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1 The Scaled Boundary Finite Element Method Lecture A L e c t u r e A1 D r. S a s s a n M o h a s s e b V i s i t i n g P r o f e s s o r M. I. T. C a m b r i d g e December 10, 2013 ETH, IBK Zürich

2 Overview Finite Element Method: FEM Boundary Element Method: BEM Scaled Boundary Finite Element Method: SBFEM Comparison of the three methods Accuracy of Scaled Boundary Finite Element Method Triangular wedge 2

3 Spatial discretisation of Finite Element Method Figure 1.9 3

4 Spatial discretisation of domain Shape functions for displacements, which are piecewise local Stiffness matrices Assembly of matrices / sparse / banded Solving system of equations Can handle large systems Inhomogeneous, anisotropic materials 4

5 Spatial discretisation of Boundary Element Method Figure

6 Boundary discretisation only Reducing the spatial dimension by one 3-D problems become 2-D, and 2-D become 1-D Requires fundamental solution, is usually complicated, exhibits singularities Shape functions for each boundary element for displacement and tractions Resulting equations are fully populated and non-symmetric Not well suited for inhomogeneous and isotropic material The conditions at infinity are satisfied rigorously 6

7 Scaled Boundary Finite Element Method. A new numerical method Problem definition: (a) bounded media (b) unbounded media Figure 2.2 7

8 Finite Elements Method No fundamental solutions required Symmetrical matrices, sparse, banded matrices Convergence by increasing number of elements Boundary Element Method Reduction of the spatial dimension by one Reduction of data preparation and computational efforts 8

9 Scaled Boundary Finite Element Method Combining advantages of FEM and BEM Reduction of partial differential equations into ordinary differential equations Analytical solution in radial direction 9

10 Developed in the last years Bounded and unbounded media Static and dynamic problems 2D and 3D problems Frequency and time domain solutions developed by Wolf and Song at EPFL Program SIMILAR downloaded from ftp://ftp.wiley.co.uk/pub/books/wolf/ and as well as staff/song.c/sbfem/similar/ and 10

11 Advantages of scaled boundary finite element method compared with those of finite element and boundary element methods Reduction of the spatial dimension by one as only the boundary is discretised with surface finite elements, reducing the data preparation and computational efforts Finite element method Boundary element method Scaled boundary finite element method Analytical solution achieved inside domain No fundamental solution required, expanding the scope of application and avoiding singular integrals Radiation condition at infinity satisfied exactly when modelling unbounded (infinite or semi-infinite) media No discretisation of free and fixed boundaries and interfaces between different materials No approximation other than that of the surface finite elements on the boundary Table

12 (ctd.): Advantages of scaled boundary finite element method compared with those of finite element and boundary element methods Symmetric dynamic-stiffness and unit-impulse response matrices for unbounded media Symmetric static-stiffness and mass matrices for bounded media (super element) Body loads processed without additional domain discretisation and thus additional approximation Straightforward calculation of stress concentrations and intensity factors based on their definition Finite element method Boundary element method Scaled boundary finite element method () () No fictitious eigenfrequencies for unbounded media Straightforward coupling by standard assemblage of structure discretised with finite elements with unbounded medium Table 14.1 (ctd.) 12

13 Infinite plate with central circular hole subjected to uniaxial tensile stress Fig Bounded model representing infinite plate in uniaxial stress field Fig Reproduced by permission of John Wiley Sons LTD 13

14 Raw (left) and recovered (right) stress as computed by scaled boundary finite element method for coarse mesh Plate 25.1 Raw (left) and recovered (right) stress as computed by finite element method for the intermediate mesh Plate 25.2 Reproduced by permission of John Wiley Sons LTD 14

15 Plate 25.1 Table

16 Cylindrical foundation embedded in half-space Fig

17 Plate 25.7 Vertical stress at 5% target error: scaled boundary finite element method (left) finite element method (right) (mesh a) 17

18 Computational efficiency and accuracy of adaptive finite element analyses (meshes a to e) and adaptive scaled boundary finite element analysis (mesh f) Table

19 Model of the trapezoidal plate 19 Ref. Extension of the scaled boundary finite element method to plate bending problems, Rolf Dieringer et al., PAMM 11, (2011)

20 Convergence study 20 Ref. Extension of the scaled boundary finite element method to plate bending problems, Rolf Dieringer et al., PAMM 11, (2011)

21 read chapter 4 Problem Statement Out-of-plane motion of wedge and truncated semi-infinite wedge of shear plate Figure

22 Equilibrium equation p: Body load per unit volume acting perpendicular to the planestress, strain relation Reformulating equation

23 Substituting equation 4.2 into equation 4.1 we get with the shear wave velocity With the surface traction τ n boundary conditions 23

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