a new meshless method to solve boundary-value problems - CiteSeerX

A NEW MESHLESS METHOD TO SOLVE BOUNDARY-VALUE PROBLEMS C. Armando Duarte J. T. Oden

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TICAM - Texas Institute for Computational and Applied Mathematics The University of Texas at Austin Taylor Hall 2.400 Austin, Texas, 78712, U.S.A.

ABSTRACT This paper presents a new family of meshless methods for the solution of boundary-value problems. In the h-p cloud method, the solution space is composed of radial basis functions associated with a set of nodes arbitrarily placed in the domain. The paper describes the construction of the h-p cloud functions using a signed partition of unity and how h, p or h-p re nements can be implemented without a mesh. The h-p cloud functions and the Galerkin method are used to solve a two dimensional boundary-value problem. Some properties of the h-p cloud functions are also discussed.

1 INTRODUCTION In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational eort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion, and separation of mesh cells to model fracture, fragmentation, free surfaces, etc. Moreover, in most computer-aided design work, the generation of an appropriate mesh constitutes, by far, the costliest portion of the computer-aided analysis of products and processes. These are among the reasons that interest in so-called meshless methods has grown rapidly in recent times. In these methods, there may be no xed connectivities among the nodes, unlike the nite element or nite dierence methods. This feature has signi cant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. Duarte (Duarte, 1995) has prepared a comprehensive review of the meshless methods 1 2

Research Assistant, TICAM. e-mail: [email protected] Professor, Director, TICAM. e-mail: [email protected]

found in the literature. The connection between the meshless methods and the corresponding underlying approximation technique were also investigated. The conclusion of that study is that, from the point of view of accuracy and eciency, in spite of the variety of the meshless methods found in the literature, all the reviewed methods have serious limitations that, in most situations, can negate some of their advantages over more reliable methods such as h-p nite element methods. This paper presents a new family of meshless methods for the solution of boundary-value problems. The h-p cloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to boundary-value problems. The method uses radial basis functions of varying size of supports and with polynomial reproducing properties of arbitrary order. The rst numerical experiments with this technique show very promising results. The paper is organized as follows: following this introduction, we discuss the construction of the h-p cloud space using a signed partition of unity. A family of functions F k;p N is de ned and some of its properties are investigated. Some illustrative examples of these functions in two dimensions are also presented. Section 3 discusses the implementation of h, p and h-p re nements in the h-p cloud context. The solution of a model problem is the subject of Section 4. Finally, in Section 5, we present conclusions, discuss the limitations of the method and directions for further research.

2 CONSTRUCTION OF THE H-P CLOUD SPACE In this section, we describe the construction of the h-p cloud functions and discuss some of their properties. One key idea used is that of a signed partition of unity. This class of functions can be used to construct linearly independent functions that have many properties in common with the global basis functions used in the nite element method like local compactness and polynomial reproducing properties. But, unlike the nite element basis functions, the functions used in the h-p cloud method can be as smooth as desired, even C 1( ) functions. And, most remarkably, there is no need to partition the domain into smaller subdomains, e.g. nite elements, to construct the h-p cloud functions. All that is needed is an arbitrarily placed set of nodes.

2.1 THE SIGNED PARTITION OF UNITY Let be an open bounded domain in RI n, n = 1; 2 or 3 and QN denote an arbitrarily chosen set of N points x 2 denoted by nodes

QN = fx ; x ; : : :; xN g; x 2

1

2

We associate with the set QN a nite open covering of in the following way: let !,  = 1; : : : ; N , denote a set of balls centered at x and with radii h chosen in such a way that TN := f!gN constitutes an open covering of

! := fy 2 RI n : kx ? y kRI n < hg

  SN ! A class of functions SN := f'gN is called a signed partition of unity subordinate to the open covering TN if it possesses the following properties: 1) ' 2 C 1(! ); 1    N 2) PN '(x) = 1; 8 x 2

=1

=1

=1

0

=1

. . . . . . . . . . . . . . . . .. . . . . . . . Figure 1: Example of an open covering in 2D. The signed partition of unity used in the h-p cloud method has also the following property: given any set PI = fP ; P ; :::; Pmg; Pj : RI n ! RI ; of m linearly independent functions containing the unity function, the signed partition of unity can be constructed such that 1

2

Pj (x) =

N X =1

Pj (x)'(x) ; x 2

(1)

The following approach is used in the h-p cloud method to build the signed partition of unity SN : Let W : RI n ! RI denote a weighting function that belongs to the space C 1(!) with the following properties: . W(y)  0 8 y 2

. W(y) := W(y ? x) where the functions W belong to the space C 1(Bh ) and Bh is a ball of radius h centered at the origin Bh = fx 2 RI n : kxkRI n < hg Next we introduce a family of functionals de ned over continuous functions de ned on by 0

0

(f; g)y :=

N X =1

W(y)f (x)g(x);

f; g : ! RI ; f; g 2 C ( )

(2)

0

Assumption 1 Given a set of m functions PI = fP ; P ; :::; Pmg; Pi : ! RI ; Pi 2 C ( ) for i = 1; : : : ; m, the weighting functions W de ned above and the functions Pi are such that 8 x 2  1

there holds

m X k=1

2

0

ak (Pk ; Pl)x  0 for l = 1; : : : ; m if and only if ak  0 for k = 1; : : : ; m:

Necessary and in some cases sucient conditions for the satisfaction of assumption 1 have been shown in (Duarte and Oden, 1995). We are now in a position to de ne the signed partition of unity used in the h-p cloud method. The function ' associated to the ball ! is de ned by

'(x) := PI T (x)A? (x)B (x)

(3)

1

where

 Aij (x) := (Pi; Pj )x; Pi ; Pj 2 PI  PI (x) := fP (x); P (x); : : :; Pm (x)gT ; 9 Pj s:t: Pj (x)  1;  B(x) := W(x)PI (x) 1

2

It can be shown that the de nition of ' given in (3) satis es the de nition of a signed partition of unity and also has the property given by (1). The proofs can be found in (Duarte and Oden, 1995).

2.2 THE FAMILIES F k;p N The most important step in the h-p cloud method is the construction of the family of functions F k;p N using the signed partition of unity SN de ned in the previous section. This class of functions can be constructed at a very low cost and has the important property that for a proper choice of the base family PI (x) we can insure that Pp  spanfF k;p N g where Pp denotes the space of polynomials of degree less or equal to p. In this section, we describe the construction of F k;p N and state some theorems concerning fundamental properties of these functions. Let Lp denote the set of tensor product Legendre polynomials Li;j;k in RI , Lijk (x) = Li(x )Lj (x )Lk (x ); 0  i; j; k  p Other sets of complete polynomials can be used as well; e.g., the smallest set of complete polynomials p. In the following SNk := f'kgN will denote a signed partition of unity that is Lk reducible for the set QN ; that is, given any element Lijk 2 Lk the following holds 8 x 2  : 3

1

2

3

=1

Lijk (x) =

N X =1

Lijk (x)'k(x)

(4)

The family of functions F k;p N is de ned by k k F k;p N = f f'(x)g [ f' Lijl (x)g : 1    N ; 0  i; j; l  p; i or j or l > k ; p  k g

(5)

The idea behind the de nition in (5) is to add, hierarchically, appropriate elements to the set SNk such that the resulting set can reproduce, as linear combinations, polynomials of degree p  k. Because of property 2) of a signed partition of unity, those elements are precisely the product of the functions 'k with the elements from the set Lp that are missing from the set Lk . For consistent results, regardless of the scale of the problem, the h-p cloud functions introduced in (5) are implemented using h-ane maps given by: F : !b ! !

F() = h + x;

where

 2!

!b := f 2 RI n : kkRI n < 1g

is a sphere of radius one and

! := fx 2 RI n : kx ? xkRI n < hg is the support of the function '. The h-p cloud function 'Lijl(x) is implemented by 

'Lijl(x) := '(x) Lb ijl  F? (x) 1



where Lb ijl() is a tensor product Legendre polynomial de ned on [?1; 1]n. The following theorems are proved in (Duarte and Oden, 1995).

Theorem 1 Let Pi ; i = 1; : : : ; m and W;  = 1; : : : ; N be the basis functions and the weighting functions used to construct the signed partition of unity SNk . Suppose that Pi ; i = 1; : : :; m 2 C l( ) and W ;  = 1; : : : ; N 2 C q( ). Then the h-p cloud functions de ned in (5) belong to the space C min l;q ( ). (

)

Theorem 2 Lp  spanfF k;p N g: 2.3 PLOT OF THE H-P CLOUD FUNCTIONS In this section, some elements from the family F k;p N are plotted for the two dimensional case. In general, there is no closed form expression for these functions. The set of functions PI used to build the signed partition of unity SNk was composed of polynomials of degree less or equal to k including the unity function P = 1. The weighting functions W were implemented using bi-splines (deBoor, 1978). These functions are piecewise polynomials and can be built with any degree of regularity. For example, quartic bi-splines are C ( ) functions. In the plots showed below quartic bi-splines were used. An uniform node arrangement with ve nodes in each direction was used to build the partition of unity. The domain was the square [?1; 1]  [?1; 1]. Figure 2(a) shows the function 'kN from the family F Nk ;p associated to a node at the origin. Figures 2(b) and 2(c) show the functions y'kN and xy'kN from the families F Nk ;p and F Nk ;p respectively. 1

3

=0

=0

=0

=0

=0

=0

1

2

3 THE H, P AND H-P VERSIONS One remarkable feature of the h-p cloud method, besides that it does not need a mesh to build the space of approximating functions, is that the implementation of h, p or h-p re nement is much easier than in the nite element method (FEM). The h version of the FEM can be implemented in several ways. One of the most successful approaches is based on the use of constrained nodes (Demkowicz et al., 1989). This technique guarantees that the h re nement at some region of the domain will not propagate throughout the entire domain only to guarantee the continuity of the solution (Demkowicz et al., 1989). In the h-p cloud method the use of constrained nodes is completely unnecessary. The implementation of the h re nement is achieved simply by inserting nodes in the regions of interest. There is no need to add extra nodes or to constraint some of them only to make the solution continuous. Figure 3(a) shows some of the balls used to build an open covering for an L shaped domain. The black dots represent the nodes and the gray

0.8 0.6

0.1 0.0 1 -0.

(a) 2-D function '

X

from the family F

k

=0;p

N

.

-1.0

0.5 1.0 1.0

0.5

0.5

0.0

0.0

Z

1.0 1.0

=0

k N

-0.5

-1.0

-1.0

-0.5

0.0

0.0

-0.5 Y

0.5

X

-1.0

Z

-0.5

0.4 0.2 0.0

Y

(b) 2-D function y'

=0

k N

from the family F

k

=0;p1

N

.

-1.0

-0.5

0.0 0.5 1.0 1.0

0.5

Y

0.0

X

-0.5

Z

-1.0

30 0.0 20 0 0. 10 0 . 0 00 0.0 10 0 . -0 0 02 -0. 0 3 0 -0.

(c) 2-D function xy'

=0

k N

from the family F

k

=0;p2

N

.

Figure 2: Examples of h-p cloud basis functions. level of the shaded areas indicates the polynomial order associated to each ball (in this example all balls have the same polynomial order). Figure 3(b) shows the h re nement of the previous discretization. New nodes were arbitrarily added and the polynomial order associated to each ball was kept xed. The p version of the method can have more than one variant. One, for example, can x the size h of the balls and increase the parameter p keeping k xed. Another possibility would be to increase simultaneously k and p. Nonetheless, mathematical analysis and numerical experiments performed by (Duarte and Oden, 1995) have shown that the rst variant is preferable (see (Duarte and Oden, 1995) for details). Figure 3(c) shows the non uniform p enrichment of the balls shown in Figure 3(a). The dierent gray levels indicate that each ball can have a dierent polynomial order associated to it regardless of the polynomial order associated to neighboring balls. In the h-p version of the method the number of nodes and the polynomial order associated to each ball are simultaneously increased.

(a) Some of the balls used to build an open covering for an L shaped domain.

(b) h re nement of the discretization shown in the Figure (a).

(c) Non uniform p enrichment of the balls shown in Figure (a).

Figure 3: The h and p versions of the cloud method.

4

CLOUD SOLUTION OF A BOUNDARY-VALUE PROBLEM

H-P

In this section, we use the techniques described in Section 2 to construct appropriate nite dimensional subspaces of functions used in the Galerkin method. The resulting approach is denoted by the h-p cloud method. We focus on the solution of a simple two dimensional boundaryvalue problem, namely, the analysis of a bar with equilateral triangular cross section subjected to torsion. The stress distribution on the cross section of the bar can be computed from the Prandtl's torsion function (x; y) (Boresi and Lynn, 1974) and is given by

xz = ;y

yz = ?;x

(6)

The other stress components are identically zero. The Prandtl's torsion function can be found solving the following boundary-value problem:

y

Find (x; y) such that

?
 = 2G  = 0

in

on @

φ=0 φ=0

where G is the shear modulus and  is the angle of twist per unit length of the bar. The domain is illustrated in Figure 4.

φ,n =0 x

φ,n =0

a/3

2a/3

Figure 4: Domain, boundary conditions and discretization used. The solution of this problem is given by (Boresi and Lynn, 1974)  p 2a   p 2a   a   = G x ? 3y ? x + 3y ? 3 x + 3 (7) 2a 3 The values G = 1=2 and a = 12 were used in the calculations. Although this is a simple boundary-value problem, it can be used to illustrate numerically the results of Theorem 1. The set PI = f1g, that is k = 0, and weighting functions W built from C splines are used to construct the signed partition of unity as described in Section 2.1. Therefore, from Theorem 1, the h-p cloud solution belongs to the space C ( ) and, consequently, all the approximate stresses are continuous throughout the entire domain. Figure 4 shows the portion of the domain discretized and the node arrangement used{ one node at each corner of the domain. The boundary conditions applied at each portion of the boundary is also indicated in Figure 4. The Dirichlet boundary conditions were imposed using Lagrange multipliers. The problem was solved using the family of cloud functions F nk ;p de ned in previous sections. The h-p cloud solution is showed in Figure 5. The total number of degrees of freedom was 31 (24 for the stiness matrix and 7 for the Lagrange multipliers). The L1 error of the h-p cloud solution is 4:43  10? . Figure 5 shows the pointwise error xz ? xzhp, where xzhp denotes the stress component computed using (6) and the h-p cloud solution. It can be observed that xzhp is continuous over the entire domain (it is indeed a C function). It should be mentioned that max jxz ? xzhpj = 0:104 and max jyz ? yzhpj = 0:120 which are regarded as very good results for such a coarse arrangement of nodes. 3

3

=0 =2

2

2

5 CONCLUSIONS AND DISCUSSION A new approach to solve boundary-value problems is discussed. The h-p cloud method has the following features: The domain does not need to be partitioned into smaller subdomains to build the approximating functions. Nonetheless, the domain may need to be partitioned somehow or covered by a cell structure (Belytschko et al., 1994) to perform numerical evaluation of functionals. In the present implementation of the h-p cloud method, we use a mesh of quadrilateral cells that exactly

ts the domain to perform the numerical integration. The only requirements on the mesh is that the cells do not overlap and that their union exactly matches the domain under analysis. Indeed, numerical integration for meshless methods is an area of active research. Z

X

Y

V3 4.99203 4.65628

5

4.32052 3.98476 3.649

4

3.31325 2.97749 2.64173 2.30598

3 2

1.97022 1.63446 1.2987

1

0.962947 0.627189 0.291432

7 2

6

1

5

0

4

-1

3

-2

2

-3

1

-4

0

0

Figure 4: h-p cloud solution. Z

X

Y

0.1

0

0.0

5

0.0

0 5

7 2

6

1

5

0

4

-1

3

-2

2

-3

1

-4

0

0 -0.

Figure 5: Pointwise error xz ? xzhp. The meshless character of the h-p cloud method makes it very attractive to solve, for example, large deformation problems, crack propagation problems and problems where fragmentation occurs. The uxes computed from the h-p cloud solution may be very smooth functions and therefore the post-processing of these quantities and other derivatives may not be required. The high regularity of the h-p cloud functions also make them attractive candidates to solve thin plate and shell problems. In the h-p cloud method, the nodes can be placed quite arbitrarily in the domain. This property can decrease substantially the cost of solving numerically many industrial problems. Also there are no xed connectivities among the nodes and therefore problems like mesh entanglement in the analysis of large deformation problems do not exist. To our knowledge, the h-p cloud method is the only meshless method where the support of the shape functions can be made of any size and at the same time the shape functions can be

constructed in such a way that polynomials of any degree can be represented as linear combination of these functions. That is, we have an h-p method in the same spirit as in the nite element method. Both h-p cloud and nite element method are very similar with regard to this point of view but the practical implementation of h-p adaptivity in the h-p cloud method is signi cantly simpler than in h-p nite element methods. Much additional work remains to be done before the h-p cloud method can be used to solve large scale industrial problems. The many advantages of the method also come with some challenging problems. The required use of Lagrange multipliers to impose Dirichlet boundary conditions increases the solution cost of the resulting system of equations since the matrix will no more be positive de nite as in the nite element method. Another major diculty is the numerical integration of the h-p cloud functions since these functions are not polynomials. Nonetheless these diculties do no seem to be unsurmontable and the potential bene ts of the method justify further research. ACKNOWLEDGMENT: The support of the CNPq of Brazil and the NSF of the USA under grant INT 9402416 is gratefully acknowledged. Author C. Armando Duarte was supported by a CNPq Graduate Fellowship and he and J. T. Oden through support of a project at TICAM sponsored by the Army Research Oce under contract DAAL03-92-G-0253.

REFERENCES Belytschko, T., Lu, Y. Y., and Gu, L. (1994). Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 37:229{256. Boresi, A. P. and Lynn, P. P. (1974). Elasticity in Engeneering mechanics. Prentice-Hall, New Jersew. deBoor, C. (1978). A Practical Guide to Splines. Springer-Verlag, New York. Demkowicz, L., Oden, J. T., Rachowicz, W., and Hardy, O. (1989). Toward a universal h-p adaptive nite element strategy, part 1. constrained approximation and data structure. Computer Methods in Applied Mechanics and Engineering, 77:79{112. Duarte, C. A. M. (1995). A review of some meshless methods to solve partial dierential equations. Technical Report 95-06, TICAM, The University of Texas at Austin. Duarte, C. A. M. and Oden, J. T. (1995). Hp clouds{a meshless method to solve boundary-value problems. Technical Report 95-05, TICAM, The University of Texas at Austin.