# Numerical methods of optimal design

Consider an optimisation problem $$\frac{\partial^2}{\partial x^2}\left( \sigma(x) \frac{\partial^2 u(x)}{\partial x^2} \right) = f(x),\; +\; \text{boundary conditions,} \;\;\; 0 \leqslant x \leqslant 1 \\ \int\limits_{0}^{1} \sigma(x) \; dx = 1, \;\;\; \sigma(x) \geqslant 0 \\ \int\limits_{0}^{1} f(x)u(x) dx \to \min\limits_{\sigma(\cdot)}$$ This is a problem of optimal 1-d beam design: $\sigma(x)$ describes thickness of the beam, $u(x)$ describes the beam deflection in point $x$ under active forces $f(x)$. The task is to choose the optimal beam form $\sigma(\cdot)$.

How to solve this problem numerically? I tried to do it in the following manner. At first I counted the solution with constant $\sigma(x) = \sigma_{0}$. Then I divided the segment $[0,1]$ into several parts and tried to find a peacewise linear function $\sigma(x) = \sigma_{1}(x)$ with joints in division points by solving the appropriate finite-dimensional optimisation problem. And I continue such process by subdividing the segment again $[0,1]$ and correcting the previous approximation of target function $\sigma(x)$. This process converges very slowly and I don't know if it converges to the global minimum. Help me please to find an appropriate algorithm.

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Do you know about the finite element method? I suspect you'll need to use elements with better continuity properties than piecewise linear functions to get better convergence. – joriki Apr 6 '12 at 16:55
Yes. It is a method of solving PDE with already given $\sigma(\cdot)$. And maybe there is some cardinally other method ? – Nimza Apr 6 '12 at 16:57