A high order control volume finite element procedure for transient heat conduction analysis of functionally graded materials

Abstract

A robust unstructured control-volume finite element method is presented for the solution of two-dimensional transient heat conduction in functionally graded materials (FGMs) with isotropic properties. The material properties at a point in the domain vary exponentially to spatial coordinates. A triangular mesh is chosen for spatial discretization and a fully implicit scheme is adapted for time discretization. Several problems are investigated and the results are successfully validated by using analytical and other numerical solutions available in the literature.

1 Introduction

Functionally graded material (FGMs) is a class of composites having continuous spatial variation of material properties. This type of material has recently been used in a severe temperature environment-a body is subjected to high transient heating and cooling within a short period of time [1, 2]. With appropriate grading profile, the thermal stress, residual stress, and stress concentration factor in the structure can be reduced when compared with those in the laminated composite structures under the same loading condition. Mostly, FGMs are manufactured from isotropic components such as metals and ceramics. FGMs have become widely used not only in thermal protection purpose but also in other areas; i.e. biomedical engineering and fuel cell [3, 4].

Transient heat conduction in FGMs has been investigated by several researchers [5–12] for better understanding of these materials. Sladek et al. [9] applied a meshless method called local boundary integral equation method (LBIEM) to analyze the transient heat conduction problem in FGM. Wang et al. [10] determined the transient temperature field in FGM plate, shell and sphere by a finite element/finite difference (FE/FD) method and also considered the effect of material properties on the temperature. Wang and Tian [11] gave more details on the above FE/FD method and demonstrated their method to two-dimensional heat transfer under transient state. Wang et al. [12] proposed the virtual boundary collocation method (VBCM) for transient heat conduction analysis in FGM.

Control volume finite element method (CVFEM) is an alternative numerical technique for the solution of FGM problems. It is a combination of numerical discretization inherited from the finite volume method and numerical interpolation of FEM and sometimes referred to as a vertex-centered finite volume method [13, 25–28]. This technique has a potential for multi-physical problems such as the conjugate heat transfer where the energy balance between solid and moving fluid needed to be considered simultaneously [35]. From authors’ point of view, CVFEM is applicable for the transient heat conduction problems in FGM where the non-homogenous material properties of FGM induce the non-linearity in heat conduction problem.

Early development of finite volume method had focused on the non-linearity in problems of fluid dynamics [13]. Therefore, the domain geometries initially modeled by finite volume method were relatively simple. The use of structured mesh based either on rectangular or body-fitted coordinates did not lead to severe errors in the numerical results. In modern applications, however, the domain geometries are much more complicated than those in the past. Consequently, it is difficult to handle these works with structured mesh solver. Unstructured mesh solver would be a promising tool for research and design in novel engineering products development. The unstructured finite volume may be roughly separated into two main groups by different definitions of control volume, the cell-centered and vertex-centered approaches. For the cell-center approach, each control volume is defined by triangular or quadrilateral mesh. The value of unknown variable, such as temperature, is referred and stored at the center of control volume. On the other hand, the vertex-centered approach starts with defining a computational node at the grid point of triangular or quadrilateral mesh and then sub-control volumes are built around it. Therefore, the control volume is built from summation of sub-control volumes. With this approach the value of unknown variable, i.e. temperature, is stored at the vertex point. Although a finite volume method have been applied on general fluid flow problems, there are some works concerning the heat conduction problem in FGM.

Early applications of the finite volume method on problems of non-homogeneous material like FGM had been carried out and reviewed by Aboudi et al. [14, 15]. They employed the two-level unit cell discretization involving generic cell and subcell levels to capture the graded material’s heterogeneous microstructure. Their approach was called high-fidelity generalized method of cells (HFGMC). Bansal and Pindera [16–20] utilized the concept of local/global stiffness matrix approach for the solution of local displacement and stress fields within the repeating unit cell. They also constructed the cell problems by a standard elasticity approach involving the direct volume-averaging of the local field equations and the satisfaction of the local continuity conditions in a surface-average sense. They had named their approach as the finite volume direct average micromechanics (FVDAM). Even though the FVDAM increased the efficiency of HFGMC, the subcell discretization used by FVDAM was limited to rectangular control volume. This method was inefficient in approximating the curved boundary and produced unrealistic stress concentrations at the interface between the materials of different elastic properties. Recently, Cavalcante [21] proposed the quadrilateral subcell discretization of parametric finite-volume theory to mitigate the negative impact of rectangular subcell mentioned above. Cavalcante et al. [22, 23] applied the parametric finite-volume theory to analyze the steady and transient thermal and stress responses of FGM cylinder subjected to the thermo-mechanical loading. Their method could improve the result of stress concentration at the material interface in an eshelby problem. Gao et al. [24] directly extended the FVDAM to quadrilateral finite volume direct average micromechanics (QFVDAM). With this technique, a system of quadrilateral subcells numbering was not restricted to an order fashion like that of Cavalcante et al. [22, 23].

In contrast to the above parametric mapping approach based on quadrilateral subcell and higher order approximation, this paper adopts the finite volume formulation based on triangular unstructured mesh with quadratic interpolation function. It is similar to the commonly used higher order interpolation in the finite element method. The objective of this paper is to present a distinguish methodology derived from CVFEM for transient two-dimensional heat conduction analysis of isotropic FGM. The accuracy and efficiency of this method will be assessed. The paper is organized as follows.

In Sect. 2, the governing equation and integral form of thermal conduction in an isotropic non-homogeneous solid are described. The control volume finite element method for spatial discretization and the time-stepping scheme for time discretization of heat conduction equation are given in Sect. 3. To demonstrate the potential and accuracy of the method, examples are analyzed and the numerical results are compared with the results from analytical solutions or other well-published numerical solutions in Sect. 4. Finally, Sect. 5 provides a summary of the major results and presents relevant conclusions.

2 Governing equation

The transient heat conduction of an isotropic body which occupies a domain Ω is considered. We employ the rectangular Cartesian coordinates, x and y, to describe the temperature distribution inside the body. The thermal boundary conditions imposed upon the body are in the form of prescribed temperature $\hat{T}$ on the boundary Γ_θ, specified heat flux $\hat{q}$ on Γ _q and convective boundary condition on Γ _c . The union of the boundaries Γ _θ , Γ _q and Γ _c comprise Γ which is the whole boundary of the domain Ω.

In a functionally graded material, all material coefficients depend on the spatial coordinates. The material properties of an isotropic material are; thermal conductivity (k = k(x, y)), density (ρ = ρ(x, y)), and specific heat per unit mass (c = c(x, y)). The governing equation for transient two-dimensional heat conduction in isotropic and continuously non-homogenous solid is given by the following partial differential equation with variable coefficients

$$ {\frac{\partial }{\partial x}}\left( {k{\frac{\partial T}{\partial x}}} \right) + {\frac{\partial }{\partial y}}\left( {k{\frac{\partial T}{\partial y}}} \right) + Q = \rho c{\frac{\partial T}{\partial t}}, $$

(1)

where T = T(x, y, t) is the temperature field and Q = Q(x, y, t) is the internal heat source per unit volume. The thermal conduction equation (1) has to be solved for prescribed boundary and initial conditions. The initial condition specifies the temperature distribution of the entire solid at time zero, that is

$$ T(x,y,0) = \bar{T}_{0} (x,y), $$

(2)

where $\bar{T}_{0} (x,y)$ is defined as a constant value or a function of the spatial coordinates, x and y. General forms of heat conduction boundary conditions are

$$ T = \hat{T}\quad {\text{on}}\;\Upgamma_{\theta } , $$

(3a)

$$ k\left( {{\frac{\partial T}{\partial x}}n_{x} + {\frac{\partial T}{\partial y}}n_{y} } \right) = \hat{q}\quad {\text{on}}\;\Upgamma_{q} , $$

(3b)

$$ k\left( {{\frac{\partial T}{\partial x}}n_{x} + {\frac{\partial T}{\partial y}}n_{y} } \right) + h\left( {T - T_{\infty } } \right) = 0\quad {\text{on}}\;\Upgamma_{c} , $$

(3c)

where $\hat{T}$ is the value of temperature at the boundary, h is the convection coefficient, T _∞ is the temperature of the surrounding medium, $\hat{q}$ is the normal heat flux that flow into or out of the domain, n _x and n _y are the components in x and y directions, respectively, of unit normal vector to the boundary Γ.

3 Control-volume finite element formulation

In this section, the weak formulation is first derived. Then, the control volume finite element method is applied to discretize the resulting integral equation from the previous step in both space and time. The discretized equation for each nodal unknown temperature can be assembled into matrices and vectors of all nodal temperatures. After applying the appropriate boundary conditions and initial condition, this system of linear algebraic equations is ready to be solved by a standard numerical technique.

3.1 Weak formulation

Firstly, Eq. 1 is integrated over a control volume Ω^cv and a time interval from t to t + Δt. Thus, integration of Eq. 1 leads to

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {{\frac{\partial }{\partial x}}\left( {k{\frac{\partial T}{\partial x}}} \right)} d\Upomega } \,dt + \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {{\frac{\partial }{\partial y}}\left( {k{\frac{\partial T}{\partial y}}} \right)} d\Upomega } \,dt + \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {Qd\Upomega\,dt} } - \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {\rho c{\frac{\partial T}{\partial t}}} d\Upomega \,dt = 0} . $$

(4)

Applying the divergence theorem to the first two term on the left-handed side of Eq. 4 results in

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma^{\text{cs}} }} {k\left( {{\frac{\partial T}{\partial x}}n_{x} + {\frac{\partial T}{\partial y}}n_{y} } \right)} d\Upgamma } \,dt + \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {Qd\Upomega } \,dt} - \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {\rho c} {\frac{\partial T}{\partial t}}d\Upomega \,dt} = 0. $$

(5)

After imposing the heat flux and heat convection boundary conditions on the left-handed side of Eq. 5, it becomes

$$ \begin{aligned} & \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma_{\text{int}} }} {k\left( {{\frac{\partial T}{\partial x}}n_{x} + {\frac{\partial T}{\partial y}}n_{y} } \right)} d\Upgamma }\, dt - \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma_{q} }} {\hat{q}d\Upgamma } \,dt - \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma_{c} }} {h\left( {T - T_{\infty } } \right)d\Upgamma } } } \,dt \\ & \quad + \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {Qd\Upomega }\,dt} - \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {\rho c{\frac{\partial T}{\partial t}}d\Upomega }\,dt} = 0. \\ \end{aligned} $$

(6)

where Γ_int denotes an internal boundary of computational domain which is the sum of control surfaces from the surrounding elements contributing to a control volume formation.

3.2 Control volume finite element method

In view of the spatial discretization, the control volume is a union of sub-control volumes inside the three-node linear triangular elements or six-node quadratic triangular elements. For instance, a sub-control volume surrounding node I is constructed by connecting midpoints of the element edges I–J and I–K to the centroid of the element named I–J–K as shown in Fig. 1a. Figure 1b illustrates six sub-control volumes within a quadratic triangular element. The formation of a control volume for quadratic triangular element is, in fact, the formation of sub-control volumes of four smaller “imitation” triangles I–L–N, J–M–L, L–M–N, and K–N–M. Figure 1c, d show two control volumes constructed from three-node and six-node triangular elements, respectively.

Fig. 1

The sub-control volume and its integration points of a control volume around each node in general a three-node triangular element, b six-node triangular element; typical shapes of control-volumes for a group of c three-node triangular elements, d six-node triangular elements

According to the concept of multiple isoparametric transformation proposed by Li et al. [29], the material properties and temperature at any point in a triangular element can be approximated by

$$ \rho = \sum\limits_{i = 1}^{n} {N_{i} \rho_{i} } ,\quad c = \sum\limits_{i = 1}^{n} {N_{i} c_{i} } ,\quad k = \sum\limits_{i = 1}^{n} {N_{i} k_{i} } , \quad T = \sum\limits_{i = 1}^{n} {N_{i} T_{i} } . $$

(7)

Here, N _i are the shape functions of the element [34] which are normally defined in terms of local coordinates over the triangular region, n is a number of nodes in a particular element, ρ _i , c _i , k _i , and T _i are the nodal values of density, specific heat, thermal conductivity, and temperature, respectively, of a particular element. Note that Eq. 7 is similar to the generalized isoparametric graded finite elements of Kim and Paulino [30]. The temperature gradients at any point in an element, appearing in the first integral of Eq. 5, can be written as

$$ {\frac{\partial T}{\partial x}} = \sum\limits_{i = 1}^{n} {{\frac{{\partial N_{i} }}{\partial x}}T_{i} } , $$

(8a)

$$ {\frac{\partial T}{\partial y}} = \sum\limits_{i = 1}^{n} {{\frac{{\partial N_{i} }}{\partial y}}T_{i} } , $$

(8b)

where ${\frac{{\partial N_{i} }}{\partial x}}$ and ${\frac{{\partial N_{i} }}{\partial y}}$ are the derivatives of shape functions at the nodal point i with respect to x and y, respectively. The midpoint rule is utilized to compute all element integrals since it gives high accuracy with little effort expended. The star symbols in Fig. 1a, b represent the integration points for determining the values of integral terms in Eq. 6. Note that the sub-control volume boundary around each node is the summation of two surfaces in the case of three-node element while it is the summation of four surfaces in the case of six-node element. Details of shape functions, their derivatives and numerical integration method are included in Appendix 1.

3.3 Time discretization

Integration with respect to time can be carried out with the following trapezoidal interpolation formula [13]:

$$ \int\limits_{t}^{t + \Updelta t} {\phi dt} = \left[ {f\phi^{t + \Updelta t} + \left( {1 - f} \right)\phi^{t} } \right]\Updelta t, $$

(9)

where f is a weighting factor between 0 and 1. In this paper, the time discretization is done through a fully implicit scheme, f = 1. Thus, the integrals of Eq. 6 can be written in discretized forms as

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma_{\text{int}} }} {k\left( {{\frac{\partial T}{\partial x}}n_{x} + {\frac{\partial T}{\partial y}}n_{y} } \right)} d\Upgamma } \,dt = \sum\limits_{j = 1}^{ng} {\left[ {k\delta S^{\text{int}} \left( {{\frac{\partial T}{\partial x}}n_{x} + {\frac{\partial T}{\partial y}}n_{y} } \right)} \right]_{j} } \Updelta t, $$

(10a)

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma_{q} }} {\hat{q}d\Upgamma } dt} = \sum\limits_{i = 1}^{nf} {\left[ {\hat{q}_{i} \delta S_{i}^{q} } \right]\Updelta t} , $$

(10b)

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upgamma_{c} }} {h\left( {T - T_{\infty } } \right)d\Upgamma } } \,dt = \sum\limits_{i = 1}^{nc} {\left[ {h_{i} \delta S_{i}^{c} \left( {T_{i}^{n} - \left( {T_{\infty } } \right)_{i}^{n} } \right)} \right]} \Updelta t, $$

(10c)

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {Qd\Upomega } \,dt} = \left( {\int\limits_{{\Upomega^{\text{cv}} }} {Qd\Upomega } } \right)\Updelta t, $$

(10d)

$$ \int\limits_{t}^{t + \Updelta t} {\int\limits_{{\Upomega^{\text{cv}} }} {\rho c{\frac{\partial T}{\partial t}}} d\Upomega \,dt = \left( {T_{i}^{n} - T_{i}^{n - 1} } \right)} \left( {\int\limits_{{\Upomega^{\text{cv}} }} {\rho c} d\Upomega } \right), $$

(10e)

where ng, nf, and nc are the number of surface segments belonging to Γ_int, Γ _q and Γ _c , respectively. δS ^int is an internal surface of sub-control volume. δS ^c _i , and δS ^q _i are the surfaces associated with heat convection, and the heat flux at boundary corresponding to node i, respectively. T ⁿ⁻¹ _i is the known temperature at the last time step t ⁿ⁻¹ of node i, T ⁿ _i is the unknown temperature at current time step t ⁿ of node i, $\left( {{\frac{\partial T}{\partial t}}} \right)^{n - 1} = \left( {{\frac{{T_{i}^{n} - T_{i}^{n - 1} }}{\Updelta t}}} \right)$ and Δt = t ⁿ − t ⁿ⁻¹ is the size of time step. A special case with assuming that there is a uniform internal heat generation rate per unit volume throughout the domain can be used to simplify Eq. 10d. The control-volume finite element approximation of heat equation can be obtained by substituting Eq. 10a through to Eq. 10e into Eq. 6,

$$ \left[ K \right]\left\{ {T^{n} } \right\} = \left\{ f \right\}. $$

(11)

The coefficient matrix [K] depends on the temperature gradients and the material properties. The right-handed side vector {f} depends on the temperature results from the previous time step, specified heat flux, internal heat generation rate and heat convection coefficient. The remaining task is solving the system of equations (Eq. 11) for the nodal values of unknown temperatures T ⁿ at current time step by using direct or iterative method. If the final time step has not reached, the last nodal temperatures are used as the current nodal temperatures for the next time step.

4 Numerical results and discussion

4.1 Assessment on predictive quality of the proposed method

The example used here is the steady heat conduction in a rectangular plate made of homogeneous material with the size of 5 m × 10 m as shown in the Fig. 2. The thermal conductivity of material is 1 W/m °C. The boundary conditions are T(x, 0) = 0, T(0, y) = 0, T(x, 10) = 100 sin (πx/10), and ∂T/∂x = 0 at x = 5. To assess the accuracy of linear and quadratic CVFEM and the effect of mesh refinement versus analytical solution, four meshes are introduced, see Fig. 3.

Fig. 2

The calculation domain and boundary conditions

Fig. 3

The computational mesh of CVFEM a 25 nodes 32 elements (linear), b 81 nodes 128 elements (linear), c 25 nodes 8 elements (quadratic), d 81 nodes 32 elements (quadratic)

The governing equation for this example is

$$ {\frac{{\partial^{2} T}}{{\partial x^{2} }}} + {\frac{{\partial^{2} T}}{{\partial y^{2} }}} = 0 $$

(12)

The exact temperature field is

$$ T\left( {x,y} \right) = 100{\frac{{\sinh \left( {{{\pi y} \mathord{\left/ {\vphantom {{\pi y} {10}}} \right. \kern-\nulldelimiterspace} {10}}} \right)\sin \left( {{{\pi x} \mathord{\left/ {\vphantom {{\pi x} {10}}} \right. \kern-\nulldelimiterspace} {10}}} \right)}}{\sinh \pi }} $$

(13)

The average error of temperature field is evaluated by

$$ E\left( {x,y} \right) = \left[ {{\frac{1}{\text{NIxNJ}}}\sum\limits_{i = 1}^{\text{NIxNJ}} {\left| {{\frac{{T_{Num} - T_{Exact} }}{{T_{Exact} }}}} \right|} } \right] \times 100\% $$

(14)

where T _Num and T _Exact are the numerical and exact values of temperature, respectively. NI and NJ are total numbers of internal nodes in x and y directions, respectively. From Table 1, the accuracy of numerical solution can be greatly improved by increasing the order of interpolation function just the same way as decreasing the element size, see Table 2.

Table 1 Average errors of various mesh sizes and orders of interpolation functions

Table 2 Comparison between exact and numerical solutions inside the calculation domain at corresponding locations

4.2 Problems on FGM material

In this section, four simple heat conduction problems in isotropic FGM are presented in order to validate and illustrate the accuracy of the proposed numerical scheme. In all problems, zero initial temperature is applied.

The first test case is about solving the transient temperature for one-dimensional heat conduction problem. A functionally graded finite square strip with a unidirectional variation of thermal conductivity and thermal diffusivity [9] is concerned. The geometry, boundary conditions and mesh are given in Fig. 4. The thermal conductivity and diffusivity are assumed in the form of exponential functions, given by

$$ k = k_{0} e^{\lambda x} , $$

(15a)

$$ \alpha = {\frac{k}{\rho c}} = \alpha_{0} e^{\lambda x} , $$

(15b)

where k ₀ = 17 W/(m °C), and α ₀ = 0.17 × 10⁻⁴m²/s.

Fig. 4

a Geometry and boundary conditions of the functionally graded finite square strip problem, b triangular mesh

Three values of exponential parameters λ = 0, 0.2, and 0.5 cm⁻¹ are used in this problem to demonstrate the non-homogeneous effect of material properties on the temperature result. In the view of boundary conditions, there is no normal heat flux on two opposite sides parallel to the y-axis while different values of temperature are specified on the two opposite sides parallel to the x-axis. The length of square strip is a = 0.04 m. The left boundary is maintained at zero temperature and the right boundary is imposed by the Heaviside step function of time, i.e., $T = \bar{T} \cdot H\left( t \right)$ with $\bar{T} = 1\;^\circ {\text{C}}$. The analytical solution of the temperature distribution in a FGM square strip at steady state can be obtained as

$$ T = \bar{T}\left[ {{\frac{{e^{ - \lambda x} - 1}}{{e^{ - \lambda a} - 1}}}} \right]. $$

(16)

When the exponential parameter is prescribed to zero, the strip has homogeneous thermal conductivity, k = k ₀ and constant thermal diffusivity, α = α ₀. The analytical solution for this special case [31] can be written as

$$ T = \bar{T}\frac{x}{a} + {\frac{2}{\pi }}\sum\limits_{n = 1}^{\infty } {\left[ {{\frac{{\bar{T}\cos \left( {n\pi } \right)}}{n}}\sin {\frac{n\pi x}{a}}} \right]} \text{e}^{{\left( { - {\frac{{\alpha n^{2} \pi^{2} t}}{{a^{2} }}}} \right)}} . $$

(17)

The accuracy of this proposed method is evaluated by comparing the numerical results with the analytical solution in Eq. 17. All of the results are obtained by using 25 nodes, 4 quadratic triangular elements, and uniform time step Δt = 1 s.

Figure 5 shows the temperature change with time which is numerically and analytically computed at three positions, x = 0.01 m, x = 0.02 m, and x = 0.03 m. Good agreement between the numerical and analytical solutions is noticed in Fig. 5. Figure 6 shows the variation of temperature with time for three λ-values at the position x = 0.02 m. The temperature distribution in material having large exponent coefficient approaches the steady state solution quicker than that in the material having small exponent coefficient. Figure 7 illustrates the distributions of temperatures inside FGM strip under different transient states. From numerical results, it is found that the temperature approaches steady state after t = 20 s.

Fig. 5

CVFEM and analytical solutions for time dependence of temperature in a homogeneous finite square strip (λ = 0) at three locations along x-axis

Fig. 6

Time dependence of temperature at the middle location of functionally graded finite square strip

Fig. 7

Temperature distribution along x-axis of functionally graded finite square strip at a t = 5 s, b t = 10 s, c t = 15 s and d t = 20 s

For the steady state condition, the analytical results from Eq. 16 are compared with the results obtained by the proposed method in Fig. 8. The numerical results are in good agreement with the analytical results. The predictive quality of CVFEM is also compared with conventional finite element method where the thermal conductivity of material is piecewise constant and represented by the value at the centroid of the element. The result suggests that the conventional FEM is not recommended for the solution of highly graded material especially in conjunction with coarse mesh, see Table 3.

Fig. 8

Comparison of CVFEM and analytical solutions for steady temperature distribution along x-axis of functionally graded finite square strip

Table 3 Comparison on predictive quality between conventional FEM and CVFEM

In the second problem, a thick-wall functionally graded cylinder with infinite length is considered, where its inner radii R ₁ is 0.08 m and outer radii R ₂ is 0.1 m. The zero initial temperature is employed in this analysis. The outer surface of the hollow cylinder has the Heaviside step time variation $T = \bar{T} \cdot H\left( t \right)$ with $\bar{T} = 1\,^\circ {\text{C}}.$ The inner surface is maintained at zero temperature. Due to symmetry in geometry and boundary conditions, only a quarter of annulus cross-section is considered. The total number of nodes and elements are 63 and 16, respectively. A uniform time step Δt = 1.0 s is used in this calculation. The geometry, boundary conditions and mesh for the second problem are depicted in Fig. 9. The thermal conductivity and diffusivity are assumed in the form of exponential functions, given by

$$ k = k_{0} e^{{\lambda \left( {r - R_{1} } \right)}} , $$

(18a)

$$ \alpha = {\frac{k}{\rho c}} = \alpha_{0} e^{{\lambda \left( {r - R_{1} } \right)}} , $$

(18b)

where k ₀ = 17 W/(m °C), α₀ = 0.17 × 10⁻⁴ m²/s. λ is a parameter dictating the non-homogeneity within the FGM cylinder. The analytical solution for homogeneous cylinder is given by [31] as follow

$$ T\left( {r,t} \right) = {\frac{{\ln \left( {r/R_{1} } \right)}}{{\ln \left( {R_{2} /R_{1} } \right)}}} - \pi \sum\limits_{n = 1}^{\infty } {{\frac{{J_{0}^{2} \left( {R_{1} \zeta_{n} } \right)U_{0} \left( {r\zeta_{n} } \right)}}{{J_{0}^{2} \left( {R_{1} \zeta_{n} } \right) - J_{0}^{2} \left( {R_{2} \zeta_{n} } \right)}}}} \text{e}^{{\left( { - \alpha \zeta_{n}^{2} t} \right)}} , $$

(19)

where

$$ U_{0} \left( {r\zeta_{n} } \right) = J_{0} \left( {r\zeta_{n} } \right)Y_{0} \left( {R_{2} \zeta_{n} } \right) - J_{0} \left( {R_{2} \zeta_{n} } \right)Y_{0} \left( {r\zeta_{n} } \right), $$

(20)

and ζ _n are roots of the following transcendental equation

$$ J_{0} (R_{1} r)Y_{0} (R_{2} r) - J_{0} (R_{2} r)Y_{0} (R_{1} r) = 0, $$

(21)

with J ₀(x) and Y ₀(x) are the zeroth order Bessel functions of first and second kind.

Fig. 9

a Geometry of the second problem and boundary conditions, b triangular mesh

Some small differences between analytical and numerical solutions are observable in Fig. 10. Figure 11 shows the temperature profiles along the radial locations of homogeneous hollow cylinder at four different time instants. Numerical results for the time variation of temperature at r = 9 cm are illustrated in Fig. 12. Figure 13 gives the temperature profiles along the radial coordinate with various values of λ.

Fig. 10

Time dependence of temperature at three radial positions in a homogenous hollow cylinder

Fig. 11

Temperature variation with the radial distance at five time levels for a homogeneous hollow cylinder

Fig. 12

Temperature variation with respect to time at r = 9 cm in a functionally graded hollow cylinder

Fig. 13

Steady state temperature distribution with radial direction for a functionally graded hollow cylinder

To illustrate the accuracy of the program in response to convective boundary, a rectangular FGM strip subjected to a non-uniform convective boundary condition in Fig. 14 is selected. The initial temperature of the plate is zero. The values of fluid temperature surrounding the bottom and the top edge of the plate are defined by function T _a f _a (x) and zero, respectively. The left and right boundaries are specified with zero normal temperature gradient. The non-dimensional parameters used in the third and fourth problems are defined as follow:

$$ \bar{x} = {x/B}, \quad \bar{L} = {L / B}, \quad \tau = {{\alpha_{o} t} / B}^{2} , \quad \alpha_{0} = {{k_{0} }/( {\rho c} )},\quad H_{a} = {{h_{a} } / {k_{a} }},\quad H_{b} = {{h_{b} } /{k_{b} }}, $$

(22)

where τ represents the non-dimensional time and α ₀ is a typical value of thermal diffusivity. The assigned values for non-dimensional parameters are given by

$$ H_{a} = 1, \quad H_{b} = 1,\quad T_{a} = 1,\quad T_{b} = 0,\quad L = 6,\quad B = 1,\quad k_{0} = 1,\quad \alpha_{0} = 1, $$

(23)

and a function f _a (x) is defined by

Fig. 14

a Geometry of the FGM plate subjected to non-uniform convection boundary conditions, b triangular mesh

$$ \begin{aligned} f_{a} \left( {\bar{x}} \right) & = \left\{ {\begin{array}{*{20}c} {1 - \left[ {\bar{x}^{\prime}/\bar{x}_{0} } \right]^{2} } \hfill & {: - \bar{x}_{0} \le \bar{x}^{\prime} \le \bar{x}_{0} } \hfill \\ 0 \hfill & {:\bar{x}^{\prime} \le - \bar{x}_{0} ,\;\bar{x}_{0} \ge \bar{x}^{\prime}} \hfill \\ \end{array} } \right.. \\ \bar{x}_{0} & = 1,\; \bar{x^{\prime}} = \bar{x} - L/2 \\ \end{aligned} . $$

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The definition of function f _b (x) is not necessary because the temperature of surrounding fluid on the top edge of a plate is prescribed to zero.

The thermal conductivity and diffusivity at any point in the FGM plate are assumed following the exponential functions as

$$ k = k_{0} e^{\lambda y/B} $$

(25a)

$$ \alpha = {\frac{k}{\rho c}} = \alpha_{0} e^{\lambda y/B} $$

(25b)

where λ represents the non-homogeneity parameter for controlling the material gradation through the thickness of the plate. In this problem, the number of equally spaced nodes along the x and y directions are 49 and 11, respectively. Initial temperature of the plate is zero and the time step is 0.01.

Figures 15 and 16 show the plot of temperature variation at the middle of the plate $\left( {\bar{x} = 3} \right)$ when τ = 0.1 and τ = 10. For the purpose of comparison, the analytical results of Ootao and Tanigawa [32] are also shown in both figures. It is notable that the steady state is reached after τ = 4. Good agreement between the analytical and proposed numerical solutions can be seen.

Fig. 15

Temperature change along the y-direction at the position $\bar{x} = 3$ when τ = 0.1

Fig. 16

Steady state temperature change along the y-direction at the position $\bar{x} = 3$

The corresponding temperature distributions on the bottom edge of the rectangular plate with different λ-values and at different time instants are depicted in Fig. 17. The high temperature zone can be noticed in the heated region near the midpoint on the bottom edge of the plate. When the parameter λ is decreased, the temperature value on the heated surface increases. Although this non-homogeneity effect can be clearly noticed at steady state, the temperature distributions of different λ-values in transient state are almost the same.

Fig. 17

Temperature distributions at different time instants on the bottom edge of the plate $\left( {\bar{y} = 0} \right)$

Finally, the third problem is considered again by assuming that the surrounding temperature on the bottom edge of the plate increased exponentially with time; viz., T _a f _a (x)g(τ), where g(τ) = 1 − e ^−γτ. From this definition, γ is a parameter dictating the rate of change in fluid temperature imposed on the bottom edge of the plate. If γ is large, the value of function g(τ) increases abruptly like a unit step function. Figure 18 plots the time histories of temperature at the centroid of the plate for different values of γ, with λ = 1. The results calculated from the present method match well with the results obtained from the meshless local Petrov–Galerkin (MLPG) method [33]. The effect of γ on temperature distributions throughout the thickness of the FGM plate with λ = 1 in a transient state (τ = 1) at the position $\bar{x} = 3$ is shown in Fig. 19.

Fig. 18

Time history of temperature at the centroid of the plate for the time-dependent heat supply with λ = 1

Fig. 19

Temperature distribution throughout the thickness of the rectangular FGM plate for the time-dependent heat supply at time τ = 1 and λ = 1

5 Summary and conclusions

The high order control-volume finite element method has been applied to analyze the transient heat conduction problems occurring in two-dimensional functionally graded solid. The material properties such as thermal conductivity, density, specific heat capacity and thermal diffusivity were assumed to vary exponentially through the thickness direction. In our validation work on homogeneous material, it has been proven that this method gave better accuracy than the first order method under the same mesh size.

Later, the predictive quality of high-order CVFEM versus conventional high-order FEM was investigated by assessing the effect of non-homogeneity of the material properties on thermal response against the analytical result under the same mesh size. The result showed that our method could provide more accurate solution than the counterpart.

The solution with time-dependent boundary condition is also illustrated to show the flexibility of numerical method in solving the problems of interest. The transient temperature solutions of material obtained from the CVFEM method agree well with the solutions from MLPG methods. Our method could also perform well with the solution of heat transfer through a hollow cylinder. It is also recommended for solution of heat transfer with steady and unsteady convective boundary conditions.

The present formulation to be used for analyzing FGMs which their properties described by other grading functions is subjected to further investigation.

Appendix 1

1.1 Shape functions, their derivatives and numerical integration method

The shape functions and their derivatives with respect to ξ and η for 3-noded triangular element are summarized in Table 4. Figure 20 shows the mapping from arbitrary 3-node triangular element to a reference triangular element. The shape functions and their derivatives with respect to ξ and η for 6-noded triangular element are summarized in Table 5. Figure 21 shows the mapping of element of arbitrary 6-node triangular element to a reference triangular element.

Table 4 The shape functions and their derivatives with respect to ξ and η for 3-noded triangular element

Fig. 20

Mapping from general triangle to a reference triangle (3 node element)

Table 5 The shape functions and their derivatives with respect to ξ and η for 6-noded triangular element

Fig. 21

Mapping from general triangle to a reference triangle (6 node element)

In the derivation of the element conductance matrix, it is necessary to establish the shape function derivatives with respect to (x, y) coordinates. By the rule of partial differentiation, we write the derivatives of the shape functions with respect to local coordinates as

$$ {\frac{{\partial N_{i} }}{\partial \xi }} = {\frac{{\partial N_{i} }}{\partial x}}\,{\frac{\partial x}{\partial \xi }} + {\frac{{\partial N_{i} }}{\partial y}}\,{\frac{\partial y}{\partial \xi }} $$

(26)

$$ {\frac{{\partial N_{i} }}{\partial \eta }} = {\frac{{\partial N_{i} }}{\partial x}}\,{\frac{\partial x}{\partial \eta }} + {\frac{{\partial N_{i} }}{\partial y}}\,{\frac{\partial y}{\partial \eta }} $$

(27)

in the algebraic form, or

$$ \left\{ {\begin{array}{*{20}c} {{\frac{{\partial N_{i} }}{\partial \xi }}} \\ {{\frac{{\partial N_{i} }}{\partial \eta }}} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{\frac{\partial x}{\partial \xi }}} & {{\frac{\partial y}{\partial \xi }}} \\ {{\frac{\partial x}{\partial \eta }}} & {{\frac{\partial y}{\partial \eta }}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {{\frac{{\partial N_{i} }}{\partial x}}} \\ {{\frac{{\partial N_{i} }}{\partial y}}} \\ \end{array} } \right\} = \left[ J \right]\left\{ {\begin{array}{*{20}c} {{\frac{{\partial N_{i} }}{\partial x}}} \\ {{\frac{{\partial N_{i} }}{\partial y}}} \\ \end{array} } \right\} $$

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in the matrix form. The matrix [J] is known as the Jacobian matrix. The Jacobian matrix components can be found explicitly in terms of the local coordinates as follows,

$$ {\frac{\partial x}{\partial \xi }} = \sum\limits_{i = 1}^{\text{npe}} {{\frac{{\partial N_{i} }}{\partial \xi }}x_{i} } ,\quad {\frac{\partial y}{\partial \xi }} = \sum\limits_{i = 1}^{\text{npe}} {{\frac{{\partial N_{i} }}{\partial \xi }}y_{i} } ,\quad {\frac{\partial x}{\partial \eta }} = \sum\limits_{i = 1}^{\text{npe}} {{\frac{{\partial N_{i} }}{\partial \eta }}x_{i} } ,{\frac{\partial y}{\partial \eta }} = \sum\limits_{i = 1}^{\text{npe}} {{\frac{{\partial N_{i} }}{\partial \eta }}y_{i} } , $$

(29)

where npe refers to the number of nodes per element.

Therefore, the required derivatives ${\frac{{\partial N_{i} }}{\partial x}}$ and ${\frac{{\partial N_{i} }}{\partial y}}$ can be obtained by

$$ \left\{ {\begin{array}{*{20}c} {{\frac{{\partial N_{i} }}{\partial x}}} \\ {{\frac{{\partial N_{i} }}{\partial y}}} \\ \end{array} } \right\} = \left[ J \right]^{ - 1} \left\{ {\begin{array}{*{20}c} {{\frac{{\partial N_{i} }}{\partial \xi }}} \\ {{\frac{{\partial N_{i} }}{\partial \eta }}} \\ \end{array} } \right\} $$

(30)

The discretized equation for heat conduction at steady state of the control volume belonging to node i is

$$ \sum\limits_{k = 1}^{\text{ns}} {\left\{ {\left( {k{\frac{\partial T}{\partial x}}} \right)n_{x} + \left( {k{\frac{\partial T}{\partial y}}} \right)n_{y} } \right\}_{k} } \Updelta A_{k} + \bar{S}_{I} \Updelta V_{I} = 0 $$

(31)

where ΔA _k represents the area of face k of the control volume covering node i; ns is the total number of faces; ΔV _I is the volume of the control volume belonging to node i; $\bar{S}_{I}$is the average heat source over the control volume; n _x and n _y are the direction cosines of a surface. For example, n _x and n _y of the face S ₁ in Fig. 22a, later referred to as S _oa , are defined as

$$ n_{x} = {\frac{{y_{o} - y_{a} }}{\Updelta A}},\quad n_{y} = {\frac{{x_{a} - x_{o} }}{\Updelta A}}, $$

(32)

where $\Updelta A = \sqrt {\left( {x_{o} - x_{a} } \right)^{2} + \left( {y_{o} - y_{a} } \right)^{2} } .$

Fig. 22

Integration points along the faces surrounding a node 1, b node 2, c node 3

Equation 31 may be rewritten in matrix form for a triangular element that relates to node i as

$$ \left[ K \right]\left\{ {T^{e} } \right\} = \left\{ f \right\} $$

(33)

where [K] is the element conductance matrix; {T ^e} is the vector containing nodal temperature values in the element; {f} is the element load vector.

1.2 The element conductance matrix and load vector

For brevity in the following expressions, the definition of Δy _mn  = y _m  − y _n and Δx _mn  = x _m  − x _n are introduced. Ω is defined as the area of an element.

3-node element:

The components of element conductance matrix and element load vector are

$$ \begin{aligned} K_{1,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{oa} }} \Updelta y_{oa} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{oa} }} \Updelta x_{oa} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{co} }} \Updelta y_{co} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{co} }} \Updelta x_{co} , \\ K_{2,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{ob} }} \Updelta y_{ob} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{ob} }} \Updelta x_{ob} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{ao} }} \Updelta y_{ao} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{ao} }} \Updelta x_{ao} , \\ K_{3,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{oc} }} \Updelta y_{oc} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{oc} }} \Updelta x_{oc} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{bo} }} \Updelta y_{bo} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{bo} }} \Updelta x_{bo} , \\ f_{1} & = - {\frac{{\bar{S}_{1} \Upomega }}{3}},f_{2} = - {\frac{{\bar{S}_{2} \Upomega }}{3}},f_{3} = - {\frac{{\bar{S}_{3} \Upomega }}{3}}. \\ \end{aligned} $$

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6-node element:

The components of element conductance matrix and element load vector are

$$ \begin{aligned} K_{1,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{ae} }} \Updelta y_{ae} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{ae} }} \Updelta x_{ae} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{fa} }} \Updelta y_{fa} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{fa} }} \Updelta x_{fa} , \\ K_{2,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{bg} }} \Updelta y_{bg} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{bg} }} \Updelta x_{bg} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{hb} }} \Updelta y_{hb} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{hb} }} \Updelta x_{hb} , \\ K_{3,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{id} }} \Updelta y_{id} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{id} }} \Updelta x_{id} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{dj} }} \Updelta y_{dj} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{dj} }} \Updelta x_{dj} , \\ K_{4,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{bh} }} \Updelta y_{bh} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{bh} }} \Updelta x_{bh} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{cb} }} \Updelta y_{cb} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{cb} }} \Updelta x_{cb} \\ & \quad \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{ac} }} \Updelta y_{ac} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{ac} }} \Updelta x_{ac} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{ea} }} \Updelta y_{ea} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{ea} }} \Updelta x_{ea} , \\ K_{5,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{dj} }} \Updelta y_{dj} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{dj} }} \Updelta x_{dj} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{cd} }} \Updelta y_{cd} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{cd} }} \Updelta x_{cd} \\ & \quad \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{bc} }} \Updelta y_{bc} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{bc} }} \Updelta x_{bc} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{gb} }} \Updelta y_{gb} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{gb} }} \Updelta x_{gb} , \\ K_{6,j} & = \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{af} }} \Updelta y_{af} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{af} }} \Updelta x_{af} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{ca} }} \Updelta y_{ca} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{ca} }} \Updelta x_{ca} \\ & \quad \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{dc} }} \Updelta y_{dc} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{dc} }} \Updelta x_{dc} + \left( {k{\frac{{\partial N_{j} }}{\partial x}}} \right)_{{S_{id} }} \Updelta y_{id} - \left( {k{\frac{{\partial N_{j} }}{\partial y}}} \right)_{{S_{id} }} \Updelta x_{id} , \\ f_{1} & = - {\frac{{\bar{S}_{1} \Upomega }}{12}},\quad f_{2} = - {\frac{{\bar{S}_{2} \Upomega }}{12}},\quad f_{3} = - {\frac{{\bar{S}_{3} \Upomega }}{12}}, \quad f_{4} = - {\frac{{\bar{S}_{4} \Upomega }}{4}},\quad f_{5} = - {\frac{{\bar{S}_{5} \Upomega }}{4}},\quad f_{6} = - {\frac{{\bar{S}_{6} \Upomega }}{4}}. \\ \end{aligned} $$

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1.3 The calculation method for integrals of an element

The integration points on the faces surrounding node i for midpoint rule integration over the faces within a 3-noded triangular element are depicted in Fig. 22 and their locations are summarized in Table 6. The integration points of midpoint rule on the faces surrounding node i within a 6-noded triangular element are depicted in Fig. 23 and their locations are summarized in Table 7.

Table 6 Integration points for 3-node element (midpoint rule)

Fig. 23

Integration points along the faces surrounding a node 1, b node 2, c node 3, d node 4, e node 5, f node 6

Table 7 Integration points for 6-node element (midpoint rule)

The surface integral can be calculated by the midpoint rule which its value equals to the product of the functional values at the center of surface and the total surface area. The locations of integration points are illustrated in Fig. 24.

Fig. 24

a Integration point for midpoint rule. b Integration points for 2-point Gauss-Quadrature rule

The Gauss quadrature rule is used to approximate the integral value of a specified function. It is a sum of the products between the weighting factors and the corresponding function values at the gauss points. The integration points for mid-point rule and 2-point Gauss quadrature rule are calculated by

$$ \left\{ {\begin{array}{*{20}c} \xi \\ \eta \\ \end{array} } \right\} = \frac{1}{2}\left( {1 - t} \right)\left\{ {\begin{array}{*{20}c} {\xi_{b} } \\ {\eta_{b} } \\ \end{array} } \right\} + \frac{1}{2}\left( {1 + t} \right)\left\{ {\begin{array}{*{20}c} {\xi_{a} } \\ {\eta_{a} } \\ \end{array} } \right\},\quad - 1 \le t \le 1 $$

(36)

where

• t = 0 for midpoint rule.

• $t = \left\{ { + {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-\nulldelimiterspace} {\sqrt 3 }}, - {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-\nulldelimiterspace} {\sqrt 3 }}} \right\}$ for 2-point Gauss quadrature rule.

The weighting factors for midpoint rule and 2-point Gauss quadrature rule are given by

• w_ip1 = 2 for midpoint rule.

• w_ip1 = w_ip2 = 1 for 2-point Gauss quadrature rule.

The line integral of F(S) over the finite length of ΔA can be simplified to

• $\int\limits_{0}^{\Updelta A} {F\left( S \right)dS \approx \left( {{\frac{\Updelta A}{2}}} \right)\left( 2 \right)\left( {f\left( 0 \right)} \right) = f\left( 0 \right)\Updelta A}$ for midpoint rule,

• $\int\limits_{0}^{\Updelta A} {F\left( S \right)dS} \approx \left( {{\frac{\Updelta A}{2}}} \right)\left( {f\left( { + {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-\nulldelimiterspace} {\sqrt 3 }}} \right) + f\left( {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\sqrt 3 }}} \right. \kern-\nulldelimiterspace} {\sqrt 3 }}} \right)} \right)$ for 2-point Gauss quadrature rule.

The integration points of 2-point Gauss quadrature rule on the faces surrounding node i within a 6-noded triangular element are summarized in Table 8.

Table 8 Integration points for 6-node element (2-point Gauss quadrature rule)