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The Cartesian Transformation Method (CTM) was proposed by Khosravifard and Hematiyan [1] for numerical integration in meshfree analysis. The main idea is to transform a domain integral into a double integral. Let us take a 2D integral as an example $ I = \int_{\Omega} F(x,y) d\Omega$, where F(x,y) is an arbitrary function being defined in … Read more
Strong formulation (or Strong form): the partial differential equation that governs the problem. For example: the equilibrium equation in solid mechanics, the Fourier’s equation for transport of thermal energy, the Fick’s law for mass transport, etc. Below is the equilibrium equation in solid mechanics, which satisfies at every point of the problem domain. $\nabla \cdot … Read more
The Gaussian quadrature for a square domain ([-1, 1] x [-1, 1]) can be conducted by a similar manner to 1D integration (see 1D Gaussian quadrature) $\int \limits_{-1}^{1} \int \limits_{-1}^{1} g(\xi,\eta) d\xi\eta \approx \sum_{i=1}^{n} \sum_{j=1}^{n} g(\xi_i, \eta_j) w_i w_j$, where $\xi$ and $\eta$ denotes the coordinate in horizontal and vertical direction, respectively. Theoretically, the number … Read more
Integration on any straight line of length $L$ can be transformed into integration on the interval [-1, 1] by a simple transformation $\int_{z_{1}}^{z_{2}} f(z) dz = \int \limits_{-1}^{1} \frac{L}{2} f(\xi) d\xi$ Note that the equation for transformation reads $z = \frac{z_2 – z_1}{2} \xi + \frac{z_1 + z_2}{2}$ The integration on interval [-1,1] can be … Read more