Results for “approximate domain boundaries” — Page 11

DOE OSTI2020

Effects of Low-Carbon Energy Adoption on Airborne Particulate Matter Concentrations With Feedbacks to Future Climate Over California

California plans to reduce emissions of long-lived greenhouse gases (GHGs) through adoption of new energy systems that will also lower concentrations of short-lived absorbing soot contained in airborne particulate matter (PM). Here we examine the direct and indirect effects of reduced PM concentrations under a low-carbon energy (GHG-Step) scenario on radiative forcing in California. Simulations were carried out using the source-oriented WRF/Chem (SOWC) model with 12 km spatial resolution for the year 2054. Furthermore, the avoided aerosol emissions due to technology advances in the GHG-Step scenario reduce ground level PM concentrations by ~8.85% over land compared to the Business-as-Usual (BAU) scenario, but changes to meteorological parameters are more modest. Top-of-atmospheric forcing predicted by the SOWC model increased by 0.15 W m -2 , surface temperature warmed by 0.001 K, and planetary boundary layer height (PBLH) increased by 2.20 cm in the GHG-Step scenario compared to the BAU scenario. PM climate feedbacks are small because the significant changes in ground level PM concentrations associated with the GHG-Step scenario are limited to the first few hundred meters of the atmosphere, with little change for the majority of the vertical column above that level. As an order-of-magnitude comparison, the long-term effects of global reductions in GHG emissions (RCP8.5–RCP4.5) lowered average surface temperature over the California study domain by approximately 0.76 K. The effects of long-lived climate pollutants such as CO 2 are much stronger than the effects of short-lived climate pollutants such as PM soot over California in the year 2054.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations

Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Finally, several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

On the wavelet optimized finite difference method

When one considers the effect in the physical space, Daubechies-based wavelet methods are equivalent to finite difference methods with grid refinement in regions of the domain where small scale structure exists. Adding a wavelet basis function at a given scale and location where one has a correspondingly large wavelet coefficient is, essentially, equivalent to adding a grid point, or two, at the same location and at a grid density which corresponds to the wavelet scale. This paper introduces a wavelet optimized finite difference method which is equivalent to a wavelet method in its multiresolution approach but which does not suffer from difficulties with nonlinear terms and boundary conditions, since all calculations are done in the physical space. With this method one can obtain an arbitrarily good approximation to a conservative difference method for solving nonlinear conservation laws.

Jameson, Leland↗

NASA NTRSDate not supplied

Effect of the Reynolds Number on the Freestream Disturbance Environment in a Mach 6 Nozzle

To understand the impact of unit Reynolds number on the acoustic disturbance field inside a high-speed wind tunnel, we use Direct Numerical Simulations (DNS) to model the turbulent boundary layers along the walls of a quasi-two-dimensional nozzle configuration. Intended as a stepping stone to fully three-dimensional simulations of freestream noise inside the NASA 20-Inch Mach 6 Wind Tunnel, the present simulations are based on periodic boundary conditions across the spanwise width of the computational domain that corresponds to about one third of the actual tunnel width. These simulations are performed at four different unit Reynolds numbers, ranging from 3.56e6 to 14.0e6 per meter. The predominantly hydrodynamic fluctuations inside the boundary layer are shown to be nearly unaffected by the presence of freestream forcing associated with the impinging acoustic radiation from the opposite wall. Thus, the Reynolds number trends associated with boundary-layer quantities are consistent with previously published DNS of flat-plate boundary layers at similar Mach numbers and wall temperature ratios. The unsteady disturbance environment within the nozzle core region is found to be approximately spatially homogeneous and purely acoustic in nature. The numerical results are used to make comparisons with tunnel noise measurements by Chou et al. [1,2]. Unlike previous comparisons involving static-pressure fluctuations based on the DNS and pitot-pressure fluctuations measured in the wind tunnel, direct comparisons involving the fluctuations in the same physical quantity, namely, the streamwise mass flux, have been reported for the first time. The predicted decrease in the root-mean-square fluctuations in pressure and mass flux with an increasing unit Reynolds number is in agreement with the measurements in the NASA 20-Inch Mach 6 Wind Tunnel. Additional details of the acoustic radiation field are quantified and should be useful toward a digital synthesis of the tunnel disturbance environment that would enable realistic simulations of the natural transition process.

Turbulence↗

NASA NTRSDate not supplied

Effect of the Reynolds Number on the Freestream Disturbance Environment in a Mach 6 Nozzle

To understand the impact of unit Reynolds number on the acoustic disturbance field inside a high-speed wind tunnel, we use Direct Numerical Simulations (DNS) to model the turbulent boundary layers along the walls of a quasi-two-dimensional nozzle configuration. Intended as a stepping stone to fully three-dimensional simulations of freestream noise inside the NASA 20-Inch Mach 6 Wind Tunnel, the present simulations are based on periodic boundary conditions across the spanwise width of the computational domain that corresponds to about one third of the actual tunnel width. These simulations are performed at four different unit Reynolds numbers, ranging from 3.56e6 to 14.0e6 per meter. The predominantly hydrodynamic fluctuations inside the boundary layer are shown to be nearly unaffected by the presence of freestream forcing associated with the impinging acoustic radiation from the opposite wall. Thus, the Reynolds number trends associated with boundary-layer quantities are consistent with previously published DNS of flat-plate boundary layers at similar Mach numbers and wall temperature ratios. The unsteady disturbance environment within the nozzle core region is found to be approximately spatially homogeneous and purely acoustic in nature. The numerical results are used to make comparisons with tunnel noise measurements by Chou et al. [1,2]. Unlike previous comparisons involving static-pressure fluctuations based on the DNS and pitot-pressure fluctuations measured in the wind tunnel, direct comparisons involving the fluctuations in the same physical quantity, namely, the streamwise mass flux, have been reported for the first time. The predicted decrease in the root-mean-square fluctuations in pressure and mass flux with an increasing unit Reynolds number is in agreement with the measurements in the NASA 20-Inch Mach 6 Wind Tunnel. Additional details of the acoustic radiation field are quantified and should be useful toward a digital synthesis of the tunnel disturbance environment that would enable realistic simulations of the natural transition process.

Acoustics↗

DOE OSTI2023

An Infinite Domain 3D Poisson Solver Based on the Barnes-Hut Algorithm

We present a domain decomposition method for the solution of the 3D Poisson equation with infinite domain boundary conditions. The method is based on an application of the Barnes-Hut tree particle scheme adapted to gridded data. Long range interactions are computed using the first two terms in the Cartesian multipole expansion of Green’s function convoluted with the charge while short range computations are performed using Hockney’s domain doubling algorithm. A standard domain decomposition strategy requires O(N 2 ) applications of Hockney’s algorithm, where N is the number of subdomains that intersect that charge support, while in the present approach only O(Nlog 2 N) such computations suffice. The discretization scheme employed is a sixth order Mehrstellen approximation of the 3D Laplace opera tor. The method exhibits satisfactory accuracy at a substantially reduced computational cost compared to the full domain decomposition Hockney’s algorithm.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2004

Adjoint Algorithm for CAD-Based Shape Optimization Using a Cartesian Method

Adjoint solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape optimization. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (geometric parameters that control the shape). More recently, emerging adjoint applications focus on the analysis problem, where the adjoint solution is used to drive mesh adaptation, as well as to provide estimates of functional error bounds and corrections. The attractive feature of this approach is that the mesh-adaptation procedure targets a specific functional, thereby localizing the mesh refinement and reducing computational cost. Our focus is on the development of adjoint-based optimization techniques for a Cartesian method with embedded boundaries.12 In contrast t o implementations on structured and unstructured grids, Cartesian methods decouple the surface discretization from the volume mesh. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin et developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation. Central to automated shape optimization algorithms is the issue of geometry modeling and control. The need to optimize complex, "real-life" geometry provides a strong incentive for the use of parametric-CAD systems within the optimization procedure. In previous work, we presented an effective optimization framework that incorporates a direct-CAD interface. In this work, we enhance the capabilities of this framework with efficient gradient computations using the discrete adjoint method. We present details of the adjoint numerical implementation, which reuses the domain decomposition, multigrid, and time-marching schemes of the flow solver. Furthermore, we explain and demonstrate the use of CAD in conjunction with the Cartesian adjoint approach. The final paper will contain a number of complex geometry, industrially relevant examples with many design variables to demonstrate the effectiveness of the adjoint method on Cartesian meshes.

Nemec, Marian↗

NASA NTRS1990

On the numerical solution of the dynamically loaded hydrodynamic lubrication of the point contact problem

The transient analysis of hydrodynamic lubrication of a point-contact is presented. A body-fitted coordinate system is introduced to transform the physical domain to a rectangular computational domain, enabling the use of the Newton-Raphson method for determining pressures and locating the cavitation boundary, where the Reynolds boundary condition is specified. In order to obtain the transient solution, an explicit Euler method is used to effect a time march. The transient dynamic load is a sinusoidal function of time with frequency, fractional loading, and mean load as parameters. Results include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency. The results are compared with the analytic solution to the transient step bearing problem with the same dynamic loading function. The similarities of the results suggest an approximate model of the point contact minimum film thickness solution.

Lim, Sang G.↗

DOE OSTI2025

Influence of simple terrain on the spatial variability of a low-level jet and wind farm performance in the AWAKEN field campaign

In wind energy research, scientific challenges are often associated with complex terrain sites, where orography, vegetation, and buildings disrupt flow uniformity. However, even sites characterized as simple terrain can exhibit significant spatial variability in wind speed, particularly during stable boundary layers (SBLs) and low-level jets (LLJs). This study investigates these terrain interactions using both simulations and observations from the American WAKe ExperimeNt (AWAKEN). We employ a multiscale Weather Research and Forecasting (WRF) model simulation, integrating mesoscale forcing in the coarse domains and representing three rows of turbines from the King Plains wind farm as generalized actuator disks (GAD) in the large-eddy simulation (LES) domains. During a nocturnal LLJ event on 3 April 2023, the downstream, wake-affected turbine rows outperformed the upstream, unwaked row by 25 %–51 %. This counterintuitive result arises from terrain-induced streamwise variations in hub-height wind speed of approximately 4 m s −1 over 5 km – equivalent to ∼50 % of the upstream reference speed. This enhancement outweighs the wake-induced reduction in mean wind speed (∼12 %) and global blockage effects reported in the literature (∼1 %–3.4 %). The multiscale simulations capture the intra-farm spatial variability in power performance observed in SCADA data. Terrain-induced vertical displacement of the LLJ, coupled with large wind shear below the jet maximum, drives the substantial streamwise acceleration within the wind farm. These findings underscore the importance of accounting for spatial variability related to terrain, even in simple landscapes, particularly during LLJ conditions. Incorporating such effects into reduced-order modeling frameworks for wind farm design and control could significantly enhance their effectiveness.

17 WIND ENERGY↗

NASA NTRS1994

Pseudospectral collocation methods for fourth order differential equations

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

Malek, Alaeddin↗

DOE OSTI2024

Simulating hindered grain boundary diffusion using the smoothed boundary method

Abstract Grain boundaries can greatly affect the transport properties of polycrystalline materials, particularly when the grain size approaches the nanoscale. While grain boundaries often enhance diffusion by providing a fast pathway for chemical transport, some material systems, such as those of solid oxide fuel cells and battery cathode particles, exhibit the opposite behavior, where grain boundaries act to hinder diffusion. To facilitate the study of systems with hindered grain boundary diffusion, we propose a model that utilizes the smoothed boundary method to simulate the dynamic concentration evolution in polycrystalline systems. The model employs domain parameters with diffuse interfaces to describe the grains, thereby enabling solutions with explicit consideration of their complex geometries. The intrinsic error arising from the diffuse interface approach employed in our proposed model is explored by comparing the results against a sharp interface model for a variety of parameter sets. Finally, two case studies are considered to demonstrate potential applications of the model. First, a nanocrystalline yttria-stabilized zirconia solid oxide fuel cell system is investigated, and the effective diffusivities are extracted from the simulation results and are compared to the values obtained through mean-field approximations. Second, the concentration evolution during lithiation of a polycrystalline battery cathode particle is simulated to demonstrate the method’s capability.

Materials Science↗

NASA NTRS1998

Three-Dimensional High-Lift Analysis Using a Parallel Unstructured Multigrid Solver

A directional implicit unstructured agglomeration multigrid solver is ported to shared and distributed memory massively parallel machines using the explicit domain-decomposition and message-passing approach. Because the algorithm operates on local implicit lines in the unstructured mesh, special care is required in partitioning the problem for parallel computing. A weighted partitioning strategy is described which avoids breaking the implicit lines across processor boundaries, while incurring minimal additional communication overhead. Good scalability is demonstrated on a 128 processor SGI Origin 2000 machine and on a 512 processor CRAY T3E machine for reasonably fine grids. The feasibility of performing large-scale unstructured grid calculations with the parallel multigrid algorithm is demonstrated by computing the flow over a partial-span flap wing high-lift geometry on a highly resolved grid of 13.5 million points in approximately 4 hours of wall clock time on the CRAY T3E.

Mavriplis, Dimitri J.↗

NASA NTRS2000

Single Airfoil Gust Response Problem

An unsteady aerodynamic code, called GUST3D, has been developed to solve the single, non-constant coefficient, inhomogeneous convective wave equation for flows with periodic vortical disturbance. The code uses a frequency-domain approach with second-order central differences and a Sommerfeld radiation condition in the far field. It has been extensively validated on model problems with analytical solutions. GUST3D requires as input certain mean flow quantities which are calculated separately by a potential flow solver. This solver calculates the mean flow using a Gothert's Rule approximation. On the airfoil surface, it uses the solution calculated by the potential code FL036. A figure shows the mean pressure along the airfoil surface. To calculate the unsteady pressure, GUST3D was run oil systematically refined grids to obtain a converged solution at each frequency. It was found that 24 points per wavelength was sufficient for convergence. The location of the outer grid boundary was also varied to check for sensitivity to the far-field boundary condition.

Scott, James R.↗

NASA NTRS1994

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Coutsias, Evangelos A.↗

NASA NTRS1986

An explicit finite-volume time-marching procedure for turbulent flow calculations

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite-volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. Control volumes are chosen so that smoothing of flow properties, typically required for stability, is not required. Different time steps are used in the different governing equations. A new pressure interpolation scheme is introduced which improves the shock capturing ability of the method. A multi-volume method for pressure changes in the boundary layer allows calculations which use very long and thin control volumes (length/height - 1000). The method is compared with two test cases. Essentially incompressible turbulent boundary layer flow in an adverse pressure gradient is calculated and the computed distributions of mean velocity and shear are in good agreement with the measurements. Transonic viscous flow in a converging diverging nozzle is calculated; the Mach number upstream of the shock is approximately 1.25. The agreement between the calculated and measured shock strength and total pressure losses is good.

Nicholson, Stephen↗

NASA NTRS1987

An explicit finite-volume time-marching procedure for turbulent flow calculations

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite-volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. Control volumes are chosen so that smoothing of flow properties, typically required for stability, is not required. Different time steps are used in the different governing equations. A new pressure interpolation scheme is introduced which improves the shock capturing ability of the method. A multi-volume method for pressure changes in the boundary layer allows calculations which use very long and thin control volumes (length/height - 1000). The method is compared with two test cases. Essentially incompressible turbulent boundary layer flow in an adverse pressure gradient is calculated and the computed distributions of mean velocity and shear are in good agreement with the measurements. Transonic viscous flow in a converging diverging nozzle is calculated; the Mach number upstream of the shock is approximately 1.25. The agreement between the calculated and measured shock strength and total pressure losses is good.

Nicholson, Stephen↗

NASA NTRS2011

Inflow/Outflow Boundary Conditions with Application to FUN3D

Several boundary conditions that allow subsonic and supersonic flow into and out of the computational domain are discussed. These boundary conditions are demonstrated in the FUN3D computational fluid dynamics (CFD) code which solves the three-dimensional Navier-Stokes equations on unstructured computational meshes. The boundary conditions are enforced through determination of the flux contribution at the boundary to the solution residual. The boundary conditions are implemented in an implicit form where the Jacobian contribution of the boundary condition is included and is exact. All of the flows are governed by the calorically perfect gas thermodynamic equations. Three problems are used to assess these boundary conditions. Solution residual convergence to machine zero precision occurred for all cases. The converged solution boundary state is compared with the requested boundary state for several levels of mesh densities. The boundary values converged to the requested boundary condition with approximately second-order accuracy for all of the cases.

Carlson, Jan-Renee↗

DOE OSTI2025

Adaptive Interface-PINNs (AdaI-PINNs) for transient diffusion: Applications to forward and inverse problems in heterogeneous media

We model transient diffusion in heterogeneous materials using a novel physics-informed neural networks framework (PINNs) termed Adaptive interface physics-informed neural networks or AdaI-PINNs (Roy et al. arXiv preprint arXiv:2406.04626, 2024). AdaI-PINNs utilize different activation functions with trainable slopes tailored to each material region within the computational domain, allowing for a fully automated and adaptive PINNs approach to model interface problems with strongly and weakly discontinuous solutions. To enhance its performance in highly heterogeneous transient diffusion systems, we prescribe a suite of robust practices, including appropriate non-dimensionalization of equations, a biased sampling method, Glorot initialization, and the hard enforcement of boundary and initial conditions. Here we evaluate the efficacy of the proposed method on several benchmark forward and inverse problems. Comparative studies on one-dimensional and two-dimensional benchmark problems reveal that the modified AdaI-PINNs outperform its unmodified counterpart, achieving root-mean-square errors that are at least two orders of magnitude better in forward problems. For inverse problems, the maximum errors in the approximated diffusion coefficients by modified AdaI-PINNs are four orders of magnitude better than those of the unmodified version. Additionally, modified AdaI-PINNs demonstrate improved stability in problems with large material mismatches.

42 ENGINEERING↗