Proper Orthogonal Decomposition for Reduced-Order Thermal Solution in Hypersonic Aerothermoelastic Simulations

Falkiewicz, Nathan; Cesnik, Carlos E. S.

doi:10.2514/1.j050701

Cited by 69 publications

(56 citation statements)

References 54 publications

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“…The thermoelastic portion of this study is a continuation of previous studies on reduced-order modeling of the heat transfer and structural dynamics problems [20][21][22]. The earliest of these [22] introduced the reduced-order thermoelastic modeling framework, which used proper orthogonal decomposition (POD) for reduction of the thermal problem and a modified modal method for reduction of the structural dynamics problem.…”

Section: Aerodynamicmentioning

confidence: 97%

“…Once the modified stiffness matrix is known at the current time instant, the system is reduced by substituting Eq. (20) into Eq. (18) and premultiplying the system by T ref ; that is,…”

Section: Reduced-order Structural Dynamic Modelmentioning

confidence: 98%

“…A subsequent work [21] employed a quasi-steady aerothermoelastic time-marching procedure to assess the effect of thermal loads on the aerodynamic forces over a control surface. A more recent work [20] described an extension of the previous papers to unsteady form and specifically addressed the use of POD with time-dependent boundary conditions. The current effort makes use of the thermoelastic portion of the framework described in [20] along with a reduced-order aerothermal model.…”

Section: Aerodynamicmentioning

confidence: 98%

“…A more recent work [20] described an extension of the previous papers to unsteady form and specifically addressed the use of POD with time-dependent boundary conditions. The current effort makes use of the thermoelastic portion of the framework described in [20] along with a reduced-order aerothermal model.…”

Section: Aerodynamicmentioning

confidence: 98%

See 3 more Smart Citations

Reduced-Order Aerothermoelastic Framework for Hypersonic Vehicle Control Simulation

Falkiewicz¹,

Cesnik²,

Crowell³

et al. 2011

AIAA Journal

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Hypersonic vehicle control system design and simulation require models that contain a low number of states. Modeling of hypersonic vehicles is complicated due to complex interactions between aerodynamic heating, heat transfer, structural dynamics, and aerodynamics. Although there exist techniques for analyzing the effects of each of the various disciplines, these methods often require solution of large systems of equations, which is infeasible within a control design and evaluation environment. This work presents an aerothermoelastic framework with reducedorder aerothermal, heat transfer, and structural dynamic models for time-domain simulation of hypersonic vehicles. Details of the reduced-order models are given, and a representative hypersonic vehicle control surface used for the study is described. The methodology is applied to a representative structure to provide insight into the importance of aerothermoelastic effects on vehicle performance. The effect of aerothermoelasticity on total lift and drag is found to result in up to an 8% change in lift and a 21% change in drag with respect to a rigid control surface for the four trajectories considered. An iterative routine is used to determine the angle of attack needed to match the lift of the deformed control surface to that of a rigid one at successive time instants. Application of the routine to different cruise trajectories shows a maximum departure from the initial angle of attack of 8%.corresponding to ith column of A C = correlation matrix c = modal coordinate of thermal proper orthogonal decomposition basis vector Cx = correlation model for kriging c p = specific heat at constant pressure d = structural modal coordinates, kriging sample point E = modulus of elasticity F = thermal load vector of full system in physical space f = generalized thermal load vector of reduced system in modal space F s = structural load vector of full system in physical space f s = generalized structural load vector of reduced system in modal space H i = coefficient matrices for integration of equations of motion h i = thickness of ith layer of thermal protection system K = thermal conductivity matrix of full system in physical space k = generalized thermal conductivity matrix of reduced system in modal space K G = geometric stiffness matrix K s = structural stiffness matrix k s = generalized stiffness matrix of reduced system in modal space k T = thermal conductivity of material K s = modified structural stiffness matrix L = aerodynamic lift, length M = thermal capacitance matrix of full system in physical space, Mach number m = generalized thermal capacitance matrix of reduced system in modal space M s = structural mass matrix of full system in physical space m s = generalized mass matrix of reduced system in modal space n = number of proper orthogonal decomposition snapshots n e = number of snapshots for kriging evaluation cases n k = number of kriging snapshots n s = number of structural parameters in reduced-order aerothermodynamic model n t = number of thermal parameters...

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Section: Aerodynamicmentioning

confidence: 97%

“…Once the modified stiffness matrix is known at the current time instant, the system is reduced by substituting Eq. (20) into Eq. (18) and premultiplying the system by T ref ; that is,…”

Section: Reduced-order Structural Dynamic Modelmentioning

confidence: 98%

Section: Aerodynamicmentioning

confidence: 98%

Section: Aerodynamicmentioning

confidence: 98%

See 2 more Smart Citations

Reduced-Order Aerothermoelastic Framework for Hypersonic Vehicle Control Simulation

Falkiewicz¹,

Cesnik²,

Crowell³

et al. 2011

AIAA Journal

Self Cite

View full text Add to dashboard Cite

show abstract

“…[11][12][13][14] The earliest of these 11 introduced the reduced-order thermoelastic modeling framework which utilized Proper Orthogonal Decomposition (POD) for reduction of the thermal problem and a modified modal method for reduction of the structural dynamics problem. A subsequent work 12 employed a quasi-steady aerothermoelastic time-marching procedure to assess the effect of thermal loads on the aerodynamic forces over a control surface.…”

Section: Ia Previous Work On Aerothermoelastic Modelingmentioning

confidence: 99%

Enhanced Modal Solutions for Structural Dynamics in Aerothermoelastic Analysis

Falkiewicz

Cesnik

2011

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference

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Hypersonic vehicle design and simulation require models that are of low order. Modeling of hypersonic vehicles is complicated due to complex interactions between aerodynamic heating, heat transfer, structural dynamics, and aerodynamics in the hypersonic regime. This work focuses on the development of efficient modal solutions for structural dynamics of hypersonic vehicle structures under transient thermal loads. The problem is outlined and aerothermoelastic coupling mechanisms are described. A previously developed reducedorder, time-domain aerothermoelastic simulation framework is used as the starting point for this study. This paper focuses on three main modeling areas: 1) The effect of aerodynamic heating on the evolution of free vibration modes shapes and frequencies is examined, 2) A surrogate modeling technique is employed for directly updating the generalized stiffness matrix and thermal loads based on the transient temperature distribution and 3) Basis augmentation techniques are employed in order to obtain more accurate solutions for the structural dynamic response. The techniques to be studied are described and applied to a representative hypersonic vehicle elevator structure. Nomenclature= vector of inputs to function C(b, X) = kriging correlation matrix C = correlation matrix c = vector of POD modal coordinates c = vector of POD modal coordinates for transformed system c p = specific heat at constant pressure d = structural modal coordinates, kriging sample point E = modulus of elasticity F T = thermal load vector of full system in physical space F S = structural load vector of full system in physical space Full = solution vector of full-order model f T = generalized thermal load vector of reduced system in modal space f S = generalized structural load vector of reduced system in modal space G(X (i) , X (j) ) = Gaussian correlation function for kriging model H i = coefficient matrices in numerical integration for structural response h = altitude h i = thickness of i-th layer of thermal protection system K T = thermal conductivity matrix of full system in physical space K G = geometric stiffness matrix K S = structural stiffness matrix * Ph.D. Candidate, K * S = modified structural stiffness matrix k * S = generalized stiffness matrix of reduced system in modal space k T = generalized thermal conductivity matrix of reduced system in modal space L = lower triangular factor in decomposition of K * S L ∞ = L ∞ error l = number of augmented mode shapes lb i = lower bound for i-th POD modal coordinate M = Mach number M T = thermal capacitance matrix of full system in physical space M S = structural mass matrix of full system in physical space m T = generalized thermal capacitance matrix of reduced system in modal space m S = generalized mass matrix of reduced system in modal space MAC i,j = modal assurance criterion value corresponding to modes i and j max i = maximum value of i-th POD modal coordinate min i = minimum value of i-th POD modal coordinate MS = margin of safety with respect to POD bounds NRMSE =...

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