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Finite Element Method by G. R. Liu and S. S. Quek 1 Finite Element Method CHAPTER 1: COMPUTATIONAL MODELLING for readers of all backgrounds G. R. Liu and S. S. Quek

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 3 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

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Finite Element Method by G. R. Liu and S. S. Quek 4 Conceptual design Modelling Physical,mathematical,computational, and operational, economical Simulation Experimental, analytical, andcomputational Analysis Photography, visual-tape, and computer graphics, visual reality Design Prototyping Testing Fabrication Virtual prototyping

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Finite Element Method by G. R. Liu and S. S. Quek 5 PHYSICAL PROBLEMS IN ENGINEERING Mechanics for solids and structures Heat transfer Acoustics Fluid mechanics Others

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Finite Element Method by G. R. Liu and S. S. Quek 6 COMPUTATIONAL MODELLING USING FEM Four major aspects: – Modelling of geometry – Meshing (discretization) – Defining material properties – Defining boundary, initial and loading conditions

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 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)

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Finite Element Method by G. R. Liu and S. S. Quek 11 A complex function is represented by piecewise linear functions

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 13 Mesh for the design of scaled model of aircraft for dynamic analysis

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Finite Element Method by G. R. Liu and S. S. Quek 14 Mesh for a boom showing the stress distribution (Picture used by courtesy of EDS PLM Solutions)

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Finite Element Method by G. R. Liu and S. S. Quek 15 Mesh of a hinge joint

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Finite Element Method by G. R. Liu and S. S. Quek 16 Axisymmetric mesh of part of a dental implant (The CeraOne abutment system, Nobel Biocare)

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 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)

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 20 Discrete system equations Principle of virtual work or variational principle – Hamiltons 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

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Finite Element Method by G. R. Liu and S. S. Quek 21 Discrete system equations Taylor series – For traditional FDM Control of conservation laws – For Finite Volume Method (FVM)

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 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)

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Finite Element Method by G. R. Liu and S. S. Quek 24 VISUALIZATION Vast volume of digital data Methods to interpret, analyse and for presentation Use post-processors 3D object representation – Wire-frames – Collection of elements – Collection of nodes

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 26 Air flow in a virtually designed building (Image courtesy of Institute of High Performance Computing)

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Finite Element Method by G. R. Liu and S. S. Quek 27 Air flow in a virtually designed building (Image courtesy of Institute of High Performance Computing)

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