Do you know of any free engineering software where I can design and test my own models?
Try CAELinux 2009. It's the ubuntu operating system, with all engineering software pre-installed (Salome, Aster, OpenFOAM, Code-Saturn, SciLab etc.) Otherwise, try Salome, I think you can install it on windows. With some add ons it can be used for solving FEM problems. Also SciLab, which is similar to Matlab.
Oh, btw CAELinux 2009 requires a 64bit PC, but you can run it directly off the DVD without having to install anything (but it is faster if it's installed).
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