Numerical Solution of Differential Equations by Isaac Fried and Werner Rheinboldt (Auth.)
By Isaac Fried and Werner Rheinboldt (Auth.)
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Extra resources for Numerical Solution of Differential Equations
Sample text
Changing the nodal values yi>yii · · · »^5 i n Fig. 1 causes the configuration of y to change and in this way the nodal values play the role of the as in Eq. 1). But, while the as in the Ritz method are not of immediate interest, the finite element weights or nodal values are of direct interest. Fig. 1 Piecewise linear displacement trial func tion over four finite elements. VJV of the same finite element position are said, in the terminology of functional anal ysis, to lie in a finite dimensional function space y.
Total Potential Energy of the Thin Elastic Beam The differential equation governing the deflection y of a thin elastic beam which is elastically supported is given in Eq. 29). , continuous and with a continuous first derivative) and that satisfies the con ditions y = 0, y' = 0 wherever y — y' = 0 occurs. These are the essential boundary conditions of the beam problem. The remaining higher-order con ditions are natural. From the definition of || · ||2 in Eq. 54) 7. INDEFINITE VARIATIONAL PRINCIPLES 43 To see that the boundary terms in Eq.
Euler-Lagrange Equations The thrust of Sections 2 and 3 was to prove that the solution to the bound ary value problem minimizes the total potential energy. Next we approach the principle of total potential energy from the other end; using the techni ques of the calculus of variations, we show that the function that minimizes 5. EULER-LAGRANGE EQUATIONS 41 n(y) in Eq. 12). Let y be the function that minimizes n{y) in Eq. 20). We perturb y by a virtual displacement sy in which ε is a scalar variable and y a continuous function of finite energy and such that y(0) = 0.