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**CHAPTER 1: COMPUTATIONAL MODELLING**

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

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**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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**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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C onceptual design Modelling Physical , mathematical , computational , and operational, economical Simulation Experimental, analytical, and computational Virtual prototyping Analysis Photography, visual - tape, and computer graphics, visual reality Design Prototyping Testing Fabrication

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**PHYSICAL PROBLEMS IN ENGINEERING**

Mechanics for solids and structures Heat transfer Acoustics Fluid mechanics Others

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**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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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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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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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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**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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**A complex function is represented by piecewise linear functions**

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

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

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Mesh of a hinge joint

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

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**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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**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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**SIMULATION Discrete system equations Equations solvers**

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

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**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

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**Discrete system equations**

Taylor series For traditional FDM Control of conservation laws For Finite Volume Method (FVM)

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**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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**For nonlinear problems, another iterative loop is needed **

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

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

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THE FINITE ELEMENT METHOD

THE FINITE ELEMENT METHOD

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