![]() ![]() But in this case, we have a problem with a “finite” number of unknowns.Ī system with a finite number of unknowns is called a discrete system.Ī system with an infinite number of unknowns is called a continuous system.įor the purpose of approximations, we can find the following relation for a field quantity \(u(x)\): If we had an infinite bar, we would have an infinite amount of unknowns ( DEGREES OF FREEDOM (DOF)). Nevertheless, we see that irrespective of the polynomial degree, the distribution over the rod is known once we know the values at the nodal points. If we choose a square approximation, the temperature distribution along the bar is much smoother. How can we predict the temperature between these points? A linear approximation is quite good, but there are better possibilities to represent the real temperature distribution. ![]() Let’s assume we know the temperature of this bar at five specific positions (Numbers 1-5 in the illustration). Consider the true temperature distribution T(x) along the bar in the picture below:įigure 2: Temperature distribution along a bar length with linear approximation between the nodal values. In order to get a better understanding of approximation techniques, we will look at a one-dimensional bar. The accuracy with which the variable changes is expressed by some approximation, for example, linear, quadratic, and cubic. These ‘certain points’ are called nodal points and are often located at the boundary of the element. This means we know values at certain points within the element but not at every point. The approximations we just mentioned are usually polynomials and, in fact, interpolations over the element(s). Combining the individual results gives us the final result of the structure. ![]() Calculations are made for every single element. To be able to make simulations, a mesh consisting of up to millions of small elements that together form the shape of the structure needs to be created. Divide and Conquer: Approximations for FEA The different colors are indicators of variable values that help predict mechanical behavior. The books by Zienkiewicz and Strang
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