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diff --git a/tutorials/module_3/6_pde.md b/tutorials/module_3/6_pde.md index 332852e..1f11b13 100644 --- a/tutorials/module_3/6_pde.md +++ b/tutorials/module_3/6_pde.md @@ -1,8 +1,28 @@ # 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: +$$ +A\frac{\partial^2u}{\partial x^2} + +B\frac{\partial^2u}{\partial x \partial y} + +C\frac{\partial^2u}{\partial y^2} + +D += 0 +$$ -## Finite Difference +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 @@ -11,11 +31,10 @@ $$ $$ Finite-different solutions -- Laplacian Difference equation +- Laplacian Difference equations in dimension $x$ and $y$: $$ \frac{\partial^2T}{\partial x^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta x^2} $$ -and $$ \frac{\partial^2T}{\partial y^2}= \frac{T_{i+1,j}-2T_{i,j}+T_{i-1,j}}{\Delta y^2} $$ @@ -56,6 +75,13 @@ MacCormack Method ## Finite-Element Method General Approach + +1. Discretization +2. Element Equations +3. Assembly +4. Boundary Conditions +5. Solutions +6. Postprocessing ### One-dimensional analysis ### Two-dimensional Analysis @@ -80,7 +106,7 @@ General Approach 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, dertermine the displacement of the springs. +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. |