tailieunhanh - The boundary element method with programming for engineers and scientists - phần 4

kết thúc vụ án DEFAULT SELECT DO N = 1,2! Xác định không có tối thiểu các điểm Gauss cần thiết IF (RonL = Rlim (N)) THEN Ngaus = N +2 EXIT END IF END DO IF (Ngaus == 0) THEN! Điểm quá gần với các Ngaus bề mặt = 5! giá trị này sẽ kích hoạt phân khu END IF CHỨC NĂNG RETURN END Ngaus | 142 The Boundary Element Method with Programming INTEGER FUNCTION Ngaus RonL ne RLIM J-------------------------------------------------------- Function returns number of Gauss points needed J to integrate a function o 1 rne J according to Eberwien et al. J-------------------------------------------------------- REAL INTENT IN RonL J R L INTEgEr INtEnT IN ne J order of Kernel 1 2 3 REAL INTENT OUT Rlim 2 J array to store values of table SELECT CASE ne CASE 1 Rlim CASE 2 Rlim CASE 3 Rlim CASE DEFAULT END SELECT DO N 1 2 J Determine minimum no of Gauss points needed IF RonL Rlim N THEN Ngaus N 2 EXIT END IF END DO IF Ngaus 0 THEN J Point is too close to the surface Ngaus 5 J this value will trigger subdivision END IF RETURN END FUNCTION Ngaus Numerical integration over one-dimensional elements In the integration of Kernel-shape function products care has to be taken because in some cases the function has a singularity or is discontinuous over the element depending on the location of Pi. Therefore we have to distinguish integration schemes for the case where Pi is one of the element nodes and where it is not. The integrals which have to be evaluated over the isoparametric element shown in Figure are for potential problems 1 1 U JNn u Pi ỉ J ỉ dệ M JNn T Pi ỉ J ỉ dệ -1 -1 NUMERICAL IMPLEMENTATION 143 where U Pi Ỉ and T Pj Ỉ are the fundamental solutions at Q Ẹ for a source at point Pi J is the Jacobian and Nn Ẹ are linear or quadratic shape functions. When point Pi is not one of the element nodes both integrals can be evaluated by Gauss Quadrature and the integrals in equation can be replaced by two sums M àĩ ỵ Nn ỉm Tfam J ệm W m-1 M AU ỵ Nn ỉm U fam J ễm Wm m 1 where the number of integration points M is determined as a function of the proximity of Pi to the integration region as explained previously. If Pi is close to the integration region a subdivision will be necessary. Figure One dimensional .