FEM/CFD Lab (ME-505) - Important Questions

Derive the weak form of the boundary value problem $-\nabla\cdot\left(k\nabla u\right)=f$ in a domain $\Omega$ with essential boundary condition $u=u_0$ on $\partial\Omega_u$ and natural boundary condition $-k\nabla u\cdot\mathbf{n}=t$ on $\partial\Omega_t$. Obtain the finite element (Galerkin) formulation using shape functions $N_i$ and show the expression for the element stiffness matrix.

For a 1D bar element of length $L$, Young's modulus $E$ and cross-sectional area $A$, obtain the element stiffness matrix using linear shape functions. Show the corresponding expression for the stiffness integrand and state how it changes for quadratic shape functions. Use the expression $k_e=\int_0^L EA\,\frac{dN^T}{dx}\frac{dN}{dx}\,dx$ as part of your derivation.

Explain the procedure to assemble the global stiffness matrix from individual element stiffness matrices for a 2D triangular finite element mesh. Describe how essential (Dirichlet) and natural (Neumann) boundary conditions are imposed in the assembled system.

Define common mesh quality metrics such as element aspect ratio and skewness. Describe residual-based a posteriori error estimators and explain how they guide adaptive mesh refinement.

Derive the weak (variational) form of the steady incompressible Navier--Stokes equations given by $\rho\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}=-\nabla p+\mu\nabla^2\mathbf{u}+\mathbf{f}$ with $\nabla\cdot\mathbf{u}=0$. Discuss why equal-order interpolation for velocity and pressure may require stabilization and outline the basic idea of the SUPG or Petrov--Galerkin stabilization.

For the steady 1D convection--diffusion equation $\rho u\,\frac{d\phi}{dx}=\frac{d}{dx}\left(\Gamma\frac{d\phi}{dx}\right)+S$, derive the finite volume discretized algebraic equation on a uniform grid using central differencing for the convective and diffusive fluxes. Discuss the stability/accuracy limitations of central differencing and state the criterion in terms of the cell Peclet number.

Compare direct solvers and iterative solvers for large sparse linear systems arising from FEM/CFD discretizations. Explain the basic principle of the Conjugate Gradient method and the role of preconditioning in improving convergence.

Explain the SIMPLE algorithm for steady incompressible flows. Starting from the discretized momentum equations (symbolically written as $A\mathbf{u}=\mathbf{H}-\nabla p$) and the continuity constraint $\nabla\cdot\mathbf{u}=0$, derive the pressure correction equation and outline the steps to update velocity and pressure until convergence.

Describe the procedures for verification and validation of CFD results. Define the $L_2$ error norm for a numerical solution $\phi_h$ against an exact solution $\phi$ and explain how the Grid Convergence Index (GCI) is used to estimate discretization uncertainty.

List and explain the essential components of a FEM/CFD lab report, including mesh description, boundary conditions, solver settings, convergence history, validation checks, and recommended figures/tables for clear presentation of results.