CHAPTER 1: COMPUTATIONAL MODELLING

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)

INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES
Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 2: INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES

CONTENTS INTRODUCTION EQUATIONS FOR THREE-DIMENSIONAL (3D) SOLIDS
Statics and dynamics Elasticity and plasticity Isotropy and anisotropy Boundary conditions Different structural components EQUATIONS FOR THREE-DIMENSIONAL (3D) SOLIDS EQUATIONS FOR TWO-DIMENSIONAL (2D) SOLIDS EQUATIONS FOR TRUSS MEMBERS EQUATIONS FOR BEAMS EQUATIONS FOR PLATES

INTRODUCTION Solids and structures are stressed when they are subjected to loads or forces. The stresses are, in general, not uniform as the forces usually vary with coordinates. The stresses lead to strains, which can be observed as a deformation or displacement. Solid mechanics and structural mechanics

Statics and dynamics Forces can be static and/or dynamic.
Statics deals with the mechanics of solids and structures subject to static loads. Dynamics deals with the mechanics of solids and structures subject to dynamic loads. As statics is a special case of dynamics, the equations for statics can be derived by simply dropping out the dynamic terms in the dynamic equations.

Elasticity and plasticity
Elastic: the deformation in the solids disappears fully if it is unloaded. Plastic: the deformation in the solids cannot be fully recovered when it is unloaded. Elasticity deals with solids and structures of elastic materials. Plasticity deals with solids and structures of plastic materials.

Isotropy and anisotropy
Anisotropic: the material property varies with direction. Composite materials: anisotropic, many material constants. Isotropic material: property is not direction dependent, two independent material constants.

Boundary conditions Displacement (essential) boundary conditions Force (natural) boundary conditions

Different structural components
Truss and beam structures

Different structural components
Plate and shell structures

EQUATIONS FOR 3D SOLIDS Stress and strain Constitutive equations
Dynamic and static equilibrium equations

Stress and strain Stresses at a point in a 3D solid:

Stress and strain Strains

Stress and strain Strains in matrix form where

Constitutive equations
s = c e or

For isotropic materials , ,

Dynamic equilibrium equations
Consider stresses on an infinitely small block

Equilibrium of forces in x direction including the inertia forces Note:

Hence, equilibrium equation in x direction Equilibrium equations in y and z directions

Dynamic and static equilibrium equations
In matrix form Note: or For static case

EQUATIONS FOR 2D SOLIDS Plane stress Plane strain

Stress and strain (3D)

Stress and strain Strains in matrix form where ,

s = c e (For plane stress) (For plane strain)

In matrix form Note: or For static case

EQUATIONS FOR TRUSS MEMBERS

Hooke’s law in 1D s = E e Dynamic and static equilibrium equations (Static)

EQUATIONS FOR BEAMS Stress and strain Constitutive equations
Moments and shear forces Dynamic and static equilibrium equations

Stress and strain Euler–Bernoulli theory

Stress and strain sxx = E exx Assumption of thin beam
Sections remain normal Slope of the deflection curve where sxx = E exx 

sxx = E exx Moments and shear forces Consider isolated beam cell of length dx

Moments and shear forces
The stress and moment

Since Therefore, Where (Second moment of area about z axis – dependent on shape and dimensions of cross-section)

Forces in the x direction Moments about point A  

Therefore,  (Static)

EQUATIONS FOR PLATES Stress and strain Constitutive equations
Moments and shear forces Dynamic and static equilibrium equations Mindlin plate

Stress and strain Thin plate theory or Classical Plate Theory (CPT)

Stress and strain Assumes that exz = 0, eyz = 0 , Therefore, ,

Stress and strain Strains in matrix form e = -z Lw where

s = c e where c has the same form for the plane stress case of 2D solids

Stresses on isolated plate cell z x y fz h xy xx xz yx yy yz O

Moments and shear forces on a plate cell dx x dy z x y O dx dy Qy My Myx Qy+dQy Myx+dMyx My+dMy Qx Mx Mxy Qx+dQx Mxy+dMxy Mx+dMx

s = c e  s = - c z Lw Like beams, Note that ,

Therefore, equilibrium of forces in z direction or Moments about A-A

 (Static) where

Mindlin plate

Mindlin plate , e = -z Lq Therefore, in-plane strains where ,

Mindlin plate Transverse shear strains Transverse shear stress

THE FINITE ELEMENT METHOD
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 3: THE FINITE ELEMENT METHOD

CONTENTS STRONG AND WEAK FORMS OF GOVERNING EQUATIONS
HAMILTON’S PRINCIPLE FEM PROCEDURE Domain discretization Displacement interpolation Formation of FE equation in local coordinate system Coordinate transformation Assembly of FE equations Imposition of displacement constraints Solving the FE equations STATIC ANALYSIS EIGENVALUE ANALYSIS TRANSIENT ANALYSIS REMARKS

STRONG AND WEAK FORMS OF GOVERNING EQUATIONS
System equations: strong form, difficult to solve. Weak form: requires weaker continuity on the dependent variables (u, v, w in this case). Weak form is often preferred for obtaining an approximated solution. Formulation based on a weak form leads to a set of algebraic system equations – FEM. FEM can be applied for practical problems with complex geometry and boundary conditions.

HAMILTON’S PRINCIPLE “Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.” An admissible displacement must satisfy: The compatibility equations The essential or the kinematic boundary conditions The conditions at initial (t1) and final time (t2)

HAMILTON’S PRINCIPLE Mathematically where L=T-P+Wf (Kinetic energy)
(Potential energy) (Work done by external forces)

FEM PROCEDURE Step 1: Domain discretization
Step 2: Displacement interpolation Step 3: Formation of FE equation in local coordinate system Step 4: Coordinate transformation Step 5: Assembly of FE equations Step 6: Imposition of displacement constraints Step 7: Solving the FE equations

Step 1: Domain discretization
The solid body is divided into Ne elements with proper connectivity – compatibility. All the elements form the entire domain of the problem without any overlapping – compatibility. There can be different types of element with different number of nodes. The density of the mesh depends upon the accuracy requirement of the analysis. The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.

Step 2: Displacement interpolation
Bases on local coordinate system, the displacement within element is interpolated using nodal displacements.

Step 2: Displacement interpolation
N is a matrix of shape functions Shape function for each displacement component at a node where

Displacement interpolation
Constructing shape functions Consider constructing shape function for a single displacement component Approximate in the form pT(x)={1, x, x2, x3, x4,..., xp} (1D)

Pascal triangle of monomials: 2D

Pascal pyramid of monomials : 3D

Enforce approximation to be equal to the nodal displacements at the nodes di = pT(xi) i = 1, 2, 3, …,nd or de=P  where ,

The coefficients in  can be found by Therefore, uh(x) = N( x) de

Sufficient requirements for FEM shape functions (Delta function property) 1. (Partition of unity property – rigid body movement) 2. 3. (Linear field reproduction property)

Step 3: Formation of FE equations in local coordinates
Since U= Nde Strain matrix e = LU Therefore,  e = L N de= B de e T Ve V c Π d B Bd ) ( 2 1 ε ò = V c T Ve e d B k ò = or where (Stiffness matrix)

Since U= Nde  or where (Mass matrix)

(Force vector)

(Hamilton’s principle)  FE Equation

Step 4: Coordinate transformation
x y x' y' Local coordinate systems Global coordinate systems (Local) (Global) where , ,

Step 5: Assembly of FE equations
Direct assembly method Adding up contributions made by elements sharing the node (Static)

Step 6: Impose displacement constraints
No constraints  rigid body movement (meaningless for static analysis) Remove rows and columns corresponding to the degrees of freedom being constrained K is semi-positive definite

Step 7: Solve the FE equations
for the displacement at the nodes, D The strain and stress can be retrieved by using e = LU and s = c e with the interpolation, U=Nd

STATIC ANALYSIS Solve KD=F for D Gauss elmination LU decomposition
Etc.

EIGENVALUE ANALYSIS [ K - li M ] fi = 0 (Homogeneous equation, F = 0)
Assume Let  (Roots of equation are the eigenvalues) [ K - li M ] fi = 0 (Eigenvector)

EIGENVALUE ANALYSIS Methods of solving eigenvalue equation
Jacobi’s method Given’s method and Householder’s method The bisection method (Sturm sequences) Inverse iteration QR method Subspace iteration Lanczos’ method

TRANSIENT ANALYSIS Structure systems are very often subjected to transient excitation. A transient excitation is a highly dynamic time dependent force exerted on the structure, such as earthquake, impact, and shocks. The discrete governing equation system usually requires a different solver from that of eigenvalue analysis. The widely used method is the so-called direct integration method.

TRANSIENT ANALYSIS The direct integration method is basically using the finite difference method for time stepping. There are mainly two types of direct integration method; one is implicit and the other is explicit. Implicit method (e.g. Newmark’s method) is more efficient for relatively slow phenomena Explicit method (e.g. central differencing method) is more efficient for very fast phenomena, such as impact and explosion.

Newmark’s method (Implicit)
Assume that Substitute into

where Therefore,

Start with D0 and Obtain using March forward in time Obtain using Obtain Dt and using

Central difference method (explicit)
(Lumped mass – no need to solve matrix equation)

Central difference method (explicit)
Find average velocity at time t = -t/2 using Find using the average acceleration at time t = 0. Find Dt using the average velocity at time t =t/2 Obtain D-t using D0 and are prescribed and can be obtained from Use to obtain assuming Obtain using Time marching in half the time step Central difference method (explicit)

REMARKS In FEM, the displacement field U is expressed by displacements at nodes using shape functions N defined over elements. The strain matrix B is the key in developing the stiffness matrix. To develop FE equations for different types of structure components, all that is needed to do is define the shape function and then establish the strain matrix B. The rest of the procedure is very much the same for all types of elements.

Finite Element Method CHAPTER 4: FEM FOR TRUSSES
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 4: FEM FOR TRUSSES

CONTENTS INTRODUCTION FEM EQUATIONS Shape functions construction
Strain matrix Element matrices in local coordinate system Element matrices in global coordinate system Boundary conditions Recovering stress and strain EXAMPLE Remarks HIGHER ORDER ELEMENTS

INTRODUCTION Truss members are for the analysis of skeletal type systems – planar trusses and space trusses. A truss element is a straight bar of an arbitrary cross-section, which can deform only in its axis direction when it is subjected to axial forces. Truss elements are also termed as bar elements. In planar trusses, there are two components in the x and y directions for the displacement as well as forces at a node. For space trusses, there will be three components in the x, y and z directions for both displacement and forces at a node.

INTRODUCTION In trusses, the truss or bar members are joined together by pins or hinges (not by welding), so that there are only forces (not moments) transmitted between bars. It is assumed that the element has a uniform cross-section.

Example of a truss structure

FEM EQUATIONS Shape functions construction Strain matrix
Element matrices in local coordinate system Element matrices in global coordinate system Boundary conditions Recovering stress and strain

Shape functions construction
Consider a truss element

Let Note: Number of terms of basis function, xn determined by n = nd - 1 At x = 0, u(x=0) = u1 At x = le, u(x=le) = u2 

(Linear element)

Strain matrix or where

Element Matrices in the Local Coordinate System
Note: ke is symmetrical Proof:

Element Matrices in the Local Coordinate System
Note: me is symmetrical too

Element matrices in global coordinate system
Perform coordinate transformation Truss in space (spatial truss) and truss in plane (planar truss)

Spatial truss (Relationship between local DOFs and global DOFs) (2x1) where , (6x1) Direction cosines

Spatial truss (Cont’d) Transformation applies to force vector as well: where

Spatial truss (Cont’d)

Spatial truss (Cont’d) Note:

Planar truss where , Similarly (4x1)

Planar truss (Cont’d)

Singular K matrix  rigid body movement Constrained by supports
Boundary conditions Singular K matrix  rigid body movement Constrained by supports Impose boundary conditions  cancellation of rows and columns in stiffness matrix, hence K becomes SPD Recovering stress and strain (Hooke’s law) x

EXAMPLE Consider a bar of uniform cross-sectional area shown in the figure. The bar is fixed at one end and is subjected to a horizontal load of P at the free end. The dimensions of the bar are shown in the figure and the beam is made of an isotropic material with Young’s modulus E. P l

EXAMPLE Exact solution of : , stress: FEM: (1 truss element) 

Remarks FE approximation = exact solution in example
Exact solution for axial deformation is a first order polynomial (same as shape functions used) Hamilton’s principle – best possible solution Reproduction property

HIGHER ORDER ELEMENTS Quadratic element Cubic element

Finite Element Method CHAPTER 5: FEM FOR BEAMS
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 5: FEM FOR BEAMS

CONTENTS INTRODUCTION FEM EQUATIONS Shape functions construction
Strain matrix Element matrices Remarks EXAMPLE

INTRODUCTION The element developed is often known as a beam element.
A beam element is a straight bar of an arbitrary cross-section. Beams are subjected to transverse forces and moments. Deform only in the directions perpendicular to its axis of the beam.

INTRODUCTION In beam structures, the beams are joined together by welding (not by pins or hinges). Uniform cross-section is assumed. FE matrices for beams with varying cross-sectional area can also be developed without difficulty.

FEM EQUATIONS Shape functions construction Strain matrix
Element matrices

Consider a beam element Natural coordinate system:

Assume that In matrix form: or

To obtain constant coefficients – four conditions At x= -a or x = -1 At x= a or x = 1

or  or

Therefore, where in which

Strain matrix Therefore, where (Second derivative of shape functions)

Element matrices Evaluate integrals

Element matrices

Remarks Theoretically, coordinate transformation can also be used to transform the beam element matrices from the local coordinate system to the global coordinate system. The transformation is necessary only if there is more than one beam element in the beam structure, and of which there are at least two beam elements of different orientations. A beam structure with at least two beam elements of different orientations is termed a frame or framework.

EXAMPLE Consider the cantilever beam as shown in the figure. The beam is fixed at one end and it has a uniform cross-sectional area as shown. The beam undergoes static deflection by a downward load of P=1000N applied at the free end. The dimensions and properties of the beam are shown in the figure. P=1000 N 0.5 m 0.06 m 0.1 m E=69 GPa =0.33

EXAMPLE Step 1: Element matrices

EXAMPLE Step 1 (Cont’d): Step 2: Boundary conditions

EXAMPLE Step 2 (Cont’d): Therefore, Kd=F where dT = [ v2 2] ,

EXAMPLE Step 3: Solving FE equation Two simultaneous equations
v2 = x 10-4 m 2 = x 10-3 rad Substitute back into first two equations of Kd=F

Remarks FE solution is the same as analytical solution
Analytical solution to beam is third order polynomial (same as shape functions used) Reproduction property

CASE STUDY Resonant frequencies of micro resonant transducer

CASE STUDY Number of 2-node beam elements Natural Frequency (Hz)
Mode 1 Mode 2 Mode 3 10 x 105 x 106 x 106 20 x 105 x 106 x 106 40 x 105 x 106 x 106 60 Analytical Calculations x 105 x 106 x 106

CASE STUDY

Finite Element Method CHAPTER 6: FEM FOR FRAMES
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 6: FEM FOR FRAMES

CONTENTS INTRODUCTION FEM EQUATIONS FOR PLANAR FRAMES
Equations in local coordinate system Equations in global coordinate system FEM EQUATIONS FOR SPATIAL FRAMES REMARKS

INTRODUCTION Deform axially and transversely.
It is capable of carrying both axial and transverse forces, as well as moments. Hence combination of truss and beam elements. Frame elements are applicable for the analysis of skeletal type systems of both planar frames (2D frames) and space frames (3D frames). Known generally as the beam element or general beam element in most commercial software.

FEM EQUATIONS FOR PLANAR FRAMES
Consider a planar frame element

Equations in local coordinate system
Combination of the element matrices of truss and beam elements From the truss element, Truss Beam (Expand to 6x6)

From the beam element, (Expand to 6x6)

+ 

Similarly so for the mass matrix and we get And for the force vector,

Equations in global coordinate system
Coordinate transformation where ,

Direction cosines in T: (Length of element)

Therefore,

FEM EQUATIONS FOR SPATIAL FRAMES
Consider a spatial frame element Displacement components at node 1 Displacement components at node 2

Combination of the element matrices of truss and beam elements

where

Coordinate transformation where ,

Direction cosines in T3

Vectors for defining location and orientation of frame element in space k, l = 1, 2, 3

Vectors for defining location and orientation of frame element in space (cont’d)

Therefore,

REMARKS In practical structures, it is very rare to have beam structure subjected only to transversal loading. Most skeletal structures are either trusses or frames that carry both axial and transversal loads. A beam element is actually a very special case of a frame element. The frame element is often conveniently called the beam element.

CASE STUDY Finite element analysis of bicycle frame

CASE STUDY 74 elements (71 nodes) Ensure connectivity Young’s modulus,
E GPa Poisson’s ratio,  69.0 0.33 74 elements (71 nodes) Ensure connectivity

CASE STUDY Horizontal load Constraints in all directions

CASE STUDY M = 20X

CASE STUDY Axial stress -9.68 x 105 Pa -6.264 x 105 Pa -6.34 x 105 Pa

Finite Element Method CHAPTER 7: FEM FOR 2D SOLIDS
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 7: FEM FOR 2D SOLIDS

CONTENTS INTRODUCTION LINEAR TRIANGULAR ELEMENTS
Field variable interpolation Shape functions construction Using area coordinates Strain matrix Element matrices LINEAR RECTANGULAR ELEMENTS Gauss integration Evaluation of me

CONTENTS LINEAR QUADRILATERAL ELEMENTS HIGHER ORDER ELEMENTS
Coordinate mapping Strain matrix Element matrices Remarks HIGHER ORDER ELEMENTS COMMENTS (GAUSS INTEGRATION)

INTRODUCTION 2D solid elements are applicable for the analysis of plane strain and plane stress problems. A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges. A 2D solid element can deform only in the plane of the 2D solid. At any point, there are two components in the x and y directions for the displacement as well as forces.

INTRODUCTION For plane strain problems, the thickness of the element is unit, but for plane stress problems, the actual thickness must be used. In this course, it is assumed that the element has a uniform thickness h. Formulating 2D elements with a given variation of thickness is also straightforward, as the procedure is the same as that for a uniform element.

2D solids – plane stress and plane strain

LINEAR TRIANGULAR ELEMENTS
Less accurate than quadrilateral elements Used by most mesh generators for complex geometry A linear triangular element:

Field variable interpolation
where (Shape functions)

Assume, i= 1, 2, 3 or

Delta function property: Therefore, Solving,

Area of triangle Moment matrix Substitute a1, b1 and c1 back into N1 = a1 + b1x + c1y:

Similarly,

where i= 1, 2, 3 J, k determined from cyclic permutation i = 1, 2 i k j j = 2, 3 k = 3, 1

Using area coordinates
Alternative method of constructing shape functions  2-3-P: Similarly,  3-1-P A2  1-2-P A3

Using area coordinates
Partitions of unity: Delta function property: e.g. L1 = 0 at if P at nodes 2 or 3 Therefore,

(constant strain element)
Strain matrix where  (constant strain element)

Element matrices Constant matrix 

Element matrices For elements with uniform density and thickness,
Eisenberg and Malvern (1973):

Element matrices Uniform distributed load:

LINEAR RECTANGULAR ELEMENTS
Non-constant strain matrix More accurate representation of stress and strain Regular shape makes formulation easy

Consider a rectangular element

where (Interpolation)

Delta function property Partition of unity

Strain matrix Note: No longer a constant matrix!

Element matrices  dxdy = ab dxdh Therefore,

Element matrices For uniformly distributed load,

Gauss integration For evaluation of integrals in ke and me (in practice) In 1 direction: m gauss points gives exact solution of polynomial integrand of n = 2m - 1 In 2 directions:

Gauss integration m xj wj Accuracy n 1 2 -1/3, 1/3 1, 1 3
2 -1/3, 1/3 1, 1 3 -0.6, 0, 0.6 5/9, 8/9, 5/9 5 4 , , , , , , 7 , , 0, , , , , , 9 6 , , , , , , , , , , 11

Evaluation of me

Evaluation of me E.g. Note: In practice, Gauss integration is often used

LINEAR QUADRILATERAL ELEMENTS
Rectangular elements have limited application Quadrilateral elements with unparallel edges are more useful Irregular shape requires coordinate mapping before using Gauss integration

Coordinate mapping Physical coordinates Natural coordinates
(Interpolation of displacements) (Interpolation of coordinates)

Coordinate mapping where ,

Coordinate mapping Substitute x = 1 into or Eliminating ,

Strain matrix or where (Jacobian matrix) Since ,

Strain matrix Therefore,
(Relationship between differentials of shape functions w.r.t. physical coordinates and differentials w.r.t. natural coordinates) Therefore, Replace differentials of Ni w.r.t. x and y with differentials of Ni w.r.t.  and 

Element matrices Murnaghan (1951) : dA=det |J | dxdh

Remarks Shape functions used for interpolating the coordinates are the same as the shape functions used for interpolation of the displacement field. Therefore, the element is called an isoparametric element. Note that the shape functions for coordinate interpolation and displacement interpolation do not have to be the same. Using the different shape functions for coordinate interpolation and displacement interpolation, respectively, will lead to the development of so-called subparametric or superparametric elements.

HIGHER ORDER ELEMENTS Higher order triangular elements
nd = (p+1)(p+2)/2 Node i, Argyris, 1968 :

HIGHER ORDER ELEMENTS Higher order triangular elements (Cont’d)
Cubic element Quadratic element

HIGHER ORDER ELEMENTS Higher order rectangular elements Lagrange type:
[Zienkiewicz et al., 2000]

HIGHER ORDER ELEMENTS Higher order rectangular elements (Cont’d)
(nine node quadratic element)

Serendipity type: (eight node quadratic element)

(twelve node cubic element)

ELEMENT WITH CURVED EDGES

COMMENTS (GAUSS INTEGRATION)
When the Gauss integration scheme is used, one has to decide how many Gauss points should be used. Theoretically, for a one-dimensional integral, using m points can give the exact solution for the integral of a polynomial integrand of up to an order of (2m-1). As a general rule of thumb, more points should be used for a higher order of elements.

COMMENTS (GAUSS INTEGRATION)
Using a smaller number of Gauss points tends to counteract the over-stiff behaviour associated with the displacement-based method. Displacement in an element is assumed using shape functions. This implies that the deformation of the element is somehow prescribed in a fashion of the shape function. This prescription gives a constraint to the element. The so- constrained element behaves stiffer than it should. It is often observed that higher order elements are usually softer than lower order ones. This is because using higher order elements gives fewer constraint to the elements.

COMMENTS ON GAUSS INTEGRATION
Two Gauss points for linear elements, and two or three points for quadratic elements in each direction should be sufficient for most cases. Most of the explicit FEM codes based on explicit formulation tend to use one-point integration to achieve the best performance in saving CPU time.

CASE STUDY Side drive micro-motor

Elastic Properties of Polysilicon
CASE STUDY 10N/m Elastic Properties of Polysilicon Young’s Modulus, E 169GPa Poisson’s ratio,  0.262 Density,  2300kgm-3 10N/m 10N/m

CASE STUDY Analysis no. 1: Von Mises stress distribution using 24 bilinear quadrilateral elements (41 nodes)

CASE STUDY Analysis no. 4: Von Mises stress distribution using 24 eight-nodal, quadratic elements (105 nodes)

CASE STUDY Analysis no. 5: Von Mises stress distribution using 192 three-nodal, triangular elements (129 nodes)

CASE STUDY Analysis no. Number / type of elements
Total number of nodes in model Maximum Von Mises Stress (GPa) 1 24 bilinear, quadrilateral 41 0.0139 2 96 bilinear, quadrilateral 129 0.0180 3 144 bilinear, quadrilateral 185 0.0197 4 24 quadratic, quadrilateral 105 0.0191 5 192 linear, triangular 0.0167

Finite Element Method CHAPTER 8: FEM FOR PLATES & SHELLS
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 8: FEM FOR PLATES & SHELLS

CONTENTS INTRODUCTION PLATE ELEMENTS SHELL ELEMENTS Shape functions
Element matrices SHELL ELEMENTS Elements in local coordinate system Elements in global coordinate system Remarks

INTRODUCTION FE equations based on Mindlin plate theory will be developed. FE equations of shells will be formulated by superimposing matrices of plates and those of 2D solids. Computationally tedious due to more DOFs.

PLATE ELEMENTS Geometrically similar to 2D plane stress solids except that it carries only transverse loads. Leads to bending. 2D equilvalent of the beam element. Rectangular plate elements based on Mindlin plate theory will be developed – conforming element. Much software like ABAQUS does not offer plate elements, only the general shell element.

(Mindlin plate theory)
PLATE ELEMENTS Consider a plate structure: (Mindlin plate theory)

PLATE ELEMENTS Mindlin plate theory: In-plane strain: where
(Curvature)

PLATE ELEMENTS Off-plane shear strain: Potential (strain) energy:
In-plane stress & strain Off-plane shear stress & strain

PLATE ELEMENTS Substituting , Kinetic energy: Substituting

PLATE ELEMENTS where ,

Shape functions Note that rotation is independent of deflection w
(Same as rectangular 2D solid) where

Shape functions where

Element matrices Substitute into  Recall that: where
(Can be evaluated analytically but in practice, use Gauss integration)

Element matrices Substitute into potential energy function
from which we obtain Note:

Element matrices (me can be solved analytically but practically solved using Gauss integration) For uniformly distributed load,

SHELL ELEMENTS Loads in all directions
Bending, twisting and in-plane deformation Combination of 2D solid elements (membrane effects) and plate elements (bending effect). Common to use shell elements to model plate structures in commercial software packages.

Elements in local coordinate system
Consider a flat shell element

Membrane stiffness (2D solid element): (2x2) Bending stiffness (plate element): (3x3)

Components related to the DOF qz, are zeros in local coordinate system. (24x24)

Membrane mass matrix (2D solid element): Bending mass matrix (plate element):

Components related to the DOF qz, are zeros in local coordinate system. (24x24)

Elements in global coordinate system
where

Remarks The membrane effects are assumed to be uncoupled with the bending effects in the element level. This implies that the membrane forces will not result in any bending deformation, and vice versa. For shell structure in space, membrane and bending effects are actually coupled (especially for large curvature), therefore finer element mesh may have to be used.

CASE STUDY Natural frequencies of micro-motor

CASE STUDY Mode Natural Frequencies (MHz)
768 triangular elements with 480 nodes 384 quadrilateral elements with 480 nodes 1280 quadrilateral elements with 1472 nodes 1 7.67 5.08 4.86 2 3 7.87 7.44 7.41 4 10.58 8.52 8.30 5 6 13.84 11.69 11.44 7 8 14.86 12.45 12.17 CASE STUDY

CASE STUDY Mode 1: Mode 2:

CASE STUDY Transient analysis of micro-motor F Node 210 x x F Node 300

CASE STUDY

Finite Element Method CHAPTER 9: FEM FOR 3D SOLIDS
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 9: FEM FOR 3D SOLIDS

CONTENTS INTRODUCTION TETRAHEDRON ELEMENT HEXAHEDRON ELEMENT
Shape functions Strain matrix Element matrices HEXAHEDRON ELEMENT Using tetrahedrons to form hexahedrons HIGHER ORDER ELEMENTS ELEMENTS WITH CURVED SURFACES

INTRODUCTION For 3D solids, all the field variables are dependent of x, y and z coordinates – most general element. The element is often known as a 3D solid element or simply a solid element. A 3D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces. At any node there are three components in the x, y and z directions for the displacement as well as forces.

TETRAHEDRON ELEMENT 3D solid meshed with tetrahedron elements

TETRAHEDRON ELEMENT Consider a four node tetrahedron element

Shape functions where Use volume coordinates (Recall Area coordinates for 2D triangular element)

Shape functions Similarly, Can also be viewed as ratio of distances
(Partition of unity) since

Shape functions (Delta function property)

Shape functions Therefore, i j l k where (Adjoint matrix) i= 1,2
(Cofactors) k = 3,4 where

Shape functions (Volume of tetrahedron) Therefore,

Strain matrix Since, Therefore, where (Constant strain element)

Element matrices where

Element matrices Eisenberg and Malvern [1973] :

Element matrices Alternative method for evaluating me: special natural coordinate system

Element matrices

Element matrices Jacobian:

Element matrices For uniformly distributed load:

HEXAHEDRON ELEMENT 3D solid meshed with hexahedron elements

Shape functions 1 7 5 8 6 4 2 z y x 3 fsz fsy fsx

Shape functions (Tri-linear functions)

Strain matrix whereby Note: Shape functions are expressed in natural coordinates – chain rule of differentiation

Strain matrix Chain rule of differentiation  where

Strain matrix Since, or

Strain matrix Used to replace derivatives w.r.t. x, y, z with derivatives w.r.t. , , 

Element matrices Gauss integration:

Element matrices For rectangular hexahedron:

Element matrices (Cont’d) where

Element matrices (Cont’d) or where

Element matrices (Cont’d) E.g.

Element matrices (Cont’d) Note: For x direction only
(Rectangular hexahedron)

Element matrices For uniformly distributed load: fsy y 5 8 6 fsz 4 7 1
2 z y x 3 fsz fsy fsx For uniformly distributed load:

Using tetrahedrons to form hexahedrons
Hexahedrons can be made up of several tetrahedrons Hexahedron made up of 5 tetrahedrons:

Using tetrahedrons to form hexahedrons
Element matrices can be obtained by assembly of tetrahedron elements Hexahedron made up of six tetrahedrons:

HIGHER ORDER ELEMENTS Tetrahedron elements 10 nodes, quadratic:

HIGHER ORDER ELEMENTS Tetrahedron elements (Cont’d) 20 nodes, cubic:

HIGHER ORDER ELEMENTS Brick elements Lagrange type: where
(nd=(n+1)(m+1)(p+1) nodes) Lagrange type: where

HIGHER ORDER ELEMENTS Brick elements (Cont’d)
Serendipity type elements: 20 nodes, tri-quadratic:

HIGHER ORDER ELEMENTS Brick elements (Cont’d) 32 nodes, tri-cubic:

ELEMENTS WITH CURVED SURFACES

CASE STUDY Stress and strain analysis of a quantum dot heterostructure
Material E (Gpa)  GaAs 86.96 0.31 InAs 51.42 0.35 GaAs cap layer InAs wetting layer InAs quantum dot GaAs substrate

CASE STUDY

CASE STUDY 30 nm 30 nm

CASE STUDY

SPECIAL PURPOSE ELEMENTS
Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 10: SPECIAL PURPOSE ELEMENTS

CONTENTS CRACK TIP ELEMENTS METHODS FOR INFINITE DOMAINS
Infinite elements formulated by mapping Gradual damping elements Coupling of FEM and BEM Coupling of FEM and SEM FINITE STRIP ELEMENTS STRIP ELEMENT METHOD

CRACK TIP ELEMENTS Fracture mechanics – singularity point at crack tip. Conventional finite elements do not give good approximation at/near the crack tip.

CRACK TIP ELEMENTS From fracture mechanics, (Near crack tip)
(Mode I fracture)

CRACK TIP ELEMENTS Special purpose crack tip element with middle nodes shifted to quarter position:

CRACK TIP ELEMENTS Move node 2 to L/4 position 
x = -0.5 (1-)x1 + (1+)(1-)x  (1+) x3 u = -0.5 (1-)u1 + (1+)(1-)u  (1+) u3 (Measured from node 1) Move node 2 to L/4 position x1 = 0, x2 = L/4, x3 = L, u1 = 0 x = 0.25(1+)(1-)L + 0.5 (1+)L  u = (1+)(1-)u2+0.5 (1+) u3

CRACK TIP ELEMENTS Simplifying, Along x-axis, x = r  where Therefore,
x = 0.25(1+)2L u= (1+)[(1-)u2+0.5u3] Along x-axis, x = r r = 0.25(1+)2L or  Note: Displacement is proportional to r u = 2(r/L) [(1-)u u3] where Note: Strain (hence stress) is proportional to 1/r Therefore,

CRACK TIP ELEMENTS Therefore, by shifting the nodes to quarter position, we approximating the stress and displacements more accurately. Other crack tip elements:

METHODS FOR INFINITE DOMAIN
Infinite elements formulated by mapping (Zienkiewicz and Taylor, 2000) Gradual damping elements Coupling of FEM and BEM Coupling of FEM and SEM

Infinite elements formulated by mapping
Use shape functions to approximate decaying sequence: In 1D: (Coordinate interpolation) 

If the field variable is approximated by polynomial, Substituting  will give function of decaying form, For 2D (3D):

Element PP1QQ1RR1 : with

Infinite elements are attached to conventional FE mesh to simulate infinite domain.

Gradual damping elements
For vibration problems with infinite domain Uses conventional finite elements, hence great versatility Study of lamb wave propagation

Attaching additional damping elements outside area of interest to damp down propagating waves

(Since the energy dissipated by damping is usually independent of ) Structural damping is defined as Equation of motion with damping under harmonic load: Since, Therefore,

Complex stiffness Replace E with E(1 + i) where  is the material loss factor. Therefore, Hence,

For gradual increase in damping, Constant factor Complex modulus for the kth damping element set Initial modulus Initial material loss factor Sufficient damping such that the effect of the boundary is negligible. Damping is gradual enough such that there is no reflection cause by a sudden damped condition.

Coupling of FEM and BEM Coupling of FEM and SEM
The FEM used for interior and the BEM for exterior which can be extended to infinity [Liu, 1992] Coupling of FEM and SEM The FEM used for interior and the SEM for exterior which can be extended to infinity [Liu, 2002]

FINITE STRIP ELEMENTS Developed by Y. K. Cheung, 1968.
Used for problems with regular geometry and simple boundary. Key is in obtaining the shape functions.

FINITE STRIP ELEMENTS (Approximation of displacement function)
(Polynomial) (Continuous series) Polynomial function must represent state of constant strain in the x direction and continuous series must satisfy end conditions of the strip. Together the shape function must satisfy compatibility of displacements with adjacent strips.

FINITE STRIP ELEMENTS Y(0) = 0, Y’’(0) = 0, Y(a) = 0 and Y’’(a) = 0
Satisfies m = , 2, 3, …, m

FINITE STRIP ELEMENTS Therefore,

FINITE STRIP ELEMENTS or where i = 1, 2, 3 ,4
The remaining procedure is the same as the FEM. The size of the matrix is usually much smaller and makes the solving much easier.

STRIP ELEMENT METHOD (SEM)
Proposed by Liu and co-workers [Liu et al., 1994, 1995; Liu and Xi, 2001]. Solving wave propagation in composite laminates. Semi-analytic method for stress analysis of solids and structures. Applicable to problems of arbitrary boundary conditions including the infinite boundary conditions. Coupling of FEM and SEM for infinite domains.

Finite Element Method CHAPTER 11: MODELLING TECHNIQUES
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 11: MODELLING TECHNIQUES

CONTENTS INTRODUCTION CPU TIME ESTIMATION GEOMETRY MODELLING MESHING
Mesh density Element distortion MESH COMPATIBILITY Different order of elements Straddling elements

CONTENTS USE OF SYMMETRY MODELLING OF OFFSETS MODELLING OF SUPPORTS
Mirror symmetry Axial symmetry Cyclic symmetry Repetitive symmetry MODELLING OF OFFSETS Creation of MPC equations for offsets MODELLING OF SUPPORTS MODELLING OF JOINTS

CONTENTS OTHER APPLICATIONS OF MPC EQUATIONS
Modelling of symmetric boundary conditions Enforcement of mesh compatibility Modelling of constraints by rigid body attachment IMPLEMENTATION OF MPC EQUATIONS Lagrange multiplier method Penalty method

INTRODUCTION Ensure reliability and accuracy of results.
Improve efficiency and accuracy.

INTRODUCTION Considerations:
Computational and manpower resources that limit the scale of the FEM model. Requirement on results that defines the purpose and hence the methods of the analysis. Mechanical characteristics of the geometry of the problem domain that determine the types of elements to use. Boundary conditions. Loading and initial conditions.

CPU TIME ESTIMATION ( ranges from 2 – 3) Bandwidth, b, affects 
- minimize bandwidth Aim: To create a FEM model with minimum DOFs by using elements of as low dimension as possible, and To use as coarse a mesh as possible, and use fine meshes only for important areas.

GEOMETRY MODELLING Reduction of a complex geometry to a manageable one. 3D? 2D? 1D? Combination? (Using 2D or 1D makes meshing much easier)

GEOMETRY MODELLING Detailed modelling of areas where critical results are expected. Use of CAD software to aid modelling. Can be imported into FE software for meshing.

MESHING Mesh density To minimize the number of DOFs, have fine mesh at important areas. In FE packages, mesh density can be controlled by mesh seeds. (Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s) Pte Ltd (s))

Element distortion Use of distorted elements in irregular and complex geometry is common but there are some limits to the distortion. The distortions are measured against the basic shape of the element Square  Quadrilateral elements Isosceles triangle  Triangle elements Cube  Hexahedron elements Isosceles tetrahedron  Tetrahedron elements

Element distortion Aspect ratio distortion Rule of thumb:

Element distortion Angular distortion

Element distortion Curvature distortion

Element distortion Volumetric distortion
Area outside distorted element maps into an internal area – negative volume integration

Element distortion Volumetric distortion (Cont’d)

Element distortion Mid-node position distortion
Shifting of nodes beyond limits can result in singular stress field (see crack tip elements)

MESH COMPATIBILITY Requirement of Hamilton’s principle – admissible displacement The displacement field is continuous along all the edges between elements

Different order of elements
Crack like behaviour – incorrect results

Different order of elements
Solution: Use same type of elements throughout Use transition elements Use MPC equations

Straddling elements Avoid straddling of elements in mesh

USE OF SYMMETRY Different types of symmetry:
Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error. Mirror symmetry Axial symmetry Cyclic symmetry Repetitive symmetry

Mirror symmetry Symmetry about a particular plane

Mirror symmetry Consider a 2D symmetric solid: u1x = 0 u2x = 0 u3x = 0
Single point constraints (SPC)

Mirror symmetry Symmetric loading Deflection = Free Rotation = 0

Mirror symmetry Anti-symmetric loading Deflection = 0 Rotation = Free

Mirror symmetry Symmetric
No translational displacement normal to symmetry plane No rotational components w.r.t. axis parallel to symmetry plane Plane of symmetry u v w x y z xy Free Fix yz zx

Mirror symmetry Anti-symmetric
No translational displacement parallel to symmetry plane No rotational components w.r.t. axis normal to symmetry plane Plane of symmetry u v w x y z xy Fix Free yz zx

Mirror symmetry Any load can be decomposed to a symmetric and an anti-symmetric load

Mirror symmetry

Mirror symmetry Dynamic problems (e.g. two half models to get full set of eigenmodes in eigenvalue analysis)

Axial symmetry Use of 1D or 2D axisymmetric elements
Formulation similar to 1D and 2D elements except the use of polar coordinates Cylindrical shell using 1D axisymmetric elements 3D structure using 2D axisymmetric elements

Cyclic symmetry uAn = uBn uAt = uBt Multipoint constraints (MPC)

Repetitive symmetry uAx = uBx

MODELLING OF OFFSETS Guidelines: , offset can be safely ignored
, offset needs to be modelled , ordinary beam, plate and shell elements should not be used. Use 2D or 3D solid elements.

MODELLING OF OFFSETS Three methods: Very stiff element Rigid element
MPC equations

Creation of MPC equations for offsets
Eliminate q1, q2, q3

Creation of MPC equations for offsets

Creation of MPC equations for offfsets
d6 = d1 +  d5 or d1 +  d5 - d6 = 0 d7 = d2 -  d4 or d2 -  d4 - d7 = 0 d8 = d or d3 - d8 = 0 d9 = d or d5 - d9 = 0

MODELLING OF SUPPORTS

MODELLING OF SUPPORTS (Prop support of beam)

MODELLING OF JOINTS Perfect connection ensured here

MODELLING OF JOINTS Mismatch between DOFs of beams and 2D solid – beam is free to rotate (rotation not transmitted to 2D solid) Perfect connection by artificially extending beam into 2D solid (Additional mass)

MODELLING OF JOINTS Using MPC equations

MODELLING OF JOINTS Similar for plate connected to 3D solid

OTHER APPLICATIONS OF MPC EQUATIONS
Modelling of symmetric boundary conditions dn = 0 ui cos + vi sin=0 or ui+vi tan =0 for i=1, 2, 3

Enforcement of mesh compatibility
Use lower order shape function to interpolate dx = 0.5(1-) d (1+) d3 dy = 0.5(1-) d (1+) d6 Substitute value of  at node 3 0.5 d1 - d d3 =0 0.5 d4 - d d6 =0

Use shape function of longer element to interpolate dx = -0.5 (1-) d1 + (1+)(1-) d  (1+) d5 Substituting the values of  for the two additional nodes d2 = 0.251.5 d 0.5 d 0.5 d5 d4 = -0.250.5 d 1.5 d 1.5 d5

In x direction, 0.375 d1 - d d d5 = 0 d d3 - d d5 = 0 In y direction, 0.375 d6- d d d10 = 0 d d8 - d d10 = 0

Modelling of constraints by rigid body attachment
d1 = q1 d2 = q1+q2 l1 d3=q1+q2 l2 d4=q1+q2 l3 Eliminate q1 and q2 (l2 /l1-1) d1 - ( l2 /l1) d2 + d3 = 0 (l3 /l1-1) d1 - ( l3 /l1) d2 + d4 = 0 (DOF in x direction not considered)

IMPLEMENTATION OF MPC EQUATIONS
(Global system equation) (Matrix form of MPC equations) Constant matrices

Lagrange multiplier method
(Lagrange multipliers) Multiplied to MPC equations Added to functional The stationary condition requires the derivatives of p with respect to the Di and i to vanish.  Matrix equation is solved

Lagrange multiplier method
Constraint equations are satisfied exactly Total number of unknowns is increased Expanded stiffness matrix is non-positive definite due to the presence of zero diagonal terms Efficiency of solving the system equations is lower

Penalty method (Constrain equations)
=1  m is a diagonal matrix of ‘penalty numbers’ Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanish Penalty matrix

Penalty method [Zienkiewicz et al., 2000] :  = constant (1/h)p+1
P is the order of element used Characteristic size of element max (diagonal elements in the stiffness matrix) or Young’s modulus

Penalty method The total number of unknowns is not changed.
System equations generally behave well. The constraint equations can only be satisfied approximately. Right choice of  may be ambiguous.

FEM FOR HEAT TRANSFER PROBLEMS
Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12: FEM FOR HEAT TRANSFER PROBLEMS

CONTENTS FIELD PROBLEMS WEIGHTED RESIDUAL APPROACH FOR FEM
1D HEAT TRANSFER PROBLEMS 2D HEAT TRANSFER PROBLEMS SUMMARY CASE STUDY

FIELD PROBLEMS General form of system equations of 2D linear steady state field problems: (Helmholtz equation) For 1D problems:

FIELD PROBLEMS Heat transfer in 2D fin Note:

FIELD PROBLEMS Heat transfer in long 2D body Note: Dx = kx, Dy =tky,
g = 0 and Q = q

FIELD PROBLEMS Heat transfer in 1D fin Note:

FIELD PROBLEMS Heat transfer across composite wall Note:

FIELD PROBLEMS Torsional deformation of bar
Note: Dx=1/G, Dy=1/G, g=0, Q=2q ( - stress function) Ideal irrotational fluid flow Note: Dx = Dy = 1, g = Q = 0 ( - streamline function and  - potential function)

FIELD PROBLEMS Accoustic problems
P - the pressure above the ambient pressure ; w - wave frequency ; c - wave velocity in the medium Note: , Dx = Dy = 1, Q = 0

WEIGHTED RESIDUAL APPROACH FOR FEM
Establishing FE equations based on governing equations without knowing the functional. (Strong form) Approximate solution: (Weak form) Weight function

WEIGHTED RESIDUAL APPROACH FOR FEM
Discretize into smaller elements to ensure better approximation In each element, Using N as the weight functions where Galerkin method Residuals are then assembled for all elements and enforced to zero.

1D HEAT TRANSFER PROBLEM
1D fin k : thermal conductivity h : convection coefficient A : cross-sectional area of the fin P : perimeter of the fin : temperature, and f : ambient temperature in the fluid (Specified boundary condition) (Convective heat loss at free end)

1D fin Using Galerkin approach, where D = kA, g = hP, and Q = hP

1D fin Integration by parts of first term on right-hand side, Using

1D fin (Strain matrix) where (Thermal conduction) (Thermal convection)
(External heat supplied) (Temperature gradient at two ends of element)

1D fin For linear elements, (Recall 1D truss element) Therefore,
for truss element (Recall stiffness matrix of truss element)

1D fin for truss element (Recall mass matrix of truss element)

1D fin or (Left end) (Right end)
At the internal nodes of the fin, bL(e) and bL(e) vanish upon assembly. At boundaries, where temperature is prescribed, no need to calculate bL(e) or bL(e) first.

1D fin When there is heat convection at boundary, E.g.
Since b is the temperature of the fin at the boundary point, b = j Therefore,

1D fin where , For convection on left side, where ,

1D fin Therefore, Residuals are assembled for all elements and enforced to zero: KD = F Same form for static mechanics problem

1D fin Direct assembly procedure or Element 1:

1D fin Direct assembly procedure (Cont’d) Element 2:
Considering all contributions to a node, and enforcing to zero (Node 1) (Node 2) (Node 3)

1D fin Direct assembly procedure (Cont’d) In matrix form:
(Note: same as assembly introduced before)

1D fin Worked example: Heat transfer in 1D fin
Calculate temperature distribution using FEM. 4 linear elements, 5 nodes

1D fin Element 1, 2, 3: , not required Element 4: , required

1D fin For element 1, 2, 3 , For element 4 ,

1D fin Heat source (Still unknown)
1 = 80, four unknowns – eliminate Q* Solving:

Composite wall Convective boundary: at x = 0 at x = H
All equations for 1D fin still applies except Recall: Only for heat convection and vanish. Therefore, ,

Composite wall Worked example: Heat transfer through composite wall
Calculate the temperature distribution across the wall using the FEM. 2 linear elements, 3 nodes

Composite wall For element 1,

Composite wall For element 2, Upon assembly,
(Unknown but required to balance equations)

Composite wall Solving:

Composite wall Worked example: Heat transfer through thin film layers
Raw material 0.2 mm h =0.01 W/cm 2 / C f = 150 0.2mm Heater k =0.1 W/cm/ 2mm Glass Iron Platinum =0.5 W/cm/ =0.4 W/cm/ Substrate Plasma 1 3 4 (1) (2) (3) 300C

Composite wall For element 1, 1 2 3 4 (1) (2) (3) For element 2,

Composite wall For element 3,

Composite wall Since, 1 = 300°C, Solving:

2D HEAT TRANSFER PROBLEM
Element equations For one element, Note: W = N : Galerkin approach

Element equations Therefore, Gauss’s divergence theorem:
(Need to use Gauss’s divergence theorem to evaluate integral in residual.) (Product rule of differentiation) Therefore, Gauss’s divergence theorem:

Element equations 2nd integral: Therefore,

Element equations

Element equations where

Element equations Define , (Strain matrix) 

Triangular elements Note: constant strain matrix (Or Ni = Li)

Triangular elements Note: (Area coordinates) E.g. Therefore,

Triangular elements Similarly, Note: b(e) will be discussed later

Rectangular elements

Rectangular elements Note: In practice, the integrals are usually evaluated using the Gauss integration scheme

Boundary conditions and vector b(e)
Internal Boundary bB(e) needs to be evaluated at boundary Vanishing of bI(e)

Need not evaluate Need to be concern with bB(e)

on natural boundary 2 Heat flux across boundary

Insulated boundary: M = S = 0  Convective boundary condition:

Specified heat flux on boundary:

For other cases whereby M, S  0

where , For a rectangular element, (Equal sharing between nodes 1 and 2)

Equal sharing valid for all elements with linear shape functions Applies to triangular elements too

for rectangular element

Shared in ratio 2/6, 1/6, 1/6, 2/6

Similar for triangular elements

Point heat source or sink
Preferably place node at source or sink

Point heat source or sink within the element
Point source/sink (Delta function) 

SUMMARY

CASE STUDY Road surface heated by heating cables under road surface

CASE STUDY Heat convection M=h=0.0034 S=ff h=-0.017 fQ*
Repetitive boundary no heat flow across M=0, S=0 Repetitive boundary no heat flow across M=0, S=0 Insulated M=0, S=0

CASE STUDY Surface temperatures: Node Temperature (C) 1 5.861 2 5.832
5.764 4 5.697 5 5.669

Finite Element Method CHAPTER 13: USING ABAQUS
for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 13: USING ABAQUS

CONTENTS ABAQUS INPUT FILE GENERAL PROCEDURES

ABAQUS INPUT FILE Model data Option block: History data Node cards
*HEADING Model of a cantilever beam with a downward force ** *NODE 1, 0. 11, 2.0 *NGEN 1, 11 *ELEMENT, TYPE=B21 1, 1, 2 *ELGEN, ELSET=RECT_BEAM 1, 10 *BEAM SECTION, ELSET=RECT_BEAM, SECTION=RECT, MATERIAL=ALU 0.025, 0.040 0., 0., -1.0 *MATERIAL, NAME=ALU *ELASTIC, TYPE=ISOTROPIC 69.E9, 0.33 *BOUNDARY 1, 1, 6, 0. *STEP, PERTURBATION *STATIC *CLOAD 11, 2, *NODE PRINT, FREQ=1 U, *NODE FILE, FREQ=1 *ELEMENT PRINT, FREQ=1 S, E *ELEMENT FILE, FREQ=1 *ENDSTEP ABAQUS INPUT FILE Node cards Element cards Model data Property cards Option block: Material cards *ELEMENT, TYPE=B23 1, 1, 2 Keyword line Boundary cards Data line Control cards Parameter Data Keyword Load cards History data Output cards

GENERAL PROCEDURES GEOMETRY DEFINITION ELEMENT PROPERTIES DEFINITION
Node Definitions (*NODE, *NGEN, *NCOPY, *NFIL, *NSET) GEOMETRY DEFINITION Element Definitions (*ELEMENT, *ELGEN, *ELCOPY, *ELSET) ELEMENT *BEAM SECTION, *SHELL SECTION, *SOLID SECTION, *MEMBRANE PROPERTIES SECTION, etc. DEFINITION MATERIAL *MATERIAL, *ELASTIC, *DE NSITY, PROPERTIES *VISCOELASTIC, *DAMPING, etc. DEFINITION BOUNDARY *BOUNDARY, *CONTACT INTERFERENCE, *CONTACT PAIR, AND INITIAL *INITIAL CONDITIONS , etc. CONDITIONS *CLOAD, *DLOAD, *DFLUX, LOADING *CECHARGE , *DECHARGE, CONDITIONS *TEMPERATURE, etc. *STATIC, *STEADY STATE DYNAMICS, ANALYSIS *PIEZOELECTRIC, *MASS DIFFUSION, TYPES *HEAT TRANSFER, etc. *NODE PRINT, *NODE FILE, *EL PRINT, OUTPUT *EL FILE, etc. REQUESTS

CHAPTER 1: COMPUTATIONAL MODELLING

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