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By Andree F. Sjoberg

A part of a chain that provides in general linguistic and anthropological examine and teaching/learning fabric on a sector of serious cultural and strategic curiosity and significance within the post-Soviet period.

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Approximate D i f f e r e n t i a l ii i]li u ll ' ,,,,,, Equations i [ Solve Algebraic E q u a t i o n s O u t p u t A p p r o x i m a t e Solution Figure _ 3. T h e finite d i f f e r e n c e m e t h o d . i i Part l Introduction 9 7 not represent as wide a range of boundary conditions on complex regions as the finite element method. In Parts III and IV, we reformulate both the finite element and the finite difference methods using the same physically based, self-referential notation. This means that the two methods can represent the same geometries and boundary conditions.

9) The desired goal of finding the equilibrium position of the bar is accomplished by solving these two equations for x~ and x2. The two components of Eq. 9 can be seen to be the equilibrium equations found by summing the moments about the left and right ends of the bar, respectively. This application of the principle of minimum potential energy has demonstrated several concepts. The use of relative coordinates has been operationally demonstrated for both the strain energy and the work function.

When the derivative expressions just formed in Eq. 3 are substituted into the functional given by Eq. 4) 24 9 Part l The next step in the procedure is to formalize the process of minimizing the functional illustrated in Fig. 2. This requires that we find the rate of change of the general functional with respect to e, the magnitude of the perturbation, and set the derivative to zero. This is accomplished by recognizing that each of the dependent variables, y, Y,x, and Y, xx, in Eqs. 4 are functions of e and by using this fact to form the total derivative of the general functional with respect to e using the chain rule.

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