Brick patterns on shells using geodesic...

Proceedings of the IASS Annual Symposium 2017“Interfaces: architecture.engineering.science”

25 - 28th September 2017, Hamburg, GermanyAnnette Bogle, Manfred Grohmann (eds.)

Brick patterns on shells using geodesic coordinatesEmil ADIELS*, Mats ANDER a, Chris J K WILLIAMSb

* Department of Architecture, Chalmers University of TechnologyGothenburg, Sweden

[email protected]

a Department of Applied Mechanics, Chalmers University of Technologyb Department of Architecture, Chalmers University of Technology

AbstractWe present two separate strategies for generating brick patterns on free form shells and vaults usinggeodesic coordinates. The brickwork is specified by a surface on which there is a geodesic coordinatesystem satisfying the condition for a constant distance between bed joints. The first strategy integratesthe generation of the geodesic coordinates in a form finding procedure derived from the geometricaland mechanical properties of a shell. The geometric and structural equations are solved using dynamicrelaxation. The second strategy can be applied on an arbitrary surface separating the form finding andbrick pattern generation enabling adaption to different constraints in the design process.

Key words: differential geometry, brick patterns, shells, brickwork, form finding, geodesic coordinates

1. Introduction

Bricks

Shell

Figure 1: Geodesic coordinates on a surfaceguarantees and equal spacing of the coordi-nate curves making it suitable as basis forbrick patterns on free form shells.

Any discussion of brickwork must include the influence of theformat and shape of the brick, and this particularly applies tobrick shells and vaults. The field of geometry and brickworkis historically not as well known or documented as the relatedtopic of stereotomy, that is the 3 dimensional cutting of stones,where the literature goes back to the 16th century in the workby Philibert de l’Orme [3]. The problem of tessellating a freeform surface with a single element type is generally very dif-ficult, if not taking the production method into account. Im-portantly, the geometry of brickwork is never exact. Tradi-tional bricks have an overall shape of a cuboid, but can be bothcurved and twisted, and can when needed be easily modifiedwith a bricklayer’s hammer or a scutch hammer, and possiblya chisel or bolster. The flexible joints, the mortar, make thestructure able to tolerate the deviance of the bricks and adaptto the global geometry. This means that the craftsmen tradi-tionally had influence on the final solution but also that thereexists a resilience in the mathematical formulation.

Old handbook drawings of patterns on vaults are often limitedto the bed joints as in figure 2. The most important geometricalconstraint is therefore that the distance between the bed joints is constant. This property can be foundin a geodesic coordinate system comprising geodesics on a surface and their orthogonal trajectories (seesection 3) which guarantees a constant length between coordinate curves on the surface. This wouldenable the placing of bricks in between the coordinate curves of such system, illustrated in figure 1.

We shall present two strategies to generate the geodesic coordinate pattern on a shell:

Copyright c© 2017 by (name(s) of the author(s) as listed above)Published by the International Association for Shell and Spatial Structures (IASS) with permission.

Proceedings of the IASS Annual Symposium 2017Interfaces: architecture.engineering.science

1. Combined form finding and pattern generation using dynamic relaxation as described in section 4.

2. Generation of patterns on an arbitrary shell from a designer specified curve and thereby separatingform and pattern as described in section 5.

2. The geometrical and structural properties of shells

Figure 2: A drawing of a saucer domeshowing various on how to place the brickpattern. Typically these brick patterns arelimited to the lines of the bed joints and notevery single brick, from Paulsson [6].

The form finding of shell structures involves the specificationof the geometry of a shell in static equilibrium subject to geo-metric constraints. The best known example is the equal meshor Chebyshev net used by Frei Otto for projects such as theMunich Olympic Stadium and the Mannheim Multihalle [5].In an alternative technique Block and Ochsendorf [2] definethe form found state of stress in a shell using Thrust NetworkAnalysis.

The geometry of the coordinate curves on a surface is specifiedby the components of the metric tensor a11, a12 = a21 and a22,using the notation in Green and Zerna [4], also known as co-efficients of the first fundamental form, E, F and G, using thenotation in Struik [7]. The state of membrane stress in a shellis specified by the components of the membrane stress tensor,n11, n12 = n21 and n22, again using the notation in Green andZerna [4].

Thus we have 6 quantities to determine, but only 3 equationsof equilibrium. We therefore need 3 more equations and in thecase of the equal mesh net they are

a11 = a22 = constant (1)

n12 = 0. (2)

Equation (2) ensures that the state of stress corresponds to forces along the lines of the mesh.

However we cannot use an equal mesh net for brickwork because we need a constant spacing betweenbed joints as shown in figure 1. If

a11a22− (a12)2

a11= constant (3)

then there is a constant distance between the curves θ 2 = constant. Green and Zerna [4] use θ 1 and θ 2

for the surface coordinates instead of the u and v which are often used in books which do not use thetensor notation, such as Struik [7].

We can arbitrarily specify that the coordinate curves are orthogonal so that

a22 = constant (4)

a12 = 0. (5)

Thus we now have 2 equations and we require one more. If we stipulate that the principal stresses areperpendicular to the bed and head joints we have

n12 = 0. (6)

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There is no absolute requirement that the principal stresses are perpendicular to the bed and head jointsin brickwork, but we have to specify some condition on stress in order to have a system of equations thatwe can solve. The obvious advantage of the principal stresses being perpendicular to the bed and headjoints is that there is no possibility of sliding along the bed joints.The only difference between equations (1) and (2) and equations (4), (5) and (6) is that in the former wehave a22 = constant, whereas in the latter we have a12 = 0. However, we shall see that this change doesmake the numerical solution more difficult.

3. Geodesic coordinates

θ1 = constantθ1 = constant

θ2 = constant

geodesics

Figure 3: Geodesic coordinates, redrawnfrom Striuk [7]. The orthogonal trajecto-ries intersect the geodesics at a right angleand equal length.

In this section we shall demonstrate the well known propertiesof a geodesic coordinate system using the notation from Greenand Zerna [4], although they do not cover this topic. Fromequations (4) and (5) we have,

0 =∂a22

∂θ 1 =∂

∂θ 1 (a2 ·a2) = 2a2 ·∂a2

∂θ 1 (7)

0 =∂a12

∂θ 2 =∂

∂θ 2 (a1 ·a2) =∂a1

∂θ 2 ·a2 +a1 ·∂a2

∂θ 2 (8)

and therefore since

∂a1

∂θ 2 =∂ 2r

∂θ 1∂θ 2 =∂a2

∂θ 1 (9)

we have

a1 ·∂a2

∂θ 2 = 0. (10)

This last equation means that the geodesic curvature of thecurves θ 1 = constant is zero, or in other words the curvesθ 1 = constant are geodesics on the surface. The shortest dis-tance between two points on a surface is a geodesic and it isa nice application of the calculus variations to show that thisis consistent with the definition of zero geodesic curvature, al-though it is intuitive in the sense that a string stretched across a smooth surface will slide sideways givingthe minimum length and zero geodesic curvature.Coordinates which satisfy (4) and (5), and therefore also (10) are known as geodesic coordinates. Gauss’sTheorema Egregium is particularly elegant in geodesic coordinates. Introducing the coefficients of thesecond fundamental form, b11, b12 = b21 and b22,

∂a2

∂θ 2 = b22a3

∂a2

∂θ 1 =1

2a11

∂a11

∂θ 2 a1 +b12a3

so that

∂ 2a2

∂θ 1∂θ 2 ·a1 =−b22b11

=∂

∂θ 2

(1

2a11

∂a11

∂θ 2

)a11 +

12a11

∂a11

∂θ 212

∂a11

∂θ 2 − (b12)2

=∂

∂θ 2

(1

2√

a11

∂a11

∂θ 2

)√

a11− (b12)2 =

∂ 2√a11

(∂θ 2)2√

a11− (b12)2.

3


Figure 4: Plan and elevations of a brick shell supported on two edges at ground level and free along the other two.Geodesic lines are shown in black

Figure 5: View showing brick pattern

Therefore

K =b11b22− (b12)

2

a11a22− (a12)2 =− 1

a22√

a11

∂ 2√a11

(∂θ 2)2 (11)

or

K =− 1w

∂ 2w∂ s2 (12)

where w is the spacing between geodesics and s is the distance measured along a geodesic. K is the

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Gaussian curvature, that is the product of the two principal curvatures. On a developable surface, suchas a plane or a cylinder, K = 0 so that Cartesian coordinates and polar coordinates are special cases ofgeodesic coordinates.

4. Numerical implementation of equilibrium combined with geodesic coordinatesFigures 4 and 5 show the results of the numerical implementation. The algorithm ensures that

1. The black lines in figure 4 are geodesics,

2. the spacing of the red lines along the black lines is constant and

3. the shell is in equilibrium under a uniform vertical load per unit plan area with the principal stressesparallel to the coordinate curves.

The structure is modelled by a system of pin ended members that are all in compression under downwardsvertical load. The equations are solved using dynamic relaxation and rather than writing out many linesof equations or pseudocode, we consider it better to include the entire code so that readers can run thecode for themselves if they so wish. The reason for this is that the code contains a number of featureswhich we found were necessary for stability, but are difficult to explain in their entirety. For example thevariable ‘loadIncrementFactor’ is there to ensure that the load does not vary too quickly as the spacingbetween geodesics changes. We also found that the relaxation factors for the different equations have tobe ‘tuned’ so that the algorithm converges. Because the structure is in compression it is unstable withnegative force densities or tension coefficients. This means that in the dynamic relaxation the nodes aremoved in the opposite direction to the out of balance force, which is equivalent to having a negativemass.

The program is written in the Processing language www.processing.org which is based on Java, butthe code would be essentially the same in C++ or C# for inclusion in some other graphic environmentsuch as OpenGL or Grasshopper.

5. Geodesic coordinates on arbitrary free form shellsThere are infinite number of ways of constructing a geodesic coordinate system on a smooth surface [7].This suggests that one can separate the form finding and pattern generation to allow freedom to designa brick pattern on an arbitrary surface. By generating the geodesics orthogonal to an arbitrary surfacecurve, i.e. becoming the first orthogonal trajectory, it should be easy to generate the rest of orthogonaltrajectories by measuring equal lengths along the geodesics. Our method for generating geodesics arebased on a series of circles, having equal radius R, with its centre on the surface contained with the normalplane of its centre. Each row of overlapping circles are connected in such a way that they intersect thesurface creating a locus of equally spaced points describing a geodesic, see figure 7. Since all points arecontained within the normal planes the curvature only contains a normal component, i.e. the geodesiccurvature is zero. The initial orthogonal trajectory is arbitrary and therefore this strategy allows to specifythe direction of the brick pattern but also to make several local patterns combined into a global pattern.For a pattern to be successful the geodesics must not cross, which we can only guess in advance, andtherefore it might even be necessary to divide a surface into several patches each with its own pattern.Our method was implemented and applied to different surfaces and is shown in figure 6. The script wasdeveloped in C# using the Rhinocommon SDK [1] to solve surface and curve intersections.

6. ConclusionsWe have developed two different strategies for generating brick patterns which satisfies the requirementfor a constant spacing between the bed joints, each having their own quality in the design. The first is

5


Figure 6: Plan and elevations of three different brick shells generated by our strategy described in section 5. Theglobal pattern can consist of one or more variations of geodesic coordinates on the surface. Since the form andpattern generation is separated it is easy to adapt to edge conditions or just making unique patterns. Depending onhow the initial orthogonal trajectory is constructed the pattern can take any direction. Closed initial curves resultsin a variation called polar geodesic coordinates, the top figure consists of three polar geodesic patterns.

6


Starting curve

Circles generating the geodesics

geodesicsOrthogonal trajectories

normal planein ri

ri+1

ni

riri-1

R

Figure 7: The generation of the geodesic coordinates is based on a initial curve from which geodesics are generatedat an right angle. A series of circles with its centre on the surface, contained in the normal plane in its centre point,intersects the surface generating a locus of points describing the geodesics. Since the points on the geodesics arecontained within a normal planes the geodesic curvature is zero.

derived from desired geometrical and structural properties making form, structural action and patternintegrated. This avoids sliding along the head joints but it also tells a unique design story. The secondstrategy disconnect the form and pattern making it easy to apply unique patterns based on visual appeal,structural or production benefits. It can be a tool for enabling conversations and cooperation betweendifferent professions in a design process.

We have not discussed the spacing of the head joints and in figure 5 the length of the bricks is shown tovary as the spacing between the geodesics varies. However it is more likely that constant length brickswould be used so that the head joints would not coincide with the geodesics.

References[1] R. M. . Associates. Rhinocommon sdk. http://developer.rhino3d.com/api/RhinoCommon/,

2017. [Online; accessed 1-April-2017].

[2] P. Block and J. Ochsendorf. Thrust network analysis: A new methodology for three-dimensionalequilibrium. Journal of the International Association for Shell and Spatial Structures, 48:167–173,2007.

[3] P. de L’Orme. Le premier tome de l’ Architecture de Philibert de de l’Orme. chez Federic Morel,1567. URL http://architectura.cesr.univ-tours.fr/traite/Images/Les1653Index.

asp.

[4] A. E. Green and W. Zerna. Theoretical elasticity. Oxford University Press and Dover, 2nd edition,1968 and 1992.

[5] E. Happold and W. Liddell. Timber lattice roof for the Mannheim Bundesgartenschau. The StructuralEngineer, 53:99–135, 1975.

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[6] G. Paulsson. Hantverkets Bok: 4, Mureri. Lindfors bokforlag, 1936.

[7] D. J. Struik. Lectures on classical differential geometry. Addison - Wesley and Dover, 2nd edition,1961 and 1988.

Appendix - Computer code listingdo ub l e [ ] [ ] [ ] coord , v e l o c i t y , f o r c e , t a n g e n t , i nP l aneNorma l ;do ub l e [ ] [ ] geodes i cForceDen , g e o d e s i c F o r c e D e n R a t e ;do ub l e [ ] [ ] o r t h o g o n a l F o r c e D e n , o r t h o g o n a l F o r c e D e n R a t e ;do ub l e [ ] [ ] s t i f f n e s s , l o a d ;i n t m, n , s a v e C y c l e ;do ub l e a , bedHeight , l e n g t h S q u a r e d , l o a d F a c t o r ;b o o l e a n s t a r t ;vo id s e t u p ( ){

m = 6 0 ;n = 120 ;coord = new d oub le [m + 1 ] [ n + 1 ] [ 3 ] ;v e l o c i t y = new d ou b le [m + 1 ] [ n + 1 ] [ 3 ] ;f o r c e = new do ub le [m + 1 ] [ n + 1 ] [ 3 ] ;t a n g e n t = new d ou b l e [m + 1 ] [ n + 1 ] [ 3 ] ;i nP l aneNorma l = new do ub le [m + 1 ] [ n + 1 ] [ 3 ] ;s t i f f n e s s = new do ub le [m + 1 ] [ n + 1 ] ;l o a d = new do ub le [m + 1 ] [ n + 1 ] ;g e o d e s i c F o r c e D e n = new do ub l e [m + 1 ] [ n ] ;g e o d e s i c F o r c e D e n R a t e = new do ub le [m + 1 ] [ n ] ;o r t h o g o n a l F o r c e D e n = new do ub le [m] [ n + 1 ] ;o r t h o g o n a l F o r c e D e n R a t e = new d oub le [m] [ n + 1 ] ;f u l l S c r e e n ( ) ;a = 5 0 0 0 . 0 ;bedHe igh t = 0 . 2 5 ∗ a / ( d ou b l e ) n ;l e n g t h S q u a r e d = bedHe igh t ∗ bedHe igh t ;l o a d F a c t o r = − 5 . 0 / ( do ub l e ) n ;f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n ; j ++){

do ub l e f a c t o r 1 = ( d oub l e ) ( i − m) / ( d ou b l e ) n ;do ub l e f a c t o r 2 = 0 . 3 ∗ ( do ub l e ) ( 2 . 0 ∗ j − n ) / ( d ou b l e ) (2 ∗ n ) ;coord [ i ] [ j ] [ 0 ] = 0 . 3 ∗ f a c t o r 1 ∗ a ;coord [ i ] [ j ] [ 1 ] = ( f a c t o r 2 − 0 . 5 ∗ ( 0 . 5 ∗ f a c t o r 1 ∗ f a c t o r 1

+ 0 . 0 3 ∗ Math . cos ( 1 . 5 ∗ PI ∗ f a c t o r 1 ∗ ( do ub l e ) n / ( do ub l e ) m) ) ∗ f a c t o r 2 ) ∗ a ;coord [ i ] [ j ] [ 2 ] = 0 . 2 ∗ a ∗ ( do ub l e ) ( j ∗ ( n − j ) )∗ ( 1 . 0 + 1 . 0 ∗ f a c t o r 1 ∗ f a c t o r 1 ) / ( d ou b l e ) ( n ∗ n ) ;

f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++) v e l o c i t y [ i ] [ j ] [ xyz ] = 0 . 0 ;}}f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n − 1 ; j ++){

g e o d e s i c F o r c e D e n [ i ] [ j ] = − 3 . 0 ;g e o d e s i c F o r c e D e n R a t e [ i ] [ j ] = 0 . 0 ;

}}f o r ( i n t i = 0 ; i <= m − 1 ; i ++){

f o r ( i n t j = 0 ; j <= n ; j ++){

o r t h o g o n a l F o r c e D e n [ i ] [ j ] = 0 . 0 ;o r t h o g o n a l F o r c e D e n R a t e [ i ] [ j ] = 0 . 0 ;

}}s t a r t = t r u e ;s a v e C y c l e = 0 ;

}vo id draw ( ){

f o r ( i n t c y c l e = 0 ; c y c l e <= 1000 ; c y c l e ++){

f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n ; j ++){

f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++) f o r c e [ i ] [ j ] [ xyz ] = 0 . 0 ;s t i f f n e s s [ i ] [ j ] = 0 . 0 ;

}}f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n ; j ++){

i n t p r e v i o u s i = i − 1 ;i n t p r e v i o u s j = j − 1 ;i f ( p r e v i o u s i < 0) p r e v i o u s i = 0 ;i f ( p r e v i o u s j < 0) p r e v i o u s j = 0 ;i n t n e x t i = i + 1 ;

8


i n t n e x t j = j + 1 ;i f ( n e x t i > m) n e x t i = m;i f ( n e x t j > n ) n e x t j = n ;d oub l e s c a l a r P r o d u c t = 0 . 0 ;d oub l e magni tudeSq = 0 . 0 ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++){

t a n g e n t [ i ] [ j ] [ xyz ] = coord [ i ] [ n e x t j ] [ xyz ] − coord [ i ] [ p r e v i o u s j ] [ xyz ] ;magni tudeSq += t a n g e n t [ i ] [ j ] [ xyz ] ∗ t a n g e n t [ i ] [ j ] [ xyz ] ;

}d oub l e magn i tude = Math . s q r t ( magni tudeSq ) ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++) t a n g e n t [ i ] [ j ] [ xyz ] /= magni tude ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++){

i nP l aneNorma l [ i ] [ j ] [ xyz ] = coord [ n e x t i ] [ j ] [ xyz ] − coord [ p r e v i o u s i ] [ j ] [ xyz ] ;s c a l a r P r o d u c t += inP laneNorma l [ i ] [ j ] [ xyz ] ∗ t a n g e n t [ i ] [ j ] [ xyz ] ;

}f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++)

inP l aneNorma l [ i ] [ j ] [ xyz ] −= t a n g e n t [ i ] [ j ] [ xyz ] ∗ s c a l a r P r o d u c t ;magni tudeSq = 0 . 0 ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++)

magni tudeSq += inP laneNorma l [ i ] [ j ] [ xyz ] ∗ i nP l aneNorma l [ i ] [ j ] [ xyz ] ;magn i tude = Math . s q r t ( magni tudeSq ) ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++) inP l aneNorma l [ i ] [ j ] [ xyz ] /= magni tude ;d oub l e c o r r e c t L o a d = l o a d F a c t o r ∗ magni tude ;d oub l e l o a d I n c r e m e n t F a c t o r = 1 . 0 e−6; / / Th i s i s t o s t o p t h e l o a d f l u c t u a t i n g t o o q u i c k l yi f ( s t a r t == t r u e ) l o a d [ i ] [ j ] = c o r r e c t L o a d ;e l s e

l o a d [ i ] [ j ] = ( 1 . 0 − l o a d I n c r e m e n t F a c t o r ) ∗ l o a d [ i ] [ j ] + l o a d I n c r e m e n t F a c t o r ∗ c o r r e c t L o a d ;i f ( i != 0 && i != m) f o r c e [ i ] [ j ] [ 2 ] += l o a d [ i ] [ j ] ;e l s e f o r c e [ i ] [ j ] [ 2 ] += 2 . 0 ∗ l o a d [ i ] [ j ] ;

}}i f ( s t a r t ) s t a r t = f a l s e ;f o r ( i n t i = 0 ; i <= m − 1 ; i ++){

f o r ( i n t j = 1 ; j <= n − 1 ; j ++){

d oub l e s c a l a r P r o d u c t = 0 . 0 ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++) s c a l a r P r o d u c t +=

( coord [ i ] [ j + 1 ] [ xyz ] − 2 . 0 ∗ coord [ i ] [ j ] [ xyz ] + coord [ i ] [ j − 1 ] [ xyz ] )∗ i nP l aneNorma l [ i ] [ j ] [ xyz ] ;

d oub l e c a r r y O v e r = 0 . 9 ;d oub l e change = − 0 .0005 ∗ s c a l a r P r o d u c t ;o r t h o g o n a l F o r c e D e n R a t e [ i ] [ j ] = c a r r y O v e r ∗ o r t h o g o n a l F o r c e D e n R a t e [ i ] [ j ] + change ;o r t h o g o n a l F o r c e D e n [ i ] [ j ] += o r t h o g o n a l F o r c e D e n R a t e [ i ] [ j ] ;i f ( o r t h o g o n a l F o r c e D e n [ i ] [ j ] > 0 . 0 ) o r t h o g o n a l F o r c e D e n [ i ] [ j ] = 0 . 0 ;i f ( i != 0 ){

o r t h o g o n a l F o r c e D e n R a t e [ i − 1 ] [ j ] = c a r r y O v e r ∗ o r t h o g o n a l F o r c e D e n R a t e [ i − 1 ] [ j ] − change ;o r t h o g o n a l F o r c e D e n [ i − 1 ] [ j ] += o r t h o g o n a l F o r c e D e n R a t e [ i − 1 ] [ j ] ;i f ( o r t h o g o n a l F o r c e D e n [ i − 1 ] [ j ] > 0 . 0 ) o r t h o g o n a l F o r c e D e n [ i − 1 ] [ j ] = 0 . 0 ;

}}}f o r ( i n t i = 0 ; i <= m − 1 ; i ++){

f o r ( i n t j = 1 ; j <= n − 1 ; j ++){

f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++){

do ub l e d e l t a = coord [ i + 1 ] [ j ] [ xyz ] − coord [ i ] [ j ] [ xyz ] ;do ub l e component = o r t h o g o n a l F o r c e D e n [ i ] [ j ] ∗ d e l t a ;f o r c e [ i ] [ j ] [ xyz ] += component ;i f ( i != m − 1)

f o r c e [ i + 1 ] [ j ] [ xyz ] −= component ;e l s e{

i f ( xyz != 0) f o r c e [ i + 1 ] [ j ] [ xyz ] −= 2 . 0 ∗ component ;}}s t i f f n e s s [ i ] [ j ] += o r t h o g o n a l F o r c e D e n [ i ] [ j ] ;i f ( i != m − 1) s t i f f n e s s [ i + 1 ] [ j ] += o r t h o g o n a l F o r c e D e n [ i ] [ j ] ;e l s e s t i f f n e s s [ i + 1 ] [ j ] += 2 . 0 ∗ o r t h o g o n a l F o r c e D e n [ i ] [ j ] ;

}}f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n − 1 ; j ++){

d oub l e t h i s L e n g t h S q u a r e d = 0 . 0 ;f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++){

do ub l e d e l t a = coord [ i ] [ j + 1 ] [ xyz ] − coord [ i ] [ j ] [ xyz ] ;t h i s L e n g t h S q u a r e d += d e l t a ∗ d e l t a ;g e o d e s i c F o r c e D e n R a t e [ i ] [ j ] = 0 . 5 ∗ g e o d e s i c F o r c e D e n R a t e [ i ] [ j ]− 0 .001 ∗ ( t h i s L e n g t h S q u a r e d − l e n g t h S q u a r e d ) / l e n g t h S q u a r e d ;

g e o d e s i c F o r c e D e n [ i ] [ j ] += g e o d e s i c F o r c e D e n R a t e [ i ] [ j ] ;do ub l e component = g e o d e s i c F o r c e D e n [ i ] [ j ] ∗ d e l t a ;f o r c e [ i ] [ j ] [ xyz ] += component ;

9


f o r c e [ i ] [ j + 1 ] [ xyz ] −= component ;}s t i f f n e s s [ i ] [ j ] += g e o d e s i c F o r c e D e n [ i ] [ j ] ;s t i f f n e s s [ i ] [ j + 1 ] += g e o d e s i c F o r c e D e n [ i ] [ j ] ;

}}f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 1 ; j <= n − 1 ; j ++){

f o r ( i n t xyz = 0 ; xyz <= 2 ; xyz ++){

i f ( i != m | | xyz != 0){

v e l o c i t y [ i ] [ j ] [ xyz ] = 0 . 9 5 ∗ v e l o c i t y [ i ] [ j ] [ xyz ] + 0 . 2 ∗ f o r c e [ i ] [ j ] [ xyz ] / s t i f f n e s s [ i ] [ j ] ;coo rd [ i ] [ j ] [ xyz ] += v e l o c i t y [ i ] [ j ] [ xyz ] ;

}}}}}background ( 2 5 5 , 255 , 2 5 5 ) ;s c a l e ( 2 . 7 ∗ ( f l o a t ) h e i g h t / ( f l o a t ) a ) ;f l o a t l e f t R i g h t = 0 . 2 ∗ ( f l o a t ) a ;f l o a t e l evDrop = 0 . 3 6 ∗ ( f l o a t ) a ;f l o a t p lanDrop = 0 . 1 6 ∗ ( f l o a t ) a ;f l o a t e l e v R i g h t = 0 . 4 5 ∗ ( f l o a t ) a ;s t r o k e W e i g h t ( 1 ) ;t r a n s l a t e ( l e f t R i g h t , p lanDrop ) ;s t r o k e ( 2 5 5 . 0 , 0 . 0 , 0 . 0 , 1 5 0 . 0 ) ;f o r ( i n t j = 0 ; j <= n ; j ++){

f o r ( i n t i = 0 ; i <= m − 1 ; i ++){

l i n e ( ( f l o a t ) coo rd [ i ] [ j ] [ 1 ] , ( f l o a t ) coord [ i ] [ j ] [ 0 ] , ( f l o a t ) coord [ i + 1 ] [ j ] [ 1 ] , ( f l o a t ) coord [ i + 1 ] [ j ] [ 0 ] ) ;l i n e ( ( f l o a t ) coo rd [ i ] [ j ] [ 1 ] , − ( f l o a t ) coo rd [ i ] [ j ] [ 0 ] , ( f l o a t ) coo rd [ i + 1 ] [ j ] [ 1 ] , − ( f l o a t ) coord [ i + 1 ] [ j ] [ 0 ] ) ;

}}s t r o k e ( 0 . 0 , 0 . 0 , 0 . 0 , 1 5 0 . 0 ) ;f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n − 1 ; j ++){

l i n e ( ( f l o a t ) coo rd [ i ] [ j ] [ 1 ] , ( f l o a t ) coord [ i ] [ j ] [ 0 ] , ( f l o a t ) coord [ i ] [ j + 1 ] [ 1 ] , ( f l o a t ) coord [ i ] [ j + 1 ] [ 0 ] ) ;i f ( i != m)

l i n e ( ( f l o a t ) coord [ i ] [ j ] [ 1 ] , − ( f l o a t ) coo rd [ i ] [ j ] [ 0 ] , ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 1 ] , − ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 0 ] ) ;}}t r a n s l a t e ( 0 . 0 , e l evDrop − planDrop ) ;s t r o k e ( 2 5 5 . 0 , 0 . 0 , 0 . 0 , 1 5 0 . 0 ) ;f o r ( i n t j = 0 ; j <= n ; j ++){

f o r ( i n t i = 0 ; i <= m − 1 ; i ++)l i n e ( ( f l o a t ) coo rd [ i ] [ j ] [ 1 ] , − ( f l o a t ) coo rd [ i ] [ j ] [ 2 ] , ( f l o a t ) coo rd [ i + 1 ] [ j ] [ 1 ] , − ( f l o a t ) coord [ i + 1 ] [ j ] [ 2 ] ) ;

}s t r o k e ( 0 . 0 , 0 . 0 , 0 . 0 , 1 5 0 . 0 ) ;f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n − 1 ; j ++)l i n e ( ( f l o a t ) coo rd [ i ] [ j ] [ 1 ] , − ( f l o a t ) coo rd [ i ] [ j ] [ 2 ] , ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 1 ] , − ( f l o a t ) coord [ i ] [ j + 1 ] [ 2 ] ) ;

}t r a n s l a t e ( e l e v R i g h t − l e f t R i g h t , p lanDrop − e levDrop ) ;s t r o k e ( 2 5 5 . 0 , 0 . 0 , 0 . 0 , 1 5 0 . 0 ) ;f o r ( i n t j = 0 ; j <= n ; j ++){

f o r ( i n t i = 0 ; i <= m − 1 ; i ++){

l i n e (− ( f l o a t ) coord [ i ] [ j ] [ 2 ] , ( f l o a t ) coo rd [ i ] [ j ] [ 0 ] , − ( f l o a t ) coo rd [ i + 1 ] [ j ] [ 2 ] , ( f l o a t ) coo rd [ i + 1 ] [ j ] [ 0 ] ) ;l i n e (− ( f l o a t ) coord [ i ] [ j ] [ 2 ] , − ( f l o a t ) coo rd [ i ] [ j ] [ 0 ] , − ( f l o a t ) coo rd [ i + 1 ] [ j ] [ 2 ] , − ( f l o a t ) coo rd [ i + 1 ] [ j ] [ 0 ] ) ;

}}s t r o k e ( 0 . 0 , 0 . 0 , 0 . 0 , 1 5 0 . 0 ) ;f o r ( i n t i = 0 ; i <= m; i ++){

f o r ( i n t j = 0 ; j <= n − 1 ; j ++){

l i n e (− ( f l o a t ) coord [ i ] [ j ] [ 2 ] , ( f l o a t ) coo rd [ i ] [ j ] [ 0 ] , − ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 2 ] , ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 0 ] ) ;i f ( i != m)

l i n e (− ( f l o a t ) coo rd [ i ] [ j ] [ 2 ] , − ( f l o a t ) coord [ i ] [ j ] [ 0 ] , − ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 2 ] , − ( f l o a t ) coo rd [ i ] [ j + 1 ] [ 0 ] ) ;}}}

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Brick patterns on shells using geodesic...

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