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At least 487 records · Page 27

Some notes on the numerical solution of shear-lag and mathematically related problems

The analysis of box beams with shear deformation of the flanges can be reduced to the solution of a differential equation. The same equation is met in other problems of stress analysis. No analytical solutions of this equation can be given for practical cases, and numerical methods of evaluation must be used. Available methods are briefly discussed. Two numerical examples show the application of the step-by-step method of integration to shear-lag problems.

Kuhn, Paul↗

The Thin Oil Film Equation

A thin film of oil on a surface responds primarily to the wall shear stress generated on that surface by a three-dimensional flow. The oil film is also subject to wall pressure gradients, surface tension effects and gravity. The partial differential equation governing the oil film flow is shown to be related to Burgers' equation. Analytical and numerical methods for solving the thin oil film equation are presented. A direct numerical solver is developed where the wall shear stress variation on the surface is known and which solves for the oil film thickness spatial and time variation on the surface. An inverse numerical solver is also developed where the oil film thickness spatial variation over the surface at two discrete times is known and which solves for the wall shear stress variation over the test surface. A One-Time-Level inverse solver is also demonstrated. The inverse numerical solver provides a mathematically rigorous basis for an improved form of a wall shear stress instrument suitable for application to complex three-dimensional flows. To demonstrate the complexity of flows for which these oil film methods are now suitable, extensive examination is accomplished for these analytical and numerical methods as applied to a thin oil film in the vicinity of a three-dimensional saddle of separation.

Brown, James L.↗

Comparison of linear inversion methods by examination of the duality between iterative and inverse matrix methods

Linear numerical inversion methods applied to atmospheric remote sounding generally can be categorized in two ways: (1) iterative, and (2) inverse matrix methods. However, these two categories are not unrelated; a duality exists between them. In other words, given an iterative scheme, a corresponding inverse matrix method exists, and conversely. This duality concept is developed for the more familiar linear methods. The iterative duals are compared with the classical linear iterative approaches and their differences analyzed. The importance of the initial profile in all methods is stressed. Calculations using simulated data are made to compare accuracies and to examine the dependence of the solution on the initial profile.

Fleming, H. E.↗

The method of lines in analyzing solids containing cracks

A semi-numerical method is reviewed for solving a set of coupled partial differential equations subject to mixed and possibly coupled boundary conditions. The line method of analysis is applied to the Navier-Cauchy equations of elastic and elastoplastic equilibrium to calculate the displacement distributions in various, simple geometry bodies containing cracks. The application of this method to the appropriate field equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling of the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by reviewing and presenting selected solutions of mixed boundary value problems in three dimensional fracture mechanics. These solutions are of great importance in fracture toughness testing, where accurate stress and displacement distributions are required for the calculation of certain fracture parameters. Computations obtained for typical flawed specimens include that for elastic as well as elastoplastic response. Problems in both Cartesian and cylindrical coordinate systems are included. Results are summarized for a finite geometry rectangular bar with a central through-the-thickness or rectangular surface crack under remote uniaxial tension. In addition, stress and displacement distributions are reviewed for finite circular bars with embedded penny-shaped cracks, and rods with external annular or ring cracks under opening mode tension. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional mechanics problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results are obtained even from the use of a relatively coarse grid.

Gyekenyesi, John P.↗

Comparison of characteristic schemes for three-dimensional, steady, isentropic flow.

The accuracy and order of three bicharacteristic schemes based on the numerical method of characteristics for three-dimensional, steady, isentropic, supersonic flow were compared based on numerical calculations. Comparisons were made for source flows and for flows in contoured axisymmetric nozzles. The first two methods were tetrahedral line networks, differing only in the manner in which the cross derivatives were evaluated. The third method was the pentahedral line network proposed by Butler. All three methods were numerically stable and produced efficient and accurate solutions. A discussion of the relative merits and disadvantages of each method is presented.

Cline, M. C.↗

The Finite Analytic Method for steady and unsteady heat transfer problems

A new numerical method called the Finite Analytical Method for solving partial differential equations is introduced. The basic idea of the finite analytic method is the incorporation of the local analytic solution in obtaining the numerical solution of the problem. The finite analytical method first divides the total region of the problem into small subregions in which local analytic solutions are obtained. Then an algebraic equation is derived from the local analytic solution for each subregion relating an interior nodal value at a point P in the subregion to its neighboring nodal values. The assembly of all the local analytic solutions thus provides the finite-analytic numerical solution of the problem. In this paper the finite analytic method is illustrated in solving steady and unsteady heat transfer problems.

Chen, C.-J.↗

Motion and interaction of decaying trailing vortices in spanwise shear wind

A simulation is presented of the drift of trailing vortices in a cross-wind near the ground by an unsteady, two-dimensional, rotational flow field with a concentration of large vorticity in vortical spots (having a finite but small effective size and finite total strength). The problem is analyzed by a combination of the method of matched asymptotic analyses for the decay of the vortical spots and the Euler solution for the unsteady rotational flow. Using the method of averaging, a special numerical method is developed in which the grid size and time step depend only on the length and velocity scales of the background flow and are independent of the effective core size of a vortical spot. The core size can be much smaller than the grid size, whereas the peak velocity in the core is inversely propertional to the spot size. Numerical results are presented to demonstrate the strong interaction between the trajectories of the vortical spots and the change of the vorticity distribution in the background flow field.

Liu, C. H.↗

Improved vortex methods for three-dimensional flows

Robust numerical methods are developed for three-dimensional incompressible vortical flows, using Lagrangian vortex elements. A successful scheme must be able to handle regions of intense vortex stretching and vortex reconnection with reasonable accuracy (without diverging). Here, consideration is given to vortex particles, also commonly called vortons or vortex sticks. The following issues are discussed: (1) use of delta-function elements and weak solutions of the vorticity equation; (2) use of smoothed elements and the choice of the smoothing function; (3) representation of viscous effects and the redistribution of element strength; and (4) conservation laws (are they satisfied?). The various proposed schemes have been tested on flows involving a strong interaction between two vortex rings.

Winckelmans, G.↗

A novel peridynamics-based approach to predict pharmaceutical tablet robustness

The pharmaceutical drug product development process can be greatly accelerated through the use of modeling and simulation techniques to predict the manufacturability and performance of a given formulation. The anticipation and possible mitigation of tablet damage due to manufacturing stresses represents a specific area of interest in the pharmaceutical industry for predicting formulation and tableting performance. While the finite element method (FEM) has been extensively used for predicting the mechanical behavior of powder material in the compaction processes, a shortcoming of the approach is the inherent difficulty to predict discontinuities (e.g., damage or cracking) within a tablet as FEM is a continuum-based approach. In this work, we propose a novel method utilizing peridynamics (PD), a numerical method that can capture discontinuities such as tablet fracture, to predict the evolution of damage and breakage in pharmaceutical tablets. The approach links (1) the finite element method – to elucidate the behavior of powders during die compaction – with (2) the peridynamics modeling technique – to model the discontinuous nature of damage and predict tablet breakage during the critical stages of unloading and ejection from the compression die. This short communication presents a proof of concept including a workflow to calibrate the linked FEM-PD simulation models. Further, it demonstrates promising results from a preliminary experimental validation of the approach. Following further development, this approach could be used to guide the optimization of compression processes through targeted changes to formulation material properties, compression process conditions, and/or tooling geometries to deliver improved process efficiency and tablet robustness.

36 MATERIALS SCIENCE↗

A Numerical Analysis of the Transient Response of an Ablation System Including Effects of Thermal Nonequilibrium, Mass Transfer and Chemical Kinetics

The differential equations governing the transient response of a one-dimensional ablative thermal protection system undergoing stagnation ablation are derived. These equations are for thermal nonequilibrium effects between the pyrolysis gases and the char layer and kinetically controlled chemical reactions and mass transfer between the pyrolysis gases and the char layer. The boundary conditions are written for the particular case of stagnation heating with surface removal by oxidation or sublimation and pyrolysis of the uncharred layer occurring in a plane. The governing equations and boundary conditions are solved numerically using the modified implicit method (Crank-Nicolson method). Numerical results are compared with exact solutions for a number of simplified cases. The comparison is favorable in each instance.

Clark, R. K.↗

Two-dimensional compressible convection extending over multiple scale heights

The theoretical description of the dynamics of a stellar convection zone is considered, taking into account one of the most basic issues by studying compressible convection extending over multiple scale heights. A revised version of a code reported by Graham (1975) is employed. Two-dimensional simulations show that nonlinear compressible convection possesses cellular structures with strong localized downward-directed plumes and broader upflows. The horizontal flows which close the circulation within the cell satisfy an approximate Bernoulli integral along a considerable portion of the horizontal trajectory. Attention is given to details regarding the numerical methods, the properties of the numerical solutions, the overall effects of compressibility on nonlinear convection, and a comparison with anelastic modal solutions.

Hurlburt, N. E.↗

Transient natural convection heat and mass transfer in a rectangular enclosure - A numerical analysis

A numerical analysis of transient heat and solute transport across a rectangular cavity with combined horizontal temperature and concentration gradients is performed by a numerical method based on the SIMPLE. Numerical results show that the average Nusselt and Sherwood numbers both decrease markedly when the solutal and thermal buoyancy forces act in the opposite directions. When the solutal and thermal buoyancy forces act in the same directions, however, the average Sherwood number increases significantly and yet the average Nusselt number decreases slightly.

Han, Samuel S.↗

Dynamic force response of spherical hydrostatic journal bearing for cryogenic applications

Hydrostatic Journal Bearings (HJB's) are reliable and resilient fluid film rotor support elements ideal to replace roller bearings in cryogenic turbomachinery. HJB' will be used for primary space-power applications due to their long lifetime, low friction and wear, large load capacity, large direct stiffness, and damping force coefficients. An analysis for the performance characteristics of turbulent flow, orifice compensated, spherical hydrostatic journal bearings (HJB's) is presented. Spherical bearings allow tolerance for shaft misalignment without force performance degradation and have also the ability to support axial loads. The spherical HJB combines these advantages to provide a bearing design which could be used efficiently on high performance turbomachinery. The motion of a barotropic liquid on the thin film bearing lands is described by bulk-flow mass and momentum equations. These equations are solved numerically using an efficient CFD method. Numerical predictions of load capacity and force coefficients for a 6 recess, spherical HJB in a LO2 environment are presented. Fluid film axial forces and force coefficients of a magnitude about 20% of the radial load capacity are predicted for the case analyzed. Fluid inertia effects, advective and centrifugal, are found to affect greatly the static and dynamic force performance of the bearing studied.

Sanandres, Luis↗

A geometric deformable model for echocardiographic image segmentation

Gradient vector flow (GVF), an elegant external force for parametric deformable models, can capture object boundaries from both sides. A new geometric deformable model is proposed that combines GVF and the geodesic active contour model. The level set method is used as the numerical method of this model. The model is applied for echocardiographic image segmentation.

NASA Program Biomedical Research and Countermeasur↗

UCLA parallel PIC framework

The UCLA Parallel PIC Framework (UPIC) has been developed to provide trusted components for the rapid construction of new, parallel Particle-in-Cell (PIC) codes. The Framework uses object-based ideas in Fortran95, and is designed to provide support for various kinds of PIC codes on various kinds of hardware. The focus is on student programmers. The Framework supports multiple numerical methods, different physics approximations, different numerical optimizations and implementations for different hardware. It is designed with "defensive" programming in mind, meaning that it contains many error checks and debugging helps. Above all, it is designed to hide the complexity of parallel processing. It is currently being used in a number of new Parallel PIC codes.

Norton, Charles D.↗

Direct Discontinuous Galerkin methods for the reacting multi-component flow equations

The Direct Discontinuous Galerkin (DDG (Liu and Yan, 2008)) method and a counterpart with Interface Correction (DDGIC (Danis and Yan, 2022)) are extended to compute diffusion terms that arise when solving the compressible multi-component flow equations in thermochemical nonequilibrium. Thermodynamic properties, transport properties, chemical reaction rates, and energy exchange terms are computed using Mutation++ (Scoggins et al., 2020). The DG method is applied on unstructured grids, where the accuracy and convergence rates can be sensitive to the numerical method chosen for parabolic terms. A method for determining the homogeneity tensor of the flow equations required for DDGIC is shown. The convergence properties of the DDG methods are studied and compared to the Interior Penalty (IP) method. A number of numerical experiments are conducted to assess the accuracy and performance of the method. The numerical results and convergence studies indicate that DDG and DDGIC provide accurate solutions and perform well for general flows in thermochemical nonequilibrium.

Diffusion↗

A study of multiple-shaker modal survey testing

The principal objective was to examine and to assess the practical value of a method of multiple-shaker sinusoidal modal vibration testing known as Asher's method. Numerical studies which simulate the application of Asher's method and a unique experimental implementation of the method were completed. Another objective of the research was to develop and to demonstrate with numerical simulation a quantitative method for determining from transfer function data the number of dominant modes of vibration in sinusoidal structural response.

Hallauer, W. L., Jr.↗

SIAM Conference on Parallel Processing for Scientific Computing, 4th, Chicago, IL, Dec. 11-13, 1989, Proceedings

Attention is given to such topics as an evaluation of block algorithm variants in LAPACK and presents a large-grain parallel sparse system solver, a multiprocessor method for the solution of the generalized Eigenvalue problem on an interval, and a parallel QR algorithm for iterative subspace methods on the CM2. A discussion of numerical methods includes the topics of asynchronous numerical solutions of PDEs on parallel computers, parallel homotopy curve tracking on a hypercube, and solving Navier-Stokes equations on the Cedar Multi-Cluster system. A section on differential equations includes a discussion of a six-color procedure for the parallel solution of elliptic systems using the finite quadtree structure, data parallel algorithms for the finite element method, and domain decomposition methods in aerodynamics. Topics dealing with massively parallel computing include hypercube vs. 2-dimensional meshes and massively parallel computation of conservation laws. Performance and tools are also discussed.

Dongarra, Jack↗