Analytical Engineering Analysis of 3D Shapes
Using volume integral($\iiint_{}^{}{f(t)}dx dy dz$) to do a AnalyticalEngineering Analysis of 3D Shapes without using mesh based FEA. Likeintegration calculates area under curve and FEA do a filled rectangularmesh or cubic mesh approximation of shapes for engineering analysis herepropose modified volume integral method to Analytical EngineeringAnalysis of 3D Shapes. Volume integral is where ∆x, ∆y, ∆z tends tozero. The difference between regular volume integral is that thefunction should change after each integration of volume integration ( 3steps of integration). For example,.
Required continuous smooth function of stress or heat in a 3D solidshape
$$= \int_{}^{}{h(G(t))\int_{}^{}{g(F(t))\int_{}^{}{f(t)dx\ dy\ dz}}}$$
Where, $$F(t) = \int_{}^{}{f(t)}$$
, $$G(t) =\int_{}^{}{g(t)}$$
If $f(t) = sin(t)$ , $g(t) = 2x$ , $h(t) = x^{2}$
then ,
$$\int_{}^{}{{(G(t))}^{2}\int_{}^{}{2F(t)\int_{}^{}{\sin{t\ dx\ dy\ dz}}}}$$
The boundary conditions are given as limits $(u(x), u(y), u(z) .. v(x),v(y), v(z) )$ of integration
$$\int_{v(x)}^{u(x)}{h(G(t))\int_{v(y)}^{u(y)}{g(F(t))\int_{v(z)}^{u(z)}{f(t)dx\ dy\ dz}}}$$
This is a new engineering analysis where a continuous plot of result ismade; unlike FEA, where discrete results are made that depends on sizeor number of elements. this will revolutionize engineering analysiswhere the results of engineering analysis are like a vector image ratherthan raster image.
In this, Upper and Lower limit can be a range; for example, 1:100 with0.1 as step size. So we get 1000 X 1000 X 1000= 1 billion total cells.Shapes can be converted to polynomials using key coordinates from CADdata and using matrix method of finding coefficients using linearalgebra; similar to solving a set of equations. Polynomials are easy tointegrate.
will the above method work?
