摘要：The explicit (semi-analytical) integration of an 8-noded plane-finite-element stiffness matrix is presented in this work. The element is superparametric having straight sides. Before to carry out the integration, the integral expressions are grouped into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are manipulated, optimized and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to write down the corresponding Fortran code. Comparisons between analytical integration and numerical integration were made. Regarding the CPU time, it was shown that the semi-analytical integration required less CPU time than numerical integration to obtain stiffness matrices. Regarding the accuracy, several tests were conducted on extremely distorted finite elements. Results shown that the analytical expressions behave very well even in these cases.
其他摘要：The explicit (semi-analytical) integration of an 8-noded plane-finite-element stiffness matrix is presented in this work. The element is superparametric having straight sides. Before to carry out the integration, the integral expressions are grouped into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are manipulated, optimized and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to write down the corresponding Fortran code. Comparisons between analytical integration and numerical integration were made. Regarding the CPU time, it was shown that the semi-analytical integration required less CPU time than numerical integration to obtain stiffness matrices. Regarding the accuracy, several tests were conducted on extremely distorted finite elements. Results shown that the analytical expressions behave very well even in these cases.
