Deterministic Numerical Methods Research Articles

The deterministic neutron transport calculation for the HTR-PM by the three-dimensional method of characteristics code ARCHER

Due to the strong geometric adaptability, the Three-Dimensional (3-D) Method Of Characteristics (MOC) is a promising candidate for 3-D whole-core high-fidelity neutron transport calculations. However, the 3-D MOC can hardly be applied to large-scale full core problems on account of the huge computational costs in memory and time. In the past, the 3-D MOC was mainly used in traditional Light Water Reactor (LWR) calculations, but rarely used for more complex core calculations. For pebble-bed High Temperature gas-cooled Reactors (HTRs), most 3-D MOC codes are out of ability to construct complex pebble beds, let alone efficient acceleration methods. The 3-D MOC code ARCHER can efficiently simulate the large-scale pebble-bed HTRs through using the Linear Source Approximation (LSA), the Coarse Mesh Finite Difference (CMFD) and the hybrid MPI-OpenMP parallel. In this work, the two-level CMFD acceleration and efficient preconditioners on Krylov subspace linear solvers are implemented in the ARCHER code in order to achieve better performance. In addition, the practical HTR-PM criticality problem is calculated by ARCHER, which is the world’s first high-fidelity neutron transport solution of large-scale pebble-bed HTRs by deterministic numerical method. It also shows that the 3-D MOC has great application potential in other complex reactors.

A robust error correction method for numerical weather prediction wind speed based on Bayesian optimization, variational mode decomposition, principal component analysis, and random forest: VMD-PCA-RF (version 1.0.0)

Abstract. Accurate wind speed prediction is crucial for the safe and efficient utilization of wind resources. However, current single-value deterministic numerical weather prediction methods employed by wind farms do not adequately meet the actual needs of power grid dispatching. In this study, we propose a new hybrid forecasting method for correcting 10 m wind speed predictions made by the Weather Research and Forecasting (WRF) model. Our approach incorporates variational mode decomposition (VMD), principal component analysis (PCA), and five artificial intelligence algorithms: deep belief network (DBN), multilayer perceptron (MLP), random forest (RF), eXtreme gradient boosting (XGBoost), light gradient boosting machine (lightGBM), and the Bayesian optimization algorithm (BOA). We first predict wind speeds using the WRF model, with initial and lateral boundary conditions from the Global Forecast System (GFS). We then perform two sets of experiments with different input factors and apply BOA optimization to tune the four artificial intelligence models, ultimately building the final models. Furthermore, we compare the aforementioned five optimal artificial intelligence models suitable for five provinces in southern China in the wintertime: VMD-PCA-RF in December 2021 and VMD-PCA-lightGBM in January 2022. We find that the VMD-PCA-RF evaluation indices exhibit relative stability over nearly a year: the correlation coefficient (R) is above 0.6, forecasting accuracy (FA) is above 85 %, mean absolute error (MAE) is below 0.6 m s−1, root mean square error (RMSE) is below 0.8 m s−1, relative mean absolute error (rMAE) is below 60 %, and relative root mean square error (rRMSE) is below 75 %. Thus, for its promising performance and excellent year-round robustness, we recommend adopting the proposed VMD-PCA-RF method for improved wind speed prediction in models.

A novel numerical method for stochastic study of fiber-reinforced composites with nanoparticles under impact loading

Nanoparticle-reinforced composites have caught attention due to their enhanced properties and wide potential use. However, the uncertainty in the distribution of nanoparticles during the manufacturing poses several challenges when deterministic numerical methods are used to model their mechanical behavior, following a macro homogeneous approach. In this work, a stochastic numerical method based on finite element modeling is proposed to simulate the inhomogeneous distribution of nanoparticles inside the composite, focusing on low-velocity impact. Each element in the finite element model is regarded as a unit cell with a constant weight fraction of nanoparticles, while the random distribution of nanoparticles is described by a variable weight fraction among elements following a Gaussian distribution. This distribution is related to the mechanical behavior of the unit cell inside the model and it was defined based on material characterization tests for different nanoparticles weight fractions. To validate the proposed model, experimental low-velocity impact tests, at different energy levels, on composites reinforced with nanoparticles (0.5 wt.%) were carried out. The numerical method was assessed by comparing experimental results in terms of mechanical response and damage phenomena. The current numerical methodology was found to contribute to uncertainty investigation based on a mixed experimental-numerical analysis.

Tribological Performance of a Composite Cold Spray for Coated Bores

The tribological performance of a thermal sprayed, mirror-like surface with localized protuberances was investigated through tribotests and computational simulation. A composite coating with a 410L steel matrix and M2 tool steel hard particles was applied by the cold spray process as a bore coating for combustion engines. The presence of protuberances promoted the quick formation of an antifriction tribofilm when tested with an SAE 0W-16 containing ZDDP and MoDTC, which significantly reduced the asperity friction in comparison to the conventional engine coated bores in reciprocating tribological tests. An in-house computational model using deterministic numerical methods was used for the mixed and hydrodynamic lubrication regime. Lubricant film thickness and friction were simulated for a piston ring versus the proposed coating. The computer simulations showed that the protuberances reduced the hydrodynamic friction by increasing the otherwise very thin oil film thickness of mirror-like surfaces. Although not intuitive, this result was caused by the reducing of the oil film shear rate.

Coupled Monte Carlo–Deterministic simulation of thermal behavior of graphene on substrate

The purpose of this paper is to study thermal effects in a graphene sheet on substrate. The temperature evolution of the phonon branches and of the crystal lattice is obtained when also the presence of the substrate is taken into account. The numerical strategy is based on the coupling of stochastic and deterministic numerical methods; Boltzmann equation for charge transport is treated by using the Direct Simulation Monte Carlo approach, whose results are treated as source terms for the phonon Boltzmann equation which is solved deterministically. The results are useful also for applications as correct design of graphene-based prototypes, when heating effects in the presence of a substrate are relevant.

XVA in a multi-currency setting with stochastic foreign exchange rates

In the present article we address the modelling and the numerical computation of the total value adjustment for European options in a multi-currency setting when the foreign exchange rates between the different involved currencies are assumed to be stochastic. Thus, we extend to a more realistic approach a previous work where constant exchange rates have been considered. New models are formulated both in terms of linear and nonlinear PDEs and expectations, the hedging arguments requiring the additional consideration of the exposure to foreign exchange risk. For the nonlinear models, Picard iteration methods are applied to the formulation in terms of expectations and compared with multilevel Picard iteration methods. In this way, we avoid the curse of dimensionality associated to the use of deterministic numerical methods (such as finite differences or finite element methods) for solving high dimensional PDEs. Some examples of option pricing problems illustrate the performance of the proposed models and numerical methods.

Regularization-free multicriteria optimization of polymer viscoelasticity model

This paper introduces a multiobjective optimization (MOP) method for nonlinear regression analysis which is capable of simultaneously minimizing the model order and estimating parameter values without the need of exogenous regularization constraints. The method is introduced through a case study in polymer rheology modeling. Prevailing approaches in this field tackle conflicting optimization goals as a monobjective problem by aggregating individual regression errors on each dependent variable into a single weighted scalarization function. In addition, their supporting deterministic numerical methods often rely on assumptions which are extrinsic to the problem, such as regularization constants and restrictions on parameter distribution, thereby introducing methodology inherent biases into the model. Our proposed non-deterministic MOP strategy, on the other hand, aims at finding the Pareto-front of all optimal solutions with respect not only to individual regression errors, but also to the number of parameters needed to fit the data, automatically reducing the model order. The evolutionary computation approach does not require arbitrary constraints on objective weights, regularization parameters or other exogenous assumptions to handle the ill-posed inverse problem. The article discusses the method rationales, implementation, simulation experiments, and comparison with other methods, with experimental evidences that it can outperform state-of-art techniques. While the discussion focuses on the study case, the introduced method is general and immediately applicable to other problem domains.

Large-Scale Heterogeneous Computing for 3D Deterministic Particle Transport on Tianhe-2A Supercomputer

Scalable parallel algorithm for particle transport is one of the main application fields in high-performance computing. Discrete ordinate method (Sn) is one of the most popular deterministic numerical methods for solving particle transport equations. In this paper, we introduce a new method of large-scale heterogeneous computing of one energy group time-independent deterministic discrete ordinates neutron transport in 3D Cartesian geometry (Sweep3D) on Tianhe-2A supercomputer. In heterogeneous programming, we use customized Basic Communication Library (BCL) and Accelerated Computing Library (ACL) to control and communicate between CPU and the Matrix2000 accelerator. We use OpenMP instructions to exploit the parallelism of threads based on Matrix 2000. The test results show that the optimization of applying OpenMP on particle transport algorithm modified by our method can get 11.3 times acceleration at most. On Tianhe-2A supercomputer, the parallel efficiency of 1.01 million cores compared with 170 thousand cores is 52%.

A separable nonlinear least squares approach for double-diode photovoltaic model parameter extraction

Parameter extraction of the double-diode photovoltaic model is a highly nonconvex optimization problem, and up until now, many metaheuristic methods have been proposed to try to avoid local minima. However, these metaheuristic methods will output different results in repeated tests and need to manually set lower and upper bounds for all parameters. In this work, a deterministic numerical method is proposed for the double-diode model parameter extraction through a separable nonlinear least squares approach. In this approach, the complexity of the optimization problem is greatly lowered by the reduction of independent parameters, and the solution can be easily obtained by deterministic search algorithms without manual preset or intervention. The proposed method is first validated on two commonly used case studies of curve data fitting, showing a comparable performance to the best reported metaheuristic methods. Then, the method is extended to parameter extraction from manufacturer datasheets, in which only three important I–V data points are available. Finally, the method is tested on a large-scale dataset containing more than one million I–V curves provided by the National Renewable Energy Laboratory (NREL). The result of the large-scale test proves the superiority of the double-diode model over the single-diode counterpart in precise modeling of photovoltaic modules.

Sequential stratified splitting for efficient Monte Carlo integration

The efficient evaluation of high-dimensional integrals is important from both theoretical and practical points of view. In particular, multidimensional integration plays a central role in Bayesian inference, statistical physics, data science, and machine learning. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in the high-dimensional setting. Consequentially, for many practical problems one must resort to approximate estimation techniques such as Monte Carlo methods. In this article, we introduce a novel sequential Monte Carlo algorithm called stratified splitting. The method provides unbiased estimates and can handle various integrand types including indicator functions, which are important for rare-event probability estimation problems. We provide rigorous analysis of the efficiency of the proposed method and present a numerical demonstration of the algorithmic performance when applied to practical application domains. Our numerical experiments suggest that the stratified splitting method is capable of delivering accurate results for a variety of integration problems while requiring reasonable computational effort.

Yansıtıcılı reaktörlerin stokastik iki-nokta reaktör kinetik denklemlerinin sayısal simülasyonu

Deterministic numerical solutions of point reactor kinetic equations give us the mean values of the neutron population and delayed neutron precursor concentrations, whereas the actual dynamical process is stochastic and the neutron population and precursor concentrations fluctuate randomly with time. In the present study, a novel stochastic model for two-point reactor kinetics equations is developed and used to analyze the dynamical behavior of the source-free strongly reflected reactors with six groups of delayed neutron precursors. To derive the Itô stochastic differential equations system corresponding to this model, the two-point reactor kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and reflected neutrons. In the case of different perturbation scenarios, both with and without the Newtonian temperature reactivity feedback effects, this system of stochastic differential equations is solved using the Euler-Murayama numerical method. It is observed that the mean response of the system is comparable with the results of other deterministic numerical methods.

The first-order scattering approximation: A closed-form extension to Beer’s law, accurate for weakly scattering media

This article applies perturbation theory to obtain an asymptotic expansion that approximates the Radiative Transfer Equation (RTE) Green’s function. The integral (source function) formulation of the RTE yields a recursive operator; applying the operator once to a point source collimated beam yields a closed-form extension to Beer’s Law. In the weakly scattering range (i.e., albedo below 0.36 or domain optical thickness below 0.85), the approximation correctly describes the energy transported in phase space for any phase function (mean squared errors below 10%). Besides the theoretical methodology’s value, the closed-form approximation allows developing a deterministic numerical method, free of directional meshes, that computes the global irradiance field in three-dimensional enclosures under different internal and boundary conditions. For the cases tested, its accuracy and versatility are comparable to a Monte Carlo method in the weakly scattering range, while its computational efficiency resembles that of the P1 approximation.

Density Difference Grid Design in a Point-Mass Filter

The paper deals with the Bayesian state estimation of nonlinear stochastic dynamic systems. The stress is laid on the point-mass filter, solving the Bayesian recursive relations for the state estimate conditional density computation using the deterministic grid-based numerical integration method. In particular, the grid design is discussed and the novel density difference grid is proposed. The proposed grid design covers such regions of the state-space where the conditional density is significantly spatially varying, by the dense grid. In other regions, a sparse grid is used to keep the computational complexity low. The proposed grid design is thoroughly discussed, analyzed, and illustrated in a numerical study.

A modified perturbation method for mathematical models with randomness: An analysis through the steady‐state solution to Burgers' partial differential equation

The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier–Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady‐state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables.

A kinetic approach of the bi-temperature Euler model

<p style='text-indent:20px;'>We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [<xref ref-type="bibr" rid="b4">4</xref>]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is independant of any parameter <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (HLL, Suliciu).

Nonlinear dynamic analysis method for large-scale single-layer lattice domes with uncertain-but-bounded parameters

Currently, nonlinear dynamic analysis for large-scale single-layer domes is commonly performed using deterministic numerical methods. However, in practical engineering cases, complex large-scale single-layer domes have many uncertain parameters that cannot be considered using deterministic methods. Therefore, there is a growing awareness that classical deterministic methods need to be extended towards the introduction of the uncertain aspects in dynamic analysis, and a non-deterministic analysis method for large-scale spatial structures is required. In this paper, a new method is presented by introducing uncertainties into the nonlinear dynamic analysis for large-scale single-layer lattice domes. The method accounts for uncertainties in material properties, structural imperfections, loads, and damping with bounds. The focus is on the treatment of uncertainty in damping and the adopted geometric shape, which is different from that of conventional approaches. Finite element dynamic analyses for sample structures with multiple sources of uncertainty subjected to dynamic loads are performed. Results show that the variability of the variables with an associated uncertainty imposes significant negative effects on the dynamic properties, dynamic demands, and safety of a dome. Uncertain damping in a structure plays the most important role in determining structural performance. The numerical results reveal the differences between conventional analysis methods with deterministic parameters used in previous practical applications and the uncertain analysis method. Finally, a parametric study is performed, and the impacts of sample size on statistical dynamic demands, single uncertain source on structural failure, and single uncertain source on damping coefficients are discussed.

Dirichlet Forms and Finite Element Methods for the SABR Model

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via ...

Forward deterministic pricing of options using Gaussian radial basis functions

The price of a fixed-term option is the expected value of the payoff at the time of maturity. When not analytically available, the option price is computed using stochastic or deterministic numerical methods. The most common approach when using deterministic methods is to solve a backward partial differential equation (PDE) such as the Black–Scholes equation for the option value. The problem can alternatively be formulated based on a forward PDE for the probability of the asset value at the time of maturity. This enables simultaneous pricing of several contracts with different payoffs written on the same underlying asset. The main drawback is that the initial condition is a (non-smooth) Dirac function. We show that by using an analytical expansion of the solution for the first part of the time interval, and applying a high-order accurate radial basis function (RBF) approximation in space, we can derive a competitive forward pricing method. We evaluate the proposed method on European call options and barrier options, and show that even for just one payoff it is more efficient than solving the corresponding backward PDE.

Predicting fatigue crack growth evolution via perturbation series expansion method based on the generalized multinomial theorem

In this study, a novel numerical calculation method is proposed to investigate the fatigue crack growth evolution in aluminum alloy sheets accounting for the measurement error. Unlike the deterministic numerical method, the initial crack length is considered to be a modified parameter with a small correction term due to the measurement error; the solution to the crack growth equation is expressed in the form of a perturbation series after introducing said small parameter. By combining the proposed perturbation series expansion method (PSEM) with the deterministic crack growth equation, a series of modified equations for predicting the crack length history are derived based on the generalized multinomial theorem. Further, by substituting the initial condition under perturbation into the modified equations, variations in crack length versus the cycle number can be obtained. The proposed method is verified by comparing numerical results with experimental data, and the results demonstrate that the proposed model is indeed feasible and effective for predicting fatigue crack growth evolution.

Statistical analysis of differential equations: introducing probability measures on numerical solutions

In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.