Introduction to Finite Element Method - strana.snu.ac.kr

1 Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Introduction to Finite Element Method Fall Semester, 2018 Hae Sung Lee Dept. of Civil and Environmental Engineering Seoul National University Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Contents 1. Introduction 2. Approximation of Functions & Variational Calculus 3. Differential Equations in One Dimension 4. Multidimensional Problems-Elasticity 5. Discretization 6. Two Dimensional Elasticity Problems 7. Various Types of elements 8. Numerical Integration 9. Convergence Criteria in Isoparameteric Element 10. Miscellaneous Topics 11. Problems with Higher Continuity Requirement - Beams 12. Mixed Formulation Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Chapter 2 Approximation of Functions and Variational Calculus Dept.

2 Of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Fundamental Considerations What is the best solution for a given problem ? Needless to say, it is the exact What is the exact solution? The solution that satisfies the governing equations as well as boundary conditions if any. How many do the exact solutions exist? Of course, In case several or infinite numbers of the exact solutions exist for a giv-en equation, we call the problem as an ill-posed problem, and have great difficulties in determining a solution of the given What if it is impossible to determine the exact solution for various reasons? We need approximate solutions. What is an approximate solution? Fundamental Questions - What is the best approximation? - How can we represent the best approximation? A Good (or Robust or Well Formulated) Approximation Should - Yield the best approximation to the exact solution for a given degree of approxima-tion.

3 - Converge to the exact solution as higher degree of approximation is employed. What is the Definition of the Best Approximation? - May be defined as the closest solution to the exact solution. - But, how close is the closest ? - Close or Far implies the distance between two spatial points. - We should define some sort of ruler to measure the distance between two elements in a function Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Normes of Functions: A measure of a function set A function set is said to be a normed space if to every f there associated a nonnegative real number f, called the norm of f, in a such way that - 0 f if and only if 0 f - ff|| for any real number . - gfgf Every normed space may be regarded as a metric space, in which the distance between any two elements in the space is measured by the defined norm. Various types of norm can be defined for a function space.

4 Among them the following norms are important. - L1 norm: VLdVff1 - L2 norm: 2/12)(2 VLdVff - H1 norm: 2/12))((1 VHdVffff Discretization - Representation of a continuously distributed quantities with some numbers. )()(1XX niiigaf )(Xf where gi are the basis functions of a function set, - Set : Collection of some objectives with the same characteristics - The basis functions should be linearly independent to each other. 0)(1 Xniiiga if and only if all 0 ia - Taylor series, Fourier series, - Approximations hmiiihniiigafgaf)()()()(11 XXXX where nm Summation Notation: Repeated indices denote summation iimiiibaba 1. Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, General Ideas for the Best Approximation Let s find out a approximate function that is closest to the given function by use of a norm defined in the function space. If this is the case, the characteristics of an approxima-tion Method depend on those of the norm used in the approximation.

5 Least Square Error(LSE) Minimization Error: hffe Minimize VhLhLdVffffe222)(21212122 mkFaKfdVgadVggfdVgdVaggdVgffdVafffakimik iVkimiVikVkVmiiikVkhVkhhk,,1for 0)()(111 Final System Equation : FKa If the basis functions are orthogonal, K becomes diagonal. Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Variation of a function - The variation of a function means a possible change in the function for the fixed x. Variational Calculus - ifiigaf then the variation of f is defined as iigaf or by definition iiaaff . - ffFaaffFaaFFfFiiii : )( - hfhf )( - hffhfh )( - , )()(dxfdaafdxdadxdfadxdfiiii - fdxdxaafafdxafdxiiii f f Variation of a function Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Minimization by Variational Calculus Min lhhdxfff02)(21)( kkklkhhlhhlhlhhaaadxafffdxfffdxffdxfff 000202)()()(21))(21()( Min kVhhadVfff possible allfor 0)(21)(2 Euler Equation MinkadxxffFfkl allfor 0),,()(0 lkklklkklkklklkkdxgfFdxdgfFgfFdxgfFgfFdx affFaffFdxaFdxxffFaa000000)()()(),,( In case the basis functions vanish at the boundary, then 0 allfor 0)(0 fFdxdfFkdxgfFdxdfFalkk lllldxffFdxdffFffFdxffFffFdxxffFf0000)() (),,()( In case the variation vanishes at the boundaries, then kkklklaaadxgfFdxdfFfdxfFdxdfFf 00)()()(.

6 Therefore, Min 0 Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Example 1 Min 212)(1)(xxdxyy subject to 11)(yxy , 22)(yxy 0))(1())(1(02/122 yydxdyydxdfFdxdfF 0))(1())(1)(1())(1())(1)(21())(1())(1(2/ 32222/122/322/122/12 yyyyyyyyyyyyyydxd baxyy 0. By applying BC, 12211212120xxyxyxxxxyyyy Example 2 Min ldxufuu02))(21()( subject to 0)0( u, 0)( lu 00))(21(2 fuufudxdfuudxdfuFdxduF Homework 1 1. Approximate a cosine function xly 2cos by polynomials based the minimization of the least square errors. Use polynomials up to the 20th order. You may use a numerical inte-gration algorithm such as Simpson s rule, the trapezoidal rule or the rectangular rule. You also need a numerical solver to solve linear simultaneous equations in the Linpack . For the accuracy of your calculation, please use double precision in your program. You should present proper discussions on your results together with some graphs that show your approximate functions and the given function.

7 2. Derive Euler equation for the following minimization function, and proper boundary condi-tions. Min ldxxfffFf0),,,()( 3. Derive the governing equation and the boundary conditions for the following problem. Min ldxwqdxwdw0222))(21()(Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Chapter 3 Elliptic Differential Equations in One Dimension Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Problems with homogenous displacement boundary conditions Problem Definition 0)()0( ,0 022 luulxfdxud Approximation Discretization miiihgau1 where 0)()0( luuhh Residuals Verbal Definition : Something left over, or resulting from Equation Residual : lxfdxudRhE 0 022 Function Residual : lxuuRhF 0 0 Error Estimator : lhhlEFRdxfdxuduudxRR0220))((2121 Least Square Error Error SquareLeast ))((21))((21))((21))((21))((210000222202 2 LSlhhlhhlhhlhhlhhRdxdxdudxdudxdudxdudxdx dudxdudxdudxdudxdudxduuudxdxuddxuduudxfd xuduu Energy Functional Total Potential Energy RRlhlhhllhhlhhlhlhlhlhlhhhhlhhRCfdxudxdx dudxduufdxdxdxdudxdudxduudxudxududxdudxd uudxfuufdxfudxuduufdxududxfdxuduu )21(21})({21)(21))((21000000220000222202 2 f Dept.

8 Of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Minimization Problems RRLSR Min Min Min hhu RR Min : Rayleigh-Ritz Method or Principle of Minimum Potential Energy - 1st Order Necessary Condition for Minimization Problem FKa mkFaKfdxgadxdxdgdxdgfdxgadaddxdxdgadxdga dadfdxdadudxdxdudxdudadamikikilkmiiliklm iiiklmiiimiiiklkhlhhkkRR ,1for 0)())(()(10100101100 0 RR: Variational principle or Principle of Virtual Work 0)()( 0)()()21(111010000000 FKaaTmkmikikikmklkmiilikklhlhhlhlhhlhlhh RRFaKafdxgadxdxdgdxdgafdxudxdxdudxudfdxu dxdxdudxdufdxudxdxdudxdu Solution Space - })( , 0)()0(|{02 ldxdxduluuuu - h : The minimization problems yield the exact solution. - h : The minimization problems yield an approximate solution. Properties of K - Symmetry :jilijljiijKdxdxdgdxdgdxdxdgdxdgK 00 - Positive Definiteness : miTjmjijijmjjlimiijmjjlimiilhaKaadxdxdgd xdgadxadxdgadxdgdxdxdu11101101200)(Kaa Dept.

9 Of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Absolute Minimum Property of Total Potential Energy ehuuu )for only holdssign equality (The )(212121)(212121212121)()()(212102000022 00000220000000000000 Cudxdxdudxdxdudxduufdxdxdxdudxdudxfdxudu dxdxdudxduufdxdxdxdudxdufdxudxdxududxduu dxdxdudxduufdxdxdxdudxdufdxudxdxdudxdudx dxdudxduufdxdxdxdudxdufdxuudxdxuuddxuudf dxudxdxdudxdueEleEleelllelleellelelellee llelelleelleleelhlhhh Weighted Residual Method mkdxfdxuddxRlhklEkk,,1for 0)(0220 - if kkg : Galerkin Method (elliptic System or self-adjoint system) FKa ,,1for 0 )(01000100000022mkfdxgadxdxdgdxdgfdxgdxa dxdgdxdgfdxgdxdxdudxdgfdxgdxdxdudxdgdxdu gdxfdxudlkimiliklklmiiiklklhklklhklhklhk k - Identical result to the Rayleigh-Ritz Method or Variational Principle - if kkg : Petrov-Galerkin Method (hyperbolic system or non-self adjoint system) Weighted Residual Method vs. Variational Principle iimiikaamk 0 ,,1for 01 hllhhhlhhlhimiimilhiiimiiufdxudxdxdudxud dxfdxududxfdxudgadxfdxudgaa p ossible allfor 0)()()()( 00022022110221 Dept.

10 Of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, Example 1 : 122 dxud, 0)1()0( uu - Exact solution : xxu21212 - First trial : 2321xaxaauh Applying BCs : 01 a, 32aa )(23xxauh , )21(3xadxduh 31)21(10211 dxxK, 61)(11021 dxxxF 21613133 aa. Therefore, uuh - Second Trial : 342321xaxaxaauh Applying BCs : 01 a, 432aaa 21)()(3423gghxxaxxau )31(),21(221xdxdgxdxdg 31)21(10211 dxxK, 54)31(102222 dxxK 21)31)(21(1022112 dxxxKK 61)(11021 dxxxF, 41)(11032 dxxxF System Equation : 0,214161542121314343 aaaa Example 2 : xdxud sin222, 0)1()0( uu - Exact Solution : xu sin - First trial : 2321xaxaauh Applying BCs : )(22xxauh , )21(2xadxduh 31)21(10211 dxxK, 4)(sin10221dxxxxF 1243122aa. Therefore, )(122xxuh ) ( hu Error = % Dept. of Civil and Environmental Eng., SNU Structural Analysis Lab. Prof. Hae Sung Lee, - Second trial : 332210xaxaxaauh Applying BCs : 00 a, 321aaa 21)()(3322gghxxaxxau )31(),21(221xdxdgxdxdg 31)21(10211 dxxK, 54)31(102222 dxxK 21)31)(21(1022112 dxxxKK 4)(sin10221dxxxxF, 6)(sin10322dxxxxF System Equation : 0,1264542121313232 aaaa ?