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# Homework 11 - BMED 2200 Modeling of Biomedical Systems...

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RLS 1 – of - 6 BMED 2200 Modeling of Biomedical Systems Spring 2008 Homework 11 solution 1. For the 2-node element, the approximation for ( ) e u x is given by Eqs. (5.20) and(5.21), so that e N is: 2 1 2 1 1 2 1 2 e e e e e e e e e e e x x x x x x x x x x x x L L ! " ! " # \$ # \$ # \$ # \$ % % % % = = & & ( ) ( ) ( ) ( ) % % * + * + * + * + , - , - N (1.1) Where 2 1 e e e L x x = ! is the length of element e. Then the force vector (the part from ( ) f x ) is given by the second of Eqs. (5.29): f dist e = ! N eT f e dx x 1 e x 2 e " = ! x 2 e ! x L e # \$ % & ( x ! x 1 e L e # \$ % & ( ) * + + + + + , - . . . . . R sin / x L # \$ % & ( dx x 1 e x 2 e " = ! R x 2 e ! x L e # \$ % & ( sin / x L # \$ % & ( x ! x 1 e L e # \$ % & ( sin / x L # \$ % & ( ) * + + + + + , - . . . . . dx x 1 e x 2 e " 2. Consider the steady state heat transfer problem of a rod of length, L, constant conductivity, a, prescribed temperature u=0 at x = 0 and L, and constant source term f(x)=-Q over the left half of the rod. x L/2 L f = -Q Temperature u = 0 Temperature u = 0 Calculate the temperature and flux results for this problem via finite elements, using two equal- length 2-node linear elements (for convenience, number your 3 mesh nodes and 2 elements left to right). Compare your finite element solution with the exact solution of the problem (suggestion: plot the exact and finite element distributions for the temperature and the flux) Solution a. Solve using two equal-length 2-node linear elements.

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Homework 11 - BMED 2200 Modeling of Biomedical Systems...

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