# Tutorial This section provides a hands-on tutorial on the basic usage of ALFE. ## Introduction ## 2D Cantilever Here is a very basic example. We can start from creating a simple 2D cantilever with 6 mm x 4 mm. It is meshed with 24 four-node plane-stress elements. The material is assumed with Young's modulus of 1 MPa and Poisson's ratio of 0.3. ### 1. Define a cantilever Define nodes and elements. The number of nodes in the two directions can be represented by `xnum`and `ynum`. ```csharp List nodes = new List(xnum * ynum); List elems = new List((xnum - 1) * (ynum - 1)); ``` Then, all nodes can be found through two for loops. ```csharp for (int i = 0; i < xnum; i++) { for (int j = 0; j < ynum; j++) { nodes.Add(new Node(i, j)); } } ``` Constructing a quadrilateral element (a pixel element) requires a nodal set and a specified material. The material can be created by the code below. ```csharp var materal = new Material(1.0, 0.3); // E = 1.0, u = 0.3 ``` Next, elements can be defined easily through inputting the **counterclockwise** vertex indices. ```csharp for (int i = 0; i < xnum - 1; i++) { for (int j = 0; j < ynum - 1; j++) { List nodalSet = new List(4) { // Counterclockwise nodes[i * ynum + j], nodes[i * ynum + (j + 1)], nodes[(i + 1) * ynum+ (j + 1)], nodes[(i + 1) * ynum + j], }; // Construct a pixel element elems.Add(new Pixel(nodalSet, material)); } } ``` ### 2. Define loads and boundary conditions Now, let's set loads and boundary conditions. So, we need to define two lists for loads and supports. The number of loads is 1 because we just apply a load on the center of the one side. Another side of the cantilever should be fixed, so the number of supports should be `ynum`. ```csharp List loads = new List(1); List supports = new List(ynum); ``` The easiest way to define supports is adding a `if` structure when defining nodes. ```csharp for (int i = 0; i < xnum; i++) { for (int j = 0; j < ynum; j++) { nodes.Add(new Node(i, j)); if (i == 0) supports.Add(new Support(j, SupportType.Fixed)); } } ``` Also, we can compute the index of the loaded node according to `ynum`. A 2D vector is used to present a force. ```csharp loads.Add(new Load(nodes.Count - (int)Math.Ceiling(ynum / 2.0), new Vector2D(0.0, -1.0))); ``` ### 3. Do FEA Before doing FEA, we need to create a model and a FE system through assembling above lists. ```csharp // Create a model Model model = new Model(2, nodes, elems, loads, supports); // Create a FE system FESystem system = new FESystem(model); ``` After that, we can initialize the system and solve it. ```csharp // Initialize the system and solve it. system.Initialize(); system.Solve(); ``` Finally, let's print all displacements ```csharp // Print the displacement information Console.WriteLine(system.DisplacementInfo()); Console.ReadKey(); ``` ### 4. Complete Code ```csharp static void Main(string[] args) { int xnum = 7; int ynum = 5; List nodes = new List(xnum * ynum); List elems = new List((xnum - 1) * (ynum - 1)); List loads = new List(1); List supports = new List(ynum); // Define nodes for (int i = 0; i < xnum; i++) { for (int j = 0; j < ynum; j++) { nodes.Add(new Node(i, j)); if (i == 0) supports.Add(new Support(j, SupportType.Fixed)); } } // Define Material Material material = new Material(1.0, 0.3); // Define elements for (int i = 0; i < xnum - 1; i++) { for (int j = 0; j < ynum - 1; j++) { List nodalSet = new List(4) { // Counterclockwise nodes[i * ynum + j], nodes[i * ynum + (j + 1)], nodes[(i + 1) * ynum+ (j + 1)], nodes[(i + 1) * ynum + j], }; // Construct a pixel element elems.Add(new Pixel(nodalSet, material)); } } // Apply the load loads.Add(new Load(nodes.Count - (int)Math.Ceiling(ynum / 2.0), new Vector2D(0.0, -1.0))); // Create a model Model model = new Model(2, nodes, elems, loads, supports); // Create a FE system FESystem system = new FESystem(model); // Initialize the system and solve it. system.Initialize(); system.Solve(); // Print the displacement information Console.WriteLine(system.DisplacementInfo()); Console.ReadKey(); } ``` **Output:** ``` ------------------- Displacement Info ------------------- 0 0 0 0 0 0 0 0 0 0 -0.122919399521718 -0.267920998495105 -0.0500423086666319 -0.227484064738565 1.36218031146597E-16 -0.223986691884282 0.0500423086666321 -0.227484064738564 0.122919399521718 -0.267920998495105 -0.162856935787937 -0.507809155006323 -0.086802959069945 -0.480393755317766 1.70459255392177E-16 -0.47401205480756 0.0868029590699452 -0.480393755317765 0.162856935787938 -0.507809155006322 -0.202833069455686 -0.735390869945012 -0.120585272382518 -0.735603485210006 8.67636025787084E-17 -0.755717881657478 0.120585272382518 -0.735603485210005 0.202833069455686 -0.735390869945011 -0.251617425739746 -0.900771587783815 -0.164473424850421 -1.00547944641779 -2.12119789659344E-16 -1.09692038996175 0.164473424850421 -1.00547944641778 0.251617425739745 -0.900771587783813 -0.145756152264201 -0.902954826576372 -0.18641612221033 -1.26756653910868 -6.15018065870458E-16 -1.59977503918858 0.186416122210329 -1.26756653910868 0.1457561522642 -0.90295482657637 0.176305974828536 -0.907973273096894 0.1868720640062 -1.23653764484316 -1.21662657260202E-15 -2.5576494344298 -0.186872064006202 -1.23653764484316 -0.176305974828538 -0.907973273096892 ``` --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. 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