0:00:00.000,0:00:05.000 Let's start with a bird's eye view on the finite-element method--FEM. 0:00:05.000,0:00:10.000 These days it's the workhorse of almost all mechanical engineers. 0:00:10.000,0:00:16.000 FEM can answer, for instance, what happens when you put a huge truck on a bridge. 0:00:16.000,0:00:21.000 To compute this deformation is application of FEM to the static case, 0:00:21.000,0:00:24.000 but FEM can also be applied in a dynamic setting, 0:00:24.000,0:00:28.000 for instance, to simulate the effect of a crash on a car body. 0:00:28.000,0:00:31.000 Here it's important to look at the process of buckling. 0:00:31.000,0:00:37.000 Whereas in the static case we're not really interested in how the truck was placed on the bridge. 0:00:37.000,0:00:39.000 It just has to be there. 0:00:39.000,0:00:43.000 I want to outline three fundamental ideas of the finite element method. 0:00:43.000,0:00:46.000 The first is discretization. 0:00:46.000,0:00:53.000 The continuous structures are approximated with the help of--guess what--finite elements-- 0:00:53.000,0:00:59.000 elements of finite size, not infinitesimal size. 0:00:59.000,0:01:05.000 When we do so the first question is which geometry these finite elements should have. 0:01:05.000,0:01:07.000 Should they be tetrahedra? 0:01:07.000,0:01:09.000 Should they be cubes? 0:01:09.000,0:01:11.000 Or should they even be curvilinear? 0:01:11.000,0:01:15.000 The second fundamental idea is that of interpolation. 0:01:15.000,0:01:20.000 Given the finite elements, how do I compute a value at an arbitrary location? 0:01:20.000,0:01:26.000 For the static case, a really fundamental idea is that of minimization 0:01:26.000,0:01:28.000 of the potential energy. 0:01:28.000,0:01:32.000 Think about a ball that rolls on a terrain of mountains and valleys. 0:01:32.000,0:01:36.000 Eventually, it's going to come to rest in a valley. 0:01:36.000,0:01:39.000 In the static case, potential energy is minimized. 0:01:39.000,0:01:42.000 Let's have a closer look at that valley. 0:01:42.000,0:01:48.000 In the Gedankenexperiment, it splices this object a little further to the left 0:01:48.000,0:01:50.000 or a little further to the right. 0:01:50.000,0:01:55.000 Then the energy stays almost the same because we're at the bottom of the valley, 0:01:55.000,0:02:01.000 which means that to displace this object in this way require no work. 0:02:01.000,0:02:03.000 This is the concept of virtual work. 0:02:03.000,0:02:07.000 For all infinitesimal displacements that are allowed-- 0:02:07.000,0:02:11.000 we can't go down and we can't go up, obviously-- 0:02:11.000,0:02:14.000 the virtual work equals 0. 0:02:14.000,0:02:19.000 In mathematics, this way of posing the problem with the help of virtual work 0:02:19.000,0:02:21.000 is called a weak form. 0:02:21.000,0:02:25.000 The strong form would be to ask for all forces to compensate. 0:02:25.000,0:02:33.000 The weak form asks for the virtual work to be equal to 0 for all allowed virtual displacements. 0:02:33.000,0:02:39.000 This weak form results in a finite number of equations that we can solve on the computer. 0:02:39.000,9:59:59.000 This finite number, however, may range in the hundred thousands or even millions.