Outline About the course Introduction Chapter 1 Introduction to Computational Methods Computational Methods (SECE) November 7, 2022 Computational Methods (SECE) Chapter 1 Outline About the course Introduction 1 About the course 2 Introduction Applications System investigation methods Scientific computing Computational Methods (SECE) Chapter 1 Outline About the course Introduction Focus: on algorithms (principles, scope, and limitations), on (efficient, stable) implementations in Matlab and C++, on numerical experiments (design and interpretation). Computational Methods (SECE) Chapter 1 Outline About the course Introduction Focus: on algorithms (principles, scope, and limitations), on (efficient, stable) implementations in Matlab and C++, on numerical experiments (design and interpretation). Goals: Knowledge of the fundamental algorithms in numerical mathematics Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms Ability to choose the appropriate numerical method for concrete problems Ability to interpret numerical results Ability to implement numerical algorithms efficiently in Matlab and C++. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Introduction Engineering: =⇒ Solving practical technical problems 1 Using scientific and mathematical tools when available, and 2 Using experience and intuition otherwise. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Introduction Engineering: =⇒ Solving practical technical problems 1 Using scientific and mathematical tools when available, and 2 Using experience and intuition otherwise. Numerical Methods: Mathematical models: provide a priori estimates of performance very desirable when prototypes or experiments are costly. Approximate solutions: for engineering applications where exact analytical solutions are not available, allowing the use of approximate numerical methods. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Quantitative Engineering Activities: Analysis and Design Analysis: Predicting the response of a system given a fixed system design and operating conditions. Gain of an electromagnetic antenna (Electrical Engineering) Power output of an electric motor (Electrical/Mechanical Engineering) 0-60 mph acceleration time of a vehicle (Mechanical Engineering) Maximum load a bridge can support (Civil Engineering) Reaction time of a chemical process (Chemical Engineering) Drag force of an airplane (Aerospace Engineering) Expected return of a product portfolio (Industrial and Operations Engineering) Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Design: Determining an ideal system design such that a desired response is achieved. Maximizing a vehicle’s fuel economy while maintaining adequate performance levels by varying vehicle design parameters. Minimizing the weight of a mountain bike while ensuring it will not fail structurally by varying frame shape and thickness. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing System investigation methods Approaches for obtaining knowledge on the properties of systems. 1 theoretical methods, 2 experimental investigations, 3 numerical simulations/scientific computation. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Theoretical methods: analytical considerations of the equations describing the problems, Only very conditionally applicable for practically relevant problems. The equations, which have to be considered for a realistic description of the processes, are usually so complex and are not solvable analytically. Simplifications, often are not valid and lead to inaccurate results (and therefore probably to wrong conclusions). More universally valid approximative formulas, as they are willingly used by engineers, usually cannot be derived from purely analytical considerations for complex systems. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Experimental investigations: aims to obtain the required system information by means of tests (with models or with real objects) using specialized apparatuses and measuring instruments. Measurements at real objects often are difficult or even impossible. For instance, the dimensions are too small or too large. Experiments are prohibited due to safety or environmental reasons. Experiments are often very expensive and time consuming. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Then what should we use? Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Then what should we use? Numerical Simulation The modern development of numerical computing is driven by the rapid increase in computer performance. Numerical results often can be obtained faster and at lower costs. Parameter variations on the computer usually are easily realizable. A numerical simulation often gives more comprehensive information due to the global and simultaneous computation of different problem-relevant quantities. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Scientific computing Is a discipline concerned with the development and study of numerical algorithms. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Scientific computing Is a discipline concerned with the development and study of numerical algorithms. Procedures in scientific computing: 1 Mathematical model: formulated in an attempt to explain and understand an observed phenomenon. 2 Algorithms: to solve the mathematical problem efficiently, accurately, and reliably. Numerical analysis may be viewed as the theory behind such algorithms. 3 Implementation: leads to questions involving programming languages, data structures, computing architectures, etc. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing Figure: 1.1 Scientific computing. Computational Methods (SECE) Chapter 1 Outline About the course Introduction Applications System investigation methods Scientific computing End of Chapter 1 Questions? Computational Methods (SECE) Chapter 1