du(l) 5 dl = U(0) = 1 and 1.) Substitute for U an unspecified trial function into governing equation, i.e. dx + = =

Transcription

1 Consider an ODE of te form: Finite Element Metod du fu g d + wit te following Boundary Conditions: U(0) and du(l) 5 dl.) Substitute for U an unspecified trial function into governing equation, i.e. ^ U U U n d U fu g R 0 for, n d + were n umber of nodes in system (or, locally, in element). U Unknown coefficients (to be determined), wic are not a function of space, in general. Known (user-specified) Basis function, wic are a function of space, but not a function of time, in general. ME 55 Finite Element Procedure (Skeleton) (Sullivan)

2 .) Force te Residual to zero in te global sense, i.e. satisfy te governing equation in te "weak form". a.) Multiply by a Weigting function: W n Wi i d i+ i i i U W fu W gw R W d for, i, n (independently) were W i Known (user-specified) Weigt function. b.) Integrate (take inner product) over global domain and force to zero. Definition of an Inner Product: f() g() f g d RWd 0 Implies tat W is ortogonal to R < ( ) > ()d or ()da or ()dv (dimensionally dependent) d i i i i U W + fu W gw R W 0 d If you can solve tis equation ten you ave solved te governing equations, at least in te weak form. ME 55 Finite Element Procedure (Skeleton) (Sullivan)

3 3.) Discretize global domain into subdomains (Elements) suc tat Σ (Elements) contiguous, non-overlapping representation of te global domain. #of Elem () () E E 4.) Select a basis function for U - a few elpful ints in tis selection are a.) select an ortogonal series basis function b.) coose a computationally efficient series E. Lagrange Polynomials applied locally on eac element n ( i) i ; n # nodes in element i ( i) 0 for all nodes not in element Assume Linear Element ( ) ( ) ( ) ( ) E Ν Ν ( ) ( ) ( ) E Ν X X ME 55 Finite Element Procedure (Skeleton) (Sullivan) 3

4 Integration Formulae for -D Linear Elements d d d d ( anyting ) ( anyting) d d d a ( ) a( ) a( ) d d and d d a ( ) a( ) a( ) d d ME 55 Finite Element Procedure (Skeleton) (Sullivan) 4

5 Assume Quadratic Element ( 3) ( )( ) 3 ( )( 3 ) ( )( 3) ( )( ) 3 3 ( )( ) ( )( ) 3 3 Ν Ν Ν Ν 3 X X X 3 ME 55 Finite Element Procedure (Skeleton) (Sullivan) 5

6 5.) Integration by parts: Required for nd order PDEs using linear basis functions; Provides a direct means to apply Boundary Conditions u dv uv v du (more generally) div(f) dv F n ds Ω S d d dw d i L W i + Wi 0 d d d d more generally, W W ( n)w ds i i + i Adding in te oter components of te ODE yields: d dw d U + f U W gw U W d d d i i i i L 0 6.) Select Weigting functions E: Coose "Galerkin" Wk k (However, for clarity retain te W symbol. It will elp wen assembling matrices later.) ote tat for eac weigting function, W i, one must assemble all contributions from eac basis function. ME 55 Finite Element Procedure (Skeleton) (Sullivan) 6

7 7.) Assemble Matrices consider only elements, and assemble all components associated wit Row I (for W i ): Ν i - Ν i Ν i + i - i i + One row of equations for eac W i ote: te local definitions of Trial and Weigting functions W 0 for all < i W 0 for all > i+ Refer to te Integration Table for all possible combinations of linear basis functions. ME 55 Finite Element Procedure (Skeleton) (Sullivan) 7

8 for row i only contributions from (i ) (i + ) d d dw d Left Elem i i Rigt Elem + 0 di dwi d d + + di+ dwi 0 + d d W 6 i i W i i i + Wi W i + + ME 55 Finite Element Procedure (Skeleton) (Sullivan) 8

9 Assembly of row I U ( + )U + U i i i+ + f Ui + f Ui+ f Ui d d L g U Wi 0 If uniform spacing ten f U U + U + U + 4U + U g ( ) ( ) i i i+ i i i+ 6 δ Ui + f 6 ote te similarities to F.D. Simpson s Rule g In general, we are forming an epression [ ]{ } { } d L A U Rs U Wi 0 d dw d i L + fw i { U } { gwi } + U Wi 0 d d d d ME 55 Finite Element Procedure (Skeleton) (Sullivan) 9

10 In a FE code: Loop over eac element and assemble te local (elemental) contributions for te Ls matri [A] and te Rs vector. After tis loop is completed, ten (and only ten) apply te boundary conditions to te set of equations. FE codes contain a lot of bookkeeping. One common mapping array is te element connectivity (or Incidence) list. It as te form: I(K,L) were L Element umber, L, E ( and E umber of Elements) K Local ode umber, K,, to te number of nodes in an element. I(K,L) global node number ( maps global node number to te local node number witin element L) ME 55 Finite Element Procedure (Skeleton) (Sullivan) 0

11 Consider te following -D eample: Element umbers Global ode umbers X L Te Element number can ave significance if using a frontal matri solver. Te ode numbering can ave significance if using a banded matri solver. ode and Element numberings ave less significance if using a sparse, iterative matri solver. Te mapping for tis eample is: I(,) 3 I(,) I(,) 4 I(,) 6 I(,3) I(,3) 4 I(,4) 7 I(,4) I(,5) 5 I(,5) 7 I(,6) 6 I(,6) 5 Te FE approimation to te governing equation is accumulated (summed) in 3 nested loops (L, I, and J) ME 55 Finite Element Procedure (Skeleton) (Sullivan)

12 for L : E (Loop over all elements) (in(,l)) (in(,l)); (Calculate element lengt) for I : (Loop for all Weigting Functions) dwd (/); if (I ) dwd - dwd; for J, (Loop for all Trial Functions) dd (/); if (J ) dd - dd; term - dd * dwd * ; term f * /6; if (I J) term * term; AL(J, I) term + term; (Local construction) Irow I(I,L); (Global Mapping) Jcol I(J,L); AG(Jcol, Irow) + AG(Jcol, Irow) + AL(J, I); (Global Matri Construction) End (End of J loop) Rs(Irow) Rs(Irow) + g * /; (Rs Construction) End (End of I loop) End (Finised wit all Elements) Everyting done at te element level! Easy to automate a) One Element (problem-specific) b) Assembly (problem-independent) ME 55 Finite Element Procedure (Skeleton) (Sullivan)

13 8) Add in te Boundary Conditions Element umbers Global ode umbers X L a.) Type BC : Satisfy Eactly U(0) (ote tis is at global node number 3 for our eample.) One less unknown in algebraic system. Remove row_3 and ten move column_3 of matri to Rs since it is known (not usually done, but possible) U Rs a a a a a a U Rs,,,3,4,,3 a a a a a a,,,3,4,,3 a a a a a U Rs 3 a 3, 3, 3,3 3,4 3, 4 4 4,3 a a a a a 4, 4, 4,3 4,4 4, a a a a a U Rs,,,3,4, U a,3 Tis process is somewat cumbersome. It can cange te bandwidt or structure of your system. It can cause oter solution-solving anomalies if one is not careful. ME 55 Finite Element Procedure (Skeleton) (Sullivan) 3

14 Cleaner process: Remove row 3 completely, i.e. place zeros in all entries. Ten place a on diagonal of [A] Put te solution (i.e. U(0)) into te Rs a a a a a,,,3,4, U Rs a a a a a U Rs U,,,3,4, U 3, 3, 3,3 3,4 3, 3 3 a a a a a U 4, 4, 4,3 4,4 4, 4 U a a a a a,,,3,4, Rs Rs Value 4 b.) Type II B.C. Satisfy approimately in {BCs} vector. Recall: du(l) 5 and for node (in our eample) te flu at dl d du(l) te boundary is 5. Also note: U Wi 5 d dl At te boundary node W i for I (global) and W i 0 for all oter nodes. Terefore to apply Type II Boundary Conditions in our eample: Rs() Rs() 5; Te entire formulation is complete. Call a matri solver and obtain te solutions for U. ME 55 Finite Element Procedure (Skeleton) (Sullivan) 4