Numerical integration

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

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