Preface - HackMD

--- tags: Mathematical modeling and optimization title: Preface --- Preface -- Modeling of an Engineering System === Mechanical engineering involves the precise understanding and application of physical principles. Historically, the development of engineering products relied heavily on empirical knowledge and personal experience, often resulting in multiple trial-and-error iterations to achieve the desired performance. However, advancements in computational technology have made it possible to simulate the functional characteristics of products with greater accuracy. With the rise in computational power, finite element simulations—whether in structural analysis, fluid dynamics, or multi-body dynamics—have become standard tools in product development. These simulations provide detailed insights that help engineers make informed decisions regarding both geometric design and functional configurations. In recent years, big data modeling has gained prominence, allowing the analysis of vast amounts of data, including logs and images. Such modeling techniques, while computationally intensive and costly—often accessible only to large enterprises or academic institutions—can yield significant insights. A thorough analysis may uncover previously unknown factors and dependencies. However, these models can also become non-physical or unrealistic if the quality of the input data is poor. When dealing with system-level analysis, the complexity increases significantly. Each component in a system has its own physical effects, contributing to the overall functionality. Additionally, there are interactions between components—both physical and functional—that add to the complexity. Gaining detailed insights into every component within such a system is often infeasible. This challenge is the focus of this book. Rather than attempting to solve every aspect of a complex system in detail, we concentrate on capturing the key phenomena of each element. Depending on the situation, some components can be simplified into time-dependent or one-dimensional subsystems. In some cases, analytical solutions for these elements may exist. By connecting these sub-solutions, we can construct a comprehensive model for a complex system. The advantage of such generic models lies in their low computational demands. By simplifying the subsystems into 0/1/2-dimensional problems, the computational cost is reduced accordingly. This modeling strategy aligns well with the current industrial focus on optimizing product performance and reducing costs. ### Cascade of simulation and modeling in the industrial environment The simulation and modeling strategies discussed above can be summarized as follows: <img src="https://live.staticflickr.com/65535/53920500227_41b9e88711_b.jpg"> __*Analytical solutions*__ are the most desirable form of mathematical problem-solving. An analytic solution is a direct formula, requiring no numerical or approximation methods. Typically, these formulas are valid only under specific conditions and for relatively simple problems, such as single physical phenomena. In complex scenarios, analytical solutions are rare and are often used only for partial analysis. It is crucial to understand the applicability of these solutions to the problem at hand. __*Data driven statistical*__ modeling is the antithesis of analytical approaches. Instead of solving equations that represent physical laws, this method analyzes data—often through linear layers—to construct a model that describes the observed data. This approach has been highly successful in areas like statistical interpretation and image recognition, where no underlying mathematical equations govern the phenomena. However, in engineering, the availability of large datasets is often limited, and the presence of erroneous measurements can lead to non-physical or unrealistic results. Thus, expert supervision is always necessary. __*Numerical simulations*__ provide another approach to modeling complex interactions. Starting from the physical principles, these simulations approximate solutions to fundamental equations, such as the conservation of mass, momentum, and energy, using numerical methods. This process involves discretizing the domain into infinitesimal elements, whose interactions reveal the macroscopic behavior of the system. Unlike analytical solutions, numerical simulations are not constrained by specific conditions but can be computationally expensive. A hybrid approach can combine numerical methods with analytical solutions. Instead of discretizing every element, __*Numerical modeling*__ uses control volumes and surfaces to describe subsystems. By simplifying non-dominant dimensions or effects, numerical methods or analytical solutions approximate the primary functions of these subsystems. These subsystems are then interconnected to account for multi-directional interactions. Typically, empirical parameters are introduced to account for uncertainties within and between systems, which can be further refined through measurements. This strategy demands a deep understanding of the underlying physical principles. If the relevant physical functions are not accurately represented in the model, the results can be physically nonsensical. ### Content of the lecture This lecture is structured to guide readers step-by-step through the skills and knowledge required to build dynamic models for mechanical systems. - Chapter I covers the basic mathematical tools, such as calculus, differential equations, and numerical methods. These tools are applied to address initial and boundary value problems using numerical methods. - Chapter II introduces the fundamental concepts of mechanical systems. For solid body dynamics, the free body diagram is the starting point for analyzing an object's forces, movements, and dynamic states. This approach also extends to continua (e.g., fluids, heat), where control volume and surface concepts provide essential insights into continuous properties. - Chapter III and IV delves into fluid mechanics, a core subject of this book. This chapter encompasses hydraulics, hydrodynamics, aerodynamics, and gas dynamics. Selected topics with practical applications will be discussed, demonstrating how simple concepts combined with numerical methods can yield powerful engineering tools. - Chapter V focuses on optimization techniques, essential for calibrating modeling parameters and finding optimal design configurations. We will begin with fundamental methods, such as the least squares method, and progress to heuristic optimization techniques, such as genetic algorithms and particle swarm optimization. This chapter serves as a bridge to practical applications, where determining the right parameters completes the engineering model. After mastering these four elements, young engineers will be equipped to analyze individual systems by identifying key functions, simplifying irrelevant dimensions, applying numerical methods to approximate solutions, and using optimization algorithms to find the best design parameters. These steps are crucial in creating engineering software tools, which can become the core technical expertise of a mid-sized enterprise. In an era of digitization, an engineering model that predicts future performance and potential issues can be invaluable for staying ahead of market trends. ### About the Author __Chi- Yao Chang, Dr.-Ing.__ <img style="float: right" width="25%" src="https://i.imgur.com/ZuJhZ8h.jpg"> - 2022 Mar. - Now Senior Development Engineer, Gas Metering, Elster-Honeywell - 2016 Oct. - 2022 Feb. Lead Engineer, CAE, inflator core and applications Aschaffenburg Inflator center, Joyson Safety Systems (Former Takata) - 2014 Feb. – 2016.Sep. Post -Doctoral research, CFD windfarm site assessment, Institute for Wind energy and Energy System technology, Fraunhofer Society - 2009 Apr. – 2014 Jan. PhD, chair of fluid mechanics and aerodynamics , Technical University Darmstadt, Germany