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diff --git a/tutorials/module_3/3_pde.md b/tutorials/module_3/3_pde.md new file mode 100644 index 0000000..2980fa7 --- /dev/null +++ b/tutorials/module_3/3_pde.md @@ -0,0 +1,200 @@ +# Partial Differential Equation +Partial differential equations are defines when two or more partial derivatives are present in an equation. Due to the widespread application in engineering, we will be looking at **second-order equations** which can be expressed in the following general form: +$$ +A\frac{\partial^2u}{\partial x^2} + +B\frac{\partial^2u}{\partial x \partial y} + +C\frac{\partial^2u}{\partial y^2} + +D += 0 +$$ + +where $A$, $B$ and $C$ are functions of both $x$ and $y$. $D$ is a a function of $x$, $y$, $u$, $\partial u / \partial x$, and $\partial u / \partial y$. Similar to our beloved quadratic formula, we can take the discriminant of the equation +$$ +\Delta = B^2 - 4 AC +$$ +Based on the discriminant we can categorize the equations into the following three categories: + +| $\Delta$ | Category | Example | +| -------- | ---------- | ------------------------ | +| - | Elliptical | Laplace equation | +| 0 | Parabolic | Heat Conduction equation | +| + | Hyperbolic | Wave equation | +## Finite Difference Methods +### Elliptic Equations +- Used for steady-state, boundary value problems +- Examples where fields where these equations are used are: steady-state heat conduction, electrostatics and potential flow + + +Description of how the Laplace equations works +$$ +\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2}=0 +$$ + +Finite-different solutions +- Laplacian Difference equations in dimension $x$ and $y$: +$$ +\boxed{\frac{\partial^2T}{\partial x^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta x^2}} +$$ +$$ +\frac{\partial^2T}{\partial y^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta y^2} +$$ + +Boundary Conditions + + +Control-Volume approach +<img + style="display: block; + margin-left: auto; + margin-right: auto; + width: 50%;" + src="control_volume_approach.png" + alt="Two Different perspsectives for developing approximate solutions of PDE: (a) finite-difference or node and (b) control-volume based"> +Computer Algorithms + +### Parabolic Equations +- Used for unstead-state, initial + boundary conditions problems +For parabolic PDE equations we also consider the change in time as well as space. + +Heat-conduction equation +Explanation of heat equation +$$ +k\frac{\partial^2T}{\partial x^2}=\frac{\partial T}{\partial t} +$$ + +Explicit methods + +Forward-time central-space (FTCS Scheme) + based on forward euler method and central difference in space. + +One dimensional Heat conduction example + We need to measure the insulation of a wall and measure the change of temperature through the + + +Simple Implicit methods +Crank-Nicolson +ADI + + +### Hyperbolic Equations +MacCormack Method +In computational fluid dynamics (CFD), the governing equations are the Navier-Stokes equations. For inviscid (no viscosity) compressible flow, these reduce to the Euler equations: +$$ +\frac{\partial u}{\partial t}+\frac{\partial f(u)}{\partial x} = 0 +$$ +where, $U$ is conserved variables. This equation is a hyperbolic PDE. + +Discretize domain: space and time + + + +Write fluxes: + + +MacCormack Algorithm +- Predictor +$$ +u^p_i = u^n_i - \frac{\Delta t}{\Delta x}(f^n_{i+1}-f^n_i) +$$ +- Corrector +$$ +u^{p+1}_i = \frac{1}{2}(u^n_i+u^p_i) - \frac{\Delta t}{2\Delta x}(f^p_{i}-f^n_{i-1}) +$$ + +Method of characteristics + +## Finite-Element Method +General Approach + +1. Discretization +2. Element Equations +3. Assembly +4. Boundary Conditions +5. Solutions +6. Post-processing +### One-dimensional analysis + + +### Two-dimensional Analysis + + + + + + + + + + + + + + + + + +## Problem 1: Finite-Element Solution of a Series of Springs + +Problem 32.4 from Numerical Methods for Engineers 7th Edition Steven C. Chapra and Raymond P. Canale + +A series of interconnected strings are connected to a fixed wall where the other is subject to a constant force F. Using the step-by-step procedure from above, determine the displacement of the springs. + +Solution: +Setup: Let's partition the system to treat each spring as an element. Thus, the system consists for 4 elements and 5 nodes. +<img + style="display: block; + margin-left: auto; + margin-right: auto; + width: 50%;" + src="example32-4.png" + alt="Diagram of system of springs example"> +Element Equations: Analyzing element 1 we get the following free body diagram: +<img + style="display: block; + margin-left: auto; + margin-right: auto; + width: 50%;" + src="example32-4element1.png" + alt="Diagram of element 1"> + +Applying Hook's law to the element we get: +$$ +F=kx +$$ +$$ +F=k(x_1-x_2) +$$ +where $(x_1-x_2)$ is how much the first spring is stretched out. +Re-writing this equation: +$$ +F_1 = kx_1 - kx_2 +$$ +$$ +F_2=-kx_1+kx_2 +$$ +$$ +\begin{bmatrix} k & -k \\ -k & k \end{bmatrix} +\begin{Bmatrix} x_{1} \\ +x_{2} \end{Bmatrix} += +\begin{Bmatrix} F_{1} \\ F_{2} \end{Bmatrix} +$$ +$$ +[k]\{x\}=\{F\} +$$ +where $[k]$ is the element property matrix or in this case element stiffness matrix. $x$ is a column vector of unknowns (in this case position of each) and $F$ is a vector column with external influence applied at the nodes. +Assembly: Once individual element equations are derived we will link them together using assembly. +$$ +[k]\{x'\}=\{F'\} +$$ +where $[k]$ is the assemblage property matrix and $\{u'\}$ and $\{F'\}$ column vectors are unknowns and external forces that are marked with primes to denote that they are an assemblage of the vectors $\{u\}$ and $\{F\}$. +$$ +\begin{bmatrix} k & -k \\ -k & k \end{bmatrix} +\begin{Bmatrix} x_{1} \\ x_2 \end{Bmatrix} +$$ + +## Problem 2: Finite + +Solve the non-dimensional transient heat conduction equation in two dimensions, which represents the transient temperature distribution in an insulated plate + +---
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