A two-field state-based Peridynamic theory for thermal contact problems

Highlights

• Stated-based Peridynamic theory is used to study the thermal contact problem.

• Differential algebraic equation is proposed to model the problem.

• Various thermal fluxes fit the proposed computational strategy.

• Good temporal accuracy and thermal condition preservation are obtained.

Abstract

Peridynamics is a non-local based method and an extension of classical continuum theory that has been proved to have the ability to solve problems involving discontinuities. This work extends the state-based Peridynamic formulation to describe heat transfer process in the adjacent regions via using the domain decomposition technique. The state-based Peridynamic heat conduction model is proposed to couple with a generalized thermal diffusion model in a two-field form, in which both the temperature and the thermal flux are treated as the primary variables. Coupled with thermal interface conditions, the proposed two-field state-based Peridynamic heat conduction naturally leads to a classical differential algebraic equation, which permits the numerical simulations of thermal contacts between various diffusion models. A unified time integration, termed as generalized single step single solve (GS4), is extended to solve the resulting differential algebraic equations. Numerical results of the simulations are reported and compared with those obtained by Finite Element Method which show that the method is promising for these applications in terms of accurately capturing the physics and preserving the interface conditions.

Introduction

Peridynamics has been widely exploited in the continuum mechanics, especially those particular problems involving size-effect, cracks, and discontinuities [1], [2]. It is a non-local extension of classical solid mechanics where the fundamental equations are reformulated in terms of integral equations instead of the differential equations such that it is well suited to the discontinuities. The non-local Peridynamic theory has also been extended to heat transfer problems. Both the bond-based [3], [4] and the state-based Peridynamic [5], [6] theories have been exploited to describe the diffusion between material points. In particular, the bond-based version is a particular case of the state-based formulation; and with certain simplifications the bond-based Peridynamic approach for heat conduction can be developed from the generalized state-based one [5].

An area of interest is determining the temperature field in the system involving thermal interface. In the ideal situation, the adjacent domains contact perfectly such that the temperature and the flux do not involve jumps across the interface. This interface of the perfect contact is classified as Perfect Interface. However, the interface between two adjacent regimes usually exhibits a thermal resistance, which poses a barrier to heat flow and leads to a temperature jump. This interface, which causes the temperature discontinuity, is classified as Kapitza Imperfect Interface. ¹ This temperature jump was observed by Kapitza [8] and also had been experimentally measured in [9], [10]. Simulating the thermal contact problems is considered to be a numerical challenge because the treatment of the temperature jump at the interface requires special techniques, which appears to be somewhat difficult in the classical finite element method (FEM). To capture the temperature discontinuity, the extended finite element method (XFEM) [11], [12], [13] was proposed to formulate the thermal contact problem. The discontinuous Galerkin method as well as the local discontinuous Galerkin finite element method have been utilized in [14]. Besides, many efforts have also been made to carry out the numerical simulation of the thermal contact of the multi-layered systems governed by the hyperbolic-type thermal diffusion models involving thermal waves [15], [16].

From the viewpoint of modeling the heat conduction process involving thermal interfaces, these existing studies mentioned above have established a solid theory and have been proved to be amenable to predict the heat transfer process. However, with the increasing demand of micro-devices, multilayered nano-scale systems and the like, the existing modeling approaches and numerical methods often lack the generality. Specifically, the adjacent domains are usually considered to be modeled by the same type of thermal flux and the size-effect influence is usually ignored. For example, [16] exploited the Cattaneo-type thermal flux to describe both the thin film and the substrate in the film-substrate structure; however, the thickness of both the film and the substrate are significantly different in general such that using the pure Cattaneo-type thermal flux to describe the entire system seems not precise; and the size-effect influence is very necessary to be considered in the model. To describe the thermal contract problem, most of the existing models [7], [17] were established by the classical single-field form, in which the temperature is the primary variable. It has been acknowledged that the single-field form for the Fourier's law governed system is described by a first-order time dependent equation of temperature while the nonFourier's system is usually a second-order time dependent equation of temperature. When the system of interest, such as the film-substrate structure, is comprised of and influenced by both the Fourier-type domain and the nonFourier-type domain, the time integration approach of the entire system has to consider the consistency between different time dependent systems (hyperbolic to parabolic). Despite difficulty of the associated time integration, most of the numerical techniques have been established on the FEM-based methods (weak form formulation, local theory), which in general needs to spatially integrate over the entire domain to get the mass, damping, and stiffness matrices to formulate the semi-discretized time dependent equation. Relevant studies can be found in [7], [11], [14], etc. However, when the orders of time derivatives in each sub-domain are inconsistent, to achieve the precise time integration of the entire domain, some dummy variables, such as second-order time derivative of temperature ($\ddot{T}$) or first-order time derivative of flux ($\dot{\mathit{q}}$) in the Fourier-type domain, have to be introduced, which is physically meaningless. Moreover, the classical FEM-based method requires a mesh of the interface and have to construct appropriate surface/interface elements [11], which is another difficult issue in the system involving complex geometries. Besides, an ill-conditioned stiffness matrix may be involved in the numerical iteration of the case with strongly heterogeneous material property [18], which usually leads to instability of the numerical simulations.

The overview described earlier indicates that there still exists a huge gap between the development of the numerical techniques and the real industrial demand of precisely modeling the thermal interface problem. In general, the numerical approach of predicting the thermal interface problem are expected to have the ability of (a) describing the non-local effect of the system and (b) convenient time stepping technique and framework without the limitation of Fourier-type (first-order time dependent system) and nonFourier-type (second-order time dependent system) of heat flux in each domain. Motivated by this, we have previously explored the development of a new time integration framework in our recently published work [19] that can basically achieve the generality of the thermal interface problem in which different sub-domains are governed by various thermal flux models. However, the numerical performance of the computational framework proposed in [19] was not clearly discussed and the numerical examples appear to limit in the one dimensional case. More importantly, the fundamental solver was limited in the local system and disguises the size-effect of the system.

In this work, the state-based Peridynamic theory is utilized to thermal contact problems in the terms of a two-field form, which treats both the temperature and the thermal flux as the primary variables. Similar to that in [19], the interface condition is treated as an algebraic equation and the combination of the two-field state-based Peridynamic heat conduction model and the interface algebraic equation naturally formulates the problem as a first-order time dependent Differential Algebraic Equation. To numerically solve the resulting differential algebraic equation, an implicit unified time integration framework is introduced based upon the Generalized Single-Step Single-Field (GS4) [20] time integration procedure and framework, which can precisely preserve the interface condition; in other words, the temperature jump caused by the imperfect interface can be ensured. The generalized thermal flux model is considered and introduced to the two-field form such that the resulting governing equation allows flexibility in that, the adjacent domains may be governed by various different thermal flux models. In addition, the proposed two-field state-based Peridynamic heat conduction model is discretized by particles in the space sense, which does not require any surface elements as required in FEM; and also has the ability of readily handling systems with complex geometries.

The advantages and numerical framework and features of the proposed method are summarized as follows:

• To the authors' knowledge, the implementation of the state-based Peridynamics for thermal contact problem has not been done. Compared with the studies, such as FEM-based theory, the state-based Peridynamic theory is acknowledged to be a non-local based method, which has been proved to be well-suited in continuum system involving discontinuities. This work can be considered as an extension of the current status of applying state-based Peridynamics in the field of heat process with thermal interfaces.

• In contrast to the classical single-field form, the proposed two-field form is formulated under the umbrella of the generalized thermal flux model such that the resulting governing equation is purely first-order time dependent equation while describing both the Fourier's thermal flux and the non-Fourier's thermal flux as well. Consequently, the associated time stepping procedure only needs to handle the first-order transient system in the present work. This procedure decreases the complexity of the problem in which each sub-domain is governed by different order time dependent system and strongly different material properties, such as Case III in Section 1.

• Furthermore, the computational features of the proposed method can not only obtain the temporal second-order accuracy with respect to temperature, temperature changing rate, and Lagrange multipliers in all the schemes generated in the GS4 time integration family, but also perfectly preserve the thermal interface conditions without any extra treatment.