Search Results for partial 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.

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.

Computational fluid dynamics (CFD) is one of the tools available for understanding and predicting the performance of thermal-fluids systems. This article qualitatively describes the basic principles of CFD. The numerical methods, such as geometry description and discretization, used to solve the CFD equations are discussed. The article also demonstrates the application of CFD to a few casting problems.

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.

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 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.

This article describes the applications of induction heat treatment of metals, including normalizing, annealing, hardening, and tempering and stress relieving. It discusses the simulation techniques of the electromagnetic and thermal processes that occur during induction heat treating. The article explains the finite-difference method, finite-element method, mutual impedance method, and boundary-element method for the numerical computation of the induction heat treating processes. It also discusses the direct and indirect coupling approaches for coupling the electromagnetic and heat-transfer problems. Modern computer simulation techniques are capable of effectively simulating electromagnetic and thermal phenomena for many processes that involve electromagnetic induction. The article considers the challenges faced by developers of modern simulation software.

Computational fluid dynamics (CFD) is reserved for computationally intensive three-dimensional simulations of thermal fluids systems where nonlinear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD techniques for their solution. It introduces discretization techniques that are used by finite-difference, finite-volume, finite-element, spectral, and some particle methods. Associated concepts of numerical stability and accuracy are also reviewed. The article describes two approaches for grid generation with complex geometries: the use of unstructured grids and the use of special differencing methods on structured grids. The article describes the four-step procedures of the CFD process: geometry acquisition, grid generation and problem specification, flow solution, and post-processing and synthesis. It provides information on the applications of the engineering CFD. Issues and directions for the engineering CFD are also described.

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.

This article provides information on the classification of various composites manufacturing processes based on similar transport processes. The composites manufacturing processes can be grouped into three categories: short-fiber suspension methods, squeeze flow methods, and porous media methods. The article presents an overview of the modeling philosophy and approach that is useful in describing composite manufacturing processes.

This article provides a discussion on probes for laboratory tests and resultant curves of industrial quenching processes. It describes the scope of the tests, and the calculation of heat-transfer coefficient (HTC) based on the tests. The article highlights the differences between the laboratory tests and characterization of industrial quenching processes. It reviews the importance of initial heat-flux density and first critical heat-flux density. The theoretical principle behind and the purpose of the temperature gradient method are discussed. The article provides information on the design of the probe, heat-extraction dynamics, and influence of wetting kinematics. It also includes discussions on the simplified 1-D temperature-distribution model, calculation of the HTC, and the finite-volume method for the heat-conduction equation.

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.

This article provides a comprehensive review and critical assessment of numerical modeling of heat and mass transfer in fusion welding. The different fusion welding processes are gas tungsten arc welding, gas metal arc welding, laser welding, electron beam welding, and laser-arc hybrid welding. The article presents the mathematical equations of mass, momentum, energy, and species conservation. It reviews the applications of heat transfer and fluid flow models for different welding processes. Finally, the article discusses the approaches to improve reliability of, and reduce uncertainty in, numerical models.

Manufacturing processes typically involve the reshaping of materials from one form to another under a set of processing conditions. This article discusses the two classification schemes of modeling for manufacturing processes, namely, on-line or off-line models and empirical, mechanistic, or deterministic models along with their important considerations. It describes the various aspects of modeling of deformation processes, casting operations, and fusion welding processes, with examples.

This article provides a brief historical perspective, a classification of metallurgical processes, basic model development efforts, and an overview of the potential future directions for the modeling of metals processing. It describes the classification of material behavior models, which can be grouped broadly into three classes: statistical, phenomenological, and mechanistic models. The article also presents an overview of the potential directions for the modeling of metals processing.