Introduction to Finite Element Methods

Preface
Glossary
	Physical Description of Processes
	Matrix– and Vector–Symbols at the Element Level
	Matrix– and Vector–Symbols at the System Level
Denomination of Elements
	Heat Conduction Elements
	Plane Stress Elements
	Kirchhoff Plate Elements
	Reissner–Mindlin Plate Elements
Table of contents
FOUNDATIONS
	1 Introduction
		1.1 Governing Equations and Approximate Solution
		1.2 General Aspects Concerning the Finite Element Method
		1.3 A Comparison of Exact Solution and Approximate Solution
			1.3.1 Analytically Exact Solutions
			1.3.2 Approximate Solutions
			1.3.3 The FE–Solution Regarding two Elements – Case 2
	2 Discretization of the Work Equation
		2.1 Principle of Virtual Displacements
			2.1.1 Virtual Work
			2.1.2 Fulfillment of Governing Equations and Boundary Conditions
			2.1.3 Fulfillment of the Conditions at the Element Intersections
		2.2 Principle of Virtual Forces
		2.3 The General Procedure to set up the Element Matrices
			2.3.1 Solution Procedure
			2.3.2 Example
			2.3.3 Matrix Notation
		2.4 Shape Functions and Convergence Criteria
			2.4.1 Shape Functions
			2.4.2 Criteria of Convergence
			2.4.3 Scaling Matrix
	3 Structure and Solution of the System of Equations
		3.1 Real and Virtual Nodal Displacements
		3.2 Equations of Condition
		3.3 Structure of the System of Equations
		3.4 Assembly of the Stiffness Matrix of the Entire System
		3.5 The Storage and Solution of the System of Equations
		3.6 The Optimization of the Bandwidth
		3.7 The Fulfillment of Dirichlet Boundary Conditions
	4 Heat Conduction
		4.1 Heat Conduction at One–Dimensional Description
			4.1.1 Governing Equations for One Dimension
			4.1.2 Boundary Conditions
			4.1.3 Weak Form of the Energy Balance Equation
			4.1.4 Example of use
		4.2 Heat Conduction Regarding Two Spatial Dimensions
			4.2.1 Governing Equations and the Weak Form
			4.2.2 Matrix Representation of the Weak Form
			4.2.3 Heat Conduction Matrix E and Operator Matrix D
			4.2.4 Linear Shape Functions Regarding the Temperature
			4.2.5 Differentiation of the Temperature Field
			4.2.6 Element Matrix
			4.2.7 Element Vector of Thermal Action
			4.2.8 Subsequent Flux Analysis
		4.3 Example of Use
	5 Membrane Structures
		5.1 Rectangular Elements Regarding Plane Stress Situation
			5.1.1 Governing Equations and Work Equations
			5.1.2 Matrix Notation of the Work Equations
			5.1.3 Elasticity Matrix E and Operator Matrix D
			5.1.4 Bi–linear Shape Functions Regarding Displacements
			5.1.5 Differentiation of Displacement Fields
			5.1.6 Element Stiffness Matrix
			5.1.7 Element Load Vector
			5.1.8 Subsequent Stress Analysis
			5.1.9 Example of Use
		5.2 Rectangular Element Comprising Modified Shear Strains
			5.2.1 Selectively Reduced Integration
			5.2.2 Example of Use
		5.3 Plane Strain
	6 Bending Structures
		6.1 Element Matrices Regarding Euler–Bernoulli Beams
			6.1.1 Governing Equations and Work Equations
			6.1.2 Matrix Notation of the Work Equation
			6.1.3 Elasticity Matrix E and Operator Matrix D
			6.1.4 Shape Functions to Describe the Deflection
			6.1.5 Differentiation of Shape Functions
			6.1.6 Element Stiffness Matrix
			6.1.7 Element Load Vector
			6.1.8 Subsequent Stress Analysis
			6.1.9 Example of Use
		6.2 Kirchhoff Plate Element with 16 DOF
			6.2.1 Governing Equations and Work Equations
			6.2.2 Matrix Notation of the Work Equation
			6.2.3 Elasticity Matrix E and Operator Matrix D
			6.2.4 Shape Functions to Describe the Deflection
			6.2.5 Differentiation of Shape Functions
			6.2.6 Element Stiffness Matrix
			6.2.7 Element Load Vector
			6.2.8 Subsequent Stress Analysis
			6.2.9 Example of Use
		6.3 Kirchhoff Plate Element with 12 DOF
		6.4 The 12 DOF Element Employing a Weak Conformity
			6.4.1 Stiffness matrix
			6.4.2 Example of Use
TRIANGULAR ELEMENTS
	7 Triangular Elements – Description of Geometry
		7.1 Local ξ–η–Coordinate System
			7.1.1 Transformation of Coordinates
			7.1.2 Transformation of Partial Derivatives
			7.1.3 The Integration of the Element Area
			7.1.4 Linear Shape Functions and Nodal Degrees of Freedom
		7.2 Description Employing Area Coordinates
			7.2.1 Transformation between Cartesian and Area Coordinates
			7.2.2 Transformation of Partial Derivatives
			7.2.3 Integration with Respect to Area Coordinates
			7.2.4 Shape Functions Employing Area Coordinates
	8 Triangular Elements to Describe Heat Conduction
		8.1 A Linear Approach Related to the ξ–η–Coordinate System
			8.1.1 Linear Shape Functions and Nodal Unknowns
			8.1.2 Differentiation of the Temperature Field
			8.1.3 Element Matrix
			8.1.4 Vector of Thermal Action
			8.1.5 Subsequent Flux Analysis
		8.2 Example of Use
	9 Triangular Elements for Membrane Structures
		9.1 Linear Shape Functions with Respect to ξ–η–Coordinates
			9.1.1 Shape Functions and Nodal Unknowns
			9.1.2 Differentiation of Displacement Fields
			9.1.3 Element Stiffness Matrix
			9.1.4 Load Vector
			9.1.5 Subsequent Stress Analysis
		9.2 Description Employing Area Coordinates
			9.2.1 Linear Shape Functions Employing Area Coordinates
		9.3 Quadratic Approach Employing ξ–η–Coordinates
			9.3.1 Quadratic Shape Functions
			9.3.2 Element Stiffness Matrix
		9.4 Quadratic Shape Functions Using Area Coordinates
			9.4.1 Stiffness Matrix
			9.4.2 Load Vector
			9.4.3 Subsequent Stress Analysis
		9.5 A Comparison of Standard Elements
	10 Triangular Elements for Kirchhoff Plates
		10.1 Choice of Shape Functions and Nodal Unknowns
		10.2 Complete Approach of the 5th Order
			10.2.1 The Process to Assemble the Element Stiffness Matrix
			10.2.2 General Polynomial Employing Area Coordinates
			10.2.3 Derivatives with Respect to x–y–Coordinates
			10.2.4 Scaling Matrix
			10.2.5 Element Stiffness Matrix
			10.2.6 Element Load Vector
			10.2.7 Subsequent Analysis for Stress Resultants
		10.3 A Plate Element with 18 DOF
		10.4 A Comparison of Standard Plate Elements
ISOPARAMETRIC ELEMENTS
	11 Numerical Integration
		11.1 Numerical Integration Using Gauss–Legendre Quadrature
		11.2 Numerical Integration Applied to Membrane Elements
			11.2.1 Shape Functions Employing Local Coordinates
			11.2.2 Derivatives with respect to local coordinates
			11.2.3 Element Stiffness Matrix and Load Vector
			11.2.4 Element Load Vector
			11.2.5 The Subsequent Stress Analysis
		11.3 Triangular Elements
			11.3.1 Triangular Elements for Membranes
			11.3.2 Triangular Elements for Plates
	12 Isoparametric Elements
		12.1 Description of the Element Geometry
		12.2 Isoparametric Elements Regarding Membranes
			12.2.1 The Stiffness Matrix and the Stress Matrix
			12.2.2 The Selectively Reduced Integration
			12.2.3 Comparison of Standard Membrane Elements
		12.3 Quadrilateral Plate Elements
			12.3.1 The 16 DOF Plate Element
			12.3.2 The 12 DOF Plate Element
HYBRID QUADRILATERAL ELEMENTS
	13 Hybrid Finite Elements
		13.1 Mixed Formulation of Governing Equations
		13.2 Mixed Formulation of Work Equations
			13.2.1 Shape Functions, Element Matrix, Load Vector
			13.2.2 Matrix Notation of the Work Equations
			13.2.3 Test
		13.3 Hybrid Discretization of Work Equations
			13.3.1 Work Equation at Element Level
			13.3.2 Stiffness Matrix and Load Vector of the Hybrid Element
	14 Hybrid–Mixed Plane Stress Elements
		14.1 Mixed Principle of Work for Plane Stress Structures
			14.1.1 Governing Equations
			14.1.2 Work Equations in Mixed Formulation
		14.2 Work Equations of a Hybrid Plane Stress Element
			14.2.1 Work Equations at Element Level
			14.2.2 Conditions of Deformation at the Element Interface
			14.2.3 Stress Conditions at the Element Interface
		14.3 Element Stiffness Matrix
		14.4 Subsequent Stress Analysis
		14.5 Linear Shape Functions
		14.6 Quadratic Shape Functions
		14.7 Linear Approaches with Coupling of Degrees of Freedom
			14.7.1 Approach Following Pian and Sumihara
			14.7.2 Approach with Transformation of Coordinates
		14.8 Convergence Behavior of the Elements
	15 Hybrid–Mixed Euler–Bernoulli Beam Elements
		15.1 Mixed Formulation Employing Forces and Displacements
			15.1.1 Governing Equations
			15.1.2 Work Equations
			15.1.3 Boundary and Element Interface Conditions
		15.2 Work Equation in Hybrid Formulation
		15.3 Element Stiffness Matrix
	16 Hybrid–Mixed Kirchhoff Plate Elements
		16.1 Mixed Principles of Work Concerning Kirchhoff Plates
			16.1.1 Governing Equations
			16.1.2 Work Equations
			16.1.3 Element Matrix and Load Vector
			16.1.4 Convergence Behavior Concerning the Mixed Element
		16.2 Hybrid–Mixed Rectangular Plate Element
			16.2.1 Work Equations
			16.2.2 Element Stiffness Matrix
			16.2.3 Convergence Behavior of the Hybrid–Mixed Element
HYBRID TRIANGULAR PLATE ELEMENTS
	17 Hybrid Triangular Plate Elements
		17.1 Cubic Approach for Triangular Plate Elements
			17.1.1 Elimination of wd at Element Level
			17.1.2 Subsequent Analysis Concerning the Stress Resultants
			17.1.3 Hybrid Elements to Ensure C1 - Conformity
		17.2 Hybrid–Displacement Elements Employing 10+3 DOF
			17.2.1 Version A Employing Lagrange Multipliers
			17.2.2 Version B Employing Rotational Degrees of Freedom
		17.3 Displacement–based Element with Weak Conformity
	18 Hybrid–Mixed Triangular Plate Elements
		18.1 Work Equations
		18.2 Approaches Related to the Deflection and the Stresses
		18.3 Element Stiffness Matrix and Load Vector
		18.4 Fulfillment of the Continuity Conditions at the Interface
		18.5 Subsequent Stress Analysis
		18.6 Test
	19 Discrete Kirchhoff –Theory Element
		19.1 The Discrete Kirchhoff Triangular Element
		19.2 Stiffness Matrix, Load Vector and Stress Analysis
		19.3 Test
	20 Benchmark Concerning Triangular Plate Elements
		20.1 Square Plate Subjected to Distributed Loading
		20.2 Trapezoidal Plate Subjected to Distributed Loading
SHEAR–DEFORMATION BEAM AND PLATE ELEMENTS
	21 Timoshenko Beam Elements
		21.1 Elements Employing Displacements as Primary Variables
		21.2 Mixed Elements
		21.3 Hybrid–Mixed Elements
		21.4 Convergence Behavior Concerning the Beam Elements
	22 Plate Elements Including Shear Deformations
		22.1 Kirchhoff and Reissner–Mindlin Theories by Comparison
		22.2 Governing Equations of the Reissner–Mindlin Theory
		22.3 Weak Formulation of the Governing Equations
		22.4 Quadrilateral Element Employing a Bi–linear Approach
			22.4.1 Approaches to Describe the Deflection and Rotations
			22.4.2 Element Stiffness Matrix and Load Vector
		22.5 Hybrid–Displacement Quadrilateral Element
			22.5.1 Element Stiffness Matrix
			22.5.2 Subsequent Stress Analysis
		22.6 Comparison of the Elements
			22.6.1 Test 1
			22.6.2 Test 2
			22.6.3 Remarks
			22.6.4 Benchmark
		22.7 Displacement–Based Triangular Element
			22.7.1 Approaches to Describe the Displacement Variables
			22.7.2 Element Stiffness Matrix and Load Vector
			22.7.3 Subsequent Stress Analysis
			22.7.4 Behabior of Convergence Concerning the Triangular Element
		22.8 Mixed Quadrilateral Element
		22.9 Hybrid–Mixed Quadrilateral Element
EVALUATION OF RESULTS
	23 Error Estimation
		23.1 The Least Squares Method
		23.2 Error Estimation by Applying the Principle of VirtualWork
		23.3 Mesh–Adaptation
	24 Quality of Elements
		24.1 The Eigenvalue Analysis
		24.2 The Locking Phenomena
		24.3 Improvement of Elements Suffering from Locking
		24.4 The Patch–Test
References
Index