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Lecture 1 test

  1. 1. Swiss Federal Institute of Technology The Finite Element Method for the Analysis of Linear Systems Prof. Dr. Michael Havbro Faber Swiss Federal Institute of Technology ETH Zurich, Switzerland Method of Finite Elements I Page 1
  2. 2. Swiss Federal Institute of Technology Contents of Today's Lecture • Motivation, overview and organization of the course • Introduction to the use of finite element - Physical problem, mathematical modeling and finite element solutions - Finite elements as a tool for computer supported design and assessment • Basic mathematical tools Method of Finite Elements I Page 2
  3. 3. Swiss Federal Institute of Technology Page 3 Motivation, overview and organization of the course • Motivation In this course we are focusing on the assessment of the response of engineering structures Method of Finite Elements I
  4. 4. Swiss Federal Institute of Technology Page 4 Motivation, overview and organization of the course • Motivation In this course we are focusing on the assessment of the response of engineering structures Method of Finite Elements I
  5. 5. Swiss Federal Institute of Technology Page 5 Motivation, overview and organization of the course • Motivation What we would like to establish is the response of a structure subject to “loading”. The Method of Finite Elements provides a framework for the analysis of such responses – however for very general problems. The Method of Finite Elements provides a very general approach to the approximate solutions of differential equations. In the present course we consider a special class of problems, namely: Linear quasi-static systems, no material or geometrical or boundary condition non-linearities and also no inertia effect! Method of Finite Elements I
  6. 6. Swiss Federal Institute of Technology Motivation, overview and organization of the course • Organisation The lectures will be given by: M. H. Faber Exercises will be organized/attended by: J. Qin By appointment, HIL E13.1 Method of Finite Elements I Page 6
  7. 7. Swiss Federal Institute of Technology Page 7 Motivation, overview and organization of the course • Organisation PowerPoint files with the presentations will be uploaded on our homepage one day in advance of the lectures http://www.ibk.ethz.ch/fa/education/ss_FE The lecture as such will follow the book: "Finite Element Procedures" by K.J. Bathe, Prentice Hall, 1996 Method of Finite Elements I
  8. 8. Swiss Federal Institute of Technology Motivation, overview and organization of the course • Overview Method of Finite Elements I Page 8
  9. 9. Swiss Federal Institute of Technology Motivation, overview and organization of the course • Overview Method of Finite Elements I Page 9
  10. 10. Page 10 Swiss Federal Institute of Technology Introduction to the use of finite element - we are only working on the basis of mathematic models! - choice of mathematical model is crucial! - mathematical models must be reliable and effective Improve mathematical model Physical problem, mathematical modeling and finite element solutions Change physical problem • Physical problem Mathematical model governed by differential equations and assumptions on -geometry -kinematics -material laws -loading -boundary conditions -etc. Finite element solution Choice of -finite elements -mesh density -solution parameters Representation of -loading -boundary conditions -etc. Assessment of accuracy of finite element Solution of mathematical model Refinement of analysis Design improvements Method of Finite Elements I Interpretation of results
  11. 11. Swiss Federal Institute of Technology Page 11 Introduction to the use of finite element • Reliability of a mathematical model The chosen mathematical model is reliable if the required response is known to be predicted within a selected level of accuracy measured on the response of a very comprehensive mathematical model • Effectiveness of a mathematical model The most effective mathematical model for the analysis is surely that one which yields the required response to a sufficient accuracy and at least costs Method of Finite Elements I
  12. 12. Page 12 Swiss Federal Institute of Technology Introduction to the use of finite element • Example Complex physical problem modeled by a simple mathematical model M = WL = 27,500 Ncm 1 W ( L + rN )3 W ( L + rN ) δ at load W = + 5 3 EI AG 6 = 0.053cm Method of Finite Elements I
  13. 13. Page 13 Swiss Federal Institute of Technology Introduction to the use of finite element • Example Detailed reference model – 2D plane stress model – for FEM ⎫ ∂τ xx ∂τ xy analysis + = 0⎪ ∂x ∂τ yx ∂y ∂τ yy ⎪ ⎬ in domain of bracket + = 0⎪ ⎪ ∂x ∂y ⎭ τ nn = 0, τ nt = 0 on surfaces except at point B and at imposed zero displacements Stress-strain relation: ⎡ ⎤ ⎢1 ν ⎡τ xx ⎤ 0 ⎥ ⎡ε xx ⎤ ⎥⎢ ⎥ E ⎢ ⎢ ⎥ τ yy ⎥ = ν 1 0 ⎥ ⎢ε yy ⎥ ⎢ 1 −ν 2 ⎢ ⎢ ⎢τ xy ⎥ 1 −ν ⎥ ⎢γ xy ⎥ ⎣ ⎦ 0 0 ⎢ ⎥⎣ ⎦ 2 ⎦ ⎣ ∂u ∂v ∂u ∂v + Strain-displacement relation: ε xx = ; ε yy = ; γ xy = ∂x ∂y ∂y ∂x Method of Finite Elements I
  14. 14. Page 14 Swiss Federal Institute of Technology Introduction to the use of finite element • Example Comparison between simple and more refined model results M = WL = 27,500 Ncm 1 W ( L + rN )3 W ( L + rN ) δ at load W = + 5 3 EI AG 6 = 0.053cm δ at load W M Method of Finite Elements I x =0 = 0.064cm = 27,500 Ncm Reliability and efficiency may be quantified!
  15. 15. Swiss Federal Institute of Technology Page 15 Introduction to the use of finite element • Observations Choice of mathematical model must correspond to desired response measures The most effective mathematical model delivers reliable answers with the least amount of efforts Any solution (also FEM) of a mathematical model is limited to information contained in the model – bad input – bad output Assessment of accuracy is based on comparisons with results from very comprehensive models – however, in practice often based on experience Method of Finite Elements I
  16. 16. Swiss Federal Institute of Technology Page 16 Introduction to the use of finite element • Observations Sometimes the chosen mathematical model results in problems such as singularities in stress distributions The reason for this is that simplifications have been made in the mathematical modeling of the physical problem Depending on the response which is really desired from the analysis this may be fine – however, typically refinements of the mathematical model will solve the problem Method of Finite Elements I
  17. 17. Swiss Federal Institute of Technology Introduction to the use of finite element • Finite elements as a tool for computer supported design and assessment FEM forms a basic tool framework in research and applications covering many different areas - Fluid dynamics - Structural engineering - Aeronautics - Electrical engineering - etc. Method of Finite Elements I Page 17
  18. 18. Swiss Federal Institute of Technology Page 18 Introduction to the use of finite element • Finite elements as a tool for computer supported design and assessment The practical application necessitates that solutions obtained by FEM are reliable and efficient however also it is necessary that the use of FEM is robust – this implies that minor changes in any input to a FEM analysis should not change the response quantity significantly Robustness has to be understood as directly related to the desired type of result – response Method of Finite Elements I
  19. 19. Page 19 Swiss Federal Institute of Technology Basic mathematical tools • Vectors and matrices Ax = b ⎡ a11 ⎢ ⎢ A = ⎢ ai1 ⎢ ⎢ ⎢ am1 ⎣ AT is the transpose of A a1i aii ⎡ x1 ⎤ ⎡ b1 ⎤ ⎢x ⎥ ⎢b ⎥ 2⎥ x=⎢ , b=⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ xn ⎦ ⎣ ⎣bm ⎦ Method of Finite Elements I a1n ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ amn ⎥ ⎦ if A = AT there is m = n (square matrix) and aij = a ji (symmetrical matrix) ⎡1 ⎢0 ⎢ I = ⎢0 ⎢ ⎢ ⎢0 ⎣ 0 0 1 0 0 1 0 0 0⎤ 0⎥ ⎥ 0 ⎥ is a unit matrix ⎥ ⎥ 1⎥ ⎦
  20. 20. Page 20 Swiss Federal Institute of Technology Basic mathematical tools • Banded matrices symmetric banded matrices aij = 0 for j > i + mA , 2mA + 1 is the bandwidth ⎡3 ⎢2 ⎢ A = ⎢1 ⎢ ⎢0 ⎢0 ⎣ 0⎤ 0⎥ ⎥ 1⎥ ⎥ 4⎥ 0 1 4 3⎥ ⎦ 2 3 4 1 Method of Finite Elements I 1 4 5 6 0 1 6 7 mA = 2
  21. 21. Page 21 Swiss Federal Institute of Technology Basic mathematical tools • Banded matrices and skylines mA + 1 ⎡3 ⎢2 ⎢ A = ⎢0 ⎢ ⎢0 ⎢0 ⎣ Method of Finite Elements I 2 0 0 0⎤ 3 0 1 0⎥ ⎥ 0 5 6 1⎥ ⎥ 1 6 7 4⎥ 0 1 4 3⎥ ⎦
  22. 22. Swiss Federal Institute of Technology Basic mathematical tools • Matrix equality A ( m × p ) = B ( n × q ) if and only if m = n, p = q, and aij = bij Method of Finite Elements I Page 22
  23. 23. Swiss Federal Institute of Technology Basic mathematical tools • Matrix addition A ( m × p ) , B ( n × q ) can be added if and only if m = n, p = q, and if C = A + B, then cij = aij + bij Method of Finite Elements I Page 23
  24. 24. Swiss Federal Institute of Technology Basic mathematical tools • Matrix multiplication with a scalar A matrix A multiplied by a scalar c by multiplying all elements of A with c B = cA bij = caij Method of Finite Elements I Page 24
  25. 25. Swiss Federal Institute of Technology Basic mathematical tools • Multiplication of matrices Two matrices A ( p × m ) and B ( n × q ) can be multiplied only if m = n C = BA m cij = ∑ air brj , C ( p × q ) r =1 Method of Finite Elements I Page 25
  26. 26. Page 26 Swiss Federal Institute of Technology Basic mathematical tools • Multiplication of matrices The commutative law does not hold, i.e. AB = CB does not imply that A = C AB ≠ BA, unless A and B commute however does hold for special cases (e.g. for B = I) The distributive law hold, i.e. Special rule for the transpose of matrix products E = ( A + B ) C = AC + BC ( AB ) T The associative law hold, i.e. G = (AB)C = A(BC) = ABC Method of Finite Elements I = BT AT
  27. 27. Swiss Federal Institute of Technology Basic mathematical tools • The inverse of a matrix The inverse of a matrix A is denoted A −1 if the inverse matrix exist then there is: AA −1 = A −1A = I The matrix A is said to be non-singular The inverse of a matrix product: ( AB ) = B-1A -1 -1 Method of Finite Elements I Page 27
  28. 28. Swiss Federal Institute of Technology Basic mathematical tools • Sub matrices A matrix A may be sub divided as: ⎡ a11 A = ⎢ a21 ⎢ ⎢ a31 ⎣ a12 a22 a32 ⎡ a11 A=⎢ ⎢ a21 ⎣ a12 ⎤ ⎥ a22 ⎥ ⎦ Method of Finite Elements I a13 ⎤ a23 ⎥ ⎥ a33 ⎥ ⎦ Page 28
  29. 29. Swiss Federal Institute of Technology Basic mathematical tools • Trace of a matrix The trace of a matrix A ( n × n ) is defined through: n tr ( A) = ∑ aii i =1 Method of Finite Elements I Page 29
  30. 30. Swiss Federal Institute of Technology Basic mathematical tools • The determinant of a matrix The determinant of a matrix is defined through the recurrence formula n det( A ) = ∑ (−1)1+ j a1 j det(A1 j ) j =1 where A1 j is the ( n − 1) × ( n − 1) matrix obtained by eliminating the 1st row and the j th column from the matrix A and where there is if A = [ a11 ] , det A = a11 Method of Finite Elements I Page 30
  31. 31. Swiss Federal Institute of Technology Basic mathematical tools • The determinant of a matrix It is convenient to decompose a matrix A by the so-called Cholesky decomposition ⎡ 1 0 0⎤ T L = ⎢l21 1 0 ⎥ A = LDL ⎢ ⎥ ⎢l31 l32 1 ⎥ ⎣ ⎦ where L is a lower triangular matrix with all diagonal elements equal to 1 and D is a diagonal matrix with components dii then the determinant of the matrix A can be written as n det A = ∏ d ii i =1 Method of Finite Elements I Page 31
  32. 32. Page 32 Swiss Federal Institute of Technology Basic mathematical tools • Tensors Let the Cartesian coordinate frame be defined by the unit base vectors ei x3 A vector u in this frame is given by e3 3 u = ∑ ui ei e2 e1 x1 x2 i =1 simply we write u = ui ei i is called a dummy index or a free index Method of Finite Elements I
  33. 33. Swiss Federal Institute of Technology Basic mathematical tools • Tensors An entity is called a tensor of first order if it has 3 components ξi in the unprimed frame and 3 components ξi' in the primed frame, and if these components are related by the characteristic law ξi' = pik ξi where pik = cos ( ei' , e k ) In the matrix form, it can be written as ξ ' = Pξ Method of Finite Elements I Page 33
  34. 34. Swiss Federal Institute of Technology Basic mathematical tools • Tensors An entity is called a second-order tensor if it has 9 components tij in the unprimed frame ' and 9 components tij in the primed frame, and if these components are related by the characteristic law ' tij = pik p jl tkl Method of Finite Elements I Page 34

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