User:Eml5526.s11.team5.srv/hw7

=Finding the static and dynamic solution of a bi-unit square using 2D LIBF=

P.D.E

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$$T div (grad u) + f = \rho \ \frac{\partial^2u}{\partial t^2} $$ $$
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 *  $$ \displaystyle                                                                     (Eq. 7.2.1)
 * }


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$$ \quad T = 4 $$ $$
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 *  $$ \displaystyle                                                                     (Eq. 7.2.2)
 * }


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$$ f (\mathbf{x}) = 1 $$ $$
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 *  $$ \displaystyle                                                                     (Eq. 7.2.3)
 * }


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$$ \quad \rho = 3 $$ $$
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 *  $$ \displaystyle                                                                     (Eq. 7.2.4)
 * }


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$$ \quad Basis \ Function \ : \ 2D \ Lagrange \ Interpolation \ Basis \ Function $$ $$
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 *  $$ \displaystyle
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Essential Boundary Condition

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$$ \quad g = 0 $$ $$
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 *  $$ \displaystyle                                                                     (Eq. 7.2.5)
 * }

Static(steady state)

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1. Find the static solution $$ \quad U^h $$ to $$ \quad 10^{-6} $$ accuracy at the center. $$
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 *  $$ \displaystyle
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Dynamic(transient)

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1. Fundamental Frequency and Fundamental Period $$
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 *  $$ \displaystyle
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<span id="(1)">
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2. Plot $$ u^h(0,t) \ vs \ t \quad  for \ t \ \epsilon \ [0,2T_1] $$ for symmetric and Non-Symmetric $$ \quad u^0 $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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3. Produce a movie of the vibrating membrane for symmetric and Non-Symmetric $$ \quad u^0 $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
 * }

Solution
Substituting the given values in the P.D.E, we get, <span id="(1)">
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$$\quad 4 \ div (grad u) \ + \ 1 \ = \ 0 $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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$$ 4 \left ( \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2}\right ) + 1 = 0$$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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$$ \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = \frac{-1}{4} $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.6)
 * }

2D LIBF

<span id="(1)">
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$$ N_I (x,y) = L_{i,m}(x).L_{j,n}(y) \quad Where \ I = i+(j-1)m $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.7)
 * }

Where,

<span id="(1)">
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$$ L_{i,m}(x) = \prod_{k=1\neq i}^{m}\frac{x-x_k}{x_i - x_k} $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }<span id="(1)">
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$$ L_{j,n}(x) = \prod_{k=1\neq j}^{m}\frac{y-y_k}{y_j - y_k} $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

Consider the Weak Form

<span id="(1)">
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$$  \tilde{m}(w,u) + \tilde{k}(w,u) = \tilde{f}(w) $$ $$
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 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.8)
 * }

where,

<span id="(1)">
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$$m(w,u) = \int\limits_\Omega {w\rho c\frac} d\Omega $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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$$ k(w,u) = \int\limits_\Omega {\nabla wK\nabla u} d\Omega  $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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$$ f(w,u) = - \int_{\Gamma h} {whd{\Gamma _h}}  + \int\limits_\Omega  {wf} d\Omega  $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
 * }

Static case
For static case, $$ \frac{\partial u}{\partial t} = 0 $$ the above equation can be written as,

<span id="(1)">
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$$ \quad K d = F $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
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where,

<span id="(1)">
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$$ {K_{ij}} = \int_ {(\nabla N_I^e)I(\nabla N_J^e)d{w^e}} $$ $$
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 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.9)
 * }

Using the above equations in MATLAB, we get,

K =  33.3257  -13.5386  -13.5386    2.8639 -13.5386  33.3257    2.8639  -13.5386       -13.5386    2.8639   33.3257  -13.5386         2.8639  -13.5386  -13.5386   33.3257

F = [0.5625  0.5625  0.5625  0.5625]^T

d = [0.0617  0.0617  0.0617  0.0617]^T

Dynamic case
For this case, It is given that the value of mass density $$ \quad \rho \ = \ 3$$

Free Vibration: Solve Eigen Value problem
The equation given to solve the Eigen value problem is,

<span id="(1)">
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$$ \mathbf{K} \mathbf{\phi} = \lambda \mathbf{M}\mathbf{\phi}  $$ $$
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 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.10)
 * }

Where,

<span id="(1)">
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$$\quad \mathbf{\phi} - Eigen \ vectors $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
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<span id="(1)">
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$$ \quad \lambda - Eigen \ values $$ $$
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 * <p style="text-align:right"> $$ \displaystyle
 * }

Using the above equation and the expression for K, F matrix, we can calculate the fundamental frequency and fundamental time period.

Fundamental Frequency =

Fundamental Time Period =

Transient Analysis with symmetric Uo
Initial conditions

<span id="(1)">
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$$u(\mathbf{x},t=0) = u^h_s(x) \ \forall \  \mathbf{x} \ \epsilon  \ \Omega  $$ $$
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 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.11)
 * }

<span id="(1)">
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$$ \dot{u} (\mathbf{x},t=0) = 0 \ \forall \  \mathbf{x} \ \epsilon  \ \Omega $$ $$
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 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.12)
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<span id="(1)">
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$$T = 4, \qquad f(\mathbf{x},t) = 0 \ in \  \Omega  $$ $$
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 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.13)
 * }

By using the above initial condition, we can produce the movie of the vibrating membrane.

Movie of the vibrating membrane
The Movie of dynamic membrane has been uploaded in the youtube and it can be seen from the link below

FE Analysis of a Dynamic Membrane - symmetric case

For T = 0,



Transient Analysis with Non-symmetric Uo
In this case, initial u value is given by,

<span id="(1)">
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$$u(\mathbf{x},t=0) = (x+1)(y+\frac{1}{2}) cos (\frac{\pi}{2}x) cos (\frac{\pi}{2}y) \ \forall \  \mathbf{x} \ \epsilon  \ \Omega$$ $$
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 * <p style="text-align:right"> $$ \displaystyle                                                                     (Eq. 7.2.14)
 * }

By using the above initial condition, we can produce the movie of the vibrating membrane.

Movie of the vibrating membrane
The Movie of dynamic membrane for the non-symmetric case, has been uploaded in the youtube and it can be seen from the link below

FE Analysis of a Dynamic membrane - Non Symmetric case