Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

Подождите немного. Документ загружается.

378 K. Gürlebeck and W. Sprößig

7.3 Shallow Water Equations

The French mathematician Barré de Saint-Vainant described the viscous, rotating

shallow water equations as follows:

∂

u +(u ·∇

)u =−2a ∧u +νΔ

u −g∇

h, (7.31)

∂

H =−(u ·∇

)H +H ∇

·u, (7.32)

H := h(t, x) −h

(t, x), (7.33)

u(0,x) = u

, (7.34)

h(0,x) = h

. (7.35)

Here H denotes the total depth of the ﬂuid, h

(t, x) determines the bottom surface

topography, 2a ∧u is the

Coriolis force, g is the gravity acceleration, ν is the dynamic

viscosity

, and u is the velocity. A good reference is the book [13].

7.3.1 Assumptions for the Formulation of St. Venant’s Equations

• The direction of the rotation axis coincides with the e

axis.

• The angular velocity |ω| is induced by Coriolis.

• The external forces reduce here to the gravity.

• ∂

p =−ρg, i.e., the vertical pressure gradient equals the buoyancy forces.

• The scale of horizontal motion is much larger than the scale of vertical motion.

• The ﬂuid is incompressible, homogeneous, has a constant density (here normal-

ized to 1).

A good reference is the dissertation by Fengler [4].

7.4 Forecasting Equations

Shallow water equations are similar to a model of forecasting equations restricted

on the sphere

∂

u +(v ·∇

)u = νΔ

u −

∇

p −2a ∧u +F in G, (7.36)

∂

T =−c

(u ·∇

)T +Q, (7.37)

∂

ρ =−(u ·∇

)ρ, (7.38)

with the

vector derivative (Günter’s gradient) ∇

,theBeltrami operator Δ

,thesur-

face gradient

∇

p,theheating rate Q, and the surface divergence ∇

·u. The vector

of outer forces F also contained the so-called

apparent gravity that is gravity reduced

by the centrifugal force. Moreover, it is assumed that ∇

·u =0 with boundary con-

ditions on ∂G =:Γ .

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 379

8 Numerical Examples

Our goal is now to show by a numerical example how good the performance of the

described methods is. For simplicity, we consider only a stationary Navier–Stokes

equation. As we have seen above, the solution of such problems is the “kernel” of

the iteration procedures. This means that it is of greatest importance to solve these

problems efﬁciently. This numerical example is taken from the paper [3], where

also Navier–Stokes equations in unbounded domains were studied. The calculations

were done by N. Faustino.

Table 1 Error estimates and convergence for Example 1

h Grid Number of iterations u

n+1

−u

 (last iteration)

15×5 ×53 4.6875 ×10

−7

2/37×7 ×73 1.8243 ×10

−7

1/29×9 ×93 1.0317 ×10

−7

1/313×13 ×13 3 4.2204 ×10

−8

1/417×17 ×17 3 2.7564 ×10

−8

Fig. 1 Error estimation for different mesh sizes during the iteration

380 K. Gürlebeck and W. Sprößig

For the domains Ω =[−2, 2]

, we use a cubic equidistant grid of (N + 1) ×

(N +1) ×(N +1), N even, points with mesh size h =

corresponding to



mh =(m

h, m

:−N/2 ≤m

≤N/2



For the stopping criteria, we use the discrete Sobolev norm, .:=.

p+3/α

with

p =4 and α =1/2, and ﬁx δ = 10

−6

as error tolerance.

Example 1 Consider the stationary Navier–Stokes in Ω =[−2, 2]

f(x)=



|, |x



Note that the right-hand side is not differentiable.

Table 1 shows the error after the third iteration for different grids. Figure 1 illus-

trates the development of the error during the iteration process. We can see that the

error decreases quickly. This is in accordance to our theoretical results. The iteration

was based on a ﬁxed-point principle. We can also observe in Example 1 that the lack

of regularity at the origin is not an obstacle for the method.

9 Conclusions

The presented methods were also applied to the solution theory of some Galpern–

Sobolev equations

, an alternative model to the three-dimensional generalized

Korteweg–de Vries equations (we refer to [9]). Meanwhile, results are published

also on a three-dimensional analogue to Airy‘s equation for waves in rigid materi-

als [17].

References

1. Bahmann, H., Gürlebeck, K., Shapiro, M., Sprößig, W.: On a modiﬁed Teodorescu transform.

Integral Transform. Spec. Funct. 12(3), 213–226 (2001)

2. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Research Notes Math.,

vol. 76. Pitman, London (1982)

3. Faustino, N., Gürlebeck, K., Hommel, A., Kähler, U.: Difference potentials for the Navier–

Stokes equations in unbounded domains. J. Differ. Equ. Appl. 12(6), 577–596 (2006)

4. Fengler, M.J.: Vector spherical harmonic and vector wavelet based non-linear Galerkin

schemes for solving the incompressible Navier–Stokes equation on the sphere. Shaker, D386

(2005)

5. Gürlebeck, K.: Grundlagen einer diskreten räumlich verallgemeinerten Funktionentheorie und

ihrer Anwendungen. Habilitationsschrift, TU Karl-Marx-Stadt (1988)

6. Gürlebeck, K., Hommel, A.: On ﬁnite difference potentials and their applications in a discrete

function theory. Math. Methods Appl. Sci. 25(16–18), 1563–1576 (2002)

7. Gürlebeck, K., Sprößig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems.

Birkhäuser, Basel (1990)

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 381

8. Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers.

Mathematical Methods in Practice. Wiley, New York (1997)

9. Gürlebeck, K., Sprößig, W.: Representation theory for classes of initial value problems with

quaternionic analysis. Math. Methods Appl. Sci. 25, 1371–1382 (2002)

10. Hung, L.U.: Finite Element Analysis of Non-Newtonian Flow. Springer, Berlin (1989)

11. Ishimura, N., Nakamura, M.: Uniqueness for unbounded classical solutions of the MHD equa-

tions. MMAS 20, 617–623 (1997)

12. Le, T.H., Sprößig, W.: On a new notion of holomorphy and its applications. Cubo Math. J.

11(1), 145–162 (2008)

13. Majda, A.: Introduction to PDE’s and Waves for Atmosphere and Ocean. Courant-Lecture

Notes, vol. 9. Courant Institute of Mathematical Sciences, New York (2003)

14. Przeworska-Rolewicz, D.: Algebraic theory of right invertible operators. Stud. Math. 48, 129–

144 (1973)

15. Rajagapol, K.R.: Mechanics of Non-Newtonian Fluids. Pitman, London (1998)

16. Ryabenskij, V.S.: The Method of Difference Potentials for Some Problems of Continuum

Mechanics. Nauka, Moscow (1987) (in Russian)

17. Schlichting, A., Sprößig, W.: Norm estimations of the modiﬁed Teodorescu transform with

application to a multidimensional equation of Airy’s type. In: Simos, Th.E., Psihoyios, G.,

Tsitouras, Ch. (eds.) Numerical Analysis and Applied Mathematics. American Institute of

Physics (AIP) Conference Series, vol. 1048 (2008)

18. Sprößig, W., Gürlebeck, K.: Representation theory for classes of initial value problems with

quaternionic analysis. Math. Methods Appl. Sci. 25, 1371–1382 (2002)

19. Sprößig, W.: Fluid ﬂow equations with variable viscosity in quaternionic setting. In: Advances

in Applied Clifford Algebras, vol. 17, pp. 259–272. Birkhäuser, Basel (2007)

20. Tasche, M.: Eine einheitliche Herleitung verschiedener Interpolationsformeln mittels der Tay-

lorschen Formel der Operatorenrechnung. Z. Angew. Math. Mech. 61, 379–393 (1981) (in

German)

Part VI

Crystallography, Holography and

Complexity

Interactive 3D Space Group Visualization

with CLUCalc and Crystallographic

Subperiodic Groups in Geometric Algebra

Eckhard M.S. Hitzer, Christian Perwass,

and Daisuke Ichikawa

Abstract The Space Group Visualizer (SGV) for all 230 3D space groups is a stan-

dalone PC application based on the visualization software CLUCalc. We ﬁrst ex-

plain the unique geometric algebra structure behind the SGV. In the second part

we review the main features of the SGV: The GUI, group and symmetry selection,

mouse pointer interactivity, and visualization options. We further introduce the joint

use with Hahn (Space-group Symmetry, 5th edn., International Tables of Crystal-

lography, vol. A, Springer, Dordrecht, 2005). In the third part we explain how to

represent the 162 so-called subperiodic groups of crystallography in geometric al-

gebra. We construct a new compact geometric algebra group representation symbol,

which allows us to read off the complete set of geometric algebra generators. For

clarity, we moreover state explicitly which generators are chosen. The group sym-

bols are based on the representation of point groups in geometric algebra by versors.

1 Introduction

Crystals are fundamentally periodic geometric arrangements of molecules. The di-

rected distance between two such elements is a Euclidean vector in R

. Intuitively

all symmetry properties of crystals depend on these vectors. Indeed, the geomet-

ric product of vectors [13], combined with the conformal model of 3D Euclidean

space [2, 15, 25–27, 38, 39, 42], yields an algebra fully expressing crystal point and

space groups [19, 22, 28–30, 33, 45]. Two successive reﬂections at (non)parallel

planes express (rotations) translations, etc. [8, 10]. This leads to a one-to-one corre-

spondence of geometric objects and symmetry operators [35] with vectors and their

products, ideal for creating a suite of interactive visualizations using CLUCalc [44]

and OpenGL [28, 29, 31, 45].

E.M.S. Hitzer (



)

Department of Applied Physics, University of Fukui, Bunkyo 3-9-1, 910-8507 Fukui, Japan

e-mail: hitzer@mech.u-fukui.ac.jp

E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,

DOI 10.1007/978-1-84996-108-0_18, © Springer-Verlag London Limited 2010

385

386 E.M.S. Hitzer et al.

For crystallographers, the subperiodic space groups [37] in 2D and 3D with only

one or two degrees of freedom for translations are also of great interest.

We begin in Sect. 2 by explaining the representation of point and space groups

in conformal geometric algebra. Next we explain the basic functions of the software

implementation, called Space Group Visualizer [46] in Sect. 3. Finally in Sect. 4

we show how to construct a new compact geometric algebra group representation

symbol for subperiodic space groups (Frieze groups, rod groups, and layer groups),

which allows us to read off the complete set of geometric algebra generators. For

clarity, we moreover state explicitly which generators are chosen.

2 Point Groups and Space Groups in Clifford Geometric

Algebra

2.1 Cartan–Dieudonné and Geometric Algebra

Clifford’s associative geometric product [13] of two vectors simply adds the inner

product to the outer product of Grassmann

ab =a ·b +a ∧b. (1)

Under this product, parallel vectors commute, and perpendicular vectors anti-

commute:



a, ax

⊥

=−x

⊥

a. (2)

This allows us to write the reﬂection of a vector x at a hyperplane through the origin

with normal a as



=−a

−1

xa, a

−1

. (3)

The composition of two reﬂections at hyperplanes whose normal vectors a, b sub-

tend the angle α/2 yields a rotation around the intersection of the two hyperplanes

by α:



=(ab)

−1

xab,(ab)

−1

. (4)

Continuing with a third reﬂection at a hyperplane with normal c according to the

Cartan–Dieudonné theorem [4, 6, 12, 18] yields the rotary reﬂections and inversions



=−(abc)

−1

xabc, x



=−i

−1

xi, i

=a ∧b ∧c, (5)

where

= means equality up to nonzero scalar factors (which cancel out in (6)). In

general the geometric product S of k normal vectors corresponds to the composition

of reﬂections to all symmetry transformations [22] of 2D and 3D crystal cell point

groups:



=(−1)

−1

xS =



−1

xS =S

−1



S, (6)

Space Group Visualization and Subperiodic Groups in GA 387

Fig. 1 Regular polygons (p = 1, 2, 3, 4, 6) and point group generating vectors a, b subtending

angles π/p shifted to center

Table 1 Geometric [22, 24] and international [20] notation for 2D point groups

Crystal Oblique Rectangular Trigonal Square Hexagonal

Geometric

21 2 3

International 1 2 mmm3m 34m 46m 6

where



S = (−1)

S is the grade involution or main involution in Clifford geometric

algebra. We call the product of invertible vectors S in (6) versor [14, 22, 24, 35, 38],

but the names Clifford monomial of invertible vectors, Clifford group element,or

Lipschitz group element are equally in use [38, 41].

2.2 Two-Dimensional Point Groups

2D point groups [22] are generated by multiplying vectors selected [28, 29, 45]asin

Fig. 1. The index p can be used to denote these groups as in Table 1. For example,

the hexagonal point group is given by multiplying its two generating vectors a, b:

6 =



a, b,R =ab,R

=−1, aR

, bR

, aR

, bR



. (7)

The rotation subgroups are denoted with bars, e.g.,

6. The identities a

= b

= 1

and R

=−1 directly correspond to relations in the group presentation [47]of6.

2.3 Three-Dimensional Point Groups

The selection of three vectors a, b, c from each crystal cell [22, 28, 29, 45]for

generating all 3D point groups is indicated in Fig. 2.Using∠(a, b) and ∠(b, c),we

can denote all 32 3D point groups (alias crystal classes) as in Table 2. For example,

the monoclinic point groups are then (international symbols of Hermann–Maugin:

2/m, m, and 2, respectively)

2 ={c,R=a ∧b =ic,i =cR,1}, 1 ={c, 1},

2 ={ic, 1}. (8)