Chapter 10 Two-Level Computation

Chapter 10 Two-Level Computation

Chapter 10 TWO-LEVEL COMPUTAnON 10.1 RECAPITULATION We have developed an extensive theory for construction of basis functions over algebraic eleme...

356KB Sizes 2 Downloads 130 Views

Chapter 10

TWO-LEVEL COMPUTAnON

10.1

RECAPITULATION We have developed an extensive theory for

construction of basis functions over algebraic elements.

The fundamental theorem of Noether and

associated algebraic geometry concepts were invaluable in the analysis.

It was demonstrated that by

appropriate choice of nodes any prescribed degree basis can be constructed for any well-set algebraic element.

Practical guidelines were presented for

numerical quadrature over elements.

The analysis is

of interest for its mathematical content and for its resolution of questions of existence and uniqueness of rational basis functions for continuous patchwork approximation over algebraically reticulated regions. The practical utility of this development has yet to be demonstrated. When performing a finite element computation, one chooses elements appropriate for the configuration being studied.

For most two-variable problems of

concern, triangles and rectangles (and less often parallelograms) are used. 314

In some situations

RATIONAL FINITE ELEMENT BASIS

isoparametric coordinates are introduced to model curved boundaries and interior interfaces.

In three

dimensions, isoparametric parallelepipeds allow considerable modeling flexibility.

There seems to be

no great need for the more general algebraic elements and associated rational basis functions. Before the discovery of isoparametric coordinates there seemed to be no great need for more general elements than triangles and parallelograms.

Now

isoparametric elements are considered indispensable for solving a wide class of problems.

Computation

sophistication rises to meet the limits of the available methodology.

It is not unreasonable to

expect that eventually problems will arise for which the wider class of algebraic elements will be essential.

For example, element geometry may be

dictated by physical considerations that preclude three- or four-sided two-dimensional elements. theory awaits such application.

The

There are computa-

tional techniques of current concern for which the more versatile algebraic elements provide a needed flexibility.

In this chapter we describe two such

techniques. 10.2

SYNTHESIS The term "synthesis" has been used in the

nuclear reactor physics literature to denote a class of computations in which the solution to a problem in n variables is approximated by linear combinations of solutions to auxiliary problems in n - 1 . variables.

The coefficients of combination are a

function of the nth variable.

A common example is

the solution of the neutron group diffusion equa315

TWO-LEVEL COMPUTATION

tions in three space dimensions (Kaplan, 1965). Let H (x,y) for m = l,2, ••• ,M be a collection of m precomputed two-dimensional basis functions. These functions are often obtained by solving two-dimensional problems characteristic of different elevations (constant value for z) of a reactor.

Let W. (z) for 1

i = 1,2, •.• ,1 be the one-variable hat functions [see Eq. (1.5)].

s

{U(x,y,z)

I

The synthesis approximation space is

I

U

2:

M

~

UimWi(z)Hm(X,y)}.

(10.1)

i=l m=l The free parameters U. (sometimes called combining 1m or mixing coefficients) are determined by the usual Ritz-Galerkin procedure to yield a "best" approximation in space S to the solution to the threedimensional problem.

By applying interior boundary

conditions on the U. , we can select subsets of the 1m H for use in each of several axial zones, thereby m reducing the number of unknowns at each elevation z .. This type of synthesis computation is extremely

1

useful and provides a flexible model for analysis of complex configurations in a far more efficient manner than would be possible with detailed threedimensional discretization.

An additional degree of

flexibility is introduced by multichannel synthesis, where a coarse finite element network is on the (x,y)-plane as shown in Fig. 10.1.

s~perimposed

Let

W~(X,y) be the wedge basis function associated with node j over element s. The trial function within the x,y region of element s is

316

RATIONAL FINITE ELEMENT BASIS

I

5

U (X,y,Z) =

M

J

s UimjWi(Z)Wj(X,y)Hm(X,y) . L L L i=l m=l j=l

(10.2)

Suppose H is obtained by solving (for each m) a m two-dimensional problem having N nodes in the (x,y)plane.

Three-dimensional discretization without

synthesis would require IN nodes.

The single-

channel synthesis approximation of Eq.(lO.l) has IM free parameters.

The multichannel mockup of Eq.

(10.2) has IMJ free parameters. In reactor computations it is not uncommon for N to exceed 10,000 and for M to be equal to (about) 5. The full freedom of (10.2) is not used in practice.

Fig. 10.1.

Superimposed coarse structure.

Some of the synthesis computation advantages are lost when MJ becomes large.

The multichannel

structure is introduced to permit a more realistic representation of gross tilts in (x,y) variation. This is accomplished even if we restrict groups of points to have the same combining coefficients. Within selected elements, we restrict U. . to be 1m)

317

TWO-LEVEL COMPUTATION

the same for all nodes j and each im.

In Fig. 10.2,

6

6

Fig. 10.2.

2

Restricted multichannel nodes.

there are four elements over which this restriction is imposed. figure.

The values for j are indicated in the

Thus the 26 nodes in the plane have only

six degrees of freedom.

We note that element 1 is

ill set but that this poses no problems. element we have 1

U (x,y,z) =

I

In this

M

II

(10.3)

i=l m=l We have eliminated the difficulty accompanying illset elements by the method of restricted variation described in Section 1.3.

The superimposed coarse-

element trial function is equal to unity over the ill-set element. The fine-element structure over which H (x,y)

m

was computed in advance is convenient for use as an integration grid when applying the Ritz-Galerkin method to obtain the discrete three-dimensional equations with the trial function of Eq.(lO.2).

For

this discretization we must evaluate W~(x,y) at each node of the integration grid. 318

J

This is easily done

HAIIUNAL FINITE ELEMENT BASIS

with the rational wedge functions.

On the other hand,

isoparametric basis functions would lead to difficulties.

These functions cannot be readily evaluated

in terms of x and y.

The rational basis functions

are ideally suited for multichannel synthesis application.

They permit a wide class of algebraic

elements.

They may be evaluated easily for integra-

tion over the fine structure.

The rational wedges

provide precisely the flexibility needed to bridge the gap between detailed three-dimensional computations and the single-channel synthesis of Eq.(lO.l). In the next section, we consider another application which is closely related to multichannel synthesis but which arises in entirely different circumstances. 10.3

COARSE MESH REBALANCING

A new

technique for accelerating convergence of

linear iterative procedures was introduced by the author (Wachspress, 1966, Chap. 9).

One interrupts

the linear iteration periodically with a variational acceleration computation in which the last iterate appears as a base function multiplied by a patchwork coarse mesh correction function.

Nakamura (1971)

subsequently analyzed use of a coarse mesh finite element representation for the mUltiplicative correction.

He and Froehlich (1967) provided more

extensive theoretical foundations for the method and performed many numerical experiments.

They called

this class of computations "coarse mesh rebalancing". The two-level structure is analogous to that of multichannel synthesis, but the motivation is quite

319

TWO-LEVEL COMPUTATION

different.

Let u be the solution to a discrete fine

mesh problem and let u{x,y) be the continuous patchwork approximation with nodal values equal to the components

of~.

Let

~t

be the approximation to u

obtained after t iterations.

Let the coarse finite

element basis functions be Wj{x,y). function

Then the trial

for the coarse mesh rebalancing is J

L

Ut{x,y)

j=l

(lO.4)

U . W. (x, y ) u (x , y) • t J J

There are J free parameters in the rebalancing computation, and the coarse representation is chosen so that these.parameters may be determined by an efficient direct method such as block Gaussian elimination (Wachspress, 1966, p. 26).

When the

"best" values for the U. are substituted into (lO.4), , J the nodal values of the resulting Ut(x,y) replace the components of resumed.

~t

and the linear iteration is

Periodic coarse mesh rebalancing has been

demonstrated to be an effective means for accelerating convergence for a wide assortment of problems. The rational basis functions permit use of much more diverse coarse mesh representations than was hitherto possible.

Physical boundaries and interfaces may be

represented by algebraic curves.

Thus availability

of general algebraic elements broadens the scope and enhances the effectiveness of coarse mesh rebalancing.

320

RATIONAL FINITE ELEMENT BASIS

10.4

CONCLUDING REMARKS This monograph has dealt primarily with the

theory of construction of rational basis functions for continuous patchwork app rox.Lma t Lon over any algebraically reticulated region.

In this last

chapter we have indicated possible applications, some of which have already been implemented.

Even

in this applications-oriented chapter the analysis has not been supported by extensive numerical studies.

such studies are not essential in a work

devoted to the laying of theoretical foundations. This is a research monograph in which further areas for study have been indicated.

Refinements,

modifications, and new concepts will undoubtedly be introduced as this research is pursued.

It is

hoped that this work has awakened a new appreciation of the interdependence of diverse pure and applied mathematics disciplines as tools in our endeavors to model natural phenomena.

321

References Aziz, A. K. (1972). "The Mathematical Foundations of the Finite Element Method." Academic Press, New York. Bocher, M. (1907). "Introduction to Higher Algebra. " MacMillan, New York. Carr, G. S. (1970). "Formulas and Theorems in Mathematics." Chelsea, Bronx, New York. Ciarlet, P. G. and Raviart, P. A. (1972a). Interpolation theory over curved elements, with applications to finite element methods, "Computer Methods in Applied Mechanics and Engineering," pp. 217-249. NOl!'th-Holland Pub!., Amsterdam. Ciarlet, P. G. and Raviart, P. A. (1972b). n General Lagrange and Hermite interpolation in R with applications to finite element methods, Archiv~

111-222.

for Rational Mechanics and Analysis

~,

pp.

Courant, R. and Hilbert, D. (1953). "Methods of Mathematical Physics," Vol. I. Interscience, New York. Coxeter, H. S. M. (1961). Geometry. " Wiley, New York. Ergatoudis, J. in plane analysis. Wales, Swansea.

"Introduction to

(1966}. Quadrilateral elements Masters thesis, University of

Faulkner, T. E. (1960). Oliver & Boyd, Edinburgh.·

"Projective Geometry."

Felippa, C. A. and Clough, R. W. (1970). The finite element method of solid mechanics, "Numerical Solution of Field Problems in Continuum Physics," Vol. II, SIAM-AMS Proc., pp. 210-252. Providence, Rhode Island.

322

RATIONAL FINITE ELEMENT BASIS

Froehlich, R. (1967). A theoretical foundation for coarse mesh variational techniques, Proc. Intern.

Conf. Res. Reactor Utile and Reactor Math., Mexico, D. F. !;J p. 219.

Fulton, W. (1969). Benjamin, New York.

"Algebraic Curves."

Hadamard, J. (1952). "Lectures on Cauchy's Problem in Linear Partial Differential Equations." Dover, New York. Herbold, R. J., Schultz, M. H., and Varga, R. S. (1969). Quadrature schemes for the numer1cal solution of boundary value problems by variational techniques, Aequ. Math. l, pp. 96-119. Hodge, w. V. D. and Pedoe, D. (1968). "Methods of Algebraic Geometry," Vols. 1 and 2. Cambridge Univ. Press, London and New York. Hoppe, V. (1973). Finite elements harmonic interpolation functions, "The of Finite Elements with Applications," J. R. Whiteman, pp. 131-142. Academic London.

with Mathematics edited by Press,

Irons, B. M. (1966). Numerical integration applied to finite element methods, Conf. on use of Digital Computers in structural Eng. univ. of Newcastle.

Jordan, W. B. (1970). Plane isoparametric structural element, KAPL Memo M-7112, UC-32,

Math and Computers TID-4500,

54th ed.

Kaplan, S. (1965). Synthesis methods in reactor analysis, "Advances in Nucl. Sci. and Eng.," Vol.III. Academic Press, New York. Macaulay, F. S. (1916). "Algebraic Theory of Modular Systems," Cambridge tracts in Math & Math Physics #19. McLeod, R. and Mitchell, A. R. (1972). The construction of basis functions for curved elements in the finite element method, J. Inst. Math App1. 10, pp.

382-393.

323

REFERENCES

McLeod, R. and Mitchell, A. R. (1975). The use of parabolic arcs in matching curved boundaries in the finite element method, J. Inst. Math. App1. (in publication) . Mitchell, A. R., Phillips, G., and Wachspress, E. L. (1971). Forbidden elements in the finite element method, J. Inst. Math. App1. ~, pp. 260-269. Muir, T. (1960). "Theory of Determinants," four vols. Dover, New York. Nakamura, S. (1971). Coarse mesh acceleration of iterative solution of neutron diffusion equations, Nuc1. Sci. and Eng. 43, pp. 116-120.

Strang, G. J. and Fix, G. (1973). "An Analysis of the Finite Element Method." Prentice Hall, Englewood Cliffs, New Jersey. synge,' J. L. (1957). "The Hypercircle in Mathematical Physics." Cambridge, London and New York. van der Waerden, B. L. (1950). "Modern Algebra," Vol. 2 (Engl. trans.). Ungar, New York. van der Waerden, B. L. (1939). "Algebraische Geometrie." Springer Pub!., New York. Varga, R. S. (1971). "Functional Analysis and Approximation Theory in Numerical Analysis." SIAM Publ., Philadelphia, Pa. Verdina, J. (1971). Point Transformations." New Jersey.

"Projective Geometry and Allyn & Bacon, Rockleigh,

Wachspress, E. L. (1966). "Iterative Solution of Elliptic Systems." Prentice Hall, Englewood Cliffs, New Jersey. Wachspress, E. L. (1971). A rational basis for function approximation, Proc. Conf. on App1. Numerical Anal~, Dundee. "Springer Verlag Lecture Notes in Math." Vol. 228, pp. 223-252.

324

RATIONAL FINITE ELEMENT BASIS

Wachspress, E. L. (1973). A rational basis for function approximation: Part II, curved sides, J.

Inst. Math. App1.

!!,

pp.

83-104.

wachspress, E. L. (1974). Algebraic geometry foundations for finite element computation, Conf. Numerical Sol. Diff. Egs., Dundee, "Springer Verlag Lecture Notes in Math." Vol. 363, pp. 177-188. Wait, R. (1971). A finite element for three dimensional function approximation, Proc. Conf. on Appl. Numerical Anal., Dundee. "Springer Verlag Lecture Notes in Math." Vol. 228, pp. 348-352. Walker, R. (1962). Dover, New York.

"Algebraic Curves."

Zienkiewicz, o. C. (1971). "The Finite Element Method in Engineering Science" (2nd ed.) . McGraw Hill, New York. Zienkiewicz, O. C. and Cheung, Y. K. (1967). "Finite Element Methods in Structural Mechanics." McGraw Hill, New York.

325