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At least 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Modeling and Simulation of Fuel Dispersal During the Loss-of-Coolant Accident

This document is the compilation of the milestone portion to a larger end of project NEUP report. The executive summary of the modeling portion is provided below: In the event of cladding rupture during a postulated LOCA in a pressurized water reactor, fuel particles, along with fission gases, can be expelled into the reactor core from the fractured fuel rod, a phenomenon referred to as fuel dispersal. The initial stage of fuel dispersal is strongly influenced by the high-pressure ejection of fuel fragments, the size and geometry of the ruptured cladding, and the depressurization history of the fuel rod during the postulated LOCA transient. Depending on the location of the burst orifice relative to the quench front, the dispersal event represents an intricate three-phase flow and heat transfer phenomenon, where high-temperature fuel particles carried by the fission gases interact with the coolant within the narrow subchannels of the fuel assemblies, inducing localized phase change. Given the unique multiphysics nature of this phenomena, the current study develops a dedicated computational framework to predict the mass distribution and cooling of dispersing fuel particles, facilitating post-accident assessment and management of the fuel assemblies. Considering the scale of nuclear reactor applications, a continuum three-fluid model is proposed for simulating the transport of solids within the reactor core. With high-temperature fuel fragments within the liquid media, nucleation sites inducing phase changes are dispersed within the flow domain. Coupled with the fact that the transient dispersal event occurs on different time scales than other three-phase flow applications, this study derives a time-averaged three-fluid flow model without losing generality. The assumptions regarding the continuum treatment of the solid phase and the modeling of fuel dispersal behavior are incorporated to simplify the governing equations and derive applicable closure relations. The computational validation of the model was conducted using adiabatic experimental results obtained from ongoing research at Oregon State University, focusing on characterizing fuel dispersal behavior during simulated LOCA conditions. Settlement characteristics of the solids, quantified by the probability distribution of equivalent particles, closely matched the probability density functions reported in experimental studies. The transport of fuel particles within a scaled 5 × 5 lattice of a pressurized-water reactor rod bundle geometry was modeled through a two-fluid Eulerian framework. The required boundary conditions were evaluated from the fuel performance code BISON in a postulated large-break LOCA scenario. The modeling framework considered solid fuel particles as granular matter, interacting with the gaseous dry steam phase and fission gases through the governing interfacial momentum exchange between the participating fluids. The simulation results provided the volume fraction of the solids obtained at the bottom surface of the enclosing tank geometry. Postulated LOCA leading to fuel dispersal phenomena involves the strong coupling between fuel thermomechanics, cladding deformation, thermal-hydraulics, and fuel particle transport. Incorporation of such a strong coupling in numerical simulation is performed by coupling the multiphysics solvers. In the case of fuel dispersal, a strong coupled simulation can be performed by coupling the BISON code for fuel performance, the TRACE code for system-level thermal hydraulics, and fuel particle transport in Multiphysics Object-Oriented Simulation Environment (MOOSE). For such intricate infrastructure, the MOOSE Framework eases the data transfer between codes. The recent version of MOOSE has incorporated the Navier-Stokes module for the fluid flow. An exploratory exercise was done to gain familiarity with finite volume capabilities in the MOOSE framework to incorporate the Spalart-Allmaras (SA) turbulence model. New finite-volume and auxiliary kernels were introduced to assemble the SA transport equation, compute turbulent viscosity, and evaluate wall distance and diagnostic turbulence terms, fully integrated with existing Navier-Stokes modules. A turbulent lid-driven cavity at a Reynolds number of approximately 10,000 is used for verification. MOOSE shows the robust solver convergence and produces the turbulent features. But it underpredicts the velocity profile and turbulent quantities, emphasizing the need to develop improved SA near-wall treatments (e.g., low-Re corrections or wall functions) as a key direction for future work.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2023

Simulating Catalysis with Realistic Pellet Geometries Using Mesoflow: A Case Study of Catalytic Propane Dehydrogenation

We present a case study of catalytic propane dehydrogenation with our open-source multiphysics solver, Mesoflow. The solver was developed to simulate reactive flow coupled to heterogeneous catalytic reactions and deactivation in the context of complex, mesoscale geometry. The method leverages cartesian block-structured adaptive mesh refinement to capture realistic catalyst microstructural features acquired directly from X-ray computed tomography data. A kinetic model for propane dehydrogenation and catalyst deactivation was developed based on temporal analysis of products (TAP) reactor experiments. The TAP reactor experiments allow for precise characterization of intrinsic kinetic reaction steps which are implemented into Mesoflow simulations to model the spatial and temporal evolution of reactants, products, and catalyst active sites. The short-term and long-term deactivation behavior is studied by using XCT data collected from fresh and aged catalyst pellets, which exhibit different microstructural features. This study employs time-splitting algorithms to connect disparate reaction and flow timescales, enabling the simulations to achieve realistic deactivation timescales on the order of minutes while the flow time-scales for small particles (100 microns) are several milliseconds. We also introduce a flexible automated python script that writes the necessary files to construct a Mesoflow simulation from user-created chemical mechanisms. We will also introduce a few new features that are added to Mesoflow such as higher order schemes, implicit chemistry integrators and the ability to run on AMD and NVIDIA graphics-processing-units.

AMReX↗
DOE OSTI2020

Understanding performance variability in standard and pipelined parallel Krylov solvers

In this work, we collect data from runs of Krylov subspace methods and pipelined Krylov algorithms in an effort to understand and model the impact of machine noise and other sources of variability on performance. We find large variability of Krylov iterations between compute nodes for standard methods that is reduced in pipelined algorithms, directly supporting conjecture, as well as large variation between statistical distributions of runtimes across iterations. Based on these results, we improve upon a previously introduced nondeterministic performance model by allowing iterations to fluctuate over time. We present our data from runs of various Krylov algorithms across multiple platforms as well as our updated non-stationary model that provides good agreement with observations. We also suggest how it can be used as a predictive tool.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Exponential time differencing for the tracer equations appearing in primitive equation ocean models

The tracer equations are part of the primitive equations used in ocean modeling and describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several equations may be added to and coupled to the dynamics system. In many relevant situations, the time-step requirements of explicit methods imposed by the transport and mixing in the vertical direction are more restrictive than those for the horizontal, and this may cause the need to use very small time steps if a fully explicit method is employed. To overcome this issue, we propose an exponential time differencing (ETD) solver where the vertical terms (transport and diffusion) are treated with a matrix exponential, whereas the horizontal terms are dealt with in an explicit way. In this work, we investigate numerically the computational speed-ups that can be obtained over other semi-implicit methods, and we analyze the advantages of the method in the case of multiple tracers.

42 ENGINEERING↗
DOE OSTI2020

Efficient and Flexible Sensitivity Matrix Computation for Adaptive Electrical Capacitance Volume Tomography

Electrical capacitance tomography is a widely used sensor modality for flow imaging in many industrial settings. Adaptive Electrical Capacitance Volume Tomography (AECVT) extends the capabilities of traditional ECT by enabling direct volumetric imaging and an improved resolution. Construction of the sensitivity matrix is a necessary step to obtain flow images. This step requires computation of the electric field inside the sensing domain, which is done via a typical field solver such as the finite element method. In this work, we present an efficient and flexible method to construct the sensitivity matrix for Adaptive Electrical Capacitance Volume Tomography (AECVT) based on individual electrode segment excitations and their judicious combination to form desired matrix elements. We illustrate how the proposed method yields the same sensitivity matrix as the traditional method but at a much lower computational cost. Once all segment contributions are obtained, we also indicate how the proposed method, unlike the traditional approach, can generate the sensitivity matrix on demand for an arbitrary combination of synthetic electrodes and obviating the need for any additional field computations. Finally, we present image reconstruction results for two different experimental scenarios where the mutual capacitance data and the corresponding sensitivity vectors are obtained through the proposed measurement combination scheme.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2021

Deterministic Calibration of MFiX-PIC, Part 1: Settling Bed

The Particle-in-cell (PIC) numerical approach for modeling granular solids in fluid flow has gained significant interest in recent years. Valued for its often shorter time-to-solution, the PIC formulation relies on modeling statistical groupings of particles called parcels in cooperation with a solids stress model to affect local solids velocity. This is in contrast to the discrete element model (DEM) where every particle in a system is modelled individually and directly coupled to local solids velocity through Newtonian mechanics. The U.S. Department of Energy (DOE), National Energy Technology Laboratory (NETL) develops and maintains Multiphase Flow with Interphase eXchanges (MFiX), a collection of open-source computational fluid dynamics (CFD) solvers. Included in the MFiX suite are traditional two-fluid model (TFM) and DEM solvers, and a recently added PIC solver (NETL, 2021). In general, PIC methodologies offer an accuracy trade-off in lieu of computational speed; and therefore, it is important to assess the credibility of MFiX-PIC simulations. For this purpose, a systematic verification, validation and uncertainty quantification (VVUQ) effort was initiated at NETL to assess the new PIC solver

42 ENGINEERING↗
DOE OSTI2025

Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation

The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. Here, to address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems—the Burgers and barotropic vorticity equation—showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.

Gauss-Newton↗
DOE OSTI2022

Three-dimensional Skyrme Hartree-Fock-Bogoliubov solver in coordinate-space representation

The coordinate-space representation of the Hartree-Fock-Bogoliubov theory is the method of choice to study weakly bound nuclei whose properties are affected by the quasiparticle continuum space. To describe such systems, we developed a three-dimensional Skyrme-Hartree-Fock-Bogoliubov solver HFBFFT based on the existing, highly optimized and parallelized Skyrme-Hartree-Fock code Sky3D. The code does not impose any self-consistent spatial symmetries such as mirror inversions or parity. The underlying equations are solved in HFBFFT directly in the canonical basis using the fast Fourier transform. To remedy the problems with pairing collapse, we implemented the soft energy cutoff and pairing annealing. The convergence of HFB solutions was improved by a sub-iteration method. The Hermiticity violation of differential operators brought by Fourier-transform-based differentiation has also been solved. Furthermore, the accuracy and performance of HFBFFT were tested by benchmarking it against other HFB codes, both spherical and deformed, for a set of nuclei, both well-bound and weakly-bound.

3D coordinate-space representation↗
DOE OSTI2022

Predicting extreme anisotropy and shape variations in impact testing of tantalum single crystals

We report recent Taylor cylinder impact tests carried out for Ta single crystals showed strong variations in dimensional changes for different crystallographic directions aligned with the cylindrical axis. In order to capture the effect of crystallography on the deformation characteristics and final shapes of the impacted cylinders, a single crystal material subroutine is adapted and embedded in the solid mechanics/dynamics Finite Element solver Abaqus to simulate the aforementioned single crystal Ta Taylor impact experiments. Details of the coupled model implementation, and insights on the role played by single crystal anisotropic flow on the deformation behavior across a broad range of strain rates and temperatures for different single crystal orientations are presented and discussed. We demonstrate the predictive capability of the adopted crystal plasticity model to capture the significant role played by crystal orientation-induced anisotropy, as well as strain hardening and adiabatic heating, on the dynamic deformation response of crystalline materials. This re-emphasizes the need of microstructure-aware models to improve the accuracy of simulations for high-consequence engineering design.

36 MATERIALS SCIENCE↗
DOE OSTI2024

An unstructured body-of-revolution electromagnetic particle-in-cell algorithm with radial perfectly matched layers and dual polarizations

A novel electromagnetic particle-in-cell algorithm has been developed for fully kinetic plasma simulations on unstructured (irregular) meshes in complex body-of-revolution geometries. The algorithm, implemented in the BORPIC++ code, utilizes a set of field scalings and a coordinate mapping, reducing the Maxwell field problem in a cylindrical system to a Cartesian finite element Maxwell solver in the meridian plane. The latter obviates the cylindrical coordinate singularity in the symmetry axis. The choice of an unstructured finite element discretization enhances the geometrical flexibility of the BORPIC++ solver compared to the more traditional finite difference solvers. Symmetries in Maxwell’s equations are explored to decompose the problem into two dual polarization states with isomorphic representations that enable code reuse. The particle-in-cell scatter and gather steps preserve charge conservation at the discrete level. Our previous algorithm (BORPIC+) discretized the E and B field components of TE Φ and TM Φ polarizations on the finite element (primal) mesh. Here, we employ a new field-update scheme. Using the same finite element (primal) mesh, this scheme advances two sets of field components independently: (1) E and B of TE Φ polarized fields, (E z , E ρ , B Φ ) and (2) D and H of TM Φ polarized fields, (D Φ , H z , H ρ ). Since these field updates are not explicitly coupled, the new field solver obviates the coordinate singularity, which otherwise arises at the cylindrical symmetric axis, ρ = 0 when defining the discrete Hodge matrices (generalized finite element mass matrices). Here, a cylindrical perfectly matched layer is implemented as a boundary condition in the radial direction to simulate open space problems, with periodic boundary conditions in the axial direction. We investigate effects of charged particles moving next to the cylindrical perfectly matched layer. We model azimuthal currents arising from rotational motion of charged rings, which produce TMΦ polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Enabling the Broader Use of MOOSE for Nuclear Energy and Other Simulation

This Final Scientific and Technical Report summarizes work performed under the Phase IIA SBIR project “Enabling the Broader Use of MOOSE for Nuclear Energy and Other Simulation” (DE-SC0020906) from August 2023 through August 2025. The objective of the Phase IIA effort was to mature and harden capabilities developed during Phase II, with the goal of enabling practical interoperability between Coreform’s isogeometric analysis (IGA) technologies and the Multiphysics Object-Oriented Simulation Environment (MOOSE), while improving robustness, performance, and scalability for complex, nuclear-relevant geometries. Over the course of Phase IIA, the project established and validated an extraction-based interoperability pathway between Coreform tools and MOOSE. A combined mesh and matrix format was defined collaboratively with MOOSE developers and integrated into the solver, enabling standard MOOSE workflows to operate on data exported from Coreform’s IGA and Flex Representation Method (FRM) pipelines. Early demonstrations validated architectural compatibility using linear solid mechanics problems, while later efforts focused on benchmark testing and external use. By the end of the project period, engineers at BWXT were able to independently set up and execute a simulation using the Coreform–MOOSE workflow and provide direct feedback that informed further refinement. In parallel, substantial effort was devoted to improving the robustness of trimmed U-spline construction for complex CAD geometries. A growing test suite of nuclear-relevant models was compiled through collaboration with multiple stakeholders and used to drive extensive bug fixing and reliability improvements. These efforts resulted in improved robustness and performance, including the addition of fallback capabilities that enhance reliability when the underlying commercial CAD kernel fails. Performance-oriented work progressed later in the project, with the development and demonstration of methods to decompose complex geometries into structured subregions and updated data representations to support more efficient solver processing. Additionally, extensive enhancements to threadsafe parallel data structures and trimming operations established a foundation for scalable processing of large assemblies. Collaboration with Sandia National Laboratories on the SGM geometric modeling kernel advanced to a functioning interface test case, positioning the workflow for future kernel integration. Overall, the Phase IIA effort successfully transitioned the project from architectural proof-of-concept to externally exercised, solver-integrated capability, while clarifying remaining technical challenges related to standardization, performance optimization, and kernel integration.

42 ENGINEERING↗
DOE OSTI2022

Using Direct Numerical Simulation of Pore-Level Events to Improve Pore-Network Models for Prediction of Residual Trapping of CO2

Direct numerical simulation and pore-network modeling are common approaches to study the physics of two-phase flow through natural rocks. For assessment of the long-term performance of geological sequestration of CO 2 , it is important to model the full drainage-imbibition cycle to provide an accurate estimate of the trapped CO 2 . While direct numerical simulation using pore geometry from micro-CT rock images accurately models two-phase flow physics, it is computationally prohibitive for large rock volumes. On the other hand, pore-network modeling on networks extracted from micro-CT rock images is computationally efficient but utilizes simplified physics in idealized geometric pore elements. This study uses the lattice-Boltzmann method for direct numerical simulation of CO 2 -brine flow in idealized pore elements to develop a new set of pore-level flow models for the pore-body filling and snap-off events in pore-network modeling of imbibition. Lattice-Boltzmann simulations are conducted on typical idealized pore-network configurations, and the interface evolution and local capillary pressure are evaluated to develop modified equations of local threshold capillary pressure of pore elements as a function of shape factor and other geometrical parameters. The modified equations are then incorporated into a quasi-static pore-network flow solver. The modified model is applied on extracted pore-network of sandstone samples, and saturation of residual trapped CO 2 is computed for a drainage-imbibition cycle. The modified model yields different statistics of pore-level events compared with the original model; in particular, the occurrence of snap-off in pore-throats is reduced resulting in a more frontal displacement pattern along the main injection direction. Compared to the original model, the modified model is in closer agreement with the residual trapped CO 2 obtained from core flow experiments and direct numerical simulation.

Kohanpur, Amir H.↗
DOE OSTI2023

A sensitivity analysis of twinning crystal plasticity finite element model using single crystal and poly crystal Zircaloy

The popularity of crystal plasticity finite element method (CPFEM) models is increasing due to their ability to predict the mechanical response of crystalline materials such as metals and metal alloys more accurately than traditional continuum mechanics models. This is since the crystal plasticity models consider the effect of atomic structure, microstructural morphology, and properties of individual grains. These CPFEM models use a large number of material parameters in order to capture the mesoscale physics which comes with the downside of the tedious calibration process. In this paper, a CPFEM code was developed to include the twinning induced grain reorientation and subsequent crystallographic slip for HPC material. The developed code is incorporated in a large-scale, parallelized nonlinear solver WARP3D. Further, a sensitivity analysis with respect to 22 material parameters was then conducted using single crystal and polycrystal representative volume element (RVE) of Zircaloy material. Loading was applied along five different crystallographic orientations for single crystal RVE and along three directions namely, rolling (RD), transverse (TD), and normal (ND) direction for polycrystal RVE. Results obtained from the sensitivity analysis were used for the calibration of material parameters for Zircaloy. Finally, developed code along with calibrated material parameters was used to investigate the effect of the hydride phase formation in Zircaloy which is a typical case observed for nuclear applications. It was found that the volume fraction of the hydride phase has a significant impact on the mechanical properties of Zircaloy.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Exploiting Power Flow Manifold to Solve AC Optimal Power Flow

AC optimal power flow has proven difficult to solve with interior point methods on GPUs. This is largely due to challenging linear algebra problems that current state of the art massively parallel linear solvers struggle with. However, the advent of Riemannian optimization techniques and the fact that the power flow equations form a smooth manifold present an alternative approach. In this talk, we present the basics of Riemannian optimization techniques in which optimization is done directly on a manifold. Then we present computational results showing that Riemannian techniques are capable of producing solutions of comparable quality as interior point methods.

AC optimal power flow↗
DOE OSTI2021

Overcoming timestep limitations in boosted-frame particle-in-cell simulations of plasma-based acceleration

Explicit electromagnetic Particle-In-Cell (PIC) codes are typically limited by the Courant- Friedrichs-Lewy (CFL) condition, which implies that the timestep multiplied by the speed of light must be smaller than the smallest cell size. In the case of boosted-frame PIC simulations of plasma-based acceleration, this limitation can be a major hinderance as the cells are often very elongated along the longitudinal direction and the timestep is thus limited by the small, transverse cell size. This entails many small-timestep PIC iterations, and can limit the potential speed-up of the boosted-frame technique. Here, by using a CFL-free analytical spectral solver, and by mitigating additional numerical instabilities that arise at large timestep, we show that it is possible to overcome traditional limitations on the timestep and thereby realize the full potential of the boosted-frame technique over a much wider range of parameters.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Merger of the Dissolved Gas Method into RELAP5-3D

The Dissolved Gas Model (DGM) was developed for Bettis Atomic Power Laboratory in their own special version of RELAP5-3D known as NUPAC by a subcontractor to. Incorporation of this important code capability in RELAP5-3D was necessary for collaboration between INL and Bettis on future code development efforts. The changes necessary to implement DGM are fundamental, changing the very solver technology at the core of RELAP5-3D, and pervasive, requiring the addition of new subprograms and data, alteration of existing data structures, and modification of more than one hundred subprograms. Moreover, NUPAC and RELAP5-3D had evolved separately and in different directions for a decade and were significantly dissimilar, so the effort was both complex and time-consuming. To ensure its correctness, a large test set comprised thousands of input cases had been constructed and had to run correctly to prove DGM was implemented correctly. A novel method for transporting and adjusting the updates for DGM into RELAP5-3D was undertaken. Tens of thousands of lines of code were either changed or added resulting in a 100% successful implementation of DGM in RELAP5-3D.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

Refactoring the elastic–viscous–plastic solver from the sea ice model CICE v6.5.1 for improved performance

This study focuses on the performance of the elastic–viscous–plastic (EVP) dynamical solver within the sea ice model, CICE v6.5.1. The study has been conducted in two steps. First, the standard EVP solver was extracted from CICE for experiments with refactored versions, which are used for performance testing. Second, one refactored version was integrated and tested in the full CICE model to demonstrate that the new algorithms do not significantly impact the physical results. The study reveals two dominant bottlenecks, namely (1) the number of Message Parsing Interface (MPI) and Open Multi-Processing (OpenMP) synchronization points required for halo exchanges during each time step combined with the irregular domain of active sea ice points and (2) the lack of single-instruction, multiple-data (SIMD) code generation. The standard EVP solver has been refactored based on two generic patterns. The first pattern exposes how general finite differences on masked multi-dimensional arrays can be expressed in order to produce significantly better code generation by changing the memory access pattern from random access to direct access. The second pattern takes an alternative approach to handle static grid properties. The measured single-core performance improvement is more than a factor of 5 compared to the standard implementation. The refactored implementation of strong scales on the Intel® Xeon® Scalable Processors series node until the available bandwidth of the node is used. For the Intel® Xeon® CPU Max series, there is sufficient bandwidth to allow the strong scaling to continue for all the cores on the node, resulting in a single-node improvement factor of 35 over the standard implementation. This study also demonstrates improved performance on GPU processors.

58 GEOSCIENCES↗
DOE OSTI2025

A high-order computational framework for particle-resolved simulations of disperse multiphase flows

This work presents a high-order numerical approach for particle-resolved simulations of disperse multiphase flows, where the Navier-Stokes equations for fluid flow are solved using a high-order spectral element method in the Eulerian framework, and the particle phase is directly simulated with a discrete element method. The coupling between particles and fluids is explicitly handled using an adapted direct-forcing immersed boundary method. Unlike the conventional schemes, a high-order barycentric Lagrange interpolation method and a Gaussian projection kernel are used to ensure accurate momentum exchange between local boundary points and surrounding fluid nodes in the framework of high-order fluid solver. Benchmark tests of increasing complexity are conducted to demonstrate the accuracy and efficiency of our method. Here, it is found that our approach exhibits an excellent convergence performance, as the fluid element/grid is refined and the number of boundary points increases. Compared to conventional low-order methods, the proposed high-order framework enables the use of substantially larger fluid elements while maintaining high accuracy in modeling fluid-particle interactions, owing to the enhanced resolution of high-order basis functions. Moreover, since the primary unknowns are stored at element or grid nodes, the high-order approach offers improved efficiency in both CPU memory usage and total computational cost.

42 ENGINEERING↗