine, and 5) The nozzle which directs exiting airflow and creates engine thrust. In the analysis, common variables such as pressure and temperature are identified by a subscript (e.g. T4) which corresponds to the thermodynamic states labeled in Figure 7. A series of assumptions are made in building the model: 1. The system operates at steady state conditions. 2. The burner is isobaric (p3=p4). 3. Turbine and compressor power are equivalent (Wc = Wt). 4. Kinetic energy is negligible except at the inlet and exit (V2= V3= V4= V5=0). 5. Potential energy changes are negligible throughout the system. 6. Combustion products are negligible, compared to air (cp(products) = cp(air)) . 7. Air is an ideal gas. 8. Enough fuel is added to reach T4 at stage 4. 9. Air that exits through the liner is negligibly faster than air entering the diffuser. (Vliner_output= Vin). 10. Inlet and outlet pressure are equal to atmospheric pressure. 7 American Institute of Aeronautics and Astronautics

Fuel B 3

Vin 1

D

2

C

Cooling air (lo)

D C B T N

4 T

5

N

Diffuser Compressor Burner (Combustor) Turbine Nozzle

Vjet 6

Figure 7. Thermodynamic model of the engine. The arrows represent the flow of air. 11. The temperature at the inlet, T1, is 298 K. A brief overview of the model is provided in the rest of this section and organized according to components. For brevity, most of the variables are defined in the nomenclature section at the beginning of the paper. Diffuser. Based on an energy balance for the diffuser and an ideal gas approximation, the temperature change across the diffuser is assumed to be small, so that the air specific heat at stage two, cp(T2), can be approximated as that of state one, cp(T1). Solving for the output temperature at state 2: 2 (1) Vin T2 = + T1 2 ⋅ c p (T1 )

Based on the definition for pressure efficiency and assuming an isentropic relationship, the pressure at state 2 can be calculated as follows: γ (2) ⎛ T1 − η D ⋅ (T1 − T2 ) ⎞ γ −1 p2 = p1 ⎜ ⎟ T1 ⎝ ⎠ Compressor Analysis. The pressure at the exit of the compressor is defined by the Compressor Pressure Ratio, CPR, designed into the system: (3) p3 = CPR ⋅ p2 and the temperature at the exit of the compressor is calculated as follows: γ −1

⎛p ⎞γ T2 ⎜ 3 ⎟ − T2 ⎝ p2 ⎠ T3 = + T2

(4)

ηC

Based on an energy balance for the compressor and an ideal gas approximation, the power utilized by the compressor is calculated as follows: (5) WC = m ⋅ [c p (T3 ) ⋅ T3 − c p (T2 ) ⋅ T2 ] Burner Analysis. An isobaric assumption implies identical entering and exiting pressures, p3 and p4, for the burner. The enthalpy of reaction is commonly approximated by the Lower Heating Value or LHV, if the temperature is within a few hundred degrees of 298 K. A more accurate approximation is represented by: (6) hPR = [c p (T4 ) − c p (T3 )] ⋅ (T4 − 298) + LHV

By definition, the rate of heat entering the burner from fuel combustion, Q , is equal to the product of the mass flow rate of the fuel and the enthalpy of the reaction. With an ideal gas approximation, the mass flow rate of the fuel can be calculated as follows: 2 2 (7) D D ( i2 ) ⋅ c p (T4 ) ⋅ T4 + (1 − i2 ) ⋅ c p (Tlo ) ⋅ Tlo − c p (T3 ) ⋅ T3 Do Do

m fuel = m

hPR + c p (T4 ) ⋅ T4


Turbine Analysis. The power output of the turbine can be calculated from an energy balance for the turbine, assuming ideal gas behavior and assuming that the power output of the turbine equals the power input to the compressor: (8) WT = WC = ( m + m fuel ) ⋅ [c p (T4 ) ⋅ T4 − c p (T5 ) ⋅ T5 ] If cp (T5) is approximated as cp(T1) and T5s is the exit temperature for an isentropic process., the temperature and pressure at the exit of the turbine can be calculated as follows: (9) WC c p (T4 ) ⋅ T4 − m + m fuel T5 = c p (T1 ) (10)

γ

⎛ T ⎞ γ −1 p5 = p4 ⎜ 5 s ⎟ ⎝ T4 ⎠

Nozzle Analysis. The exiting temperature and velocity of the nozzle can be calculated from an energy balance as follows: γ −1 (11) ⎡ ⎤ γ p ⎛ ⎞ 6 ⎥ T6 = T5 − η D ⋅ ⎢T5 − T5 ⎜ ⎟ ⎢ ⎝ p5 ⎠ ⎥

⎣

V jet =

⎦

2 ⋅ [c p (T5 ) ⋅ T5 − c p (T6 ) ⋅ T6 ]

(12)

Thrust Calculation. Based on a momentum balance of the system, excluding the air that passes through the combustor liner, the thrust created by the engine can be calculated as follows: 2 2 (13) D D Fthrust = ( i2 ⋅ m + m fuel ) ⋅ V jet − i2 ⋅ m ⋅ Vin Do Do Minimum Combustor Length. We used a scaling formula and data about an existing combustor to create a function for determining the minimum combustor length for complete combustion. A JT9D annular combustor was selected as a reference, and its values were used for scaling in this formula, including air pressure and temperature at stages 3 and 4, respectively, of Ptotal ,3 = 2.179 MPa and Ttotal ,4 = 1591.7 K. −1.51

Lmin ∝

Ptotal ,3

(14)

Ttotal ,4

Validation. For validation, the system model coded in Matlab was compared with well known models and programs including NASA’s EngineSim34 that takes inputs and produces a thrust output for engine designs. Our model agrees with the reference models within ± 2.5%--an acceptable range for present purposes. Subsystem (Materials) Level Model. With gases undergoing combustion in the chamber, the liner is exposed to the highly elevated combustion temperature, T4, and experiences significant thermal stresses. The liner is also subjected to the pressure of combustion, estimated to be 2.18 MPa, which contributes to the mechanical stresses. As the combustion temperature increases, the cell walls become thicker to conduct heat to the convective passageways and to withstand the increased thermal stresses. Therefore, there is a tradeoff between the mass or total volume of material, Vliner, in the combustor liner and the combustion temperature, T4. Feasible solutions are those for which the effective von Mises stress due to thermal and mechanical loading in any cell wall does not exceed the yield strength of the base material at the elevated temperature in that cell wall. The liner material is assumed to be a Molybdenum-Silicon-Boride alloy with several advantageous characteristics, including high melting temperatures (around 2273K or higher), low density (8810 kg/m3), high oxidation resistance, and high yield strength at both room temperature and elevated temperature (1500 MPa at 300K and 400 MPa at 1650K). Its Young’s modulus, thermal conductivity, and coefficient of thermal expansion are assumed to be 327 GPA, 100 W/m-K and 6E-6 m/m-K, respectively.35,31


Figure 8. Combustor liner topology (rectangular configuration).

To evaluate the stresses in the combustor liner, a thermostructural analysis is conducted with the commercially available finite element analysis software ANSYS.36 The finite element analysis of the liner is divided into two sequential steps. The first step is to perform a CFD analysis to determine the temperature distribution in the cell walls, accounting for conduction and convection. The second step is to perform a thermo-elastic structural finite element analysis to determine the stresses in the cell walls due to thermal and mechanical loading, using the temperature profile from the previous analysis as input. As shown in Figure 8, the combustor liner section has four external surfaces: an outer surface, an inner surface exposed to combustion, and two sides that interface with adjacent slices. For thermal analysis, the sides and outer surface of the slice are assumed to be insulated. In the analyses, the combustion temperature is varied in the range of 1400 – 2200K. The temperature actually applied to the inner surface of the liner is reduced by 300K to simulate the effect of a thin layer of protective ceramic coating. Turbulent flow is assumed within the combustor liner cells with an inlet temperature of 300K and a mass flowrate of cooling air of 105 kg/s. The ducts and cell walls are meshed with fluid142 elements. For the structural analysis, boundary conditions include symmetric boundary conditions applied to the sides of the slice, a pressure of 2.18 MPa applied to the inner surface, and atmospheric pressure applied on the outer surface. For each analysis, a minimum of 45 elements are utilized along the length of the liner and 2 elements through each cell wall, meshed with solid45 elements. To ensure consistent results, the mesh density is refined until there is no significant change (within 0.1%) in the resulting temperature and stress values. C. STEP 3: Generate Targets or Approximate Models for Coupled Parameters As presented in Step 1 and Figure 5, the mode of communication from the system-level team to the subsystem (materials) design team is via targets, constants, and approximate models. In this case, the system-level designer fixes the values for Do, p3, p4, and m —coupled parameters that depend on aspects of the system that are not designed in the present example. The values are recorded in Table 1. The system-level designer assigns targets for the coupled variables with the most influence on system-level performance— Vliner and T4—and allows the materials level designer to assign values freely to the coupled parameters (Di and Tlo) that have relatively weak influence on system-level performance. Finally, the system-level designer creates an approximate model for the minimum combustor length required for complete combustion, Lmin, as a function of coupled parameters. Table 1 summarizes the information related to the communication from the system to the subsystem level. Table 1. Information passed from the system-level team to the material design team. •

• • •

Fixed values in the shared design space 743 kg/s ) Flow rate ( m Outer diameter (Do) 1m 2.18 MPa Combustor chamber pressure (p3 and p4) Target for Combustor Exit Temperature (T4): 2200 K Target for Liner Volume (Vliner): 0.25m3 Approximation model for minimum combustor length constraint: Lmin = (1.45494) - (0.00042)T4 + (0.16012) Di − (0.10185) Di2


D. STEP 4: Use Targets, Approximate Models, and Analysis Tools to Create a Family of Solutions The task for the materials level designer is to utilize the targets, constants, and approximate model in Table 1 to generate a family of Pareto solutions for the materials level problem. The solutions are to be communicated to the system-level designer who will choose a solution that meets her goals as closely as possible. The materials level objectives are to minimize the volume of material in the liner, Vliner, with a target of 0.25 m3 and to maximize the combustion temperature, T4, with a target of 2200 K. These objectives are conflicting in nature. The materials designer assigns different weights to each of the objectives so that a Pareto set of solutions can be generated. Accordingly, the materials design objective function in Figure 5 becomes: + − (15) Z = W d +W d ss

1

1

2

2

where W1 and W2 are the weights on Vliner and T4, respectively. The materials level design problem is solved by coupling ANSYS with iSIGHT design integration software.32 iSIGHT runs ANSYS iteratively by adjusting its APDL input file, executing the ANSYS analysis, and parsing its output file. The design problem is solved initially using an exploratory, multi-island genetic algorithm followed by a generalized reduced gradient (GRG) algorithm for local search. Further local searches are performed with the GRG algorithm, starting from the solution from the previous cycle, until convergence is achieved. The results are validated by executing the genetic algorithm again, in combination with a sequential quadratic programming (SQP) algorithm. The resulting set of Pareto solutions are shown in Figures 9 and 10. The pre-determined targets are to reach a combustion temperature of 2200K and a volume of 0.25 m3, as recorded in Table 1, and the applied weights are (1,0), (0,1), (0.5,0.5), (0.75,0.25), and (0.25,0.75) for W1 and W2, respectively. In Figure 9, the solutions are plotted with combustion temperature (T4) on the abscissa and the inverse of volume (1/Vliner) on the ordinate axis. The solutions demonstrate a clear Pareto curve such that it is not possible to improve either objective without worsening the other. A schematic of each solution is illustrated in Figure 10, along with accompanying values for weights, Di, L, T4, and Vliner. The Pareto set of solutions obtained with the GRG algorithm are validated using the SQP algorithm. A comparison of the Pareto curves is shown in Figure 11. The curves show good agreement between the SQP and GRG algorithms and validate the numerical results.

Volume Inverse (in m-3)

Pareto Plot - Combustion Temperature Vs Inverse of Volume - GRG 4 3.5 (1729, 0.277) 3 2.5 2 1.5 1 0.5 0 1700 1750 1800

(1963, 0.283) (2016, 0.297) (2028, 0.343) (2049, 0.393)

1850

1900

1950

2000

2050

2100

Combustion Temperature (K)

Figure 9. Pareto set of solutions in terms of temperature and inverse volume.


W1 (on T4) = 1.0 W2 (on Vliner) = 0 T4 (Combustion temp) = 2049 K Vliner = 0.393 m3 Di = 0.635 m L = 0.810 m

W1 (on T4) = 0.5 W2 (on Vliner) = 0.5 T4 = 2016 K Vliner = 0.297 m3 Di = 0.649 m L = 0.771 m

W1 (on T4) = 0.75 W2 (on Vliner) = 0.25 T4 = 2028 K Vliner = 0.343 m3 Di = 0.672 m L = 0.737 m

All-at-once (AAO) W1 (on T4) = 0.25 W1 (on T4) = 0 (Section 3.7 Validation) W2 (on Vliner) = 0.75 W2 (on Vliner) = 1.0 T4 = 2028 K T4 = 1963 K T4 = 1729 K Vliner = 0.32 m3 Vliner = 0.277 m3 Vliner = 0.283 m3 Di = 0.677 m Di = 0.680 m Di = 0.680 m L = 0.730 m L = 0.755 m L = 0.792 m Figure 10. Pareto set of design configurations for combustor liner.

Inverse Volume in inverse cubic meter

Comparison of Pareto curves for GRG and SQP 5

Pareto Curve using SQP

4 3

Pareto Curve using GRG

2 1 0 1700

1800

1900

2000

2100

Combustion Temperature (K)

Figure 11. Validation of Pareto points using SQP algorithm. E. STEP 5: Evaluate and Select Solutions The information received from the subsystem design team consists of a Pareto set of feasible combustor liner designs with a variety of values for the conflicting objectives of minimizing the material volume in the combustor liner (Vliner) and maximizing the combustion temperature (T4). The system designer’s task is to select the solution from the set that achieves system-level goals for thrust/weight ratio as closely as possible and serves as the final design. To complete Step 5, the system-level designer does not need to know all of the details of each of the materials level solutions; only the coupled parameter values they represent. 12 American Institute of Aeronautics and Astronautics

In order to choose from among these Pareto optimal points, the system-level designer compares engine thrustto-weight ratios for each. The engine thrust-to-weight ratio is a benchmark for measuring the merit of an engine design. Higher thrust is desirable because it allows a more powerful engine. Lower weight is desirable because it means less fuel is used throughout an entire flight. A typical research and development goal of aircraft engine companies is to increase this ratio.33 Combustor liner material volume and density is used to calculate mass and weight for the thrust-to-weight ratio, and the weight of all other engine components is assumed to be constant. The ratio for each member of the Pareto set is listed in Table 2. As shown in Table 2, there is a broad range of thrust/weight ratios for the Pareto solutions, with solution 5 performing particularly poorly due to exceptionally low thrust. Solution 3 exhibits the highest thrust/weight ratio with solutions 2 and 4 exhibiting ratios within 5% of solution 3. Table 2. Engine thrust-to-weight ratios for the Pareto set of materials level solutions. SOLUTION NUMBER

1 2 3 4 5

WEIGHT ON TEMP (T4) 100% 75% 50% 25% 0%

WEIGHT ON VOLUME (VLINER) 0% 25% 50% 75% 100%

THRUST (KN)

COMBUSTOR MASS (KG)

COMBUSTOR WEIGHT (KN)

ENGINE THRUST / WEIGHT

120704.42 126541.84 114177.39 107583.68 38085.412

3461.76 3020.22 2614.41 2494.78 2439.72

34.306 29.930 25.908 24.723 24.177

3.518 4.270 4.451 4.395 1.575

F. STEP 6: Propagate Design Information Throughout the System The final step is to propagate the selection of Solution Number 3 throughout the system so that the entire design can be documented in terms of both coupled and uncoupled parameters. G. Validation The effectiveness of the method is evaluated by comparing it with a benchmark design method called the all-atonce (AAO) approach. In the AAO approach, the two local design problems are combined into a single systemwide design problem. An integrated design problem is often much more computationally expensive to solve and fails to provide many of the benefits of the flexibility-based approach cited in Section 1. However, it provides an excellent benchmark because the resulting solutions reflect the extent to which it is possible to achieve system-level goals by designing the entire system at once. The AAO design problem is formulated as shown in Figure 12 by merging the system-level and materials-level design problems in Figure 5. In Table 3, the thrust/weight ratio for the Pareto solutions (Solutions 1, 2, 3, 4, and 5) is compared to the AAO solution. The final column, containing efficiency (in %), was calculated by considering the AAO scenario as the benchmark. For example, efficiency of Solution 3 would be: Efficiency (%) = 4.45/4.55 = 97.75%. By comparing the results, we observe that with only 5 Pareto solutions, the flexibility-based method leads to solutions within 2.25% of the benchmark AAO solution. In fact, 3 of the solutions lie within 7% of the benchmark value. However, such satisfactory results are not guaranteed for each Pareto solution. Some solutions, such as Solutions 1 and 2 in Table 3, compare unfavorably with the benchmark solution, differing by more than 65% in the case of Solution 2. From these observations, we conclude that the overall quality of the solution is likely to increase, from the system perspective, as the number of Pareto solutions increases. Of course, the trend depends on the number of coupled parameters and the nature of the system-level design space (e.g., the complexity of the relationship between the coupled parameters and system-level objectives in terms of discontinuities, nonlinearities, and degree of multimodality). The computational expense of the flexibility-based approach is proportional to the number of Pareto solutions generated. However, this expense is offset by two competing factors: (1) an integrated design problem typically takes much longer to solve because of the increased number of design variables and the execution of coupled analyses sequentially, and (2) the integrated approach requires extensive iterations across scales, whereas the flexibility-based approach requires only one. In an actual design problem, these iterations between design teams would be associated with potentially extensive wait times and delays as collaborating designers make time in their


All-At-Once Design Problem Given ANSYS thermostructural model of the cellular combustor liner Thermodynamic model for system (Sect 3.2.1) with T4, Tlo, Di as inputs Mass model for system with Vliner as input Values of fixed parameters (Do, P3, P4, T3, m ) Find

L , h h h h , Offset Satisfy Constraints and Bounds 1

,

2

,

3

,

4

σ ≤ σ y (T )

0.02 m ≤ DISTi ≤ 0.06 m

0.5 m ≤ Di ≤ 0.68 m

DIST = h1 − 0.5 Di

0.005 m ≤ Offset ≤ 0.01 m

DIST1,2,3 = hi +1 − hi

0.6 m ≤ L ≤ 0.85 m

DIST4 = 0.5 Do − h4

L ≥ Lmin

Do , p3 , p4 , m fixed by system designer

Goals

thrust / weight

( thrust / weight )target

+ d1− − d1+ = 1

Minimize

Zs = f ( d1− )

Figure 12. All-at-once design problem formulation. Table 3. Comparison of Pareto solutions with AAO benchmark solution. SOLUTION NUMBER

AAO 1 2 3 4 5

WEIGHT ON TEMP. AT LINER 100% 0% 50% 75% 25%

WEIGHT ON VOLUME 0% 100% 50% 25% 75%

ENGINE THRUST/WEIGHT (KN/KG) 4.55 3.51 1.57 4.45 4.27 4.39

EFFICIENCY (%)

Benchmark 77.23 34.58 97.75 93.77 96.51

schedules to revisit the design problem in each iteration. Furthermore, the flexibility-based approach offers the benefits cited in Section 1, including the capability of keeping individual designers in the loop, accommodating analysis and synthesis activities that cannot be automated easily, and the general decentralization of the design process.

IV. Closure A flexibility-based approach for solving multiscale design problems is presented. A central feature of the method is the exchange of families or Pareto sets of solutions between collaborating designers operating on different scales. The Pareto solutions provide flexibility for coordinating interdependent design problems that because the sets of solutions embody a variety of values for coupled parameters and a spectrum of tradeoffs between them. In contrast, the conventional approach of exchanging single, point solutions typically leads to extensive iterations across scales because collaborating designers lack the flexibility to choose a satisfactory solution from a variety of options. Accordingly, the latter approach seems to be better suited for problems with fully automated analysis and synthesis that are more amenable to repeated iteration; whereas the flexibility-based approach seems more appropriate for collaboration between designer who are actively involved in the multiscale design process with roles that may include problem formulation, on-going model building and verification, and solution generation and validation. 14 American Institute of Aeronautics and Astronautics

The effectiveness of the approach is illustrated with a multiscale design example involving system-level gas turbine engine design and mesoscale materials design of cellular materials for the engine’s combustor liner. With only a few Pareto solutions, the flexibility-based approach yields results for the example problem that differ from benchmark solutions by less than 2.5%. On-going work involves extending the approach and investigating its effectiveness for problems with more than two scales, applications in industrial settings, and more complex design spaces with nonlinear, multimodal, and discontinuous relationships between coupled parameters and system-level performance.

Acknowledgments We gratefully acknowledge support from the University of Texas at Austin, National Science Foundation grant DMI-0632766, and the Schlumberger Corporation.

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