[gnuplot_preamble,"set key bottom"]
plot2d([legendre_p(0, x), legendre_p(1, x), legendre_p(2, x),legendre_p(3, x), legendre_p(4, x), legendre_p(5, x)], [x, 0, 2], [y, -2, 5], [legend,"P_0 (x) = 1","P_1 (x) = x","P_2 (x) = (1/2)*(3*x^2 - 1)", "P_3 (x) = (1/2)*x*(5*x^2 - 3)","P_4 (x) = (1/8)*(35*x^4 - 30*x^2 + 3)","P_5 (x) = (1/8)*x*(63*x^4 - 70*x^2 + 15)" ], [ylabel,"P_n (x)"], [gnuplot_preamble,"set key bottom"]);
[xlabel,"expr"]
[ylabel,"expr"]
[legend,"expr1","expr2"]
plot2d(func, x, [xlabel,"expr"], [ylabel,"expr"], [legend,"expr1","expr2"])
The OpenFOAM® Foundation is a non-stock corporation, incorporated on 5th August 2011 in Delaware, USA. It is a nonprofit organization whose “specific objectives and purposes... shall be to promote and manage the free open source distribution of the OpenFOAM software” (Bylaws, Article II). The Foundation has taken on the guardianship of the OpenFOAM software, previously undertaken by OpenCFD Ltd. Through its bylaws, it formalises the commitment, begun by OpenCFD in 2004, to ensure OpenFOAM is free and open source only. It is supported by SGI and lists Mark Barrenechea (CEO, SGI) and Henry Weller (creator of OpenFOAM) amongst its Board of Directors.
The Foundation distributes the current repository version of OpenFOAM (and will distribute future ‘version’ releases of OpenFOAM) under the GNU general public licence (GPL). The GPL gives users the freedom to modify and redistribute the software and a guarantee of continued free use — as long as the terms of the GPL are adhered to. There are two elements to the GPL that provide protection against exploitation by companies including OpenFOAM within non-free and/or closed source software products. First, when any modified version of the software is redistributed, the source code must also be made available by the distributor. Secondly, any modified version can only legally be distributed open source under the GPL and software that links intimately enough to OpenFOAM has to be distributed under the GPL as well.
People and organisations contribute to OpenFOAM on the understanding that their contributions become part of the public commons of free software and are not exploited by producers of non-free software. The Foundation exists to ensure this, adopting a suitably strong, open source license, i.e. the GPL. It will manage the software code base and promote the software, e.g. through its website www.openfoam.org, whose content will grow over time. OpenFOAM Foundation 15th August 2011
The OpenFOAM® (Open Field Operation and Manipulation) CFD Toolbox is a free, open source CFD software package produced by OpenCFD Ltd. It has a large user base across most areas of engineering and science, from both commercial and academic organisations. OpenFOAM has an extensive range of features to solve anything from complex fluid flows involving chemical reactions, turbulence and heat transfer, to solid dynamics and electromagnetics. It includes tools for meshing, notably snappyHexMesh, a parallelised mesher for complex CAD geometries, and for pre- and post-processing. Almost everything (including meshing, and pre- and post-processing) runs in parallel as standard, enabling users to take full advantage of computer hardware at their disposal.
By being open, OpenFOAM offers users complete freedom to customise and extend its existing functionality, either by themselves or through support from OpenCFD. It follows a highly modular code design in which collections of functionality (e.g. numerical methods, meshing, physical models, …) are each compiled into their own shared library. Executable applications are then created that are simply linked to the library functionality. OpenFOAM includes over 80 solver applications that simulate specific problems in engineering mechanics and over 170 utility applications that perform pre- and post-processing tasks, e.g. meshing, data visualisation, etc.
The OpenFOAM Team produces the OpenFOAM® open source CFD toolbox and distributes it through the OpenFOAM Foundation. The OpenFOAM Team was formed at OpenCFD Ltd, a company established in 2004 to coincide with the release of its OpenFOAM software under general public licence. The company was founded by: Henry Weller, the creator of OpenFOAM and its chief developer from inception to the present day; Chris Greenshields; and, Mattijs Janssens. In 2011, the team moved to SGI Corp, following its acquisition of OpenCFD Ltd.
The OpenFOAM Team has unrivalled experience with OpenFOAM. Over the past 7 years they have successfully provided services to numerous science/engineering companies, consultancies and universities. Developments: written over half a million of lines of code. Support: delivered over 10,000 hours of support. Training: delivered over 100 training courses to over 1000 OpenFOAM users. The OpenFOAM Team is Henry Weller, Chris Greenshields, Mattijs Janssens, Andy Heather, Sergio Ferraris, Jenya Collings, Gijs Wierink and Laurence McGlashan.
diff(expr, x)
legendre_p(n, x)
legendre_q(n, x)
assoc_legendre_q(n, m, x)
assoc_legendre_q(n, m, x)
legendre_p(n, m, x)
plot2d([legendre_p(0, x), legendre_p(1, x), legendre_p(2, x),legendre_p(3, x), legendre_p(4, x), legendre_p(5, x)], [x, -2, 2], [y, -2, 2], [legend,"P_0 (x) = 1","P_1 (x) = x","P_2 (x) = (1/2)*(3*x^2 - 1)", "P_3 (x) = (1/2)*x*(5*x^2 - 3)","P_4 (x) = (1/8)*(35*x^4 - 30*x^2 + 3)", "P_5 (x) = (1/8)*x*(63*x^4 - 70*x^2 + 15)" ], [ylabel,"P_n (x)"], [gnuplot_preamble,"set key bottom"]);
BURG is basically a boot loader for Linux based on GRUB. Burg has a highly configurable menu system which works in both text and graphic mode. Simply put, BURG can be extensively customized and there are a number of good quality BURG themes available all over the web for free. Lets find out the best among them and also see how to install BURG and the following BURG themes in Ubuntu 11.10.I like this because it looks like a nice way to handle multiple installations of Linux and other OSs such as Mac and Windows.
\usepackage{caption}
singlelinecheck=false
\captionsetup[subfloat]{singlelinecheck=false}
labelsep=period
labelfont=sc
\captionsetup{labelsep=period, labelfont=sc}
\usepackage[labelsep=period, labelfont=sc]{caption}
\begin{figure}
\centering
\subfloat[]{\label{fig:fig_10a_axial_velocity_for_alpha_30_at_4_z_loc_w_1_over_1} \includegraphics[scale = 0.8]{fig_10a_axial_velocity_for_alpha_30_at_4_z_loc_w_1_over_1.png}}
\subfloat[]{\label{fig:fig_10b_axial_velocity_for_alpha_45_at_4_z_loc_w_1_over_1} \includegraphics[scale = 0.8]{fig_10b_axial_velocity_for_alpha_45_at_4_z_loc_w_1_over_1.png}}
\caption{Axial velocity distribution shown at several axial positions and divergence angles of (a) $ 30^{\circ} $ and (b) $ 45^{\circ} $.} \label{fig:fig_10_axial_vel}
\end{figure}
\begin{figure}
\centering
\captionsetup[subfloat]{singlelinecheck=false}
\subfloat[]{\label{fig:fig_10a_axial_velocity_for_alpha_30_at_4_z_loc_w_1_over_1} \includegraphics[scale = 0.8]{fig_10a_axial_velocity_for_alpha_30_at_4_z_loc_w_1_over_1.png}}
\subfloat[]{\label{fig:fig_10b_axial_velocity_for_alpha_45_at_4_z_loc_w_1_over_1} \includegraphics[scale = 0.8]{fig_10b_axial_velocity_for_alpha_45_at_4_z_loc_w_1_over_1.png}}
\captionsetup{labelsep=period, labelfont=sc}
\caption{Axial velocity distribution shown at several axial positions and divergence angles of (a) $ 30^{\circ} $ and (b) $ 45^{\circ} $.} \label{fig:fig_10_axial_vel}
\end{figure}
f(x) := sin(x) + cos(x)
plot2d(f(x), [x, 0, 7], [y, -2, 2]);
f(x) := sin(x) - cos(x)
plot2d(f(x), [x, 0, 7], [y, -2, 2]);
plot2d(bessel_y (0, x), [x, 0, 20], [y, -2, 1]);
plot2d(bessel_y (1, x), [x, 0, 20], [y, -2, 1]);
plot2d(bessel_y (2, x), [x, 0, 20], [y, -2, 1]);
plot2d([bessel_y (0, x), bessel_y (1, x), bessel_y (2, x)], [x, 0, 20]);
plot2d(bessel_j (0, x), [x, 0, 20]);
plot2d(bessel_j (1, x), [x, 0, 20]);
plot2d(bessel_j (1, x), [x, 0, 20]);
plot2d([bessel_j (0, x), bessel_j (1, x), bessel_j (2, x)], [x, 0, 20]);
bessel_j (index, expr)
bessel_y (index, expr)
bessel_i (index, expr)
bessel_k (index, expr)
\dfrac{\dfrac{\partial^2 C_1}{\partial R^2}}{\dfrac{\partial C_1}{\partial R}}
\dfrac{\hspace{0.25 cm}\dfrac{\partial^2 C_1}{\partial R^2} \hspace{0.25 cm}}{\hspace{0.25 cm} \dfrac{\partial C_1}{\partial R} \hspace{0.25 cm}}
\dfrac{\hspace{0.25 cm}\dfrac{\partial^2 C_1}{\partial R^2} \hspace{0.25 cm}}{\dfrac{\partial C_1}{\partial R}}
\dfrac{\dfrac{\partial^2 C_1}{\partial R^2}}{\hspace{0.25 cm} \dfrac{\partial C_1}{\partial R} \hspace{0.25 cm}}
\tiny
\scriptsize
\footnotesize
\small
\normalsize
\large
\Large
\LARGE
\huge
\Huge
\begin{align}
\dfrac{\partial^2 T \left( x, t + \dfrac{\Delta t}{2} \right)}{\partial x^2} = &\dfrac{1}{2} \left[ \dfrac{T \left( x + \Delta x, t + \Delta t \right) - 2 T \left( x, t + \Delta t \right) + T \left( x - \Delta x, t + \Delta t \right)}{\Delta x^2} \right. \nonumber \\ &\left. + \dfrac{T \left( x + \Delta x, t \right) - 2 T \left( x, t \right) + T \left( x - \Delta x, t \right)}{\Delta x^2} \right] + O \left( \Delta x^2 \right)
\end{align}
\begin{equation}
\begin{align}
\dfrac{\partial^2 T \left( x, t + \dfrac{\Delta t}{2} \right)}{\partial x^2} = &\dfrac{1}{2} \left[ \dfrac{T \left( x + \Delta x, t + \Delta t \right) - 2 T \left( x, t + \Delta t \right) + T \left( x - \Delta x, t + \Delta t \right)}{\Delta x^2} \right. \\ &\left. + \dfrac{T \left( x + \Delta x, t \right) - 2 T \left( x, t \right) + T \left( x - \Delta x, t \right)}{\Delta x^2} \right] + O \left( \Delta x^2 \right)
\end{align}
\end{equation}
\big(
\Big(
\bigg(
\Bigg(
\underbrace{t + \dfrac{\Delta t}{2}}_{t_0}
\underbrace{t + \dfrac{\Delta t}{2}}_{\grave{t}_0}
\underbrace{t + \dfrac{\Delta t}{2}}_{\acute{t}_0}
\underbrace{t + \dfrac{\Delta t}{2}}_{\text{\`{t}}_0}
\underbrace{t + \dfrac{\Delta t}{2}}_{\text{\'{t}}_0}
\underbrace{t + \dfrac{\Delta t}{2}}_{\textit{\`{t}}_0}
\underbrace{t + \dfrac{\Delta t}{2}}_{\textit{\'{t}}_0}
The cause of this problem is the APIC (Advanced Programmable Interrupt Controller), on which Windows depends. You must enable this feature in the virtual machine settings.
Right click on the virtual machine in VirtualBox console window.
Select "Settings..." Click "System" tab on the left and check "Enable IO APIC" item.
