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At least 577 records · Page 32

Status of the MARS code

This report describes major features of the most recent version of the MARS code as well as ongoing developments. The list of features includes various options for geometry models, a beam line builder based on MADX code, import of geometry models in GDML format, use of structured and unstructured meshes for scoring purposes, an update to the recent TENDL library for a number of projectiles at low energies (up to 250 MeV), and a recently implemented method to calculate spatial distribution of residual dose in a single computer run without an intermediate source. Examples of the code application to various projects are presented as well.

Rakhno, Igor [Fermilab] (ORCID:0000000265828058)↗

A Parallel Cartesian Approach for External Aerodynamics of Vehicles with Complex Geometry

This workshop paper presents the current status in the development of a new approach for the solution of the Euler equations on Cartesian meshes with embedded boundaries in three dimensions on distributed and shared memory architectures. The approach uses adaptively refined Cartesian hexahedra to fill the computational domain. Where these cells intersect the geometry, they are cut by the boundary into arbitrarily shaped polyhedra which receive special treatment by the solver. The presentation documents a newly developed multilevel upwind solver based on a flexible domain-decomposition strategy. One novel aspect of the work is its use of space-filling curves (SFC) for memory efficient on-the-fly parallelization, dynamic re-partitioning and automatic coarse mesh generation. Within each subdomain the approach employs a variety reordering techniques so that relevant data are on the same page in memory permitting high-performance on cache-based processors. Details of the on-the-fly SFC based partitioning are presented as are construction rules for the automatic coarse mesh generation. After describing the approach, the paper uses model problems and 3- D configurations to both verify and validate the solver. The model problems demonstrate that second-order accuracy is maintained despite the presence of the irregular cut-cells in the mesh. In addition, it examines both parallel efficiency and convergence behavior. These investigations demonstrate a parallel speed-up in excess of 28 on 32 processors of an SGI Origin 2000 system and confirm that mesh partitioning has no effect on convergence behavior.

Aftosmis, M. J.↗

Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates

In this paper, we present new Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials. Two different approaches are used to discretize the evolution equations – the first one is the true volume approach and the second one is the area-weighted approach. For the true volume approach, the DG equations are derived using the true 3D volume that is consistent with the geometry conservation law (GCL). The Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem using surfaces areas for axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based on the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, the Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem in 2D Cartesian geometry. This area-weighted approach, in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. Finally, a suite of test problems are calculated to demonstrate stable mesh motion, the expected second order accuracy of these methods, and that the new area-weighted DG method preserves spherical symmetry on 1D radial flow problems with equal-angle polar meshes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Reassessment of uncertainties for the PU-MET-INTER-002 benchmark - 14432

Uncertainty for the PU-MET-INTER-002 benchmark of ZPR-6/10 was unusually large for a ZPR assembly and was dominated by the uncertainties in matrix tube pitch and composition uncertainties for plutonium, graphite and stainless steel plates. The original uncertainty analysis for PU-MET-INTER-002 was limited by computer resources, software and the nuclear data available in 2002, all of which have been improved since. An as-built Monte Carlo model of ZPR-6/10 was used to quantify components of the transformation from the detailed model to a simplified RZ benchmark in 2002. Compositional uncertainties were estimated based upon TWODANT calculations of the RZ benchmark model (20 group, S{sub 8} - P{sub 1}). MC{sup 2}-2 was used with END/B-V.2 data to generate homogenized cross section data for TWODANT. The MC{sup 2}-2 calculations did not account for lattice cell spatial heterogeneity which was driven by the manual process required to generate ∼80 modifications to the deterministic model and by the software limitations of MC{sup 2}-2. Improvements to DIF3D-VARIANT, development of the MC{sup 2}-3 and BuildZPRModel codes, and better computer resources make accurate 3D transport calculations of reactivity changes practical today. BuildZPRModel imports the as-built geometry and creates a 'drawer' homogenized DIF3D model where each radial DIF3D mesh is the exact size of a single ZPR matrix tube and drawer. A MC{sup 2}-3 input is constructed for each drawer to produce homogenized cross section data where the spatial cell heterogeneity is captured by solving a 1D MOC transport calculation on the ultrafine energy group level. Collapsing 3D drawers into 1D MC{sup 2}-3 drawer models follows conventional procedures and is done by BuildZPRModel. MCNP6 was used to re-compute relevant geometry components of the uncertainty using ENDF/B-VII.0 data with the as-built model. Changes in the geometry components compared with earlier results are insignificant. DIF3D calculations of the compositional uncertainties were performed using 33, 70, 116, 230, 425, and 703 group energy structures and selectively combined with P{sub 3}-P{sub 7} transport approximations and P{sub 1} and P{sub 3} scattering kernels. 3D diffusion theory calculations were also performed for comparison. Many composition uncertainty components were essentially unchanged with the updated methodology. However, the dominant experimental uncertainty for PU-MET-INTER-002 is the composition of stainless steel. The new calculations reduce the stainless steel uncertainty from ∼115 pcm to ∼70 pcm which significantly reduces the total uncertainty for PU-MET-INTER-002. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Fuel Injection Dynamics and Composition Effects on RDE Performance

Rotating detonation engines (RDEs) provide a promising route to substantially increasing cycle efficiency in stationary gas turbines. Much of this increase relies on the ability to achieve consistent pressure gain within the combustor. In particular, the design of injectors that feed fuel and air into the detonation channel plays a crucial role. Such injectors have to ensure proper mixing of fuel and oxidizer, while minimizing backflow of detonation products into the feed plenums, and reduce susceptibility to the complex wave structures that exist within the combustor. From a practical perspective, such RDEs also need to operate with variable fuel composition. When fuel mixtures with components that possess vastly different oxidation pathways and time-scales are used, there could be additional losses through deflagrative burning instead of detonation-driven heat release. Such sensitivity to the complete flow path is akin to the physics of thermoacoustic instabilities in conventional gas turbines. In this sense, RDEs pose a unique research challenge: the performance of the device relies on the small-scale heat release process, which is highly dependent on the flow interactions within the full-scale system. As a result, canonical flow configurations, instrumented with detailed diagnostics or modeled using high-fidelity tools, but only focus on the small-scale processes will not contain the key system-level interactions. At the same time, macroscopic measurements and models that only capture system-level performance will not provide insight into the key sources of pressure losses. These couplings and sensitivities provide a formidable challenge to both experimental and simulation studies of the effects Thus, a joint experimental/computational program designed specifically to address these challenges was undertaken in this program. The focus of this program was on two key topics: a) the interaction between injector flow and the overall wave dynamics within the combustor, and b) the deflagration/detonation structure in multi-component fuels that are of practical interest. Both topics involve interaction of small-scale heat release processes with the geometry-dependent wave structure. Studies focused on the study of full-scale RDE systems, based on a 6-inch conventional annular geometry. Experimentally RDEs were studied using a combination of diagnostics. A combination of optical diagnostics and aero-thermo-acoustic analysis based on a combination of spectral and mode decomposition analysis was used to identify the dynamics of the detonation wave and other secondary waves that exist in the system. These studies have helped the identification and investigation of injector and detonation dynamics arising from coupling, and how they affect RDE mixing, detonation structure, operability and performance. Performance of RDEs was investigated through thrust stand measurements, which was used to evaluate the effective pressure gain generated by the system through the concept of equivalent available pressure. Optical diagnostics were developed and implemented to investigate the distribution of heat release, across the detonation wave. Novel optical diagnostics of NIR imaging was also developed and applied to investigate the high temperature / high pressure distribution across the detonation wave. In order to complement the experiments, the computational tools were geared to simulate the full experimental setup. GPU-based acceleration of the models and computations were developed to enable rapid simulation of the full system. In addition, the use of adaptive mesh refinement, and unstructured grid formulation, enabled the investigation of realistic geometries studied in the laboratory. The simulations produced a wealth of detail on the structure of the detonation wave under different operating conditions. Emphasis was placed on quantifying mixture pre-burning and the impact on wave propagation and structure.

03 NATURAL GAS↗

Convergence study of wakefield simulations with GdfidL and ECHO3D

The interaction of charged particle beams with vacuum chamber components gives rise to electromagnetic wakefields, whose frequency-domain representation is known as beam coupling impedance. Geometric impedance arising from discontinuities and transitions in the vacuum chamber is the focus of this study. Minimizing this impedance is essential to mitigate adverse collective effects in modern storage rings operating with high-intensity particle beams. Accurate and reliable impedance simulations is a key factor of the vacuum chamber design. This paper presents the results of a convergence study of two widely used electromagnetic solvers, GdfidL and ECHO3D, applied to key vacuum-chamber components of the National Synchrotron Light Source II (NSLS-II) storage ring. Detailed comparisons are performed for several geometries, including flange absorbers, RF bellows, button-type beam position monitors, and an in-vacuum undulator (IVU). The results show notable differences in convergence and computational efficiency between the two codes. While GdfidL provides highly resolved results and serves as a common benchmark tool, ECHO3D yields consistent results with coarser meshes, significantly reducing simulation time and memory demands. Simulations with a full-geometry IVU model demonstrate that simplified taper-transition models can miss important impedance contributions. In conclusion, these findings provide practical guidelines for efficient and accurate impedance modeling to optimize design of vacuum chamber components for accelerators.

36 MATERIALS SCIENCE↗

A Local Incident Flux Response Expansion Transport Method for Coupling to the Diffusion Method in Cylindrical Geometry

A local incident flux response expansion transport method is developed to generate transport solutions for coupling to diffusion theory codes regardless of their solution method (e.g., fine mesh, nodal, response based, finite element, etc.) for reactor core calculations in both two-dimensional (2-D) and three-dimensional (3-D) cylindrical geometries. In this approach, a Monte Carlo method is first used to precompute the local transport solution (i.e., response function library) for each unique transport coarse node, in which diffusion theory is not valid due to strong transport effects. The response function library is then used to iteratively determine the albedo coefficients on the diffusion-transport interfaces, which are then used as the coupling parameters within the diffusion code. This interface coupling technique allows a seamless integration of the transport and diffusion methods. The new method retains the detailed heterogeneity of the transport nodes and naturally constructs any local solution within them by a simple superposition of local responses to all incoming fluxes from the contiguous coarse nodes. A new technique is also developed for coupling to fine-mesh diffusion methods/codes. The local transport method/module is tested in 2-D and 3-D pebble-bed reactor benchmark problems consisting of an inner reflector, an annular fuel region, and a controlled outer reflector. It is found that the results predicted by the transport module agree very well with the reference fluxes calculated directly by MCNP in both benchmark problems.

99 GENERAL AND MISCELLANEOUS↗

Effects of Mesh Irregularities on Accuracy of Finite-Volume Discretization Schemes

The effects of mesh irregularities on accuracy of unstructured node-centered finite-volume discretizations are considered. The focus is on an edge-based approach that uses unweighted least-squares gradient reconstruction with a quadratic fit. For inviscid fluxes, the discretization is nominally third order accurate on general triangular meshes. For viscous fluxes, the scheme is an average-least-squares formulation that is nominally second order accurate and contrasted with a common Green-Gauss discretization scheme. Gradient errors, truncation errors, and discretization errors are separately studied according to a previously introduced comprehensive methodology. The methodology considers three classes of grids: isotropic grids in a rectangular geometry, anisotropic grids typical of adapted grids, and anisotropic grids over a curved surface typical of advancing layer grids. The meshes within the classes range from regular to extremely irregular including meshes with random perturbation of nodes. Recommendations are made concerning the discretization schemes that are expected to be least sensitive to mesh irregularities in applications to turbulent flows in complex geometries.

Diskin, Boris↗

Computerized Generation and Simulation of Meshing and Contact of New Type of Novikov-Wildhaber Helical Gears

In this report, we propose a new geometry for low-noise, increased-strength helical gears of the Novikov-Wildhaber type. Contact stresses are reduced as a result of their convex-concave gear tooth surfaces. The gear tooth surfaces are crowned in the profile direction to localize bearing contact and in the longitudinal direction to obtain a parabolic function of transmission errors. Such a function results in the reduction of noise and vibrations. Methods for the generation of the proposed gear tooth surfaces by grinding and hobbing are considered, and a tooth contact analysis (TCA) computer program to simulate meshing and contact is applied. The report also investigates the influence of misalignment on transmission errors and shift of bearing contact. Numerical examples to illustrate the developed approaches are proposed. The proposed geometry was patented by Ford/UIC (Serial Number 09-340-824, pending) on June 28, 1999.

Litvin, Faydor L.↗

Spur gears: Optimal geometry, methods for generation and Tooth Contact Analysis (TCA) program

The contents of this report include the following: (1) development of optimal geometry for crowned spur gears; (2) methods for their generation; and (3) tooth contact analysis (TCA) computer programs for the analysis of meshing and bearing contact on the crowned spur gears. The method developed for synthesis is used for the determination of the optimal geometry for crowned pinion surface and is directed to reduce the sensitivity of the gears to misalignment, localize the bearing contact, and guarantee the favorable shape and low level of the transmission errors. A new method for the generation of the crowned pinion surface has been proposed. This method is based on application of the tool with a surface of revolution that slightly deviates from a regular cone surface. The tool can be used as a grinding wheel or as a shaver. The crowned pinion surface can also be generated by a generating plane whose motion is provided by an automatic grinding machine controlled by a computer. The TCA program simulates the meshing and bearing contact of the misaligned gears. The transmission errors are also determined.

Litvin, Faydor L.↗

A subdivision-stabilized B-spline mixed material point method

Subjected to external loadings, polymeric materials, e.g., biological tissues, hydrogels, and elastomers, may undergo extreme, nearly incompressible, (self-)contact deformations. For numerical modeling employing mesh-based techniques such as the finite element method (FEM), these deformations pose significant challenges due to large distortions in the deformed geometry, accuracy issues stemming from volumetric locking effects, and increased computational cost from complex contact searches. As an alternative to mesh-based methods, the material point method (MPM), a continuum-based particle technique, is gaining attention for its ability to handle extreme distortions and capture no-slip contact without added cost. For nearly incompressible material behaviors, while mixed formulations can address locking effects by treating displacements and pressure as independent fields, they can suffer from numerical instabilities close to the incompressibility limit due to the violation of the inf-sup condition, leading to inaccurate nodal pressure solutions. Here we propose an efficient and stable mixed B-spline material point method with highest achievable regularity for quasi-compressible polymeric materials. Using the two-scale relation of B-splines, we introduce a subdivision-stabilization for the two-field mixed MPM and obtain numerically stable, oscillation-free nodal solutions with equal-order interpolations with optimal regularity. Building on the Eulerian-Lagrangian nature of MPM, a previously-converged solution framework is adopted to mitigate issues related to cell-crossing and numerical fracture artifact present in standard MPM. We assess the stability and accuracy of the developed mixed MPM at large deformations for soft materials through the benchmark Cook’s membrane problem. Additionally, we test the robustness of the proposed MPM by modeling several examples, including the compression and indentation of a circular block into a quasi-compressible substrate and the twisting deformation of a rectangular block. The findings demonstrate the MPM’s capabilities for modeling practical soft material applications.

36 MATERIALS SCIENCE↗

DNS and LES of turbulent flow in a closed channel featuring a pattern of hemispherical roughness elements

Three mesh adaptivity algorithms were developed to facilitate and expedite the use of the CADIS and FW-CADIS hybrid Monte Carlo/deterministic techniques in accurate full-scale neutronics simulations of fusion energy systems with immense sizes and complicated geometries. First, a macromaterial approach enhances the fidelity of the deterministic models without changing the mesh. Second, a deterministic mesh refinement algorithm generates meshes that capture as much geometric detail as possible without exceeding a specified maximum number of mesh elements. Finally, a weight window coarsening algorithm decouples the weight window mesh and energy bins from the mesh and energy group structure of the deterministic calculations in order to remove the memory constraint of the weight window map from the deterministic mesh resolution. The three algorithms were used to enhance an FW-CADIS calculation of the prompt dose rate throughout the ITER experimental facility and resulted in a 23.3% increase in the number of mesh tally elements in which the dose rates were calculated in a 10-day Monte Carlo calculation. Additionally, because of the significant increase in the efficiency of FW-CADIS simulations, the three algorithms enabled this difficult calculation to be accurately solved on a regular computer cluster, eliminating the need for a world-class super computer.

Nuclear Science & Technology↗

Automatic Mesh Adaptivity for Hybrid Monte Carlo/Deterministic Neutronics Modeling of Fusion Energy Systems

Three mesh adaptivity algorithms were developed to facilitate and expedite the use of the CADIS and FW-CADIS hybrid Monte Carlo/deterministic techniques in accurate full-scale neutronics simulations of fusion energy systems with immense sizes and complicated geometries. First, a macromaterial approach enhances the fidelity of the deterministic models without changing the mesh. Second, a deterministic mesh refinement algorithm generates meshes that capture as much geometric detail as possible without exceeding a specified maximum number of mesh elements. Finally, a weight window coarsening algorithm decouples the weight window mesh and energy bins from the mesh and energy group structure of the deterministic calculations in order to remove the memory constraint of the weight window map from the deterministic mesh resolution. The three algorithms were used to enhance an FW-CADIS calculation of the prompt dose rate throughout the ITER experimental facility and resulted in a 23.3% increase in the number of mesh tally elements in which the dose rates were calculated in a 10-day Monte Carlo calculation. Additionally, because of the significant increase in the efficiency of FW-CADIS simulations, the three algorithms enabled this difficult calculation to be accurately solved on a regular computer cluster, eliminating the need for a world-class super computer.

Hybrid Monte Carlo/deterministic↗

Investigation of advanced counterrotation blade configuration concepts for high speed turboprop systems. Task 4: Advanced fan section aerodynamic analysis

The purpose of this study is the development of a three-dimensional Euler/Navier-Stokes flow analysis for fan section/engine geometries containing multiple blade rows and multiple spanwise flow splitters. An existing procedure developed by Dr. J. J. Adamczyk and associates and the NASA Lewis Research Center was modified to accept multiple spanwise splitter geometries and simulate engine core conditions. The procedure was also modified to allow coarse parallelization of the solution algorithm. This document is a final report outlining the development and techniques used in the procedure. The numerical solution is based upon a finite volume technique with a four stage Runge-Kutta time marching procedure. Numerical dissipation is used to gain solution stability but is reduced in viscous dominated flow regions. Local time stepping and implicit residual smoothing are used to increase the rate of convergence. Multiple blade row solutions are based upon the average-passage system of equations. The numerical solutions are performed on an H-type grid system, with meshes being generated by the system (TIGG3D) developed earlier under this contract. The grid generation scheme meets the average-passage requirement of maintaining a common axisymmetric mesh for each blade row grid. The analysis was run on several geometry configurations ranging from one to five blade rows and from one to four radial flow splitters. Pure internal flow solutions were obtained as well as solutions with flow about the cowl/nacelle and various engine core flow conditions. The efficiency of the solution procedure was shown to be the same as the original analysis.

Crook, Andrew J.↗

Interface capturing simulations of bubble population effects in PWR subchannels

As the computational power of high-performance computing (HPC) facilities grows, so too does the feasibility of using first principle based simulation to study turbulent two-phase flows within complex pressurized water reactor (PWR) geometries. Direct numerical simulation (DNS), integrated with an interface capturing method, allows for the collection of high-fidelity numerical data using advanced analysis techniques. The research presented here employs the massively parallel, finite-element based, unstructured mesh code, PHASTA, to simulate a set of two-phase bubbly flows through PWR subchannel geometries including auxiliary structures (spacer grids and mixing vanes). The main objective of the presented work is to analyze bubble dynamics and turbulence interactions at varying bubble concentrations to support the development of advanced two-phase flow closure models. Turbulent two-phase flows in PWR subchannels were simulated at hydraulic Reynolds numbers of 81,000 with bubble concentrations of 3%–15% by gas volume fraction (768–3928 resolved bubbles, respectively) and compared against a 1% void fraction case (262 bubbles) that had been previously simulated. The finite element mesh utilized for the study at higher bubble concentrations was composed of 1.55 billion elements, compared to the previous study which employed 1.11 billion elements, ensuring all turbulence scales and individual bubbles within the flow are fully resolved. For each case, the resolved initial bubble size was 0.65 mm in diameter (resolved with 25 grid points across the diameter). The simulations were analyzed to find flow features such as the mean velocity profile, bubble relative velocity and the effect of the bubbles on the turbulent conditions.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

High Order Wall-Modeled Large-Eddy Simulation on Mixed Unstructured Meshes

In the present study, an algebraic equilibrium wall model previously developed for hexahedral elements is extended to handle mixed meshes including prismatic, tetrahedral, and pyramidal elements in the context of a discontinuous high-order method. This extension is needed for complex geometries, for which high-order mixed elements (e.g., tetrahedral and pyramidal elements) are often necessary near solid walls to avoid meshing challenges. Various design decisions are discussed to achieve the best performance on massively parallel CPU/GPU architectures for a production-level high-order large-eddy simulation solver based on the flux reconstruction/correction procedure via reconstruction method, hpMusic. The extension to other elements is first evaluated using a benchmark channel flow problem at various Reynolds numbers. After that, flow over the NASA high-lift Common Research Model (CRM-HL) from the 4th AIAA High-lift Prediction Workshop is computed to further test the new implementation. Computational results at the third- and fourth-order accuracies are compared with experimental data.

Engineering↗

Applications of Space-Filling-Curves to Cartesian Methods for CFD

The proposed paper presents a variety novel uses of Space-Filling-Curves (SFCs) for Cartesian mesh methods in 0. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, most are applicable on general body-fitted meshes -both structured and unstructured. We demonstrate the use of single O(N log N) SFC-based reordering to produce single-pass (O(N)) algorithms for mesh partitioning, multigrid coarsening, and inter-mesh interpolation. The intermesh interpolation operator has many practical applications including warm starts on modified geometry, or as an inter-grid transfer operator on remeshed regions in moving-body simulations. Exploiting the compact construction of these operators, we further show that these algorithms are highly amenable to parallelization. Examples using the SFC-based mesh partitioner show nearly linear speedup to 512 CPUs even when using multigrid as a smoother. Partition statistics are presented showing that the SFC partitions are, on-average, within 10% of ideal even with only around 50,000 cells in each subdomain. The inter-mesh interpolation operator also has linear asymptotic complexity and can be used to map a solution with N unknowns to another mesh with M unknowns with O(max(M,N)) operations. This capability is demonstrated both on moving-body simulations and in mapping solutions to perturbed meshes for finite-difference-based gradient design methods.

Aftosmis, Michael J.↗

Applications of Space-Filling-Curves to Cartesian Methods for CFD

This paper presents a variety of novel uses of space-filling-curves (SFCs) for Cartesian mesh methods in CFD. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, many are applicable on general body-fitted meshes-both structured and unstructured. We demonstrate the use of single theta(N log N) SFC-based reordering to produce single-pass (theta(N)) algorithms for mesh partitioning, multigrid coarsening, and inter-mesh interpolation. The intermesh interpolation operator has many practical applications including warm starts on modified geometry, or as an inter-grid transfer operator on remeshed regions in moving-body simulations Exploiting the compact construction of these operators, we further show that these algorithms are highly amenable to parallelization. Examples using the SFC-based mesh partitioner show nearly linear speedup to 640 CPUs even when using multigrid as a smoother. Partition statistics are presented showing that the SFC partitions are, on-average, within 15% of ideal even with only around 50,000 cells in each sub-domain. The inter-mesh interpolation operator also has linear asymptotic complexity and can be used to map a solution with N unknowns to another mesh with M unknowns with theta(M + N) operations. This capability is demonstrated both on moving-body simulations and in mapping solutions to perturbed meshes for control surface deflection or finite-difference-based gradient design methods.

Aftosmis, M. J.↗