diff options
Diffstat (limited to 'tutorials/module_3/6_pde.md')
| -rw-r--r-- | tutorials/module_3/6_pde.md | 200 |
1 files changed, 0 insertions, 200 deletions
diff --git a/tutorials/module_3/6_pde.md b/tutorials/module_3/6_pde.md deleted file mode 100644 index 2980fa7..0000000 --- a/tutorials/module_3/6_pde.md +++ /dev/null @@ -1,200 +0,0 @@ -# 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 - ----
\ No newline at end of file |