ࡱ> CEB[@ cdbjbj44 9sViVi"D@@@@HHH8 d==| `$$$%q+q+q+'3)3)3)3Ir38<$'>Ry@<9,*|q+,,<@@$%<=1=1=1,@8$%'3=1,'3=1=12x2%p TjHy. 2'3 =0==2A0A2@@@@A2Dq+>+,=1+$+q+q+q+<<H!1HFinite Element Method Consider an ODE of the form:  EMBED Equation.DSMT4  with the following Boundary Conditions:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  1.) Substitute for U an unspecified trial function into governing equation, i.e.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  where n = Number of nodes in system (or, locally, in element). Uj = Unknown coefficients (to be determined), which are not a function of space, in general. Nj = Known (user-specified) Basis function, which are a function of space, but not a function of time, in general. 2.) Force the Residual to zero in the global sense, i.e. satisfy the governing equation in the "weak form". a.) Multiply by a Weighting function:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  where Wi = Known (user-specified) Weight function. b.) Integrate (take inner product) over global domain and force to zero. Definition of an Inner Product:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Implies that W is orthogonal to R < ( ) > =  EMBED Equation.DSMT4  (dimensionally dependent)  EMBED Equation.DSMT4  If you can solve this equation then you have solved the governing equations, at least in the weak form. 3.) Discretize global domain into subdomains (Elements) such that S (Elements) = contiguous, non-overlapping representation of the global domain.  EMBED Equation.DSMT4  4.) Select a basis function for  - a few helpful hints in this selection are a.) select an orthogonal series basis function b.) choose a computationally efficient series Ex. Lagrange Polynomials applied locally on each element  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Assume Linear Element (N = 2)   EMBED MSDraw.Drawing.8.1  Assume Quadratic Element (N = 3)  EMBED Equation.DSMT4   EMBED MSDraw.Drawing.8.1  5.) Integration by parts: Required for  order PDEs using linear basis functions; Provides a direct means to apply Boundary Conditions  EMBED Equation.DSMT4   EMBED Equation.DSMT4  more generally,  EMBED Equation.DSMT4  Adding in the other components of the ODE yields:  EMBED Equation.3  6.) Select Weighting functions Ex: Choose "Galerkin"  EMBED Equation.DSMT4  (However, for clarity retain the W symbol. It will help when assembling matrices later.) Note that for each weighting function, Wi, one must assemble all contributions from each basis function Nj. 7.) Assemble Matrices consider only 2 elements, and assemble all components associated with Row I (for Wi):  EMBED MSDraw.Drawing.8.1  One row of equations for each Wi Note: the local definitions of Trial and Weighting functions  EMBED Equation.DSMT4  for all  EMBED Equation.DSMT4  = 0 for all  EMBED Equation.DSMT4   for row i only contributions from  EMBED Equation.DSMT4  Left Elem Right Elem  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  + 0 =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  +  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  = 0 +  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  + 0 =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  +  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  = 0 +  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  +  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  Assembly of row I  EMBED Equation.DSMT4  If uniform spacing then h1 = h2 = h  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  +  EMBED Equation.DSMT4  Simpsons Rule = g Note the similarities to F.D. In general, we are forming an expression  EMBED Equation.DSMT4   EMBED Equation.3  In a FE code: Loop over each element and assemble the local (elemental) contributions for the Lhs matrix [A] and the Rhs vector. After this loop is completed, then (and only then) apply the boundary conditions to the set of equations. FE codes contain a lot of bookkeeping. One common mapping array is the element connectivity (or Incidence) list. It has the form: IN(K,L) where L = Element Number, L = 1, NE ( and NE = Number of Elements) K = Local Node Number, K = 1, 2, to the number of nodes in an element. IN(K,L) = global node number ( maps global node number to the local node number within element L) Consider the following 1-D example:  The Element number can have significance if using a frontal matrix solver. The Node numbering can have significance if using a banded matrix solver. Node and Element numberings have less significance if using a sparse, iterative matrix solver. The mapping for this example is: IN(1,1) = 3 IN(2,1) = 1 IN(1,2) = 4 IN(2,2) = 6 IN(1,3) = 1 IN(2,3) = 4 IN(1,4) = 7 IN(2,4) = 2 IN(1,5) = 5 IN(2,5) = 7 IN(1,6) = 6 IN(2,6) = 5 The FE approximation to the governing equation is accumulated (summed) in 3 nested loops (L, I, and J) Do L = 1, NE (Loop over all elements) h = x(in(2,L)) x(in(1,L)) (Calculate element length) Do I = 1, 2 (Loop for all Weighting Functions) dWdx = (1/h) if (I = 1) dWdx = - dWdx Do J = 1, 2 (Loop for all Trial Functions) dNdx = (1/h) if (J = 1) dNdx = - dNdx term1 = - dNdx * dWdx * h term2 = f * h/6 if (I = J) term2 = 2 * term2 AL(J, I) = term1 + term2 (Local 2x2 construction) Irow = IN(I,L) (Global Mapping) Jcol = IN(J,L) AG(Jcol, Irow) += AL(J, I) (Global Matrix Construction) End Do (End of J loop) Rhs(Irow) += g * h/2 (Rhs Construction) End Do (End of I loop) End Do (Finished with all Elements) Everything done at the element level! Easy to automate a) One Element (problem-specific) b) Assembly (problem-independent) 8) Add in the Boundary Conditions  a.) Type 1 BC : Satisfy Exactly U(0) = 1 (Note this is at global node number 3 for our example.) One less unknown in algebraic system. Remove row_3 and then move column_3 of matrix to Rhs since it is known (not usually done, but possible)   EMBED Equation.DSMT4  This process is somewhat cumbersome. It can change the bandwidth or structure of your system. It can cause other solution-solving anomalies if one is not careful. Cleaner process: Remove row 3 completely, i.e. place zeros in all entries. Then place a 1 on diagonal of [A] Put the solution (i.e. U(0)=1) into the Rhs  EMBED Equation.DSMT4  b.) Type II B.C. Satisfy approximately in {BCs} vector. Recall:  EMBED Equation.DSMT4  and for node 2 (in our example) the flux at the boundary is 5. Also note:  EMBED Equation.DSMT4  At the boundary node Wi = 1 for I = 2 (global) and Wi = 0 for all other nodes. Therefore to apply Type II Boundary Conditions in our example: Rhs(2) = Rhs(2) 5 The entire formulation is complete. 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