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By Wolfgang Bangerth

Textual content compiled from the fabric offered through the second one writer in a lecture sequence on the division of arithmetic of the ETH Zurich through the summer time time period 2002. thoughts of 'self-adaptivity' within the numerical resolution of differential equations are mentioned, with emphasis on Galerkin finite point versions. Softcover.

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The coordinate origin is in the 50 middle of the bar. The end points of the bar have coordinates x= -L/2 and x=L/2. The natural or local coordinate is defined as ~=2x/L. It means that the end points of the bar have the natural coordinates ~1=-1 and ~2=1. They are dimensionless coordinates. Another basic characteristic of these coordinates is that they are always in the direction of the bar, regardless of the bar orientation in the global (x, y, z) coordinates. The coordinate of any point along the bar is expressed in terms of the coordinates of the end points.

In this case of the beam element the homogeneous differential equation is: 28 ~w = ax4 o. 14) satisfies this differential equation. That is the reason why the element matrix and the final results derived by the beam element are exact. e. a refined element, with a displacement function of a higher order is assumed, the final results of such an element will not be exact. 2. 2. e. mixed variables, and consequently the name of the method: mixed FEM. One can say that the method is based on the application of variational principles which contain potential and complementary energy components.

The potential energy can be expressed in terms of the bending and twisting moments. ~ 2 2 VMXMy + (1 +v) ~xy]dxdY. 13) This expression depends also on the nodal displacements wi' ax; and ay ;. By going deeply into the essence of the problem, one can come to the conclusion that by variation of the potential energy on the normal displacements w; shear forces in direction of the varied displacement are obtained. In addition, by variation of the rotations nodal moments in direction of those rotations should be obtained.

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