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Consider that we are finding a solution for a nonlinear equation: $f(u) = 0$. The Newton-Raphson (N-R) iterative scheme is as follows: Iteration 0: select an initial guess $u_{0}$ for the solution, see Figure 1. Iteration 1: calculate the value of $f$ at $u_{0}$: $f(u_{0})$, and the value of the first order derivative of $f$ … Read more
Let us consider an elastic deformable body $\Omega$. The discrete form of equilibrium equation should take the following form $\mathbf{K} \mathbf{u} = \mathbf{F}$, where $\mathbf{K}$ is the stiffness matrix, $\mathbf{F}$ is the vector of (external) load, and $\mathbf{u}$ is the vector of nodal displacements. For simplicity, we assume that $\Omega$ is a two dimensional rectangular … Read more
Strong formulation of the governing (partial differential equation) equation of steady-state transfer conduction $ \nabla \cdot \mathbf{\mathbf{k} \nabla T} + Q = 0 $, (1) in which $T$ is the temperature; $\mathbf{k} = diag(k_{xx}, k_{yy}, k_{zz})$ is the tensor of thermal conductivity; $Q$ is the heat sink/heat source. Boundary conditions are given by $T = … Read more
Shape functions (defined in natural coordinates) $N_{1} = \frac{1}{4}(1 – \xi)(1 – \eta)$ $N_{2} = \frac{1}{4}(1 + \xi)(1 – \eta)$ $N_{3} = \frac{1}{4}(1 + \xi)(1 + \eta)$ $N_{4} = \frac{1}{4}(1 – \xi)(1 + \eta)$ The above shape functions are in fact the Lagrangian interpolation Note that we define the element as a square of size … Read more