Using the Scilab software and the reference data, write a program that determines: a) The temperatures at the inner nodal points in the system, the position vectors of the inner nodal points, and the von Neumann stability coefficient as a function of the dimensions of the plate, the element size, the temperatures at the boundaries, the thermal diffusivity of the material, the defined time-step, the total run time, and a vector containing the temperatures at the inner nodes for the initial conditions of the system by applying the BTCS method. $[T,x,y,r] = f (L_x, L_y, h, T(0,y,t), T(L_x, y, t), T(x, 0, t), T(x, L_y, t), a, \Delta t, t_{end}, T(x, y, 0))$ b) The temperatures at the inner nodal points in the system, the position vectors of the inner nodal points, and the von Neumann stability coefficient as a function of the dimensions of the plate, the element size, the temperatures at the boundaries, the thermal diffusivity of the material, the defined time-step, the total run time, and a vector containing the temperatures at the inner nodes for the initial conditions of the system by applying the FTCS method. $[T,x,y,r] = f (L_x, L_y, h,T(0, y, t), T(L_x, y, t), T(x, 0, t), T(x, L_y, t), a, \Delta t, t_{end}, T(x, y, 0))$ c) The surface graph that shows the variation of the temperature with respect to the position in the plate and a graph that shows the heat map of the temperatures in the plate for each time-step. The program should also be able to store the snapshots of the graphs corresponding to each time-step in the simulation. Results The results will be based on the stored snapshots of the graphs that must be generated using the program for each case. Additionally, a video needs to be produced that shows the evolution of the system using the snapshots for each case.
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Adi S.
A machine contains the grid of wires shown in the accompanying sketch. At the seven indicated points, the temperature is kept fixed at the given values (in $^{\circ} \mathrm{C}$ ). Consider the temperatures $T_{1}(t), T_{2}(t),$ and $T_{3}(t)$ at the other three mesh points. Because of heat flow along the wires, the temperatures $T_{i}(t)$ changes according to the formula \[ T_{i}(t+1)=T_{i}(t)-\frac{1}{10} \sum\left(T_{i}(t)-T_{\mathrm{adj}}(t)\right) \], where the sum is taken over the four adjacent points in the grid and time is measured in minutes. For example, \[ \begin{aligned} T_{2}(t+1)=& T_{2}(t)-\frac{1}{10}\left(T_{2}(t)-T_{1}(t)\right)-\frac{1}{10}\left(T_{2}(t)-200\right) \\ &-\frac{1}{10}\left(T_{2}(t)-0\right)-\frac{1}{10}\left(T_{2}(t)-T_{3}(t)\right) \end{aligned} \] Note that each of the four terms we subtract represents the cooling caused by heat flowing along one of the wires. Let $\vec{x}(t)=\left[\begin{array}{l}T_{1}(t) \\ T_{2}(t) \\ T_{3}(t)\end{array}\right]$. a. Find a $3 \times 3$ matrix $A$ and a vector $\vec{b}$ in $\mathbb{R}^{3}$ such that \[ \vec{x}(t+1)=A \vec{x}(t)+\vec{b} \]. b. Introduce the state vector \[ \vec{y}(t)=\left[\begin{array}{c} T_{1}(t) \\ T_{2}(t) \\ T_{3}(t) \\ 1 \end{array}\right] \], with a "dummy" 1 as the last component. Find a $4 \times 4$ matrix $B$ such that \[ \vec{y}(t+1)=B \vec{y}(t) \] (This technique for converting an affine system into a linear system is introduced in Exercise $35 ;$ see also Exercise $32 .$ ) c. Suppose the initial temperatures are $T_{1}(0)=$ $T_{2}(0)=T_{3}(0)=0 .$ Using technology, find the temperatures at the three points at $t=10$ and $t=30$ What long-term behavior do you expect? d. Using technology, find numerical approximations for the eigenvalues of the matrix $B$. Find an eigenvector for the largest eigenvalue. Use the results to confirm your conjecture in part (c).
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A thin metal triangular plate (as pictured) has its three edges held at constant temperatures Ta = 110°C, Tb = 85°C, and Tc = 35°C. When the temperature of the plate reaches equilibrium, the temperature of the plate at an internal grid point is approximately the average of the different temperatures of the plate at the surrounding four grid points. Formulate a system of three linear equations that can be solved to determine the internal temperatures t1, t2, and t3. Write the system as an augmented matrix, and then input this matrix using Maple's Matrix command (make sure that all elements of the augmented matrix are written as whole numbers or fractions here, do not use decimals). The augmented matrix is: Reduce the augmented matrix to row-echelon or reduced row-echelon form and hence determine the approximate temperatures t1, t2, and t3 in degrees Celsius to two decimal places. t1 = ( ) degrees Celsius, to 2 decimal places) t2 = ( ) degrees Celsius, to 2 decimal places) t3 = ( ) degrees Celsius, to 2 decimal places)
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