A parallel multi-domain solution methodology applied to nonlinear thermal transport problems in nuclear fuel pins☆
Introduction
Many real world engineering problems involve multiple coupled nonlinear physical processes that occur both within and across several interacting physical domains. Robust, accurate, and efficient three dimensional simulations for some of these complex problems pose significant challenges that require a combination of powerful numerical algorithms, efficient parallel implementations, and massive computing resources to tackle. These challenges include developing the numerical methods and the parallel software infrastructure for coupling physical phenomena that occur on the surface and within the interior of physical domains, coupling structured and unstructured mesh calculations, coupling models with different discretizations, and using tightly coupled solution methods to solve certain coupled physics problems and loosely coupled approaches for others. Developing such simulation capabilities is a nontrivial task.
In this article, we will focus primarily on one such complex application where all of the features outlined above are present: thermal transport in nuclear fuel rods. However, we will also devote some effort to describing the parallel code infrastructure that was developed to provide the necessary meshing, discretization, linear algebra, linear and nonlinear solvers, physics modules (conservation laws and constitutive models), material property databases, and parallelization mechanisms for simulating this application in the hope that it will be beneficial to the broader scientific community.
A nuclear fuel assembly consists of several hundred nuclear fuel rods (shown in Fig. 1) bound together by spacer grids. While some of the rod locations are reserved for instrumentation and safety, most of the rods contain nuclear fuel. Each individual nuclear fuel rod in turn consists of several hundred nearly cylindrical nuclear fuel pellets (each with a height to diameter ratio of approximately one) stacked one on top of another to form a long column enclosed within a metal tube called the clad. Heat is generated within the pellets by nuclear fission and is distributed within the pellets and clad via a diffusive process. There is thermal contact (modeled as a convective process) between neighboring pellets and between the pellets and the clad. Each fuel rod is cooled with water flowing axially up the outer surface of the clad.
Modeling the heat transfer, along with other physics, leads to a very high-aspect ratio problem with many inter-dependent domains. Traditional nuclear fuel simulation eliminates the computational challenge by approximating the heat transfer as entirely radial and neglecting the axial and azimuthal components, which are only coupled through the coolant temperature and other simplified physics [2], [3]. Recent efforts to develop advanced modeling and simulation tools for nuclear fuel rods [4], [5], which include simulating full three-dimensional fuel rods with resolved pellets, have relied upon standard solution and preconditioning strategies that do not necessarily take advantage of the physics and geometry of the problem. This manuscript will not however address the challenges of structural dynamics and the associated feedback on heat transfer.
With respect to specific work related to heat transfer within nuclear fuel rods, there are several existing efforts to develop parallel codes that model three-dimensional heat transfer within nuclear fuel rods, including PLEIADES/ALCYONE [4], [6], MOOSE/Bison [4], and BACO [7]. These codes are all focused on the integration of the many physics required for modeling nuclear fuel performance in steady-state and transients to improve the underlying material science, including fracture/contact mechanics, fission gas generation and release, and corrosion chemistry.
We have developed an efficient scalable parallel simulation framework for solving such multi-domain, multi-physics problems and have used it to solve the specific nuclear fuel problem described above. Within our particular application a nonlinearly consistent Jacobian Free Newton Krylov (JFNK) method is used (though the ability to use alternative solution methods also exists) across all the domains for each fuel rod. Physics-based preconditioning is used to accelerate the solution process and the multi-domain (pellets and clad) aspect of the problem is leveraged in developing methods that minimize communication as well as avoid the formation of full matrices over the whole domain.
In the next section, we present a mathematical description of the problem under consideration. Section 3 will describe the finite element discretization of the models in Section 2. The algorithms used to solve the resulting nonlinear system of equations are described in Section 4. The computational framework that was used in this work is briefly described in Section 5. Section 6 reports on numerical experiments performed to verify and validate our code and test its parallel scalability. Section 7 provides details on coupling to reduced order flow models, coupling to oxide growth models on the exterior clad, and parallel full assembly simulations that couple thermal transport components on unstructured meshes with a structured mesh radiation transport code. The paper ends with a few concluding remarks.
Section snippets
Model
A 3D fuel rod domain, Ω, is modeled as consisting of the union of N pellet subdomains, , , a clad subdomain , and a gap region , i.e., the global domain . The number and geometric complexity of fuel pellets in fuel rods can vary significantly; from simple cylinders to the complex pellet geometries shown later in this manuscript and from a few pellets in an experimental rod to more than 400 pellets in a commercial nuclear fuel rod. In our numerical
Domain discretization
As can be seen from Fig. 2, Fig. 3, the clad subdomain and the pellet subdomains are bounded, connected volumes with piecewise smooth curved boundaries. For the purposes of this paper is in general approximated by a polyhedral domain during the mesh generation process. is partitioned into a set, , of non-overlapping general hexahedral elements, which are geometrically conforming, i.e.:
- •
.
- •
The intersection of two elements, and
Solution strategy
Several nonlinear solution strategies could be used to solve the nonlinear system of algebraic equations described in Section 3. The fact that our problems often involve several hundred subproblems each discretized over a separate physical domain makes solution methods that do not require formation of the full Jacobian matrix attractive. In particular we choose to use a Jacobian-Free Newton Krylov (JFNK) method [12] for our nonlinear solver. The efficiency of a JFNK method when applied to a
Computational infrastructure
Several computational tools are necessary for performing simulations such as the one described in this paper; we developed the Advanced Multi-Physics (AMP) [20], [21] package for this purpose. AMP is a complete system for simulating stationary and time dependent, multi-domain, coupled physics problems. AMP consists of several software components. Each component is designed to provide a uniform consistent interface which interacts with other components, and developers of other components are
Numerical experiments
A suite of numerical experiments were defined to verify the accuracy of the thermal transport capability of AMP for a multi-domain problem that is based on the geometry and materials of nuclear fuel. Independent studies were performed to verify the accuracy of the solution using the method of manufactured solutions (Section 6.2), evaluate the accuracy of the code with respect to experimental data and a well characterized code used for regulatory analysis (Section 6.3), and evaluate the
Fuel assembly modeling
In Section 5 the components of the AMP multi-physics infrastructure that enabled the development of nonlinearly consistent multi-domain thermal transport calculations that form the main process of this paper were described. Here we illustrate further multi-physics capabilities of AMP by describing further extensions of the fuel rod modeling capability. Since our focus is on solution and coupling methodology and due to space limitations we will concentrate on the relevant coupling aspects with
Conclusions
Many real world engineering problems involve the complex interaction between many bodies, in a nonlinear manner. From mesh generation to predicting results, modeling these large complex systems presents significant computational challenges. An efficient parallel, multi-domain solution methodology has been developed and implemented to solve these systems by leveraging the natural decomposition of the problem associated with the individual domains. This methodology has been demonstrated for
Acknowledgements and access
The AMP (Advanced Multi-Physics) code is distributed with a modified BSD license and accessible either by contacting the corresponding author or through the Radiation Safety Information Computational Center (RSICC) at Oak Ridge National Laboratory, with an RSICC license, as CCC-793. The development of AMP, and the nuclear fuel performance application built upon it, was funded by the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program of the U.S. Department of Energy Office of
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This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).