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DOE OSTI2020

An extended/generalized phase-field finite element method for crack growth with global-local enrichment

In this paper, an extended/generalized finite element method (XFEM/GFEM) for simulating quasistatic crack growth based on a phase-field method is presented. The method relies on approximations to solutions associated with two different scales: a global scale, that is, structural and discretized with a coarse mesh, and a local scale encapsulating the fractured region, that is, discretized with a fine mesh. A stable XFEM/GFEM is employed to embed the displacement and damage fields at the global scale. The proposed method accommodates approximation spaces that evolve between load steps, while preserving a fixed background mesh for the structural problem. In addition, a prediction-correction algorithm is employed to facilitate the dynamic evolution of the confined crack regions within a load step. Several numerical examples of benchmark problems in two- and three-dimensional quasistatic fracture are provided to demonstrate the approach.

multiscale↗

DOE OSTI2014

Independent dose per monitor unit review of eight U.S.A. proton treatment facilities

Purpose: Compare the dose per monitor unit at different proton treatment facilities using three different dosimetry methods. Methods: Measurements of dose per monitor unit were performed by a single group at eight facilities using 11 test beams and up to six different clinical portal treatment sites. These measurements were compared to the facility reported dose per monitor unit values. Results: Agreement between the measured and reported doses was similar using any of the three dosimetry methods. Use of the ICRU 59 N{sub D,w} based method gave results approximately 3% higher than both the ICRU 59 N{sub X} and ICRU 78 (TRS-398) N{sub D,w} based methods. Conclusions: Any single dosimetry method could be used for multi-institution trials with similar conformity between facilities. A multi-institutional trial could support facilities using both the ICRU 59 N{sub X} based and ICRU 78 (TRS-398) N{sub D,w} based methods but use of the ICRU 59 N{sub D,w} based method should not be allowed simultaneously with the other two until the difference is resolved.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2018

Finite element analysis of compliant cladding and base metal systems

This report describes a roadmap towards a design method for high temperature nuclear cladded components that does not require long term testing of clad materials. Molten Salt Reactors (MSRs) will operate using corrosive fuel or coolant salts. The existing Class A metallic materials qualified for ASME Boiler and Pressure Vessel Code Section III, Division 5 design are not optimal for corrosion resistance when exposed to MSR salts. Corrosion could then severely limit the service life of MSRs. One option to alleviate this could be to Code qualify new corrosion resistant materials. However, this is a lengthy and expensive process and MSR developers have a near term need for a design solution that enables reactor designs with long service lives. An alternative would be to use cladded components – overlay some existing Class A material with a thin layer of some corrosion resistant material. This report describes the outlines of a design strategy that would enable this approach. The strategy relies on approximate design analysis methods for certain types of clad/base systems that avoids requiring long-term mechanical properties for the clad materials. This would allow the use of cladded components in the near term, without requiring long testing programs. However, it will limit the types of clad materials that can be used. This report describes the design framework, demonstrates the effectiveness of the approximate analysis methods, and confirms the approach using full scale finite element analysis of prototypical cladded high temperature components. It also describes a roadmap towards a complete design method and surveys available clad materials and bonding methods for suitably, considering only mechanical and not environmental effects.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2022

Exponential time differencing scheme for mass transport and depletion in molten salt reactors

This work extends the capability previously shown for addressing the problem of computing depletion and mass transport calculations in molten salt reactors (MSRs) by calculating matrix exponentials. Additional algorithms are implemented to compute the matrix exponential and the action of the matrix exponential on a matrix. These algorithms include two methods based on the Pade approximation, a Taylor series method, and three methods based on Cauchy's integral formula. In addition to the added matrix exponential solvers, a variable-order total variation diminishing scheme is applied to the convective flux approximation to provide enhanced accuracy. Finally, a simplified MSR problem is shown for each of the exponential time differencing solvers along with classical backwards differencing integrators. The results show excellent convergence for exponential time differencing methods. Computation time is a key element for selecting the optimal solver in these problems, and this work shows that Pade and Cauchy-based solvers may provided the fastest and most accurate solutions. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2017

Multi-fidelity stochastic collocation method for computation of statistical moments

We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in . By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Combining Sparse Approximate Factorizations with Mixed-precision Iterative Refinement

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Communication: Improved pair approximations in local coupled-cluster methods

In local coupled cluster treatments the electron pairs can be classified according to the magnitude of their energy contributions or distances into strong, close, weak, and distant pairs. Different approximations are introduced for the latter three classes. In this communication, an improved simplified treatment of close and weak pairs is proposed, which is based on long-range cancellations of individually slowly decaying contributions in the amplitude equations. Benchmark calculations for correlation, reaction, and activation energies demonstrate that these approximations work extremely well, while pair approximations based on local second-order Møller-Plesset theory can lead to errors that are 1-2 orders of magnitude larger.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Modeling material interfaces with hybrid adhesion method

A molecular dynamics simulation approach is presented to approximate layered material structures using discrete interatomic potentials through classical mechanics and the underlying principles of quantum mechanics. This method isolates the energetic contributions of the system into two pure material layers and an interfacial region used to simulate the adhesive properties of the diffused interface. The strength relationship of the adhesion contribution is calculated through small-scale separation calculations and applied to the molecular surfaces through an inter-layer bond criterion. By segregating the contributions into three regions and accounting for the interfacial excess energies through the adhesive surface bonds, it is possible to model each material with an independent potential while maintaining an acceptable level of accuracy in the calculation of mechanical properties. This method is intended for the atomistic study of the delamination mechanics, typically observed in thin-film applications. Therefore, the work presented in this paper focuses on mechanical tensile behaviors, with observations in the elastic modulus and the delamination failure mode. To introduce the hybrid adhesion method, we apply the approach to an ideal bulk copper sample, where an interface is created by disassociating the force potential in the middle of the structure. Various mechanical behaviors are compared to a standard EAM control model to demonstrate the adequacy of this approach in a simple setting. In addition, we demonstrate the robustness of this approach by applying it on (1) a Cu-Cu 2 O interface with interactions between two atom types, and (2) an Al-Cu interface with two dissimilar FCC lattices. These additional examples are verified against EAM and COMB control models to demonstrate the accurate simulation of failure through delamination, and the formation and propagation of dislocations under loads. Finally, the results conclude that by modeling the energy contributions of an interface using hybrid adhesion bonds, we can provide an accurate approximation method for studies of large-scale mechanical properties, as well as the representation of various delamination phenomena at the atomic scale.

36 MATERIALS SCIENCE↗

DOE OSTI2014

Analyzing the errors of DFT approximations for compressed water systems

We report an extensive study of the errors of density functional theory (DFT) approximations for compressed water systems. The approximations studied are based on the widely used PBE and BLYP exchange-correlation functionals, and we characterize their errors before and after correction for 1- and 2-body errors, the corrections being performed using the methods of Gaussian approximation potentials. The errors of the uncorrected and corrected approximations are investigated for two related types of water system: first, the compressed liquid at temperature 420 K and density 1.245 g/cm{sup 3} where the experimental pressure is 15 kilobars; second, thermal samples of compressed water clusters from the trimer to the 27-mer. For the liquid, we report four first-principles molecular dynamics simulations, two generated with the uncorrected PBE and BLYP approximations and a further two with their 1- and 2-body corrected counterparts. The errors of the simulations are characterized by comparing with experimental data for the pressure, with neutron-diffraction data for the three radial distribution functions, and with quantum Monte Carlo (QMC) benchmarks for the energies of sets of configurations of the liquid in periodic boundary conditions. The DFT errors of the configuration samples of compressed water clusters are computed using QMC benchmarks. We find that the 2-body and beyond-2-body errors in the liquid are closely related to similar errors exhibited by the clusters. For both the liquid and the clusters, beyond-2-body errors of DFT make a substantial contribution to the overall errors, so that correction for 1- and 2-body errors does not suffice to give a satisfactory description. For BLYP, a recent representation of 3-body energies due to Medders, Babin, and Paesani [J. Chem. Theory Comput. 9, 1103 (2013)] gives a reasonably good way of correcting for beyond-2-body errors, after which the remaining errors are typically 0.5 mE{sub h} ≃ 15 meV/monomer for the liquid and the clusters.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2014

Comparing deflection measurements of a magnetically steerable catheter using optical imaging and MRI

Purpose: Magnetic resonance imaging (MRI) is an emerging modality for interventional radiology, giving clinicians another tool for minimally invasive image-guided interventional procedures. Difficulties associated with endovascular catheter navigation using MRI guidance led to the development of a magnetically steerable catheter. The focus of this study was to mechanically characterize deflections of two different prototypes of the magnetically steerable catheterin vitro to better understand their efficacy. Methods: A mathematical model for deflection of the magnetically steerable catheter is formulated based on the principle that at equilibrium the mechanical and magnetic torques are equal to each other. Furthermore, two different image based methods for empirically measuring the catheter deflection angle are presented. The first, referred to as the absolute tip method, measures the angle of the line that is tangential to the catheter tip. The second, referred to the base to tip method, is an approximation that is used when it is not possible to measure the angle of the tangent line. Optical images of the catheter deflection are analyzed using the absolute tip method to quantitatively validate the predicted deflections from the mathematical model. Optical images of the catheter deflection are also analyzed using the base to tip method to quantitatively determine the differences between the absolute tip and base to tip methods. Finally, the optical images are compared to MR images using the base to tip method to determine the accuracy of measuring the catheter deflection using MR. Results: The optical catheter deflection angles measured for both catheter prototypes using the absolute tip method fit very well to the mathematical model (R{sup 2} = 0.91 and 0.86 for each prototype, respectively). It was found that the angles measured using the base to tip method were consistently smaller than those measured using the absolute tip method. The deflection angles measured using optical data did not demonstrate a significant difference from the angles measured using MR image data when compared using the base to tip method. Conclusions: This study validates the theoretical description of the magnetically steerable catheter, while also giving insight into different methods and modalities for measuring the deflection angles of the prototype catheters. These results can be used to mechanically model future iterations of the design. Quantifying the difference between the different methods for measuring catheter deflection will be important when making deflection measurements in future studies. Finally, MR images can be used to reliably measure deflection angles since there is no significant difference between the MR and optical measurements.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2018

Numerical similarity solution for a variable coefficient K(m, n) equation

A technique for finding numerical similarity solutions to an initial boundary value problem (IBVP) for generalized K(m, n) equations is described. The equation under consideration is nonlinear and has variable coefficients. The original problem is transformed with the aid of Lie symmetries to an initial value problem (IVP) for a nonlinear third-order ordinary differential equation. The existence and uniqueness of the solution are examined, and the problem is consequently solved with the aid of a finite-difference scheme for various values of the governing parameters. In lieu of an exact symbolic solution, the scheme is validated by comparing the numerical solutions with the approximate analytic solutions obtained with the aid of the method of successive approximations in their region of validity. The accuracy, efficiency, and consistency of the scheme are demonstrated. Numerical solutions to the original initial boundary value problem are constructed for selected parameter values with the aid of the transforms. The qualitative behavior of the solutions as a function of the governing parameters is analyzed, and it is found that the examined IBVPs for generalized K(m, n) equations with variable coefficients that are functions of time, do not admit solitary wave or compacton solutions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Template-based automatic breast segmentation on MRI by excluding the chest region

Purpose: Methods for quantification of breast density on MRI using semiautomatic approaches are commonly used. In this study, the authors report on a fully automatic chest template-based method. Methods: Nonfat-suppressed breast MR images from 31 healthy women were analyzed. Among them, one case was randomly selected and used as the template, and the remaining 30 cases were used for testing. Unlike most model-based breast segmentation methods that use the breast region as the template, the chest body region on a middle slice was used as the template. Within the chest template, three body landmarks (thoracic spine and bilateral boundary of the pectoral muscle) were identified for performing the initial V-shape cut to determine the posterior lateral boundary of the breast. The chest template was mapped to each subject's image space to obtain a subject-specific chest model for exclusion. On the remaining image, the chest wall muscle was identified and excluded to obtain clean breast segmentation. The chest and muscle boundaries determined on the middle slice were used as the reference for the segmentation of adjacent slices, and the process continued superiorly and inferiorly until all 3D slices were segmented. The segmentation results were evaluated by an experienced radiologist to mark voxels that were wrongly included or excluded for error analysis. Results: The breast volumes measured by the proposed algorithm were very close to the radiologist's corrected volumes, showing a % difference ranging from 0.01% to 3.04% in 30 tested subjects with a mean of 0.86% ± 0.72%. The total error was calculated by adding the inclusion and the exclusion errors (so they did not cancel each other out), which ranged from 0.05% to 6.75% with a mean of 3.05% ± 1.93%. The fibroglandular tissue segmented within the breast region determined by the algorithm and the radiologist were also very close, showing a % difference ranging from 0.02% to 2.52% with a mean of 1.03% ± 1.03%. The total error by adding the inclusion and exclusion errors ranged from 0.16% to 11.8%, with a mean of 2.89% ± 2.55%. Conclusions: The automatic chest template-based breast MRI segmentation method worked well for cases with different body and breast shapes and different density patterns. Compared to the radiologist-established truth, the mean difference in segmented breast volume was approximately 1%, and the total error by considering the additive inclusion and exclusion errors was approximately 3%. This method may provide a reliable tool for MRI-based segmentation of breast density.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2019

The conforming virtual element method for polyharmonic problems

Here, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form $(–Δ)^pu = f , p ≥ 1$. More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in suitable norms.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Nonadiabatic Quantum Molecular Dynamics in Dense Manifolds of Electronic States

Most nonadiabatic molecular dynamics methods require the determination of a basis of adiabatic or diabatic electronic states at every time step, but in dense manifolds of electronic states, such approaches become intractable. A notable exception is Ehrenfest molecular dynamics, which can be implemented without explicit determination of such a basis but suffers from unphysical behavior when propagation on a mean-field potential energy surface (PES) does not accurately reflect the true dynamics on multiple electronic states. Here we introduce the multiple cloning for dense manifolds of states (MCDMS) method, a systematically improvable approximation to the multiple cloning method. MCDMS avoids both the mean-field PES problem and the need to compute the full electronic spectrum. This is achieved by reformulating multiple cloning to use a subspace of approximate eigenstates constructed from the time-dependent Ehrenfest electronic wave function. Finally, by application to model systems, we show that this approach allows a substantial reduction in the size of the required electronic basis without significant loss in accuracy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2019

Fractional dynamics of an erbium-doped fiber laser model

In this paper we investigate the model of the time-fractional dynamics of an erbium-doped fiber laser model (TFDEFL) with Liouville–Caputo (LC), Caputo–Fabrizio–Caputo (CFC) and Atangana–Baleanu–Caputo (ABC) time-fractional derivatives. We employ the homotopy analysis transform method (HATM) to calculate approximate solutions for the TFDEFL model. This method gives the solution in the form of a rapidly convergent series that can ensure the convergence in solving the resultant series. We study the convergence analysis of HATM by computing the interval of convergence through the h-curves, the residual error function and the average residual error, respectively. We also show the effectiveness and accuracy of this method by comparing the approximate solutions based upon the LC, CFC and ABC time-fractional derivatives.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Perspective on Kramers symmetry breaking and restoration in relativistic electronic structure methods for open-shell systems

Without rigorous symmetry constraints, solutions to approximate electronic structure methods may artificially break symmetry. In the case of the relativistic electronic structure, if time-reversal symmetry is not enforced in calculations of molecules not subject to a magnetic field, it is possible to artificially break Kramers degeneracy in open shell systems. This leads to a description of excited states that may be qualitatively incorrect. Despite this, different electronic structure methods to incorporate correlation and excited states can partially restore Kramers degeneracy from a broken symmetry solution. For single-reference techniques, the inclusion of double and possibly triple excitations in the ground state provides much of the needed correction. Formally, however, this imbalanced treatment of the Kramers-paired spaces is a multi-reference problem, and so methods such as complete-active-space methods perform much better at recovering much of the correct symmetry by state averaging. Using multi-reference configuration interaction, any additional corrections can be obtained as the solution approaches the full configuration interaction limit. A recently proposed “Kramers contamination” value is also used to assess the magnitude of symmetry breaking.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Acceleration of diffusive molecular dynamics simulations through mean field approximation and subcycling time integration

Highlights: • Two numerical methods are presented to accelerate DMD simulations. • The methods greatly reduce the computation time without much loss of accuracy. • The accelerated DMD model is capable of simulating long-term mass transport at atomistic length scales. • Demonstrated for the propagation of an atomically sharp phase boundary over 30 seconds. - Abstract: Diffusive Molecular Dynamics (DMD) is a class of recently developed computational models for the simulation of long-term diffusive mass transport at atomistic length scales. Compared to previous atomistic models, e.g., transition state theory based accelerated molecular dynamics, DMD allows the use of larger time-step sizes, but has a higher computational complexity at each time-step due to the need to solve a nonlinear optimization problem at every time-step. This paper presents two numerical methods to accelerate DMD based simulations. First, we show that when a many-body potential function, e.g., embedded atom method (EAM), is employed, the cost of DMD is dominated by the computation of the mean of the potential function and its derivatives, which are high-dimensional random variables. To reduce the cost, we explore both first- and second-order mean field approximations. Specifically, we show that the first-order approximation, which uses a point estimate to calculate the mean, can reduce the cost by two to three orders of magnitude, but may introduce relatively large error in the solution. We show that adding an approximate second-order correction term can significantly reduce the error without much increase in computational cost. Second, we show that DMD can be significantly accelerated through subcycling time integration, as the cost of integrating the empirical diffusion equation is much lower than that of the optimization solver. To assess the DMD model and the numerical approximation methods, we present two groups of numerical experiments that simulate the diffusion of hydrogen in palladium nanoparticles. In particular, we show that the computational framework is capable of capturing the propagation of an atomically sharp phase boundary over a time window of more than 30 seconds. The effects of the proposed numerical methods on solution accuracy and computation time are also assessed quantitatively.

36 MATERIALS SCIENCE↗

DOE OSTI2018

Communication-efficient Distributed Solutions to a System of Linear Equations with Laplacian Sparse Structure

Two communication-efficient distributed algorithms are proposed to solve a system of linear equations Ax = b with Laplacian sparse A. A system of linear equations with Laplacian sparse A can be found in many applications, e.g., the power flow problems and other network flow problems. The first algorithm is based on the gradient descent method in optimization and the agents only share two parts of the system state instead of that of the whole system state, which saves significant communication. The two parts shared by every agent through a communication link are the state information of its own and its neighbors connected by the communication link. The second method is obtained from an approximation of the Newton method, which converges faster. It requires twice as much communication as the first one but is still communicationefficient due to the low dimension of each part shared between agents. The convergence at a linear rate of both methods is proved. A comprehensive comparison of the convergence rate, communication burden, and computation costs between the methods is made. Finally, a simulation is conducted to show the effectiveness of both methods.

Wang, Peng↗