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Let's start with a bird's eye view on the finite-element method--FEM.
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These days it's the workhorse of almost all mechanical engineers.
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FEM can answer, for instance, what happens when you put a huge truck on a bridge.
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To compute this deformation is application of FEM to the static case,
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but FEM can also be applied in a dynamic setting,
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for instance, to simulate the effect of a crash on a car body.
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Here it's important to look at the process of buckling.
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Whereas in the static case we're not really interested in how the truck was placed on the bridge.
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It just has to be there.
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I want to outline three fundamental ideas of the finite element method.
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The first is discretization.
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The continuous structures are approximated with the help of--guess what--finite elements--
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elements of finite size, not infinitesimal size.
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When we do so the first question is which geometry these finite elements should have.
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Should they be tetrahedra?
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Should they be cubes?
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Or should they even be curvilinear?
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The second fundamental idea is that of interpolation.
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Given the finite elements, how do I compute a value at an arbitrary location?
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For the static case, a really fundamental idea is that of minimization
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of the potential energy.
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Think about a ball that rolls on a terrain of mountains and valleys.
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Eventually, it's going to come to rest in a valley.
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In the static case, potential energy is minimized.
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Let's have a closer look at that valley.
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In the Gedankenexperiment, it splices this object a little further to the left
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or a little further to the right.
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Then the energy stays almost the same because we're at the bottom of the valley,
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which means that to displace this object in this way require no work.
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This is the concept of virtual work.
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For all infinitesimal displacements that are allowed--
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we can't go down and we can't go up, obviously--
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the virtual work equals 0.
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In mathematics, this way of posing the problem with the help of virtual work
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is called a weak form.
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The strong form would be to ask for all forces to compensate.
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The weak form asks for the virtual work to be equal to 0 for all allowed virtual displacements.
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This weak form results in a finite number of equations that we can solve on the computer.
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This finite number, however, may range in the hundred thousands or even millions.