Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction to Finite Element Methods UNIT I. Numerical Methods – Definition and Advantages  Definition: Methods that seek quantitative approximations.

Similar presentations


Presentation on theme: "Introduction to Finite Element Methods UNIT I. Numerical Methods – Definition and Advantages  Definition: Methods that seek quantitative approximations."— Presentation transcript:

1 Introduction to Finite Element Methods UNIT I

2 Numerical Methods – Definition and Advantages  Definition: Methods that seek quantitative approximations to the solutions of mathematical problems  Advantages:

3 What is a Numerical Method – An Example Example 1:

4 What is a Numerical Method – An Example Example 1:

5 What is a Numerical Method – An Example Example 2:

6 What is a Numerical Method – An Example Example 2:

7 What is a Numerical Method – An Example Example 3:

8 What is a Finite Element Method      

9 Discretization 1-D2-D 3-D?-D Hybrid

10 Approximation Numerical Interpolation Non-exact Boundary Conditions

11 Applications of Finite Element Methods  Structural & Stress Analysis  Thermal Analysis  Dynamic Analysis  Acoustic Analysis  Electro-Magnetic Analysis  Manufacturing Processes  Fluid Dynamics

12 Lecture 2 Review

13 Matrix Algebra Row and column vectors Addition and Subtraction – must have the same dimensions Multiplication – with scalar, with vector, with matrix Transposition – Differentiation and Integration

14 Matrix Algebra Determinant of a Matrix: Matrix inversion - Important Matrices diagonal matrix identity matrix zero matrix eye matrix

15 Numerical Integration Calculate: Newton – Cotes integration Trapezoidal rule – 1 st order Newton-Cotes integration Trapezoidal rule – multiple application

16 Numerical Integration Calculate: Newton – Cotes integration Simpson 1/3 rule – 2 nd order Newton-Cotes integration

17 Numerical Integration Calculate: Gaussian Quadrature Trapezoidal Rule : Gaussian Quadrature : Chooseaccording to certain criteria

18 Numerical Integration Calculate: Gaussian Quadrature 2pt Gaussian Quadrature 3pt Gaussian Quadrature Let:

19 Numerical Integration - Example Calculate: Trapezoidal rule Simpson 1/3 rule 2pt Gaussian quadrature Exact solution

20 Linear System Solver Solve: Gaussian Elimination: forward elimination + back substitution Example:

21 Linear System Solver Solve: Gaussian Elimination: forward elimination + back substitution Pseudo code: Forward elimination: Back substitution: Do k = 1, n-1 Do i = k+1,n Do j = k+1, n Do ii = 1, n-1 i = n – ii sum = 0 Do j = i+1, n sum = sum +

22 Finite Element Analysis (F.E.A.) of 1-D Problems UNIT II

23 Historical Background Hrenikoff, 1941 – “frame work method” Courant, 1943 – “piecewise polynomial interpolation” Turner, 1956 – derived stiffness matrice for truss, beam, etc Clough, 1960 – coined the term “finite element” Key Ideas: - frame work method piecewise polynomial approximation

24 Axially Loaded Bar Review: Stress: Strain: Deformation: Stress: Strain: Deformation:

25 Axially Loaded Bar Review: Stress: Strain: Deformation:

26 Axially Loaded Bar – Governing Equations and Boundary Conditions Differential Equation Boundary Condition Types prescribed displacement (essential BC) prescribed force/derivative of displacement (natural BC)

27 Axially Loaded Bar –Boundary Conditions Examples fixed end simple support free end

28 Potential Energy Elastic Potential Energy (PE) - Spring case - Axially loaded bar - Elastic body x Unstretched spring Stretched bar undeformed: deformed:

29 Potential Energy Work Potential (WE) P f f: distributed force over a line P: point force u: displacement A B Total Potential Energy Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable.

30 Potential Energy + Rayleigh-Ritz Approach P f A B Example: Step 1: assume a displacement field  is shape function / basis function n is the order of approximation Step 2: calculate total potential energy

31 Potential Energy + Rayleigh-Ritz Approach P f A B Example: Step 3:select a i so that the total potential energy is minimum

32 Galerkin’s Method P f A B Example: Seek an approximation so In the Galerkin’s method, the weight function is chosen to be the same as the shape function.

33 Galerkin’s Method P f A B Example:

34 Finite Element Method – Piecewise Approximation x u x u

35 FEM Formulation of Axially Loaded Bar – Governing Equations Differential Equation Weighted-Integral Formulation Weak Form

36 Approximation Methods – Finite Element Method Example: Step 1: Discretization Step 2: Weak form of one element P2P2 P1P1 x1x1 x2x2

37 Approximation Methods – Finite Element Method Example (cont): Step 3: Choosing shape functions - linear shape functions l x1x1 x2x2 x    

38 Approximation Methods – Finite Element Method Example (cont): Step 4: Forming element equation Let, weak form becomes E,A are constant

39 Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 1: Element 1: Element 2: Element 3:

40 Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Assembled System:

41 Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 2: Element connectivity table Element 1Element 2Element global node index (I,J) local node (i,j)

42 Approximation Methods – Finite Element Method Example (cont): Step 6: Imposing boundary conditions and forming condense system Condensed system:

43 Approximation Methods – Finite Element Method Example (cont): Step 7: solution Step 8: post calculation

44 Summary - Major Steps in FEM Discretization Derivation of element equation weak form construct form of approximation solution over one element derive finite element model Assembling – putting elements together Imposing boundary conditions Solving equations Postcomputation

45 Exercises – Linear Element Example 1: E = 100 GPa, A = 1 cm 2

46 Linear Formulation for Bar Element x=x 1 x=x 2  2 2 1 1  x=x 1 x= x 2 u1u1 u2u2 f(x) L = x 2 -x 1 u x

47 Higher Order Formulation for Bar Element 1 3 u1u1 u3u3 u x u2u2 214 u1u1 u4u4 2 u x u2u2 u3u3 3 1n u1u1 unun 2 u x u2u2 u3u3 3 u4u4 …………… 4

48 Natural Coordinates and Interpolation Functions Natural (or Normal) Coordinate: x=x 1 x= x 2  =-1  =1 x    =-1  =1 1 2   =-1  =   =-1  =1 3

49 Quadratic Formulation for Bar Element  =-1  =0  =1   

50 Quadratic Formulation for Bar Element u1u1 u3u3 u2u2 f(x) P3P3 P1P1 P2P2  =-1  =0  =1

51 Exercises – Quadratic Element Example 2: E = 100 GPa, A 1 = 1 cm 2 ; A 1 = 2 cm 2

52 Some Issues Non-constant cross section: Interior load point: Mixed boundary condition: k

53 Finite Element Analysis (F.E.A.) of I-D Problems – Applications

54 Plane Truss Problems Example 1: Find forces inside each member. All members have the same length. F

55 UNIT II

56 Arbitrarily Oriented 1-D Bar Element on 2-D Plane Q 2, v 2  P 2, u 2 Q 1, v 1 P 1, u 1

57 Relationship Between Local Coordinates and Global Coordinates

58

59 Stiffness Matrix of 1-D Bar Element on 2-D Plane Q 2, v 2  P 2, u 2 Q 1, v 1 P 1, u 1

60 Arbitrarily Oriented 1-D Bar Element in 3-D Space  x,  x,  x are the Direction Cosines of the bar in the x-y-z coordinate system xx x xx xx y z

61 Stiffness Matrix of 1-D Bar Element in 3-D Space xx x xx xx y z

62 Matrix Assembly of Multiple Bar Elements Element I I I

63 Matrix Assembly of Multiple Bar Elements Element I I I

64 Matrix Assembly of Multiple Bar Elements Apply known boundary conditions

65 Solution Procedures u 2 = 4FL/5AE, v 1 = 0

66 Recovery of Axial Forces Element I I I

67 Stresses inside members Element I I I

68 Lecture 5 FEM of 1-D Problems: Applications

69 Torsional Shaft Review Assumption: Circular cross section Shear stress: Deformation: Shear strain:

70 Finite Element Equation for Torsional Shaft

71 Bending Beam Review Normal strain: Pure bending problems: Normal stress: Normal stress with bending moment: Moment-curvature relationship: Flexure formula: x y M M

72 Bending Beam Review Deflection: Sign convention: Relationship between shear force, bending moment and transverse load: q(x) x y + - M M M + - V V V

73 Governing Equation and Boundary Condition Governing Equation Boundary Conditions Essential BCs – if v or is specified at the boundary. Natural BCs – if or is specified at the boundary. { 0

74 Weak Formulation for Beam Element Governing Equation Weighted-Integral Formulation for one element Weak Form from Integration-by-Parts (1 st time)

75 Weak Formulation Weak Form from Integration-by-Parts (2 nd time) V(x 2 ) x = x 1 M(x 2 ) q(x) y x x = x 2 V(x 1 ) M(x 1 ) L = x 2 -x 1

76 Weak Formulation Weak Form Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1

77 Ritz Method for Approximation Let w(x)=  i (x), i = 1, 2, 3, 4 Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 where

78 Ritz Method for Approximation Q3Q3 x = x 1 y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 Q4Q4

79 Ritz Method for Approximation Q3Q3 x = x 1 y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 Q4Q4

80 Selection of Shape Function The best situation is Interpolation Properties

81 Derivation of Shape Function for Beam Element – Local Coordinates How to select  i ??? and where Let Find coefficients to satisfy the interpolation properties.

82 Derivation of Shape Function for Beam Element How to select  i ??? e.g. Let Similarly

83 Derivation of Shape Function for Beam Element In the global coordinates:

84 Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1 x=x 2 x=x 1 1 3 3 2 2 4 4

85 Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1

86 Finite Element Analysis of 1-D Problems - Applications F L L L Example 1. Governing equation: Weak form for one element where

87 Finite Element Analysis of 1-D Problems Example 1. Approximation function: 3 3 2 2 1 1 4 4 x=x 1 x=x 2

88 Finite Element Analysis of 1-D Problems Example 1. Finite element model: P 1, v 1 P 2, v 2 P 3, v 3 P 4, v 4 M 1,  1 M 2,  2 M 3,  3 M 4,  4 I II III Discretization:

89 Matrix Assembly of Multiple Beam Elements Element I I

90 Matrix Assembly of Multiple Beam Elements Element I I

91 Solution Procedures Apply known boundary conditions

92 Solution Procedures

93 Shear Resultant & Bending Moment Diagram

94 Plane Flame Frame: combination of bar and beam E, A, I, L Q 1, v 1 Q 3, v 2 Q 2,  1 P 1, u 1 Q 4,  2 P 2, u 2

95 Finite Element Model of an Arbitrarily Oriented Frame  x y  x y

96 local global

97 Plane Frame Analysis - Example Rigid Joint Hinge Joint Beam II Beam I Beam Bar F FF F

98 Plane Frame Analysis P 1, u 1 P 2, u 2 Q 2,  1 Q 4,  2 Q 1, v 1 Q 3, v 2

99 Plane Frame Analysis P 1, u 2 Q 3, v 3 Q 2,  2 Q 4,  3 Q 1, v 2 P 2, u 3

100 Plane Frame Analysis

101

102 Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction UNIT IV

103 Heat Transfer Mechanisms  Conduction – heat transfer by molecular agitation within a material without any motion of the material as a whole.  Convection – heat transfer by motion of a fluid.  Radiation – the exchange of thermal radiation between two or more bodies. Thermal radiation is the energy emitted from hot surfaces as electromagnetic waves.

104 Heat Conduction in 1-D Heat flux q: heat transferred per unit area per unit time (W/m 2 ) Q: heat generated per unit volume per unit time C: mass heat capacity Governing equation: Steady state equation:  : thermal conductivity

105 Thermal Convection Newton’s Law of Cooling

106 Thermal Conduction in 1-D Dirichlet BC: Boundary conditions: Natural BC: Mixed BC:

107 Weak Formulation of 1-D Heat Conduction ( Steady State Analysis ) Governing Equation of 1-D Heat Conduction

108 Formulation for 1-D Linear Element Let f2f2 x1x1 x2x2 1 2 T1T1 x T2T2 f1f1 x2 x2 x1 x1 1T1 1T1 2T2 2T2

109 Formulation for 1-D Linear Element Let w(x)=  i (x), i = 1, 2

110 Element Equations of 1-D Linear Element f2f2 x1x1 x2x2 1 2 T1T1 x T2T2 f1f1

111 1-D Heat Conduction - Example A composite wall consists of three materials, as shown in the figure below. The inside wall temperature is 200 o C and the outside air temperature is 50 o C with a convection coefficient of h = 10 W(m 2.K). Find the temperature along the composite wall. t1t1 t2t2 t3t3 x

112 Thermal Conduction and Convection- Fin Objective: to enhance heat transfer dx t x w Governing equation for 1-D heat transfer in thin fin where

113 Fin - Weak Formulation ( Steady State Analysis ) Governing Equation of 1-D Heat Conduction

114 Formulation for 1-D Linear Element Let w(x)=  i (x), i = 1, 2

115 Element Equations of 1-D Linear Element f2f2 x=0 x=L 1 2 T1T1 x T2T2 f1f1

116 Lecture 7 Finite Element Analysis of 2-D Problems

117 2-D Discretization Common 2-D elements:

118 2-D Model Problem with Scalar Function - Heat Conduction Governing Equation in  Boundary Conditions Dirichlet BC: Natural BC: Mixed BC:

119 Weak Formulation of 2-D Model Problem Weighted - Integral of 2-D Problem Weak Form from Integration-by-Parts -----

120 Weak Formulation of 2-D Model Problem Green-Gauss Theorem where n x and n y are the components of a unit vector, which is normal to the boundary .

121 Weak Formulation of 2-D Model Problem Weak Form of 2-D Model Problem EBC: Specify T(x,y) on  NBC: Specify on  where is the normal outward flux on the boundary  at the segment ds.

122 FEM Implementation of 2-D Heat Conduction – Shape Functions Step 1: Discretization – linear triangular element T1T1 T2T2 T3T3 Derivation of linear triangular shape functions: Let Interpolation properties Same

123 FEM Implementation of 2-D Heat Conduction – Shape Functions linear triangular element – area coordinates T1T1 T2T2 T3T3 A3A3 A1A1 A2A2 11 22 33

124 Interpolation Function - Requirements Interpolation condition Take a unit value at node i, and is zero at all other nodes Local support condition  is zero at an edge that doesn’t contain node i. Interelement compatibility condition Satisfies continuity condition between adjacent elements over any element boundary that includes node i Completeness condition The interpolation is able to represent exactly any displacement field which is polynomial in x and y with the order of the interpolation function

125 Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element 1 1 2 2 3 3 4 4 Note: The local node numbers should be arranged in a counter-clockwise sense. Otherwise, the area Of the element would be negative and the stiffness matrix can not be formed. Let

126 Weak Form of 2-D Model Problem Assume approximation: and let w(x,y)=  i (x,y) as before, then where FEM Implementation of 2-D Heat Conduction – Element Equation

127

128 Assembly of Stiffness Matrices

129 Imposing Boundary Conditions The meaning of q i :

130 Imposing Boundary Conditions Equilibrium of flux: FEM implementation: Consider

131 Calculating the q Vector Example:

132 2-D Steady-State Heat Conduction - Example 0.6 m 0.4 m A B C D AB and BC: CD: convection DA: x y

133 Finite Element Analysis of Plane Elasticity

134 Review of Linear Elasticity Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition. Stress: measurement of force intensity with 2-D

135 Review of Linear Elasticity Traction (surface force) : Equilibrium – Newton’s Law

136 Review of Linear Elasticity Strain: measurement of intensity of deformation Generalized Hooke’s Law

137 Plane Stress and Plane Strain Plane Stress - Thin Plate:

138 Plane Stress and Plane Strain Plane Strain - Thick Plate: Plane Stress: Plane Strain: Replace E by and by

139 Equations of Plane Elasticity Governing Equations (Static Equilibrium) Constitutive Relation (Linear Elasticity) Strain-Deformation (Small Deformation)

140 Specification of Boundary Conditions EBC: Specify u(x,y) and/or v(x,y) on  NBC: Specify t x and/or t y on  where is the traction on the boundary  at the segment ds.

141 UNIT V

142 Weak Formulation for Plane Elasticity where are components of traction on the boundary 

143 Finite Element Formulation for Plane Elasticity Let where and

144 Constant-Strain Triangular (CST) Element for Plane Stress Analysis Let

145 Constant-Strain Triangular (CST) Element for Plane Stress Analysis

146 4-Node Rectangular Element for Plane Stress Analysis Let

147 4-Node Rectangular Element for Plane Stress Analysis For Plane Strain Analysis: and

148 Loading Conditions for Plane Stress Analysis

149 Evaluation of Applied Nodal Forces

150

151 Element Assembly for Plane Elasticity A B

152 A B

153 Comparison of Applied Nodal Forces

154 Discussion on Boundary Conditions Must have sufficient EBCs to suppress rigid body translation and rotation For higher order elements, the mid side nodes cannot be skipped while applying EBCs/NBCs

155 Plane Stress – Example 2

156 Plane Stress – Example 3

157 Evaluation of Strains

158 Evaluation of Stresses Plane Stress Analysis Plane Strain Analysis

159 Finite Element Analysis of 2-D Problems – Axi- symmetric Problems

160 Axi-symmetric Problems Definition: A problem in which geometry, loadings, boundary conditions and materials are symmetric about one axis. Examples:

161 Axi-symmetric Analysis Cylindrical coordinates: quantities depend on r and z only 3-D problem 2-D problem

162 Axi-symmetric Analysis

163 Axi-symmetric Analysis – Single-Variable Problem Weak form: where

164 Finite Element Model – Single-Variable Problem Ritz method: where Weak form where

165 Single-Variable Problem – Heat Transfer Heat Transfer: Weak form where

166 3-Node Axi-symmetric Element 1 2 3

167 4-Node Axi-symmetric Element a b   r z

168 Single-Variable Problem – Example Step 1: Discretization Step 2: Element equation

169 Time-Dependent Problems

170 In general, Key question: How to choose approximate functions? Two approaches:

171 Model Problem I – Transient Heat Conduction Weak form:

172 Transient Heat Conduction let: and ODE!

173 Time Approximation – First Order ODE Forward difference approximation - explicit Backward difference approximation - implicit

174 Time Approximation – First Order ODE  - family formula: Equation

175 Time Approximation – First Order ODE Finite Element Approximation

176 Stability of – Family Approximation Stability  Example

177 FEA of Transient Heat Conduction  - family formula for vector:

178 Stability Requirment where Note: One must use the same discretization for solving the eigenvalue problem.

179 Transient Heat Conduction - Example

180

181

182

183

184

185

186 Model Problem II – Transverse Motion of Euler- Bernoulli Beam Weak form: Where:

187 Transverse Motion of Euler-Bernoulli Beam let: and

188 Transverse Motion of Euler-Bernoulli Beam

189 ODE Solver – Newmark’s Scheme where Stability requirement: where

190 ODE Solver – Newmark’s Scheme Constant-average acceleration method (stable) Linear acceleration method (conditional stable) Central difference method (conditional stable) Galerkin method (stable) Backward difference method (stable)

191 Fully Discretized Finite Element Equations

192 Transverse Motion of Euler-Bernoulli Beam

193

194

195


Download ppt "Introduction to Finite Element Methods UNIT I. Numerical Methods – Definition and Advantages  Definition: Methods that seek quantitative approximations."

Similar presentations


Ads by Google