Search Results for linear 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.

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 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 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.

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.

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.

Impact wear can be defined as the wear of a solid surface that is due to percussion, which is a repetitive exposure to dynamic contact by another solid body. This article discusses the volume (or mass) removal of material either at or under engineering contact stress levels and outlines a rational, semi-empirical impact wear theory. It illustrates a linear wear mechanism that occurs in print heads and repetitive impacts that take place in metallic machine contacts. The article concludes with information on plotting a wear curve for an originally plane, massive carbon steel machine platen subjected to repetitive compound impact by a hard, nonwearing spherical-ended steel alloy component.

This article presents various equations that are essential for the modeling of both single-phase and multiphase profiles. It includes the fundamental laws of diffusion, along with its equations and solutions. The article provides information on the series of applications that illustrate how various diffusional processes can be modeled.

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 discusses the deformation and viscoelastic characteristics of plastics as polymeric materials, focusing on the test methods used for the evaluation of their mechanical properties, methods available for analytically predicting the deformation response of polymers, and the effect of viscoelasticity on the test methods used. Two common ways of evaluating viscoelasticity of plastics are by means of creep experiments and dynamic mechanical experiments. Graphic or tabular analysis of test data, time-temperature superposition, and empirical correlation methods are commonly employed for analytical prediction of deformation characteristics of polymers.

This article discusses the conditions for collapse in center-cracked panels and describes the energy criterion for fracture. Measurement of toughness of any material by means of tensile and crack test is discussed. The procedures to be followed for linear elastic fracture mechanics cases are reviewed, along with elastic-plastic fracture mechanics and plastic fracture mechanics procedures with the aid of residual strength diagram. The article also explains the geometry factors needed to determine the toughness of materials.

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.

This article presents typical problems encountered in the analysis of experimental creep and creep-rupture data and the possible solutions to these drawbacks. It provides information on planning the test and creep strain/time relationships. The exponential creep equation and the rational polynomial creep equation are discussed. The article also describes the dependence of stress and temperature on equation parameters and explains the lot-centered regression analysis.

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.

Torsion tests can be carried out on most materials, using standard specimens, to determine mechanical properties such as modulus of elasticity in shear, yield shear strength, ultimate shear strength, modulus of rupture in shear, and ductility. This article discusses the torsional deformation of prismatic bars of circular cross-section and torsional response of prismatic bars of noncircular cross-section. It analyzes the elastic deformation, plastic deformation, and the effect of strain rate on plastic deformation. The article describes the theory of anisotropy in plastic torsion and the various components of a torsion testing machine. These include drive system, test section, torque and rotational displacement transducers, and rigid frame.

There are several main sources of thermophysical property data that provide the most authoritative and comprehensive compilations of critically and systematically evaluated data that are presently available. This article provides thermophysical property data to assist in the materials properties selection for the simulation of casting processes. The measurements of thermophysical property are difficult due to high temperatures and the reactivity of some alloys. The article discusses the strategies adopted to minimize the effects of high temperatures and reactivity of alloys. It presents the thermophysical properties of pure metals and some commercial alloys and tabulates the enthalpy of fusion and solidus and liquidus temperatures for various alloys of commercial interest. The article also lists the density, thermal conductivity, surface tension, and viscosity for some commercial alloys.