18.11 Numerical integration

@Chandrupatla2002

@Chandrupatla2002

@Chandrupatla2002

@Chandrupatla2002

@Chandrupatla2002

@Chandrupatla2002

\[\int_x f(x) \; dx \approx \displaystyle \sum_{i=1}^n w_i f(x_i)\]

\[\iint_A f(x,y) \; dA \approx \displaystyle \sum_{i=1}^n \sum_{j=1}^m w_i w_j f(x_{ij}, y_{ij})\]

\[\iiint_V f(x,y) \; dV \approx \displaystyle \sum_{i=1}^n \sum_{j=1}^m \sum_{k=1}^o w_i w_j w_k f(x_{ijk}, y_{ijk}, z_{ijk})\]

Thus, we evaluate the FEA integrals numerically. Typically, this integration is exact. (The number of integration points used is typically sufficient to integrate the assumed order of the polynomial. Sometimes reduced integration is used. Exactness is not guaranteed but the cost is lower and (occasionally) the error in integration is beneficial for certain numerical issues inherent in the FEA process.