A Jacobian-free Newton Krylov method for mortar-discretized thermomechanical contact problems

Abstract

Multibody contact problems are common within the field of multiphysics simulation. Applications involving thermomechanical contact scenarios are also quite prevalent. Such problems can be challenging to solve due to the likelihood of thermal expansion affecting contact geometry which, in turn, can change the thermal behavior of the components being analyzed. This paper explores a simple model of a light water reactor nuclear fuel rod, which consists of cylindrical pellets of uranium dioxide (UO₂) fuel sealed within a Zircalloy cladding tube. The tube is initially filled with helium gas, which fills the gap between the pellets and cladding tube. The accurate modeling of heat transfer across the gap between fuel pellets and the protective cladding is essential to understanding fuel performance, including cladding stress and behavior under irradiated conditions, which are factors that affect the lifetime of the fuel.

The thermomechanical contact approach developed here is based on the mortar finite element method, where Lagrange multipliers are used to enforce weak continuity constraints at participating interfaces. In this formulation, the heat equation couples to linear mechanics through a thermal expansion term. Lagrange multipliers are used to formulate the continuity constraints for both heat flux and interface traction at contact interfaces. The resulting system of nonlinear algebraic equations are cast in residual form for solution of the transient problem. A Jacobian-free Newton Krylov method is used to provide for fully-coupled solution of the coupled thermal contact and heat equations.

Introduction

Within a typical light water nuclear reactor fuel rod, a column of cylindrical UO₂ fuel pellets is enclosed by a protective metal cladding that serves to separate the pellets from the reactor coolant and to prevent the release of fission products into the coolant. One important design requirement for a reactor fuel is to ensure the integrity of the cladding over the lifetime of the fuel under operating conditions. The cladding integrity can be impacted by many issues including cladding stress, strain, and deformation; all of which can alter the susceptibility of the cladding to damage. The stress state within the cladding is strongly influenced by the forces applied to the cladding by the pellets [22], which is influenced by the thermal and radiation conditions the pellets have been subjected to during their lifetime. One very important effect governing the temperature at the center of the pellet is the heat transfer between the pellet and the cladding across the helium filled gap that separates them. Fig. 1(a) shows a thermal calculation of a section of a fuel rod, and Fig. 1(b) and (c) show the relationship between the pellet and cladding, where the gap has been magnified for clarity.

As the fuel ages, the pellets swell both radially and axially, and the cladding “creeps down” onto the swelling pellet stack, such that the pellets and cladding eventually come into contact. Upon initial heating, the pellets fracture due to thermal stress into a set of fragments that modify their geometry. Fracture is ignored in this paper but will be considered at a later date. Even with this simplification, the problem is challenging due to the coupling between the thermal and mechanical aspects of the problem and the relative motion between the pellet and cladding.

Fuel performance calculations typically employ a Lagrangian mechanics formulation due to the history dependence of the mechanics in both the pellet and cladding. It is also necessary to accurately treat the heat transfer across the narrow gap between the pellet and cladding surfaces. The annulus is long and narrow, it is unlikely that Rayleigh numbers larger than ten are present. This paper thus assumes that conduction is the dominant heat transfer mechanism across the gap. The pellets swell both axially and radially over time and act to close the gap, in conjunction with creepdown of the cladding. The axial expansion is troublesome from a modeling perspective. A fuel rod is composed of hundreds of pellets with the lowermost pellet supported; this axial expansion is additive up the stack such that the top pellet moves vertically a significant distance with respect to the cladding. Any mesh within the gap would quickly shear and become useless. Further, as can be inferred from Fig. 1(c), a given mesh face on the pellet “sees” one or more faces on the cladding with this face to face relationship varying with the transient behavior of the system. Further complicating the gap problem is the need to accurately calculate the transient volume of the gap, the pressure of the gas contained in the gap, and the plenum volume and conditions at the top of the fuel rod. The plenum pressure changes appreciably over time and should be considered in the model of the fuel behavior.

In lieu of meshing the gap, the proposed method employs a mortar finite element approach to develop a common integration space between the pellet and cladding. A mortar finite element approach was selected here due to potential advantages of the approach when contacting surfaces are nearly flat, its excellent convergence properties and accurate resolution of contact stresses on interfaces [9]. The mortar method is also relatively immune from the condition called “locking,” or an overconstraint of the contact interface conditions. Puso and Laursen [17] discuss examples of “locking” that can occur when two-pass node-on-segment contact approaches are used and compares these results with a mortar approach.

The aging of a reactor fuel is best described as consisting of long periods of quasi-steady operation punctuated by short transient events. The initial transient occurs upon warmup of the fuel on reactor startup; the fission process heats up the fuel until it comes into thermomechanical equilibrium with the coolant. As the fuel ages beyond this initial transient, the fission process causes slow changes in the thermomechanics of the fuel until it is spent. Occasional reactor power cycles may occur that disturb the equilibrium of the fuel. As the life of the fuel is measured in terms of many months or years, implicit time integration is likely the best approach to model fuel performance. Thus, this paper develops an implicit fully coupled approach to address these concerns.

There are several efforts that are examining three dimensional high fidelity calculations of reactor fuel performance. Recent examples include the sophisticated coupled approaches employed by PLEIADES [13] and the BISON code [15]. The BISON study employed a simple boundary condition for the thermal conditions external to the pellet and did not consider mechanical contact. This work develops a more sophisticated approach that could be employed in codes like BISON; this thermomechanical contact method is specifically designed for tightly-coupled implicit solutions that employ Jacobian-free solution methods.

There is a comprehensive basis of experience in the use of mortar methods for contact (e.g. [9], [11], [12], [17], [18]), and in the solution of contact problems using Newton’s method [10], [19], [26], [28], to name a few. There is appreciable work in the area of thermomechanical contact, most of which employs an operator splitting approach [8], [26].

The approach proposed here solves fully coupled thermomechanics in the solid bodies within the domain; the cylindrical pellets and the cladding annular region around the pellet. Heat conduction through helium inside the gap is approximated using a model for gap conductance that is valid for all gap distances (including completely closed). The model uses a mortar method to solve the contact constraint problems for both heat transfer across the gap and mechanical contact as the gap closes. The mechanical contact model is modified from traditional approaches to account for an evolving gas pressure in the gap; this pressure is treated using a time-varying normal traction applied to the elements on both sides of the gap. The resulting two-dimensional fully coupled transient system is then solved using a Jacobian-free Newton Krylov solution method [3]. To the author’s knowledge, the formulation of the problem in this manner involving the JFNK solution strategy, is unique.

The remainder of the paper is organized into four sections; Section 2 summarizes the thermomechanical model and the contact discretization, Section 3 gives an overview of the solver and Newton regularization strategy, Section 4 presents the performance of this approach, and Section 5 presents conclusions and future plans.