Single-Variable Problems in 2D

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Time-Dependent Problems: FEM Formulations Coupled Formulation $$ u(x,t) \approx u^e(x, t) = \sum_{j=1}^n u_j \psi_j(x, t) $$ Decoupled Formulation $$ u(x, t) \approx u^e(x, t) = \sum_{j=1}^n u_j(t) \psi_j(x) $$ Two Steps of Decoupled Formulation $$ \text{compute } \{u\}_{s+1} \text{ using } \{u\}_s, \{u\}_{s-1}, \dots $$ $$ u(x,t_s) \approx u^e(x,t_s) = \sum_{j=1}^n u_j^e(t_s) \psi_j(x) $$ Differential Equation for Time-Dependent Problems $$ -a \frac{\partial}{\partial x} \left( \frac{\partial u}{\partial x} \right) + b \frac{\partial^2}{\partial x^2} \left( \frac{\partial^2 u}{\partial x^2} \right) + c_0 u + c_1 \frac{\partial u}{\partial t} + c_2 \frac{\partial^2 u}{\partial t^2} = f(x,t) $$ Weak Form of the Differential Equation $$ \int_{x_a}^{x_b} \left( a \frac{\partial w}{\partial x} \frac{\partial u}{\partial x} + b \frac{\partial^2 w}{\partial x^2} \frac{\partial^2 u}{\partial x^2} + c_0 w u + c_1 w \frac{\partial u}{\partial t} + c_2 w \frac{\partial^2 u}{\partial t^2} - w f \right) dx - Q_1 w(x_a) - Q_3 w(x_b) - \tilde{Q}_2 \left( \frac{\partial w}{\partial x} \right)_{x_b} - \tilde{Q}_4 \left( \frac{\partial w}{\partial x} \right)_{x_a} = 0 $$ $$ [K]\{u\} + [M^1]\{\dot{u}\} + [M^2]\{\ddot{u}\} = \{F\} $$ $[K] = [K^1] + [K^2] + [M^0]$ $M_{ij}^0 = \int_{x_a}^{x_b} c_0 \psi_i \psi_j dx$ $M_{ij}^1 = \int_{x_a}^{x_b} c_1 \psi_i \psi_j dx$ $M_{ij}^2 = \int_{x_a}^{x_b} c_2 \psi_i \psi_j dx$ $K_{ij}^1 = \int_{x_a}^{x_b} a \frac{d\psi_i}{dx} \frac{d\psi_j}{dx} dx$ $K_{ij}^2 = \int_{x_a}^{x_b} b \frac{d^2\psi_i}{dx^2} \frac{d^2\psi_j}{dx^2} dx$ $F_i = \int_{x_a}^{x_b} \psi_i f dx + \tilde{Q}_i$ Parabolic Equations (Time Approximation) $$ a \frac{du}{dt} + bu = f(t) $$ $$ (1-\alpha)\dot{u}_s + \alpha\dot{u}_{s+1} = \frac{u_{s+1}-u_s}{\Delta t_{s+1}} \quad \text{for } 0 \le \alpha \le 1 $$ $$ u_{s+\alpha} = (1-\alpha)u_s + \alpha u_{s+1} $$ $$ [a + \alpha \Delta t_{s+1} b]u_{s+1} = [a - (1-\alpha) \Delta t_{s+1} b]u_s + \Delta t_{s+1} [\alpha f_{s+1} + (1-\alpha) f_s] $$ $$ u_{s+1} = \frac{a - (1-\alpha) \Delta t_{s+1} b}{a + \alpha \Delta t_{s+1} b} u_s + \frac{\Delta t_{s+1} [\alpha f_{s+1} + (1-\alpha) f_s]}{a + \alpha \Delta t_{s+1} b} $$ $$ |A| \le 1 \implies \left| \frac{a - (1-\alpha) \Delta t_{s+1} b}{a + \alpha \Delta t_{s+1} b} \right| \le 1 $$ Fully Discretized Finite Element Equations (Parabolic) $$ [M]\{\dot{u}\} + [K]\{u\} = \{F\} $$ $$ \Delta t_{s+1} [(1-\alpha)\{\dot{u}\}_s + \alpha\{\dot{u}\}_{s+1}] = \{u\}_{s+1} - \{u\}_s $$ $$ [K]_{s+1}\{u\}_{s+1} = [\tilde{K}]_s\{u\}_s + \{\tilde{F}\}_{s,s+1} $$ $[\tilde{K}]_{s+1} = [M] + \alpha_1 [K]_{s+1}$ $[\tilde{K}]_s = [M] - \alpha_2 [K]_s$ $\{\tilde{F}\}_{s,s+1} = \Delta t_{s+1} [\alpha \{F\}_{s+1} + (1-\alpha) \{F\}_s]$ $\alpha_1 = \alpha \Delta t_{s+1}$ $\alpha_2 = (1-\alpha) \Delta t_{s+1}$ Mass Lumping Row-Sum Lumping $$ M_{ii}^L = \sum_{j=1}^n \int_{x_a}^{x_b} \rho \psi_i \psi_j dx = \int_{x_a}^{x_b} \rho \psi_i dx $$ Linear element: $[M^L] = \frac{\rho h_e}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ Quadratic element: $[M^L] = \frac{\rho h_e}{6} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ Proportional Lumping $$ M_{ii}^L = \alpha \int_{x_a}^{x_b} \rho \psi_i \psi_i dx $$ $$ \alpha = \frac{\int \rho dx}{\sum_i \int \rho \psi_i \psi_i dx} $$ $$ (\Delta t_{cri})_L = h \left( \frac{2\rho}{E} \right)^{1/2} > (\Delta t_{cri})_C $$ Hyperbolic Equations (Time Approximation) $$ [K]\{u\} + [C]\{\dot{u}\} + [M]\{\ddot{u}\} = \{F\} $$ $$ \{u(0)\} = \{u_0\}, \{\dot{u}(0)\} = \{\dot{v}_0\} $$ $$ \{u\}_{s+1} = \{u\}_s + \Delta t \{\dot{u}\}_s + (\Delta t)^2 \left( \frac{1}{2}-\gamma \right) \{\ddot{u}\}_s + (\Delta t)^2 \gamma \{\ddot{u}\}_{s+1} $$ $$ \{\dot{u}\}_{s+1} = \{\dot{u}\}_s + \Delta t (1-\alpha) \{\ddot{u}\}_s + \Delta t \alpha \{\ddot{u}\}_{s+1} $$ $$ [\tilde{K}]_{s+1}\{u\}_{s+1} = [\tilde{K}]_s\{u\}_s + \{\tilde{F}\}_{s,s+1} $$ $[\tilde{K}]_{s+1} = [K]_{s+1} + a_3[M]_{s+1} + a_6[C]_{s+1}$ $\{\tilde{F}\}_{s,s+1} = \{F\}_{s+1} + [M]_{s+1}\{A\}_s + [C]_{s+1}\{B\}_s$ $\{A\}_s = a_3\{u\}_s + a_4\{\dot{u}\}_s + a_5\{\ddot{u}\}_s$ $\{B\}_s = a_6\{u\}_s + a_7\{\dot{u}\}_s + a_8\{\ddot{u}\}_s$ $a_3 = \frac{1}{\gamma (\Delta t)^2}$, $a_4 = \frac{1}{\gamma \Delta t}$, $a_5 = \frac{1}{2\gamma} - 1$ $a_6 = \frac{\alpha}{\gamma \Delta t}$, $a_7 = \frac{\alpha}{\gamma} - 1$, $a_8 = \frac{\alpha}{2\gamma} (\Delta t) - (\Delta t)$ $\{\ddot{u}\}_{s+1} = a_3(\{u\}_{s+1} - \{u\}_s) - a_4\{\dot{u}\}_s - a_5\{\ddot{u}\}_s$ $\{\dot{u}\}_{s+1} = \{\dot{u}\}_s + a_1\{\ddot{u}\}_s + a_2\{\ddot{u}\}_{s+1}$ $a_1 = (1-\alpha)\Delta t$, $a_2 = \alpha\Delta t$ Eigenvalue Problems $$ A(u) = \lambda B(u) $$ $$ \frac{d^2 u}{dx^2} = \lambda u(x) $$ Formulation of Eigenvalue Problems Axial Motion of a Bar $$ \rho A \frac{\partial^2 u}{\partial t^2} - \frac{\partial}{\partial x} \left( EA \frac{\partial u}{\partial x} \right) = 0 $$ $$ u(x,t) = U(x)T(t) $$ $$ T(t)=e^{-i\omega t} $$ $$ -\rho A \omega^2 U - \frac{d}{dx} \left( EA \frac{dU}{dx} \right) = 0 $$ Finite Element Formulation for Eigenvalue Problems $$ -\frac{d}{dx} \left( a \frac{dU}{dx} \right) + c(x)U(x) = \lambda c_0(x)U(x) $$ $$ \int_{x_a}^{x_b} \left( a \frac{dw}{dx} \frac{dU}{dx} + c w U - \lambda c_0 w U \right) dx - Q_1 w(x_a) - Q_n w(x_b) = 0 $$ $$ [K^e]\{u^e\} - \lambda [M^e]\{u^e\} = \{Q^e\} $$ $K_{ij}^e = \int_{x_a}^{x_b} \left( a(x) \frac{d\psi_i}{dx} \frac{d\psi_j}{dx} + c(x)\psi_i\psi_j \right) dx$ $M_{ij}^e = \int_{x_a}^{x_b} c_0(x)\psi_i\psi_j dx$ Natural Vibration of Beams (Euler-Bernoulli Beam Theory) $$ \rho A \frac{\partial^2 w}{\partial t^2} - \rho I \frac{\partial^4 w}{\partial t^2 \partial x^2} + EI \frac{\partial^4 w}{\partial x^4} = 0 $$ $$ w(x,t) = W(x)e^{-i\omega t} $$ $$ \lambda = \omega^2 $$ $$ \frac{d^2}{dx^2} \left( EI \frac{d^2 W}{dx^2} \right) - \lambda \left( \rho A W - \rho I \frac{d^2 W}{dx^2} \right) = 0 $$ $$ ([K^e] - \omega^2 [M^e]) \{\Delta^e\} = \{Q^e\} $$ $$ \Delta^e = \begin{pmatrix} w_1 \\ \theta_1 \\ w_2 \\ \theta_2 \end{pmatrix} $$ Stiffness Matrix $[K^e]$: $$ [K^e] = \frac{EI}{L^3} \begin{pmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{pmatrix} $$ Consistent Mass Matrix $[M^e]$: $$ [M^e] = \frac{\rho A L}{420} \begin{pmatrix} 156 & 22L & 54 & -13L \\ 22L & 4L^2 & 13L & -3L^2 \\ 54 & 13L & 156 & -22L \\ -13L & -3L^2 & -22L & 4L^2 \end{pmatrix} + \frac{\rho I}{30L} \begin{pmatrix} 36 & 3L & -36 & 3L \\ 3L & 4L^2 & -3L & -L^2 \\ -36 & -3L & 36 & -3L \\ 3L & -L^2 & -3L & 4L^2 \end{pmatrix} $$ Lumped Mass Matrix $[M^L]$: $$ [M^L] = \frac{\rho A L}{2} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $$ Stability (Buckling) of Beams (EBT) $$ \frac{d^2}{dx^2} \left( EI \frac{d^2 W}{dx^2} \right) + N^0 \frac{d^2 W}{dx^2} = 0 $$ $$ \lambda = N^0 $$ $$ [K^e]\{\Delta^e\} - N^0 [G^e]\{\Delta^e\} = \{Q^e\} $$ Geometric Stiffness Matrix $[G^e]$: $$ [G^e] = \frac{1}{30L} \begin{pmatrix} 36 & 3L & -36 & 3L \\ 3L & 4L^2 & -3L & -L^2 \\ -36 & -3L & 36 & -3L \\ 3L & -L^2 & -3L & 4L^2 \end{pmatrix} $$ Frame Elements Natural Vibration: $[K^e]\{\Delta^e\} - \omega^2 [M^e]\{\Delta^e\} = \{Q^e\}$ Buckling: $[K^e]\{\Delta^e\} - N^0 [G^e]\{\Delta^e\} = \{Q^e\}$ $[K^e] = [T]^T [\tilde{K}^e] [T]$ $[M^e] = [T]^T [\tilde{M}^e] [T]$ $[G^e] = [T]^T [\tilde{G}^e] [T]$ Single-Variable Problems in Two Dimensions General Second-Order Equation $$ A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u + G = 0 $$ Weak Form Formulation $$ \int_{\Omega} w \left( A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u + G \right) d\Omega = 0 $$ $$ \int_{\Omega} w A \frac{\partial^2 u}{\partial x^2} d\Omega = \oint_{\partial \Omega} w A \frac{\partial u}{\partial x} n_x ds - \int_{\Omega} A \frac{\partial w}{\partial x} \frac{\partial u}{\partial x} d\Omega - \int_{\Omega} w \frac{\partial A}{\partial x} \frac{\partial u}{\partial x} d\Omega $$ Common Two-Dimensional Problem Types Poisson Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x,y)$ Laplace Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ Convection-Diffusion Equation: $k \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) - u_x \frac{\partial u}{\partial x} - u_y \frac{\partial u}{\partial y} = Q$ Helmholtz Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + k^2 u = 0$ Finite Element Discretization for 2D Problems $$ u^e(x,y) = \sum_{j=1}^{n_e} u_j^e N_j(x,y) $$ Element Equations (Example: Poisson's Equation) $$ \int_{\Omega} \left( \frac{\partial w}{\partial x} \frac{\partial u}{\partial x} + \frac{\partial w}{\partial y} \frac{\partial u}{\partial y} \right) d\Omega = - \int_{\Omega} w f d\Omega + \oint_{\partial \Omega} w \frac{\partial u}{\partial n} ds $$ $$ [K^e]\{u^e\} = \{F^e\} $$ $K_{ij}^e = \int_{\Omega_e} \left( \frac{\partial N_i}{\partial x} \frac{\partial N_j}{\partial x} + \frac{\partial N_i}{\partial y} \frac{\partial N_j}{\partial y} \right) d\Omega$ $F_i^e = - \int_{\Omega_e} N_i f d\Omega + \int_{\Gamma_e} N_i q_n ds$ For a 3-node linear triangular element (constant $k_x, k_y$): $$ [K^e] = \frac{1}{4A} \left( k_x \begin{pmatrix} b_1^2 & b_1b_2 & b_1b_3 \\ b_2b_1 & b_2^2 & b_2b_3 \\ b_3b_1 & b_3b_2 & b_3^2 \end{pmatrix} + k_y \begin{pmatrix} c_1^2 & c_1c_2 & c_1c_3 \\ c_2c_1 & c_2^2 & c_2c_3 \\ c_3c_1 & c_3c_2 & c_3^2 \end{pmatrix} \right) $$ $A$ is the element area $b_i = y_j - y_k$ $c_i = x_k - x_j$ For a 4-node linear quadrilateral element (isoparametric formulation): $$ [K^e] = \int_{-1}^{1} \int_{-1}^{1} ([B_x]^T k_x [B_x] + [B_y]^T k_y [B_y]) \det(J) d\xi d\eta $$ Where $[B_x] = \frac{\partial \{N\}}{\partial x}$ and $[B_y] = \frac{\partial \{N\}}{\partial y}$ are row vectors of shape function derivatives with respect to global coordinates. $$ \{F^e\} = \int_{-1}^{1} \int_{-1}^{1} \{N\}^T f \det(J) d\xi d\eta + \oint_{\partial \Omega} \{N\}^T q_n ds $$ Shape Functions for 2D Elements Triangular Elements (3-node linear): $N_i = L_i$ $L_i = \frac{1}{2A} (a_i + b_i x + c_i y)$ $a_i = x_j y_k - x_k y_j$ $b_i = y_j - y_k$ $c_i = x_k - x_j$ Quadrilateral Elements (4-node bilinear): $N_1 = \frac{1}{4}(1-\xi)(1-\eta)$ $N_2 = \frac{1}{4}(1+\xi)(1-\eta)$ $N_3 = \frac{1}{4}(1+\xi)(1+\eta)$ $N_4 = \frac{1}{4}(1-\xi)(1+\eta)$ Isoparametric Formulation $$ x = \sum N_i(\xi,\eta) x_i $$ $$ y = \sum N_i(\xi,\eta) y_i $$ $$ \begin{pmatrix} \frac{\partial N_i}{\partial \xi} \\ \frac{\partial N_i}{\partial \eta} \end{pmatrix} = [J] \begin{pmatrix} \frac{\partial N_i}{\partial x} \\ \frac{\partial N_i}{\partial y} \end{pmatrix} \quad \text{where } [J] = \begin{pmatrix} \frac{\partial x}{\partial \xi} & \frac{\partial y}{\partial \xi} \\ \frac{\partial x}{\partial \eta} & \frac{\partial y}{\partial \eta} \end{pmatrix} $$ $$ \begin{pmatrix} \frac{\partial N_i}{\partial x} \\ \frac{\partial N_i}{\partial y} \end{pmatrix} = [J]^{-1} \begin{pmatrix} \frac{\partial N_i}{\partial \xi} \\ \frac{\partial N_i}{\partial \eta} \end{pmatrix} $$ $$ d\Omega = dx dy = \det(J) d\xi d\eta $$