﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:05.00,Default,,0000,0000,0000,,Let's start with a bird's eye view on the finite-element method--FEM. Dialogue: 0,0:00:05.00,0:00:10.00,Default,,0000,0000,0000,,These days it's the workhorse of almost all mechanical engineers. Dialogue: 0,0:00:10.00,0:00:16.00,Default,,0000,0000,0000,,FEM can answer, for instance, what happens when you put a huge truck on a bridge. Dialogue: 0,0:00:16.00,0:00:21.00,Default,,0000,0000,0000,,To compute this deformation is application of FEM to the static case, Dialogue: 0,0:00:21.00,0:00:24.00,Default,,0000,0000,0000,,but FEM can also be applied in a dynamic setting, Dialogue: 0,0:00:24.00,0:00:28.00,Default,,0000,0000,0000,,for instance, to simulate the effect of a crash on a car body. Dialogue: 0,0:00:28.00,0:00:31.00,Default,,0000,0000,0000,,Here it's important to look at the process of buckling. Dialogue: 0,0:00:31.00,0:00:37.00,Default,,0000,0000,0000,,Whereas in the static case we're not really interested in how the truck was placed on the bridge. Dialogue: 0,0:00:37.00,0:00:39.00,Default,,0000,0000,0000,,It just has to be there. Dialogue: 0,0:00:39.00,0:00:43.00,Default,,0000,0000,0000,,I want to outline three fundamental ideas of the finite element method. Dialogue: 0,0:00:43.00,0:00:46.00,Default,,0000,0000,0000,,The first is discretization. Dialogue: 0,0:00:46.00,0:00:53.00,Default,,0000,0000,0000,,The continuous structures are approximated with the help of--guess what--finite elements-- Dialogue: 0,0:00:53.00,0:00:59.00,Default,,0000,0000,0000,,elements of finite size, not infinitesimal size. Dialogue: 0,0:00:59.00,0:01:05.00,Default,,0000,0000,0000,,When we do so the first question is which geometry these finite elements should have. Dialogue: 0,0:01:05.00,0:01:07.00,Default,,0000,0000,0000,,Should they be tetrahedra? Dialogue: 0,0:01:07.00,0:01:09.00,Default,,0000,0000,0000,,Should they be cubes? Dialogue: 0,0:01:09.00,0:01:11.00,Default,,0000,0000,0000,,Or should they even be curvilinear? Dialogue: 0,0:01:11.00,0:01:15.00,Default,,0000,0000,0000,,The second fundamental idea is that of interpolation. Dialogue: 0,0:01:15.00,0:01:20.00,Default,,0000,0000,0000,,Given the finite elements, how do I compute a value at an arbitrary location? Dialogue: 0,0:01:20.00,0:01:26.00,Default,,0000,0000,0000,,For the static case, a really fundamental idea is that of minimization Dialogue: 0,0:01:26.00,0:01:28.00,Default,,0000,0000,0000,,of the potential energy. Dialogue: 0,0:01:28.00,0:01:32.00,Default,,0000,0000,0000,,Think about a ball that rolls on a terrain of mountains and valleys. Dialogue: 0,0:01:32.00,0:01:36.00,Default,,0000,0000,0000,,Eventually, it's going to come to rest in a valley. Dialogue: 0,0:01:36.00,0:01:39.00,Default,,0000,0000,0000,,In the static case, potential energy is minimized. Dialogue: 0,0:01:39.00,0:01:42.00,Default,,0000,0000,0000,,Let's have a closer look at that valley. Dialogue: 0,0:01:42.00,0:01:48.00,Default,,0000,0000,0000,,In the Gedankenexperiment, it splices this object a little further to the left Dialogue: 0,0:01:48.00,0:01:50.00,Default,,0000,0000,0000,,or a little further to the right. Dialogue: 0,0:01:50.00,0:01:55.00,Default,,0000,0000,0000,,Then the energy stays almost the same because we're at the bottom of the valley, Dialogue: 0,0:01:55.00,0:02:01.00,Default,,0000,0000,0000,,which means that to displace this object in this way require no work. Dialogue: 0,0:02:01.00,0:02:03.00,Default,,0000,0000,0000,,This is the concept of virtual work. Dialogue: 0,0:02:03.00,0:02:07.00,Default,,0000,0000,0000,,For all infinitesimal displacements that are allowed-- Dialogue: 0,0:02:07.00,0:02:11.00,Default,,0000,0000,0000,,we can't go down and we can't go up, obviously-- Dialogue: 0,0:02:11.00,0:02:14.00,Default,,0000,0000,0000,,the virtual work equals 0. Dialogue: 0,0:02:14.00,0:02:19.00,Default,,0000,0000,0000,,In mathematics, this way of posing the problem with the help of virtual work Dialogue: 0,0:02:19.00,0:02:21.00,Default,,0000,0000,0000,,is called a weak form. Dialogue: 0,0:02:21.00,0:02:25.00,Default,,0000,0000,0000,,The strong form would be to ask for all forces to compensate. Dialogue: 0,0:02:25.00,0:02:33.00,Default,,0000,0000,0000,,The weak form asks for the virtual work to be equal to 0 for all allowed virtual displacements. Dialogue: 0,0:02:33.00,0:02:39.00,Default,,0000,0000,0000,,This weak form results in a finite number of equations that we can solve on the computer. Dialogue: 0,0:02:39.00,9:59:59.99,Default,,0000,0000,0000,,This finite number, however, may range in the hundred thousands or even millions.