Numerical Efficiency Research Articles

Constrained sensitivity filtering technique for topology optimization with lower computational expense

AbstractConstrained Sensitivity Filtering Method (CSFM) has been proposed to save computational expenses in topology optimization. The sensitivity filtering technique is widely adopted to avoid numerical instabilities. Because the filtered sensitivity values are recalculated from the normalized density values and the sensitivity values from within a fixed range of neighborhoods, the values are sometimes completely different from the original ones. Thus, the idea of the CSFM is to control the change of filtered sensitivity values based on the previous iterations. By controlling the changes of filtered sensitivity values within certain limitation, optimal layouts can be obtained with lower compliance and the computational expenses can also be reduced in comparison to that by conventional topology optimization. The computational expense with CSFM topology optimization could be reduced by up to 85%. The numerical examples established that CSFM topology optimization has improved the numerical efficiency and effectiveness.

Statistical Analysis of Fixed Mini-Batch Gradient Descent Estimator

We study here a fixed mini-batch gradient descent (FMGD) algorithm to solve optimization problems with massive datasets. In FMGD, the whole sample is split into multiple nonoverlapping partitions. Once the partitions are formed, they are then fixed throughout the rest of the algorithm. For convenience, we refer to the fixed partitions as fixed mini-batches. Then for each computation iteration, the gradients are sequentially calculated on each fixed mini-batch. Because the size of fixed mini-batches is typically much smaller than the whole sample size, it can be easily computed. This leads to much reduced computation cost for each computational iteration. It makes FMGD computationally efficient and practically more feasible. To demonstrate the theoretical properties of FMGD, we start with a linear regression model with a constant learning rate. We study its numerical convergence and statistical efficiency properties. We find that sufficiently small learning rates are necessarily required for both numerical convergence and statistical efficiency. Nevertheless, an extremely small learning rate might lead to painfully slow numerical convergence. To solve the problem, a diminishing learning rate scheduling strategy can be used. This leads to the FMGD estimator with faster numerical convergence and better statistical efficiency. Finally, the FMGD algorithms with random shuffling and a general loss function are also studied. Supplementary materials for this article are available online.

Integrated Three‐Dimensional Geological and Numerical Groundwater Model Development

AbstractConceptual site model (CSM) development is a foundational step for numerical groundwater model construction. Tools for reviewing and combining disparate data to develop a CSM have been furthered through software that efficiently integrates multiple data streams, especially when a large amount of information is available at complex sites. Three‐dimensional (3D) interfaces for data visualization and geological modeling are becoming commonly used to compile data, support development of a robust CSM, and create effective visuals that enable project stakeholders to understand site complexity. This article describes a 3D interface workflow applied to a geological and numerical groundwater modeling case study where numerical modeling (i.e., MODFLOW family of codes) was used to assess remedy effectiveness. Specifically, the numerical model was used to overview an in situ solidification (ISS) application, where soil mixing will be conducted with cement, for possible adverse effects to an existing groundwater treatment system. 3D geological modeling allowed for the existing numerical groundwater flow model to be reconstructed to include previously overlooked data on buried infrastructure, which led to a detailed depiction of high‐permeability channels and a more accurate representation of site conditions. Modeling showed that the ISS remedy would not adversely impact the existing groundwater treatment system. Integration of existing data within a 3D interface led to hydrogeological insights that would have been difficult to obtain from two‐dimensional visualization of information, improved numerical modeling efficiency, and informed site remedy decisions. Additionally, the 3D interface created effective visuals that increased internal and external project team conceptual understanding.

Spherical Grid Creation and Modeling Using the Galerkin Compiler GC_Sphere

The construction of spherical grids is, to a large extent, a question of organized programming. Such grids come in the form of rhomboidal/triangular grids and hexagonal grids. We are here mainly interested in Local-Galerkin high-order schemes and consider the classical fourth-order o4 method for comparison. High-order Local-Galerkin schemes imply sparse grids in a natural way, with an expected saving of computer runtime. Sparse grids on the sphere are described for rhomboidal and hexagonal cells. They are obtained by not using some of the full grid points. Technical problems and grid organization will be discussed with the purpose of reaching fully realistic applications. We present the description of a programming concept allowing people, using different programming styles at different locations, to work together. The concept of geometric files is introduced. Such geometric files can be offered for downloading and are supposed to allow Local-Galerkin methods to be introduced into an existing model with little effort. When the geometric files are known, the solution on a spherical grid is equivalent to the limited-area Galerkin solutions on the (irregular) plane grids on the patches. The proposed grids can be used with spectral elements (SE) and the Local-Galerkin methods o2o3 and o3o3. The latter offer an increased numerical efficiency which, in a toy model test, resulted in a ten-times-reduced computer run time.

A Modified Gradient Method for Distributionally Robust Logistic Regression over the Wasserstein Ball

In this paper, a modified conjugate gradient method under the forward-backward splitting framework is proposed to further improve the numerical efficiency for solving the distributionally robust Logistic regression model over the Wasserstein ball, which comprises two phases: in the first phase, a conjugate gradient descent step is performed, and in the second phase, an instantaneous optimization problem is formulated and solved with a trade-off minimization of the regularization term, while simultaneously staying in close proximity to the interim point obtained in the first phase. The modified conjugate gradient method is proven to attain the optimal solution of the Wasserstein distributionally robust Logistic regression model with nonsummable steplength at a convergence rate of 1/T. Finally, several numerical experiments to validate the effectiveness of theoretical analysis are conducted, which demonstrate that this method outperforms the off-the-shelf solver and the existing first-order algorithmic frameworks.

Wigner Gaussian dynamics: Simulating the anharmonic and quantum ionic motion

The atomic motion controls important features of materials, such as thermal transport, phase transitions, and vibrational spectra. However, the simulation of ionic dynamics is exceptionally challenging when quantum fluctuations are relevant (e.g., at low temperatures or with light atoms) and the energy landscape is anharmonic. In this work, we formulate the Time-Dependent Self-Consistent Harmonic Approximation (TDSCHA) in the Wigner framework, paving the way for the efficient computation of the nuclear motion in systems with sizable quantum and thermal anharmonic fluctuations. Besides the improved numerical efficiency, the Wigner formalism unveils the classical limit of TDSCHA and provides a link with the many-body perturbation theory of Feynman diagrams. We further extend the method to account for the non-linear couplings between phonons and photons, responsible, e.g., for a nonvanishing Raman signal in high-symmetry Raman inactive crystals, firstly discussed by Rasetti and Fermi. We benchmark the method in phase III of high-pressure hydrogen ab initio. The nonlinear photon-phonon coupling reshapes the IR spectra and explains the high-frequency shoulder of the H2 vibron observed in experiments. The Wigner TDSCHA is computationally cheap and derived from first principles: it is unbiased by assumptions on the phonon-phonon and phonon-photon scattering and does not depend on empirical parameters. Therefore, the method can be adopted in unsupervised high-throughput calculations.

Fractional-Order Control Techniques for Renewable Energy and Energy-Storage-Integrated Power Systems: A Review

The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and greenhouse emissions. The control system of renewable energy units and energy storage systems has a high effect on their performance and absolutely on the efficiency of the total power network. Classical controllers are based on integer-order differentiation and integration, while the fractional-order controller has tremendous potential to change the order for better modeling and controlling the system. This paper presents a comprehensive review of the energy system of renewable energy units and energy storage devices. Various papers are evaluated, and their methods and results are presented. Moreover, the mathematical fundamentals of the fractional-order method are mentioned, and the various studies are categorized based on different parameters. Various definitions for fractional-order calculus are also explained using their mathematical formula. Different studies and numerical evaluations present appropriate efficiency and accuracy of the fractional-order techniques for estimating, controlling, and improving the performance of energy systems in various operational conditions so that the average error of the fractional-order methods is considerably lower than other ones.

Accelerated second-order hydrodynamic load calculation on semi-submersible floaters

A rapid method for calculation of second-order hydrodynamic wave loads on semi-submersible platforms is developed and validated against radiation–diffraction theory. The method is based on slender-body theory and builds on modal truncation of the quadratic transfer function (QTF). The semi-submersible floater is split into individual members and the existing theory for vertical cylindrical columns is extended to include the heave force. Further expressions for the surge, heave and pitch load on the horizontal pontoons are derived and implemented. The accuracy of the method is assessed by comparison to radiation–diffraction results using the Pinkster approximation. We find that the slender-body approximation for the column surge force is most accurate for small values of the diameter-to-draft ratio. For the three sea states considered, this error is below 10% for diameter-to-draft ratios less than 0.2.Error analysis is provided for the column heave and pitch and the pontoon loads. For all members, application of 128 modes in the QTF approximation is found sufficient to accurately represent the full slender-body QTF solution.Next, the first- and second-order loads on the full floater under different sea state conditions are compared to radiation–diffraction theory. With 128 modes, the second-order loads are obtained 2500 times faster than with conventional approaches with error levels of 22% for surge, 10% for pitch and zero error for heave. The surge error is discussed and linked to the small draft of the columns. The numerical efficiency of the method allows the consideration of second-order loads in the first stages of the design and optimisation of semi-submersible floaters.

Improved multi-relaxation time thermal pseudo-potential lattice Boltzmann method with multi-block grid and complete unit conversion for liquid–vapor phase transition

In recent years, the thermal pseudo-potential lattice Boltzmann method (LBM) has been widely adopted in numerical simulations of liquid–vapor phase transition systems. However, the unit conversion for thermal pseudo-potential LBM remains incomplete and elusive, and the numerical simulation efficiency is limited by the uniformly single-block grid. In this paper, the dimensionless evolution equations of thermal pseudo-potential LBM with multi-relaxation time operators are derived for the convenience of adopting real physical parameters and improvement of computational efficiency. The energy equation is re-derived and improved for enhanced accuracy and convenience of numerical calculation. Additionally, a more accurate Martin–Hou equation of state for cryogen is adopted and a modified term for surface tension coefficient is improved to confirm that the surface tension coefficient is grid independent. Moreover, a three-layer boundary structure for the coarse grid is proposed to introduce the multi-block grid into the thermal pseudo-potential LBM for taking into account the intermolecular force and internal heat source term. The aforementioned works improve the thermal pseudo-potential LBM and enable efficient and accurate simulation of the liquid–vapor phase transition within the three-dimensional structure with real physical parameters of a specific working fluid. Finally, numerical simulations are adopted to validate the efficiency and accuracy of the proposed improvements for simulating liquid–vapor phase transition.

Assessment of turbulence models and mesh types for the simulation of multi-body separation by coupling the CFD and rigid body dynamics

Turbulence models are considered one of the most important sources of numerical errors in the CFD studies on different aeronautical problems. The mesh type can also affect the numerical accuracy and computational efficiency. In this paper, various simulations on the store separation from an aircraft model in the transonic flow have been carried out by coupling the CFD and the rigid body dynamics on the overset mesh. The influence of S-A model, Standard k-epsilon model, SST k-omega model and inviscid model on the simulation results is examined. Different mesh types of both unstructured mesh and structured mesh as component part in overset are also considered. Grid-independence study are first carried out. The effects of different turbulence models and mesh types are then examined by comparing the trajectories, velocities and Eulerian angles. Recommendations are given to get better results for the simulation of multi-body separation problem.

Derivation of the Penman–Monteith Equation with the Thermodynamic Approach. II: Numerical Solutions and Evaluation

A review of the derivation of the Penman–Monteith equation with the thermodynamic approach of Monteith is presented in a companion manuscript. The resultant set of equations (expressed in terms of latent heat flux, lf, sensible heat flux, qf, final air temperature, Ta, and the slope parameter related to the saturation vapor pressure curve, Δ) represents a coupled system. Thus, a pair of alternative numerical solutions, with different levels of complexity, were developed and evaluated in the study reported here. Results showed that the alternative models (labeled as model 1 and 2) produced outputs that are essentially identical and also in close agreement with a reference solution. Intercomparison of the alternative models based on the criteria of numerical efficiency and robustness suggests that each model represents a comparable alternative to the other to estimate evaporation. However, owing to its simplicity, model 1 was selected for further consideration. A comparison of the outputs of model 1 with those of the conventional model (i.e., the approach widely used to evaluate the Penman–Monteith set of equations), based on data sets covering a range of evaporation conditions, showed that the difference in the approaches implemented in the two models has a significant effect on estimates of qf, a limited effect on lf, and a negligible effect on Ta. Notably, the results also showed that the mean absolute residual for latent heat flux, lf (i.e., the mean of the absolute residuals between estimates obtained with model 1 and the conventional model) is relatively small (only about 8.2%), suggesting that differences between lf estimates computed with model 1 and the conventional model should generally be within the margin of error of the conventional model.

Linear well posedness of regularized equations of sea-ice dynamics

The viscous–plastic equations (VPE) of Hibler [J. Geophys. Res. 82(27), 3932–3938 (1977)] are widely adopted and used in Earth system models to represent sea-ice drift due to surface winds, ocean currents, and internal stresses. However, it has been reported by various investigators, at least in one space dimension, that both Hibler’s original equations and their variant using a pressure replacement are ill posed in divergent flow regimes. Especially, Guba et al. [J. Phys. Oceanogr. 43(10), 2185–2199 (2013)] shows that both variants are ill-posed when the flow divergence exceeds a minimum threshold and their results seem to extend to two dimensions when a tensile cut-off is used. In particular, Hibler uses a Heaviside function cut-off for the viscosity coefficients of the VPE’s to avoid a singularity at infinity. Lemieux et al. [J. Comput. Phys. 231(17), 5926–5944 (2012)] regularized the Heaviside function by a hyperbolic tangent for numerical efficiency. Here, we show that, for periodic data, the linearized one-dimensional regularized VPE’s, in which the Heaviside function is replaced with a hyperbolic tangent, is well posed in the case of Hibler’s original equations. Moreover, we prove that the linearization procedure, for the regularized equations, is consistent, in the sense that the residual converges to zero that the perturbation of the solutions goes to zero, in suitable norms.

Phase-field modeling of austenitic steels used in turbines

The performance in hydro-electric turbine casting and repair requires understanding of how process parameters and chemistry selection affect solidification microstructures. The aim of this study is to provide a quantitative phase-field formulation for process-microstructure relationships that seeks to model stainless steels. We have developed a phase-field model to simulate austenitic stainless steel solidification under experimental thermal histories. To this end we look at a pseudo-binary approximations for numerical efficiency. The pseudo-binary formulation is underpinned by the alloying element equivalent value, a metallurgical tool used to analyze the microstructural impact of “minor” alloying elements in stainless steels. For model validation we develop thin wall casting experiments to measure the thermal history and chemistry controlled microstructure. The models incorporate a thermodynamic parameterization and are linked to a thermal-phase transformation model which represents the experimentally measured thermal history. The results display a good agreement with the primary branch spacing and cellular to dendritic transition of the casting experiments. These models and software provide the basis for future expansion to include more complex microstructures.

Numerically efficient two-step CFD simulation model of flow regime and heat transfer for hot water shower sterilizers

This work was carried out to create a numerically efficient validated simulation model for Hot Water Shower (SWS) sterilizers. The goal of the work was to achieve a more homogeneous temperature and water distribution inside the sterilizers developed in the future. A lab-scale test bench of an SWS sterilizer with a configurable water volume flow rate and temperature was designed to validate the Computational Fluid Dynamics (CFD) model that was created. The heating and cooling phases at water volume flow rates of 20 and 40m3/h per m2 spraying surface were analyzed. Additionally, the flow and film regime were studied. The CFD model was split into two simulation steps to significantly improve the numerical efficiency. The novel two-step approach developed is computationally 1800 times faster as compared with a complete sterilization cycle than a state-of-the-art approach. The simulated temperature of the test load was in good accordance with the experimental measurements. The flow regime was largely accurately replicated. Overall, the created validated CFD model is suitable for upscaling to an industry-sized SWS sterilizer due to its numerical efficiency.

Data-Driven High-to-Low for Coarse Grid System Thermal Hydraulics

Traditional one-dimensional system thermal-hydraulic analysis has been widely applied in the nuclear industry for licensing purposes because of its numerical efficiency. However, such tools have inherently limited opportunities for modeling multiscale multidimensional flows in large reactor enclosures. Recent interest in three-dimensional coarse grid (CG) simulations has shown their potential in improving the predictive capability of system-level analysis. At the same time, CGs do not allow one to accurately resolve and capture turbulent mixing and stratification, whereas implemented in CG solvers relatively simple turbulence models exhibit large model form uncertainties. Therefore, there is a strong interest in further advances in CG modeling techniques. In this work, two high-to-low data-driven (DD) methodologies (and their combination) are explored to reduce grid and model-induced errors using a case study based on the Texas A&M upper plenum of a high-temperature gas-cooled reactor facility. The first approach relies on the use of a DD turbulence closure [eddy viscosity predicted by a neural network (NN)]. A novel training framework is suggested to consider the influence of grid cell size on closure. The second methodology uses a NN to predict velocity errors to improve low-fidelity results. Both methodologies and their combination have shown the potential to improve CG simulation results by using data with higher fidelity.

Multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration using preconditioned Krylov subspace method for multigroup neutron Boltzmann transport equation.

A multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration [accelerated steady discrete unified gas kinetic scheme (SDUGKS)] is proposed to improve the convergence of the original SDUGKS for an optically thick system in solving the multigroup neutron Boltzmann transport equation (NBTE) to analyze the distribution of fission energy in the reactor core. In the accelerated SDUGKS, by solving the coarse mesh macroscopic governing equations (MGEs) derived from the moment equations of the NBTE, the numerical solutions of the NBTE on fine meshes at the mesoscopic level can be rapidly obtained from the prolongation of the coarse mesh solutions of the MGE. Furthermore, the use of the coarse mesh can greatly reduce the computational variables and improve the computational efficiency of the MGE. The biconjugate gradient stabilized Krylov subspace method with the modified incomplete LU preconditioner and the lower-upper symmetric-Gauss-Seidel sweeping method are implemented to solve the discrete systems of the macroscopic coarse mesh acceleration model and mesoscopic SDUGKS to further improve the numerical efficiency. Numerical solutions validate good numerical accuracy and high acceleration efficiency of the proposed accelerated SDUGKS for the complicated multiscale neutron transport problems.

A new simplified distortional hardening model for nonlinear strain paths

The material behaviors under nonlinear strain paths have been successfully explained through distortional hardening model in which the yield surface is mainly rotating and contracting as plastic deformation occurs. In order to apply to industrial applications, it is beneficial for any constitutive model including distortional hardening model to have simplicity of equation, which is related to numerical efficiency, as well as accuracy in describing material behaviors. This study was mainly motivated to construct a simple form of equation for distortional hardening model with reasonable accuracy as the conventional distortional hardening models. In this paper, a new simplified distortional hardening model is proposed by considering one of the pioneering works for distortional hardening (Tozawa, 1978) and definition of homogeneous function. Any yield function could be used for initial anisotropy and asymmetry yielding. In addition, with the combination of isotropic hardening model for consistency condition, the new model could explain both isotropic and anisotropic hardening under proportional and non-proportional loading paths. The proposed model reasonably described the material behaviors including Basuchinger effect, permanent softening, nonlinear transient behavior, and yield surface contraction. Finally, the availability in finite element simulation has been identified from the springback prediction in U-draw bending.

Three-dimensional numerical simulation, energy efficiency and economic benefit estimation of oil shale in situ pyrolysis process

Oil shale in situ conversion processing by the downhole burner heating technology (DBHT) has been technically proven as a field pilot project in the Songliao Basin, Northeast China. In this study, the three-dimensional numerical simulations of oil shale in situ pyrolysis with the hexagonal well pattern were conducted based on the idea of DBHT and the actual oil shale reservoir in Songliao Basin to examine the temperature field and thermal processes. The effects of different well spacings (30, 35, 40, and 45 m), gas injection rates (5, 10, 15, and 20 m s−1), and fracture spacings (2.5 and 3.5 m) on the thermal evaluation process were analyzed. The relative energy efficiency and economic viability of operating an oil shale in situ retort in various cases were evaluated. The results showed that increasing well spacing could enlarge the thermal field and improve the ultimate hydrocarbon production of oil shale, but the excessive well spacing caused the heating time to be too long. An increase in the gas injection rate can significantly boost heat transmission efficiency. The energy efficiency comparison of various cases showed that smaller well spacing was conducive to making full use of energy. The results of economic efficiency estimation indicated that a high recovery of shale oil (>80%) would be very difficult to be made with economy, but optimal conditions can be developed in some cases, determined mainly by the price of shale oil. We anticipate that this study can provide new ideas for designing the oil shale in situ conversion processing.

On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves

This paper presents a novel theoretical framework in the Hamiltonian theory of nonlinear surface gravity waves. The envelope of surface elevation and the velocity potential on the free water surface are introduced in the framework, which are shown to be a new pair of canonical variables. Using the two envelopes as the main unknowns, coupled envelope evolution equations (CEEEs) are derived based on a perturbation expansion. Similar to the high-order spectral method, the CEEEs can be derived up to arbitrary order in wave steepness. In contrast, they have a temporal scale as slow as the rate of change of a wave spectrum and allow for the wave fields prescribed on a computational (spatial) domain with a much larger size and with spacing longer than the characteristic wavelength at no expense of accuracy and numerical efficiency. The energy balance equation is derived based on the CEEEs. The nonlinear terms in the CEEEs are in a form of the separation of wave harmonics, due to which an individual term is shown to have clear physical meanings in terms of whether or not it is able to force free waves that obey the dispersion relation. Both the nonlinear terms that can only lead to the forcing of bound waves and those that are capable of forcing free waves are demonstrated, in the case of the latter through the analysis of the quartet and quintet resonant interactions of linear waves. The relations between the CEEEs and two other existing theoretical frameworks are established, including the theory for a train of Stokes waves up to second order in wave steepness (Fenton, ASCE J. Waterway Port Coastal Ocean Engng, vol. 111, issue 2, 1985, pp. 216–234) and a semi-analytical framework for three-dimensional weakly nonlinear surface waves with arbitrary bandwidth and large directional spreading by Li & Li (Phys. Fluids, vol. 33, issue 7, 2021, 076609).

An Accurate Phase Interface Locating Algorithm for Pore-Scale Two-Phase Interfacial Flows

Direct numerical simulation of pore-scale flow in porous media at pore scale is a fast developing technique to investigate pore-scale flow behaviors. However, with the decrease of the spatial scale, the interfacial tension of the two-phase interface will have an important impact on the two-phase flow processes. As a result, spurious currents near the interface with this method have an important impact on the numerical accuracy and computational efficiency, seriously affecting the numerical prediction of pore-scale fluid flow. Ghost cell method can greatly reduce the spurious currents near the interface, but it needs to locate the phase interface accurately. In this work, a new method using constant velocity spiral to approximate the two-phase interface is proposed. The method not only considers the curvature of the phase interface but also considers the influence of the curvature change on the phase interface, which greatly improves the ability of the phase interface position. The new method can improve the accuracy of interfacial tension treatment and then improve the prediction ability of volume of fluid in high interfacial tension driven flow. Numerical simulations of capillary rising, droplet spreading on a plane, and bubble rising show that the method is accurate and has a strong engineering application prospect.