MECN4039A Tools III — Lectures 1 & 2: Study Notes, Flashcards, and Practice Test Summary & Study Notes

📘 Lecture 1 — Introduction and Basic Concepts

Course structure & assessment This lecture outlines a project-based course linked to engineering streams (Aero, Ind, Mech) and lists specialist components such as CFD, FEA, and Discrete Event Simulation. Assessment is broken into multiple components (tutorials, assignments, exam-equivalent parts) each contributing percent weights to the final grade.

Why numerical methods? Numerical methods let us approximate physical systems that are not analytically tractable by converting continuous models into discrete, computable operations. Implementations target basic chip operations (add/subtract) and trade analytical exactness for computational tractability.

Precision and rounding Digital computers store numbers in binary with finite precision: single precision (~7 decimal digits) and double precision (~16 decimal digits). Be aware of truncation and rounding errors which accumulate during repeated operations.

Randomness and PRNGs Pseudo-random numbers are generated deterministically from a seed using algorithms like the linear congruential generator with the general form $X_{n+1} = (a X_n + c) \bmod m$. PRNGs are deterministic and periodic, but are suitable for stochastic modelling when used appropriately.

Statistics and distributions Stochastic processes are modelled probabilistically. Transformations of uniform random variables produce common distributions. For example, the Box–Muller transform gives standard normal samples from uniforms using $Z_0 = \sqrt{-2\ln U_1}\cos(2\pi U_2)$ and $Z_1 = \sqrt{-2\ln U_1}\sin(2\pi U_2)$.

Other numerical operations Common operations include quadrature (numerical integration) using Newton–Cotes (trapezoidal, Simpson) or Gaussian rules. Differentiation uses finite-difference schemes (forward, central, backward) and linearisation approximates nonlinear behaviour with local linear models to improve computational efficiency.

Matrices and performance Discretised physical models produce large, often very sparse matrices (~lots of zeros). Solver choices include direct methods (LU, Cholesky) which are exact in finite arithmetic but costly, and iterative methods (Jacobi, Conjugate Gradient, GMRES) which exploit sparsity and are memory efficient for large systems.

Ordinary differential equations (ODEs) Many physical systems are modelled by ODEs. The typical approach is to convert higher-order ODEs into first-order systems by introducing state variables (e.g., $x_1 = x$, $x_2 = x'$) and then solve via matrix methods or standard integrators. Similar ideas extend to PDE discretisation.

📗 Lecture 2 — Intermediate Concepts and General Workflow

Boundary conditions A boundary condition specifies known values of the solution in space/time. Common types are Dirichlet (specify the exact variable value at the boundary) and Neumann (von Neumann) (specify the flux or derivative normal to the boundary). Poor specification yields poor results (GIGO).

Discretisation and resolution Discretisation breaks continuous mathematical elements into smaller parts (spatial, temporal or both). Resolution is the minimum detail captured at that discretisation. Increasing resolution increases computational cost. Biased discretisation uses finer resolution where high gradients occur.

General workflow for computational modelling A typical workflow: identify phenomenon → select solution approach → define domain → specify solver and linearisation → define boundary conditions → solve → monitor and refine → verify & validate → iterate → post-process → interpret and act. Monitoring intermediate indicators is important to ensure convergence and correctness.

Domain definition The domain is the computational space/time region representing the quantity of interest. It should be large enough to include relevant effects but as small as possible to reduce computation time.

Solver specification and tuning Solvers convert calculus-based governing equations into algebraic systems via discretisation and linearisation, then solve iteratively. Modelling approximates real behaviour; tuning adjusts model parameters for better agreement with reference data.

Residuals, error, and monitors A residual is the difference between predicted function values and known ones (mathematical consistency). Error compares simulation results to the true or reference solution. Low residuals indicate consistency but do not guarantee correctness. Solution monitoring tracks indicators (residual norms, physical quantities) for convergence.

Verification vs validation Verification asks “did I build the model correctly?” (compare to analytical/theoretical or alternative-solutions). Validation asks “did I build the correct model?” (compare to experimental or published physical data). Both processes are needed for confidence: solutions may be verified but invalid or vice versa.