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The boundary element method with programming for engineers and scientists - phần 8

Phương pháp thứ hai là tương tự như phương pháp tiếp cận theo phương pháp phần tử hữu hạn. Ở đây, chúng ta xây dựng một "ma trận độ cứng", K, của từng vùng, các hệ số trong đó là các chất khử tạp chất hoặc tractions do nhiệt độ / chuyển vị đơn vị. | 344 The Boundary Element Method with Programming Ĩ Í 3 0 c M3 o 3 E 1 0E6 V 0 0 3 3 0 3 X 5 0 ty 10 Figure 12.4 Cantilever beam The expected error for the discontinuous displacement at common nodes of adjacent elements nodes is less then 0.1 . Figure 12.5 Vertical displacements Discont 5 Elem ------- Cont 5 Elem Discont 3 Elem Cont 3 Elem ------ Analytical ------- 12.2.5 Test Example - Multiple Regions This example is a cube with a distributed boundary load of 10 KN m2 on the top of the cube. The geometry is shown in Figure 12.6 and the material parameters for all regions are E 1000kN m2 v 0. For the purpose of demonstrating the corner problem the cube is subdivided into four regions. Region 1 and 2 is discretised with 8 linear elements. Region 3 and 4 consists of 6 linear elements. The points B and D of regions 3 and 4 are corner nodes. These points are located at the interface between regions and therefore need special attention. The calculation is done two times first with the program prog111 which uses continuous elements and then with the program prog111_discont the discontinuous version of the multi-region program. If we compare the tractions at interface elements in Figures 12.7 12.8 with 12.9 at the interface between regions we CORNERS AND CHANGING GEOMETRY 345 see that the value that should be constant fluctuates widely if continuous elements are used. 10 KN m2 Figure 12.6 Vertical Displacements distance x m Figure 12.7 Tractions tx at the boundary of regions 3 and 4 along the line ABC If discontinuous elements are used the tractions which are now evaluated at points slightly inside show no fluctuation and only a small jump which is due to coarseness of the mesh. Indeed the diagram in Figure 12.7 indicates a gross violation of equilibrium tx continuous tx discontinuous 346 The Boundary Element Method with Programming conditions if continuous elements are used because for V 0 the tractions should be equal to zero everywhere. 12 10 8 6 4 2 0 1 2 distance x m