0.6 Tensors, lagrange methods, and hausdorff measure  (Page 2/2)

For derivations and explanations of these equations, see [2] and [3].

Note that this doesn't mean that any function like u(x) satisfying the equations is a minimizer. In fact, any stationary function for I(u) will satisfy the Euler-Lagrange equations. However, the point for us is that if we can create some sort of functional describing possible truss structures given a set of balanced forces, then we can employ the Euler-Lagrange equations to see which of these truss structures are "stationary," and thus possibly cost minimizers for supporting the given set of point forces.

The next question one might ask is how to construct such a functional for a finite set of point forces.

Using Lagrange Multipliers to Model a Finite System of Point Forces

The Lagrange multiplier method is used to find the maximum and minimum of a function subject to a set of constraints. For example, one could find the maximum and minimum values of the function f(x,y) = 6x + 8y subject to the constraint g(x,y) = x ² + y ² - 1 = 0 (e.g. maximize and minimize the function on the unit circle). Our goal is to minimize the cost of an admissible truss structure subject to constraints regarding the application of a finite number of point forces. For example, given a balanced system of point forces in the plane,

Unknowns :

1. Weights of each possible beam in the structure with wij = wji

2. Li multipliers to balance the x component of the applied forces

3. Mi multipliers to balance the y component of the applied forces

Constraints = balance of force components in each coordinate direction at each point of application. These equations are of the form:

At a point, a0 :

(x-component force, y-component force) = Li [sum of the x-components of the weights of each beam connected at a0] + Mi[sum of the y-components of the weights of each beam connected at a0 ]

As one might imagine, this problem can become complicated to solve in cases which lack symmetry. In the usual basic equilateral triangle example (with a point also at the centroid) that we used, there were still 4 constraint equations (one for each point of application) with an x-component part, y-component part, and all the unknown weights.

One should note that any solution for the simultaneous equations will generate an admissible truss structure. However, solving these equations simultaneously is a difficult problem in higher dimensions, or even when there are just more than four points and forces, none of which are symmetrically arranged. Overall, this is usually an impractical approach for finding a cost-minimizing truss structure given a set of balanced point forces.

Intro to Hausdorff Measure

Definition : If U is a nonempty subset of ${\Re }^n$ , the diameter of U is said to be $\left|U\right|$ = sup $\left\{\right|x-y|$ , x,y $\in U\}$

Definition : Fix $\delta >$ 0. If E $\subset \cup U_i$ where i = 1,..., $\infty$ and $\left|U\right|$ $\le \delta$ , we say that the collection of $U_i$ 's is a $\delta$ -cover of E.

Definition : Let E $\subset R^n$ . For every $\delta >$ 0 and $\alpha \ge$ 0, $\delta$ , $\alpha$ $\in \Re$ , let $H_{\delta }^{\alpha }$ (E) = inf $\{\sum |U_i{|}^{\alpha }$ , E $\subset \cup U_i$ , $|U_i|\le \delta \}$ where i = 1,..., $\infty$ . Then, we get the $\alpha$ -dimensional Hausdorff measure of E by letting $\delta$ go to 0 as we take the sum over all $\delta$ -covers.

So, $H^{\alpha }$ (E) = ${lim}_{\delta \to 0}$ $H_{\delta }^{\alpha }$ (E) = sup $\left\{H_{\delta }^{\alpha }\right\}$ among all $\delta >$ 0 since $H_{\delta }^{\alpha }$ increases as $\delta$ decreases.

One important basic result in developing the notion of Hausdorff measure is that if $\delta$ is strictly less than the distance between two positively separated sets, E and F, then no set in a $\delta$ -cover of $E\cup F$ can intersect both E and F. This means that for some $\alpha >$ 0,

$H_{\delta }^{\alpha }(E\cup F)$ = $H_{\delta }^{\alpha }\left(E\right)$ + $H_{\delta }^{\alpha }\left(F\right)$

Now, suppose we have two values for $\alpha$ , s and t, with s<t and we want to consider the Hausdorff measure of a set E $\subset R^n$ . Then, $H_{\delta }^s$ (E) $\ge$ $\delta s-t$ $H_{\delta }^t$ (E) $\Rightarrow$ if $H_{\delta }^t$ (E)>0, then $H_{\delta }^s$ (E) is infinite as $\delta$ goes to 0.

From this, we can see that there is a unique value for $\alpha$ , called the Hausdorff dimension , d, of E such that $H^{\alpha }$ (E) = $\infty$ if 0 $\le \alpha <dimE$ and $H^{\alpha }$ (E) = 0 if dimE $<\alpha <\infty$ For all of the above in the Hausdorff measure section, see [4].

So given a set E, how do we calculate its Hausdorff dimension?

Definition : Let f:A $\to$ B (and for our purposes, assume, A,B $\subset$ $R^n$ ). The function, f, is said to Holder continuous of exponent $\gamma$ if: sup $\frac{\left|f\left(x\right)-f\left(y\right)\right|}{{\left|x-y\right|}^{\gamma }}$ $<$ $\infty$ among all x,y $\in A$ .

Proposition : If f:A $\to$ B is onto and Holder continuous for some $\gamma$ , then dim(B) $\le$ $\frac{1}{\gamma }$ dim(A), where these are the Hausdorff dimensions of A and B.

Proof : Let $\cup$ $U_i$ , i = 1,..., $\infty$ be a $\delta$ -covering of A for some $\delta >$ 0. Then, f( $\cup$ $U_i$ ) covers B and by the Holder continuity condition, we have that for each $U_i$ , C $|A\cap U_I{|}^{\gamma }$ $>$ $|f(A\cap U_i)|$ , where the bars denote the diameters of those sets and C is a constant.

This implies that for any $\alpha$ , $C^{\frac{\alpha }{\gamma }}$ $|A\cap U_I{|}^{\alpha }$ $>$ $|f(A\cap U_i){|}^{\frac{\alpha }{\gamma }}$

Taking this sum over all $ı$ , we get that $\sum |f(A\cap U_i){|}^{\frac{\alpha }{\gamma }}$ $<$ $C^{\frac{\alpha }{\gamma }}$ $\sum |A\cap U_i{|}^{\alpha }$ $<$ $C^{\frac{\alpha }{\gamma }}$ $\sum |U_i{|}^{\alpha }$ $\Rightarrow$ $H^{\frac{\alpha }{\gamma }}$ (B) $<$ $C^{\frac{\alpha }{\gamma }}$ $H^{\alpha }$ (A)

Now, suppose the $\alpha$ is larger than dim(A). This means that $H^{\alpha }$ (A) = 0 so that $H^{\alpha }$ (B) = 0 as well. Decrease $\alpha$ until it reaches dim(A). Then, dim(B) $\le$ $\frac{1}{\gamma }$ dimA, as desired.

We can use these measures in the context of the Michell problem. Developing the problem in new ways might allow one to use different techniques to find minimal cost structures among admissible configurations.

References

[1] A brief introduction to tensors and their properties,

http://www.brown.edu/departments/engineering/courses/en221/notes/tensors/tensors.htm

[2] Introduction to the Calculus of Variations Imperial College Press, 1992.

[3] Euler-Lagrange equation, http://farside.ph.utexas.edu/teaching/3361/fluidhtml/node181.html, April 2012

[4] K.J. Falconer, The Geometry of Fractal Sets Cambridge University Press, 1985.

[5] Wilfred, Gango. Michell trusses and existence of lines of principal action. 2004.