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

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

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

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

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

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

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.

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

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.

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)

A complex function is represented by piecewise linear functions

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.

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

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

Mesh of a hinge joint

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

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

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)

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

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

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

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

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)

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

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

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

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