A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies

doi:10.1016/j.ijthermalsci.2015.10.017

International Journal of Thermal Sciences

Volume 100, February 2016, Pages 430-437

https://doi.org/10.1016/j.ijthermalsci.2015.10.017 Get rights and content

Highlights

•
We formulated FS-PIM to deal with the nonlinear heat transfer analysis of composite structures.
•
Both nonlinear steady and transient heat transfer problems are investigated.
•
The FS-PIM models perform much better than the standard FEM.

Abstract

A face-based smoothed point interpolation method (FS-PIM) is formulated to solve nonlinear heat transfer analysis of composite structures. For this method, the problem domain is first discretized using tetrahedral elements, and the face-based smoothing domains are further formed based on the faces of the tetrahedral meshes. Then, the field functions are approximated using PIM shape functions, which are constructed by different polynomial basis. Finally, the smoothed Galerkin weak form was utilized to obtain the discretized system equations. Numerical examples with different kinds of boundary conditions are investigated to verify the validity of the present method. The obtained results show that the presented FS-PIM schemes are very effective, when dealing with the nonlinear heat transfer analysis of composite structures.

Introduction

In the past few years, composite structures have been widely applied in civilian industries, which can improve the performance of structures and provide significant weight saving. Usually, composite structures work under complex thermal loads and hence, the relevant heat transfer analysis is of great importance in many practical engineering areas [1], [2], [3].

So much research has been done in recent yeas, focusing on solving the heat transfer problems [4], [5], [6]. There is no doubt that when the geometries, material models and boundary conditions are complex, it is usually very difficult to find analytical solutions. Therefore, numerical methods are widely employed to deal with the related analysis by researchers. For instance, Liu and Tan [7] employed the meshless local Petrov–Galerkin (MLPG) method to deal with the heat transfer analysis. Hematiyan et al. [8] used the boundary element method (BEM) to analyze transient heat transfer problems, in which the problem domain is discretized using boundary elements. Khosravifard et al. [9] proposed an improved meshless radial point interpolation method (RPIM) for the analysis of transient heat conduction problems. Ching and Chen [10] employed the MLPG method to investigate the transient thermal response of two-dimensional solids. Singh et al. [11] utilized the element free Galerkin (EFG) method to obtain the numerical solution of heat transfer problems, in which the approximation function is constructed from a set of scattered nodes.

Among all these methods, the meshfree methods have been found to be a dominant numerical method, such as reproducing kernel particle (RKP) method [12], diffuse element method (DEM) [13], point interpolation method (PIM) [14], etc. The meshfree point interpolation method is formulated using the Galerkin weak form, in which shape functions are constructed through interpolation based on a set of nodes in a local support domain. The PIM shape functions constructed by such procedure possess the Delta function property, which can permit the straightforward imposition of essential boundary conditions. Using shape functions generated by PIM, Liu et al. have applied the smoothing technique [15], [16], [17], [18], [19], [20], [21] to the PIM. Then, a node-based smoothed point interpolation method (NS-PIM) [22], [23] and an edge-based smoothed point interpolation method (ES-PIM) [24] are proposed and extended to deal with heat transfer problems based on the generalized gradient smoothing operation, in which a generalized gradient smoothing technique [25], [26] is utilized to establish the smoothed Galerkin weak form and the quality of solutions can be improved effectively.

Based on these previous studies, a face-based smoothed point interpolation method (FS-PIM) is further formulated and extended to deal with nonlinear heat transfer analysis of three dimensional (3D) composite structures. The discretized system equations are derived using the smoothed Galerkin weak form. It is assumed that the material in each portion is homogeneous, and all portions of structures are perfectly bonded together without clearance or sliding. The material parameters are considered to be temperature-dependent. Due to nonlinearity, the Newton–Raphson iteration method is utilized to deal with the analysis. Numerical examples with various kinds of boundary conditions are presented to verify the validity of different FS-PIM schemes for the nonlinear heat transfer analysis of composite structures.

Section snippets

Thermal governing equations and boundary conditions

It has been assumed that the material obeys Fourier's Law of heat conduction. The differential equations governing steady and transient heat conduction are given as follows, respectively $(k_{x} \frac{\partial {}^{2}T}{\partial x^{2}} + k_{y} \frac{\partial {}^{2}T}{\partial y^{2}} + k_{z} \frac{\partial {}^{2}T}{\partial z^{2}}) + q_{v} = 0$ $(k_{x} \frac{\partial {}^{2}T}{\partial x^{2}} + k_{y} \frac{\partial {}^{2}T}{\partial y^{2}} + k_{z} \frac{\partial {}^{2}T}{\partial z^{2}}) + q_{v} = ρ c \frac{\partial T}{\partial t}$ where T is the temperature, q_v is the internal heat generation, ρ is the density, c is the specific heat, k_x, k_y and k_z are the thermal conductivities in the x, y, z directions, respectively. The initial condition and a set of thermal

Construction of PIM shape functions

Consider a problem domain with a set of arbitrarily scattered points x_i (i = 1, 2, …, n), n is the number of nodes in the local support domain. For an interest point x, the approximation of temperature function T can be expressed as $T (x) = \sum_{i = 1}^{n} P_{i} (x) a_{i} = P^{T} (x) a$ where P(x) is monomial basis function, n is the number of field nodes in the local support domain and a is the unknown coefficient yet to be determined.

The polynomial basis of first order and second order is utilized for three-dimensional

Results and discussion

Nonlinear steady, transient heat transfer and thermal elastic analysis of three-dimensional composite structures are presented here to illustrate the validity of FS-PIM schemes with various kinds of boundary conditions. For comparison, the standard FEM codes are also developed to evaluate the same problems in FORTRAN, using the same meshes as the FS-PIM schemes. In order to study the convergence, the thermal equivalent energy and thermal equivalent energy error norm are defined as $E_{T} = \int_{V} ɛ^{T} k_{0} ɛ ⅆ V$ $E_{e} =$

Conclusion

In this paper, a face-based smoothed point interpolation method (FS-PIM) is formulated to deal with the heat transfer analysis of composite structures with nonlinearity. Formulations based on the smoothed Galerkin weak form are presented. Different support nodes selection schemes are discussed and several numerical examples with different kinds of boundary conditions are then investigated to examine validity of the present FS-PIM schemes. From these studies, several remarks can be made as

Acknowledgments

The support of National Science Foundation of China (11472101), Key Project Supported by the Education Department of Henan Province (15A460002) and Priority Academic Program Development of Jiangsu Higher Education Institutions are gratefully acknowledged.

Nomenclature

h_c: convection coefficient, W/(m² °C)
k: thermal conductivity, W/(m °C)
N: number of faces
n₀: unit outward normal component
q₀: heat flux, W/m²
T₀: initial temperature, °C
T_k: known temperature, °C
T_∞: environmental temperature, °C

Greek symbols

φ: shape functions
c: specific heat
ρ: density, kg/m³
Ω: problem domain
Γ: global or local boundary
P: monomial basis function
P_m: moment matrix
T_s: vector of nodal function values

Subscripts and superscripts

i, j: tensor indices
k: smoothing domain for face k

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Citation Excerpt :
Fortunately, Liu et al.[10, 11] proposed a G space theory and the weakened weak (W2) formulation. By applying strain smoothing technique [12, 13], a class of smoothed finite element models (S-FEMs)[14, 15] is further developed based on these formulations, such as the cell-based finite element method (CS-FEM) [16-18], the node-based finite element method (NS-FEM) [19-21], the edge-based smoothed finite element (ES-FEM) [22-24], the face-based finite element method (FS-FEM) [25, 26]. Besides, in recent years, some improved numerical methods have been proposed, such as the alpha finite element method (α-FEM) [27], the stable node-based smoothed finite element method (SNS-FEM) [28-31], the gradient weighted finite element method (GW-FEM) [32, 33] and so on.
In this work, an edge-based smoothed finite element method (ES-FEM) is embedded into the commercial software ABAQUS by using user defined element subroutines (Implicit/UEL, Explicit/VUEL). Unstructured meshes (i.e. triangular and tetrahedral elements) are employed in the framework due to their good geometric adaptability and the advantages of being automatically generated in the software. In ES-FEM, smoothing domains are further constructed associated with edges of triangular or tetrahedral elements. Linear and no-linear material models are included for static, free vibration, and large deformation analysis. Several numerical examples for predicting displacements, eigenvalues, and forging forming process are implemented. Results illustrated that the ES-FEM by using UEL and VUEL presents preciser than traditional finite element method (FEM), temporally stable, and can overcome the difficulties such as mesh distortion when the FEM is solving large deformation problems such as forging. It is valuable to embed the excellent numerical models, for example ES-FEM, into the commercial software since engineers can use it conveniently without professional knowledge. In this work, the detailed procedures are provided for the implementation of ES-FEM in UEL and VUEL, including pre-processing, post-processing and subroutine writing processes.

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A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies

Highlights

Abstract

Introduction

Section snippets

Thermal governing equations and boundary conditions

Construction of PIM shape functions

Results and discussion

Conclusion

Acknowledgments

Nomenclature

Greek symbols

Subscripts and superscripts

Int. J. Heat Mass Transfer

Int. J. Therm. Sci.

Int. J. Therm. Sci.

Eng. Anal. Bound. Elem.

J. Therm. Sci.

Int. J. Therm. Sci.

Appl. Math. Comput.

Appl. Math. Modell.

Int. J. Heat Mass Transfer

Finite Elem. Anal. Des.

Int. J. Heat Mass Transfer

Eng. Anal. Bound. Elem.

Int. J. Therm. Sci.

Int. J. Heat Mass Transfer