Finite Element Method (FEM) Different from the finite difference method (FDM) described earlier, the FEM introduces approximated solutions of the variables at every nodal points, not their derivatives as has been done in the FDM. The region of interest is subdivided into small regions that are called finite elements. It will then be assumed that some predetermined function ( such as (x,y,z,t)) in terms of dependent variables (such as the spatial and time coordinates, x,y,x,z & t) can be used to replace the dependent variable (T(x,y,z,t)at the node (with unknown coefficients (ai) to be determined). N T ( x, y , z , t ) aii ( x, y , x, t ), where N is the total number of i nodal points in the domain. This function has to satisfy both the governing equation (heat diffusion equation for heat transfer problem, for example) at every nodal points and the boundary condition at every exterior nodal points. By substitute this function into every points we can obtain a system of algebraic equations in terms of the unknown coefficients (ai). This system of equations can then be solved using standard numerical schemes described before. Finite Element Example Determine the temperature distribution of the flat plate as shown below using finite element analysis. Assume one-dimensional heat transfer, steady state, no heat generation and constant thermal conductivity. The two surfaces of the plate are maintained at constant temperatures of 100C and 0C, respectively. T=100C T=0C
x2=3/2 x1=1/2 x3=5/2 1 2 3 First, divide the plate into three elements (1,2 & 3). The temperatures of these three elements are represented by their nodal temperatures T1,T2 & T3, respectively. Next, assume the temperature is a function of its coordinate: T(x)=Ax2+Bx+C. A,B & C are three constants. Finally, determine the constants using the governing equation and all corresponding boundary conditions. L=3 Example (cont.) To simplify the solution, we can apply the governing equation first: 2 2
T 2T 0, 2 2 ( Ax 2 Bx C ) 2 A 0 x x Therefore, A=0 for all nodal temperature functions. This is no surprise for us since we know the steady state, no generation, 1-D heat transfer should have a linear temperature distribution. Therefore: T(x)=Bx+C and the three nodal equations are: T1=Bx1+C=(B/2)+C, T2=Bx2+C=(3B/2)+C, T3= Bx3+C=(5B/3)+C Therefore, there are only two constants to be solved and they can be determined using the two boundary conditions. At the left-side surface, the temperature is a constant 100C and there is a constant heat transfer into the element 1 and the same amount of the heat is transferred to the element 2 since there can be no heat accumulation inside the element to satisfy the steady state condition. q(left surface to element 1) = q(element 1 to element 2) Example (cont.) 100 T1 T1 T2 100 T1 T1 T k k , x1 0 x2 x1 1/ 2 1
B 3B 3T1 T2 200, 3 C C 200 2 2 0( B ) 2(C ) 200: The first equation zero The second equation can be determined by using the boundary condition on the other side of the plate: k T2 T3 T 0 T2 T3 T3 k 3 , x3 x2 3 x3 1 1/ 2 5B 3B 3T3 T2 0, 3 C
C 0 2 2 6 B 2C 0 : The second equation Example (cont.) The finite element matrix for the constants B & C is: 0 2 B 200 6 2 C 0 100 Solve using any numerical scheme: B=, C 100 3 100 T ( x ) 100 x 3 This equation satisfies both boundary conditions: T(x=0)=100C and T(x=3)=0 C. For most finite element problems, we have to use thousands or even millions of elements in order to resolve as much detailed information as possible. Therefore, a fast numerical solver for the matrix (system of equations) is necessary to obtain satisfactory results. The use of numerical scheme has been discussed previously when we introduce the finite difference method.
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