GEOMETRICAL MODELING PACI(AGEoden/Dr._Oden_Reprints/... · 2008-04-09 · GEOMETRICAL MODELING...

GEOMETRICAL MODELING PACI(AGE

L.Demkowicz~ A.Bajer*, K.Banas*The Texas Institute of Computationallviechanics

The University of Texas at AustinAustin, Texas 78712

Austin, August 1992

Abstract

This document provides a short description of the first edition of a package formodeling 2-D and 3-D manifolds using multiblock technique. The update includesin particular implicitly parametrized triangles and rectangles allowing for modelingobjects resulting from intersections of surfaces in lIl.3.

Acknowledgement: The support of this work by the Office of Naval Research under GrantN00014-92-J-1l61 is gratefully acknowledged.

'On leave from the Section of Applied Mathematics, Technical University of Cracow, ul. Warszawska 24,31-155, Krakow, Poland.

1

Contents

1 Theory Manual 1

1.1 Main assumptions of the code 1

1.2 Data structure. . 3

1.3 Catalog of points 6

1.3.1 Regular point (labcl=l) 6/

1.3.2 Implicit point (label=50) . 6

1.4 Catalog of curves . . . . . . . . . 7

1.4.1 Segment of straight line (label=l) . 7

1.4.2 Quarter of a circle (label= -1) . 7

1.4.3 Segment of a circle (label= -2) . 8

1.4.4 Implicit curve (label=50) . 8

1.5 Catalog of triangles . . . . . . . . 10

1.5.1 Plane triangle (label=l) . 10

1.5.2 Spherical triangle (octant of a sphere, label= -1) . 10

1.5.3 Circular triangle (quarter of a circular disk, label= -2) 11

1.5.4 Part of a spherical triangle (label= -3) . . . . . . . . . 13

1.5.5 Implicit triangle (label=50) 13

1.5.6 Implicit spherical triangle by area coordinates (labcl= -4) . 18

1.6 Catalog of rectangles . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6.1 Bilinear quadrilateral (label= 1) . . . . . . . . . . . . . . . 20

1.6.2 Transfinite interpolation with linear blending functions quadrilateral(label= 2) . . . . . . . . . . . . . 20

1.6.3 Cylindrical rectangle (label= -1) . 20

1.6.4 Implicit rectangle (label=50) . . 21

1.7 Catalog of triangular prisms . . . . . . 24

1.7.1 Linear x linear prism (label= 1) 211

1.7.2 Triangular shell (label= -1) .. 24

1.8 Catalog of rectangular prisms . .

1.8.1 Trilinear prism (label= 1) .

1.8.2 Rectangular shell (label= -1).1.9 Catalog of surfaces .

1.9.1 Plane normal to a given vector and passing through a point

1.9.2 Plane passing through three points

1.9.3 Sphere .... o'

1.9.4 Cylinder

2 User Manual

2.1 Input deck file.

2.2 Output interface

3 Example Manual

3.1 Example 1 - a house

3.2 Example 2 - a sphere

3.3 Example 3 - a cylinder

3.4 Example 4 - a spherical shell .

3.5 Example 5 - a cylindrical shell

3.6 Example 6 - a cylinder with spherical incaps shell

3.7 Example 7 - a mockO model with rings

3.8 Example 8 - a mockO model with tower

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1 Theory Manual

1.1 Main assumptions of the code

Geometrical entities of interest are classified into the following classes:

• points,

• (segments of) curves, I

• triangles,

• rectangles (quadrilaterals),

• triangular prisms (blocks),

• rectangular (quadrilateral) prisms (blocks).

All quantities falling into the different classes are prescribed in one, global cartesian systemof (physical) coordinates Xi, i = 1, ... , N, with N = 2 for planar and N = 3 for threedimensional problems.

Each of the classes has a coresponding catalog of objects. For example, the simplestclass of objects, the points, contains usual geometrical points uniquely defined by settingjust their physical coordinates, but it will also include points with an additionally specifiedplane allowing for a construction of curves, triangles and quadrilaterals meeting at the pointand tangent to the plane.

Mathematically, each of the geometrical objects is identified with its correspondingparametrization. More specifically, by a curve for instance, we mean a transformation Xc

defined on the reference segment of line [0,1] into]RN, N = 2,3

In the same way we define the remaining entities. The reference objects are displayed inFig. 1. If two or more maps are used to parametrize a specific geometric figure, we shalldistinguish between them as separate objects.

Conceptually, the parametrizations are classified into two classes:

• explicit parametrizations

1

1

o 1

I

1 I ,

01 1 • ~1

1

I*--//

II

~--- 1

/

Figure 1: Reference triangle, rectangle, triangular, and rectangular prisms.

2

• implicit parametrizations

In the first case, a map is defined explicitly by coding a specific (not necessarily simple)formula. The simplest examples include objects which are characterized uniquely by entitiesof lower dimension and a specific interpolation rule. For instance, a segment of straight lineis uniquely defined by setting its endpoints, a plane triangle, by identifying its edges (andusing the linear interpolation rule), etc.

In the second case, the maps are defined implicitly, by specifying (in general, nonlinear)systems of equations for coordinates x,y,z. In order to assess the value of the map forsome specific choice of reference coordinates, the system has to be solved using typicallyNewton-Raphson iterations.

The structure of the code is open. New definitions can be easily added to the catalogs,enriching the existing capabilities of the code.

One extra matter must be very strongly emphasized - the compatibility of parametriza-tions. Let us use an example to explain it. Fig. 2 displays the reference triangle used todefine various triangles from the catalog of triangles and a corresponding (curvilinear) tri-angle in the physical space. In turn, each of the triangle sides, a curve, is defined throughits own parametrization defined on the reference interval [0, 1]. 'With

• Xt - the map defining the triangle,

• Xc - the map defining the curve,

• ~c - the standard map from [0,1] onto the corresponding side of the reference triangle,depending on the orientation of the side alone,

there must be:Xc = Xt 0 ec

In the case depicted in Fig. 2 for instance, we have

1.2 Data structure

e E [0,1]

(1.1)

(1.2)

A geometrical object consists of points, curves, triangles, rectangles and, in the case of 3-Dmanifolds, also of triangular and/or rectangular prisms.

3

~2

1

o 1 a 1

Figure 2: Compatibility of parametrizations.

4

Within each of the classes, the items are denumerated using consequtive integers startingfrom 1. The total number of each of the entities is specified by the following variables:

• NRPOINT - number of points

• NRCURVE - number of curves

• NRTRIAN - number of triangles

• NRRECTA - number of rectangles

• NRBTRIA - number of triangular prisms

• N RBRECT - number of rectangular prisms

For each of the geometrical entities a (variable) number of both integer and real parametersdefining the geometry of the item and its relative (topological) position wrt other entities,must be remembered. These parameters are stored in two data base arrays:

• IGEOMTR - for integer parameters

• RGEOMTR - for real parameters

Note that only double precision reals are used in the code!

As the number of parameters vary for different items, the first entries in both IGEOA1TRand RGEOMTR must be stored explicitly. This is done using the following arrays:

• NPOINT(2,MAXPO) for points

• NCURVE(2,MAXCU) for curves

• NTRIAN(2,MAXTR) for triangles

• NRECTA (2,A1AXRE) for rectangles

• NBTRIA (2,MAXBT) for triangular prisms

• NBRECT(2,NfAXBR) for rectangular prisms

5

For example, for a triangle with number no, the first corresponding integer variable is storedin IGEOMTR(it), where it = NTRIAN(l,no) and the first real parameter is stored in RGE-OMTR(it), where it = NTRIAN(2,no). Order of both integer and rcal paramcters and theirdefinitions for a particular kind of a geometrical entity is precisely defined and can be foundin the corresponding catalog. As usually, parameters AIAX PO, ... , j\1 AX B R denote theanticipated maximum number of points, curves etc. and are defined through the" parameter"FORTRAN statement.

For the sake of implicit par<l;µ1etrizations, additional information for dealing with surfacesmust be stored. The related variables include

• NRSURF - number of surfaces

• NSURF(2,MAXSU) contains addresses to the first corresponding data in IGEOA1TRand RGEOMTR

with MAXSU the maximum number of surfaces anticipated.

1.3 Catalog of points

The following integer parametrs are common for all kinds of points:

• label defining the particular kind of point,

• number of curves meeting at the point,

• numbers of the curves meeting at the point listed in an arbitrary order.

1.3.1 Regular point (label=l)

Integer parameters: noneReal parameters:

• Xl"", XN - physical coordinates of the point

1.3.2 Implicit point (label=50)

Integer parameters:

6

• NS1, NS2, NS3 - numbers of surfaces intersecting at the point

Real parameters:

• X, y, Z - a starting point for Newton-Raphson iterations

1.4 Catalog of curves

The following integer parametei·s are common for all kinds of curves:

• label defining the particular kind of curve,

• endpoints numbers,

• number of triangles and rectangles meeting along the curve,

• +/- nicknames of the figures meeting along the curve, where the nicknames arcdefined as:

for triangles, and

numberofthetriangle * 10 + 1

numberoftherectangle * 10 + 2

(1.3)

(1.4)

for rectangles (quadrilaterals).The plus sign indicates that the orientation of the figure is consistent with the orien-tation of the curve, otherwise the sign is negative.

1.4.1 Segment of straight line (label=l)

Integer parameters: noneReal parameters: none

1.4.2 Quarter of a circle (label= -1)


• x, y, z - coordinates of the center of a circle

7

~he parametrization map

1.4.3 Segment of a circle (label= -2)


• x, y, z - coordinates of the centre of a circle

1.4.4 Implicit curve (label=50)I

Integer parameters:

• NSl, NS2, NS3, NS4 - numbers of surfaces defining the curve (see Fig. 3)

Real parameters: none

Parametrization

Denoting the surface equations by

epi (x, y, z) = 0 , i = 1, ... , 4

(X(~),Y(O,Z(O)T, ~ E [0,1]is defined by the following nonlinear system of equations

{

ept (x, y, z) = 0

ep2(x, y, z) = 0

(1 - e)ep3(X, y, z) + eep4(X, y, z) = 0

Differentiating the above system with reference coordinate e, we obtain

aepl ax aept ay aept az--+--+--=0ax oe oy oe oz oe

Oep2ox Oep2ay Oep2oz--+--+--=0ax oe ay o~ oz o~

_ (1_ t) [O'P3 ox Oep3oy 0'P3 Oz]'P3+ ., ox ae + oy o~ + oz o~

t [oep't ox oep.. oy Oep1oz] _ 0+ ep4 +., ax a~ + ay a~ + az a~ -

8

NS2

Figure 3, Implicit curve. Numbers of surfaces defiuing the curve.

9

Once the coordinates x, y, z are determined, the above lineal'system is solved for derivative

( )

ToX oy ozoe oe' o~

1.5 Catalog of triangles

The following integer parameters are common for all kinds of triangles:I

• label defining the particular kind of triangle,

• numbers of curves constituting sides of the triangle with positive sign if the orien-tation of the curve is consistent with that for the triangle and negative sign otherwise;

and in the 3-D case additionally:

• nicknames of the prisms adjacent to the triangle with the nicknames defined asfollows

for triangular prisms, and

numberoftheprism * 10 + 1

numberoftheprism * 10 + 2

(1.5)

(1.6)

for the rectangular prisms. Note that all prisms adjacent to a triangle must be trian-gular prisms!

1.5.1 Plane triangle (label=l)


Note that all sides of the triangle must be segments of straight line.

1.5.2 Spherical triangle (octant of a sphere, label= -1)

Integer parameters: noneReal parameters: noneNote that all sides of the triangle must be quarters of cil'des (label= -1)

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Parametrization

Given octant of a sphere, shown in Fig. 4, we construct the corresponding parametrization,mapping the reference triangle onto the octant; by introducing auxiliary spherical coordinates0, 'l/J and considering composition of two singular maps

• inverse map to the map mapping (O,~) x (-~, 0) in the (0, 'l/J) plane onto the referencetriangle

• the usual spherical coordinates parametrization

x~ = r cos 'l/J cos 0x~ = r cos 'l/J sin (}

I . ./.X3 = rSln 'I-'

(1. 7)

(1.8)

It can be checked that the resulting composition is al with bounded derivatives. By com-bining the map with a rigid rotation, we can parametrize an arbitrary octant of a sphere.

1.5.3 Circular triangle (quarter of a circular disk, label= -2)


Note that two sides of the triangle must be segments of straight line (label=l) and oneside must be quarter of a circle (label= -1)

Parametrization

Given a quadrant of a circle shown in Fig. 5, we introduce the auxiliary domain (0, ~) x (0, R)in polar coordinates (),r and consider two singular maps

• transformation mapping ((), r) coordinates onto the reference triangle:~ _ r 28<,,1 - "R--;-6 = fl(1- 2:)

11

(1.9)

1

1

o

,77".

2 I

Xl1

X~

Figure 4: Parametrization of a spherical triangle

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• the usual polar coordinates parametrization

x~ = r cos ()

x~ = l' sinO

x~ = 0

(1.10)

By combining the inverse of the first map with the second map and next with a rigid rotationin 11l3, we can parametrize quadrant of a circular disc, arbitrary located in IC?

/

1.5.4 Part of a spherical triangle (label= -3)


Note that the sides of the triangle must be listed using the order specified in Fig. 6.

Parametrization

In the same way as with the octant of a sphere, the transformation is composed of composetwo singular maps:

• map mapping reference triangle onto (O,~) x ((3,~) plane

o =

t/J =

11"~1 .----.1.~1 + ~2

(!3 - ~) . (el + e2) + ~

• the usual spherical coordinates parametrization

x ~ = r cos t/J cos 0

x~ = r cos t/J sin 0

x~ = r sin t/J

1.5.5 Implicit triangle (label=50)

Integer parameters:

13

Xl3

1

o

o

I

Xl1

1

r

Tr2

R

Figure 5: Parametrization of a circular triangle

14

7r I i2

X'

~2

~

1l NC1

~~Ir"I)\I

01 ~1I r

X'

1

./2

X'1

{3 , ,

Lo IT

2

Figure 6: Parametrization of part of a spherical triangle. Note the denumeration of curvesNC1, NC2, NC3 defining the triangle.

15

• NS1, NS2, NS3, NS4 - numbers of surfaces defining the triangle (see Fig. 7)


Parametrization


<pj(x,y,z) = 0 , i = 1, ... ,4/

the parametrization map(x (et,6), y (6,6), z (6,6))T

is defined by the following nonlinear system of equations

<PI ( X, y, z) = 0

(1- I(e))<p2(x,y,z) + J(e)<f'4(X,y,Z) = 0

6 [(1 - It (el + e2)) <ps(x, y, z) + ft (el + e2) <r'3(X, y, z)]

+6 [(1 - 12 (6 + 6)) <ps(x, y, z) + h (6 + 6) <P3(X, y, z)] = 0where

• 'Ps(x, y, z) is the equation of the degeneratcd sphcre with centre at the first vertex ofthe triangle

<Ps(X, y, z) = (x - xd2 + (y - vd2 + (z - zd2

with Xt, YI, Zl) coordinates of the first vertex

.e=!6-6+!2 6 + e2 2

• I(e), 11(e), 12(e) are stretching functions determined by requesting the compatibilityof the triangle parametrization with the existing, specified parametrization of its sides:

(1 - I(e))<P2 (x~(e)) + J(e)<P4 (x~(e)) = 0

(1 - II (ed)<ps (x~(ed) + II (6)<P3 (x~(ed) = 0

(1 - 12(e2))'Ps (X~(e2)) + 12(e2)<P3 (X~(e2)) = 0

with x~(ed, x~(e), X~(e2) being the paramctrizations of the sides.

As in the case of the implicit curve definition, the two systems of equations are differentiatedwith respect to the reference coordinates to yield a linear system of equations for derivativesof x,y,z with el,6.

16

Figure 7: Implicit triangle. Numbering of surfaces defining the triangle.

17

1.5.6 Implicit spherical triangle by area coordinates (label= -4)


Parametrization

The area coordinates for a point Xl on the octant of a sphere are defined as

PiAi = P

where Pi are the areas of the curvilinear triangles determined by geodetics passing throughXl, compo Fig. 8, and P is the total area of the triangle

With cp and X being the usual spherical coordinates the (nontrivial) formulas for PI and P2

read as follows:

r.p + . ( sin1/J ) . ( sinr.p·sin1/J )arCSIn - arCSInJSin21/J + cos2 cp cos21/J JSin21/J + cos2 r.p cos21/J

. ( cosr.p·sin1/J ) . ( sin1/J )r.p + arCSIn - arCSInJSin21/J + cos21/J sin2 r.p JSin21/J + cos21/J sin2 <p

Relating the area coordinates to the reference coordinates in the usual way

7r

=62"

the system of equations is solved for r.p and X which in turn determine coordinates xL i =1,2,3 and, upon a possible translation and/or notation, the physical coordinates x, y, Z.

As usual in differentiating the equations, one obtains the corresponding system of equa-tions for derivatives of x, y, z with the reference coordinates.

1.6 Catalog of rectangles

The following integer parameters are common for all kinds of rectangles:

18

Figure 8: Implicit spherical triangle by area coordinates.

19

• label defining the particular kind of rectangle,

• numbers of curves constituting sides of the rectangle with positive sign if the orien-tation of the curve is consistent with that for the rectangle and negative sign otherwise;

and in the 3-D case additionally:

• nicknames of the prisms adjacent to the rectangle.

1.6.1 Bilinear quadrilateral (label=l)


Note that all sides of the quadrilateral must be segments of straight line.

1.6.2 Transfinite interpolation with linear blending functions quadrilateral (la-bel= 2)

Formulas for the curves constituting sides of the quadrilateral are extended to the wholereference rectangle using the classical transfinite interpolation and linear blending functions.Integer parameters: noneReal parameters: none

1.6.3 Cylindrical rectangle (label= -1)


Note that two opposite sides of the quadrilateral must be segments of a straight lineand another two sides must be quarters of circles (label= -1).

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Parametrization

The usual, cylindrical coordinates parametrization (comp Fig. 9)

x~ = rcos (i6)

x~ = rsin (%6)I

x~ = H6

is combined with an appropiate rigid rotation in I1l3.

1.6.4 Implicit rectangle (label=50)

Integer parameters:

(1.11)

• NS1, NS2) NS3) NS4) NS5 - members of surfaces defining the rectangle (see Fig. 10).


Parametrization


'Pj(X) = 0, i = 1, ... ,5

the parametrization map

x(6,6)is defined by the following system of equations:

<Pl(X)

(1 - e2)

+6(1 - 6)

~1

o{(I - fl(~d)' <Ps(x) + fl(~l)' <P3(X)}

{(I - 12(6))· <Ps(x) + !2(el)' <P3(X)} = 0

{(I - h(e2))' <P2(X) + h(e2)' <P4(X)}

{(I - f4(~2))' <P2(X) + fl(~2)' <P4(X)} = 0

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1 I i

o 1

Figure 9: Parametrization of a cylindrical rectangle.

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~(X,y,z)=o

4'2 ( X , Y , Z) =a

~(X,y,z)=o

~(X,y,z)=o

<P (X,y,Z)=O1

Figure 10: Implicit rectangle.

23

where fi(~), i = 1, ... ,4 are the stretching functions determined by requesting the com-patibility of the rectangle parametrization with the existing, specified parametrization of itssides:

with

(1 - fl(~d)

(1 - f2(6))

(1 - h(6))

(1 - f4(~2))

CPs (X~(el)) + fl(el)

CPs (x~(6)) + 12(6)

CP2 (x~(6)) + f3(6), 2

CP2 (Xc(~2)) + h(~2)

CP3 (x~) = 0

CP3 (x~) = 0

CP4 (x~) = 0

CP4 (x~) = 0

being the parametrizations of the sides.

1.7 Catalog of triangular prisnls

The following integer parameters are common for all kinds of triangular prisms:

• label defining the particular kind of prism,

• nicknames of figures constituting sides of the prism, with positive sign if the ori-entation of the figure is consistent with the outward normal unit vector for the prismand negative sign otherwise.

1.7.1 Linear x linear prism (label=l)


Note that both bases must be plane triangles and lateral sides are necessary bilinearquadrilaterals.

1.7.2 Triangular shell (label= -1)


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The parametrization of the prism is obtained by a simple linear interpolation of thecorresponding parametrizations for the bases of the prism

By taking in particular, for instance, identical parametrizations of spherical triangles withdifferent radii, one obtains a parametrization of the corresponding spherical shell with con-stant thickness, etc.

1.8 Catalog of rectangular prisms

The following integer parameters are common for all kinds of rectangular prisms:

• label defining the particular kind of prism,

• nicknames of figures constituting sides of the prism, with positive sign if the ori-entation of the figure is consistent with the outward normal uni t vector for the prismand negative sign otherwise.

1.8.1 Trilinear prism (label=l)


Note that all the sides must be bilinear quadrilaterals.

1.8.2 Rectangular shell (label= -1)


The parametrization of the prism is obtained by a simple linear interpolation of thecorresponding parametrization for the bases of the prism

By taking in particular, for instance, identical parametrization of cylindrical rectangles (label= - 1) with different radii, one obtains a parmetrization of the corresponding cylindrical shellwith constant thickness, etc.

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1.9 Catalog of surfaces

1.9.1 Plane normal to a given vector and passing through a point


• xo, Yo, Zo - coordinates of the point

• a, b, c - components of the normal vector

Equation:(x - xo)a + (y - yo)b + (z - zo)c = 0

1.9.2 Plane passing through three points


• xl, YI, ZI - coordinates of the first point (xd

• X2, Y2, Z2 - coordinates of the second point (X2)

• X3, Y3, Z3 - coordinates of the third point (X3)

Equation:

where

1.9.3 Sphere


• Xo, Yo, Zo - coordinates of the centre of the sphere

• r - radius of the sphere

Equation:

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1.9.4 Cylinder


• Xo, Yo, Zo - coordinates of the centre point for the base of the cylinder

• dl, d2, d3 - a vector parallel to the axis of the cylinder

• r - radius of the cylinder I

Equation:

where:VI = A . (:v - :vol

with transformation matrix A whose third column coincides with the unit vector of vectord(d1, d2, d3) and the first two are some vectors orthogonal to d and orthogonal to each other.

2 User Manual

2.1 Input deck file

All data corresponding to a particular manifold, prepared by the user, are read from fileinput using the following format:

NDIM,MANDIM - dimension of the space (2 or 3) and manifold (2 or 3),

NRSURFACE - the total number of surfaces.

For each of the surfaces a separate data follows in the format prescribed in the catalog ofsurfaces.

Example

A sphere with coordinates (-1, 2, 1.5) and radius 4.

3 (label identifying a sphere)

-1,2,1.5 (coordinates of the centre of the sphere)

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4 (radius of the sphere)

The input file continues:

NRPOINT - the total number of points

For each of the points a separate data follows in the format prescribed in the catalog ofpoints.

I

Example

A regular point with coordinates (-5,0,1.2) and curved with numbers 1,8,10,14, meeting atthe point.

1 (label identifying regular points),

4 (number of curves meeting at the point),

1,8,10,14 (numbers of the curves),

-5,0,1.2 (coordinates of the point).


NRCURVE - the total number of curves

For each of the curves a separate data follows in the format prescri bed in the catalog ofcurves.

Example

A segment of straight line connecting points with numbers 5 and 6 with triangles number5,11 and rectangles 7 and 18 meeting along the line.

1 (label identifying segments of straight line),

5,6 (endpoints numbers),

4 (total number of triangles and ractangles metting along the curve),

-51,111,-72,182 (nicknames of the triangles and rectangles with orientation).

28


NRTRIAN - the total number of triangles

For each of the triangles a separate data follows in the format prescribed in the catalog oftriangles.

Example

2-D case, a plane triangle with curves number 6,11,18 constituting its sides, all with orien-tation consistent with that of the triangle.

1 (label identifying plane triangles),

6,-11,18 (numbers of curves constituting the sides of the triangle with orientation).

Example

3-D case, a plane triangle with curves number 6,11,18 constituting its sides and traingularprisms 9 and 12 adjacent to the triangle. Curve number 11 has an orinetation opposite tothat of the tiaringle, otherwise the orientations arc consistent.


6,-11,18 (numbers of curves constituting the sides of the triangle with negative sign indi-cating opposite orientations),

91,121 (nicknames of the prisms adjacent to the triangle).

Example

3-D case, a plane triangle with curves number 6,11,18 constituting its sides and only onetriangular prism number 13 adjacent to the triangle (lying on the surface of a 3-D mani-fold being modeled). Curve number 11 has an orientation opposite to that of the triangle,otherwise the orientations are consistent.


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6,-11,18 (numbers of curves constituting the sides of the triangle with negative sign indi-cating opposite orientations),

131 (nickname of the single prism adjacent to the triangle).


NRRECTA - the total number of rectangles

For each of the rectangles a separate data follows in the format prescribed in the catalog ofrectangles.

Example

2-D case, a bilinear quadrilateral with curves number 7,12,9 and 13 constituting its sides.Out of the four curves only curve 9 has an orientation consistent with that of the rectangle.

1 (label identifying bilinear quadrilaterals),

-7,-12,9,-13 (numbers of curves constituting the sides of the rectangle, negative signs indi-cate curve orientation opposite to the orientation of the quadrilateral).

Example

3-D case, a bilinear quadrilateral with curves number 7,12,9 and 13 constituting its sidesand two prisms adjacent to the quadrilateral, a triangular prism number 8 and a rectangularprism number 6. All curves have an orientation opposite to the orientation of the rectangle.

1 (label identifying bilinear quadrilaterals),

-7,-12,-9,-13 (numbers of curves constituting the sides of the rectangle, negative signs in-dicate curve orientation opposite to the orientation of the quadrilateral),

81,62 (nicknames of the prisms adjacent to the rectangle).

End of the input file for a 2-D case. In a 3-D case the input file continues as follows:

NRBTRIA - the total number of triangular prisms

For each of the prisms a separate data follows in the format prescribed in the catalog oftriangular prisms.

30

Example

A linear x linear prism with triangles number 1 and 13 constituting its' bases and rcctanglesnumber 5,9,14, constituting its lateral sides, where only triangle 1 and rectangles 5,9 haveorientation consistent with the outward normal unit vector for the prism.

1 (label identifying linear x linear prisms),

11,-131,52,92,-141 (nicknames of the trianglcs nad rectanglcs constituting the sides of theprism with negative sign~ indicating orientation opposite to the outward unit vectorfor the prism).


NRBRECT - the total number of rectangular prisms

For each of the prisms a separate data follows in the format prescribed in the catalog ofrectangular prisms.

Example

A trilinear prism with rectangles number 1 and 13 constituting its bases and rectanglesnumber 5,9,14 and 15 constituting its lateral sides. Orientation of rectangles 1,5 and 9 isconsistent with the outward normal unit vector for the prism, for the remaining rectanglesis opposite.

1 (label identifying trilinear prisms),

12,-132,52,92,-142,-152 (nicknames of the rectangles constituting the sides of the prismwith negative signs indicating orientation opposite to the outward unit vector for theprism).

End of the input file for a 3-D case.

2.2 Output interface

The output interface includes a number of routines listed below and allowing for the cal-culation of such geometrical quantities as physical coordinates of points lying on a specificgeometrical entity and their derivatives wrt parameters defining the object.

31

subroutine point(No, X)

input:

No - number of a point

output:

X - coordinates of the point /

subroutine curve(No,Xi, X,Dxdxi)

input:

No - number of a curve

Xi - a parameter specifying a point on the curve (between 0 and 1)

output:

X - coordinates of the point

Dxdxi - derivatives of the coordinates wrt to the parameter

subroutine trian(No,Xi, X,Dxdxi)

input:

No - number of a traingle

Xi - parameters specifying a point in a reference triangle

output:


Dxdxi - derivatives of the coordinates wrt to the parameters

32

subroutine recta(No,Xi, X,Dxdxi)

input:

No - number of a rectangle

Xi - parameters specifying a point in a reference square

output:



subroutine btrian(No,Xi, X,Dxdxi)

input:

No - number of a triangular prism

Xi - parameters specifying a point in the reference prism

output:



subroutine brecta(No,Xi, X,Dxdxi)

input:

No - number of a rectangular prism

Xi - parameters specifying a point in the reference prism

output:



33

3 Example Manual

3.1 Example 1 - a house

The following is the input file for a two-block structure shown in Fig. 11.

3,3 dimension of the space and manifold

10 number of points

1 point 1 label

3 number of curves meeting at point 1

1,4,5 the curves numbers

4.,0.,0. point 1 coordinates

1 point 2 label




1 point 3 label



O.,2. ,0. point 3 coordinates

34

1.07

26

X3

10

17- / . r"\16

0, 1.5

1

1

9I

@ I.... _./

,/

,-(j) CDl,,/

,/,/

,/

./

5

5

• 1 points numbers

lines numbers

figures numbers

Figure 11: Example 1 - a house. Problem data

35

1 point 4 label




1 point 5 label


5,9,12,13 the curves numbers

4.,0.,1.5 point 5 coordinates

1 point 6 label




1 point 7 label




1 point 8 label



36


1 point 9 label



1.,1. ,3. point 5 coordinates

1 point 10 label




17 number of curves

1 curve 1 label

1,2 endpoints of curve 1

2 number of figures meeting along the curve

12,32 nicknames of the figures

1 curve 2 label



-12,-42 nicknames of the figures

:37

1 curve 3 label



-12,52 nicknames of the figures

1 curve 4 label



12,-62 nicknames of the figures

1 curve 5 label




1 curve 6 label




1 curve 7 label


38



1 curve 8 label




1 curve 9 label



-32,22,11 nicknames of the figures

1 curve 10 label



42,-22,-72 nicknames of the figures

1 curve 11 label



-52,-22,21 nicknames of the figures

39

1 curve 12 label



62,22,82 nicknames of the figures

1 curve 13 label




1 curve 14 label




1 curve 15 label




1 curve 16 label



40


1 curve 17 label


2 number of figures meeting along the curveI


2 number of triangles

1 triangle 1 label

9,14,-13 sides numbers

11,0 adjacent prisms

1 triangle 2 label

11,16,-15 sides numbers


8 number of rectangles

1 rectangle 1 label

1,-2,-3,4 sides numbers


41

1 rectangle 2 label

9,-10,-11,12 sides numbers


1 rectangle 3 label

1,6,-9,-5 sides numbers


1 rectangle 4 label

-2,7,10,-6 sides numbers


1 rectangle 5 label



1 rectangle 6 label



1 rectangle 7 label



42

1 rectangle 8 label



1 number of triangular prisms I

1 triangular prism 1 label

-21,11,-22,72,82 sides numbers

1 number of rectangular prisms

1 rectangular prism 1 label

-12,22,32,42,-52,-62 sides numbers

3.2 Example 2 - a sphere

See file input sphere.

3.3 Example 3 - a cylinder

See file input cylinder.

3.4 Example 4 - a spherical shell

See file input sphe7'ical shell.

43

44

...... '..

Figure 13: Example 3 - a cylinder

45

-46

3.5 Exanlple 5 - a cylindrical shell

See file input cylind1'ical shell.

3.6 Exaluple 6 - a cylinder with spherical incaps shell

See file input cylinder with incaps shell.

l

3.7 Exaluple 7 - a 1110ckOl1l0del with rings

See Fig. 17 for geometric data and input mockO rings.

3.8 Example 8 - a mockO luodel with tower

See Fig. 19 for geometric data and input mockO tower.

,.p.C/.)

.,r

Figure 15: Example 5 - a cylindrical shell J:y

49

C.no

9.31

0.58

I. 29.09 .133.93

Thickness =0.021

Figure 17: Data for a mockO model with rings.

- 4.36

51

~tv

I .. 29.09

33.93

Thickness=O.021

Figure 19: Data for a mockO model with tower.

_I

4.36

53

-v'"t:loSo....:.:::uoS~

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GEOMETRICAL MODELING PACI(AGEoden/Dr._Oden_Reprints/... · 2008-04-09 · GEOMETRICAL MODELING...

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