3 generate the element sti ness and mass matrices and

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Unformatted text preview: subspaces of HE and H01, respectively however, this may not be the case because of errors introduced in approximating the essential boundary conditions and/or the domain . These e ects will also have to be appraised (cf. Section 7.3). Choosing a basis for S N , we write U and V in the form of (3.3.1). The variational problem is written as a sum of contributions over the elements and the element sti ness and mass matrices and load vectors are generated. For the model problem (3.1.1) this would involve solving N X e=1 Ae(V U ) ; (V f )e; < V >e] = 0 8V 2 S0N (3.3.2a) 18 Multi-Dimensional Variational Principles 5 s 6 n θ u=α 4 pu n+ γu = β 7 3 8 1 2 U y 7 x 12345678 1 2 3 4 5 6 K=7 8 8 4 K e , le 1 0 11 00 11 00 1 0 11 00 11 00 1 0 11 00 11 00 1 0 11 00 11 00 1 2 3 4 5 6 7 l= 8 1 0 1 0 1 0 1 0 1 0 Figure 3.3.1: Two-dimensional domain having boundary @ = @ E @ N with unit normal n discretized by triangular nite elements. Schematic representation of the assembly of the element sti ness matrix Ke and element load vector le into the global sti ness matrix K and load vector l. where Ae(V U ) = ZZ e (VxpUx + Vy pUy + V qU )dxdy (3.3.2b) 3.3. Overview of the Finite Element Method (V f )e = ZZ 19 V fdxdy (3.3.2c) e < V >e= Z V ds (3.3.2d) @ e \@ ~ N is the domain occupied by element e, and N is the number of elements in the mesh. The boundary integral (3.3.2d) is zero unless a portion of @ e coincides with the boundary of the nite element domain @ ~ . Galerkin formulations for self-adjoint problems such as (3.1.6) lead to minimum problems in the sense of Theorem 3.1.1. Thus, the nite element solution is the best solution in S N in the sense of minimizing the strain energy of the error A(u ; U u ; U ). The N strain energy of the error is orthogonal to all functions V in SE as illustrated in Figure 3.3.2 for three-vectors. e 1 0u 1 0 1 0 H1 E 1 0 1 0U 1 0 SN E N 1 Figure 3.3.2: Subspace SE of HE illustrating the \best" approximation property of the solution of Galerkin's method. 4. Assemble the global sti ness and mass matrices and load vector. The element sti ness and mass matrices and load vectors that result from evaluating (3.3.2b-d) are added directly into global sti ness and mass matrices and a load vector. As depicted in Figure 3.3.1, the indices assigned to unknowns associated with mesh entities (vertices as shown) determine the correct positions of the elemental matrices and vectors in the global sti ness and mass matrices and load vector. 20 Multi-Dimensional Variational Principles 5. Solve the algebraic system. For linear problems, the assembly of (3.3.2) gives rise to a system of the form dT (K + M)c ; l] = 0 (3.3.3a) where K and M are the global sti ness and mass matrices, l is the global load vector, cT = c1 c2 ::: cN ]T (3.3.3b) dT = d1 d2 ::: dN ]T : (3.3.3c) and Since (3.3.3a) must be satis ed for all choices of d, we must have (K + M)c = l: (3.3.4) For the model problem (3.1.1), K + M will be sparse and positive de nite. With proper treatment of the boundary conditions, it will also be symmetric (cf. Chapter 5). Each step in the nite element solution will be examined in greater detai...
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