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CE 620 CE 620 FINITE ELEMENT METHOD Yogesh M. Desai Department - - PowerPoint PPT Presentation
CE 620 CE 620 FINITE ELEMENT METHOD Yogesh M. Desai Department - - PowerPoint PPT Presentation
CE 620 CE 620 FINITE ELEMENT METHOD Yogesh M. Desai Department of Civil Engineering Indian Institute of Technology Bombay Indian Institute of Technology Bombay Powai, Mumbai - 400076 Instructor Yogesh M. Desai Office Room No. 126 Civil
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Introduction
- Many
problems in engineering and applied science are governed by differential
- r integral equations.
- Due
to complexities in geometry, ti d b d diti i properties and boundary conditions in most real-world problems, an exact solution cannot be obtained. solution cannot be obtained.
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Introduction Introduction
Finite element method is an approximate numerical method for solving problems of engineering and mathematical sciences. Useful for problems with complicated geometries, external influences and g properties for which analytical solutions are not available.
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OBJECTIVES / LEARNING OUTCOMES OBJECTIVES / LEARNING OUTCOMES
- Understanding of different semi-analytical /
numerical methods to solve a variety of problems
- Understanding of general steps of FEM
- Understanding of finite element formulations
- Ability to derive equations related to FEA of various1 D
- Ability to derive equations related to FEA of various1-D ,
2-D and 3-D problems
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OBJECTIVES / LEARNING OUTCOMES OBJECTIVES / LEARNING OUTCOMES
- Understanding of advantages and disadvantages of the
FEM
- Exposure to computer implementation of the FEM
Exposure to computer implementation of the FEM
- Ability to do FE analysis independently with proper
interpretation of results interpretation of results
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Course Contents
Introduction Introduction Overview of various methods to solve integral and differential equations 5 – 8 Lectures Variational Calculus Basics of Finite Element Methods Local and Global Finite Element Methods Application of FEM to solve various 1-D, 2-D and 3-D Problems 12 – 16 Lectures 2 D and 3 D Problems
- C0 Continnum
- C1 Continnum
- C1 Continnum
- Convergence and Error Estimation
- Iso-parametric formulation
- Numerical integration
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Course Contents
Concept of Sub-structuring Conditions of Symmetry / Anti-symmetry 3 – 5 Lectures Computer Implementation of FEM Application of FEM to Time Dependent Problems 4 – 6 Lectures Problems Partial FEM Exposure to Hybrid FEM Total Lectures ~ 28 28 (~ 42 Hrs)
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Assessment Scheme
Assignments and Term Projects :20 % Mid - Term Exam :30 % (as per time table) End - Term Exam :50 % (as per time table)
Notes: (1) Bring calculator to all the lecture sessions. ( ) g (2) 80% Attendance is required.
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Brief History
It is difficult to document the exact origin of the FEM, because the basic concepts have evolved i d f 150
- ver a period of 150 or more years.
Hrennikoff [1941] – Framework method for Hrennikoff [1941] Framework method for elasticity problems Courant [1943] - Variational form L [1947 1953] Fl ibilit d Stiff Levy [1947, 1953] - Flexibility and Stiffness Argyris [1955] - Energy Theorems and Structural Analysis Analysis Turner, Clough, Martin and Topp [1956]
- Stiffness Method
Cl h [1960] T d “Fi it El t ” Clough [1960] - Termed “Finite Elements”
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Brief History
In early 1960s, engineers used the method
Brief History
In early 1960s, engineers used the method for approximate solution of problems in stress analysis, fluid flow, heat transfer, and other areas. Th fi t b k th FEM b Zi ki i d The first book on the FEM by Zienkiewicz and Chung was published in 1967.
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How can FEM Help ?
- Can be applied to a variety of fields like
structural mechanics, aerospace engineering, structural mechanics, aerospace engineering, geotechnical engineering, fluid mechanics, hydraulic and water resource engineering, mechanical engineering, nuclear engineering, electrical and electronics engineering, metall rgical chemical and en ironmental metallurgical, chemical and environmental engineering, meteorology and bioengineering, etc etc.
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- Easily applied to complex, irregular-shaped
Easily applied to complex, irregular shaped
- bjects composed of several different
properties and having complex boundary conditions and external influences. A li bl t t d t t ( t ti ) ti
- Applicable to steady-state (static), time
dependent as well as characteristic value problems problems.
- Applicable to linear as well as nonlinear
pp problems.
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Fi it El t M th d i A i t N i l Finite Element Method is an Approximate Numerical Method to Solve Problems of Engineering and Mathematical Sciences. Any given problem reduces to
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CE 620 : FINITE ELEMENT METHOD Assignment No. 1 (Due on January 15, 2013)
- Q. 1 Matrices [K] 33, [T]35, and {q}51 are defined as
[K] = [K]T = 10 5 4 5 10 2 4 2 10 ; [ T] = 3 2 1 1 0.5 0.5 1 4 1 0.2 3 2 {q} = { 3 0 -2 1 -4}T Compute matrix [M1] to [M6] appearing in the following expression (a) [M1] = [K]-1 (b) [M2] = 3[K]-1 +4[K] (c) [M3] = [K] [T] (d) [M4] =5[K]-1 [T] {q} (e) [M5] = [T]T [K] [T] (f) [M6] = 2 1 {q}T [T]T [K][T]{q}
- Q. 2 Given
2 2 3
3 . Find [ ] and [ ] [ ] . 2 d a t t a b c a dt dt t
- Q. 3 Given
1 [ ] 2
T T
q K q q f where
11 12 13 1 1 12 22 23 2 2 13 23 33 3 3
[ ] ; ; k k k q f K k k k q q f f k k k q f Derive equations arising from
1 2 3
; ; q q q
- Q. 4 Matrix equation [K] {q} = {f} is given where
[ K] = 6 7 9 2 13 7 18 6 10 6 9 6 42 15 11 2 10 15 8 6 13 6 11 6 24 {q} = [ q 1 q 2 q 3 q 4 q 5 ] T ; {f} = [100 70 -50 150 -35 ] T Compute {q}51 by employing Gauss elimination algorithm.