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NASA NTRS1977

Solution of transient optimization problems by using an algorithm based on nonlinear programming

A new algorithm is presented for solution of dynamic optimization problems which are nonlinear in the state variables and linear in the control variables. It is shown that the optimal control is bang-bang. A nominal bang-bang solution is found which satisfies the system equations and constraints, and influence functions are generated which check the optimality of the solution. Nonlinear optimization (gradient search) techniques are used to find the optimal solution. The algorithm is used to find a minimum time acceleration for a turbofan engine.

Teren, F.↗

NASA NTRS1978

A split band-Cholesky equation solving strategy for finite element analysis of transient field problems

The paper describes the split-Cholesky strategy for banded matrices arising from the large systems of equations in certain fluid mechanics problems. The basic idea is that for a banded matrix the computation can be carried out in pieces, with only a small portion of the matrix residing in core. Mesh considerations are discussed by demonstrating the manner in which the assembly of finite element equations proceeds for linear trial functions on a triangular mesh. The FORTRAN code which implements the out-of-core decomposition strategy for banded symmetric positive definite matrices (mass matrices) of a coupled initial value problem is given.

Cooke, C. H.↗

NASA NTRS1993

On the generalized VIP time integral methodology for transient thermal problems

The paper describes the development and applicability of a generalized VIrtual-Pulse (VIP) time integral method of computation for thermal problems. Unlike past approaches for general heat transfer computations, and with the advent of high speed computing technology and the importance of parallel computations for efficient use of computing environments, a major motivation via the developments described in this paper is the need for developing explicit computational procedures with improved accuracy and stability characteristics. As a consequence, a new and effective VIP methodology is described which inherits these improved characteristics. Numerical illustrative examples are provided to demonstrate the developments and validate the results obtained for thermal problems.

Mei, Youping↗

NASA NTRS1987

Numerical studies of the thermal design sensitivity calculation for a reaction-diffusion system with discontinuous derivatives

The aim of this study is to find a reliable numerical algorithm to calculate thermal design sensitivities of a transient problem with discontinuous derivatives. The thermal system of interest is a transient heat conduction problem related to the curing process of a composite laminate. A logical function which can smoothly approximate the discontinuity is introduced to modify the system equation. Two commonly used methods, the adjoint variable method and the direct differentiation method, are then applied to find the design derivatives of the modified system. The comparisons of numerical results obtained by these two methods demonstrate that the direct differentiation method is a better choice to be used in calculating thermal design sensitivity.

Hou, Jean W.↗

NASA NTRS1983

An eigenvalue method for solving transient heat conduction problems

The eigenvalue method, which has been used by researchers in structure mechanics, is applied to problems in heat conduction. Its formulation is decribed in terms of an examination of transient heat conduction in a square slab. Taking advantage of the availability of the exact solution, we compare the accuracy and other numerical properties of the eigenvalue method with those of existing numerical schemes. The comparsion shows that, overall, the eigenvalue method appears to be fairly attractive. Furthermore, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed and retained to yield reasonably high accuracy. Greater savings are attained in the computation time for a transient problem with long time duration and a large computational domain.

Shih, T. M.↗

DOE OSTI2020

Time-stepping DPG formulations for the heat equation

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for computations and software implementations. The sole examination of time-stepping DPG formulations was performed by Führer, Heuer, and Gupta, who applied Rothe’s method to an ultraweak formulation of the heat equation to develop an implicit time-stepping scheme; their work emphasized theoretical results, including error estimates in time and space. Here, we follow Führer, Heuer, and Gupta in examining the heat equation; our focus is on numerical experiments, examining the stability and accuracy of several formulations, including primal as well as ultraweak, and explicit as well as implicit and Crank–Nicolson time-stepping schemes. We are additionally interested in communication-avoiding algorithms, and we therefore include a highly experimental formulation that places all the trace terms on the right-hand side of the equation.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1966

New computer program solves wide variety of heat flow problems

Boeing Engineering Thermal Analyzer /BETA/ computer program uses numerical methods to provide accurate heat transfer solutions to a wide variety of heat flow problems. The program solves steady-state and transient problems in almost any situation that can be represented by a resistance-capacitance network.

Almond, J. C.↗

NASA NTRS1989

Assessment of numerical methods for the solution of fluid dynamics equations for nonlinear resonance systems

The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.

Przekwas, A. J.↗

NASA NTRS1982

Transient thermal stress problem for a circumferentially cracked hollow cylinder

The transient thermal stress problem for a hollow elasticity cylinder containing an internal circumferential edge crack is considered. It is assumed that the problem is axisymmetric with regard to the crack geometry and the loading, and that the inertia effects are negligible. The problem is solved for a cylinder which is suddenly cooled from inside. First the transient temperature and stress distributions in an uncracked cylinder are calculated. By using the equal and opposite of this thermal stress as the crack surface traction in the isothermal cylinder the crack problem is then solved and the stress intensity factor is calculated. The numerical results are obtained as a function of the Fourier number tD/b(2) representing the time for various inner-to-outer radius ratios and relative crack depths, where D and b are respectively the coefficient of diffusivity and the outer radius of the cylinder.

Nied, H. F.↗

DOE OSTI2022

Transient MOC with frequency transform and DSA on unstructured mesh

We present an implementation of the transient method of characteristics (MOC) with isotropic time derivatives, accelerated by diffusion synthetic acceleration (DSA). The fully implicit frequency transform method is used to solve the transient problem with analytic precursor integration. The code works on meshes composed of almost any of the commonly used non-curvilinear finite element types, and can handle the deformation of geometry in time-dependent transport calculations. We present results of a continuous Fourier analysis for the transient multigroup DSA problem, and representative benchmarking results are presented for the C5G7-TD benchmark in 2D showing reasonable performance and agreement compared to other codes. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1969

Engineering thermal analyzer /BETA 2/

Computer program uses numerical methods to provide accurate heat transfer solutions to a wide variety of heat flow problems. This highly versatile program will solve steady state and transient problems in almost any situation that can be presented by a resistance-capacitance network.

Scates, J. H.↗

NASA NTRS1995

A theoretical analysis of steady-state photocurrents in simple silicon diodes

A theoretical analysis solves for the steady-state photocurrents produced by a given photo-generation rate function with negligible recombination in simple silicon diodes, consisting of a uniformly doped quasi-neutral region (called 'substrate' below) adjacent to a p-n junction depletion region (DR). Special attention is given to conditions that produce 'funneling' (a term used by the single-eventeffects community) under steady-state conditions. Funneling occurs when carriers are generated so fast that the DR becomes flooded and partially or completely collapses. Some or nearly all of the applied voltage, plus built-in potential normally across the DR, is now across the substrate. This substrate voltage drop affects substrate currents. The steady-state problem can provide some qualitative insights into the more difficult transient problem. First, it was found that funneling can be induced from a distance, i.e., from carriers generated at locations outside of the DR. Secondly, it was found that the substrate can divide into two subregions, with one controlling substrate resistance and the other characterized by ambipolar diffusion. Finally, funneling was found to be more difficult to induce in the p(sup +)/n diode than in the n(sup +)/p diode. The carrier density exceeding the doping density in the substrate and at the DR boundary is not a sufficient condition to collapse a DR.

Edmonds, L.↗

NASA NTRS1976

Solutions to divergence form equations using the method of partial implicitization

Partial implicitization is well suited for use on vector-processing computers since the method is an explicit unconditionally stable numerical technique. So far equations cast in nondivergence form have been used, though equations cast in the divergence form would have inherently greater accuracy. Since the method of partial implicitization is applicable only to relaxing problems to their steady state and is not applicable to true transient problems, a simple mathematical manipulation can be carried out which will produce the desired partial implicitization solution to divergence form equations.

Graves, R. A., Jr.↗

NASA NTRS2004

Sensitivity Equation Derivation for Transient Heat Transfer Problems

The focus of the paper is on the derivation of sensitivity equations for transient heat transfer problems modeled by different discretization processes. Two examples will be used in this study to facilitate the discussion. The first example is a coupled, transient heat transfer problem that simulates the press molding process in fabrication of composite laminates. These state equations are discretized into standard h-version finite elements and solved by a multiple step, predictor-corrector scheme. The sensitivity analysis results based upon the direct and adjoint variable approaches will be presented. The second example is a nonlinear transient heat transfer problem solved by a p-version time-discontinuous Galerkin's Method. The resulting matrix equation of the state equation is simply in the form of Ax = b, representing a single step, time marching scheme. A direct differentiation approach will be used to compute the thermal sensitivities of a sample 2D problem.

Hou, Gene↗

NASA NTRS1983

Transient thermal stress problem for a circumferentially cracked hollow cylinder

The paper is concerned with the transient thermal stress problem for a long hollow circular cylinder containing an internal axisymmetric circumferential edge crack that is suddenly cooled from inside. It is assumed that the transient thermal stress problem is quasi-static, i.e., the inertial effects are negligible. Also, all thermoelastic coupling effects and the possible temperature dependence of the thermoelastic constants are neglected. The problem is considered in two parts. The first part is the evaluation of transient thermal stresses in an uncracked cylinder; the second part is the isothermal perturbation problem for the cracked cylinder in which the crack surface tractions, equal and opposite to the thermal stresses obtained from the first problem, are the only external loads. The superposition of the two solutions gives results for the cracked cylinder.

Nied, H. F.↗

DOE OSTI2023

An implicit-in-time DPG formulation of the 1D1V Vlasov-Poisson equations

Efficient solution of the Vlasov equation, which can be up to six-dimensional, is key to the simulation of many difficult problems in plasma physics. The discontinuous Petrov-Galerkin (DPG) finite element methodology provides a framework for the development of stable (in the sense of Ladyzhenskaya–Babuška–Brezzi conditions) finite element formulations, with built-in mechanisms for adaptivity. While DPG has been studied extensively in the context of steady-state problems and to a lesser extent with space-time discretizations of transient problems, relatively little attention has been paid to time-marching approaches. In the present work, we study a first application of time-marching DPG to the Vlasov equation, using backward Euler for a Vlasov-Poisson discretization. We demonstrate adaptive mesh refinement for two problems: the two-stream instability problem, and a cold diode problem. Furthermore, we believe the present work is novel both in its application of unstructured adaptive mesh refinement (as opposed to block-structured adaptivity, which has been studied previously) in the context of Vlasov-Poisson, as well as in its application of DPG to the Vlasov-Poisson system. We also discuss extensive additions to the Camellia library in support of both the present formulation as well as extensions to higher dimensions, Maxwell equations, and space-time formulations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Deterministic modelling of the SPERT IV reactor transients using the multi-physics capability of WIMS

The ANSWERS WIMS reactor physics code is being developed for whole core multi-physics modelling. The established neutronics capability for lattice calculations has recently been extended to be suitable for whole core modelling of Small Modular Reactors (SMRs). A whole core transport, SP3 or diffusion flux solution is combined with fuel assembly resonance shielding. An integrated thermal hydraulic solver permits temperature and density variations to feedback to the neutronics calculation. This capability can be applied to both steady state and time dependent transient problems. Nuclear reactor design and safety case development requires assessment of a range of reactor transients, to inform both normal operation limitations and accident scenario analysis. This paper presents new methodology developed in WIMS to couple the core neutronics to the integrated core thermal hydraulics solver for the simulation of reactor transients in whole core models. To support the validation of the multi-physics capability of WIMS, this capability has been applied to the reactivity insertion transient experiments performed in the plate fuelled SPERT-IV reactor. This study employs WIMS using the whole core solver MERLIN, which calculates the time-dependent flux distribution and magnitude, coupled to ARTHUR, which solves for the time-dependent thermal hydraulics solution, and provides thermal feedback via the cross sections generated by GEOM, which performs resonance shielding calculations and generates cross section data for the plate geometry. This spatial kinetics model, with dynamic cross section generation, allows for variations in the temperature and neutron flux profile with time.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

NASA NTRS1975

NASTRAN solutions of problems described by simultaneous parabolic differential equations

NASTRAN solution techniques are shown for a numerical analysis of a class of coupled vector flow processes described by simultaneous parabolic differential equations. To define one physical problem type where equations of this form arise, the differential equations describing the coupled transfers of heat and mass in mechanical equilibrium with negligible mass average velocity are presented and discussed. Also shown are the equations describing seepage when both electrokinetic and hydrodynamic forces occur. Based on a variational statement of the general problem type, the concepts of scalar transfer elements and parallel element systems are introduced. It is shown that adoptation of these concepts allows the direct use of NASTRAN's existing Laplace type elements for uncoupled flow (the heat transfer elements) for treating multicomponent coupled transfer. Sample problems are included which demonstrate the application of these techniques for both steady-state and transient problems.

Mason, J. B.↗