Computational methods in aeronautics

The objective of the course is to introduce students to the fundamentals of computer-based simulation and analysis of aerospace engineering problems. Students are first acquainted with the theoretical foundations of computational methods and then gain practical experience through their implementation on a computer by solving problems from various aerospace disciplines (aerodynamics, flight mechanics, aircraft structures, etc.). The course is designed so that selected representative problems are addressed comprehensively—from the formulation of the mathematical model to its numerical solution and analysis of results. Particular emphasis is placed on understanding the capabilities of computers in solving engineering problems of varying complexity, modeled by algebraic, integral, and differential equations.

Upon successful completion of the course, the student is able to: • explain the theoretical foundations and limitations of key numerical methods (accuracy, stability, convergence, etc.); • formulate mathematical models of aerospace problems (aerodynamics, flight mechanics, aircraft structures) in the form of algebraic, integral, and differential equations; • select an appropriate numerical method with respect to the type of problem, required accuracy, and available computational resources; • implement basic numerical algorithms in a programming environment (e.g., solving systems of equations, numerical integration, methods for ordinary differential equations); • apply finite difference methods and the finite element method (FEM) to simple engineering problems and interpret the obtained results; • verify and validate numerical results (error analysis, comparison with analytical solutions/reference data, convergence studies); • analyze the sensitivity of solutions to input parameters and discuss the reliability of results; • integrate knowledge from mathematics, programming, mechanics, and fluid mechanics in solving specific aerospace problems; • document and clearly present the modeling process, numerical solution, and analysis of results.

Theoretical teaching covers the fundamental concepts of applying computational methods to solving engineering problems in aerospace engineering. An introductory part addresses the general approach to solving engineering problems using computers, including all phases—from problem definition, formulation of mathematical and numerical models, algorithm development, to implementation, testing, verification, and interpretation of results. Special attention is devoted to: • fundamentals of programming in the MATLAB environment (data types, operators, control structures, functions, and scripts); • solving problems modeled by algebraic equations, with applications in gas dynamics and standard atmosphere models; • fundamental concepts of function approximation (Taylor polynomials, interpolation, splines, curve fitting); • numerical integration (rectangle rules, trapezoidal and Simpson’s rule, multiple integrals); • numerical solution of ordinary differential equations (Euler method, Runge–Kutta methods, error, stability, and stiffness analysis); • fundamentals of the finite element method (FEM), including discretization, formation of the stiffness matrix, application of boundary conditions, and solution of large systems of equations. Theoretical instruction is supported by the analysis of typical aerospace problems such as gas flow, flight dynamics, and structural stress analysis.

Practical teaching is carried out through computer-based exercises in the MATLAB environment and is focused on applying theoretical knowledge through the development and implementation of numerical algorithms. It includes: • implementation of basic programming structures and development of simple functions and scripts; • development of programs for calculating standard atmosphere parameters and one-dimensional flow problems (isentropic, Rayleigh, Fanno flow, shock waves); • application of approximation and interpolation methods to real datasets; • implementation of numerical integration and its application to the calculation of geometric characteristics and aircraft performance (e.g., takeoff run); • development and testing of programs for solving ordinary differential equations and simulation of aircraft motion (takeoff, maneuvers, specific motion cases); • application of the finite element method to simple structures (2D and 3D trusses), including system assembly, solution, and stress analysis; • verification, validation, and analysis of obtained results, as well as their visualization. Practical instruction is organized so that students independently carry out the entire process—from problem formulation to the development of a functional computational solution and interpretation of results.

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Computer classroom "Simlab" - 452 Z. Petrović, S. Stupar, Computer-Aided Design – Finite Difference Method, Faculty of Mechanical Engineering, Belgrade Cvetković, A., Radojević, S., MATLAB 1, Faculty of Mechanical Engineering, Belgrade Lecture notes. Exercise materials. Internet resources

Total assigned hours: 75

New material: 21
Elaboration and examples (recapitulation): 9

Auditory exercises: 15
Laboratory exercises: 0
Calculation tasks: 10
Seminar paper: 5
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 5
Review and grading of the project: 0
Test: 5
Test: 0
Final exam: 5

Activity during lectures: 0
Test/test: 30
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 40
Project: 0
Final exam: 30
Requirement for taking the exam (required number of points): 0

Cvetković, A., Radojević, S., MATLAB 1, Faculty of Mechanical Engineering, Belgrade; Z. Petrović, S. Stupar, Computer-Aided Design – Finite Difference Method, Faculty of Mechanical Engineering, Belgrade