# Thermal and Stress analysis with the Finite Element Method by Petr Krysl

By Petr Krysl

This textbook presents an available and self-contained description of the Galerkin finite aspect approach for the 2 very important versions of continuum mechanics, brief warmth conduction and elastodynamics, from formula of the governing equations to implementation in Matlab. The assurance follows an intuitive procedure: 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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Later on we will see examples of other mass matrices that results from inaccurate numerical quadrature rules. 4). ) In general we will not be able to satisfy the initial conditions exactly by the chosen trial function. For instance, let us say we wanted to set the initial deflection to a sinusoidal curve. If we use the piecewise linear test function of the form we have discussed so far, we cannot match a smooth sinusoidal curve. ). So, N for instance we can set the initial deflection in the trial function w(x, 0) = i=1 N i (x)wi (0) by computing the degrees of freedom from the interpolation condition as N ¯ (xj ) = w(xj , 0) = W N N N i (xj )wi (0) = Ni (xj )w(i) (0) = i=1 i=1 δji w(i) (0) = w(j) (0), i=1 In words, the degree of freedom at node j is set to the value of the prescribed deflection at node j.

Exercise 14. 3x+1/3 on a two-element mesh on the interval 0 ≤ x ≤ 1. Discuss the approximation of the derivative by the interpolating function. Solution: For variety we will set up the mesh as shown: elements 1:(3,2), 2:(1,3). The interpolating function is constructed element-by-element. The basis functions are associated with nodes, basis function Nj with node j. 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.

4. The broken line cannot be represented as a linear combination of linear functions that are all defined on the whole interval 0 ≤ x ≤ L (only two such functions are linearly independent, and these functions cannot represent the corners in the broken line). Therefore, we have to describe the piecewise linear curve interval by interval. The interpolant may be written as a linear combination of basis functions. In one dimension, the piecewise linear basis function is called the hat function. The six functions that are shown in Fig.