Santosh Chavan, profile picture
Uploaded bySantosh Chavan
PPT, PDF7,762 views

Finite element method

The document provides an introduction to the finite element method (FEM). It discusses how FEM can be used to obtain approximate solutions to boundary value problems in engineering. It outlines the general steps involved, including preprocessing (defining the model), solution/processing (computing unknown values), and postprocessing (analyzing results). Examples of FEM applications include structural analysis, fluid flow, heat transfer, and more. The key aspects of FEM include discretizing the domain into simple elements, choosing shape functions to approximate variations within each element, and assembling the element equations into a global system of equations to solve.

Embed presentation

Downloaded 93 times
Introduction to Finite Element Method Dr. Santosh Chavan Assistant Professor Department of Mechanical Engineering
Scope, Purpose, and Scale Small-scale, single part optimization Large-scale simulation, e.g. vehicle crash-worthiness These may both employ the same software, but with significantly different models!
Large-Scale Simulation, Full Vehicle Models Part of the CAE Domain Such analyses and simulations likely include multi-physics and media-structure interaction
Actual Geometry Derived Idealized Geometry Eng. Analysis. Model (FEM) Simplify Idealize De-Feature Pave Mesh Discretize Analysis Model Creation (FEA) A mechanical engineer, a structural engineer, and a piping engineer may each require different forms of geometry capture.
Geometry Representation Same Object ... Multiple/Different Forms of Geometry Capture 1D Line (Curv e) 3D Solid (Volume) 2D Surface (Shell)
Exploded View - Same Geometry, Different Data Capture 1D Line (Curve) 3D Solid (Volume) 2D Surface (Shell)
INTRODUCTION  The finite element method (FEM), sometimes referred to as finite element analysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering.  Simply stated, a boundary value problem is a mathematical problem in which one or more dependent variables must satisfy a differential equation everywhere within a known domain of independent variables and satisfy specific conditions on the boundary of the domain.  Boundary value problems are also sometimes called field problems. The field is the domain of interest and most often represents a physical structure.  The field variables are the dependent variables of interest governed by the differential equation. The boundary conditions are the specified values of the field variables (or related variables such as derivatives) on the boundaries of the field.  Depending on the type of physical problem being analyzed, the field variables may include physical displacement, temperature, heat flux, and fluid velocity to name only a few.
A GENERAL PROCEDURE FOR FINITE ELEMENT ANALYSIS • Certain steps in formulating a finite element analysis of a physical problem are common to all such analyses, whether structural, heat transfer, fluid flow, or some other problem. • These steps are embodied in commercial finite element software packages. The steps are described as follows. PREPROCESSING The preprocessing step is, quite generally, described as defining the model and includes  Define the geometric domain of the problem.  Define the element type(s).  Define the material properties of the elements.  Define the geometric properties of the elements (length, area, and the like).  Define the element connectivity's (mesh the model).  Define the physical constraints (boundary conditions).  Define the loadings.
SOLUTION / PROCESSING • During the solution phase, finite element software assembles the governing algebraic equations in matrix form and computes the unknown values of the primary field variable(s). • The computed values are then used by back substitution to compute additional, derived variables, such as reaction forces, element stresses, and heat flow. POSTPROCESSING Analysis and evaluation of the solution results is referred to as postprocessing. Check equilibrium. Calculate factors of safety. Plot deformed structural shape. Animate dynamic model behavior. Produce color-coded temperature plots. While solution data can be manipulated many ways in postprocessing, the most important objective is to apply sound engineering judgment in determining whether the solution results are physically reasonable.
EXAMPLES OF FINITE ELEMENT ANALYSIS
1. Mathematical Model Physical Problems Mathematical Model Solution Identify control variables Assumptions (empirical law) (1) Modeling (2) Types of solution Exact Eq. Approx. Eq. Exact Sol. ◎ ◎ Approx. Sol. ◎ ◎ Eq. Sol.
(3) Methods of Solution method
(3) Method of Solution A. Classical methods They offer a high degree of insight, but the problems are difficult or impossible to solve for anything but simple geometries and loadings. B. Numerical methods (I) Energy: Minimize an expression for the potential energy of the structure over the whole domain. (II) Boundary element: Approximates functions satisfying the governing differential equations not the boundary conditions. (III) Finite difference: Replaces governing differential equations and boundary conditions with algebraic finite difference equations. (IV) Finite element: Approximates the behavior of an irregular, continuous structure under general loadings and constraints with an assembly of discrete elements.
2. Finite Element Method (1) Definition FEM is a numerical method for solving a system of governing equations over the domain of a continuous physical system, which is discretized into simple geometric shapes called finite element. Continuous system Time-independent PDE Time-dependent PDE Discrete system Linear algebraic eq. ODE
(2) Discretization Modeling a body by dividing it into an equivalent system of finite elements interconnected at a finite number of points on each element called nodes. Continuous system Discretization system System Discretization Nodes Finite Elements
3. Historical Background
4. Analytical Processes of Finite Element Method (1) Structural stress analysis problem A. Conditions that solution must satisfy a. Equilibrium b. Compatibility c. Constitutive law d. Boundary conditions Above conditions are used to generate a system of equations representing system behavior. B. Approach a. Force (flexibility) method: internal forces as unknowns. b. Displacement (stiffness) method: nodal disp. As unknowns. For computational purpose, the displacement method is more desirable because its formulation is simple. A vast majority of general purpose FE software's have incorporated the displacement method for solving structural problems.
(2) Analysis procedures of linear static structural analysis A. Build up geometric model a. 1D problem line b. 2D problem surface c. 3D problem solid
B. Construct the finite element model a. Discretize and select the element types (a) element type 1D line element 2D element 3D brick element (b) total number of element (mesh) 1D: 2D: 3D:
b. Select a shape function 1D line element: u=ax+b c. Define the compatibility and constitutive law d. Form the element stiffness matrix and equations (a) Direct equilibrium method (b) Work or energy method (c) Method of weight Residuals e. Form the system equation Assemble the element equations to obtain global system equation and introduce boundary conditions
C. Solve the system equations a. elimination method Gauss’s method (Nastran) b. iteration method Gauss Seidel’s method Displacement field strain field stress field D. Interpret the results (postprocessing) a. deformation plot b. stress contour
5. Applications of Finite Element Method Structural Problem Non-structural Problem Stress Analysis - truss & frame analysis - stress concentrated problem Buckling problem Vibration Analysis Impact Problem Heat Transfer Fluid Mechanics Electric or Magnetic Potential
Basic Steps of Solving FEM 1. Derive an equilibrium equation from the potential energy equation in terms of material displacement. 2. Select the appropriate finite elements and corresponding interpolation functions. Subdivide the object into elements. 3. For each element, re-express the components of the equilibrium equation in terms of interpolation functions and the element’s node displacements. 4. Combine the set of equilibrium equations for all the elements into a single system and solve the system for the node displacements for the whole object. 5. Use the node displacements and the interpolation functions of a particular element to calculate displacements (or other quantities) for points within the element.
Steps in FEM 1. Discretize and Select Element Type 2. Select a Displacement Function 3. Define Strain/Displacement and Stress/Strain Relationships 4. Derive Element Stiffness Matrix & Eqs. 5. Assemble Equations and Introduce B.C’s 6. Solve for the Unknown Displacements 7. Solve for Element Stresses and Strains 8. Interpret the Results 24
Typical Application of FEM • Structural/Stress Analysis • Fluid Flow • Heat Transfer • Electro-Magnetic Fields • Soil Mechanics • Acoustics 25
Advantages • Irregular Boundaries • General Loads • Different Materials • Boundary Conditions • Variable Element Size • Easy Modification • Dynamics • Nonlinear Problems (Geometric or Material) 26
End of Chapter One
Definitions for this section For an element, a stiffness matrix is a matrix such that where relates local coordinates and nodal displacements to local forces of a single element. Bold denotes vector/matrices. k̂ f̂ = k̂ d̂, k̂ x̂,ŷ,ẑ ( ) d̂ f̂ 28
Spring Element k 1 2 L x̂ f̂1x̂ ,d̂1x̂ f̂2x̂ ,d̂2x̂ 29
Definitions f̂1x̂ local nodal force d̂1x̂ displacement node k - spring constant x̂ local coordinate direction f̂2x̂ local nodal force d̂2x̂ displacement node These are scalar values 30
Stiffness Relationship for a Spring ˆ f1x f̂2x ì í ï î ï ü ý ï þ ï = k11 k12 k21 k22 é ë ê ê ù û ú ú d̂1x d̂2x ì í ï î ï ü ý ï þ ï 31
Steps in Process 1) Discretize and Select Element Type 2) Select a Displacement Function 3) Define Strain/Displacement and Stress/Strain Relationships 4) Derive Element Stiffness Matrix & Eqs. 5) Assemble Equations and Introduce B.C.’s 6) Solve for the Unknowns (Displacements) 7) Solve for Element Stresses and Strains 8) Interpret the Results 32
Step 1 - Select the Element Type L T k 1 2 x̂ T d̂1x̂ d̂2x̂ 33
Step 2 - Select a Displacement Function Assume a displacement function Assume a linear function. Number of coefficients = number of d-o-f Write in matrix form. û û = a1 + a2x̂ û = 1 x̂ [ ] a1 a2 ì í î ü ý þ 34
û(0) = a1 + a2 (0) = d̂1x = a1 û(L) = a1 + a2 (L) = d̂2 = a2L + d̂1x û Express as function of and d̂1x d̂2x a2 = d̂2x - d̂1x L Solve for a2 : 35
û = a1 + a2x̂ Substituting back into: Yields: û = d̂2x - d̂1x L æ è ç ç ö ø ÷ ÷x̂ + d̂1x 36
In matrix form: û = 1- x̂ L x̂ L é ë ê ù û ú d̂1x d̂2x ì í ï î ï ü ý ï þ ï û = N1 N2 [ ] d̂1x d̂2x ì í ï î ï ü ý ï þ ï Where : N1 =1- x̂ L and N2 = x̂ L 37
Shape Functions N1 and N2 are called Shape Functions or Interpolation Functions. They express the shape of the assumed displacements. N1 =1 N2 =0 at node 1 N1 =0 N2 =1 at node 2 N1 + N2 =1 Recall Fourier Transform, in which the basis functions are sinusoidal functions. Here, the bases are linear functions! 38
1 2 N1 L 39
1 2 N2 L 40
2 1 N1 N2 L 41
Step 3 - Define Strain/Displacement and Stress/Strain Relationships T = kd d = û(L)- û(0) d = d̂2x - d̂1x T - tensile force  - total elongation Here is where physics comes into play! 42
Step 4 - Derive the Element Stiffness Matrix and Equations f̂1x = -T f̂2x = T T = - f̂1x = k d̂2x - d̂1x ( ) T = f̂2x = k d̂2x - d̂1x ( ) f̂1x = k d̂1x - d̂2x ( ) f̂2x = k d̂2x - d̂1x ( ) 43
ˆ f1x ˆ f2 x ì í ï î ï ü ý ï þ ï = k -k -k k é ë ê ù û ú d̂1x d̂2x ì í ï î ï ü ý ï þ ï k̂ é ë ù ûº k -k -k k é ë ê ù û ú Stiffness Matrix 44
Step 5 - Assemble the Element Equations to Obtain the Global Equations and Introduce the B.C. K [ ] = k̂(e) é ë ù û e=1 N å F { } = f̂(e) { } e=1 N å Note: not simple addition! An example later. (e) indicates “element” index 45
Step 6 - Solve for Nodal Displacements Solve : K [ ] d { } = F { } 46
Step 7 - Solve for Element Forces Once displacements at each node are known, then substitute back into element stiffness equations to obtain element nodal forces. 47
Two Spring Assembly k1 1 2 k2 1 2 3 x F3x F2x 48
For element 1: f̂1x f̂3x ì í ï î ï ü ý ï þ ï = k1 -k1 -k1 k1 é ë ê ù û ú d̂1x d̂3x ì í ï î ï ü ý ï þ ï For element 2 : ˆ f3x f̂2x ì í ï î ï ü ý ï þ ï = k2 -k2 -k2 k2 é ë ê ù û ú d̂3x d̂2x ì í ï î ï ü ý ï þ ï 49
Elements 1 and 2 remain connected at node 3. This is called the continuity or compatibility requirement. d3x (1) = d3x (2) = d3x Continuity/Compatibility Condition 50
Assemble Global force matrix F3x = f̂3x (1) + ˆ f3x (2) F2x = f̂2x (2) F1x = f̂1x (1) This is just adding. 51
F3x = -k1d1x + k1d3x + k2d3x - k2d2x F2x = -k2d3x + k2d2x F1x = k1d1x - k1d3x In the matrix form: F1x F2x F3x ì í ï î ï ü ý ï þ ï = k1 0 -k1 0 k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú d1x d2x d3x ì í ï î ï ü ý ï þ ï or F [ ]= K [ ] d { } Substitution will give you these equations 52
Global Force Matrix: Global Displacement Matrix: F1x F2x F3x ì í ï î ï ü ý ï þ ï d1x d2x d3x ì í ï î ï ü ý ï þ ï Global Stiffness Matrix: k1 0 -k1 0 k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú But, this way is really cumbersome! Computers can’t do this either! 53
Assembly of [K] - An Alternative Method k1 1 2 k2 1 2 3 x F3x F2x 54
For element 1: f̂1x f̂3x ì í ï î ï ü ý ï þ ï = k1 -k1 -k1 k1 é ë ê ù û ú d̂1x d̂3x ì í ï î ï ü ý ï þ ï For element 2 : ˆ f3x f̂2x ì í ï î ï ü ý ï þ ï = k2 -k2 -k2 k2 é ë ê ù û ú d̂3x d̂2x ì í ï î ï ü ý ï þ ï Recall that 55
Assembly of [K] - An Alternative Method node 1 3 [k(1) ] = k1 -k1 -k1 k1 é ë ê ê ù û ú ú node 3 2 [k(2) ] = k2 -k2 -k2 k2 é ë ê ê ù û ú ú Insert row and column 2 with zeros Flip row, flip columns, and insert row 1 with zeros 1 3 3 2 56
Expand Local [k] matrices to Global Size k1 1 0 -1 0 0 0 -1 0 1 é ë ê ê ê ù û ú ú ú d̂1x (1) d̂2x (1) d̂3x (1) ì í ï ï î ï ï ü ý ï ï þ ï ï = f1x (1) f2x (1) f3x (1) ì í ï î ï ü ý ï þ ï k2 0 0 0 0 1 -1 0 -1 1 é ë ê ê ê ù û ú ú ú d̂1x (2) d̂2x (2) d̂3x (2) ì í ï ï î ï ï ü ý ï ï þ ï ï = f1x (2) f2x (2) f3x (2) ì í ï î ï ü ý ï þ ï Computers can do this! 57
Net Force f1x (1) 0 f3x (1) ì í ï î ï ü ý ï þ ï + 0 f2x (2) f3x (2) ì í ï î ï ü ý ï þ ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï 58
k1 1 0 -1 0 0 0 -1 0 1 é ë ê ê ê ù û ú ú ú d̂1x (1) d̂2x (1) d̂3x (1) ì í ï ï î ï ï ü ý ï ï þ ï ï +k2 0 0 0 0 1 -1 0 -1 1 é ë ê ê ê ù û ú ú ú d̂1x (2) d̂2x (2) d̂3x (2) ì í ï ï î ï ï ü ý ï ï þ ï ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï 59
d̂1x (1) d̂2x (1) d̂3x (1) ì í ï ï î ï ï ü ý ï ï þ ï ï = d̂1x (2) d̂2x (2) d̂3x (2) ì í ï ï î ï ï ü ý ï ï þ ï ï = d1x d2x d3x ì í ï î ï ü ý ï þ ï Compatibility Displacements of the shared nodes are equal. 60
k1 0 -k1 0 k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú d1x d2x d3x ì í ï î ï ü ý ï þ ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï 61 Assembling the Components
Boundary Conditions • Must Specify B.C’s to prohibit rigid body motion. • Two type of B.C’s – Homogeneous - displacements = 0 – Nonhomogeneous - displacements = nonzero value 62
Homogeneous B.C’s • Delete row and column corresponding to B.C. • Solve for unknown displacements. • Compute unknown forces (reactions) from original (unmodified) stiffness matrix. 63
k1 1 2 k2 1 2 3 x F3x F2x 64
k1 0 -k1 0 k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú 0 d2x d3x ì í ï î ï ü ý ï þ ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï k2 -k2 -k2 k1 + k2 é ë ê ù û ú d2x d3x ì í î ü ý þ = F2x F3x ì í î ü ý þ F1x = -k1 d3x Example: Homogeneous BC, d1x=0 65
66 Reference books
Thank You

More Related Content

PPTX
FEM and it's applications
byChetan Mahatme
 
PPT
Introduction to FEA
bySakthivel Murugan
 
PPTX
Introduction of Finite Element Analysis
byMuthukumar V
 
PDF
Finite Element Method
byTahir Khan
 
PPT
Intro to fea software
bykubigs
 
PPT
Finite Element Analysis - UNIT-1
bypropaul
 
PPTX
Finite element analysis
bySonal Upadhyay
 
PPTX
Finite Element Method
byMuhammad Haris
 
FEM and it's applications
byChetan Mahatme
 
Introduction to FEA
bySakthivel Murugan
 
Introduction of Finite Element Analysis
byMuthukumar V
 
Finite Element Method
byTahir Khan
 
Intro to fea software
bykubigs
 
Finite Element Analysis - UNIT-1
bypropaul
 
Finite element analysis
bySonal Upadhyay
 
Finite Element Method
byMuhammad Haris
 

What's hot

PPTX
Finite Element Methods
byDr.Vikas Deulgaonkar
 
PDF
Introduction to finite element analysis
byTarun Gehlot
 
PDF
Introduction to Finite Element Method
bySyedAbdullaZain
 
PPT
An Introduction to the Finite Element Method
byMohammad Tawfik
 
PPTX
Finite Element Analysis of Truss Structures
byMahdi Damghani
 
PPTX
FEA good practices presentation
byMahdi Damghani
 
PPT
Finite Element Analysis - UNIT-3
bypropaul
 
PPTX
Lec9 finite element_beam_structures 1
byMahdi Damghani
 
PPT
Introduction to finite element method(fem)
bySreekanth G
 
PDF
Finite Element Methode (FEM) Notes
byZulkifli Yunus
 
PDF
Me2353 finite-element-analysis-lecture-notes
byAmit Ghongade
 
PPT
Finite Element Analysis - UNIT-4
bypropaul
 
PPT
Introduction to FEA and applications
byPadmanabhan Krishnan
 
PDF
5. stress function
byYASWANTH BHAIRAVABHOTLA
 
PPSX
Non linear analysis
byYuva Raj
 
PDF
Application of Boundary Conditions to Obtain Better FEA Results
byKee H. Lee, P.Eng.
 
PDF
FEM: Introduction and Weighted Residual Methods
byMohammad Tawfik
 
PPTX
Introduction to FEA
byKishan Sharma
 
PPTX
Finite element method
byMevada Maulik
 
PPTX
INTRODUCTION TO FINITE ELEMENT ANALYSIS
byAchyuth Peri
 
Finite Element Methods
byDr.Vikas Deulgaonkar
 
Introduction to finite element analysis
byTarun Gehlot
 
Introduction to Finite Element Method
bySyedAbdullaZain
 
An Introduction to the Finite Element Method
byMohammad Tawfik
 
Finite Element Analysis of Truss Structures
byMahdi Damghani
 
FEA good practices presentation
byMahdi Damghani
 
Finite Element Analysis - UNIT-3
bypropaul
 
Lec9 finite element_beam_structures 1
byMahdi Damghani
 
Introduction to finite element method(fem)
bySreekanth G
 
Finite Element Methode (FEM) Notes
byZulkifli Yunus
 
Me2353 finite-element-analysis-lecture-notes
byAmit Ghongade
 
Finite Element Analysis - UNIT-4
bypropaul
 
Introduction to FEA and applications
byPadmanabhan Krishnan
 
5. stress function
byYASWANTH BHAIRAVABHOTLA
 
Non linear analysis
byYuva Raj
 
Application of Boundary Conditions to Obtain Better FEA Results
byKee H. Lee, P.Eng.
 
FEM: Introduction and Weighted Residual Methods
byMohammad Tawfik
 
Introduction to FEA
byKishan Sharma
 
Finite element method
byMevada Maulik
 
INTRODUCTION TO FINITE ELEMENT ANALYSIS
byAchyuth Peri
 

Similar to Finite element method

PPTX
FEM.pptx on enginiiering design and coading
bypandeyjii2424
 
PDF
IRJET- Stress – Strain Field Analysis of Mild Steel Component using Finite El...
byIRJET Journal
 
PPTX
Introduction of finite element analysis1
byssuser2209b4
 
PDF
Weighted residual.pdf
byvenkateshanthirugnan
 
PPTX
FEA: Analytical Methods.pptx
byDrDineshDhande
 
PPTX
chapter 1- introduction to finite element method
byAberaAbebe14
 
PPT
Introduction to Finite Element Methods - basics
byVishnuVardhan909561
 
PDF
fem-ppt-introduction-sachin-new.pdf
byJAAgent
 
PPTX
Finite Element Method.pptx
byHassanShah396906
 
PPTX
Unit-I--FEA
byMuthukumar V
 
PDF
Lecture on Introduction to finite element methods & its contents
byMesayAlemuTolcha1
 
PPTX
Fea unit 1
bys Kumaravel
 
PPTX
Fea unit 1
byDevalakshmanperumal1991
 
PPTX
Introduction to finite element analysis (FEA)
byDEEPANDURAIK1
 
PDF
Introduction_to_exament_Methods.pdf
byMustafaELALAMI
 
PPTX
tuttuturtCH 1 and 2 FEM ppt finite el.pptx
byozelalemshiferaw
 
PDF
Lecture 01 - Introduction to Finite EM.pdf
byNorbertsanJeba
 
PPT
FEM_PPT.ppt
byFlimMovie
 
PPT
INTRODUCTION TO FINITE ELEMENT METHODS PPT
byVishnuVardhan909561
 
PDF
282165203-Module-1-Introduction-to-FEM.pdf
bySilviViv
 
FEM.pptx on enginiiering design and coading
bypandeyjii2424
 
IRJET- Stress – Strain Field Analysis of Mild Steel Component using Finite El...
byIRJET Journal
 
Introduction of finite element analysis1
byssuser2209b4
 
Weighted residual.pdf
byvenkateshanthirugnan
 
FEA: Analytical Methods.pptx
byDrDineshDhande
 
chapter 1- introduction to finite element method
byAberaAbebe14
 
Introduction to Finite Element Methods - basics
byVishnuVardhan909561
 
fem-ppt-introduction-sachin-new.pdf
byJAAgent
 
Finite Element Method.pptx
byHassanShah396906
 
Unit-I--FEA
byMuthukumar V
 
Lecture on Introduction to finite element methods & its contents
byMesayAlemuTolcha1
 
Fea unit 1
bys Kumaravel
 
Fea unit 1
byDevalakshmanperumal1991
 
Introduction to finite element analysis (FEA)
byDEEPANDURAIK1
 
Introduction_to_exament_Methods.pdf
byMustafaELALAMI
 
tuttuturtCH 1 and 2 FEM ppt finite el.pptx
byozelalemshiferaw
 
Lecture 01 - Introduction to Finite EM.pdf
byNorbertsanJeba
 
FEM_PPT.ppt
byFlimMovie
 
INTRODUCTION TO FINITE ELEMENT METHODS PPT
byVishnuVardhan909561
 
282165203-Module-1-Introduction-to-FEM.pdf
bySilviViv
 

Recently uploaded

PDF
Software Engineering : Process Model
byGunjalSanjay
 
PPTX
Module 1- foundation of data science IT25201.pptx
byRSUMATHY1
 
PPTX
FROM MEASUREMENT TO MEANING : Metrology-Driven Battery Testing for Reliable ...
byRunjhunDutta
 
PDF
Software Engineering :Software Engineering Practice
byGunjalSanjay
 
PDF
Software Engineering Perspective Process Models
byGunjalSanjay
 
PDF
Introduction to Software , Product and Process
byGunjalSanjay
 
PDF
LS029B3SX06A 2.9 Inch TFT LCD 2K Resolution 2160X2160 For AR VR
bySyluxdisplay
 
PDF
PROBLEM SLOVING AND PYTHON PROGRAMMING UNIT 2.pdf
byA R SIVANESH M.E., QIP-PG (Cyber Security)., (Ph.D)
 
PPTX
TechSprint WinterHack || After an intense hackathon filled with creativity, ...
bybajpaitusharoffon678
 
PPTX
Industry 4.0- fourth industrial revolution Presentation.pptx
byrameswari15491
 
PDF
Software Engineering : Nature of Software
byGunjalSanjay
 
PDF
FPGA Fabric and Synthesis All Parts Combined
byAmr Adel
 
PPTX
Energy system modelling to support national energy and climate policy making ...
byIEA-ETSAP
 
PPTX
Tips for Designing Flex Circuits for Medical Applications
byEpec Engineered Technologies
 
PDF
Reality Check Deploying Computer Vision and LLMs at the Edge
byICS
 
PPTX
An Early Production Facility (EPF) is a modular, temporary system for rapidly...
byDrAdelSalem1
 
PDF
TRUSTWORTHY AI: BUILDING SAFE AND ETHICAL MACHINE LEARNING SYSTEMS
byAdri Jovin
 
PDF
RVV Basic Terms and Analogy - (en/zhTW) - DannyJiang
byDanny Jiang
 
PPTX
1.-Static-sensor-characteristics,sensors and actuators ;SDU,denmark.pptx
byMohammadTabish20
 
PPTX
The Half-Life of Preventive Maintenance: Why PM Compliance Fails & How to Res...
byMaintWiz Technologies Private Limited
 
Software Engineering : Process Model
byGunjalSanjay
 
Module 1- foundation of data science IT25201.pptx
byRSUMATHY1
 
FROM MEASUREMENT TO MEANING : Metrology-Driven Battery Testing for Reliable ...
byRunjhunDutta
 
Software Engineering :Software Engineering Practice
byGunjalSanjay
 
Software Engineering Perspective Process Models
byGunjalSanjay
 
Introduction to Software , Product and Process
byGunjalSanjay
 
LS029B3SX06A 2.9 Inch TFT LCD 2K Resolution 2160X2160 For AR VR
bySyluxdisplay
 
PROBLEM SLOVING AND PYTHON PROGRAMMING UNIT 2.pdf
byA R SIVANESH M.E., QIP-PG (Cyber Security)., (Ph.D)
 
TechSprint WinterHack || After an intense hackathon filled with creativity, ...
bybajpaitusharoffon678
 
Industry 4.0- fourth industrial revolution Presentation.pptx
byrameswari15491
 
Software Engineering : Nature of Software
byGunjalSanjay
 
FPGA Fabric and Synthesis All Parts Combined
byAmr Adel
 
Energy system modelling to support national energy and climate policy making ...
byIEA-ETSAP
 
Tips for Designing Flex Circuits for Medical Applications
byEpec Engineered Technologies
 
Reality Check Deploying Computer Vision and LLMs at the Edge
byICS
 
An Early Production Facility (EPF) is a modular, temporary system for rapidly...
byDrAdelSalem1
 
TRUSTWORTHY AI: BUILDING SAFE AND ETHICAL MACHINE LEARNING SYSTEMS
byAdri Jovin
 
RVV Basic Terms and Analogy - (en/zhTW) - DannyJiang
byDanny Jiang
 
1.-Static-sensor-characteristics,sensors and actuators ;SDU,denmark.pptx
byMohammadTabish20
 
The Half-Life of Preventive Maintenance: Why PM Compliance Fails & How to Res...
byMaintWiz Technologies Private Limited
 

Finite element method

  • 1.
    Introduction to Finite ElementMethod Dr. Santosh Chavan Assistant Professor Department of Mechanical Engineering
  • 2.
    Scope, Purpose, andScale Small-scale, single part optimization Large-scale simulation, e.g. vehicle crash-worthiness These may both employ the same software, but with significantly different models!
  • 3.
    Large-Scale Simulation, FullVehicle Models Part of the CAE Domain Such analyses and simulations likely include multi-physics and media-structure interaction
  • 4.
    Actual Geometry Derived Idealized Geometry Eng. Analysis. Model(FEM) Simplify Idealize De-Feature Pave Mesh Discretize Analysis Model Creation (FEA) A mechanical engineer, a structural engineer, and a piping engineer may each require different forms of geometry capture.
  • 5.
    Geometry Representation Same Object... Multiple/Different Forms of Geometry Capture 1D Line (Curv e) 3D Solid (Volume) 2D Surface (Shell)
  • 6.
    Exploded View -Same Geometry, Different Data Capture 1D Line (Curve) 3D Solid (Volume) 2D Surface (Shell)
  • 7.
    INTRODUCTION  The finiteelement method (FEM), sometimes referred to as finite element analysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering.  Simply stated, a boundary value problem is a mathematical problem in which one or more dependent variables must satisfy a differential equation everywhere within a known domain of independent variables and satisfy specific conditions on the boundary of the domain.  Boundary value problems are also sometimes called field problems. The field is the domain of interest and most often represents a physical structure.  The field variables are the dependent variables of interest governed by the differential equation. The boundary conditions are the specified values of the field variables (or related variables such as derivatives) on the boundaries of the field.  Depending on the type of physical problem being analyzed, the field variables may include physical displacement, temperature, heat flux, and fluid velocity to name only a few.
  • 8.
    A GENERAL PROCEDUREFOR FINITE ELEMENT ANALYSIS • Certain steps in formulating a finite element analysis of a physical problem are common to all such analyses, whether structural, heat transfer, fluid flow, or some other problem. • These steps are embodied in commercial finite element software packages. The steps are described as follows. PREPROCESSING The preprocessing step is, quite generally, described as defining the model and includes  Define the geometric domain of the problem.  Define the element type(s).  Define the material properties of the elements.  Define the geometric properties of the elements (length, area, and the like).  Define the element connectivity's (mesh the model).  Define the physical constraints (boundary conditions).  Define the loadings.
  • 9.
    SOLUTION / PROCESSING •During the solution phase, finite element software assembles the governing algebraic equations in matrix form and computes the unknown values of the primary field variable(s). • The computed values are then used by back substitution to compute additional, derived variables, such as reaction forces, element stresses, and heat flow. POSTPROCESSING Analysis and evaluation of the solution results is referred to as postprocessing. Check equilibrium. Calculate factors of safety. Plot deformed structural shape. Animate dynamic model behavior. Produce color-coded temperature plots. While solution data can be manipulated many ways in postprocessing, the most important objective is to apply sound engineering judgment in determining whether the solution results are physically reasonable.
  • 10.
    EXAMPLES OF FINITEELEMENT ANALYSIS
  • 11.
    1. Mathematical Model Physical Problems Mathematical Model Solution Identifycontrol variables Assumptions (empirical law) (1) Modeling (2) Types of solution Exact Eq. Approx. Eq. Exact Sol. ◎ ◎ Approx. Sol. ◎ ◎ Eq. Sol.
  • 12.
    (3) Methods ofSolution method
  • 13.
    (3) Method ofSolution A. Classical methods They offer a high degree of insight, but the problems are difficult or impossible to solve for anything but simple geometries and loadings. B. Numerical methods (I) Energy: Minimize an expression for the potential energy of the structure over the whole domain. (II) Boundary element: Approximates functions satisfying the governing differential equations not the boundary conditions. (III) Finite difference: Replaces governing differential equations and boundary conditions with algebraic finite difference equations. (IV) Finite element: Approximates the behavior of an irregular, continuous structure under general loadings and constraints with an assembly of discrete elements.
  • 14.
    2. Finite ElementMethod (1) Definition FEM is a numerical method for solving a system of governing equations over the domain of a continuous physical system, which is discretized into simple geometric shapes called finite element. Continuous system Time-independent PDE Time-dependent PDE Discrete system Linear algebraic eq. ODE
  • 15.
    (2) Discretization Modeling abody by dividing it into an equivalent system of finite elements interconnected at a finite number of points on each element called nodes. Continuous system Discretization system System Discretization Nodes Finite Elements
  • 16.
  • 17.
    4. Analytical Processesof Finite Element Method (1) Structural stress analysis problem A. Conditions that solution must satisfy a. Equilibrium b. Compatibility c. Constitutive law d. Boundary conditions Above conditions are used to generate a system of equations representing system behavior. B. Approach a. Force (flexibility) method: internal forces as unknowns. b. Displacement (stiffness) method: nodal disp. As unknowns. For computational purpose, the displacement method is more desirable because its formulation is simple. A vast majority of general purpose FE software's have incorporated the displacement method for solving structural problems.
  • 18.
    (2) Analysis proceduresof linear static structural analysis A. Build up geometric model a. 1D problem line b. 2D problem surface c. 3D problem solid
  • 19.
    B. Construct thefinite element model a. Discretize and select the element types (a) element type 1D line element 2D element 3D brick element (b) total number of element (mesh) 1D: 2D: 3D:
  • 20.
    b. Select ashape function 1D line element: u=ax+b c. Define the compatibility and constitutive law d. Form the element stiffness matrix and equations (a) Direct equilibrium method (b) Work or energy method (c) Method of weight Residuals e. Form the system equation Assemble the element equations to obtain global system equation and introduce boundary conditions
  • 21.
    C. Solve thesystem equations a. elimination method Gauss’s method (Nastran) b. iteration method Gauss Seidel’s method Displacement field strain field stress field D. Interpret the results (postprocessing) a. deformation plot b. stress contour
  • 22.
    5. Applications ofFinite Element Method Structural Problem Non-structural Problem Stress Analysis - truss & frame analysis - stress concentrated problem Buckling problem Vibration Analysis Impact Problem Heat Transfer Fluid Mechanics Electric or Magnetic Potential
  • 23.
    Basic Steps ofSolving FEM 1. Derive an equilibrium equation from the potential energy equation in terms of material displacement. 2. Select the appropriate finite elements and corresponding interpolation functions. Subdivide the object into elements. 3. For each element, re-express the components of the equilibrium equation in terms of interpolation functions and the element’s node displacements. 4. Combine the set of equilibrium equations for all the elements into a single system and solve the system for the node displacements for the whole object. 5. Use the node displacements and the interpolation functions of a particular element to calculate displacements (or other quantities) for points within the element.
  • 24.
    Steps in FEM 1.Discretize and Select Element Type 2. Select a Displacement Function 3. Define Strain/Displacement and Stress/Strain Relationships 4. Derive Element Stiffness Matrix & Eqs. 5. Assemble Equations and Introduce B.C’s 6. Solve for the Unknown Displacements 7. Solve for Element Stresses and Strains 8. Interpret the Results 24
  • 25.
    Typical Application ofFEM • Structural/Stress Analysis • Fluid Flow • Heat Transfer • Electro-Magnetic Fields • Soil Mechanics • Acoustics 25
  • 26.
    Advantages • Irregular Boundaries •General Loads • Different Materials • Boundary Conditions • Variable Element Size • Easy Modification • Dynamics • Nonlinear Problems (Geometric or Material) 26
  • 27.
  • 28.
    Definitions for thissection For an element, a stiffness matrix is a matrix such that where relates local coordinates and nodal displacements to local forces of a single element. Bold denotes vector/matrices. k̂ f̂ = k̂ d̂, k̂ x̂,ŷ,ẑ ( ) d̂ f̂ 28
  • 29.
    Spring Element k 1 2 L x̂ f̂1x̂,d̂1x̂ f̂2x̂ ,d̂2x̂ 29
  • 30.
    Definitions f̂1x̂ local nodalforce d̂1x̂ displacement node k - spring constant x̂ local coordinate direction f̂2x̂ local nodal force d̂2x̂ displacement node These are scalar values 30
  • 31.
    Stiffness Relationship fora Spring ˆ f1x f̂2x ì í ï î ï ü ý ï þ ï = k11 k12 k21 k22 é ë ê ê ù û ú ú d̂1x d̂2x ì í ï î ï ü ý ï þ ï 31
  • 32.
    Steps in Process 1)Discretize and Select Element Type 2) Select a Displacement Function 3) Define Strain/Displacement and Stress/Strain Relationships 4) Derive Element Stiffness Matrix & Eqs. 5) Assemble Equations and Introduce B.C.’s 6) Solve for the Unknowns (Displacements) 7) Solve for Element Stresses and Strains 8) Interpret the Results 32
  • 33.
    Step 1 -Select the Element Type L T k 1 2 x̂ T d̂1x̂ d̂2x̂ 33
  • 34.
    Step 2 -Select a Displacement Function Assume a displacement function Assume a linear function. Number of coefficients = number of d-o-f Write in matrix form. û û = a1 + a2x̂ û = 1 x̂ [ ] a1 a2 ì í î ü ý þ 34
  • 35.
    û(0) = a1+ a2 (0) = d̂1x = a1 û(L) = a1 + a2 (L) = d̂2 = a2L + d̂1x û Express as function of and d̂1x d̂2x a2 = d̂2x - d̂1x L Solve for a2 : 35
  • 36.
    û = a1+ a2x̂ Substituting back into: Yields: û = d̂2x - d̂1x L æ è ç ç ö ø ÷ ÷x̂ + d̂1x 36
  • 37.
    In matrix form: û= 1- x̂ L x̂ L é ë ê ù û ú d̂1x d̂2x ì í ï î ï ü ý ï þ ï û = N1 N2 [ ] d̂1x d̂2x ì í ï î ï ü ý ï þ ï Where : N1 =1- x̂ L and N2 = x̂ L 37
  • 38.
    Shape Functions N1 andN2 are called Shape Functions or Interpolation Functions. They express the shape of the assumed displacements. N1 =1 N2 =0 at node 1 N1 =0 N2 =1 at node 2 N1 + N2 =1 Recall Fourier Transform, in which the basis functions are sinusoidal functions. Here, the bases are linear functions! 38
  • 39.
  • 40.
  • 41.
  • 42.
    Step 3 -Define Strain/Displacement and Stress/Strain Relationships T = kd d = û(L)- û(0) d = d̂2x - d̂1x T - tensile force  - total elongation Here is where physics comes into play! 42
  • 43.
    Step 4 -Derive the Element Stiffness Matrix and Equations f̂1x = -T f̂2x = T T = - f̂1x = k d̂2x - d̂1x ( ) T = f̂2x = k d̂2x - d̂1x ( ) f̂1x = k d̂1x - d̂2x ( ) f̂2x = k d̂2x - d̂1x ( ) 43
  • 44.
    ˆ f1x ˆ f2 x ì í ï î ï ü ý ï þ ï = k -k -kk é ë ê ù û ú d̂1x d̂2x ì í ï î ï ü ý ï þ ï k̂ é ë ù ûº k -k -k k é ë ê ù û ú Stiffness Matrix 44
  • 45.
    Step 5 -Assemble the Element Equations to Obtain the Global Equations and Introduce the B.C. K [ ] = k̂(e) é ë ù û e=1 N å F { } = f̂(e) { } e=1 N å Note: not simple addition! An example later. (e) indicates “element” index 45
  • 46.
    Step 6 -Solve for Nodal Displacements Solve : K [ ] d { } = F { } 46
  • 47.
    Step 7 -Solve for Element Forces Once displacements at each node are known, then substitute back into element stiffness equations to obtain element nodal forces. 47
  • 48.
    Two Spring Assembly k1 12 k2 1 2 3 x F3x F2x 48
  • 49.
    For element 1: f̂1x f̂3x ì í ï î ï ü ý ï þ ï = k1-k1 -k1 k1 é ë ê ù û ú d̂1x d̂3x ì í ï î ï ü ý ï þ ï For element 2 : ˆ f3x f̂2x ì í ï î ï ü ý ï þ ï = k2 -k2 -k2 k2 é ë ê ù û ú d̂3x d̂2x ì í ï î ï ü ý ï þ ï 49
  • 50.
    Elements 1 and2 remain connected at node 3. This is called the continuity or compatibility requirement. d3x (1) = d3x (2) = d3x Continuity/Compatibility Condition 50
  • 51.
    Assemble Global forcematrix F3x = f̂3x (1) + ˆ f3x (2) F2x = f̂2x (2) F1x = f̂1x (1) This is just adding. 51
  • 52.
    F3x = -k1d1x+ k1d3x + k2d3x - k2d2x F2x = -k2d3x + k2d2x F1x = k1d1x - k1d3x In the matrix form: F1x F2x F3x ì í ï î ï ü ý ï þ ï = k1 0 -k1 0 k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú d1x d2x d3x ì í ï î ï ü ý ï þ ï or F [ ]= K [ ] d { } Substitution will give you these equations 52
  • 53.
    Global Force Matrix:Global Displacement Matrix: F1x F2x F3x ì í ï î ï ü ý ï þ ï d1x d2x d3x ì í ï î ï ü ý ï þ ï Global Stiffness Matrix: k1 0 -k1 0 k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú But, this way is really cumbersome! Computers can’t do this either! 53
  • 54.
    Assembly of [K]- An Alternative Method k1 1 2 k2 1 2 3 x F3x F2x 54
  • 55.
    For element 1: f̂1x f̂3x ì í ï î ï ü ý ï þ ï = k1-k1 -k1 k1 é ë ê ù û ú d̂1x d̂3x ì í ï î ï ü ý ï þ ï For element 2 : ˆ f3x f̂2x ì í ï î ï ü ý ï þ ï = k2 -k2 -k2 k2 é ë ê ù û ú d̂3x d̂2x ì í ï î ï ü ý ï þ ï Recall that 55
  • 56.
    Assembly of [K]- An Alternative Method node 1 3 [k(1) ] = k1 -k1 -k1 k1 é ë ê ê ù û ú ú node 3 2 [k(2) ] = k2 -k2 -k2 k2 é ë ê ê ù û ú ú Insert row and column 2 with zeros Flip row, flip columns, and insert row 1 with zeros 1 3 3 2 56
  • 57.
    Expand Local [k]matrices to Global Size k1 1 0 -1 0 0 0 -1 0 1 é ë ê ê ê ù û ú ú ú d̂1x (1) d̂2x (1) d̂3x (1) ì í ï ï î ï ï ü ý ï ï þ ï ï = f1x (1) f2x (1) f3x (1) ì í ï î ï ü ý ï þ ï k2 0 0 0 0 1 -1 0 -1 1 é ë ê ê ê ù û ú ú ú d̂1x (2) d̂2x (2) d̂3x (2) ì í ï ï î ï ï ü ý ï ï þ ï ï = f1x (2) f2x (2) f3x (2) ì í ï î ï ü ý ï þ ï Computers can do this! 57
  • 58.
  • 59.
    k1 1 0 -1 00 0 -1 0 1 é ë ê ê ê ù û ú ú ú d̂1x (1) d̂2x (1) d̂3x (1) ì í ï ï î ï ï ü ý ï ï þ ï ï +k2 0 0 0 0 1 -1 0 -1 1 é ë ê ê ê ù û ú ú ú d̂1x (2) d̂2x (2) d̂3x (2) ì í ï ï î ï ï ü ý ï ï þ ï ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï 59
  • 60.
  • 61.
    k1 0 -k1 0k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú d1x d2x d3x ì í ï î ï ü ý ï þ ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï 61 Assembling the Components
  • 62.
    Boundary Conditions • MustSpecify B.C’s to prohibit rigid body motion. • Two type of B.C’s – Homogeneous - displacements = 0 – Nonhomogeneous - displacements = nonzero value 62
  • 63.
    Homogeneous B.C’s • Deleterow and column corresponding to B.C. • Solve for unknown displacements. • Compute unknown forces (reactions) from original (unmodified) stiffness matrix. 63
  • 64.
  • 65.
    k1 0 -k1 0k2 -k2 -k1 -k2 k1 + k2 é ë ê ê ê ù û ú ú ú 0 d2x d3x ì í ï î ï ü ý ï þ ï = F1x F2x F3x ì í ï î ï ü ý ï þ ï k2 -k2 -k2 k1 + k2 é ë ê ù û ú d2x d3x ì í î ü ý þ = F2x F3x ì í î ü ý þ F1x = -k1 d3x Example: Homogeneous BC, d1x=0 65
  • 66.
  • 67.