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The solution of twodimensional steady and transient fluid flow problem by the truly meshless local PetrovGalerkin (MLPG) method has been addressed in the present article. The unknown function of velocity u(x) is approximated by moving least square approximant u^{h}(x). The essential boundary condition is imposed both by the direct and penalty function methods. Fourth order spline weight function, monomial basis function and a set of nonconstant coefficients are used to construct the approximants. The twolevel q method is employed for temporal discretization. The results obtained by the MLPG method are compared with the analytical solution and also with the benchmark method results and found to be in the excellent agreement.
steadystate analysis, transient analysis, incompressible fluid flow, direct method of interpolation, penalty function method, meshless local petrovgalerkin method
Finite element method (FEM) has been established as a powerful numerical technique to address the complex problems of different engineering domains; still it is encountered with certain limitations including the mesh creation, which is a costly and time consuming task than the assembly and solution of the FE equations. Also, there are certain class of problems for which the FEM is not a suitable technique to address. To overcome these problems, a large number of meshless methods have been evolved in the recent years. Some of these methods are: smoothed particle hydrodynamics (SPH) method [13], diffuse element method [4], local boundary integral equation (LBIE) method [56], element free Galerkin (EFG) method [78], finite point method (FPM) [9], MLPG method [1012], point interpolation method (PIM) [13], radial point interpolation method (RPIM) [1416] and meshfree weakstrong form (MWSF) methods [17], etc.
However, among all the meshfree methods, the MLPG method has become quite popular due to its successful acceptability in various fields of engineering [1823]. In this method the domain discretization originates from a weak form over a local subdomain of arbitrary shape which is located completely inside the global domain. It constructs the global stiffness matrix through the integration over local subdomains, naturally. Integration over local domains removes the need of any background mesh over entire domain. Thus, the proposed method does not need any mesh either at the stage of interpolation or at the stage of integration; hence, it can appropriately be called a truly meshfree method.
The MLPG method works on PetrovGalerkin formulation i.e. it picks up test and trial functions from different function spaces. This feature provides the flexibility to formulate the discretization method in different ways. However, there is a strong base of available literature [2431] that proves that the MLPG method is one of the promising methods to solve complex engineering problems including fluid flow problems efficiently and effectively.
In the present article, the MLPG method is used to solve the steady and transient fluid flow problems in twodimensional space. Direct method of interpolation (DM) and penalty function methods (PM) [32] are used to enforce the essential boundary conditions (EBC). C^{++} codes have been developed to model the problem. The results obtained by the MLPG method are compared with that of the analytical and benchmark solutions.
The MLPG method utilizes the MLS scheme [33], which consists of three components: a completely monomial basis function, a set of coefficients that depends on node position and a weight function associated with each node. The unknown function u(x) is approximated by MLS approximant u^{h}(x) as,
${{u}^{h}}(\mathbf{x})=\sum\limits_{j=1}^{m}{{{p}_{j}}(\mathbf{x}){{a}_{j}}(\mathbf{x})}\equiv {{\mathbf{p}}^{T}}(\mathbf{x})\mathbf{a}(\mathbf{x})$ (1)
where p^{T}(x) = (p_{1 }(x), p_{2 }(x),……, p_{m }(x)) is a complete monomial basis. The coefficient vector a(x) is determined by minimizing a weighted discrete L_{2}norm defined as,
$J=\sum\limits_{i=1}^{n}{w}(\mathbf{x},{{\mathbf{x}}_{i}}){{[{{u}^{h}}(\mathbf{x})u({{\mathbf{x}}_{i}})]}^{2}}=\sum\limits_{i=1}^{n}{w}(\mathbf{x},{{\mathbf{x}}_{i}}){{[{{\mathbf{p}}^{\mathbf{T}}}\mathbf{(}{{\mathbf{x}}_{\mathbf{i}}}\mathbf{)a(x)}u({{\mathbf{x}}_{i}})]}^{2}}$ (2)
where w(x, x_{i}) is a weight function, u(x_{i}) = u_{i}is the nodal parameter of the field variable at node x_{i} and n is the number of nodes in the support domain of x for which the weight function w(x,x_{i})≠0. The stationarity of J with respect to a(x) results in the following linear system:
$\mathbf{A(x)a(x)=B(x)u}$ (3)
The above equation can be written as,
$\mathbf{a}(\mathbf{x})={{\mathbf{A}}^{1}}\mathbf{(x)B(x)u}$ (4)
where matrices A(x) and B(x) are defined as,
$\mathbf{A(x)}=\sum\limits_{i=1}^{n}{w}(\mathbf{x},{{\mathbf{x}}_{i}})\mathbf{p(}{{\mathbf{x}}_{\mathbf{i}}}\mathbf{)}{{\mathbf{p}}^{\mathbf{T}}}\mathbf{(}{{\mathbf{x}}_{\mathbf{i}}}\mathbf{)}$ (5)
$\begin{align} & \mathbf{B(x)}=[w(\mathbf{x},{{\mathbf{x}}_{1}})\mathbf{p(}{{\mathbf{x}}_{1}}\mathbf{)},w(\mathbf{x},{{\mathbf{x}}_{2}})\mathbf{p(}{{\mathbf{x}}_{2}}\mathbf{)},... \\ & ....,w(\mathbf{x},{{\mathbf{x}}_{n}})\mathbf{p(}{{\mathbf{x}}_{n}}\mathbf{)}] \\\end{align}$ (6)
Substituting a(x) in Eqn. (1), the MLS approximant is obtained as,
${{u}^{h}}(\mathbf{x})=\sum\limits_{i=1}^{n}{{{\Phi }_{i}}{{u}_{i}}}=\mathbf{\Phi (x)}u$ (7)
The choice of weight function affects the resulting approximation u^{h}(x). The weight function w(x, x_{i}) is nonzero over a small domain in the neighbourhood of node x_{i}. Therefore, the selection of appropriate weight function is essential in the MLPG method. The fourth order spline weight function [34] is used in the present analysis and can be written as,
$w(\mathbf{x},{{\mathbf{x}}_{\mathbf{i}}})=\left\{ \begin{matrix} 16{{d}^{2}}+8{{d}^{3}}3{{d}^{4}} & \text{if }0\le d\le 1 \\ 0 & \text{if }d>1 \\ \end{matrix} \right.$ (8)
where $d_{i}=\left\xx_{i}\right\$, i.e. the Euclidean distance from node x_{i} to point x.
MLS shape functions do not possess Kronecker delta function property; hence, imposition of EBC is not an easy task.
An incompressible fluid flowing through a long and uniform duct is considered in Fig. 1.
Figure 1. Model crosssection of the fluid flowing through a duct
The momentum equation for incompressible fluid flowing through a long and uniform duct is given as,
$\mu (\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}})\frac{\partial p}{\partial z}=\rho \frac{\partial u}{\partial t}$ (9)
The initial and essential boundary conditions are,
$\left. \begin{align} & at\text{ }t=\text{ }0,u={{u}_{ini}} \\ & at\text{ }{{\Gamma }_{1}},\text{ }{{u}_{top\text{ }surface}}=\text{ }{{u}_{{{\Gamma }_{1}}}}=\text{ }0 \\ & at\text{ }{{\Gamma }_{2}},\text{ }{{u}_{left\text{ }surface}}=\text{ }{{u}_{{{\Gamma }_{2}}}}=\text{ }0 \\ & at\text{ }{{\Gamma }_{3}},\text{ }{{u}_{bottom\text{ }surface}}=\text{ }{{u}_{{{\Gamma }_{3}}}}=\text{ }0 \\ & at\text{ }{{\Gamma }_{4}},\text{ }{{u}_{right\text{ }surface}}=\text{ }{{u}_{{{\Gamma }_{4}}}}=\text{ }0 \\\end{align} \right\}$ (10)
The weighted residual formation for Eqn. (9) in local domain Ω_{Q} can be expressed as,
$\int\limits_{{{\Omega }_{Q}}}{v\left\{ \mu (\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}})\frac{\partial p}{\partial z}\rho \frac{\partial u}{\partial t} \right\}}d\Omega =0$ (11)
The weak form of the Eqn. (11) is obtained as,
$\int_{\Gamma }{v}\mu (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y})d\Omega \int\limits_{{{\Omega }_{Q}}}{\left\{ \mu (\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}) \right\}}d\Omega \int\limits_{{{\Omega }_{Q}}}{\left\{ v\frac{\partial p}{\partial z} \right\}}d\Omega \int\limits_{{{\Omega }_{Q}}}{\left\{ \rho v\frac{\partial u}{\partial t} \right\}}d\Omega +\int_{\Gamma }{\alpha (u\overline{u})}d\Omega =0$ (12)
3.1 Enforcement of boundary conditions by the Penalty function method (PM)
The penalty parameter can be expressed by α and is equal to 1 x 10^{10}.
$\int_{{{\Gamma }_{1}}}{v}\mu (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y})d\Omega +\int_{{{\Gamma }_{2}}}{v}\mu (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y})d\Omega +\int_{{{\Gamma }_{3}}}{v}\mu (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y})d\Omega +\int_{{{\Gamma }_{4}}}{v}\mu (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y})d\Omega \int\limits_{{{\Omega }_{Q}}}{\left\{ \mu (\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}) \right\}}d\Omega \int\limits_{{{\Omega }_{Q}}}{\left\{ v\frac{\partial p}{\partial z} \right\}}d\Omega $
$\int\limits_{{{\Omega }_{Q}}}{\left\{ \rho v\frac{\partial u}{\partial t} \right\}}d\Omega +\alpha \int\limits_{{{\Gamma }_{1}}}{(u{{u}_{{{\Gamma }_{1}}}}})vd\Gamma +\alpha \int\limits_{{{\Gamma }_{2}}}{(u{{u}_{{{\Gamma }_{2}}}}})vd\Gamma +\alpha \int\limits_{{{\Gamma }_{3}}}{(u{{u}_{{{\Gamma }_{3}}}}})vd\Gamma +\alpha \int\limits_{{{\Gamma }_{4}}}{(u{{u}_{{{\Gamma }_{4}}}}})vd\Gamma =0$ (13)
The unknown function, u, at any instant of time t, is approximated by MLS scheme as follows:
${{u}^{h}}(\mathbf{x})=\sum\limits_{i=1}^{{{n}_{s}}}{{{\Phi }_{i}}}{{u}_{i}}=\mathbf{\Phi }u$ (14)
The following relations are obtained:
$\mathbf{C}u'+\mathbf{K}u=\mathbf{F}$ (15)
${{K}_{ij}}=\int_{{{\Gamma }_{1}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega +\int_{{{\Gamma }_{2}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega +\int_{{{\Gamma }_{3}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega +\int_{{{\Gamma }_{4}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega \int\limits_{{{\Omega }_{Q}}}{\left\{ \mu (\frac{\partial {{\phi }_{j}}}{\partial x}\frac{\partial {{v}_{i}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y}\frac{\partial {{v}_{i}}}{\partial y}) \right\}}d\Omega $
$+\int\limits_{{{\Gamma }_{1}}}{\alpha {{v}_{i}}{{\phi }_{j}}}d{{\Gamma }_{1}}+\int\limits_{{{\Gamma }_{2}}}{\alpha {{v}_{i}}{{\phi }_{j}}}d{{\Gamma }_{2}}+\int\limits_{{{\Gamma }_{3}}}{\alpha {{v}_{i}}{{\phi }_{j}}}d{{\Gamma }_{3}}+\int\limits_{{{\Gamma }_{4}}}{\alpha {{v}_{i}}{{\phi }_{j}}}d{{\Gamma }_{4}}$ (16)
${{C}_{ij}}=\int\limits_{{{\Omega }_{Q}}}{\rho {{v}_{i}}{{\Phi }_{j}}\text{d}\Omega }$ (17)
${{F}_{i}}=\int\limits_{{{\Omega }_{Q}}}{\left\{ {{v}_{i}}\frac{\partial p}{\partial z} \right\}}d\Omega +\int\limits_{{{\Gamma }_{1}}}{\alpha {{v}_{i}}{{u}_{{{\Gamma }_{_{1}}}}}}d{{\Gamma }_{1}}+\int\limits_{{{\Gamma }_{2}}}{\alpha {{v}_{i}}{{u}_{{{\Gamma }_{2}}}}}d{{\Gamma }_{2}}+\int\limits_{{{\Gamma }_{3}}}{\alpha {{v}_{i}}{{u}_{{{\Gamma }_{3}}}}}d{{\Gamma }_{3}}+\int\limits_{{{\Gamma }_{4}}}{\alpha {{v}_{i}}{{u}_{{{\Gamma }_{4}}}}}d{{\Gamma }_{4}}$ (18)
3.2 Enforcement of boundary conditions by the direct method of interpolation (DM)
${{K}_{ij}}=\left\{ \begin{matrix} {{\Phi }_{j}} & \text{if }{{x}_{i}}\in \Gamma \\ \left[ \begin{align} & \int_{{{\Gamma }_{1}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega +\int_{{{\Gamma }_{2}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega + \\ & \int_{{{\Gamma }_{3}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega +\int_{{{\Gamma }_{4}}}{{{v}_{i}}}\mu (\frac{\partial {{\phi }_{j}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y})d\Omega  \\ & \int\limits_{{{\Omega }_{Q}}}{\left\{ \mu (\frac{\partial {{\phi }_{j}}}{\partial x}\frac{\partial {{v}_{i}}}{\partial x}+\frac{\partial {{\phi }_{j}}}{\partial y}\frac{\partial {{v}_{i}}}{\partial y}) \right\}}d\Omega \\\end{align} \right] & {{x}_{i}}\notin \Gamma \\\end{matrix} \right.$ (19)
${{C}_{ij}}=\left\{ \begin{matrix} \int\limits_{{{\Omega }_{Q}}}{\rho {{v}_{i}}{{\Phi }_{j}}\text{d}\Omega } & \text{ if }{{x}_{i}}\notin \Gamma \\ 0 & \text{ otherwise} \\\end{matrix} \right.$ (20)
${{F}_{i}}=\left\{ \begin{align} & \overline{\text{u}}\text{ if }{{x}_{i}}\in \Gamma \text{ } \\ & \int\limits_{{{\Omega }_{Q}}}{\left\{ {{v}_{i}}\frac{\partial p}{\partial z} \right\}}d\Omega \text{ }otherwise \\\end{align} \right.$ (21)
Temporal discretization has been addressed by twolevel q method [35] as,
$[\mathbf{C}+\theta \Delta t\mathbf{K}]{{\mathbf{T}}^{{{n}^{'}}+1}}=[\mathbf{C}+(\theta 1)\Delta t\mathbf{K}]{{\mathbf{T}}^{n}}^{'}+\Delta t\mathbf{F}$ (22)
Where, n' denotes the time level.
In this section, viscous incompressible fluid flowing through a long and uniform duct is simulated by the MLPG method. Both, steadystate and transient analysis are performed. The fluid properties such as viscosity and density are assumed to be constant.
The model and its data used for the steadystate and transient analysis of fluid flowing through a long duct are shown in Table 1 and Table 1 respectively.
Table 1. Data for twodimensional steadystate and transient fluid flow
Parameter 
Value of the Parameter 
D 
0.10 m 
H 
0.10 m 
∂p/∂z 
4000 N/m^{2}/m 
ρ 
1000 kg/m^{3} 
μ 
2.5 Ns/m^{2} 
u_{s} 
0.0 m/s 
u_{in}_{i} 
0.0 m/s 
∆t 
0.01 s 
The accuracy of the results in twodimensional analysis, obtained by the MLPG method, largely depends upon the values of parameters likeorder of basis function (m), number of internal nodes required in the support domain for the interpolation of point of interest (N_{SN}), number of nodes required on the boundary of support domain for the interpolation of point of interest (N_{SB}) and number of nodes required to calculate average nodal spacing (N_{L}); so, there is a strong need to identify the optimum range of these parameters before initialization.
According to Liu and Gu [34], for twodimensional cases, the average nodal spacing can be calculated by,
${{N}_{L}}=\frac{\sqrt{{{A}_{S}}}}{\sqrt{{{n}_{{{A}_{S}}}}}1}$ (23)
where, A_{S} is the area of the estimated support domain; n_{AS} is the number of nodes covered by the estimated domain with the area of A_{S}.
4.1 Steadystate analysis
A study has been conducted for the steadystate problem to identify the optimum range of parameters at the centre of duct (x= H/2, y= D/2) by calculating the relative errors with respect to the analytical solution [8].
The optimum range of parameters for steadystate analysis has been identified as 7 for N_{SN}, 7 for N_{SB} and 6 for N_{L} respectively at 121 and 324 nodes for m=3.
Figure 2. Variation of relative error for N_{SN} by taking PM and DM for the enforcement of EBC
Figure 3. Variation of relative error for N_{SB} by taking PM and DM for the enforcement of EBC
Figure 4. Variation of relative error for N_{L} by taking PM and DM for the enforcement of EBC
Table 2. Comparison of MLPG results with analytical solution under steadystate conditions at x= H/2, y= D/2 of the duct crosssection
S. No. 
Number of nodes 
Analytical solution result 
MLPG results 

v, m/s 
% Error 

PM 

1 
121 
1.1802 
1.1801 
0.0085 
2 
324 
1.1972 
1.4404 

DM 

1 
121 
1.1802 
1.1822 
0.1695 
2 
324 
1.1975 
1.4659 
Table 2 shows an error estimate for the MLPG method at the centre of the duct crosssection. The maximum error obtained by the PM and DM has been found to be 1.44% and 1.46% respectively at 324 nodes. However, this error is 0.0085% and 0.1695% respectively at 121 nodes; hence, the results obtained by the MLPG method at 121 nodes are more accurate than 324 nodes. Also, the PM is found to be more accurate than DM.
4.2 Transient analysis
The analysis of transient model has been carried out in this section by the MLPG method by using the PM and DM at 121 and 324 nodes respectively. A study to identify the optimized range of parameters have been conducted and explored as m=3, N_{SN}= 7, N_{SB}=7 and N_{L}=6 for 121 nodes and m=3, N_{SN}= 7, N_{SB}=7 and N_{L}=5 for 324 nodes respectively.
Figure 5. Transient twodimensional analysis of fluid flow problem at x= H/4, y= D/4
Figure 6. Velocity of fluid at x/H = 0.0125 obtained by the MLPG method using PM
Figure 7. Velocity of fluid at x/H = 0.025 obtained by the MLPG method using PM
Figure 8. Velocity of fluid at x/H = 0.050 obtained by the MLPG method using PM
Figure 9. Velocity of fluid at x/H = 0.0125 obtained by the MLPG method using DM
Figure 10. Velocity of fluid at x/H = 0.025 obtained by the MLPG method using DM
Figure 11. Velocity of fluid at x/H = 0.050 obtained by the MLPG method using DM
A comparison of the MLPG results and the benchmark method results [8] at x= H/4, y= D/4 for 121 and 324 node numbers respectively are demonstrated in Fig. 5. The results obtained by the MLPG results are found to be in good agreement with the benchmark method results; also, the figure demonstrates the superiority of the PM over DM.
Figs. 68 show the progression of fluid velocity with time at different locations along the crosssection of the duct till the attainment of steadystate by the MLPG method for 121 and 324 nodes respectively.
The EBC is imposed by the PM. The velocity is found to be least near the walls of the duct and most at the centre of the crosssection i.e at x/H = 0.050.
Figs. 911 show the progression of fluid velocity with time at different locations along the crosssection of the duct till the attainment of steadystate by the MLPG method for 121 and 324 nodes respectively. The EBC is imposed by the DM. The velocity is found to be least near the walls of the duct and most at the centre of the crosssection i.e at x/H = 0.050.
It is because of the boundary conditions imposed on the four boundaries of the duct crosssection (refer Eqn. 10). At and near the surfaces the velocity of fluid is assumed to be negligible; however, the fluid velocity apart from the surface gradually increases and reaches to maximum at the centre due to no or less effect of boundary conditions. It can also be observed that after certain period of time the timeprofiles attain the steadystate.
The MLPG method, due to its flexibility in formulation is employed to investigate the fluid flow problems. Careful selection of parameters yield the accurate results. In the steadystate and transient twodimensional fluid flow problem the EBC is imposed both by the DM and PM respectively. Twolevel q method has been employed in both the problems for temporal discretization. The fourth order spline weight function has been used to interpolate the test function. C^{++} codes have been developed to model the sample problems. The results obtained by the model problems have been compared with that of the analytical and benchmark solutions and found to be in very good agreement. The fluid velocity is found to be highest at the centre of the crosssection of the duct. Finally, it can be concluded that the MLPG method can be a good choice to solve the fluid flow though a long and uniform duct with meshing and remeshing of computational domain.
A 
Crosssectional area of duct, m^{2} 
a(x) 
Vector of coefficients 
A(x) 
Weighted momentum matrix 
D 
Bredth of duct, m 
F 
Load vector 
H 
Length of duct, m 
K 
Stiffness matrix 
p(x) 
Basis function 
∆t 
Time step, s 
u_{f} 
Fluid velocity, m/s 
u_{s} 
Fluid velocity along the surface of duct, m/s 
u_{i} 
Nodal parameter of u at x=x_{i} 
u_{ini} 
Initial velocity of fluid, m/s 
u(x) 
Unknown scalar function of a field variable 
v 
Test function for MLPG method 
w 
Weight function 
w_{i} 
Gauss weights 
x 
Coordinate in x direction 
y 
Coordinate in y direction 
Greek symbols 

a 
Penalty parameter 
b 
thermal expansion coefficient, K^{1} 
Γ 
Boundary of global domain 
Γ_{1} 
Boundary of the top surface of the duct 
Γ_{2} 
Boundary of the left surface of the duct 
Γ_{3} 
Boundary of the bottom surface of the duct 
Γ_{3} 
Boundary of the right surface of the duct 
μ 
Dynamic viscosity, Ns/m^{2} 
ρ 
Density of material, kg/m^{3} 
Φ 
MLS shape function 
Ω 
Domain 
Ω_{Q} 
Local domain 
∂Ω_{Q} 
Boundary of local domain 
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