## Finite Element Analysis with Error Estimators: An by J. E. Akin

By J. E. Akin

This key textual content is written for senior undergraduate and graduate engineering scholars. It gives you a whole advent to finite point tools and to automated edition (error estimation) that would permit scholars to appreciate and use FEA as a real engineering instrument. it's been in particular built to be available to non-mathematics scholars and gives the single entire textual content for FEA with mistakes estimators for non-mathematicians. blunders estimation is taught on approximately 1/2 all FEM classes for engineers at senior undergraduate and postgraduate point; no different current textbook for this industry covers this subject. *

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**Example text**

061, respectively. These compare well with the previous results. If the second boundary condition had been applied other than at x = 0 then we would have a more complicated relation between the Φ. For example, assume we move the boundary condition to x = 0. 5. Then evaluating the approximate solution there yields u(0. 5) = 0. 5 Φ1 + 0. 25 Φ2 + 0. 125 Φ3 which is called a linear constraint equation on Φ, or a multipoint constraint (MPC). 2 User software interfaces in MODEL Chapter 2, Mathematical preliminaries 39 other words we would have to solve the weighted residual algebraic system subject to a linear constraint.

Results Quadratures Solution Geometry Coordinates ! ! ! ! ! ! ! ... Partial Notations List COORD = SPATIAL COORDINATES OF ELEMENT’S NODES DGH = GLOBAL DERIVATIVES OF INTERPOLATION FUNCTIONS G = GEOMETRIC INTERPOLATION FUNCTIONS H = SCALAR INTERPOLATION FUNCTIONS LT_FREE = NUMBER OF DEGREES OF FREEDOM LT_GEOM = NUMBER OF GEOMETRY NODES LT_PARM = DIMENSION OF PARAMETRIC SPACE LT_QP = NUMBER OF QUADRATURE POINTS LT_N = NUMBER OF NODES PER ELEMENT N_SPACE = DIMENSION OF PHYSICAL SPACE PT = QUADRATURE COORDINATES WT = QUADRATURE WEIGHTS ...

The weak form given by Eq. 14 is also referred to as the dual problem. If both the original weak form and the dual problem are solved it is possible to compute both an upper bound and a lower bound of the error in the approximation. Having both bounds is not always worth the extra computational cost. To illustrate how to classify the boundary conditions, or to establish a dual problem, consider the model differential equation d 2u L(u) = x ∈ ]0, 1[ dx 2 has the inner product 1 1 d2 u < v, L(u) > = dx.