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NASA NTRS1992

Observations on computational methodologies for use in large-scale, gradient-based, multidisciplinary design

Various computational methodologies relevant to large-scale multidisciplinary gradient-based optimization for engineering systems design problems are examined with emphasis on the situation where one or more discipline responses required by the optimized design procedure involve the solution of a system of nonlinear partial differential equations. Such situations occur when advanced CFD codes are applied in a multidisciplinary procedure for optimizing an aerospace vehicle design. A technique for satisfying the multidisciplinary design requirements for gradient information is presented. The technique is shown to permit some leeway in the CFD algorithms which can be used, an expansion to 3D problems, and straightforward use of other computational methodologies.

Newman, P. A.↗

NASA NTRS1992

Observations on computational methodologies for use in large-scale, gradient-based, multidisciplinary design incorporating advanced CFD codes

How a combination of various computational methodologies could reduce the enormous computational costs envisioned in using advanced CFD codes in gradient based optimized multidisciplinary design (MdD) procedures is briefly outlined. Implications of these MdD requirements upon advanced CFD codes are somewhat different than those imposed by a single discipline design. A means for satisfying these MdD requirements for gradient information is presented which appear to permit: (1) some leeway in the CFD solution algorithms which can be used; (2) an extension to 3-D problems; and (3) straightforward use of other computational methodologies. Many of these observations have previously been discussed as possibilities for doing parts of the problem more efficiently; the contribution here is observing how they fit together in a mutually beneficial way.

Newman, P. A.↗

NASA NTRS1991

Sensitivity analysis, approximate analysis, and design optimization for internal and external viscous flows

A gradient-based design optimization strategy for practical aerodynamic design applications is presented, which uses the 2D thin-layer Navier-Stokes equations. The strategy is based on the classic idea of constructing different modules for performing the major tasks such as function evaluation, function approximation and sensitivity analysis, mesh regeneration, and grid sensitivity analysis, all driven and controlled by a general-purpose design optimization program. The accuracy of aerodynamic shape sensitivity derivatives is validated on two viscous test problems: internal flow through a double-throat nozzle and external flow over a NACA 4-digit airfoil. A significant improvement in aerodynamic performance has been achieved in both cases. Particular attention is given to a consistent treatment of the boundary conditions in the calculation of the aerodynamic sensitivity derivatives for the classic problems of external flow over an isolated lifting airfoil on 'C' or 'O' meshes.

Taylor, Arthur C., III↗

NASA NTRS1996

Observations Regarding Use of Advanced CFD Analysis, Sensitivity Analysis, and Design Codes in MDO

Observations regarding the use of advanced computational fluid dynamics (CFD) analysis, sensitivity analysis (SA), and design codes in gradient-based multidisciplinary design optimization (MDO) reflect our perception of the interactions required of CFD and our experience in recent aerodynamic design optimization studies using CFD. Sample results from these latter studies are summarized for conventional optimization (analysis - SA codes) and simultaneous analysis and design optimization (design code) using both Euler and Navier-Stokes flow approximations. The amount of computational resources required for aerodynamic design using CFD via analysis - SA codes is greater than that required for design codes. Thus, an MDO formulation that utilizes the more efficient design codes where possible is desired. However, in the aerovehicle MDO problem, the various disciplines that are involved have different design points in the flight envelope; therefore, CFD analysis - SA codes are required at the aerodynamic 'off design' points. The suggested MDO formulation is a hybrid multilevel optimization procedure that consists of both multipoint CFD analysis - SA codes and multipoint CFD design codes that perform suboptimizations.

Newman, Perry A.↗

NASA NTRS1993

Formulation for Simultaneous Aerodynamic Analysis and Design Optimization

An efficient approach for simultaneous aerodynamic analysis and design optimization is presented. This approach does not require the performance of many flow analyses at each design optimization step, which can be an expensive procedure. Thus, this approach brings us one step closer to meeting the challenge of incorporating computational fluid dynamic codes into gradient-based optimization techniques for aerodynamic design. An adjoint-variable method is introduced to nullify the effect of the increased number of design variables in the problem formulation. The method has been successfully tested on one-dimensional nozzle flow problems, including a sample problem with a normal shock. Implementations of the above algorithm are also presented that incorporate Newton iterations to secure a high-quality flow solution at the end of the design process. Implementations with iterative flow solvers are possible and will be required for large, multidimensional flow problems.

Hou, G. W.↗

NASA NTRS1994

Multidisciplinary design optimization using genetic algorithms

Multidisciplinary design optimization (MDO) is an important step in the conceptual design and evaluation of launch vehicles since it can have a significant impact on performance and life cycle cost. The objective is to search the system design space to determine values of design variables that optimize the performance characteristic subject to system constraints. Gradient-based optimization routines have been used extensively for aerospace design optimization. However, one limitation of gradient based optimizers is their need for gradient information. Therefore, design problems which include discrete variables can not be studied. Such problems are common in launch vehicle design. For example, the number of engines and material choices must be integer values or assume only a few discrete values. In this study, genetic algorithms are investigated as an approach to MDO problems involving discrete variables and discontinuous domains. Optimization by genetic algorithms (GA) uses a search procedure which is fundamentally different from those gradient based methods. Genetic algorithms seek to find good solutions in an efficient and timely manner rather than finding the best solution. GA are designed to mimic evolutionary selection. A population of candidate designs is evaluated at each iteration, and each individual's probability of reproduction (existence in the next generation) depends on its fitness value (related to the value of the objective function). Progress toward the optimum is achieved by the crossover and mutation operations. GA is attractive since it uses only objective function values in the search process, so gradient calculations are avoided. Hence, GA are able to deal with discrete variables. Studies report success in the use of GA for aircraft design optimization studies, trajectory analysis, space structure design and control systems design. In these studies reliable convergence was achieved, but the number of function evaluations was large compared with efficient gradient methods. Applicaiton of GA is underway for a cost optimization study for a launch-vehicle fuel-tank and structural design of a wing. The strengths and limitations of GA for launch vehicle design optimization is studied.

Unal, Resit↗

NASA NTRS1990

Multi-disciplinary optimization of aeroservoelastic systems

Efficient analytical and computational tools for simultaneous optimal design of the structural and control components of aeroservoelastic systems are presented. The optimization objective is to achieve aircraft performance requirements and sufficient flutter and control stability margins with a minimal weight penalty and without violating the design constraints. Analytical sensitivity derivatives facilitate an efficient optimization process which allows a relatively large number of design variables. Standard finite element and unsteady aerodynamic routines are used to construct a modal data base. Minimum State aerodynamic approximations and dynamic residualization methods are used to construct a high accuracy, low order aeroservoelastic model. Sensitivity derivatives of flutter dynamic pressure, control stability margins and control effectiveness with respect to structural and control design variables are presented. The performance requirements are utilized by equality constraints which affect the sensitivity derivatives. A gradient-based optimization algorithm is used to minimize an overall cost function. A realistic numerical example of a composite wing with four controls is used to demonstrate the modeling technique, the optimization process, and their accuracy and efficiency.

Karpel, Mordechay↗

NASA NTRS1992

Multidisciplinary optimization of aeroservoelastic systems using reduced-size models

Karpel, Mordechay↗

NASA NTRS1995

Sensitivity Analysis of Wing Aeroelastic Responses

Design for prevention of aeroelastic instability (that is, the critical speeds leading to aeroelastic instability lie outside the operating range) is an integral part of the wing design process. Availability of the sensitivity derivatives of the various critical speeds with respect to shape parameters of the wing could be very useful to a designer in the initial design phase, when several design changes are made and the shape of the final configuration is not yet frozen. These derivatives are also indispensable for a gradient-based optimization with aeroelastic constraints. In this study, flutter characteristic of a typical section in subsonic compressible flow is examined using a state-space unsteady aerodynamic representation. The sensitivity of the flutter speed of the typical section with respect to its mass and stiffness parameters, namely, mass ratio, static unbalance, radius of gyration, bending frequency, and torsional frequency is calculated analytically. A strip theory formulation is newly developed to represent the unsteady aerodynamic forces on a wing. This is coupled with an equivalent plate structural model and solved as an eigenvalue problem to determine the critical speed of the wing. Flutter analysis of the wing is also carried out using a lifting-surface subsonic kernel function aerodynamic theory (FAST) and an equivalent plate structural model. Finite element modeling of the wing is done using NASTRAN so that wing structures made of spars and ribs and top and bottom wing skins could be analyzed. The free vibration modes of the wing obtained from NASTRAN are input into FAST to compute the flutter speed. An equivalent plate model which incorporates first-order shear deformation theory is then examined so it can be used to model thick wings, where shear deformations are important. The sensitivity of natural frequencies to changes in shape parameters is obtained using ADIFOR. A simple optimization effort is made towards obtaining a minimum weight design of the wing, subject to flutter constraints, lift requirement constraints for level flight and side constraints on the planform parameters of the wing using the IMSL subroutine NCONG, which uses successive quadratic programming.

Issac, Jason Cherian↗

NASA NTRS1993

A new approach to mixed H2/H infinity controller synthesis using gradient-based parameter optimization methods

In the past few years, the mixed H(sub 2)/H-infinity control problem has been the object of much research interest since it allows the incorporation of robust stability into the LQG framework. The general mixed H(sub 2)/H-infinity design problem has yet to be solved analytically. Numerous schemes have considered upper bounds for the H(sub 2)-performance criterion and/or imposed restrictive constraints on the class of systems under investigation. Furthermore, many modern control applications rely on dynamic models obtained from finite-element analysis and thus involve high-order plant models. Hence the capability to design low-order (fixed-order) controllers is of great importance. In this research a new design method was developed that optimizes the exact H(sub 2)-norm of a certain subsystem subject to robust stability in terms of H-infinity constraints and a minimal number of system assumptions. The derived algorithm is based on a differentiable scalar time-domain penalty function to represent the H-infinity constraints in the overall optimization. The scheme is capable of handling multiple plant conditions and hence multiple performance criteria and H-infinity constraints and incorporates additional constraints such as fixed-order and/or fixed structure controllers. The defined penalty function is applicable to any constraint that is expressible in form of a real symmetric matrix-inequity.

Ly, Uy-Loi↗

NASA NTRS1993

A homotopy algorithm for digital optimal projection control GASD-HADOC

The linear-quadratic-gaussian (LQG) compensator was developed to facilitate the design of control laws for multi-input, multi-output (MIMO) systems. The compensator is computed by solving two algebraic equations for which standard closed-loop solutions exist. Unfortunately, the minimal dimension of an LQG compensator is almost always equal to the dimension of the plant and can thus often violate practical implementation constraints on controller order. This deficiency is especially highlighted when considering control-design for high-order systems such as flexible space structures. This deficiency motivated the development of techniques that enable the design of optimal controllers whose dimension is less than that of the design plant. A homotopy approach based on the optimal projection equations that characterize the necessary conditions for optimal reduced-order control. Homotopy algorithms have global convergence properties and hence do not require that the initializing reduced-order controller be close to the optimal reduced-order controller to guarantee convergence. However, the homotopy algorithm previously developed for solving the optimal projection equations has sublinear convergence properties and the convergence slows at higher authority levels and may fail. A new homotopy algorithm for synthesizing optimal reduced-order controllers for discrete-time systems is described. Unlike the previous homotopy approach, the new algorithm is a gradient-based, parameter optimization formulation and was implemented in MATLAB. The results reported may offer the foundation for a reliable approach to optimal, reduced-order controller design.

Collins, Emmanuel G., Jr.↗

NASA NTRS1997

Efficient Gradient-Based Shape Optimization Methodology Using Inviscid/Viscous CFD

The formerly developed preconditioned-biconjugate-gradient (PBCG) solvers for the analysis and the sensitivity equations had resulted in very large error reductions per iteration; quadratic convergence was achieved whenever the solution entered the domain of attraction to the root. Its memory requirement was also lower as compared to a direct inversion solver. However, this memory requirement was high enough to preclude the realistic, high grid-density design of a practical 3D geometry. This limitation served as the impetus to the first-year activity (March 9, 1995 to March 8, 1996). Therefore, the major activity for this period was the development of the low-memory methodology for the discrete-sensitivity-based shape optimization. This was accomplished by solving all the resulting sets of equations using an alternating-direction-implicit (ADI) approach. The results indicated that shape optimization problems which required large numbers of grid points could be resolved with a gradient-based approach. Therefore, to better utilize the computational resources, it was recommended that a number of coarse grid cases, using the PBCG method, should initially be conducted to better define the optimization problem and the design space, and obtain an improved initial shape. Subsequently, a fine grid shape optimization, which necessitates using the ADI method, should be conducted to accurately obtain the final optimized shape. The other activity during this period was the interaction with the members of the Aerodynamic and Aeroacoustic Methods Branch of Langley Research Center during one stage of their investigation to develop an adjoint-variable sensitivity method using the viscous flow equations. This method had algorithmic similarities to the variational sensitivity methods and the control-theory approach. However, unlike the prior studies, it was considered for the three-dimensional, viscous flow equations. The major accomplishment in the second period of this project (March 9, 1996 to March 8, 1997) was the extension of the shape optimization methodology for the Thin-Layer Navier-Stokes equations. Both the Euler-based and the TLNS-based analyses compared with the analyses obtained using the CFL3D code. The sensitivities, again from both levels of the flow equations, also compared very well with the finite-differenced sensitivities. A fairly large set of shape optimization cases were conducted to study a number of issues previously not well understood. The testbed for these cases was the shaping of an arrow wing in Mach 2.4 flow. All the final shapes, obtained either from a coarse-grid-based or a fine-grid-based optimization, using either a Euler-based or a TLNS-based analysis, were all re-analyzed using a fine-grid, TLNS solution for their function evaluations. This allowed for a more fair comparison of their relative merits. From the aerodynamic performance standpoint, the fine-grid TLNS-based optimization produced the best shape, and the fine-grid Euler-based optimization produced the lowest cruise efficiency.

Baysal, Oktay↗

NASA NTRS1995

Contribution to the optimal shape design of two-dimensional internal flows with embedded shocks

We explore the practicability of optimal shape design for flows modeled by the Euler equations. We define a functional whose minimum represents the optimality condition. The gradient of the functional with respect to the geometry is calculated with the Lagrange multipliers, which are determined by solving a co-state equation. The optimization problem is then examined by comparing the performance of several gradient-based optimization algorithms. In this formulation, the flow field can be computed to an arbitrary order of accuracy. Finally, some results for internal flows with embedded shocks are presented, including a case for which the solution to the inverse problem does not belong to the design space.

Iollo, Angelo↗

NASA NTRS1993

Multidisciplinary Design Techniques Applied to Conceptual Aerospace Vehicle Design

Multidisciplinary design optimization (MDO) is an emerging discipline within aerospace engineering. Its goal is to bring structure and efficiency to the complex design process associated with advanced aerospace launch vehicles. Aerospace vehicles generally require input from a variety of traditional aerospace disciplines - aerodynamics, structures, performance, etc. As such, traditional optimization methods cannot always be applied. Several multidisciplinary techniques and methods were proposed as potentially applicable to this class of design problem. Among the candidate options are calculus-based (or gradient-based) optimization schemes and parametric schemes based on design of experiments theory. A brief overview of several applicable multidisciplinary design optimization methods is included. Methods from the calculus-based class and the parametric class are reviewed, but the research application reported focuses on methods from the parametric class. A vehicle of current interest was chosen as a test application for this research. The rocket-based combined-cycle (RBCC) single-stage-to-orbit (SSTO) launch vehicle combines elements of rocket and airbreathing propulsion in an attempt to produce an attractive option for launching medium sized payloads into low earth orbit. The RBCC SSTO presents a particularly difficult problem for traditional one-variable-at-a-time optimization methods because of the lack of an adequate experience base and the highly coupled nature of the design variables. MDO, however, with it's structured approach to design, is well suited to this problem. The result of the application of Taguchi methods, central composite designs, and response surface methods to the design optimization of the RBCC SSTO are presented. Attention is given to the aspect of Taguchi methods that attempts to locate a 'robust' design - that is, a design that is least sensitive to uncontrollable influences on the design. Near-optimum minimum dry weight solutions are determined for the vehicle. A summary and evaluation of the various parametric MDO methods employed in the research are included. Recommendations for additional research are provided.

Olds, John Robert↗

NASA NTRS1995

Wing-section optimization for supersonic viscous flow

To improve the shape of a supersonic wing, an automated method that also includes higher fidelity to the flow physics is desirable. With this impetus, an aerodynamic optimization methodology incorporating thin-layer Navier-Stokes equations and sensitivity analysis had been previously developed. Prior to embarking upon the wind design task, the present investigation concentrated on testing the feasibility of the methodology, and the identification of adequate problem formulations, by defining two-dimensional, cost-effective test cases. Starting with two distinctly different initial airfoils, two independent shape optimizations resulted in shapes with similar features: slightly cambered, parabolic profiles with sharp leading- and trailing-edges. Secondly, the normal section to the subsonic portion of the leading edge, which had a high normal angle-of-attack, was considered. The optimization resulted in a shape with twist and camber which eliminated the adverse pressure gradient, hence, exploiting the leading-edge thrust. The wing section shapes obtained in all the test cases had the features predicted by previous studies. Therefore, it was concluded that the flowfield analyses and sensitivity coefficients were computed and fed to the present gradient-based optimizer correctly. Also, as a result of the present two-dimensional study, suggestions were made for the problem formulations which should contribute to an effective wing shape optimization.

Item, Cem C.↗

NASA NTRS1995

Wing-section optimization for supersonic viscous flows

The recent interest in High Speed Civil Transport (HSCT) has resulted in renewed research studies of optimized supersonic cruise transport configurations. Incorporation of flow viscosity effects in the design process of such a supersonic wing is currently under investigation. This may lead to more accurate problem formulations and, in turn, greater aerodynamic efficiency than can be obtained by the traditional, inviscid, linear theories. In this context, for a design code to be a candidate for a complex optimization problem, such as three-dimensional viscous supersonic wing design, it should be validated using simpler building-block shapes. To optimize the shape of a supersonic wing, an automated method that also includes higher fidelity to the flow physics is desirable. With this impetus, an aerodynamic optimization methodology incorporating Navier-Stokes equations and sensitivity analysis had been previously developed. Prior to embarking upon the wing design task, the present investigation concentrated on testing the flexibility of the methodology, and the identification of adequate problem formulations, by defining two-dimensional, cost-effective test cases. Starting with two distinctly different initial airfoils, two independent shape optimizations resulted in shapes with very similar features. Secondly, the normal section to the subsonic portion of the leading edge, which had a high normal angle-of-attack, was considered. The optimization resulted in a shape with twist and camber, which eliminated the adverse pressure gradient, hence, exploiting the leading-edge thrust. The wing section shapes obtained in all the test cases had the features predicted by previous studies. Therefore, it was concluded that the flowfield analyses and the sensitivity coefficients were computed and fed to the present gradient-based optimizer correctly. Also, as a result of the present two-dimensional study, suggestions were made for problem formulations which should contribute to an effective wing shape optimization.

Item, Cem C.↗

NASA NTRS1997

Three-Dimensional Viscous Alternating Direction Implicit Algorithm and Strategies for Shape Optimization

A gradient-based shape optimization based on quasi-analytical sensitivities has been extended for practical three-dimensional aerodynamic applications. The flow analysis has been rendered by a fully implicit, finite-volume formulation of the Euler and Thin-Layer Navier-Stokes (TLNS) equations. Initially, the viscous laminar flow analysis for a wing has been compared with an independent computational fluid dynamics (CFD) code which has been extensively validated. The new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4 with coarse- and fine-grid based computations performed with Euler and TLNS equations. The influence of the initial constraints on the geometry and aerodynamics of the optimized shape has been explored. Various final shapes generated for an identical initial problem formulation but with different optimization path options (coarse or fine grid, Euler or TLNS), have been aerodynamically evaluated via a common fine-grid TLNS-based analysis. The initial constraint conditions show significant bearing on the optimization results. Also, the results demonstrate that to produce an aerodynamically efficient design, it is imperative to include the viscous physics in the optimization procedure with the proper resolution. Based upon the present results, to better utilize the scarce computational resources, it is recommended that, a number of viscous coarse grid cases using either a preconditioned bi-conjugate gradient (PbCG) or an alternating-direction-implicit (ADI) method, should initially be employed to improve the optimization problem definition, the design space and initial shape. Optimized shapes should subsequently be analyzed using a high fidelity (viscous with fine-grid resolution) flow analysis to evaluate their true performance potential. Finally, a viscous fine-grid-based shape optimization should be conducted, using an ADI method, to accurately obtain the final optimized shape.

Pandya, Mohagna J.↗

NASA NTRSDate not supplied

Aerothermal Shape Optimization of Actively-Cooled Battery Packs using Conjugate Heat Transfer

Thermal management for battery is important for electric aircraft because battery temperature is critically important to vehicle safety, and it also has direct impact on the efficiency of the battery system. Because ambient air is a readily available resource for aircraft, this paper considers an active cooling concept with forced convection of ambient air through the battery pack. Conjugate heat transfer analysis is used to solve the coupled aero-thermal problem, which consists of a finite-volume computational fluid dynamics solver for the fluid domain, and a conduction heat transfer solver for the solid domain. A mixed Neumann and Dirichlet boundary condition is developed for the fluid-solid interface, which allows the solid domain to completely submerge in the fluid domain. A gradient-based optimization method is adopted, and the discrete adjoint approach implemented in DAFoam is used to efficiently compute the gradients. The aero-thermal coupling for primal analysis and gradient computation is handled using the OpenMDAO-based MPhys framework. A constant heat source is prescribed for the battery cells, and the battery shape (design variable) is optimized to minimize cooling pump power and battery weight (composite objective function) while keeping the battery temperature below a threshold (constraint). The optimized design achieves a 44.6% and 1.5% reduction in the cooling pump power and battery weight, respectively, and the maximal temperature constraint is satisfied. This work has the potential to reduce battery-pack weight, improve performance, and reduce the weight of thermal management systems for electric vertical take-off and landing aircraft.

thermal management↗