Search Results for differential equations

This article describes the models of physical phenomena involving solution of differential equations such as ordinary or partial differential equations. It reviews the basic concepts of differential calculus and tabulates the expansion of functions into power series. A table of linear partial differential equations is also presented.

Several methods are developed for the numerical solution of partial differential equations, namely, meshed-solution methods such as the finite-element method (FEM), finite-difference method, and boundary-element method; and numerical algorithms consisting of so-called meshed-solution methods. This article introduces the methods of so-called meshed solutions, with an emphasis on the FEM. It presents some basic differential equations that are used to model the responses of structures, components, processes, or systems with emphasis on continuum mechanics. The article provides an outline on the mathematical principles of solving differential equations. It also reviews linear structural problems to illustrate the concept of the FEMs.

Electromagnetic problem solutions are based on the macroscopic theory of the continuous model for the electromagnetic field (EMF). It is described by a system of integral or partial differential equations for five vector quantities, namely, electric field strength, electric flux density, current density, magnetic field strength, and magnetic flux density. This article describes the behavior of the EMF by Maxwell's equations in integral or differential forms. It discusses the definition of potentials; methods of mathematical modeling; boundary conditions; and energy, power density, and electromagnetic forces.

Corrosion modeling is an essential benchmarking element for the selection and life prediction associated with the introduction of new materials or processes. These models are most naturally expressed in terms of differential equations or in other nonexplicit forms of mathematics. This article discusses the principles and applications of various models developed for understanding the corrosion mechanism. These models include mechanistic models, including Pourbaix model, thermophysical module, electrochemical module, and ion association model; risk-based models; and knowledge models. The risk-based model and knowledge models are illustrated with examples for better understanding. The article also describes boundary-element modeling and pitting corrosion fatigue models.

Computational fluid dynamics (CFD) provides an efficient, alternate, virtual approach for simulating and analyzing quenching processes with an impact on component design, manufacturing process, and quality. This article provides domain insights for quenching researchers and CFD practitioners for the modeling of the industrial quenching process and for supporting the diverse multifunctional needs in an industry, ranging from primary metallurgical companies (steel, aluminum, and other alloys), original equipment manufacturers, engineering companies, captive and commercial heat treating facilities, quench system manufacturers, and quench fluid suppliers. It describes the governing differential equations for the fluid flow and heat-transfer phenomena during quenching. The article also discusses different modeling categories to determine a CFD methodology for quenching.

This article focuses on the various assumptions involved in the numerical modeling of welds, including the geometry of the welded structure and the weld joint, thermal stress, strain, residual stress, and the microstructure in the heat-affected and fusion zones.

This article is a comprehensive collection of formulas and numerical solutions, addressing many heat-transfer scenarios encountered in welds. It provides detailed explanations and dimensioned drawings in order to discuss the geometry of weld models, transfer of energy and heat in welds, microstructure evaluation, thermal stress analysis, and fluid flow in the weld pool.

This article reviews the classical models for the pseudo-steady-state temperature distribution of the thermal field around moving point and line sources. These include thick- and thin-plate models and the medium-thick-plate model. The analytical solutions to the differential heat flow equation under conditions applicable to fusion welding are provided. The article also provides an overview of the factors affecting heat flow in a real welding situation using the analytical modeling approach because this makes it possible to derive relatively simple equations that provide the required background for an understanding of the temperature-time pattern.

This article begins with information on fundamental Laplace transformation rules. Some general theorems concerning operations on transforms are provided. The article also discusses the applications of Laplace transforms. It concludes with a table that lists the values of Laplace transforms.

Design and analysis of electrical/electronic systems and components tends to be different from the corresponding process for most mechanical and hydraulic systems. This article provides an overview of three overlapping phases followed in electrical design: functional, electrical, and physical. It also presents information on the simulation and testing carried out to evaluate the design behavior.

Designing of induction heating, or, generally electro technological installations, requires mathematical modeling for solving problems related to various physical phenomena, including electromagnetic (EM), thermal, mechanical, fluidic, and metallurgical fields. This article focuses on the solution of Maxwell's equations (MEs) and provides some basic information regarding the heat transfer and fluid equations, because these physical phenomena usually are strongly coupled to magnetic and electric fields. The solutions are usually obtained by using specific numerical methods such as finite-element method, finite difference method, boundary-element method or volume-integral method, and direct-solution method. The article also discusses the typical structure of commercial codes (preprocessor, solver, and postprocessor) to solve field problems mainly in finite-element method.

This article focuses on approximate closed-form analytical methods, such as slab and upper bound methods, used for forward and inverse design of metal forming problems. Selected examples of application of these methods to metal forming processes are also discussed.

Computational fluid dynamics (CFD) is a computationally intensive three-dimensional simulation of thermal fluids systems where non-linear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD techniques. It introduces some common techniques for discretizing the fluid-flow equations and methods for solving the discrete equations. These include finite-difference methods, finite-element methods, spectral methods, and computational particle methods. The article describes the approaches for grid generation with complex geometries. It discusses the four-step procedures used in the CFD process for engineering design: geometry acquisition, grid generation and problem specification, flow solution, and post-processing and synthesis. The article also provides information on the engineering applications of the CFD. It concludes with a discussion on issues and directions for engineering CFD.

This article is a comprehensive collection of fluid dynamic equations for properties of fluids, fluid statics, fluid motion, dimensional analysis, and boundary layer flow. It presents equations for analyzing problems in fluid mechanics, continuity equation, momentum equation, and energy equation for solving various problems related to fluid dynamics.

This article is a comprehensive collection of formulas, tables, and analytical solutions, addressing hundreds of heat-transfer scenarios encountered in science and engineering. With detailed explanations and dimensioned drawings, the article demonstrates how to set up and solve real-world problems, accounting for material properties, environmental variables, boundary and state conditions, and the primary modes of heat transfer: conduction, convection, and radiation. The article also includes reference data and provides closed-form solutions for common heat-transfer applications such as insulated pipes, cooling fins, radiation shields, and composite structures and configurations.

This article is a comprehensive collection of formulas, tables, and analytical solutions, addressing hundreds of heat-transfer scenarios encountered in science and engineering. It also demonstrates how to set up and solve real-world problems, while accounting for material properties, environmental variables, boundary and state conditions, and the primary modes of heat transfer: conduction, convection, and radiation.

This article provides an overview of experimental, analytical, and numerical tools for temperature evaluation of dry and lubricated systems. It describes the analytical methods and numerical techniques for frictional heating and temperature estimation, as well as viscous heating in full-film lubrication. The article also discusses the viscous heating temperature measurements and numerical analysis of viscous heating.