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NASA NTRS1992

Probabilistic finite elements for fatigue and fracture analysis

Attenuation is focused on the development of Probabilistic Finite Element Method (PFEM), which combines the finite element method with statistics and reliability methods, and its application to linear, nonlinear structural mechanics problems and fracture mechanics problems. The computational tool based on the Stochastic Boundary Element Method is also given for the reliability analysis of a curvilinear fatigue crack growth. The existing PFEM's have been applied to solve for two types of problems: (1) determination of the response uncertainty in terms of the means, variance and correlation coefficients; and (2) determination the probability of failure associated with prescribed limit states.

Belytschko, Ted↗

NASA NTRS2013

A Floating Node Method for the Modelling of Discontinuities Within a Finite Element

This paper focuses on the accurate numerical representation of complex networks of evolving discontinuities in solids, with particular emphasis on cracks. The limitation of the standard finite element method (FEM) in approximating discontinuous solutions has motivated the development of re-meshing, smeared crack models, the eXtended Finite Element Method (XFEM) and the Phantom Node Method (PNM). We propose a new method which has some similarities to the PNM, but crucially: (i) does not introduce an error on the crack geometry when mapping to natural coordinates; (ii) does not require numerical integration over only part of a domain; (iii) can incorporate weak discontinuities and cohesive cracks more readily; (iv) is ideally suited for the representation of multiple and complex networks of (weak, strong and cohesive) discontinuities; (v) leads to the same solution as a finite element mesh where the discontinuity is represented explicitly; and (vi) is conceptually simpler than the PNM.

Pinho, Silvestre T.↗

DOE OSTI2017

The Parareal Method for Solving Space-Dependent Neutron Kinetics Equations in Hexagonal-z Geometry

Neutron kinetics plays an important role in reactor safety and analysis. The backward Euler method is the most widely used time integration method in the calculation of space-dependent nuclear reactor kinetics. Xia (2006) used this method to discretize the derivative of flux, and adopted the analytic integration to discretize the delayed neutron precursor concentrations. Diagonally Implicit Runge-Kutta (DIRK) method owns high accuracy and excellent stability, and it was applied in the rectangle core for calculating core the nuclear reactor kinetics by Wenbo Zhao (2013). Recently, DIRK was applied to the neutron kinetics for hexagonal-z geometry application (CAI, 2016). As solving the neutron kinetics equations is very time-consuming and the number of available cores continues to increase with parallel architectures evolving, parallel algorithms need to be designed in order to effectively utilize the available resources. However, it is difficult to parallel in time axis because the later moment is strongly dependent on the previous moment. Some methods have been proposed to parallel in time algorithm over the last decades. One of the famous methods to realize time parallelism is the Parareal algorithm (Lions, 2001), which offers a promising approach to increase the degree of concurrency. More than one decade from this method first proposed, this method has been studied and applied in the many fields of applied mathematics, physics, chemistry, parallel computing, etc. Fischer (2014 applied the Parareal method in the field of the Navier-Stokes Equations. Baudron (2014) combined the Parareal method with finite element method to solve the neutron kinetics equations, where the Parareal method was implemented based on the implicit Euler method or method. In this paper, the Parareal method is studied in the hexagonal-z geometry with the help of DIRK developed for neutron kinetics equations (CAI, 2016)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2015

Discrete Modeling of Early-Life Thermal Fracture in Ceramic Nuclear Fuel

Fracturing of ceramic fuel pellets heavily influences performance of light water reactor (LWR) fuel. Early in the life of fuel, starting with the initial power ramp, large thermal gradients cause high tensile hoop and axial stresses in the outer region of the fuel pellets, resulting in the formation of radial and axial cracks. Circumferential cracks form due to thermal gradients that occur when the power is ramped down. These thermal cracks cause the fuel to expand radially, closing the pellet/cladding gap and enhancing the thermal conductance across that gap, while decreasing the effective conductivity of the fuel in directions normal to the cracking. At lower length scales, formation of microcracks is an important contributor to the decrease in bulk thermal conductivity that occurs over the life of the fuel as the burnup increases. Because of the important effects that fracture has on fuel performance, a realistic, physically based fracture modeling capability is essential to predict fuel behavior in a wide variety of normal and abnormal conditions. Modeling fracture within the context of the finite element method, which is based on continuous interpolations of solution variables, has always been challenging because fracture is an inherently discontinuous phenomenon. Work is underway at Idaho National Laboratory to apply two modeling techniques model fracture as a discrete displacement discontinuity to nuclear fuel: The extended finite element method (XFEM), and discrete element method (DEM). XFEM is based on the standard finite element method, but with enhancements to represent discontinuous behavior. DEM represents a solid as a network of particles connected by bonds, which can arbitrarily fail if a fracture criterion is reached. This paper presents initial results applying the aforementioned techniques to model fuel fracturing. This work has initially focused on early life behavior of ceramic LWR fuel. A coupled thermal-mechanical XFEM method that includes discontinuities in both temperature and displacement fields at crack locations has been developed and is being applied to thermal fracture of LWR fuel. A DEM model of coupled heat conduction and solid mechanics has been developed and used to simulate random initiation and propagation of thermally driven cracks during initial power cycles. This DEM model predicts the formation of realistic radial cracking patterns during power rise and circumferential cracks as power is ramped down. These initial results are very encouraging, and these techniques are expected to provide improved understanding of fuel behavior in a wide variety of conditions.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2015

Cutting Tetrahedra by Node Identifiers

This report briefly outlines an algorithm for dividing a tetrahedron intersected by a planar interface into conforming sub-tetrahedra. The problem of conformal decomposition of tetrahedral meshes arises in enriched finite element methods; in particular, we are concerned with the Conformal Decomposition Finite Element Method (CDFEM) and variants of the eXtended Finite Element Method (XFEM). The algorithm presented is based on the paper How to Subdivide Pyramids, Prisms and Hexahedra into Tetrahedra by Dompierre, Labbe, Vallet, and Camarero (1999), and here is applied and extended to the problem of fully defining and tracking all geometric features of the sub-tetrahedra generated when a tetrahedron is cut by a planar surface.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

High Fidelity CFD Simulations Supporting the KP-FHR

Kairos Power, LLC, is developing its version of the Fluoride-cooled High-temperature Reactor, the KP-FHR. The design uses a pebble bed core with fluoride salt as a coolant. The pebbles used in the KP-FHR have a diameter of 4 cm, with a shell fuel region where TRISO particles are embedded. A Pebble bed core design is adopted by several Gen IV reactors, They boast many benefits, such as fuel integrity, highly efficient heat transfer, and passive safety. However, it is challenging to accurately predict temperature and flow inside a pebble bed. Traditional approaches use the porous media model, which regards the pebble bed as a continuous medium, but with different temperature fields representing different levels, such as the fluid temperature, pebble surface temperature, and pebble center temperature. Empirical heat transfer correlations are adopted to calculate the heat transfer coefficient between different phases. However, empirical correlations are usually validated with experimental data, which usually lacks detail inside the pebble bed. The available experimental data is also generally at a high Reynolds number, which falls outside of the conditions of KP-FHR. Explicit computational fluid dynamics (CFD) simulations of randomly packed pebble beds have only become feasible recently. This is thanks to the rapid development of computational power and scalable algorithms. In this work, we used the Spectral Element Method (SEM) CFD code NekRS to simulate the randomly packed pebble bed in a cylindrical container. NekRS, which is the GPU variant of Nek5000, but refactored to utilize the computational power of GPUs using the OCCA library to run on hybrid architecture high performance computing systems. It was initially developed with the libParamunal library, but truncated and tuned for large-scale turbulence simulation. As a result, the SEM reaches higher precision with the same degrees of freedom by using a high-order Lagrange polynomial basis distributed on Gauss-Lobatto-Legendre quadrature inside each element, compared to lower-order methods, such the Finite Volume Method and Finite Element Method. The report is divided into five parts. We start with a general discussion of the pebble bed reactor, along with a specific investigation into the KP-FHR. The second part presents the numerical methodology. In the third part, we study a modular pebble bed with 1741 pebbles in a container of 7 pebble-diameter radius. Beyond LES simulations done by NekRS, we also leveraged the thermal radiation model in OpenFOAM to study heat transfer under no-forced-flow scenarios. Then, in the fourth part we simulated a pebble bed similar to the size of the Hermes Test Reactor. The total number of pebbles is in these simulations is 34,374. The container radius is 14 pebble-diameters. Finally, the report concludes in part five, with a discussion of future work.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

NASA NTRS1993

A variational method for finite element stress recovery and error estimation

A variational method for obtaining smoothed stresses from a finite element derived nonsmooth stress field is presented. The method is based on minimizing a functional involving discrete least-squares error plus a penalty constraint that ensures smoothness of the stress field. An equivalent accuracy criterion is developed for the smoothing analysis which results in a C sup 1-continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis. Application of the smoothing analysis to residual error estimation is also demonstrated.

Tessler, A.↗

NASA NTRS1990

Application of a finite element algorithm for high speed viscous flows using structured and unstructured meshes

A higher-order streamline upwinding Petrov-Galerkin finite element method is employed for high speed viscous flow analysis using structured and unstructured meshes. For a Mach 8.03 shock interference problem, successive mesh adaptation was performed using an adaptive remeshing method. Results from the finite element algorithm compare well with both experimental data and results from an upwind cell-centered method. Finite element results for a Mach 14.1 flow over a 24 degree compression corner compare well with experimental data and two other numerical algorithms for both structured and unstructured meshes.

Vemaganti, Gururaja R.↗

NASA NTRS1997

Dynamic Electrothermal Model of a Sputtered Thermopile Thermal Radiation Detector for Earth Radiation Budget Applications

The Clouds and the Earth's Radiant Energy System (CERES) is a program sponsored by the National Aeronautics and Space Administration (NASA) aimed at evaluating the global energy balance. Current scanning radiometers used for CERES consist of thin-film thermistor bolometers viewing the Earth through a Cassegrain telescope. The Thermal Radiation Group, a laboratory in the Department of Mechanical Engineering at Virginia Polytechnic Institute and State University, is currently studying a new sensor concept to replace the current bolometer: a thermopile thermal radiation detector. This next-generation detector would consist of a thermal sensor array made of thermocouple junction pairs, or thermopiles. The objective of the current research is to perform a thermal analysis of the thermopile. Numerical thermal models are particularly suited to solve problems for which temperature is the dominant mechanism of the operation of the device (through the thermoelectric effect), as well as for complex geometries composed of numerous different materials. Feasibility and design specifications are studied by developing a dynamic electrothermal model of the thermopile using the finite element method. A commercial finite element-modeling package, ALGOR, is used.

Weckmann, Stephanie↗

DOE OSTI2018

Design and convergence analysis of the conforming virtual element method for polyharmonic problems

In the recent years, there has been a tremendous interest to numerical methods that approximate partial differential equations (PDEs) on computational meshes with arbitrarily-shaped polygonal/polyhedral (polytopal, for short) elements. A nonexhaustive list of such methods include the Mimetic Finite Difference method [4, 9, 21, 24, 28, 33, 35, 62, 64–67], the Polygonal Finite Element Method [68, 73, 74], the polygonal Discontinuous Galerkin Finite Element Methods [5, 7, 8, 10, 13–15, 18, 36, 38, 39, 41, 50], the Hybridizable Discontinuous Galerkin and Hybrid High-Order Methods [46–48, 51], the Gradient Discretization method [49, 53, 54], the Finite Volume Method [52], and the BEM-based FEM [71]. An alternative approach that also proved to be very successful is provided by the Virtual Element method (VEM), which was originally proposed in [19] for the numerical treatment of second-order elliptic problems [43, 44], and readily extended to linear and nonlinear elasticity [22, 25, 56], plate bending problems [37], Cahn-Hilliard equation [3], Stokes equations [2], Laplace-Beltrami equation [55], Darcy-Brinkam equation [76], discrete topology optimization problems [6], fracture networks problems [29], eigenvalue problems [59, 75]. The mixed virtual element formulation was proposed in [20, 34]; the nonconforming Virtual element formulations was proposed for second-order elliptic problems in [16], and later extended to general advection-reaction-diffusion problems, Stokes equation, the biharmonic problems, the iegenvalue problem, and the Schrodinger equation in [11, 30, 42, 45, 58, 80]. The p- and hp-version of the VEM were developed in [26, 27] and efficient multigrid methods for the resulting linear system of equations in [12]. A posteriori error estimates can be found in [40].

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2004

Higher-Order Finite Elements for Computing Thermal Radiation

Two variants of the finite-element method have been developed for use in computational simulations of radiative transfers of heat among diffuse gray surfaces. Both variants involve the use of higher-order finite elements, across which temperatures and radiative quantities are assumed to vary according to certain approximations. In this and other applications, higher-order finite elements are used to increase (relative to classical finite elements, which are assumed to be isothermal) the accuracies of final numerical results without having to refine computational meshes excessively and thereby incur excessive computation times. One of the variants is termed the radiation sub-element (RSE) method, which, itself, is subject to a number of variations. This is the simplest and most straightforward approach to representation of spatially variable surface radiation. Any computer code that, heretofore, could model surface-to-surface radiation can incorporate the RSE method without major modifications. In the basic form of the RSE method, each finite element selected for use in computing radiative heat transfer is considered to be a parent element and is divided into sub-elements for the purpose of solving the surface-to-surface radiation-exchange problem. The sub-elements are then treated as classical finite elements; that is, they are assumed to be isothermal, and their view factors and absorbed heat fluxes are calculated accordingly. The heat fluxes absorbed by the sub-elements are then transferred back to the parent element to obtain a radiative heat flux that varies spatially across the parent element. Variants of the RSE method involve the use of polynomials to interpolate and/or extrapolate to approximate spatial variations of physical quantities. The other variant of the finite-element method is termed the integration method (IM). Unlike in the RSE methods, the parent finite elements are not subdivided into smaller elements, and neither isothermality nor other unrealistic physical conditions are assumed. Instead, the equations of radiative heat transfer are integrated numerically over the parent finite elements by use of a computationally efficient Gaussian integration scheme.

Gould, Dana C.↗

NASA NTRS1994

Comparison of Gap Elements and Contact Algorithm for 3D Contact Analysis of Spiral Bevel Gears

Three dimensional stress analysis of spiral bevel gears in mesh using the finite element method is presented. A finite element model is generated by solving equations that identify tooth surface coordinates. Contact is simulated by the automatic generation of nonpenetration constraints. This method is compared to a finite element contact analysis conducted with gap elements.

Bibel, G. D.↗

DOE OSTI2014

Electromagnetic Extended Finite Elements for High-Fidelity Multimaterial Problems LDRD Final Report

Surface effects are critical to the accurate simulation of electromagnetics (EM) as current tends to concentrate near material surfaces. Sandia EM applications, which include exploding bridge wires for detonator design, electromagnetic launch of flyer plates for material testing and gun design, lightning blast-through for weapon safety, electromagnetic armor, and magnetic flux compression generators, all require accurate resolution of surface effects. These applications operate in a large deformation regime, where body-fitted meshes are impractical and multimaterial elements are the only feasible option. State-of-the-art methods use various mixture models to approximate the multi-physics of these elements. The empirical nature of these models can significantly compromise the accuracy of the simulation in this very important surface region. We propose to substantially improve the predictive capability of electromagnetic simulations by removing the need for empirical mixture models at material surfaces. We do this by developing an eXtended Finite Element Method (XFEM) and an associated Conformal Decomposition Finite Element Method (CDFEM) which satisfy the physically required compatibility conditions at material interfaces. We demonstrate the effectiveness of these methods for diffusion and diffusion-like problems on node, edge and face elements in 2D and 3D. We also present preliminary work on h -hierarchical elements and remap algorithms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1989

A finite element based method for solution of optimal control problems

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of elemental values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Also noteworthy among characteristics of the finite element formulation is the fact that in the algebraic equations which contain costates, they appear linearly. Thus, the remaining equations can be solved iteratively without initial guesses for the costates; this reduces the size of the problem by about a factor of two. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal is to evaluate the feasibility of this approach for real-time guidance applications. To this end, a simplified two-stage, four-state model for an advanced launch vehicle application is presented which is suitable for finite element solution.

Bless, Robert R.↗

DOE OSTI2018

Hermite analogs of the lowest order Raviart–Thomas mixed method for convection–diffusion equations

The Raviart–Thomas mixed finite-element method of the lowest order (Raviart and Thomas in mixed finite-element methods for second-order elliptic problems, Lecture Notes in mathematics, Springer, New York, 1977) commonly known as the RT{sub 0} method, is a well-established and popular numerical tool to solve diffusion-like problems providing flux continuity across inter-element boundaries. Douglas and Roberts extended the method to the case of more general second-order boundary-value problems including the convection–diffusion equations (cf. this journal Douglas in Comput Appl Math 1:91–103; 1982). The main drawback of these methods, however, is the poor representation of the primal variable by piecewise constant functions. The Hermite analog of the RT{sub 0} method for treating pure diffusion phenomena proposed in Ruas (J Comput Appl Math, 246:234–242; 2013) proved to be a valid alternative to attain higher order approximation of the primal variable while keeping intact the matrix structure and the quality of the discrete flux variable of the original RT{sub 0} method. Non-trivial extensions of this method are studied here that can be viewed as Hermite analogs of the two Douglas and Roberts’ versions of the RT{sub 0} method, to solve convection–diffusion equations. A detailed convergence study is carried out for one of the Hermite methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1984

On the solution of problems involving impact type loading

The response of beam and plate members to pulse and impact loading is investigated, and numerical methods are analyzed and compared. The equations of motion, the finite difference and finite element methods, the time integration/Runge-Kutta techniques, and material modeling are discussed in detail. It is found that both the finite difference and the finite element methods could be accurately employed to discretize the spatial variation in the displacements. Central differences or a fourth order Runge-Kutta algorithm could be used for the time integration. The total energy of the system would give the stability and accuracy of the solution. Results showed that the finite element method provided better efficiency in obtaining accurate solution than the finite difference method when a scalar processor is used. However, the finite difference method was more efficient on the vector processor. Therefore, the most efficient method of solution depends on the type of computer present for the analysis.

Moyer, E. T., Jr.↗

DOE OSTI2018

A non-conforming finite volume element method for the two-dimensional Navier–Stokes/Darcy system

We consider the discretization of the stationary Navier–Stokes/Darcy system in a two-dimensional domain by the non-conforming finite volume element method. We use the standard formulation of the Navier–Stokes/Darcy system in the primitive variables and take as approximation space the non-conforming P{sub 1} elements for velocity and piezometric head and piecewise constant elements for the hydrostatic pressure. We prove that the unique solution of the non-conforming finite volume element method converges to the true solution with optimal order for velocity and piezometric head in discrete H{sup 1} norm and for pressure in discrete L{sup 2} norm, respectively. Finally, some numerical experiments are presented to validate our theoretical results.

42 ENGINEERING↗

DOE OSTI2024

Meshfree Methods

Meshfree methods have undergone substantial development and have received much attention in the last two decades. This new family of numerical methods is designed to inherit the main advantages of the finite element method such as compact supports of shape functions and good approximation properties while, at the same time, overcome the main disadvantages of the finite element method caused by the mesh dependence. The meshfree methods share a common feature that no mesh is needed and shape functions are constructed from sets of points, thus eliminating the need for time consuming mesh generation. The most significant advantage of meshfree methods is the flexibility in customizing approximation functions for desired regularity and for capturing essential physics and features of the particular problems of interest. Adaptivity formulation and multiple-scale solution strategies also can be implemented with relative ease. It has become clear that the meshfree methods provide considerable advantages over the conventional finite element methods in solving problems involving moving discontinuities, evolving material interfaces, multiple-scale phenomena, large material distortion and structural deformation, and fracture and damage processes. This Chapter gives an overview of many classes of meshfree methods, with more detailed discussions on Smoothed Particle Hydrodynamics (SPH), the Reproducing Kernel Particle Method (RKPM), Peridynamics (PD), the Material Point Method (MPM), as well as their applications in various challenging engineering problems.2

Chen, Jiun-Shyan↗