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Equivalence of perturbation method, Newtons method, and adjoint-perturbation method for solving boundary value problems from variational calculus treatment of optimal control problems
Oscillating wing aerodynamic load boundary value problem reduction to sequence of steady lifting- surface problems
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Finite-difference Navier-Stokes calculations for unsteady, three-dimensional, incompressible, viscous flows induced by initial vorticity distributions are presented and discussed in this paper. The initial vorticity distributions are assumed to be embedded in a flow field of infinite extent that is quiescent at infinity. These vorticity distributions are typical of vortex rings and other closed vortical tubes or structures. Such structures are important elements in fluid flows such as jets, atmospheric convection and the far-field wakes of aircraft; studies of their interaction may aid in an understanding of complex fluid flows. The calculations employ a method recently proposed by Ting to approximate the infinite-domain boundary value problem with a finite boundary computational domain, and this method is shown to yield accurate three-dimensional results for reasonable expenditures of computer time. Because of the efficiency of the boundary condition technique and the resulting Navier-Stokes code, a 16-bit minicomputer with virtual memory was capable of performing the calculations for the unsteady motion of two obliquely colliding vortex rings. The results of these calculations are presented in the paper.
Underintegrated methods are investigated with respect to their stability and convergence properties. The focus was on identifying regions where they work and regions where techniques such as hourglass viscosity and hourglass control can be used. Results obtained show that underintegrated methods typically lead to finite element stiffness with spurious modes in the solution. However, problems exist (scalar elliptic boundary value problems) where underintegrated with hourglass control yield convergent solutions. Also, stress averaging in underintegrated stiffness calculations does not necessarily lead to stable or convergent stress states.
The paper presents an application of adaptive finite element methods to the modeling of low-cycle continuum damage and life prediction of high-temperature components. The major objective is to provide automated and accurate modeling of damaged zones through adaptive mesh refinement and adaptive time-stepping methods. The damage modeling methodology is implemented in an usual way by embedding damage evolution in the transient nonlinear solution of elasto-viscoplastic deformation problems. This nonlinear boundary-value problem is discretized by adaptive finite element methods. The automated h-adaptive mesh refinements are driven by error indicators, based on selected principal variables in the problem (stresses, non-elastic strains, damage, etc.). In the time domain, adaptive time-stepping is used, combined with a predictor-corrector time marching algorithm. The time selection is controlled by required time accuracy. In order to take into account strong temperature dependency of material parameters, the nonlinear structural solution a coupled with thermal analyses (one-way coupling). Several test examples illustrate the importance and benefits of adaptive mesh refinements in accurate prediction of damage levels and failure time.
Thermodynamics and higher-order fluid theories - boundary value problems for unsteady simple shear flow
A new inverse method for the aerodynamic design of airfoils is presented for subcritical flows. The pressure distribution in this method can be prescribed as a function of the arclength of the still unknown body. It is shown that this inverse problem is mathematically equivalent to solving only one nonlinear boundary value problem subject to known Dirichlet data on the boundary.
A numerical algorithm is presented for solving laminar, steady, supersonic interacting boundary-layer flows for quasi-three-dimensional configurations. The interaction problem is treated as a boundary-value problem and a salient feature of the scheme is the direct implementation of the downstream boundary condition. Solutions are presented for axisymmetric and swept (yawed) compression ramps for both adiabatic and heat transfer conditions over a Mach number range of 2-6. The results are in good agreement with experimental data and existing theories for axisymmetric cases. For the swept (yawed) configurations, lack of experimental data makes a direct comparison impossible, but the present solutions are found to be in qualitative agreement with earlier studies. In addition it is shown that the trends obtained here for the sweep effects are well predicted by a simple extension of the two-dimensional asymptotic theory.
Quasi-optimum control technique based on Hamiltonian to solve two-point boundary value problem for nonlinear system
Rarefied gas dynamics - solution of boundary value problem of boltzmann equation at arbitrary knudsen number by transfer equations
Matrix algebra solution, using coordinate systems transformations, to boundary value problem for structures composed of shell elements
From a stochastic control perspective, the Schrödinger bridge is a density-valued continuous curve parameterized by time that connects a given pair of initial and terminal probability densities via minimum effort controlled Brownian motion. The control-affine Schrödinger bridge extends this idea to a generic control-affine Itô diffusion, possibly with an additive state cost. Here, in this letter, we recast the necessary conditions of optimality for the control-affine Schrödinger bridge problem as a two point boundary value problem for a quantum mechanical Schrödinger PDE with complex potential. This complex-valued potential is a generalization of the real-valued Bohm potential in quantum mechanics. Our derived potential is akin to the optical potential in nuclear physics where the real part of the potential encodes elastic scattering (transmission of wave function), and the imaginary part encodes inelastic scattering (absorption of wave function). The key takeaway is that the process noise that drives the evolution of probability densities induces an absorbing medium in the evolution of wave function. These results make new connections between control theory and non-equilibrium statistical mechanics through the lens of quantum mechanics.
Invariant imbedding equations for transport problems derived and applied to boundary value and initial value problems
A new inverse method for aerodynamic design of airfols is presented for subcritical flows. The pressure distribution in this method can be prescribed as a function of the arc length of the as-yet unknown body. This inverse problem is shown to be mathematically equivalent to solving only one nonlinear boundary value problem subject to known Dirichlet data on the boundary. The solution to this problem determines the airfoil, the freestream Mach number, and the upstream flow direction. The existence of a solution to a given pressure distribution is discussed. The method is easy to implement and extremely efficient. A series of results for which comparisons are made with the known airfoils is presented.
A new inverse method for aerodynamic design of subcritical airfoils is presented. The pressure distribution in this method can be prescribed in a natural way, i.e. as a function of arclength of the as yet unknown body. This inverse problem is shown to be mathematically equivalent to solving a single nonlinear boundary value problem subject to known Dirichlet data on the boundary. The solution to this problem determines the airfoil, the free stream Mach number M(sub x) and the upstream flow direction theta(sub x). The existence of a solution for any given pressure distribution is discussed. The method is easy to implement and extremely efficient. We present a series of results for which comparisons are made with the known airfoils.
Equitriangular integral transform for solving boundary value problems in viscous flow and heat transfer
Solutions and parametric computer study for class of boundary value problems of time dependent temperature distribution in two layer, composite slab