50.34.139.236: /* Bolt thrust estimations for various pistol/revolver cartridges */

2014-12-30T02:05:35Z

Bolt thrust estimations for various pistol/revolver cartridges

en>Francis Flinch: /* Calculating bolt thrust */ internal ref

2013-05-09T11:44:17Z

Calculating bolt thrust: internal ref

← Older revision		Revision as of 13:44, 9 May 2013
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	~~They call~~ the ~~author Boris~~. As a ~~girl what I absolutely love~~ is ~~to fence and I've been doing it for~~ a ~~while~~. ~~The job he's been occupying for years is an invoicing officer and his salary is really~~ [~~http~~://~~pinterest~~.~~com~~/~~search~~/~~pins~~/?q=~~worthwhile worthwhile~~]. ~~Georgia is where our home is~~. ~~If you want to find out more~~ the ~~look~~ at ~~his website~~: ~~http~~://~~www~~.~~alcoholysociedad~~.~~org~~/~~publicaciones~~/~~descargas~~/~~canada~~-~~goose~~-~~outlet.html~~<br><br>~~Take a look~~ at ~~my homepage .~~.. [http://www.~~alcoholysociedad~~.~~org~~/~~publicaciones~~/~~descargas~~/~~canada~~-~~goose~~-~~outlet~~.~~html Canada Goose En España~~]		In mathematics, the '''Kronecker sum of discrete Laplacians''', named after [[Leopold Kronecker]], is a discrete version of the [[separation of variables]] for the continuous [[Laplacian]] in a [[Cuboid#Rectangular_cuboid\|rectangular cuboid]] domain.

			==General form of the Kronecker sum of discrete Laplacians==

			In a general situation of the [[separation of variables]] in the discrete case, the multidimensional [[Discrete Laplace operator\|discrete Laplacian]] is a [[Kronecker_product#Kronecker_sum_and_exponentiation\|Kronecker sum]] of 1D discrete Laplacians.

			===Example: 2D discrete Laplacian on a regular grid with the homogeneous Dirichlet boundary condition===

			Mathematically, using the [[Kronecker_product#Kronecker_sum_and_exponentiation\|Kronecker sum]]:

			:<math>L = \mathbf{D_{xx}}\oplus\mathbf{D_{yy}}=\mathbf{D_{xx}}\otimes\mathbf{I}+\mathbf{I}\otimes\mathbf{D_{yy}}, \,</math>

			where <math>\mathbf{D_{xx}} </math> and <math>\mathbf{D_{yy}} </math> are 1D discrete Laplacians in the ''x''- and ''y''-directions, correspondingly, and <math>\mathbf{I} </math> are the identities of appropriate sizes. Both <math>\mathbf{D_{xx}} </math> and <math>\mathbf{D_{yy}} </math> must correspond to the case of the homogeneous [[Dirichlet boundary condition]] at end points of the ''x''- and ''y''-intervals, in order to generate the 2D discrete Laplacian ''L'' corresponding to the homogeneous [[Dirichlet boundary condition]] everywhere on the boundary of the rectangular domain.

			Here is a sample [[OCTAVE]]/[[MATLAB]] code to compute ''L'' on the regular 10×15 2D grid:
			<source lang="matlab">
			nx = 10; % number of grid points in the x-direction;
			ny = 15; % number of grid points in the y-direction;
			ex = ones(nx,1);
			Dxx = spdiags([ex -2*ex ex], [-1 0 1], nx, nx); %1D discrete Laplacian in the x-direction ;
			ey = ones(ny,1);
			Dyy = spdiags([ey, -2*ey ey], [-1 0 1], ny, ny); %1D discrete Laplacian in the y-direction ;
			L = kron(Dyy, speye(nx)) + kron(speye(ny), Dxx) ;
			</source>

			==Eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid==

			Knowing all [[eigenvalue]]s and [[eigenvector]]s of the factors, all [[eigenvalue]]s and [[eigenvector]]s of the [[Kronecker product]] can be [[Kronecker_product#Spectrum\|explicitly calculated]]. Based on this, [[eigenvalue]]s and [[eigenvector]]s of the [[Kronecker_product#Kronecker_sum_and_exponentiation\|Kronecker sum]]
			can also be explicitly calculated.

			The [[eigenvalue]]s and [[eigenvector]]s of the standard [[Central_difference#Higher-order_differences\|central difference approximation of the second derivative]] on an interval for traditional combinations of boundary conditions at the interval end points are [[Eigenvalues and eigenvectors of the second derivative\|well known]]. Combining these expressions with the formulas of [[eigenvalue]]s and [[eigenvector]]s for the [[Kronecker_product#Kronecker_sum_and_exponentiation\|Kronecker sum]], one can easily obtain the required answer.

			===Example: 3D discrete Laplacian on a regular grid with the homogeneous Dirichlet boundary condition===

			:<math>L = \mathbf{D_{xx}}\otimes\mathbf{I}\otimes\mathbf{I}+\mathbf{I}\otimes\mathbf{D_{yy}}\otimes\mathbf{I}+\mathbf{I}\otimes\mathbf{I}\otimes\mathbf{D_{zz}}, \,</math>

			where <math>\mathbf{D_{xx}},\,\mathbf{D_{yy}}</math> and <math>\mathbf{D_{zz}} </math> are 1D discrete Laplacians in every of the 3 directions, and <math>\mathbf{I} </math> are the identities of appropriate sizes. Each 1D discrete Laplacian must correspond to the case of the homogeneous [[Dirichlet boundary condition]], in order to generate the 3D discrete Laplacian ''L'' corresponding to the homogeneous [[Dirichlet boundary condition]] everywhere on the boundary. The eigenvalues are

			:<math> \lambda_{jx,jy,jz} =
			-\frac{4}{h_x^2} \sin\left(\frac{\pi j_x}{2(n_x + 1)}\right)^2
			-\frac{4}{h_y^2} \sin\left(\frac{\pi j_y}{2(n_y + 1)}\right)^2
			-\frac{4}{h_z^2} \sin\left(\frac{\pi j_z}{2(n_z + 1)}\right)^2
			</math>

			where <math> j_x=1,\ldots,n_x,\,j_y=1,\ldots,n_y,\,j_z=1,\ldots,n_z,\,</math>, and the corresponding eigenvectors are

			:<math>v_{ix,iy,iz,jx,jy,jz} =
			\sqrt{\frac{2}{n_x+1}} \sin\left(\frac{i_x j_x \pi}{n_x+1}\right)
			\sqrt{\frac{2}{n_y+1}} \sin\left(\frac{i_y j_y \pi}{n_y+1}\right)
			\sqrt{\frac{2}{n_z+1}} \sin\left(\frac{i_z j_z \pi}{n_z+1}\right)
			</math>

			where the multi-index <math>{jx,jy,jz}</math> pairs the eigenvalues and the eigenvectors, while the multi-index
			<math>{ix,iy,iz}</math> determines the location of the value of every eigenvector at the [[regular grid]]. The boundary points, where
			the homogeneous [[Dirichlet boundary condition]] are imposed, are just outside of the grid.

			==Available software==
			An [[OCTAVE]]/[[MATLAB]] code http://www.mathworks.com/matlabcentral/fileexchange/27279-laplacian-in-1d-2d-or-3d is available under a [[BSD License]], which computes the sparse matrix of the 1, 2D, and 3D negative Laplacians on a rectangular grid for combinations of Dirichlet, Neumann, and Periodic boundary conditions using [[Kronecker sum]]s of discrete 1D Laplacians. The code also provides the exact [[eigenvalue]]s and [[eigenvector]]s using the explicit formulas given above.

			[[Category:Operator theory]]
			[[Category:Matrix theory]]
			[[Category:Numerical differential equations]]
			[[Category:Finite differences]]

en>Francis Flinch: /* Bolt thrust estimations for various rifle cartridges */ added info

2012-07-18T09:53:04Z

Bolt thrust estimations for various rifle cartridges: added info

New page

They call the author Boris. As a girl what I absolutely love is to fence and I've been doing it for a while. The job he's been occupying for years is an invoicing officer and his salary is really [http://pinterest.com/search/pins/?q=worthwhile worthwhile]. Georgia is where our home is. If you want to find out more the look at his website: http://www.alcoholysociedad.org/publicaciones/descargas/canada-goose-outlet.html<br><br>Take a look at my homepage ... [http://www.alcoholysociedad.org/publicaciones/descargas/canada-goose-outlet.html Canada Goose En España]

Bolt thrust - Revision history

50.34.139.236: /* Bolt thrust estimations for various pistol/revolver cartridges */

en>Francis Flinch: /* Calculating bolt thrust */ internal ref

en>Francis Flinch: /* Bolt thrust estimations for various rifle cartridges */ added info