# chap4-1 - CHAPTER 4: Isoparametric Elements and Solution...

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1 CHAPTER 4: Isoparametric Elements and Solution Techniques NODE NUMBERING How global stiffness coefficients K ij are stored strongly influences computer storage requirement and program execution speed. Storage format depends on how nodes are numbered. Shown is a six-node structure. Say, each node has one dof only. Element stiffness matrices look like

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2 Assembly Element stiffness matrices are assembled by placing their entries in the proper rows and columns of the global stiffness matrix K . Element 2-4, for example, is connected to nodes 2 and 4. Hence, its stiffness coefficients are placed in rows 2 and 4 of K . Coefficients c and d are thus added because elements 1-2 and 2-4 collectively resist a force applied to node 2. ( Think that there are two parallel springs at node 2, one from each neighboring element.)
3 Bandwidth The assembled K has non-zero areas as shown in figure K is banded , i.e., all coefficients beyond a certain distance from the diagonal are zero. With good node numbering, nonzero coefficients cluster in a narrow band along the diagonal. The bandwidth is the width of this band There may be zeros within the band; there are only zeros outside. Here “ half bandwidth ”is±3.±

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4 Poor Node-Numbering There are many zeros within the skyline . These zeros would have to be stored in the computer with most storage schemes. When the equilibrium equation Kd=R is solved by a direct solver such as Gaussian elimination, such zeros become nonzero. . Undesirable : round-off error increases! The “ half bandwidth ”is±6± in± this case.
5 Sparse Matrices and storage In practice global stiffness matrices contain many zeros.

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## This note was uploaded on 06/06/2011 for the course EAS 4240 taught by Professor Peterifju during the Spring '08 term at University of Florida.

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chap4-1 - CHAPTER 4: Isoparametric Elements and Solution...

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