Future high-speed flight vehicles will be exposed to long-duration aerodynamic heating. The resulting thermal expansion can result in significant nonlinear deformation. As a result, detailed simulations of long portions of the vehicle's trajectory must be performed to account for path-dependent structural response. Recent developments in nonlinear reduced-order models have demonstrated orders-of-magnitude reductions in computation time for acoustic response prediction simulations. Coupling of these structural nonlinear models with reducedorder thermal models could result in similar savings when applied to long-duration quasistatic thermoelastic simulation. This paper describes the application of reduced-order methods to perform long duration thermoelastic simulations. The methods are applied to an example problem representing a ramp panel on a high-speed vehicle. Results show good agreement of predicted displacements compared to a full-order finite element simulation.
NomenclatureA r ijk = cubic stiffness coefficient in the nonlinear stiffness function B r ij = quadratic stiffness coefficient in the nonlinear stiffness function D, D' = data matrix of linear displacements and its approximation D T , D T ' = data matrix of thermal fields and its approximation E = Young's Modulus f, f T , f = force vectors: external, thermal, and modal K , K T = linear stiffness matrices: at a reference temperature and at the thermal field (physical space) K, K t = modal linear stiffness matrix at a reference temperature and the tangent modal stiffness K ΔT , K ΔT (i) = modal linear stiffness matrices at the thermal field and at the i-th basis K1 , K2 = quadratic and cubic stiffness matrices, respectively (physical space) M = mass matrix N = number of degrees-of-freedom P = number of time intervals S, S' = diagonal matrix of singular values and a truncated version S T , S T ' = diagonal matrix of singular values and a truncated version (thermal) U, U' = left hand eigenvector matrix in the singular value decomposition (SVD) and a truncated version U T , U T ' = left hand eigenvector matrix in the SVD and a truncated version (thermal) u i , v i = left and right eigenvectors from the SVD V, V' = right hand eigenvector matrix in the SVD and a truncated version V T , V T ' = right hand eigenvector matrix in the SVD and a truncated version (thermal) e = approximation error based on the Frobenius norm h = panel thickness m, n, p = number of thermal basis vectors, structural basis vectors, and membrane basis vectors q = vector of structural modal coordinates t = time w, w L = vector of structural displacements and the linear version = matrix of displacement basis vectors = vector of nonlinear structural stiffness functions = matrices of nonlinear structural stiffness functions 1 Senior Aerospace Engineer, Structural Sciences Center, AFRL/RBSM. 2 = matrix of normal mode shapes for the structure = matrix of thermal basis = coefficient of thermal expansion i = displacement basis vector fq = variations of the mod...