This article provides a summary of the overall development of the finite element method (FEM) and its contribution to the materials forming industry. It presents an overview of FEM methodologies and applications in the order of their usage in typical manufacturing (bulk forming process) process sequence: primary materials processing, hot forging and cold forming, and product assembly. The article discusses the material fracture and dies stress analysis and presents the optimization techniques used in 2-D and 3-D preform die design.

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.

This article provides a summary of the overall development of the finite element method (FEM) and its contribution to the materials forming industry. It focuses on the overall philosophy and evolution of the FEM for solving bulk forming issues. A number of applications of FEM are presented in the order they would be used in a typical manufacturing process sequence: primary materials processing, hot forging and cold forming, and product assembly. The article discusses four FEM modules: the deformation model, the heat-transfer model, the microstructural model, and the carbon diffusion model. The article also covers material fracture and die stress analysis and reviews optimization of the design of forming processes.

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 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.

Simulation programs are becoming more effective tools in reducing the need for physical testing and the avoidance of costly downstream problems by solving the problems upfront in the early development stage. This article provides a brief review of the history and applied analysis of simple forming operations. It focuses on metal stamping simulation based on the finite-element methods or model (FEM) with emphasis on software tools using the three-dimensional FEM technology. The article discusses two aspects of particular importance in finite-element analysis of sheet forming and springback analysis: the type of solution algorithm/governing equation and the type of element. The article provides information on various models for material yield criteria.

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 summarizes many approaches that are used to simulate relaxation of bulk residual stresses in components. It presents analytical examples to highlight the complexity of residual stress and strain distributions observed in simple geometries, with ideal material behavior and trivial loading and boundary conditions. The article discusses approximate and advanced solution techniques that can be employed in practice for simulation of residual stress relief: finite-difference method and finite-element method. It also describes advanced techniques applicable to transient creep, advanced constitutive models, and complicated stress and temperature loading histories.

Ultrasound is an ideal modality for nondestructive evaluation (NDE) because it enables the interior of objects to be assessed without the safety and access issues associated with radiography. This article summarizes the history of array usage in NDE and its relationship to medical applications. It discusses the mathematics behind classical beamforming, full matrix capture, and total focusing methods of imaging. The article shows how ultrasonic array data can be simulated by direct numerical methods (most commonly using finite-element methods), analytical methods, or hybrid methods. It also considers various methods of comparing the performance of arrays and imaging algorithms. The article provides a comparison of various advanced and nonlinear imaging algorithm and looks at some practical industrial applications of arrays. It concludes with some future perspectives for arrays in NDE.

This article introduces the fundamental sources of residual stresses common to most manufacturing processes. It explains the effect of material removal on residual stresses and distortions in a part. The article assists the reader in making a choice between trial-and-error and computer-simulation approaches for the control of residual stresses. It provides a summary of the commonly used techniques of measuring residual stresses. The article also describes the finite element method for predicting residual stresses caused by various manufacturing processes. It concludes with a discussion on the operations involved in thermal and mechanical stress-relief methods.

This article describes a method to predict mechanical properties of cast iron materials and illustrates how to use the predictions in computer-aided tools for the analysis of castings subjected to load. It outlines some ways to predict the hardness and elastic modulus of cast iron without going into dislocation theory. The article discusses modeling of hardness in cast iron based on a regular solution equation in which the properties of each phase depend on chemical composition and coarseness. It describes the evaluation of material parameters from the tensile stress-strain curve. The article concludes with an illustration of a finite-element method (FEM) model containing heterogeneous mechanical properties using local material definitions.

This article discusses the numerical simulation of the forming of aluminum alloy sheet metals. The macroscopic and microscopic aspects of the plastic behavior of aluminum alloys are reviewed. The article presents constitutive equations suitable for the description of aluminum alloy sheets. It explains testing procedures and analysis methods that are used to measure the relevant data needed to identify the material coefficients. The article describes the various formulations of finite element methods used in sheet metal forming process simulations. Stress-integration procedures for both continuum and crystal-plasticity mechanics are also discussed. The article also provides various examples that illustrate the simulation of aluminum sheet forming.

This article focuses on four industrial additive manufacturing approaches that are used to create polymer parts. The first section focuses on material extrusion, providing information on lumped-parameter material flow models and higher-fidelity models developed to estimate temperature distribution. The second section covers polymer powder-bed sintering/ fusion, discussing the different levels of scale used to address modeling and the impact of process settings: thermodynamics at the powder-bed surface, consolidation of adjacent particles in the fusion process, and fusion and molecular-level behavior within particles. The third section on vat photopolymerization (VPP) discusses two primary approaches to modeling VPP processes, namely a lumped-parameter approach to estimate cured regions in the vat, known as the Jacobs model, and a high-fidelity, continuum approach that uses finite-element methods. The final section is devoted to material jetting, focusing on simulations used to study droplet generation at the nozzle and droplet impact.

The goal of micromechanics and analysis is to use the predictive methodology to develop tailored composites and also to make accurate predictions of their performance in service. This article reviews results derived from micromechanics analyses, based on finite-element method of unidirectional fiber reinforced metal matrix composites (MMCs). It discusses the elastic deformation and elastic-plastic deformation analysis of discontinuously reinforced MMCs. The article provides an overview of analysis of strength, fatigue, and fracture toughness for macromechanics fiber-reinforced and discontinuous reinforced composites.

This article outlines several polycrystal formulations commonly applied for the simulation of plastic deformation and the prediction of deformation texture. It discusses the crystals of cubic and hexagonal symmetry that constitute the majority of the metallic aggregates used in technological applications. The article defines the basic kinematic tensors, reports their relations, and presents expressions for calculating the change in crystallographic orientation associated with plastic deformation. It surveys some of the polycrystal models in terms of the relative strength of the homogeneous effective medium (HEM). The article analyzes the anisotropy predictions of rolled face-centered-cubic and body centered-cubic sheets and presents simulations of the axial deformation of hexagonal-close-packed zirconium. The applications of polycrystal constitutive models to the simulation of complex forming operations, through the use of the finite element method, are also presented.

This article provides an overview of cavitation erosion with a specific focus on the estimation of mass loss. It describes the mechanisms of cavitation erosion and the types of laboratory devices to evaluate the resistance to cavitation erosion of materials. The laboratory devices include rotating disks, vibratory devices, cavitating liquid jets, and high-speed cavitation tunnels. The article discusses materials selection and surface protection to prevent cavitation erosion. It reviews the fluid-structure interaction that plays a role in cavitation erosion particularly for compliant materials. The article provides information on the numerical prediction of cavitation erosion damage by the finite element method (FEM).

This article discusses two basic forms of extrusion: cold and hot. It provides information on three types of extrusion processes, namely, direct extrusion, reverse extrusion, and hydrostatic extrusion. The article also discusses the mechanics, analysis, tooling and die design of extrusion as well as thermodynamics. The finite-element method suitable for simulation of metal forming processes is explained. The article examines the extrusion defects that are divided into three different categories including surface, subsurface, and internal type. It includes information on friction and lubrication modeling of extrusion processes. The article also discusses the fundamentals of extrusion technology of titanium alloys and aluminum. It concludes with information on two forms of wear in extrusion, namely, adhesive and abrasive wear.

Finite element analysis is a computer-based numerical method for solving engineering problems in bodies of user-defined geometry. This article introduces the important issues of finite elements (especially accuracy and efficiency) in a nonacademic manner. It describes the Rayleigh-Ritz procedure for solving structural problems based on the principle of virtual work. The article discusses continuum elements, such as hexahedra, pentahedra, tetrahedra, quadrilaterals, and triangles, commonly used in three- or two-dimensional domains. It considers structural elements such as beam element, plate element, shell element, and elbow element. The article presents three examples to illustrate the types of problems that can be addressed and the decisions that must be made when using finite element analysis.

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.

Modeling will help reduce machining problems and thereby enable more rapid introduction of high-performance materials and components. This article discusses the technical needs of aircraft engine and airframe structural components and modeling of heat-treat-induced residual stress by finite-element residual-stress analysis. It describes the two-dimensional (2-D) and three-dimensional (3-D) procedures involved in finite-element residual-stress analysis. The article deals with the 2-D and 3-D machining distortion validation on engine-disk-type components. It describes methods for obtaining machining-induced residual stresses, including detailed finite-element analysis of the cutting process, the simple fast-acting mechanistic model, and the semi-empirical linear stress model. The article concludes with information on the modeling benefits and implementation of modeling in a production environment.