Invited Paper Engineering Mechanics, Iowa State University,€¦ · Department of Aerospace Engineering and Engineering Mechanics, Iowa State University, /owa Introduction The discretization

Invited Paper

Spline assisted grid optimization scheme for

the boundary element method

A.K. Mitra, L.R. Salazar, M.L. Sawyer

Department of Aerospace Engineering and

Engineering Mechanics, Iowa State University,

/owa

Introduction

The discretization of the integral equations in the boundary-element methodinvolves two steps. Firstly, the boundary of the domain is discretized intoelements, and secondly, certain functions are discretized in terms of thenodal values and approximating polynomials. The discretized geometrymust closely resemble the original shape of the domain, and the approxi-mate representation of the functions must also resemble the actual behavior.Since, the accuracy of the solution depends on the geometric as well as thefunctional discretization, a grid optimization scheme is complete only whenit addresses both of these issues.

The optimization of the geometric discretization is simple, because theshape of the domain is known a priori, and the quality of the grid could beeasily gauged by a straightforward comparison of the shapes of the originaland the discretized domains. However, the functional discretization requiresthe knowledge of the behavior of certain functions, and that knowledgemay not be available. As a result, many grid optimization schemes relyon error indicators and error estimators[l-8]. An alternative scheme hasbeen proposed by Mitra[9]. This scheme is suitable for problems posed inpolygonal domains. The geometric part of the optimization is ignored andunnecessary in this scheme, because the boundary consists of the straightsides of the polygon. The functional discretization is based on estimates ofthe behavior of the functions on the boundary. These estimates are obtainedfrom an eigenfunction analysis. The advantages of this scheme are that itdoes not require any error calculation, and the optimized grid is obtainedeven before the problem is solved. However, the disadvantage of this schemeis that the eigenfunction analysis cannot be performed for a domain withsmooth and curved boundaries.

Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

644 Boundary Elements

In this paper, a new scheme is proposed in which we have the same ad-vantage as that of the scheme of Ref.[9], i.e., the error indicators need notbe calculated, and in addition, the inapplicability of the scheme of Ref.[9]for curved boundaries is eliminated. For a greater success in grid optimiza-tion, the geometric and functional discretizations should go together in acoordinated fashion. Otherwise, a conflict may arise on a part of the bound-ary with the geometry requiring small elements and the functional behaviorallowing large elements or vice versa. The geometry will require small el-ements when the local curvature of the boundary is large, and similarly,the functional behavior will require small elements when the curvature ofthe function is large. From this standpoint, both of these factors can beconsidered simultaneously with the largest of the various curvatures locallycontrolling the grid.

The local curvatures of the geometry can be calculated even before theproblem is solved, but the calculation of the curvatures of the functions canbe performed only after an approximate solution is obtained. From thisaspect, the present scheme is a two step grid optimization scheme. Onebegins with a grid optimized with respect to the geometry, solves the prob-lem on this grid to obtain an estimate of the functional behavior, calculatesthe curvatures of all the functions, optimizes the grid with respect to thegeometry and the functions, and finally solves the problem a second timeon this overall optimized grid. In this paper, the combined geometric andfunctional scheme is presented for Laplace problems in two-dimensions. Theoutput from this scheme is a set of straight elements with linear variationof the functions on each element. Such arrangement of elements is chosen,because (i) it is computationally least expensive, and (ii) exact analyticintegration can be performed.

The information about a smooth curved boundary is conveniently pro-vided to a boundary-element solver in a discrete fashion, and the functionalinformation is available from the solver in a discrete fashion. Hence, in thefirst step of the algorithm, the discrete data are interpolated by a smoothfunction for easy evaluation of derivatives. Magnitudes of these derivativesare directly related to the local curvature which controls the size of the el-ements. In the following section, the method of interpolation using cubicsplines is presented.

Cubic Splines

Consider the boundary of a domain that is composed of several piecewisesmooth curves. For the sake of this discussion, consider one of these curvesthat is defined by Nj number of definition points (z;,yi). These definitionpoints are different from the ultimate grid points. One starts off with a largenumber of definition points, and the number of definition points is usually


Boundary Elements 645

much larger than the number of grid points. For each curve, we define aboundary fitted arc-length variable, say s. The direction of travel, clockwiseor counter-clockwise, on these curves during the integration depends onwhether we are solving an internal or an external problem. This directionof travel automatically fixes the origin of 3. The value of s can be calculatedfrom the expression

a,+i = aj + {(«,-+! - xj)> + (y;+i - y,-)'}* , ; = 1, 2, • • • , N< - 1 (1)

with si = 0. Now, two ordered pairs {xj,Sj]j = 1,2, • • • , Nj} and {%/_,, Sj\j =1, 2, • • • , Nd} can be formed. In these ordered pairs, x and y are consideredto be functions of s. When only the geometric optimization is considered,the optimization is controlled by x and y only. However, when geometricand functional optimizations are considered together, the potential u andflux q will be included in the list in addition to x and y.

Consider one generic ordered pair {Fj,Sj]j = 1,2, •••,̂ } and fit aspline S(s) such that

5(3,-) = Fj,j = l,2,...,N+ (2)

In each interval {sj < s < Sj+i; j = 1, 2, • • • , TVj — 1}, a cubic spline denotedby Sj is expressed as

S,(s) = Fj + S\3j)(a - aj) + C,(s - 3j)> + £3(3 - atf (3)

where S'(sj) is the unknown first derivative of S(s) at 3 = 3j. The constantsCi and 63 are obtained by enforcing the conditions

%+i) = F,+i, (4)

s;(̂,) = s'(W (5)

The expressions for Ci and 63 are

- Sj)* Sj+i

,-F̂ ) (̂3,) +03 = - - — H -- - - — — . (()

*

The unknown first derivative S'(sj) is calculated by enforcing the conditionof continuity of the second derivative

5j'(j,-+i) = s;Vi(3,+o- (8)

This continuity condition yields the relation



The continuity of the second derivative, Eq.(8), cannot in general be en-forced at the two end-points of a segment. Thus, Nd — 2 number of equa-tions of type (9) can only be written. However, Eq.(9) contains Nd numberof unknown first derivatives, and one will fall short of two equations. Thetwo necessary additional conditions are derived by employing different ap-proaches for different possibilities.

Case I:When the boundary of the domain consists of only one curved segment andthis segment wraps around and join smoothly with itself, the continuityof first and second derivatives are required at the joining point. In thatsituation, one writes the two additional conditions as

_̂,WJ (10)

and

S'(3l) = S'(3Nj. (11)

When the boundary of the domain consists of several segments that do notjoin smoothly with each other, or when the boundary consists of only onesegment that wraps around but does not join smoothly with itself, the twofollowing conditions of continuity of third derivative at two selected pointsare enforced.

(12)

and

After the ordered pair {F,, 3j\j = 1, 2, • • • , Nd} is interpolated by usingthe cubic spline, the next task of discretization can be considered.

Tolerance and Optimization

Consider a spline interpolation S(s) that is defined in a finite interval. Wewish to approximate this function by an assembly of linear functions. Thenumber of these linear pieces is not fixed, and the knots are also unknown.The goal is to calculate the number of pieces and the knot positions by



demanding a certain level of accuracy. We begin our analysis by assumingthat 5(3) is quadratic. Then, within a sub-interval [s*, Jfc+i], S(s) can be

exactly expandable as

S(3) = 5(3,) + S'(3k)(i ~ 3k) + S"(3.)(3 - Srf (14)

where 5' and 5" are the first and second derivative, respectively. In thesame interval, the linear approximation L(s ) can be written as

L(s) = 3(3,) + '~'" [5(3»+i) - S(*k)}. (15)•Sfc+1 - -Sfc

The difference between 5(3) and L(s) is the error in the approximation.It can be easily shown that the error is maximum at the mid-point of theinterval; and the maximum value e^ax is given by

Denoting the length of the interval by h, one can write

(17)

In this equation, if one inserts the value of the allowable error or thetolerance for e,nax, the required element length can be calculated. As men-tioned before, the element length is primarily controlled by the curvature ofthe function that is being interpolated. Large curvature means large secondderivative, and large second derivative demands small element length.

Equation(17) is based on the assumption that 5(3) is quadratic. How-ever, it can be extended to include higher order functions by following theanalysis given by Powell[10, page 262]. This extended expression is of the

form

f 18)^

For a given tolerance, by employing Eq.(18), a finite interval can bebroken up into elements in a marching process. The marching process beginsfrom one end of the interval. The second derivative at this end is calculatedby differentiating the spline interpolation. An element length is calculatedfrom this derivative by using Eq.(17). Using this element length, the otherend point of the element is located. The second derivative at the secondend point is now calculated. Now, we have two second derivatives availableat the two end points of the element. These two derivatives are insertedin Eq.(18), and an improved element length is calculated. The computercode of this research goes through a maximum number of five successive



improvements of the length of each element. After the first element lengthis obtained, the process marches on to the second element, and continuesuntil the whole interval is covered with elements.

The Algorithm

The algorithm is developed in such a way that the optimization schemeremains uncoupled from the boundary-element solver, and can be run in-dependently. In the following paragraphs, the optimization scheme is de-scribed for one curved segment. When there are several curved segments,this scheme should be employed to each of these segments and one segmentat a time.

The optimization scheme begins with NJ number of definition pointsdefining the geometry. , From this data, two ordered pairs {xj,3j\j —1,2, • • • , ./V<f} and {yj,Sj]j = 1,2, • • • , A/j} are created. Two spline inter-polations x(s] and y(s) are derived from these ordered pairs. For two splinefunctions, the first estimate of the step size is computed from

and the step size is improved by utilizing the equation

The subscript 1 with c denotes the specified tolerance for this first level ofgeometric optimization.

Consider a grid, optimized with respect to geometry alone, that containsN\ number of nodes. This grid is then utilized to solve the problem by usinga boundary-element solver for Laplace equation.

The output from the solver is then combined with the specified boundaryconditions to form four vectors each of length N\. These four vectors arethe sets of Cartesian coordinates of the nodes, the potential at the nodes,and the flux at the nodes. The Cartesian coordinates are then utilizedto calculate the arc length coordinate s\ and four ordered pairs, namely,[xj,Sj]j = 1,2, •••,#,,}, {yj,3j\j = l,2,--.,JVd}, {uj,Sj]j = 1,2, • • • , N*},and {qjj3j]j — 1,2,- •• , TVj}, are constructed. Among these ordered pairs,•u and q are the potential and the flux, respectively. Four spline interpo-lations are now obtained from these pairs, and the grid is optimized withrespect to all the four variables. The chosen tolerance for this second levelof optimization is £2- This tolerance could be same as or different from thetolerance c% chosen in the first level. However, one must not begin witha high value of c% and demand a lower €2- The optimized grid from the



second level with say TV] number of nodes is then utilized to obtain the final

solution of the problem.

Before we present the results for the complete process, we would testthe accuracy of the spline assisted optimization process by considering the

geometric optimization only.

Geometric Optimization

For the testing of the geometric optimization, a 2:1 ellipse is considered.We begin with 25 definition points on each quarter of the ellipse. Thesepoints are distributed in such a way that their x-coordinates are at equalintervals. This distribution is chosen due to convenience, and it is nota requirement. The optimized location of the nodes are obtained fromgeometric consideration only. The optimization program is executed for therange 0.001 < e\ < 0.01. In order to measure the closeness of the discretizedboundary to the original boundary, we define an error parameter as

- Apply

Pell(21)

Ov.*-Ul

0.010

0.008

0.006

0.004

0.002

0.0000.000 0.002 0.004 0.006 0.008 0.010 0.012

Tolerance

Figure 1: (O) Variation of tolerance (ci) with error (8A). ( ) Line of unit

slope.



where A^u is the area of the ellipse, Apoiy is the area of the polygon afterdiscretization, and P^n is the perimeter of the ellipse. In Fig.(l), 8A isplotted against the chosen tolerance Ci.

The staircase pattern of this plot needs some explanation. In somesituations, the grid optimizer meets the specified tolerance level not byadding nodes, but by repositioning the same number of nodes. AlthoughSA decreases due to this repositioning, but this decrease is very small. Insome other situations, the optimizer cannot meet the tolerance criteriononly by repositioning, and it is forced to add one extra node. When thishappens SA drops abruptly. However, one important observation is that forall chosen tolerance values 8A < £1- In other words, the error in geometrydiscretization is guaranteed to be less than the chosen tolerance. This factis emphasized by drawing the broken line with unit slope in Fig.(l).

In Figs. (2-3) the two grids after geometry optimization with two dif-ferent values of g% are shown. The crowding of the nodes at the nose of theellipse is notable.

1.50

1.25

1.00

0.75

0.50

0.25

0.000.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

Figure 2: Geometry optimization of quarter-ellipse with Ci = 0.01.

ResultsAs described before, the optimization is performed in two levels with toler-ances £1 and £2- The error in the final solution depends on both of thesetolerances. However, one must follow the requirement that e\ < €2- Inorder to examine the relationship between the tolerances and the error inthe solution, three problems with known solutions are examined. The errorin the numerical solution is gauged by an error parameter defined as


8 =

Boundary Elements

A I /saw - / I <#

651

(22)

where / is the exact solution, /BEM is the boundary-element solution, andintegration is over the boundary of the domain. Further, the variable /,in this expression of the error parameter, is the unknown flux on Dirichletboundaries or the unknown potential on Neumann boundaries. The quan-tity 8 is a measure of the integrated and normalized absolute error.

1.50

1.25

1.00

>» 0.7

0.50

0.25k

o.oo-0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

Figure 3: Geometry optimization of quarter-ellipse with e\ = 0.001.

1.0-

Oc.0

0.0 -

r* c _

1 n —- 1 .U

Da

a

aH

I

Q

1a

a

D

m

__jl_

ti

Q

a

B

ai

m

iaD

aQ

EX

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

X

Figure 4: Geometry optimization of Rankine oval with &i = 0.01.



Example 1A Rankine oval with source strength 3 and distance from origin to sourceof 1.5 is placed in a uniform flow. The number of definition points on theoval is 61. The grid after geometry optimization with &i — 0.01 is shownin Fig.(4), and the grid after combined geometry and function optimizationwith £2 = 0.01 is shown in Fig.(5).

1 n -l.U

Oc _.3

On «

Oc _.3

In —.U

ELQQfLBHa

Ea

m

•

a

D

•

"C

a

• 1

_a_

'

" ,

a

•

" B

ta

B _ _Qa

, GQD.~~ru~v~~

3

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

X

Figure 5: Combined geometry and function optimization of Rankine ovalwith ei = 0.01.

The oval problem is solved for various values of the tolerance, £1, andthe ratio £2/^1- For each case, the error parameter, 6, is calculated. InFig. (6), 8 is plotted against £2- A line with unit slope is also shown in thisfigure. One notable fact in this figure is that the data points cluster aroundthe line with unit slope. This shows that the tolerance that is specified evenbefore the problem is solved provides a good upper bound for the integratedboundary error in the final solution.

In Table(l), the variation of the number of nodes withall of these cases, z\ = £2-

is shown. In

Table 1: Variation of number of nodes with tolerance c% = £2-

Tolerance Number of Nodes

0.010.0080.0060.0040.0020.001

283136446186



0.012

0.010

2 0.008UJ

*"O

0.006

0.004

0.002

<

I

8//X/

> *i c

I/'/.£

1 '

AX

9'jx'7 O

A,'X. / ^^ 0 I

tD

]

x/X i//A 'a

XXXXX

a

0.000 0.002 0.004 0.006 0.008 0.010 0.012

Tolerance

Figure 6: Rankine oval problem; variation of boundary error, 6, with toler-ance, 62, for various ratios, £2/^1. (°) ratio = 1; (o) ratio = 5; (A) ratio =

10.

1.50

1.25

1.00

0.75

0.50

0.25

0.00-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

Figure 7: Ellipse at 25°; combined geometry and function optimization with

E2 = 0.01.



Example 2Potential flows with angles of attack of 0° and 25° past a 2:1 ellipse areconsidered next. All the calculations begin with 100 definition points onthe ellipse.

In Fig.(7), the combined geometry and function optimized grid is shownfor the ellipse at an angle of attack of 25°. For this optimization £2 is setto 0.01. When this grid is compared with the geometry optimized gridof Fig.(2), we find that for only geometry optimization, 7 nodes on thequarter ellipse are sufficient; whereas the combined geometry and functionoptimization requires 9 nodes on the quarter ellipse. The two additionalnodes are required for proper modeling of the acceleration of the flow aroundthe nose of the ellipse.

In Figs.(8-9), 8 is plotted against €2 for various values of the ratio Cg/Ci-In all the cases, the specified tolerance provides an excellent bound for theultimate integrated boundary error.

LU

c3Offl

n m n -

Onno -

n nnfi -

n r\r\A -

n nnoU.OUZ

/ |

xXX/

XXXX

-,_ J,/

'

Q '1 •

/

, . l

,/'

'

0.000 0.002 0.004 0.006 0.008 0.010 0.012

Tolerance

Figure 8: Ellipse at 0°; variation of boundary error, 8, with tolerance, 62,for various ratios, 62/^1- (O) ratio = 1; (o) ratio = 5; (A) ratio = 10.

"DiscussionThe proposed grid optimization algorithm does not require any error cal-culation. It can satisfy the needs for appropriate geometry and functionalmodeling. The algorithm can calculate the requirements on the element sizeimposed by either the geometry or the function and identify the strongercontrolling factor. The tolerance, that is chosen during the optimizationand even before the final solution is obtained, provides an excellent esti-



mate of the ultimate error expected in the final solution. This enables theuser to be assured of certain level of accuracy before the solver program is

executed.

The optimization algorithm is developed as a stand alone program un-coupled from the boundary-element solver. As a consequence, users canadd the power of optimization without modifying their solvers. Extensionof the algorithm to other areas, e.g., elastostatics, is straightforward. Thefirst level of geometry optimization will remain the same; and in the secondlevel, the optimization should be performed with respect to the geometry,the two components of traction, and the two components of displacement.

ok.UJ

flj"Oc3OCO

0.012-

0.010 "

0.008 -

r\ nncO.OUb

0.004-

///// „ 1

1 /i //s1

1

///s/

[.'

////

Ii:• fi /s 8 <i

/

/

i ^ '

/

i3

0.000 0.002 0.004 0.006 0.008 0.010 0.012

Tolerance

Figure 9: Ellipse at 25°; variation of boundary error, 8, with tolerance, e?,for various ratios, EI/CI. (O) ratio = 1; (o) ratio = 5; (A) ratio = 10.

One fact that became clear during this research, but was not incorpo-rated, was that for consistency, one must normalize the variables beforeoptimization. The other factor, that may speed-up the optimization pro-cess and retain symmetries of the geometry, is the utilization of dynamicprogramming technique in the element length calculations. These improve-ments will be reported in the future.

References

1. Alarcon, E., Reverter, A., and Molina, J., Comput. Struct., 20, p.151,

1985.

2. Ingber, M.S. and Mitra, A.K.,/n*. J. Num. Meth. Engrg., 23, p.2121,

1986.



3. Cerrolaza, M.Y. and Alarcon, E., Comm. Appi Num. Meth., 3,p.335, 1987.

4. Rencis, J.J. and Mullen, R.L., . Comp. Mech., 3, p.309, 1988.

5. Leal, R.P. and Mota Scares, C.A., Comput. Struct., 30, p.841, 1988.

6. Rank, E., Int. J. Num. Meth. Engrg., 28, p.1335, 1989.

7. Rencis, J.J. and Jong, K.Y., Comp. Meth. Appl. Math., 73, p.295,1989.

8. higher, M.S. and Mitra, A.K., Engrg. Anal. Bound. Elmnt., 9, p.13,1992.

9. Mitra, A.K., J. Comp. Phys., 100, p.246, 1992.

10. Powell, M.J.D., Approximation Theory and Methods, Cambridge Uni-versity Press, New York, 1981.


Documents

Invited Paper Engineering Mechanics, Iowa State University,€¦ · Department of Aerospace Engineering and Engineering Mechanics, Iowa State University, /owa Introduction The discretization