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
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