By Petr Krysl
This textbook presents an obtainable and self-contained description of the Galerkin finite aspect strategy for the 2 vital versions of continuum mechanics, temporary warmth conduction and elastodynamics, from formula of the governing equations to implementation in Matlab. The insurance follows an intuitive process: the salient positive aspects of every preliminary boundary price challenge are reviewed, together with a radical description of the boundary stipulations; the strategy of weighted residuals is utilized to derive the discrete equations; and transparent examples are brought to demonstrate the tactic.
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Extra info for Thermal and Stress analysis with the Finite Element Method
Nf . , Nf . 14) i=1 that makes it clear we have converted the original BVP to a linear-algebra problem of a system of coupled linear differential equations. The matrices are rectangular, because the number of rows 56 Thermal and Stress Analysis with the FEM is the number of free degrees of freedom, whereas the number of columns is the total number of degrees of freedom. 15) i=Nf +1 where the terms on the right-hand side account for the so-called prescribed-displacement loads (loads due to prescribed motion of the supports).
For element 2 we don’t get to assemble any load vector since at neither node of that element the deflections are known. As a result, the right-hand side load vector is identical to the one derived previously. 14 Partitioned form We will change the rules of the game, temporarily, by allowing reactions to make their appearance in the weighted residual statement. e. 32) where we have introduced the definition of the reaction R0 = −P w′ (0). , N ). Since we haven’t eliminated the reaction R0 by the design of the test functions, it will be present in the resulting equations as an additional unknown.
The coefficients of the linear combination wj need to be determined from the interpolating conditions. For the finite element basis functions this is a breeze because of the Kronecker delta property +1, if j = k; 0, otherwise. Nj (xk ) = This gives wh (xk ) = Nj (xk )wj = N1 (xk )w1 + . . + Nk (xk )wk + . . 9 Piecewise linear basis functions 31 so that wk = g(xk ). 0000 The construction of the linear combination is depicted in this figure: And here are the interpolated (solid line) and interpolating (dashed line) functions.