Talk:Muhamad akbar fuad

Question 1: Can you derive finite element equations from 1D PDE governing equations using weighted residual in a very simple way along with an example?

Certainly! Deriving the finite element equations from a 1D partial differential equation (PDE) using the weighted residual method is a great way to understand the basics of the finite element method (FEM). Let’s go through it step-by-step with a simple example.

Problem Setup: 1D PDE Example Consider the 1D steady-state heat conduction equation with no internal heat generation, as shown below: d/dx (k du/dx)=0 ;for x∈[0,L]

Step 1: Formulate the Weak Form (Weighted Residual Form) To derive the finite element equations, we start with the weak form of the governing equation. This involves:

  1.Multiplying the differential equation by a test function v(x),
  2.Integrating over the domain,
  3.Applying integration by parts to reduce the order of derivatives on u.

Step 2: Discretize the Domain and Approximate u(x) Now, we approximate u(x) using finite element shape functions. Suppose we divide the domain into two linear elements, and approximate u(x) as: u(x) ≈ u1*N1(x) + u2*N2(x) where: N1(x) and N2(x) are linear shape functions, u1 and u2 are the nodal values of u at nodes 1 and 2.

Step 3: Apply the Weighted Residual Method to Each Element For each element, we substitute the approximated u(x) and v(x) (chosen as the shape functions) into the weak form.

  1.Evaluate the integrals
  2.Compute the stiffness matrix entries for each element
  3.Assemble the Global System of Equations
  4.Apply Boundary Conditions

Summary This procedure gives you a system of linear equations:

K∙U=F

where: K is the global stiffness matrix, U is the vector of unknown nodal values, F is the force vector, which includes contributions from the Neumann boundary condition. This example demonstrates the key steps in using the weighted residual method to derive finite element equations from a 1D PDE. Let me know if you need further clarification on any specific step!