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NASA NTRS1997

An Application of the Difference Potentials Method to Solving External Problems in CFD

Numerical solution of infinite-domain boundary-value problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the original infinite-domain formulation must be altered and/or augmented so that on one hand the solution is not changed (or changed slightly) and on the other hand the finite discrete formulation becomes available. One widely used approach to constructing such discretizations consists of truncating the unbounded original domain and then setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The role of the ABC's is to close the truncated problem and at the same time to ensure that the solution found inside the finite computational domain would be maximally close to (in the ideal case, exactly the same as) the corresponding fragment of the original infinite-domain solution. Let us emphasize that the proper treatment of artificial boundaries may have a profound impact on the overall quality and performance of numerical algorithms. The latter statement is corroborated by the numerous computational experiments and especially concerns the area of CFD, in which external problems present a wide class of practically important formulations. In this paper, we review some work that has been done over the recent years on constructing highly accurate nonlocal ABC's for calculation of compressible external flows. The approach is based on implementation of the generalized potentials and pseudodifferential boundary projection operators analogous to those proposed first by Calderon. The difference potentials method (DPM) by Ryaben'kii is used for the effective computation of the generalized potentials and projections. The resulting ABC's clearly outperform the existing methods from the standpoints of accuracy and robustness, in many cases noticeably speed up the multigrid convergence, and at the same time are quite comparable to other methods from the standpoints of geometric universality and simplicity of implementation.

Ryaben 'Kii, Victor S.↗

DOE OSTI2021

Thermal conductivity modeling of monodispersed microspheres using discrete element method

Particle beds are widely used in various systems and processes, such as particle heat exchangers, granular flow reactors, and additive manufacturing. Accurate modeling of the thermal conductivity of particle beds and understanding of their heat transfer mechanisms are important. However, previous models were based on simple cubic packing of particles, which could not accurately represent the actual heat transfer processes under certain conditions. Here, we examine the effect of the packing structure on the thermal conductivity of particle beds. We use monodispersed silica microspheres with average particle sizes ranging from 23 to 330 μm as a model material. We employ a transient hot-wire technique to measure the thermal conductivity of the particle beds with packing density of 43%–57% within a temperature range of room temperature to 500 °C and under N2 gaseous pressures of 20–760 Torr. We then use a discrete element method (DEM) to obtain the realistic packing structure of the particles, which is then fed into a finite-element model (FEM) to calculate the thermal conductivity, with the consideration of solid conduction, gas conduction, and radiation heat transfer. Our results show that the thermal conductivity model based on the more realistic random packing structure derived from the DEM shows better agreement with the experimental data compared to that based on the simple cubic-packing structure. The combined DEM and FEM methodology can serve as a useful tool to predict the effective thermal conductivity of particle beds and to quantify different heat transfer mechanisms under various conditions.

14 SOLAR ENERGY↗

DOE OSTI2021

Evolution of microstructures in radiation fields using a coupled binary-collision Monte Carlo phase field approach

The simulation of radiation effects in materials broadly falls into two categories. At short time and length scales lies the modeling of primary radiation damage, such as point defect creation, energy deposition, and ballistic mixing. This is followed by the modeling at longer time scales of thermally activated microstructure evolution and defect reactions, such as recombination, clustering, and coarsening. The binary collision Monte Carlo method is an established, numerically efficient method for the computation of primary radiation damage. Conversely, the phase field method is a state of the art method for microstructure evolution on longer time and length scales. Here we present a concurrent coupling of these two methods, overcoming the difference between the discrete object Monte Carlo paradigm for primary radiation damage and the continuum field variable approach for microstructure evolution. The coupling is bidirectional, in which the microstructure evolution in the MOOSE finite element frame- work provides the spatial scattering data set for the charged particle trans- port and receives point defect, mass transport, and heat source terms from the simulated collision cascades that contribute to the field variable evolution. The concurrent coupling scheme is implemented in the code Magpie and demonstrated by investigating patterning for a model irradiated immiscible binary alloy. The results from the coupled binary collision Monte Carlo/phase field simulations reproduce the results of analytical models for phase separation, phase mixing, and patterning, supporting the approach and indicating its utility for modeling real materials systems.

36 MATERIALS SCIENCE↗

NASA NTRS1981

Unconditional instability of inflow dependent boundary conditions in difference approximations to hyperbolic systems

The stability of finite difference approximations to initial boundary hyperbolic systems is studied. As is well known, a proper specification of boundary conditions for such systems is essential for their solutions to be well defined. A discrete analogue of the above is proved - if the numerical boundary conditions are consistent with an inflow part of the problem, they render the overall computation unstable. An example of the inviscid gasdynamics equations is considered.

Tadmor, E.↗

NASA NTRS1983

The unconditional instability of inflow-dependent boundary conditions in difference approximations to hyperbolic systems

Tadmor, E.↗

NASA NTRS1995

Micromechanical models for textile composites

Numerical and analytical micromechanical models are presented to predict the thermoelastic behavior of a textile composite. In the numerical model, the unit-cell is discretized with finite elements, and periodic boundary conditions are imposed between opposite faces of the unit-cell. For a thin textile composite, stress gradients effects through the thickness are demonstrated. The consequent difference in the stiffness and strength behavior of thick and thin composites are discussed. The numerical model is implemented to predict 3-D thermo-elastic constants for a thick textile composite, and the plate thermo-mechanical properties for a thin textile composite. The numerical model is extended to compute the thermal residual microstresses due to processing to predict the composite failure envelopes. An analytical model - Selective Averaging Method (SAM) - is proposed, which is based on a judicious combination of stiffness and compliance averaging to estimate the 3-D elastic constants. Both the models are tested and verified for several examples by comparing the stiffness properties with elasticity solutions and available results.

Sankar, Bhavani V.↗

NASA NTRS1999

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

The technical details are summarized below: Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. . An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing.

Grossman, Bernard↗

DOE OSTI2022

Krylov subspace recycling for evolving structures

Krylov subspace recycling is a powerful tool when solving a long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these series appear naturally, as typically hundreds or thousands of optimization steps are needed with only small changes in the geometry. In this setting, however, applying Krylov subspace recycling can be a difficult task. As the geometry evolves, in general, so does the finite element mesh defined on or representing this geometry, including the numbers of nodes and elements and element connectivity. This is especially the case if re-meshing techniques are used. As a result, the number of algebraic degrees of freedom in the system changes, and in general the linear system matrices resulting from the finite element discretization change size from one optimization step to the next. Changes in the mesh connectivity also lead to structural changes in the matrices. In the case of re-meshing, even if the geometry changes only a little, the corresponding mesh might differ substantially from the previous one. Obviously, this prevents any straightforward mapping of the approximate invariant subspace of the linear system matrix (the focus of recycling in this work) from one optimization step to the next; similar problems arise for other selected subspaces. In this paper, we present an algorithm to map an approximate invariant subspace of the linear system matrix for the previous optimization step to an approximate invariant subspace of the linear system matrix for the current optimization step, for general meshes. This is achieved by exploiting the map from coefficient vectors to finite element functions on the mesh, combined with interpolation or approximation of functions on the finite element mesh. We demonstrate the effectiveness of our approach numerically with several proof of concept studies for a specific meshing technique.

42 ENGINEERING↗

NASA NTRS1992

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

Yu, Sheng-Tao↗

DOE OSTI2023

Modeling diffusion and types I-V sorption of water vapor in heterogeneous systems

Here, a sorption and diffusion model is developed using a mobile-immobile decomposition of the vapor concentration. Henry’s mode represents the mobile species which diffuses through the solid, while Langmuir and pooling modes represent possible immobilization processes that cause nonlinearities in sorption capacity. The model can simulate the five classical sorption types and can incorporate additional dynamics compared to models based on thermodynamic equilibrium. Additionally, the framework allows for discontinuities in material properties encountered in multi-material systems. The resulting non-linear coupled equations are solved by employing a finite element method to discretize in space, and a backward Euler method to discretize in time. The discretized system of equations is solved via a Newton-type iteration scheme at each time-step. Four different materials are parameterized for the model, which are then used in examples to demonstrate the model’s ability to capture complex sorption processes in multi-material systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1986

Estimates of the bolometric albedos and radiation balance of Uranus and Neptune

Models possessing an upper haze layer of finite optical depth and a lower cloud layer of infinite optical depth at discrete altitudes are used to bound the wavelength-averaged phase integrals and bolometric albedos of Uranus and Neptune. The models differ in the assumed value of the particles' single scattering phase function and the wavelength dependence of the haze optical depth. A range of phase functions, from the isotropic to those characterizing Titan, Jupiter, and Saturn atmosphere particles, are discussed. The results obtained imply that the meteorological regimes in the observable atmospheres of Uranus and Neptune may differ considerably; internal heat flux could play a much more important role for Neptune than for Uranus.

Pollack, J. B.↗

NASA NTRS1986

Vertical structure of aerosols and clouds in the atmospheres of Uranus and Neptune: Implications for their heat budgets

Models possessing an upper haze layer of finite optical depth and a lower cloud layer of infinite optical depth at discrete altitudes are used to bound the wavelength-averaged phase integrals and bolometric albedos of Uranus and Neptune. The models differ in the assumed value of the particles single scattering phase function and the wavelength dependence of the haze optical depth. A range of phase functions, from the isotropic to those characterizing Titan, Jupiter, and Saturn atmosphere particles, are discussed. The results obtained imply that the meteorological regimes in the observable atmospheres of Uranus and Neptune may differ considerably; internal heat flux could play a much more important role for Neptune than for Uranus.

Pollack, James B.↗

NASA NTRS1992

A fast direct solver for a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of two-dimensional separable elliptic equations on the sphere is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wavenumber and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

Moorthi, Shrinivas↗

NASA NTRS2009

Ablative Thermal Response Analysis Using the Finite Element Method

A review of the classic techniques used to solve ablative thermal response problems is presented. The advantages and disadvantages of both the finite element and finite difference methods are described. As a first step in developing a three dimensional finite element based ablative thermal response capability, a one dimensional computer tool has been developed. The finite element method is used to discretize the governing differential equations and Galerkin's method of weighted residuals is used to derive the element equations. A code to code comparison between the current 1-D tool and the 1-D Fully Implicit Ablation and Thermal Response Program (FIAT) has been performed.

Dec John A.↗

NASA NTRS1975

Stress analysis and stress intensity factors for finite geometry solids containing rectangular surface cracks

The line method of analysis is applied to the Navier-Cauchy equations of elastic equilibrium to calculate the displacement field in a finite geometry bar containing a variable depth rectangular surface crack under extensionally applied uniform loading. The application of this method to these equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. Using the obtained displacement field, normal stresses and the stress intensity factor variation along the crack periphery are calculated for different crack depth to bar thickness ratios. Crack opening displacements and stress intensity factors are also obtained for a through-thickness, center cracked bar with variable thickness. The reported results show a considerable potential for using this method in calculating stress intensity factors for commonly encountered surface crack geometries in finite solids.

Gyekenyesi, J. P.↗

DOE OSTI2025

Window convolution of the galaxy clustering bispectrum

In galaxy survey analysis, the observed clustering statistics do not directly match theoretical predictions but rather have been processed by a window function that arises from the survey geometry including the sky footprint, redshift-dependent background number density and systematic weights. While window convolution of the power spectrum is well studied, for the bispectrum with a larger number of degrees of freedom, it poses a significant numerical and computational challenge. In this work, we consider the effect of the survey window in the tripolar spherical harmonic decomposition of the bispectrum and lay down a formal procedure for their convolution via a series expansion of configuration-space three-point correlation functions, which was first proposed by Sugiyama et al. (2019). We then provide a linear algebra formulation of the full window convolution, where an unwindowed bispectrum model vector can be directly premultiplied by a window matrix specific to each survey geometry. To validate the pipeline, we focus on the Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1) luminous red galaxy (LRG) sample in the South Galactic Cap (SGC) in the redshift bin 0.4 ≤ z ≤ 0.6. We first perform convergence checks on the measurement of the window function from discrete random catalogues, and then investigate the convergence of the window convolution series expansion truncated at a finite of number of terms as well as the performance of the window matrix. This work highlights the differences in window convolution between the power spectrum and bispectrum, and provides a streamlined pipeline for the latter for current surveys such as DESI and the Euclid mission.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1995

Nonlinear wave propagation using three different finite difference schemes (category 2 application)

Three common finite difference schemes are used to examine the computation of one-dimensional nonlinear wave propagation. The schemes are studied for their responses to numerical parameters such as time step selection, boundary condition implementation, and discretization of governing equations. The performance of the schemes is compared and various numerical phenomena peculiar to each is discussed.

Pope, D. Stuart↗

NASA NTRS1997

Mechanical Characterization and Micromechanical Modeling of Woven Carbon/Copper Composites

The present investigation examines the in-plane mechanical behavior of a particular woven metal matrix composite (MMC); 8-harness (8H) satin carbon/copper (C/Cu). This is accomplished via mechanical testing as well as micromechanical modeling. While the literature is replete with experimental and modeling efforts for woven and braided polymer matrix composites, little work has been done on woven and braided MMC's. Thus, the development and understanding of woven MMC's is at an early stage. 8H satin C/Cu owes its existence to the high thermal conductivity of copper and low density and thermal expansion of carbon fibers. It is a candidate material for high heat flux applications, such as space power radiator panels. The experimental portion of this investigation consists of monotonic and cyclic tension, compression, and Iosipescu shear tests, as well as combined tension-compression tests. Tests were performed on composite specimens with three copper matrix alloy types: pure Cu, Cu-0.5 weight percent Ti (Cu-Ti), and Cu-0.7 weight percent Cr (Cu-Cr). The small alloying additions are present to promote fiber/matrix interfacial bonding. The analytical modeling effort utilizes an approach in which a local micromechanical model is embedded in a global micromechanical model. This approach differs from previously developed analytical models for woven composites in that a true repeating unit cell is analyzed. However, unlike finite element modeling of woven composites, the geometry is sufficiently idealized to allow efficient geometric discretization and efficient execution.

Bednarcyk, Brett A.↗