By Petr Krysl
Version of a taut cord -- the strategy of Galerkin -- Statics and dynamics examples for the cord version -- Boundary stipulations for the version of a taut twine -- version of warmth conduction -- Galerkin strategy for the version of warmth conduction -- Steady-state warmth conduction ideas -- temporary warmth conduction strategies -- increasing the library of aspect varieties -- Discretization mistakes, mistakes keep an eye on, and convergence -- version of elastodynamics -- Galerkin formula for elastodynamics -- Finite components for precise three-D difficulties -- examining the stresses -- aircraft pressure, airplane tension, and axisymmetric versions -- Consistency + balance = convergence
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Extra resources for A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA
With this setup we start computing. There are two test functions, NK = N The trial function for the mnemonic mesh has the form w(x) = N 1 w1 + N 2 1 and NM = N 2 . 29) Now that we have computed the elementwise load vector what do we do with it? We take a dose of reality and note that the degrees of freedom at nodes K, M are really not 1 and 2, but (K) and (M ). Therefore the components of the elementwise load vector are in fact contributions to L(K) and L(M) . To assemble the elementwise load vectors we use the so-called element equation arrays.
The physical unit of the deflection w(x) is [length], and since the basis functions themselves are non-dimensional, the degrees of freedom wi must have the physical units of [length]. Evaluating w(x) at the node k, we obtain N w(xk ) = Ni (xk )wi , i=1 28 Thermal and Stress Analysis with the FEM where the crucial expression is Ni (xk ): by definition, the basis function Nk has value +1 at xk , while all other functions Ni , i = k are zero at xk . 23) where the symbol δik is known as the Kronecker delta δik = 1, if i = k; 0, otherwise.
8 1 To compare their derivatives we just differentiate both the interpolated and interpolating function. 3. The derivative of the interpolating function needs to be again computed element-by-element. Within the extent of element 1 the interpolating function derivative is wh′ (x) = N3′ (x)w3 + N2′ (x)w2 where N3′ (x) = 1/(x3 − x2 ), N2′ (x) = 1/(x2 − x3 ) 34 Thermal and Stress Analysis with the FEM and the coefficients w3 , w2 are as computed above. Consequently we get wh′ (x) = w2 − w3 = −1/5 x2 − x3 that is rise-over-run as expressed from the geometry of a straight line.
A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA by Petr Krysl