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DOE OSTI2018

Boundary node Petrov–Galerkin method in solid structures

Based on the interpolation of the Lagrange series and the Finite Block Method (FBM), the formulations of the Boundary Node Petrov–Galerkin Method (BNPGM) are presented in the weak form in this paper and their applications are demonstrated to the elasticity of functionally graded materials, subjected to static and dynamic loads. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate to the normalized coordinate with 8 seeds for two-dimensional problems. The first-order partial differential matrices of boundary nodes are obtained in terms of the nodal values of the boundary node, which can be utilized to determine the tractions on the boundary. Time-dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin’s inversion method is applied to determine the physical values in the time domain. Illustrative numerical examples are given and comparison has been made with the analytical solutions, the Boundary Element Method (BEM) and the Finite Element Method (FEM).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A new method approximating offset curve by Bézier curve using parallel derivative curves

In this paper, we present a new method for the offset approximation of an n-th degree Bézier curve by an n+1-th degree Bézier curve using parallel derivative curves. Our method is based on the Gauss map approximation of the n-th degree Bézier curve by the n+1-th degree Bézier curve whose derivative curve is parallel to that of the n-th degree Bézier curve. Thus, the convolution of the Gauss map approximation curve and n-th Bézier curve is a polynomial of degree n+1, which is our offset approximation of the n-th degree Bézier curve. Our approximation method has two advantages. One is that the offset approximation method yields a non-rational (polynomial) curve which is obtained by the simple sum of two Bézier curves. The other is that the offset approximation curve satisfying the endpoint interpolation of the given offset curve automatically has the same tangent lines as the offset curve at both endpoints. We illustrate our offset approximation method with some numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A numerical approach for solving a class of variable-order fractional functional integral equations

This paper proposes new discretization techniques to estimate variable-order fractional integral operators based on the piecewise integro quadratic spline interpolation. The proposed methods are modified to solve a class of variable-order fractional functional integral equations. Moreover, we investigate the performance of the proposed methods by solving the variable-order fractional pantograph and Emden–Fowler functional integral equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Plasma equation of state accounting for ion core volume

Highlights: •Thermodynamically consistent Saha model accounting for non-point ion cores is constructed. •The applicability range of the model is expanded on superhigh densities and moderate temperatures. •Experimental verification of the model is performed which has demonstrated that the achieved accuracy is about 1% for pressures larger than 11 Mbar. -- Abstract: Equation of state for gaseous plasma is well described by the Saha model. In the present paper, we include accounting for finite ion core volumes into this model. This improvement allows to expand the Saha model on superhigh densities and moderate temperatures where plasma can be considered liquid. In this domain, thermodynamics of the Saha model is essentially close to the one of the Tomas–Fermi model which is conventionally used for condensed matter. To construct wide-range equation of state, we imply interpolation of the two mentioned models providing rigorous thermodynamic consistency. Comparison with experimental data on shock adiabats shows good agreement for a very wide range of pressures.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Application of the Redlich-Kister expansion for estimating the density of molten fluoride psuedo-ternary salt systems of nuclear industry interest

The development of next-generation molten salt reactors relies on accurate knowledge of the thermophysical properties of the candidate coolant and fueled molten salts. These thermophysical properties include density, viscosity, thermal conductivity, and heat capacity. Because of difficulties in measuring thermophysical properties of molten salts, there are many gaps in the current state of thermophysical property knowledge of these salts, particularly those that contain actinides or beryllium. Therefore, leveraging modeling techniques to estimate unknown molten salt thermophysical properties and guide future experimental measurements has high value for the nuclear industry. Here, the densities of molten fluoride pseudo-ternary salt systems, which are of interest to the nuclear industry, were estimated using Redlich-Kister expansion and Muggianu interpolation techniques. The pseudo-ternary systems considered for estimation in this study were NaF-LiF-ZrF 4 , LiF-BeF 2 -ZrF 4 , LiF-BeF 2 -ThF 4 , NaF-LiF-BeF 2 , NaF-KF-BeF 2 , NaF-ZrF 4 -UF 4 , and NaF-BeF 2 -UF 4 . This Redlich-Kister estimation approach accounts for nonideal mixing behavior based on pseudo-binary subsystem interaction parameters determined from experimentally measured pseudo-binary system density data sets. The Redlich-Kister estimation was compared with the method of additive molar volumes, which assumes ideal mixing. Additionally, the Redlich-Kister approach was used to determine previously unknown binary and ternary interaction parameters based on experimentally measured density data sets for select pseudo-ternary salt systems. The results of this study show improvement in density estimation using the Redlich-Kister approach for all systems considered compared with estimation by additive molar volumes. Furthermore, this analysis allowed for the estimation of nonideal density behavior in experimentally unstudied ZrF 4 -UF 4 and BeF 2 -UF 4 , as well as the quantification of ternary interaction in NaF-LiF-ZrF 4 , NaF-BeF 2 -UF 4 , and NaF-ZrF 4 -UF 4 .

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold

This paper introduces a surrogate modeling scheme based on Grassmannian manifold learning to be used for cost-efficient predictions of high-dimensional stochastic systems. The method exploits subspace-structured features of each solution by projecting it onto a Grassmann manifold. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. The method utilizes a solution clustering approach in order to identify regions of the parameter space over which solutions are sufficiently similarly such that they can be interpolated on the Grassmannian. In this clustering, the reduced-order solutions are partitioned into disjoint clusters on the Grassmann manifold using the eigen-structure of properly defined Grassmannian kernels and, the Karcher mean of each cluster is estimated. Then, the points in each cluster are projected onto the tangent space with origin at the corresponding Karcher mean using the exponential mapping. For each cluster, a Gaussian process regression model is trained that maps the input parameters of the system to the reduced solution points of the corresponding cluster projected onto the tangent space. Using this Gaussian process model, the full-field solution can be efficiently predicted at any new point in the parameter space. In certain cases, the solution clusters will span disjoint regions of the parameter space. In such cases, for each of the solution clusters we utilize a second, density-based spatial clustering to group their corresponding input parameter points in the Euclidean space. The proposed method is applied to two numerical examples. Here, the first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.

42 ENGINEERING↗
DOE OSTI2024

A quasi-meshfree method for constructing boundary-aware reproducing bases on geometrically complex domains using manifold geodesics

In many applications, physical domains are geometrically complex making it challenging to perform coarse-scale approximation. A defeaturing process is often used to simplify the domain in preparation for approximation and analysis at the coarse scale. In this paper, a methodology is presented for constructing a coarse-scale reproducing basis on geometrically complex domains given an initial fine-scale mesh of the fully featured domain. The initial fine-scale mesh can be of poor quality and extremely refined. The construction of the basis functions begins with a coarse-scale covering of the domain and generation of weighting functions with local support. Manifold geodesics are used to define distances within the local support for general applicability to non-convex domains. Conventional moving least squares is used to construct the coarse-scale reproducing basis. Applications in quasi-interpolation and linear elasticity are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Assessment and uncertainty analysis of groundwater risk

Highlights: • Two different sets of index evaluation systems for shallow and deep groundwater were built respectively. • Ordinary kriging was adopted to execute the interpolation estimation. • The synthetic weights were adopted merging the AHP method and PCA method. • The uncertainty of groundwater risk was analyzed by indicator kriging. Groundwater with relatively stable quantity and quality is commonly used by human being. However, as the over-mining of groundwater, problems such as groundwater funnel, land subsidence and salt water intrusion have emerged. In order to avoid further deterioration of hydrogeological problems in over-mining regions, it is necessary to conduct the assessment of groundwater risk. In this paper, risks of shallow and deep groundwater in the water intake area of the South-to-North Water Transfer Project in Tianjin, China, were evaluated. Firstly, two sets of four-level evaluation index system were constructed based on the different characteristics of shallow and deep groundwater. Secondly, based on the normalized factor values and the synthetic weights, the risk values of shallow and deep groundwater were calculated. Lastly, the uncertainty of groundwater risk assessment was analyzed by indicator kriging method. The results meet the decision maker's demand for risk information, and overcome previous risk assessment results expressed in the form of deterministic point estimations, which ignore the uncertainty of risk assessment.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Does the air pollution model influence the evidence of socio-economic disparities in exposure and susceptibility?

Highlights: • Daily exposure to ozone may vary across air pollution models, considerably. • The exposure model influenced disparity of exposure across socio-economic strata. • The air pollution model does not affect disparity of susceptibility. • Multiple air pollution models should be used to assess socio-economic disparities. Studies assessing socio-economic disparities in air pollution exposure and susceptibility are usually based on a single air pollution model. A time stratified case-crossover study was designed to assess the impact of the type of model on differential exposure and on the differential susceptibility in the relationship between ozone exposure and daily mortality by socio-economic strata (SES) in Montreal. Non-accidental deaths along with deaths from cardiovascular and respiratory causes on the island of Montreal for the period 1991–2002 were included as cases. Daily ozone concentration estimates at partictaipants’ residence were obtained from the five following air pollution models: Average value (AV), Nearest station model (NS), Inverse-distance weighting interpolation (IDW), Land-use regression model with back-extrapolation (LUR-BE) and Bayesian maximum entropy model combined with a land-use regression (BME-LUR). The prevalence of a low household income (< 20,000/year) was used as socio-economic variable, divided into two categories as a proxy for deprivation. Multivariable conditional logistic regressions were used considering 3-day average concentrations. Multiplicative and additive interactions (using Relative Excess Risk due to Interaction) as well as Cochran's tests were calculated and results were compared across the different air pollution models. Heterogeneity of susceptibility and exposure according to socio-economic status (SES) were found. Ratio of exposure across SES groups means ranged from 0.75 [0.74–0.76] to 1.01 [1.00–1.02], respectively for the LUR-BE and the BME-LUR models. Ratio of mortality odds ratios ranged from 1.01 [0.96–1.05] to 1.02 [0.97–1.08], respectively for the IDW and LUR-BE models. Cochran's test of heterogeneity between the air pollution models showed important heterogeneity regarding the differential exposure by SES, but the air pollution model was not found to influence heterogeneity regarding the differential susceptibility. The study showed air pollution models can influence the assessment of disparities in exposure according to SES in Montreal but not that of disparities in susceptibility.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Workflow-based automatic processing for Internet of Floating Things crowdsourced data

Data from sensors incorporated into mobile devices, such as networked navigational sensors, can be used to capture detailed environmental information. We describe here a workflow and framework for using sensors on boats to construct unique new datasets of underwater topography (bathymetry). Starting with a large number of measurements of position, depth, etc., obtained from such an Internet of Floating Things, we illustrate how, with a specialized protocol, data can be communicated to cloud resources, even when using delayed, intermittent, or disconnected networks. We then propose a method for automatic sensor calibration based on a novel reputation approach. Sampled depth data are interpolated efficiently on a cloud computing platform in order to provide a continuously updated bathymetric database. Our prototype implementation uses the FACE-IT Galaxy workflow engine to manage network communication and exploits the computational power of GPGPUs in a virtualized cloud environment, working with a CUDA-parallel algorithm, for efficient data processing. We report on an initial evaluation involving data from a sailing vessel in Italian coastal waters.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A multiscale MDCT image-based breathing lung model with time-varying regional ventilation

A novel algorithm is presented that links local structural variables (regional ventilation and deforming central airways) to global function (total lung volume) in the lung over three imaged lung volumes, to derive a breathing lung model for computational fluid dynamics simulation. The algorithm constitutes the core of an integrative, image-based computational framework for subject-specific simulation of the breathing lung. For the first time, the algorithm is applied to three multi-detector row computed tomography (MDCT) volumetric lung images of the same individual. A key technique in linking global and local variables over multiple images is an in-house mass-preserving image registration method. Throughout breathing cycles, cubic interpolation is employed to ensure C{sub 1} continuity in constructing time-varying regional ventilation at the whole lung level, flow rate fractions exiting the terminal airways, and airway deformation. The imaged exit airway flow rate fractions are derived from regional ventilation with the aid of a three-dimensional (3D) and one-dimensional (1D) coupled airway tree that connects the airways to the alveolar tissue. An in-house parallel large-eddy simulation (LES) technique is adopted to capture turbulent-transitional-laminar flows in both normal and deep breathing conditions. The results obtained by the proposed algorithm when using three lung volume images are compared with those using only one or two volume images. The three-volume-based lung model produces physiologically-consistent time-varying pressure and ventilation distribution. The one-volume-based lung model under-predicts pressure drop and yields un-physiological lobar ventilation. The two-volume-based model can account for airway deformation and non-uniform regional ventilation to some extent, but does not capture the non-linear features of the lung.

42 ENGINEERING↗
DOE OSTI2013

Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes

This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k{sub eff}, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k{sub eff} with directional dependence. General error estimators are derived for any given functional of the flux and applied to k{sub eff} to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k{sub eff} goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2013

Numerical stability of relativistic beam multidimensional PIC simulations employing the Esirkepov algorithm

Rapidly growing numerical instabilities routinely occur in multidimensional particle-in-cell computer simulations of plasma-based particle accelerators, astrophysical phenomena, and relativistic charged particle beams. Reducing instability growth to acceptable levels has necessitated higher resolution grids, high-order field solvers, current filtering, etc. except for certain ratios of the time step to the axial cell size, for which numerical growth rates and saturation levels are reduced substantially. This paper derives and solves the cold beam dispersion relation for numerical instabilities in multidimensional, relativistic, electromagnetic particle-in-cell programs employing either the standard or the Cole–Karkkainnen finite difference field solver on a staggered mesh and the common Esirkepov current-gathering algorithm. Good overall agreement is achieved with previously reported results of the WARP code. In particular, the existence of select time steps for which instabilities are minimized is explained. Additionally, an alternative field interpolation algorithm is proposed for which instabilities are almost completely eliminated for a particular time step in ultra-relativistic simulations.

43 PARTICLE ACCELERATORS↗
DOE OSTI2014

A consistent and conservative scheme for MHD flows with complex boundaries on an unstructured Cartesian adaptive system

The numerical simulation of Magnetohydrodynamics (MHD) flows with complex boundaries has been a topic of great interest in the development of a fusion reactor blanket for the difficulty to accurately simulate the Hartmann layers and side layers along arbitrary geometries. An adaptive version of a consistent and conservative scheme has been developed for simulating the MHD flows. Besides, the present study forms the first attempt to apply the cut-cell approach for irregular wall-bounded MHD flows, which is more flexible and conveniently implemented under adaptive mesh refinement (AMR) technique. It employs a Volume-of-Fluid (VOF) approach to represent the fluid–conducting wall interface that makes it possible to solve the fluid–solid coupling magnetic problems, emphasizing at how electric field solver is implemented when conductivity is discontinuous in cut-cell. For the irregular cut-cells, the conservative interpolation technique is applied to calculate the Lorentz force at cell-center. On the other hand, it will be shown how consistent and conservative scheme is implemented on fine/coarse mesh boundaries when using AMR technique. Then, the applied numerical schemes are validated by five test simulations and excellent agreement was obtained for all the cases considered, simultaneously showed good consistency and conservative properties.

42 ENGINEERING↗
DOE OSTI2014

A one-time truncate and encode multiresolution stochastic framework

In this work a novel adaptive strategy for stochastic problems, inspired from the classical Harten's framework, is presented. The proposed algorithm allows building, in a very general manner, stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems, from unsteady to discontinuous solutions. Its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach, but with an extension to uncertainty quantification problems. The present strategy permits to build numerical scheme with a higher accuracy with respect to other classical uncertainty quantification techniques, but with a strong reduction of the numerical cost and memory requirements. Moreover, the flexibility of the proposed approach allows to employ any kind of probability density function, even discontinuous and time varying, without introducing further complications in the algorithm. The advantages of the present strategy are demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account, such as discontinuous and unsteady custom-defined probability density functions. In addition to algebraic and ordinary differential equations, numerical results for the challenging 1D Kraichnan–Orszag are reported in terms of accuracy and convergence. Finally, a two degree-of-freedom aeroelastic model for a subsonic case is presented. Though quite simple, the model allows recovering some physical key aspect, on the fluid/structure interaction, thanks to the quasi-steady aerodynamic approximation employed. The injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix. All the numerical results are compared with respect to classical Monte Carlo solution and with a non-intrusive Polynomial Chaos method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Accurate gradient approximation for complex interface problems in 3D by an improved coupling interface method

Most elliptic interface solvers become complicated for complex interface problems at those “exceptional points” where there are not enough neighboring interior points for high order interpolation. Such complication increases especially in three dimensions. Usually, the solvers are thus reduced to low order accuracy. In this paper, we classify these exceptional points and propose two recipes to maintain order of accuracy there, aiming at improving the previous coupling interface method [26]. Yet the idea is also applicable to other interface solvers. The main idea is to have at least first order approximations for second order derivatives at those exceptional points. Recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point, whenever this is possible. Recipe 2 is to flip domain signatures and introduce a ghost state so that a second-order method can be applied. This ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface. The original state is recovered by a post-processing using nearby states and jump conditions. The choice of recipes is determined by a classification scheme of the exceptional points. The method renders the solution and its gradient uniformly second-order accurate in the entire computed domain. Numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions, and a real molecule ( (1D63)) which is double-helix shape and composed of hundreds of atoms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

In this paper we present a time-parallel algorithm for the 3D neutrons calculation of a transient model in a nuclear reactor core. The neutrons calculation consists in numerically solving the time dependent diffusion approximation equation, which is a simplified transport equation. The numerical resolution is done with finite elements method based on a tetrahedral meshing of the computational domain, representing the reactor core, and time discretization is achieved using a θ-scheme. The transient model presents moving control rods during the time of the reaction. Therefore, cross-sections (piecewise constants) are taken into account by interpolations with respect to the velocity of the control rods. The parallelism across the time is achieved by an adequate use of the parareal in time algorithm to the handled problem. This parallel method is a predictor corrector scheme that iteratively combines the use of two kinds of numerical propagators, one coarse and one fine. Our method is made efficient by means of a coarse solver defined with large time step and fixed position control rods model, while the fine propagator is assumed to be a high order numerical approximation of the full model. The parallel implementation of our method provides a good scalability of the algorithm. Numerical results show the efficiency of the parareal method on large light water reactor transient model corresponding to the Langenbuch–Maurer–Werner benchmark.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Local time–space mesh refinement for simulation of elastic wave propagation in multi-scale media

This paper presents an original approach to local time–space grid refinement for the numerical simulation of wave propagation in models with localized clusters of micro-heterogeneities. The main features of the algorithm are –the application of temporal and spatial refinement on two different surfaces; –the use of the embedded-stencil technique for the refinement of grid step with respect to time; –the use of the Fast Fourier Transform (FFT)-based interpolation to couple variables for spatial mesh refinement. The latter makes it possible to perform filtration of high spatial frequencies, which provides stability in the proposed finite-difference schemes. In the present work, the technique is implemented for the finite-difference simulation of seismic wave propagation and the interaction of such waves with fluid-filled fractures and cavities of carbonate reservoirs. However, this approach is easy to adapt and/or combine with other numerical techniques, such as finite elements, discontinuous Galerkin method, or finite volumes used for approximation of various types of linear and nonlinear hyperbolic equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗