two math apps

8/12/2019 Two Math Apps

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Two Math Apps

Jayanta Majumder


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Geometric CAD

Engineering calculations- Stress distribution

- Fluid flow

- Cast solidification

- Layout optimisation

- Digital mock-up

- Heat transfer

Free Geometric CAD tools:

http://www.123dapp.com/

(Dont end up being a user)
http://www.123dapp.com/http://www.123dapp.com/


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Representation of 3d objectsB-Rep = Geometry + Topology

Calculation engines :

- Function minimisation

- Root finding

- Linear algebra


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Shapes come from math

functions

(x,y,z) = ( 4+0.3*sin(7*u)+cos(5*v))*cos(u)*cos(v),

4+0.3*sin(7*u)+cos(5*v))*cos(u)*sin(v),

2*(4+0.3*sin(7*u)+cos(5*v))*sin(u) )

(x,y,z) = ((3+0.2*sin(6*(u+v)+1))*cos(u)*cos(v),

(3+0.2*sin(6*(u+v)+1))*cos(u)*sin(v),

(3+0.2*sin(6*(u+v)+1))*sin(u))

In practice the functions are

mostly polynomials in

entertainment and industrial

product design and proceduralfor machine design


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3D printing from equations

using Mathematicahttp://www.ms.unimelb.edu.au/~segerman/papers/3d_printed_visualisation.pdf


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Realism of Rendering : State of the

artSkillset :

Numerical robustness CS/Algorithms in general

Numerical stability

Understanding limitations

of floating point

computation

GEOMETRY


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Empirical Models :

Models based on data

Empirical models i.e. the models that help build and maintain empires

e.g. models of cannon-ball trajectory.

Modern empires like Apple, Microsoft, Google, Exxon, Shell, Dow runtheir operations based on empirical models of their plants and processes.

[Citation needed]

in fact the said etymology is false.


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Empirical vs. Mechanistic Models

It is more satisfying to find mechanistic solution to a problem because itgives more reach and is less parochial.

Todays empirical model can instigate tomorrows mechanistic model

e.g. Balmer-series model for hydrogen-emission lines were explained ~30

years later by Niels Bohrs model.


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But that doesnt always happen

Titius-Bode planetary distance model (1772)

D(n) = ( 4 3 2)/10 AU

where n =, 0, 1, 2, 3, 4, ..

It was fairly accurate for

planets observable in 1772, but

doesnt work for Neptune and

Pluto.

No mechanistic significance

has been found.

l net

Actual distance

from the Sun

(in AU)

Distance by

Titus-Bode

Formula

Mercury0.387 0.4

Venus0.723 0.7

Earth1 1

Mars1.52 1.6

Asteroid Belt!3.0 2.8

Jupiter5.2 5.2

Saturn9.55 10

Uranus 19.2 19.6


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Method of building empirical models

Connecting dots, but in higher dimensions.

Hypothesize a function of the input variables in terms of a few freeparameters.

Tune the free parameters for which the aggregate of errors is as low as

possible. The tuning is a maxima/minima problem.

How to come up with model functions?

- Some mechanistic reasoning

- Random trials

- Understanding of shapes of terms

- Study of the residue distribution and residue contributors

- Sum of sigmoids and radial functions (neural networks)


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Linear Model

In this special case, the tuning turns out to be the problem of solving a systemof linear equations.

In general, we minimise the following function over the possible values of thefree parameters vector a :

() =

(; )

Linear model has the special structure that :

;

()()

Here , , , are terms involving the ith input vector.

Please note that the model is linear only in terms of its free parameters, but theterms can be arbitrarily non-linear.


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Model Accuracy Criteria


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A 1-variable model with level

snapped data

100 ( )Where t is the immersed time as measured in days.

Here () 1 log( +) in which S is chosen to be 100.


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THANKS!

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two math apps

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