Key words: mesh free; whole derivative; Cartesian frame. Chinese ... in the subdomain (d) derive the value and derivatives value of the aimed node, and then.
Applied Mathematics and Mechanics (English Edition), 2006, 27(1):1–6 c Editorial Committee of Appl. Math. Mech., ISSN 0253-4827
MESH FREE METHOD BASED ON LOCAL CARTESIAN FRAME LIU Gao-lian(
),
∗
)
LI Xiao-wei(
(Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P.R.China) (Contributed by LIU Gao-lian)
Abstract: A new mesh free method proposed by the authors was presented, in which the derivatives at each node were constructed using whole derivative formulas through the nodes selected around the node using local Cartesian frame in an autonomous manner, so that without any element it could be considered as a completely mesh free method. The method was tested with a numerical example, and reliable solution was obtained with high accuracy and efficiency. Key words: mesh free; whole derivative; Cartesian frame Chinese Library Classification: O357.5 2000 Mathematics Subject Classification: 76S05;74K99 Digital Object Identifier (DOI):10.1007/s 10483-006-0101-1
Introduction Many problems, like crack propagation, are characterized by a continuous change in the geometry of the domain under consideration. It is troublesome and expensive to use conventional method( such as finite element method or finite difference method) to analyze these problems because of the continuous remeshing of the domain to avoid the breakdown of the calculation due to excessive mesh distortion. So, in order to effectively and efficiently solve these problems, in recent years, researchers have mainly focused on the development of socalled meshless (or mesh free) method, computation of which is done on a node-by-node basis without the creation of finite elements, and a number of methods have been recently proposed in the literature accordingly[1−8]. The methodology of the meshless (or mesh free) method consists of following steps: (a) deploy nodes arbitrarily in the domain under analysis, (b) select some nodes from the domain around each aimed node, (c) construct a approximation function of the unknown parameter in the subdomain (d) derive the value and derivatives value of the aimed node, and then (e)solve the governing equations. The step (b) and step (c) are very important, which usually determine the complexity of the operation and the accuracy of the numerical results. Most of the methods presented in the references implement the approach circling a subdomain around each aimed nodes in a prescribed radius and constructing approximation function through the values of all the nodes in the subdomain. As known, it is very difficult to set suitable radiuses during circling subdomain if the nodes are deployed promiscuously and not easy to construct approximation functions in a generalized form because of the variety of the number of nodes in different subdomain. Therefore, the authors are motivated by above deficiency to explore and develop an other approach in this researching area. ∗ Received Nov.23,2004; Revised Sep.21,2005 Project supported by the National Natural Science Foundation of China (No.10372055) and the Shanghai Leading Academic Discipline Project (No.Y0103) Corresponding author LI Xiao-wei, Associate Professor, Doctor, E-mail: xwli@staff.shu.edu.cn
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LIU Gao-lian and LI Xiao-wei
In this paper, a new mesh free method is presented in which the derivatives at each aimed node are constructed using whole derivative formulas through the values of the nodes selected around the aimed node using local Cartesian frame in an autonomous manner. Elements are completely ignored in this method, and the techniques can be carried out simply in a generalized form, so the shortcomings in the previous methods can be overcome . The method has been tested with a numerical example, and reliable solution was obtained with high accuracy and efficiency.
1
Discretization Methodology
In order to show the discretization methodology, we take the following Poisson equation as a sample, ∂2φ ∂2φ + 2 = Q(x, y). (1) ∂x2 ∂y In the mesh free method, it is necessary to approximate the value and the derivative values of one node using the values of its nearest nodes. So an important task is to select nodes for each aimed node, and we will describe the process of selecting nodes in the current approach. Firstly, the nodes are arbitrarily deployed in the domain under analysis . Then we select node O as a aimed node at which the derivatives will be constructed, and introduce a Cartesian frame Oxy whose origin is superposed on the node O, shown in Fig.1. The node 1 is selected if it is in the forth quadrant and is closest to the x-axis and closest to the node O but not on the x-axis, of course, the node 1 can not be the node O itself. The node 2 is selected if it is in the first quadrant and is closest to the x-axis closest to the node O but not on the x-axis. The node 3 is selected if it is in the first quadrant and is closest to the y-axis and closest to the node O but not on the y-axis. And the other nodes 3 to 8 are selected in similar principle. The same operation is carried out for all aimed nodes in Fig.1 The Cartesian frame Oxy on the aimed node O and the selected the inner of the whole domain. So, the work of nodes selecting nodes can be finished with recording the numbers of the 8 nodes for each aimed node. 2 Take the term ∂∂xφ2 as an example to demonstrate the process of discretization. Now we introduce a accessorial point 1 which is the intersect point of the segment 12 and the xaxis and a accessorial point 6 which is the intersect point of the segment 56 and the x-axis. 2 Therefore, the discretization form of the term ∂∂xφ2 can be written as following: ∂ 2φ ax φ1 + bx φ0 + cx φ6 = , ∂x2 h1 + h2
(2a)
where ax = h21 , bx = − h21 − h22 , cx = h22 , h1 = 1 0, h2 = 6 0, so, h1 + h2 = 6 1 . Since the points 1 and 6 are accessorial points but not the true nodes, the terms φ1 and φ6 in the equation (2a) should be removed. In order to remove the term φ1 , the following formulas are implemented, dφ ∂φ ∂φ + tg β2 , = dx 02 ∂x ∂y
Mesh Free Method Based on Local Cartesian Frame
dφ ∂φ ∂φ + tg β1 . = dx 01 ∂x ∂y
3
Note βi is the angle between the segment 0i and x-axis, i = 1, 2, · · · , 8. Then we can get 1 ∂φ dφ dφ = − , (2b1) ∂y tg β1 − tg β2 dx 01 dx 02 ∂φ 1 dφ dφ = − tg β2 tg β1 . ∂x tg β1 − tg β2 dx 02 dx 01 Discretizing Eq.(2b2), result in the formulation φ1 − φ0 φ2 − φ0 h1 − tg β1 tg β2 . φ1 − φ0 = tg β2 − tg β1 x1 − x0 x2 − x0
(2b2)
(2c)
Similarly, we can get the formulation
φ5 − φ0 φ6 − φ0 h2 φ − φ0 = − tg β5 tg β6 . tg β6 − tg β5 x5 − x0 x6 − x0 6
(2d)
Substituting Eqs.(2c) and (2d) into Eq.(2a) to remove φ1 and φ6 , completes the discretiza2 tion form of ∂∂xφ2 , ∂2φ φ1 − φ0 φ2 − φ0 1 ax h 1 = {(ax + bx + cx )φ0 + [tg β2 − tg β1 ] ∂x2 h1 + h2 tg β2 − tg β1 x1 − x0 x2 − x0 φ5 − φ0 φ6 − φ0 cx h 2 − [tg β6 − tg β5 ]}. tg β6 − tg β5 x5 − x0 x6 − x0 The discretization form of
∂2φ ∂y 2
can be derived in the same process
∂2φ φ3 − φ0 φ4 − φ0 1 ay h 3 = {(ay + by + cy )φ0 + [tg β4 − tg β3 ] ∂y 2 h3 + h4 tg β4 − tg β3 y3 − y0 y4 − y0 φ7 − φ0 φ8 − φ0 ch4 − [tg β8 − tg β7 ]}, tg β8 − tg β7 y7 − y0 y8 − y0 where ay =
2 h3 ,
by = − h23 −
(3)
2 h 4 , cy
=
2 h4 ,
(4)
h3 = 3 0, h4 = 8 0, So, h3 + h4 = 8 3 .
Substituting Eqs.(3) and (4) into Eq.(1), results in the final discretization equation, 8
ai φi + a0 φ0 = Q(x, y),
i=1
where a0
=
1 ax h 1 −1 −1 {(ax + bx + cx ) + [tg β2 − tg β1 ] h1 + h2 tg β2 − tg β1 x1 − x0 x2 − x0 cx h 2 −1 −1 − [tg β6 − tg β5 ]} tg β6 − tg β5 x5 − x0 x6 − x0 −1 −1 1 ay h 3 + {(ay + by + cy ) + [tg β4 − tg β3 ] h3 + h4 tg β4 − tg β3 y3 − y0 y2 − y0 −1 −1 cy h 4 − [tg β8 − tg β7 ]}, tg β8 − tg β7 y5 − y0 y6 − y0
(5)
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LIU Gao-lian and LI Xiao-wei
a1 =
ax h 1 1 1 tg β2 , h1 + h2 tg β2 − tg β1 x1 − x0
a2 =
−ax h1 1 1 tg β1 , h1 + h2 tg β2 − tg β1 x2 − x0
a3 =
ay h 3 1 1 tg β3 , h3 + h4 tg β4 − tg β3 y3 − y0
a4 =
−ay h3 1 1 tg β3 , h3 + h4 tg β4 − tg β3 y4 − y0
a5 =
−cx h2 1 1 tg β6 , h1 + h2 tg β6 − tg β5 x5 − x0
a6 =
cx h 2 1 1 tg β5 , h1 + h2 tg β6 − tg β5 x6 − x0
a7 =
−cy h4 1 tg β8 f rac1y7 − y0 , h3 + h4 tg β8 − tg β7
a8 =
cy h 4 1 1 tg β7 . h3 + h4 tg β8 − tg β7 y8 − y0
Numerical Examples
A flow field over a cylinder in a straight groove shown in Fig.5 was simulated using current mesh free method. The governing equation is written as ∂2ψ ∂ 2ψ + = 0, ∂x2 ∂y 2
(6)
where ψ is the stream function. Boundary conditions are as follows: on left boundary, ∂ψ ∂x = 0; on right boundary, ∂ψ ∂x = 0; on upper boundary, ψ = H/2; on symmetrical line and surface of cylinder, ψ = 0. This problem has accurate solution,
2πy H sh sin H H πx πy ,
ψ=y− 2 2 − cos 2π ch H H 2
πa
(7)
where H denotes the width of the groove and a denotes the radius of the cylinder. In this example, we set H=8.0, a=1.0. The nodes are arbitrarily deployed in the whole domain, as shown in Fig.2. The contours of ψ obtained using current mesh free method are shown in Fig.2. The Pressure distributions are shown in Fig.3, and the good agreement between the simulated result and the analytical solution demonstrates the accuracy of the current method.
Mesh Free Method Based on Local Cartesian Frame
Fig.2
Fig.3
3
5
Nodes distribution and contours of ψ
The pressure distribution on the cylinder surface
Conclusions
The mesh free method in this paper is characterized with two techniques, one is that the process of selecting nodes for each aimed node using local Cartesian frame in an autonomous manner, and another is that the derivatives at each aimed node are constructed using whole derivative formulas through the values of the nodes selected around the aimed node. The two techniques are theoretically simple and can be carried out easily in a generalized form, so can solve the deficiency of the previous mesh free methods in the same researching region. The numerical example show that the two technique are effective and efficient.
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