The Finite Element Method

Transcription

1 The Finite Element Method A Practical Course G. R. Liu and S. S. Quek Chapter 1: Computational modeling An overview 1

2 CONTENTS INTRODUCTION PHYSICAL PROBLEMS IN ENGINEERING COMPUTATIONAL MODELLING USING FEM Geometry modelling Meshing Material properties specification Boundary, initial and loading conditions specification SIMULATION Discrete system equations Equation solvers VISUALIZATION 2

3 Design process for an advanced engineering system C onceptual design Modelling Physical, mathematical, computational, and operational, economical Simulation Experimental, analytical, and computational Analysis Photography, visual tape, and computer graphics, visual reality Virtual prototyping Design Prototyping Testing Fabrication 3

4 INTRODUCTION Design process for an engineering system Major steps include computational modelling, simulation and analysis of results. Process is iterative. Aided by good knowledge of computational modelling and simulation. FEM: an indispensable tool 4

5 PHYSICAL PROBLEMS IN ENGINEERING Mechanics for solids and structures Heat transfer Acoustics Fluid mechanics Others 5

6 COMPUTATIONAL MODELLING USING FEM Four major aspects: Modelling of geometry Meshing (discretization) Defining material properties Defining boundary, initial and loading conditions 6

7 Modelling of geometry Points can be created simply by keying in the coordinates. Lines/curves can be created by connecting points/nodes. Surfaces can be created by connecting/rotating/ translating the existing lines/curves. Solids can be created by connecting/ rotating/translating the existing surfaces. Points, lines/curves, surfaces and solids can be translated/rotated/reflected to form new ones. 7

8 Modelling of geometry Use of graphic software and preprocessors to aid the modelling of geometry Can be imported into software for discretization and analysis Simplification of complex geometry usually required 8

9 Modelling of geometry Eventually represented by discretized elements Note that curved lines/surfaces may not be well represented if elements with linear edges are used. 9

10 Meshing (Discretization) Why do we discretize? Solutions to most complex, real life problems are unsolvable analytically Dividing domain into small, regularly shaped elements/cells enables the solution within a single element to be approximated easily Solutions for all elements in the domain then approximate the solutions of the complex problem itself (see analogy of approximating a complex function with linear functions) 10

11 A complex function is represented by piecewise linear functions F ( x ) Unknown function of field variable Unknown discrete values of field variable at nodes elements nodes x 11

12 Meshing (Discretization) Part of preprocessing Automatic mesh generators: an ideal Semi-automatic mesh generators: in practice Shapes (types) of elements Triangular (2D) Quadrilateral (2D) Tetrahedral (3D) Hexahedral (3D) Etc. 12

13 Mesh for the design of scaled model of aircraft for dynamic analysis 13

14 Mesh for a boom showing the stress distribution (Picture used by courtesy of EDS PLM Solutions) 14

15 Mesh of a hinge joint 15

16 Axisymmetric mesh of part of a dental implant (The CeraOne abutment system, Nobel Biocare) 16

17 Property of material or media Type of material property depends upon problem Usually involves simple keying in of data of material property in preprocessor Use of material database (commercially available) Experiments for accurate material property 17

18 Boundary, initial and loading conditions Very important for accurate simulation of engineering systems Usually involves the input of conditions with the aid of a graphical interface using preprocessors Can be applied to geometrical identities (points, lines/curves, surfaces, and solids) and mesh identities (elements or grids) 18

19 SIMULATION Two major aspects when performing simulation: Discrete system equations Principles for discretization Problem dependent Equations solvers Problem dependent Making use of computer architecture 19

20 Discrete system equations Principle of virtual work or variational principle Hamilton s principle Minimum potential energy principle For traditional Finite Element Method (FEM) Weighted residual method PDEs are satisfied in a weighted integral sense Leads to FEM, Finite Difference Method (FDM) and Finite Volume Method (FVM) formulations Choice of test (weight) functions Choice of trial functions 20

21 Discrete system equations Taylor series For traditional FDM Control of conservation laws For Finite Volume Method (FVM) 21

22 Equations solvers Direct methods (for small systems, up to 2D) Gauss elimination LU decomposition Iterative methods (for large systems, 3D onwards) Gauss Jacobi method Gauss Seidel method SOR (Successive Over-Relaxation) method Generalized conjugate residual methods Line relaxation method 22

23 Equations solvers For nonlinear problems, another iterative loop is needed For time-dependent problems, time stepping is also additionally required Implicit approach (accurate but much more computationally expensive) Explicit approach (simple, but less accurate) 23

24 VISUALIZATION Vast volume of digital data Methods to interpret, analyze and for presentation Use post-processors 3D object representation Wire-frames Collection of elements Collection of nodes 24

25 VISUALIZATION Objects: rotate, translate, and zoom in/out Results: contours, fringes, wire-frames and deformations Results: iso-surfaces, vector fields of variable(s) Outputs in the forms of table, text files, xy plots are also routinely available Visual reality A goggle, inversion desk, and immersion room 25

26 Air flow in a virtually designed building (Image courtesy of Institute of High Performance Computing) 26

27 Air flow in a virtually designed building (Image courtesy of Institute of High Performance Computing) 27