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finite difference method
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Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005876
EISBN: 978-1-62708-167-2
...-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. computer...
Abstract
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.
Book Chapter
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005431
EISBN: 978-1-62708-196-2
... Abstract 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...
Abstract
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.
Book Chapter
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... 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...
Abstract
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.
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005533
EISBN: 978-1-62708-197-9
.... 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...
Abstract
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.
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005506
EISBN: 978-1-62708-197-9
... 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...
Abstract
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.
Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005897
EISBN: 978-1-62708-167-2
... Abstract Estimation of process parameters for selective heating and heat treating of simple- and complex-shaped workpieces in induction hardening can be accurately carried out using numerical simulation techniques such as the finite-element analysis and the finite-different method. Along...
Abstract
Estimation of process parameters for selective heating and heat treating of simple- and complex-shaped workpieces in induction hardening can be accurately carried out using numerical simulation techniques such as the finite-element analysis and the finite-different method. Along with the significant benefits of modern numerical simulations, it is important to be able to use rough estimation techniques to develop a general understanding of the critical parameters of a particular induction heating system. This article discusses such numerical techniques for estimating the critical parameters: workpiece power estimation; estimation of electrical and thermal efficiency of the coil; and frequency selection for heating solid cylinders, tubes, pipes, slabs, plates, strips, and rectangular workpieces.
Image
Published: 01 October 2014
Fig. 17 Probabilistic approach, which captures process variations during the carburizing operation and predicts quality variations in terms of case-depth variations. FDM, finite-difference method. Source: Ref 42
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Book Chapter
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005233
EISBN: 978-1-62708-187-0
... to solve the CFD equations are discussed. The article also demonstrates the application of CFD to a few casting problems. casting applications computational fluid dynamics discretization finite-difference method finite-element method finite-volume method numerical methods thermal-fluids...
Abstract
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.
Series: ASM Desk Editions
Publisher: ASM International
Published: 01 December 1998
DOI: 10.31399/asm.hb.mhde2.a0003176
EISBN: 978-1-62708-199-3
..., and the molding method used. Designers can predict the effect of the design on the structure of the final part using solidification simulation models, namely finite element and finite difference models, and rapid prototyping. The article concludes with a short note on how the quality is assured in the foundry...
Abstract
This article provides general guidelines for casting design to provide progressive solidification, minimize heat concentration, eliminate cores, and prevent distortion. Casting design also affects tolerances. Casting tolerances depend on the alloy being poured, the size of the casting, and the molding method used. Designers can predict the effect of the design on the structure of the final part using solidification simulation models, namely finite element and finite difference models, and rapid prototyping. The article concludes with a short note on how the quality is assured in the foundry.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002444
EISBN: 978-1-62708-194-8
... 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...
Abstract
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.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002446
EISBN: 978-1-62708-194-8
... that contains Δ b i . Significant disadvantages of the finite difference method exist when it is used with CAE analysis tools, namely, computational expense and lack of accuracy. Performance measures and thus the system response must be evaluated N + 1 times to compute the design sensitivity...
Abstract
This article discusses tools that are used for the systematic optimization of engineering designs. It focuses on the practical application of optimization technology in a computer-aided engineering environment. The article presents numerical optimization algorithms and provides some background on how these algorithms make decisions when searching for the optimal design. It also provides information on structural optimization, topology optimization, materials processing optimization, multidisciplinary optimization, and global optimization.
Series: ASM Handbook
Volume: 4A
Publisher: ASM International
Published: 01 August 2013
DOI: 10.31399/asm.hb.v04a.a0005789
EISBN: 978-1-62708-165-8
... or Dirichlet's boundary condition for the heat-conduction equation ( Eq 10 ) on [0, r n ]. Solve this problem, for example, by using the implicit finite-difference method ( Ref 19 , 20 ). In each time step, simple iterations can be used to adjust all thermal properties to new temperatures. Extend...
Abstract
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.
Series: ASM Handbook
Volume: 13A
Publisher: ASM International
Published: 01 January 2003
DOI: 10.31399/asm.hb.v13a.a0003596
EISBN: 978-1-62708-182-5
... methods such as finite-element methods, finite-difference methods, or boundary-element methods are applied to analyze the ECM process. A single equation has been derived ( Ref 6 ) for numerical methods analysis. It is applicable simultaneously to both the cases, that is, for f = 0 and f ≠ 0. Equation...
Abstract
This article discusses four subsystems of the electrochemical machining (ECM) system: power source, electrolyte cleaning and supply system, tool and tool-feed system, and workpiece and workpiece-holding system. It describes the theory of ECM and provides information on the electrolytes used in ECM. The article reviews the methods associated with workpiece shape prediction. The procedures and integrated approach for the tool design in ECM are discussed. The article also explains the process control, capabilities, and the limitations of ECM. It concludes with information on the applications of ECM.
Book Chapter
Series: ASM Handbook Archive
Volume: 11
Publisher: ASM International
Published: 01 January 2002
DOI: 10.31399/asm.hb.v11.a0003544
EISBN: 978-1-62708-180-1
... Abstract This article describes three design-life methods or philosophies of fatigue, namely, infinite-life, finite-life, and damage tolerant. It outlines the three stages in the process of fatigue fracture: the initial fatigue damage leading to crack initiation, progressive cyclic growth...
Abstract
This article describes three design-life methods or philosophies of fatigue, namely, infinite-life, finite-life, and damage tolerant. It outlines the three stages in the process of fatigue fracture: the initial fatigue damage leading to crack initiation, progressive cyclic growth of crack, and the sudden fracture of the remaining cross section. The article discusses the effects of loading and stress distribution on fatigue cracks, and reviews the fatigue behavior of materials when subjected to different loading conditions such as bending and loading. The article examines the effects of load frequency and temperature, material condition, and manufacturing practices on fatigue strength. It provides information on subsurface discontinuities, including gas porosity, inclusions, and internal bursts as well as on corrosion fatigue testing to measure rates of fatigue-crack propagation in different environments. The article concludes with a discussion on rolling-contact fatigue, macropitting, micropitting, and subcase fatigue.
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005503
EISBN: 978-1-62708-197-9
... a finite-difference method is used in the z -direction (thickness), making use of the same finite elements in the x - y plane for solving Eq 3 . The finite-difference method (FDM) is a relatively efficient and simple numerical method for solving differential equations. In this method, the physical...
Abstract
This article focuses on the axisymmetric 2.5-dimensional approach used in metal powder injection molding (PIM) simulations. It describes three stages of PIM simulations: filling, packing, and cooling. The article discusses the process features of numerical simulation of PIM, such as filling and packing analysis, cooling analysis, and coupled analysis between filling, packing, and cooling stages. It explains the experimental material properties and verification for filling, packing, and cooling stages in the PIM simulations. The article presents simulation results from some of the 2.5-dimensional examples to demonstrate the usefulness of the computer-aided engineering analysis and optimization capability of the PIM process.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002443
EISBN: 978-1-62708-194-8
... ) and: (Eq 11) B = ( 2 P L 3 ) / ( 27 π 4 E I ) Comparing this new solution to the analytical solution, deflections are everywhere within a ±1% band of the analytical solution, and peak stress differs from the accepted solution by 8%. The finite element method...
Abstract
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.
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005587
EISBN: 978-1-62708-174-0
... analysis of welds. Solving the Energy Equation The energy equation defined previously can now be solved for a specific welding situation. Finite-element methods (FEMs) have been the method of choice for stress analysis. Finite-difference methods (FDMs) have been the method of choice for fluid flow...
Abstract
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.
Series: ASM Handbook
Volume: 6
Publisher: ASM International
Published: 01 January 1993
DOI: 10.31399/asm.hb.v06.a0001480
EISBN: 978-1-62708-173-3
... the method of choice for stress analysis. Finite-difference methods (FDM) have been the method of choice for fluid flow. Whether the FDM or FEM method is best is an old argument. In rough terms, FEM has been used more frequently for complex geometries and for stress analysis, whereas FDM has been used more...
Book Chapter
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005540
EISBN: 978-1-62708-197-9
... include ABAQUS Explicit and LS-DYNA. The essence of the FEM, as opposed to other discrete treatments such as the finite-difference method, lies in an equivalent work principle. Continuous displacements within an element are represented by a small number of nodal displacements (and possibly other...
Abstract
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.
Series: ASM Handbook
Volume: 18
Publisher: ASM International
Published: 31 December 2017
DOI: 10.31399/asm.hb.v18.a0006385
EISBN: 978-1-62708-192-4
... 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...
Abstract
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.
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