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A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.

Approximate and exact numerical integration of the gas dynamic equations

A highly accurate approximation and a rapidly convergent numerical procedure are developed for two dimensional steady supersonic flow over an airfoil. Examples are given for a symmetric airfoil over a range of Mach numbers. Several interesting features are found in the calculation of the tail shock and the flow behind the airfoil.

Lewis, T. S.

Implicit numerical integration for periodic solutions of autonomous nonlinear systems

A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.

Thurston, G. A.

Surrogate models for development of unconventional shale reservoirs by an integrated numerical approach of hydraulic fracturing, flow and geomechanics, and machine learning

We develop well-completion surrogate models by taking an integrated workflow of hydraulic fracturing, flow, geomechanics, and machine learning simulation. There are three steps in the proposed workflow. First, history-matching processes are conducted with the field data including pumping and production data for characterization. Second, full-physics simulation is performed with various parameters of the field development (e.g., cluster spacing, clusters per stage, pumping rates and times, amount of proppant, and well spacing) to generate multiple simulation results by changing the parameters of the completion design with well-known hydraulic fracturing, reservoir, geomechanics simulators to calculate fracture geometry, reservoir depressurization, induced stress changes. The workflow is demonstrated over a field in the Southern Midland Basin. Here, we take two completion scenarios: a single well case followed by a multi-well case. Finally, a Long Short-Term Memory (LSTM) machine learning algorithm is employed to create surrogate models that can replicate the full-physics simulation results. Furthermore, results show that the trained models applied in the single well and multi-well cases for a particular geological system can provide good accuracy close to those provided by full-physics simulations. Specifically, the site-specific surrogate models can predict fracture parameters (length, height, and surface area) and cumulative production accurately with computational efficiency, suggesting our proposed workflow can be used as a pragmatic tool for expediting the well completion optimization process.

Geomechanics

Long-Time Numerical Integration of the Three-Dimensional Wave Equation in the Vicinity of a Moving Source

We propose a family of algorithms for solving numerically a Cauchy problem for the three-dimensional wave equation. The sources that drive the equation (i.e., the right-hand side) are compactly supported in space for any given time; they, however, may actually move in space with a subsonic speed. The solution is calculated inside a finite domain (e.g., sphere) that also moves with a subsonic speed and always contains the support of the right-hand side. The algorithms employ a standard consistent and stable explicit finite-difference scheme for the wave equation. They allow one to calculate tile solution for arbitrarily long time intervals without error accumulation and with the fixed non-growing amount of tile CPU time and memory required for advancing one time step. The algorithms are inherently three-dimensional; they rely on the presence of lacunae in the solutions of the wave equation in oddly dimensional spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady artificial boundary conditions to be used for the numerical simulation of waves propagating with finite speed over unbounded domains.

Ryabenkii, V. S.

Sensitivity of inelastic response to numerical integration of strain energy

The exact solution to the quasi-static, inelastic response of a cantilever beam of rectangular cross section subjected to a bending moment at the tip is obtained. The material of the beam is assumed to be linearly elastic-linearly strain-hardening. This solution is then compared with three different numerical solutions of the same problem obtained by minimizing the total potential energy using Gaussian quadratures of two different orders and a Newton-Cotes scheme for integrating the strain energy of deformation. Significant differences between the exact dissipative strain energy and its numerical counterpart are emphasized. The consequence of this on the nonlinear transient responses of a beam with solid cross section and that of a thin-walled beam on elastic supports under impulsive loads are examined.

Kamat, M. P.

SIRTF controller simulation - Instability masked by numerical integration

In the course of a simulation study of a candidate design for the Space Infrared Telescope Facility (SIRTF), an unusual phenomenon was observed. The uncompensated control system was unstable, but a numerical simulation with the fixed-step-size classical fourth-order Runge-Kutta method gave a stable response. This phenomenon is described in the setting in which it occurred. The Runge-Kutta simulation model is analyzed as a discrete linear system and shown to be stable, thus corroborating the numerical results.

Rajan, N.

A prefiltering version of the Kalman filter with new numerical integration formulas for Riccati equations

A prefiltering version of the Kalman filter is derived for both discrete and continuous measurements. The derivation consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements. This prefiltering version of the Kalman filter easily handles numerical problems associated with rapid transients and ill-conditioned Riccati matrices. Therefore, the derived technique for extrapolating the Riccati matrix from one time to the next constitutes a new set of integration formulas which alleviate ill-conditioning problems associated with continuous Riccati equations. Furthermore, since a time segment of continuous measurements is converted into a single discrete measurement, Potter's square root formulas can be used to update the state estimate and its error covariance matrix. Therefore, if having the state estimate and its error covariance matrix at discrete times is acceptable, the prefilter extends square root filtering with all its advantages, to continuous measurement problems.

Womble, M. E.

Numerical integration of nearly-Hamiltonian systems

The reported investigation is concerned with the solution of systems of differential equations which are derived from a Hamiltonian function in the extended phase space. The problem selected involves a one-dimensional perturbed harmonic oscillator. The van der Pol equation considered has an exact asymptotic value for its amplitude. Comparisons are made between a numerical solution and a known analytical solution. In addition to the van der Pol problem, known solutions regarding the restricted problem of three bodies are used as examples for perturbed Keplerian motion. The extended phase space Hamiltonian discussed by Stiefel and Scheifele (1971) is considered. A description is presented of two canonical formulations of the perturbed harmonic oscillator.

Bond, V. R.

Static, stability, and dynamic analysis of shells of revolution by numerical integration - A comparison

Recent innovations in digital computer technology have enabled engineers to analyze shell structures of complex configurations without unduly restrictive approximations. An attempt is made to compare the various programs now generally available from the point of view of the advantages of the relative technique utilized, as well as the programmed state of the art. Many of the comparisons are based on the sample problems solved by the STARS-2 system of programs. These examples indicate both the structural detail which can be analyzed by, and the analytical capabilities available in, the numerical shell-of-revolution programs. All advantages and differences are demonstrated by use of solutions for realistic shell problems in the areas of statics, stability, vibrations, and dynamic response of shells subjected to time-dependent loadings.

Svalbonas, V.

On numerical integration and computer implementation of viscoplastic models

Due to the stringent design requirement for aerospace or nuclear structural components, considerable research interests have been generated on the development of constitutive models for representing the inelastic behavior of metals at elevated temperatures. In particular, a class of unified theories (or viscoplastic constitutive models) have been proposed to simulate material responses such as cyclic plasticity, rate sensitivity, creep deformations, strain hardening or softening, etc. This approach differs from the conventional creep and plasticity theory in that both the creep and plastic deformations are treated as unified time-dependent quantities. Although most of viscoplastic models give better material behavior representation, the associated constitutive differential equations have stiff regimes which present numerical difficulties in time-dependent analysis. In this connection, appropriate solution algorithm must be developed for viscoplastic analysis via finite element method.

Chang, T. Y.

Numerical integration of second order differential equations

Performance characteristics of higher order approximations of Runge-Kutta type are analyzed, and performance predictors for time required on machine and for error size are developed. Technique is useful in evaluating system performance, analyzing material characteristics, and designing inertial guidance and nuclear instrumentation and materials.

Shanks, E. B.

The calculation of electromagnetic fields in the Fresnel and Fraunhofer regions using numerical integration methods

Some results obtained with a digital computer program written at Goddard Space Flight Center to obtain electromagnetic fields scattered by perfectly reflecting surfaces are presented. For purposes of illustration a paraboloidal reflector was illuminated at radio frequencies in the simulation for both receiving and transmitting modes of operation. Fields were computed in the Fresnel and Fraunhofer regions. A dual-reflector system (Cassegrain) was also simulated for the transmitting case, and fields were computed in the Fraunhofer region. Appended results include derivations which show that the vector Kirchhoff-Kottler formulation has an equivalent form requiring only incident magnetic fields as a driving function. Satisfaction of the radiation conditions at infinity by the equivalent form is demonstrated by a conversion from Cartesian to spherical vector operators. A subsequent development presents the formulation by which Fresnel or Fraunhofer patterns are obtainable for dual-reflector systems. A discussion of the time-average Poynting vector is also appended.

Schmidt, R. F.