By Petr Krysl

ISBN-10: 981256876X

ISBN-13: 9789812568762

Version of a taut cord -- the tactic of Galerkin -- Statics and dynamics examples for the twine version -- Boundary stipulations for the version of a taut cord -- version of warmth conduction -- Galerkin strategy for the version of warmth conduction -- Steady-state warmth conduction suggestions -- brief warmth conduction ideas -- increasing the library of aspect forms -- Discretization blunders, errors regulate, and convergence -- version of elastodynamics -- Galerkin formula for elastodynamics -- Finite parts for precise 3-D difficulties -- studying the stresses -- aircraft pressure, airplane tension, and axisymmetric versions -- Consistency + balance = convergence

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**Extra info for A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA**

**Example text**

We say basis function NK is associated with node K. e. associated with node 3): as shown in the Fig. 5, it is nonzero over two elements, 2 and 3; zero everywhere else. e. N3 (x3 )), which is shared by elements 2 and 3. Choosing N3 (x3 ) = 1 has certain advantages, which will be introduced momentarily. Using the concept of Lagrange interpolation polynomials, we may write the function N3 within element 2 as N3 (x) = x − x2 , x3 − x2 x2 ≤ x ≤ x3 , and within element 3 as x − x4 N3 (x) = , x3 ≤ x ≤ x4 .

7) 0 η(x)µw(x, ¨ t) dx = 0 , 0 Here we will also use the trick of setting the test function to zero at the boundary where the reaction (−P w′ (0)) is unknown so that we get L η(x)rB (x, t) dx + η(L)rF (t) = 0 − L 0 L η ′ (x)P w′ (x, t) dx + 0 L η(x)q(x, t) dx + η(L)FL (t) − η(x)µw(x, ¨ t) dx = 0 , 0 where we require η(0) = 0. 4 Mass matrix and load vector 55 At this point the only changes with respect to the statics case are the presence of time in these expressions, and the additional term L η(x)µw(x, ¨ t) dx 0 As we will show next, this term will lead to a mass matrix and an inertial force.

And since a linear function is defined uniquely by two points, we may suspect that wh (x) can exactly (without error) interpolate (or as we say “reproduce”) an arbitrary linear function. In fact a little bit of symbolic manipulation confirms this: We form symbolically the two functions g(x) and wh (x) >> syms A B C x1 x2 x real g=A*x^2+B*x+C; w1 =subs(g,x,x1); w2 =subs(g,x,x2); wh=(x-x2)/(x1-x2)*w1+(x-x1)/(x2-x1)*w2; and then we compute the error simplify(g-wh) ans = A*(x - x1)*(x - x2) The error term is a quadratic expression in x, and it is zero when A = 0.