www.ietdl.org Published in IET Science, Measurement and Technology Received on 31st May 2008 Revised on 11th August 2008 doi: 10.1049/iet-smt:20080082
Special Issue – selected papers from CEM 2008
ISSN 1751-8822
Imposing boundary conditions in the meshless local Petrov– Galerkin method A.R. Fonseca1 S.A. Viana2 E.J. Silva1 R.C. Mesquita1 1
Universidade Federal de Minas Gerais, Av. Ant^onio Carlos, 6627 Pampulha, Belo Horizonte, MG, Brazil Suzlon Energia Eo´lica do Brasil Ltda, Rua Eduardo Sabo´ia, 399 Papicu, Fortaleza, CE, Brazil E-mail:
[email protected] 2
Abstract: A particular meshless method, named meshless local Petrov – Galerkin is investigated. To treat the essential boundary condition problem, an alternative approach is proposed. The basic idea is to merge the best features of two different methods of shape function generation: the moving least squares (MLS) and the radial basis functions with polynomial terms (RBFp). Whereas the MLS has lower computational cost, the RBFp imposes in a direct manner the essential boundary conditions. Thus, dividing the domain into different regions a hybrid method has been developed. Results show that it leads to a good trade-off between computational time and precision.
1
Introduction
Physical problems described by partial differential equations are usually solved using numerical methods. Traditional methods use a mesh or grid discretisation of the domain, which is obtained through the connectivity among nodes. The most common methods are the finite element method (FEM) and finite difference method (FDM). However, these methods are somewhat limited. For instance, FDM uses a grid (structured mesh), that does not fit well to complex geometries such as those with curvatures or with very different geometric feature sizes. These characteristics may lead to discretisation errors or to meshes, that are too refined because of small features in the geometric input. On the other hand, FEM uses an unstructured mesh which is able to fit complex geometries. However, the task of mesh building is not always simple, especially for complex 3D geometries, where generating a mesh with satisfactory quality is still a research problem. Meshfree methods do not need any connectivity structure among nodes avoiding the mesh generation problem. It is a very attractive choice, especially when solving problems that involve movement, boundary and shape deformation. However, because of the absence of a mesh a few drawbacks are introduced such as neighbour search, local matrix inversions and other computations that ultimately IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387 – 394 doi: 10.1049/iet-smt:20080082
increase the computational cost. Consequently, in order to increase the efficiency of the methods, special techniques should be employed [1]. Among several meshless methods, some of the most used by the electromagnetic community are: element free Galerkin (EFG) [1 – 4], meshless local Petrov – Galerkin (MLPG) [2, 5] and smooth particle hydrodynamics (SPH) [6]. All these methods solve the problem constructing the approximation using patches that cover the domain. Each patch has an associated node where the unknown value is defined. SPH was one of the first meshfree methods to be developed. It is based on an integral formulation of a function. The Dirac’s delta function is approximated by a kernel function W and the integral is performed over a set of particles [6]. Initially, SPH was formulated to solve astrophysics problems [7], but lately it is mostly used to solve problems of fluid dynamics [6]. Recently, the SPH has been adapted to solve time-domain electromagnetic problems, generating a method known as smoothed particle electromagnetics (SPEM) [8]. In SPEM, the particles of the SPH are divided into two groups: electric and magnetic particles. The former shall keeps the information about the electrical field components, whereas the latter keeps the information about magnetic field components. The SPH interpolation is then applied to the time dependent Maxwell equations [8]. SPH has many advantages in 387
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www.ietdl.org computations: it is simple to program, suitable for large deformation computation, and so on. On the other hand, SPH has some inherent drawbacks: the essential boundary conditions are not satisfied, and its interpolation procedure lacks consistency or completeness. Several techniques have been developed in the last years aimed at improving its performance and eliminating pathologies in numerical computations [6]. EFG was developed by Belytschko et al. [4] in 1994. The method is based on the diffuse elements method, which was the first Galerkin type of meshless technique. Initially, both methods were developed to solve mechanical problems, and later on applied to computational electromagnetics [1, 3]. EFG is considered a meshfree method in the sense that no connectivity among nodes is necessary. Nevertheless, to perform the numerical integration EFG requires an auxiliary grid that must involve the entire domain. This grid does not depend upon the node distribution and can be chosen to be regular; therefore it is far simpler than the typical FEM mesh. Like FEM, because of the Galerkin method the trial and shape functions must be from the same space. moving least square (MLS) approximation technique is commonly chosen for constructing trial and shape functions [2]. As it will be presented in Section 2, the MLPG uses a procedure named local weak form. In this technique, each node defines its own sub-domain that is, independent from the remaining. Those sub-domains are allowed to have any size and shape and they eliminate the need of a grid for numerical integration. Furthermore, the Petrov – Galerkin method allows the trial and shape functions to be chosen from different spaces. These two features make this method very flexible and distinguishes MLPG as a truly meshfree method. The meshless techniques are still under development and much attention has been given to overcoming some of their drawbacks. For instance, when solving boundary value problems, the imposition of the essential boundary conditions may be a problem since some of the meshless shape functions do not always satisfy the Kronecker-delta condition. Over the last decade, some authors have proposed the use of techniques such as the Lagrange multiplier, the penalty method and FEM coupling in an attempt to overcome this drawback [2]. Nevertheless, the use of such techniques may bring about undesired minor issues, such as the increased number of unknowns in the system because of the Lagrange multiplier technique and the uncertainty involved in finding a suitable value for the penalty parameter when using the penalty method. This latter one may also lead to a poor conditioning of the stiffness matrix [2]. In this paper, we propose an alternative approach to impose the essential boundary conditions in the MLPG method. The proposed method is discussed in the following sections. 388 & The Institution of Engineering and Technology 2008
2
MLPG method formulation
The MLPG method differs from the other meshless techniques based on the Galerkin method because the Petrov – Galerkin approach uses trial and test function from different spaces [5]. By using this approach, it is possible to solve a boundary value problem as several localised problems, instead of a single global problem as in the EFG method [3]. To illustrate the MLPG formulation, we will consider a twodimensional electrostatic problem in an arbitrary domain V, with Dirichlet boundary condition for the electric potential V ¼ Vu on G ¼ @V (Fig. 1). The approximated solution Vh is sought using the weighted residual method and the essential boundary condition is imposed using the penalty method as ð
ð
h
W (r (1rV ))dV þ a V
G
W (V h Vu )dG ¼ 0
(1)
where W is the test function, 1 the electric permittivity of the medium and a the penalty parameter with typical values from 104 to 108 [9]. The approximated solution Vh at the node k is defined as Vkh ¼
n X
fki Vi
(2)
i¼1
where fki is the value of the shape function of the node i at the node k, n the number of nodes neighbouring k that are used to build the approximation and Vi the value of V at the node i. The test function W is chosen to have compact support which defines a closed sub-domain Vq centred at each node, as shown in Fig. 1. Consequently, (1) can now be solved locally [5], in each Vq ð
h
ð
W (r (1rV ))dV þ a Vq
Gq
W (V h Vu )dG ¼ 0
(3)
In meshless methods, the shape functions in (2) can be defined using different procedures such as the MLS, the
Figure 1 Domain representation IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387– 394 doi: 10.1049/iet-smt:20080082
www.ietdl.org kernel method, radial basis functions (RBF) or polynomial approximations, among others. According to Atluri and Shen [5], any of the above methods can be used in the MLPG. Atluri also suggests possible functions that can be used to define W, such as the Heaviside step function or the MLS weight function. In this work, W is chosen to be the Heaviside function, that is W ¼ 1, on Vq and W ¼ 0, outside Vq , which leads to [10] ð
ð
h
(1rV ) n^ dG þ a Gq
Gq
(V h Vu )dG ¼ 0
(4)
where nˆ is the normal unitary vector over Gq . Since r W ¼ 0, only integration on the segments of the boundary of the subdomains Vq needs to be evaluated in (4). In fact, our choice of weight function is based on Atluri’s work [5] where it is shown that among several weight functions the Heaviside function provides accurate results.
3 Shape functions and interpolation In a broad sense, shape functions are interpolation functions [2]. The approximation function V h is obtained by taking the sum of the known values at some points weighted by their respective shape functions. Therefore at any point of the domain, the approximated value of the function V h is given by h
V (x) ¼
n X
fi Vi
(5)
i¼1
where Vi is the fictitious nodal value of the unknown function at xi , n the number of known points and fi the shape function of xi . Generally, shape functions have compact support, influencing only the nearby nodes on a small region of the entire domain. Fig. 2 depicts the influence domain of a given node j defined by Rf . Note that the influence domain overlaps the node i integration sub-domain (Viq), which has a radius RV, the support of Heaviside function. Once shape functions have compact support, in the
integration scheme it is necessary to consider only nodes in which the influence domains overlap the integration subdomain being evaluated. In Fig. 2, shape functions for all nodes outside the search box, also called support domain for node i [2], with radius equal to Rf þ RV, do not influence the sub-domain defined by node i. Therefore the nodes marked by light gray are not considered on the evaluation of the sum in (5), for the node i.
3.1 Radial basis functions with polynomial terms RBFs with compact support are a particular type of RBFs. These functions, unlike the multiquadrics and the thin plate spline, present a support that decreases until it vanishes as the distance from a central point increases. The Wendland RBF [2] defined by (6) is an example of RBF with compact support 8 < (1 r)6 [6 þ 36r þ 82r 2 þ 72r 3 r,1 R(d ) ¼ þ 30r 4 þ 5r 5 ]; : 0 r .¼ 1
(6)
where d is the distance from the centre point and r ¼ d/R a normalised distance, with R assumed to be the RBF support radius. Fig. 3a shows the RBF given in (6) centred on point (5,5), considering r e R 2 and R ¼ 5. Fig. 3b shows its partial derivative with respect to x. An inherent characteristic of the RBF functions is the lack of completeness, that is, they do not form complete basis to fit functions [2]. To overcome this drawback, the RBF is combined with polynomial terms V h (x) ¼ Rt (x)a þ pt (x)b
(7)
where Rt (x) ¼ [R1(x) R2(x) . . . Rn(x)] is a vector of the values of all the n RBF’s at point x that are not zero (the RBF’s of the neighbouring nodes), p t (x) ¼ [ p1(x) p2(x) . . . pm(x)] is a vector containing monomials of a complete polynomial basis with m terms and a ¼ [a1 a2 . . . an]t e b ¼ [b1 b2 . . . bm]t are unknown coefficient vectors. To find a and b that best fit the curve, a linear system of equations is generated from (7) where it is enforced that the approximation V h(xi) ¼ Vi . Then, the matrix equation is given by V ¼ R0 a þ Pb
(8)
where V ¼ [V1 V2 . . . Vn]t is a vector whose elements are the nodal values of the function, R0 ¼ [R(x1) R(x2) . . . R(xn)]t is an n n matrix and P ¼ [ p(x1) p(x2) . . . p(xn)]t is an n m matrix. From (5) and (8), it is possible to obtain [3] Figure 2 Neighbours’ search box IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387 – 394 doi: 10.1049/iet-smt:20080082
fi (x) ¼ Rt (x)Sai þ pt (x)Sbi
(9) 389
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Figure 3 Wendland RBF a R(d ) b @R(d )/@x
where Sai is the ith column of Sa given by 1 1 Sa ¼ R1 0 [1 PS b ] ¼ R0 R0 PS b
(10)
and a(x) ¼ [a1 (x) a2 (x) am (x)]t is a vector of unknown coefficients which are determined by minimising the following functional
and Sbi is the ith column of Sb given by J (a(x)) ¼ 1 t 1 Sb ¼ [P t R1 0 P] P R0
(11)
It is easy to show that the RBFp shape functions possess the Kronecker delta function property; in the calculation of Sa and Sb , the R0 matrix is symmetric, positive defined, and so it has an inverse. P tR21 0 P is symmetric and if n m it will be full rank and it will also have an inverse [2]. Then, Sa and Sb in (10) and (11) can be determined, and also fi(x) in (9) which will be linearly independent. From (5) and from the fact that V h(xP i) ¼ Vi [imposed by (8)], we conclude that V h (xj ) ¼ Vj ¼ ni¼1 fi (xj )Vi , 8j. Letting t t Vs ¼ V1 , V2 , . . . , Vi , . . . , Vn ¼ 0, 0, . . . , Vi , . . . , 0 (12) and substituting in (5), we have at x ¼ xj , that V h (xj ) ¼ Vj ¼
n X
fk (xj )Vk ¼ fi (xj )Vi
(15)
i¼1
where w is a weight function, commonly called a kernel function. If w has compact support like (6), those points xi closer to the interpolation point x will have greater influence on its interpolation. Note that the vector a is not constant; it depends on the point x being evaluated. This does not happen on RBFp, where the obtained coefficients for each fi are constant. To minimise (15), @J =@a ¼ 0 is taken and considering (5), we obtain [2]
fi (x) ¼ pt (x)[A1 Bi ]
(16)
(13) W ¼ diag(w(jx x1 j) w(jx x2 j) w(jx xn j))
when i ¼ j it leads to Vi ¼ fi(xi)Vi which leads to fi (xi) ¼ 1 and fi (xj ) ¼ 0 when i = j because of the fact that the fi (x) are linearly independent. Note that the condition n m implies that a minimum number of points are needed to ensure that the RBFp works correctly.
3.2 Moving least square In MLS method, the function approximation is given by (14)
where pt (x) ¼ [p1 (x) p2 (x) pm (x)] is a vector containing monomials of a complete polynomial basis with m terms 390 & The Institution of Engineering and Technology 2008
w(jx xi j)[pt (xi )a(x) Vi ]2
where Bi is the ith column of B ¼ P t W (m n) and A ¼ BP (m m), where W is a diagonal matrix (n n) given by
k¼1
V h (x) ¼ pt (x)a(x)
n X
(17)
when comparing both interpolation methods, one should note that in the MLS it is necessary to invert an m m matrix for each integration point over the sub-domain boundary. For the RBFp, only one n n matrix inversion for all integration points is needed. Usually, n, the number of neighbours that influence an integration sub-domain, is bigger than the polynomial base order, m. Then, a single inversion of an n n matrix has a bigger computational cost than many inversions of m m matrices. In this work, the polynomial basis for all methods is pðxÞ ¼ [1 x y]t such that m ¼ 3. As shown in Section 5, n can have values as big as 64. IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387– 394 doi: 10.1049/iet-smt:20080082
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Mixed MLPG formulation
Among the approximation techniques commonly used in meshless methods, the RBF is one of the few that will lead to a shape function that satisfies the Kronecker delta. However, as we have shown, its computational cost is higher if compared with the MLS approach. Considering that, this paper proposes a mixed formulation for the MLPG. It uses the Wendland RBF with polynomial terms of Section 3.1 to define the approximation for the nodes along the boundaries (boundary nodes) with Dirichlet boundary conditions and those close to them, and the MLS for the remaining nodes (inner nodes). Fig. 4 shows the node classification. The main problem consists in defining a procedure of how to mix the shape functions when a node has neighbours of the two types: inner and boundary nodes. As a first tentative (mixed_1), the shape function associated with a node is chosen based on its classification as shown in Fig. 5a. The black nodes are classified as boundary nodes, and so its related shape function will always be the Wendland RBF with polynomial terms, independent of the classification of the node that defines the integration sub-domain. On the other hand, the white nodes are classified as inner nodes and are supposed to use MLS functions. As a second approach (mixed_2), the classification takes into account only the node that defines the integration subdomain to define the shape function for all neighbouring nodes. This means that, when building the approximation for node m, the shape functions of its neighbouring nodes are defined by the RBF (Fig. 5b). Conversely, when building the approximation for node j, the shape functions
of its neighbouring nodes will be defined by the MLS. Note that using this strategy a node may have two different shape functions associated with it, depending on how the sub-domain is being integrated. Both approaches eliminate the need of using the penalty method to enforce essential boundary conditions.
5
Results
To validate both approaches of the mixed method, the problem shown in Fig. 6a is considered. Boundaries defined by y ¼ 10 m and x ¼ 10 m have @V =@^n ¼ 0, a Neumann boundary condition, where nˆ is a normal unitary vector. Fig. 6b shows the mesh used by a finite element program to solve the problem. The minimal distance between vertices is h ¼ 0.5 m. The same problem was solved by the three MLPG approaches using as nodes the same vertices of the FEM mesh. The results were compared with the solution of FEM using a dense mesh with h ¼ 0.05 m. This solution is considered a good approximation of the exact solution. Fig. 7 shows the potential values for all the considered methods along the segment defined by points A and B (Fig. 6) with RV ¼ 0:6, Rf ¼ 1:2 and a ¼ 106 . As we can see, mixed_1 formulation provides an unacceptable result. Actually, mixed_1 method combines totally different shape functions in (5) for the same sub-domain integration. The consequence is that it no more satisfies the consistency criterion given by a partition of unit [11] n X
fi ¼ 1
(18)
i¼1
Hence, this approach is not suitable to merge shape functions and will not be considered in the examples that follow. Mixed_2 uses only one type of shape function for the same integration sub-domain; sub-domains are independent of each other, and so (18) is satisfied.
Figure 4 Node classification
The percentage errors along segment AB are shown in Fig. 8. The results show that close to point B, with a physical singularity, errors are bigger, but all MLPG-based
Figure 5 Mixed formulation shape functions a Mixed_1 b Mixed_2
IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387 – 394 doi: 10.1049/iet-smt:20080082
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Figure 6 Test problem 1
Figure 8 Percentage error along segment AB
Figure 7 Potential along segment AB
Figure 9 Test problem 2
methods are more accurate than the FEM for the same number of nodes. This is because of the higher order of the meshless shape functions. Far from point B, all methods have similar orders of errors. As can be seen, the errors for mlpg_rbf and mixed_2 formulations are essentially the same. Differences among the results of these formulations were insignificant for all the tests made. The percentage error for each node was calculated using
To measure the average percentage error for this problem, we consider
V Vidense FEM error% ¼ 100 abs i dense FEM Vi
! (19)
As expected, the value for the last point is zero, and so the percentage error (19) makes no sense. The absolute error at this point for mlpg_mls was 0.4586. The other methods have no error on the Dirichlet boundary because those values are directly imposed. A second example is shown in Fig. 9a. Fig. 9b shows the result using the mixed formulation for a regular 15 15 grid of nodes. The analytical solution for this problem is given by [12] 1 sinðnpx=10Þ sinh npy=10 40 X V (x, y) ¼ p n¼1,3,5,... n sinh(np) 392 & The Institution of Engineering and Technology 2008
(20)
error% ¼ 100
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 P n1 ni¼1 Vi Viexact Vmax
(21)
where n is the total number of nodes scattered over the domain. Using (21), the values of the average error for the methods were 4.76% (MLS), 2.71% (RBF) and 2.75% (Mixed_2). Fig. 10a shows the solution for test problem 2 using MLS for shape functions with RV ¼ Rf ¼ 0:5 and a ¼ 106 . As can be seen, the error is high because of oscillations in the boundary. This occurs because using the penalty method the Dirichlet boundary conditions are not directly defined but they have to be calculated. The abrupt potential variations on the top corners lead to low-precision results by the penalty method close to those regions. Fig. 10b shows the exact solution for problem 2. To obtain an idea of the computational costs of more real problems, where the number of internal nodes is much higher than the number of nodes near the boundary, a grid of 101 101 nodes was distributed in the domain of the problem 2. Table 1 shows the relative times for assembling the linear system. Boldface denotes simulations where the mixed formulation obtained better time performance than RBFp. IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387– 394 doi: 10.1049/iet-smt:20080082
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Figure 10 Problem 2 solution a MLPG with MLS, with RV ¼ Rf ¼ 0.5 and a ¼ 106 b Exact solution
Table 1 Comparison among methods, based on relative times integration points ¼ 4 n
integration points ¼ 20
MLS
RBFp
Mixed
MLS
RBFp
Mixed
16
1.13
1.00
1.10
3.05
1.48
2.86
25
1.32
1.46
1.30
3.77
2.54
3.53
36
1.51
2.35
1.63
4.57
4.30
4.52
49
1.70
4.32
2.33
5.65
7.91
6.15
64
2.00
7.28
3.06
6.84
13.20
8.05
n is the number of neighbour nodes used to build the shape function approximation in a sub-domain. As the number of neighbours n increases, the RBFp processing time increases fast because of the n n matrix inversion (Section 3). On the other hand, the total number of integration points affects the MLS performance since an m m matrix inversion is done for each point. However, usually just a few number of integration points is enough to obtain good precision. Mixed_2 formulation always presents an intermediate processing time between MLS and RBFp. If the number of nodes close to Dirichlet boundary is much smaller than the number of inner nodes, the mixed_2 processing time will be near the MLS time. In conclusion, the proposed mixed_2 formulation is generally faster than the RBFp formulation with better precision than the MLS formulation.
6
Conclusions
The MLPG method presents some issues to impose Dirichlet boundary conditions when using shape functions that do not satisfy the Kronecker delta property. To solve this problem, this paper proposed mixed formulations that combine two types of shape functions. The first is the MLS, which is fast to compute, and was used inside the domain. The second is the Wendland RBF with polynomial terms, which IET Sci. Meas. Technol., 2008, Vol. 2, No. 6, pp. 387 – 394 doi: 10.1049/iet-smt:20080082
eliminates the need for any special techniques to impose the Dirichlet boundary conditions, and was used to define the approximation for the nodes along the boundaries. Two approaches to mix shape functions were tried. The first one considered only node classification to define the shape functions on each node. This approach provided unacceptable results because it does not satisfy the partition of unity condition and for this reason, it was discarded. The second one considered as the shape function the one related to the node that defines the integration subdomain. This approach can provide accurate results with the advantage of fast computation for interior nodes. For the presented problems, all MLPG solutions have the same magnitude error inside the domain. The MLS presented oscillations on the boundaries whereas RBF and the mixed formulation did not. From the results, it can be seen that the mean errors obtained using the proposed mixed method are close to the errors using RBFp, which are smaller than when using MLS. The mixed formulation becomes interesting when the number of internal nodes is much bigger than that of boundary nodes. Table 1 shows that there are problems instances (integration 393
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element methods’, Comput. Model Eng. Sci., 2002, 3, pp. 11– 51
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[6] LIU G.R., LIU M.B.: ‘Smoothed particle hydrodynamics – a meshfree particle method’ (World Scientific Publishing Company, 2003)
Acknowledgments
The authors would like to thank the Brazilian agency CNPq for financial support.
8
References
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[12] SADIKU M.N.O.: ‘Elements of electromagnetics’ (Oxford University Press, 2001, 4th edn., 2006)
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