Polynomials, shape finding procedures and reverse mathematics and graphics are presented here that improve the down time and quality of the manufacturing process.

1. Polynomial Formulations, Shape
Finding Procedures and Reverse
Mathematics for CAD/CAM and
Prototype Manufacturing Applications
Prof. Padmanabhan Krishnan
School of Mechanical Engineering
VIT, Vellore 632014.
Email: padmanabhan.k@vit.ac.in

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Graphic FE Design of Lord Shiva

3. Contents
• Nature and Symmetry
• Nature and Mathematics
• Orders of Polynomials And Their Importance
• Shape Finding Procedures
• Biomimetics
• Reverse Mathematics for Design and Forming
• Applications In Prototypes and Manufacturing
Operations

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Nature and Symmetry

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Nanoclays
• Silicates layers separated by an interlayer
or gallery.
• Silicates layers are ~ 1 nm thick, 300 nm
to microns laterally.
• Polymers as interlayers.
• Tailor structural, optical properties.

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Nanoclays
Structure of phyllosilicates
( Source: Dubois, Alexandre 2000)

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Wood: A natural, fiber-reinforced
composite
Cell walls: layered cellulose microfibrils (linear chains of glucose residues,
degree of polymerization  5000 – 10000,  40-50 % w/w of dry wood
depending on species), bound to matrix of hemicellulose and lignin

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Cellulose Nanocrystals (II)
Cellulose microfibrils are secreted by certain non-photosynthetic
bacteria (e.g. Acetobacter xylinum), and from the mantle of sea-
squirts (“tunicates”) (e.g. Ciona intestinalis)
These highly pure forms are free from lignin/hemicelluloses;
fermentation of glucose is a possible microbial route to large-scale
cellulose production.
Adult sea-squirts
Nanocrystalline cellulose whiskers, from acid
hydrolysis of bacterial cellulose. Image courtesy of
Profs. W.T. Winter and M. Roman, Dept. of Chemistry,
SUNY-ESF, and Dept. of Wood Science and Forest
Products at Virginia Tech.

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Cellulose Nanocrystals (I)
Cellulose (linear chains of glucose residues), bound to matrix of lignin
and hemicellulose, comprises  40-50 % w/w of dry wood
Individual fibers have major dimensions ~ 1-3 mm, consisting of spirally
wound layers of microfibrils bound to lignin-hemicellulose matrix;
microfibrils contain crystalline domains of parallel cellulose chains;
individual crystalline domains ~ 5-20 nm in diameter, ~ 1-2 m in length
Nanocrystalline domains separable from amorphous regions by
controlled acid hydrolysis (amorphous regions degrade more rapidly)
Crystalline domain elastic modulus (longitudinal) ~ 150 GPa: compare
martensitic steel ~ 200 GPa, carbon nanotubes ~ 103 GPa
Suggests possible role for cellulose nanocrystals as a renewable, bio-
based, low-density, reinforcing filler for polymer-based nanocomposites

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NaCl Structure

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CsCl Structure

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HCP Close Packing
Examples: Mg, Cd, Zn,Be

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Left and right crystals !

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The Sunflower Radial Symmetry
Sunflowers boast radial symmetry and an interesting type of numerical
symmetry known as the Fibonacci sequence. The Fibonacci sequence is 1, 2, 3,
5, 8, 13, 21, 24, 55, 89, 144, and so on (each number is determined by adding
the two preceding numbers together).

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Bi Lateral and Glide Plane Symmetry

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Flowers
Hypogynous Perigynous Epigynous

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Flowers
Polysepalous
Gamosepalous
United Sepals

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Flowers
• Polypetalous ( four fold, five fold, many fold)
• Gamopetalous ( united)
• Dentists use terms like peridontal, epi, ortho, prostho and
endo in dentistry
Polypetalous 4 fold Polypetalous , multiple. Gamopetalous, united

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Botanist Jamili Nais measuring
Rafflesia flower, Rafflesia keithii, Mt
Kinabalu National Park, Sabah
World’s biggest flower has
five fold symmetry
Ho-Mg-Zn Quasicrystal
Five fold symmetry
Ordered but not periodic

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Star Fish and Sea Urchin
Star fish has n=5, meaning a five fold symmetry and sea urchin
Shown here has n= 9. Both have rotational symmetry

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The Nautilus Spiral Symmetry
In addition to plants, some animals, like the nautilus, exhibit Fibonacci
numbers. For instance, the shell of a nautilus is grown in a “Fibonacci spiral.”
The spiral occurs because of the shell’s attempt to maintain the same
proportional shape as it grows outward. Ibn n the case of the nautilus, this
growth pattern allows it to maintain the same shape throughout its whole life
(unlike humans, whose bodies change proportion as they age).

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Faunal Bi lateral Symmetry

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The Peacock’s Bilateral Symmetry
The feathered male peacock has perfect symmetry !

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A push pull symmetry is shown here of a goat

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Near Perfect Mirror Image of Milky Way
As we’ve seen, symmetry and mathematical patterns exist almost everywhere
we look—but are these laws of nature limited to our planet alone? Apparently
not. Having recently discovered a new section on the edges of the Milky Way
Galaxy, astronomers now believe that the galaxy is a near-perfect mirror image
of itself.

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Nature and Mathematics

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The Human Brain
Neo cortex
Visual Maths, FEM, SVG, Mathematics of Graphics, Lyrical Music, Quantitative
Pattern Recognition, Modular Assembly are all Demanding on both the sides !

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Division and multiplication
• We learn division from Mitosis and
meiosis, cell division and duplication!

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Order of Equations in Nature
• The first from Classical Physics is the fundamental law of energy
radiation known as Wien's law. It is based on wavelengths to the
fifth power. This means energy as photons is based on wavelengths
to the fifth power! Atomic and nuclear emitted energy has a proven
fifth power basis; just as the 5SPACE formula has.
• After Einstein failed in his attempt to unify Gravity with
Electromagnetism, Kaluza & Klein [K&K] proved that if space-time
had a fifth dimension in which Charge was the angular momentum in
that fifth dimension. Gravity and Electromagnetism could indeed be
unified with General Relativity…………………………………………
• More equations in the next slides !

30. Laws of Nature
• Mass= Volume X ρ ( Density)
• Heat Generated in a circuit= I 2 R, Current
squared X Resistance.
• Gravitation has a square rule !
• Area of a Circle = πr2, square rule !
• Quadratic polynomials , Lagrange !
• Volume = r 3 , cube of side or a X b X c
• Mechanical power of swimming ꭃ the cube of
velocity !
8/17/2022 30

31. Laws of Nature
• Hermite polynomials solve beams !
• Stefan- Boltzman : Energy radiated per unit area
is fourth power of Temperature.
• Step work: Work need to transition from one
stance of limb to another increases with fourth
power of step length.
• Cylinder flow: Steady flow around a buoyant
finite cylinder within a rapidly rotating core has a
component proportional to the inverse fifth
power;
8/17/2022 31

32. Laws of Nature
• ……rotating rapidly at right angles to the
earth's gravity and containing a rigid
cylindrical float is found under the assumption
of small viscosity.
• Lightning from cumulonimbus clouds :
Lightning power goes up with the 5th power
of the cloud size. The 5th power of the cold-
cloud depth is proportional to the stored static
electric energy and to the charging rate in the
convective cloud.
•8/17/2022 32

33. Laws of Nature
• According to Brahm's law (sometimes
called Airy's law), the maximum mass of
objects that may be moved away by a river
is proportional to the sixth power of the
river flow speed.
• Van der Waals forces : Van der Waals
forces are proportional to the inverse
seventh power of distance, so they act
only over short distances.
8/17/2022 33

34. Laws of Nature
• Corrosion in river flow: Corrosion of iron
and steels by acid river water is
proportional to 7.5th power of the flow
velocity.
• Jet noise: Lighthill's eighth power law
states that the acoustic power generated
by a jet engine is proportional to the eighth
power of the jet speed.

35. Laws of Nature
• Carbon flow from ocean: Flow of carbon from
the upper ocean to the atmosphere is
proportional to the ninth power of the mass of
carbon in the upper ocean.
• Stellar carbon cycle reaction rate: In energy-
generation reactions for fusion in stellar interiors
in which hydrogen is converted into carbon, the
reaction rate is proportional to anything from
Temperature 13 in hot massive stars to T20 in
stars like the Sun. In red super giants it can be
up to T 40 .
8/17/2022 35

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Exponential Behaviour
Like in sound attenuation with
unattenuated initial value Ao,
attenuation coefficient (k) and
distance or depth (t)

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Visual Mathematics
Cartesian equation:
y4 - x4 + a y2 + b x2 = 0
Polar equation (Special case):
r = √[(25 - 24tan2(θ))/(1 - tan2(θ))]
Polar equation:
r2 = a2θ
Parametric Cartesian equation:
x = a sin(nt + c), y = b sin(t)

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Visual Mathematics
Cartesian equation:
y(x2 + a2) = a3
or parametrically:
x = at, y = a/(1 + t2) Polar equation:
r = a sin(kθ) Cartesian equation:
y = x cot(πx/2a)
Polar equation:
r = 2aθ/(πsin(θ))
Lived during 1719-1799 in
Bologna, before Karl Fredrich
Gauss who Lived during 1777-1855.

39. Statistical Reliability
39
(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure The Weibull distribution describes the fraction of the samples
that fail at any given applied stress. It will be interesting to see the
Weibul distribution plots for multiscale materials.
1- Probability (Failure) = P ( Reliability )

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Visual Mathematics
Polar equation:
r = a/θ
Polar equation:
rp = ap cos(pθ)
A point-based
representation of the
mathematical surface:

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Visual Mathematics
ParametricPlot3D= [{r
Cos[[Theta]], r Sin[[Theta]], r^2},
{r, 0, 1}, {[Theta], 0, 2 [Pi]}]
A Parametric 3D Design

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Finite Element Analysis of Human
Facial Expressions
Think of FEM as the best non destructive test method that you can ever conceive
And refine the precision and bias in your approach with out wasting Nature’s resources.
It is possible in the near future !

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Nature and Physics

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Mechanically efficient shapes

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Microstructural Efficiency

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A Tribute to Marutsakha!
And Chiranjilal Verma’s Student !

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Shivkar Bapuji Talpade
His documents are preserved in
HAL, Bangalore, and Parle, Mumbai.
His flying prototype was a huge toy
To his relatives. Their kids played
Sitting in it in the courtyard. No
Indian followed him for decades…..
Till the late comers arrived…!

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Wright Brothers and the Dragon Fly
It took ~100 to 350 million years of evolution to power the insect flight !
The dragon fly

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Indian ornithoptors of dragonflies

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Catastrophe Theory
• Thom referred to events like a heart beat, a
buckling of beam, a stock market crash, a riot or
a tornado as catastrophes. He proved that all
catastrophic events In our four dimensional
world are combinations of seven elementary
catastrophes. In higher dimensions the number
quickly approaches infinity. The cusp, elliptical
umbulicus (belly button) are all examples.
• Topological attractors !

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CATASTROPHE THEORY
Catastrophe theory is a special branch of
dynamical systems theory. Continuous
phenomena are easily dealt with. A change in
one quantity produces a predictable change in
another. Mathematical functions that represent
continuous events can be graphed curves. The
governing equations for such phenomena and
extensive mathematical methods of solving
them have been developed.

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Rene Thom 1923-2002

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On the other hand, “non linear” equations generally
have no exact solutions , and the erratic events
associated with them are harder to describe or predict.
The new science of chaos, is opening up new ways to
deal with such events.

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Buckling Types and Topology
To topologically formulate all the types of
buckling through master/parent equations and
scaling that yield all the other formulations in
order to analyze the buckling and post
buckling behaviour of the most common and
important material structures. To construct
buckling mechanism maps. To validate
topology through finite element methods and
prove the correspondence in the pattern
recognition of buckling load-deflection plots.

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TOPOLOGICAL GEOMETRIES
 Consider the equation, y = K x (P-x) (1)
Where, y is the load, x is the deflection and K and P are
constants.
Where,
K = 0.5,1,1.5,2,2.5,3,3.5,4,4.5,5 and
P = 0.5,1,1.5,2,2.5,3,3.5,4,4.5,5
x increases from 0.1 in the increments of 0.05.

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Choosing K=1.5 and P=1 makes the equation y = 1.5x
(1-x), which can be iterated with an arbitrary x value,
calculating the resulting y value, substituting that y value
back as x , and calculating another y value, and so on. This
method known as the attractor method. Thus two plots are
possible, (x,y) and (y,x) as y is obtained from x, substituted
back as x. When these new formulations are iterated and
plotted in some variable domains, they provide all the
necessary information on the various types of buckling
behaviour of material systems. They replicate the various
buckling load deflection patterns obtained from material
structures. This is the attractor method !

58. Regular Co-ordinates
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.05 0.1 0.15 0.2
x
y
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
x
y
Linear Non-linear
x=.15
y=.5(1.5-x)
x=0.1
y=0.5x (4-x)

59. 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.2 0.4 0.6
x
y
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
x
y
Bi-modal buckling Snap back buckling
x=.45
y=1.5x (.5-x)
x=.3
y=2.5x (1-x)

60. 0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0 0.2 0.4 0.6 0.8
x
y
0
0.5
1
1.5
2
2.5
0 1 2 3
x
y
Snap through buckling Hinged arch
x=0.4
y=3x (1-x)
x=.2
y=2x (2-x)

61. FEM MODEL OF HOLLOW
CYLINDER

62. Outer diameter = 158mm
Inner diameter = 138mm
Height = 900mm
Poisson’s ratio = 0.29
Young’s Modulus = 2.15e5 N/mm2
The element used for this model is Solid 186.The
applied pressure is 0.430N/m2. For this analysis large
deformation was set ON and also Arc length solution
was turned ON.

63. FEM METHOD
x: 0-2,y: 0-2.5
TOPOLOGICAL METHOD
x: 0-2, y: 0-2.5
0
0.5
1
1.5
2
2.5
3
0 1 2 3
x
y
x=0.1
1.25y=0.5x (4-x)

64. VALIDATION
As the topological optimization is valid only when
correspondence with actual material structures exists, the same
was carried out using the FEM buckling and post buckling
analyses for various cases involving buckling and post-
buckling behaviour like snap through and snap back. The
ANSYS programme was used to carry out the buckling
analyses. These analyses replicate the buckling behaviour
obtained through iterations of formulations, thereby
confirming the validity and physical significance of a
topological equivalence in simplifying the buckling problem of
material structures in general.

65. LINEAR
Eccentric Column

66. Eccentric Column-FEM MODEL

67. LINEAR BUCKLING
x: 0-0.157
y: 0-0.15
x: 0-0.104
y: 0-0.15
FEM METHOD

68. x: 0-0.13
y: 0-0.15
x: 0-0.12
y: 0-0.15
FEM METHOD

69. 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.05 0.1 0.15 0.2
Y
Y*1.2
Y*1.4
Y*1.6
y*0.9
TOPOLOGICAL METHOD x: 0-0.15
y: 0-0.16

70. NON-LINEAR
Plate

71. Plate-FEM MODEL

72. NON-LINEAR BUCKLING
x: 0-2
y: 0-5.6
x: 0-1.9
y: 0-2.8
FEM METHOD

73. x: 0-2
y: 0-4.5
x: 0-2
y: 0-3.6 FEM METHOD

74. 0
1
2
3
4
5
6
7
8
0 1 2 3
Y
Y*1.5
Y*2.5
Y*3.5
y*3.8
TOPOLOGICAL METHOD x: 0-2
y: 0-8

75. BI-MODAL BUCKLING
Two coaxial tubes, the inner
one of steel and cross-
sectional area As, and the
outer one of Aluminum alloy
and of area Aa, are
compressed between heavy,
flat end plates, as shown in
figure. Assuming that the end
plates are so stiff that both
tubes are shortened by exactly
the same amount.

76. Pipe-FEM MODEL

77. BI-MODAL BUCKLING
x: 0.2-1
y: 0-0.32
x: 0.2-1
y: 0-0.19
FEM METHOD

78. x: 0.24-1
y: 0-0.215
x: 0.24-1.04
y: 0-0.215
FEM METHOD

79. 0.332
0.333
0.334
0.335
0.336
0.337
0.338
0.339
0.34
0.341
0.342
0 0.2 0.4 0.6 0.8 1 1.2 1.4
y,x
y,1.2x
y,1.4x
y,1.6x
y,1.5x
TOPOLOGICAL METHOD x: 0.3-1.2
y: 0.333-0.342

80. HINGED SHELL
A hinged cylindrical
shell is subjected to a
vertical point load (P)
at its center.
hinged cylindrical shell

81. Hinged cylindrical shell-FEM MODEL

82. SNAP BACK BUCKLING
x: 0-1.65
y: 0-1.1
x: 0-1.7
y: 0-0.85
FEM METHOD

83. x: 0-1.65
y: 0-1
x: 0-1.65
y: 0-1.2
FEM METHOD

84. 0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
y,x
1.4y,1.05x
1.6y,1.06x
1.8y,1.08x
2y,1.2x
TOPOLOGICAL METHOD x: 0.25-1.75
y: 0.3-1.1

85. SNAP THROUGH BUCKLING
x: 0-1.25
y: 0-1.6
x: 0-1.3
y: 0-1.6
FEM METHOD

86. x: 0-1.22
y: 0-1.6
x: 0-1.2
y: 0-1.6
FEM METHOD

87. 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.28 1.3 1.32 1.34 1.36 1.38
y
0.8y
0.6y
0.4y
0.2y
Topological Method x: 1.3-1.365
y: 0.1-1.4

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BIOMIMETICS

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Caterpillars and ICs
The SOP, SSOP and other dual in line packages are all carterpillar designs !

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Microsystems and Nanosystems
Packaging Roadmap
Many of the encapsulated ICs
These days come with bio resins
And their composites !

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Box fish designs
The ostracion cubicus, also known as the boxfish
has a rather large body, but is able to swim very fast
because of its low co-efficient of drag and rigid
exoskeleton. The Mercedes designers began modelling
a new vehicle after the boxfish that proved to be one of the automobiles with lowest co-
efficient of drag ever tested.

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Biomimetics
The Enthiran Robot The Ornithopter

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Biomimetics-The Mouse
It is not that easy to design an ergonomic mouse ! Most off them hurt
By the time you finish modelling a differential gear !

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Nature Inspired Buildings
The leaf building of Brazil Snail stairs in Barcelona

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The Double Helix
The helix cutter , twist and the helical
ratio drill are all manifestations of the DNA !
Fig. 2. (a) twist drill; (b) ratio drill.

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The Pineapple
The pinapple cutter has edges that run both clockwise
and counterclockwise and is used to reduce the delaminations
In layered materials, as any cutter can be rotated in only one
direction . The counterclockwise cutting edge trails and
reduces the delaminations that are otherwise caused by
Single edge end milling cutters and drills.

97. Reverse Mathematics (RM) for
Design and Manufacturing
8/17/2022 97

98. 8/17/2022 98

99. Nature is where design
begins !
Design
is defined as the
complete information
required to produce a
product or render
service
- Anonymous

100. Reverse Mathematics (RM)
• From a shape or form , the governing equations
or the parametric equations could be derived
and modifications made to that, if any, to
reproduce a variant form or shape.
• Though MATLAB can produce 2D, 3D drawings
and shapes governed by equations, it cannot do
the reverse pattern.
• We need a reverse software to recognise
drawings, plots and shapes in the form of
equations in an editable and spontaneous form.
8/17/2022 100

101. Reverse Mathematics
• Reverse mathematics, especially algebra,
should be interfaced with a CAM module to
produce such shapes in Casting, Moulding, RP,
Additive Manufacturing or Machining.
• So it becomes easier to produce variants from
primitive shapes or replicate a shape through
mathematical recognition and scaling.
• Topological optimization and Repair can also be
done to the models.
• Complex shapes can be produced interactively
or passively, thereby reducing man-hours.
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102. Applications In Prototypes and
Manufacturing Operations
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103. Geometric Modelling
• Primitives and Feature Based Modelling
with Reverse Mathematics or Algebra!
• Polynomial, Constitutive, Parametric
Designs
• Scalable Vector Graphics can be used
with XML tools for a more defined image,
picture or model.
• Topological and other Repairs !
8/17/2022 103

104. Rapid Prototyping
• 3D drawing from equations !
• Stereolithography
• Selective Laser Sintering ( SLS)
• Layered Object Modelling ( LOM)
• Fused Deposition Modelling ( FDM)
• Any other Additive Manufacturing
Techniques !
8/17/2022 104

105. Data Preparation and NC
• Numerical Control ( NC) using part
programming !
• Generic machine independent formatPost
processor MCD ( Machine Control Data ).
• Model captured and converted to equations
through RM, or created through equations,
feature designed and repaired through
equations, forms a set of data.
8/17/2022 105

106. Data Preparation and NC
• GDM ( Geometric Destructive Modelling )
like gouging can be understood and
programmed using the same language of
RM and RA.
• Eg: Tool path generation, for removal of
geometry.
• A paradigm change in part programming !
• Reduces down / lead time in product
realization
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107. Advantages of RM/RA
• Improves knowledge and information library on
reverse mathematics.
• Significantly reduces down /lead time in
producing 2D or 3D drawings
• CAM interface is easier and RM/RA can be used
depending upon the part .
• Can be programmed for moulding, casting, RP,
AM and Machining .
• Significantly reduces downtime for product
visualization and realization.
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108. 8/17/2022 108
The Rig Veda
There were impregnations, There
were powers,
There was energy below, There was
impulse above
-Rig Veda, Existence, 10.129.5
Krishi Mandala ( Agricultural Applications)

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Rig Veda on Infinity
pûrnamadah pûrnamidam pûrnât
pûrnamudacyate pûrnâsya
pûrnamadaya pûrnamevâvasishyate
From infinity is born infinity.
When infinity is taken out of infinity,
only infinity is left over.