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partial differential equations

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Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005453
EISBN: 978-1-62708-196-2
... Abstract 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...
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...
Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005877
EISBN: 978-1-62708-167-2
... Abstract 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...
Image
Published: 01 December 2009
Fig. 10 Structure of a chemical vapor deposition (CVD) reactor simulation. PDE, partial differential equation More
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
... of the enthalpy, thermal flux, and a distributed heat-source term, S (J/m 3 · s), the energy equation in differential form is: (Eq 3) H − ∇ · q − S = 0 or, in terms of temperature, it is: (Eq 4) ρ C p T ˙ − ∇ · ( − κ ∇ T ) − S = 0 This is a parabolic partial...
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
... This is a parabolic partial differential equation. The essential parts of any such equation are the boundary conditions, the initial conditions, and the parameters such as specific heat, C p , thermal conductivity, κ, and heat source per unit volume, S . The boundary conditions can be either essential (prescribed...
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
... problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure, and temperature) is transformed into an algebra problem of solving a large system of simultaneous linear equations for discrete unknowns that represent the state...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002445
EISBN: 978-1-62708-194-8
... conductors and their spacings. These are best described by partial differential equations in space and time, rather than by the ordinary differential equations in time that are satisfactory for discrete circuits. These partial differential equations cannot be solved analytically for most geometrical...
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
... model. Usually, the mathematical model consists of continuous partial differential equations valid everywhere over the domain of the problem, while the numerical solution represents a discretized distribution of the actual values of fields (the numerical solution is not continuous; it exists only...
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
... and process design Background and History Computational fluid dynamics has as its objective the numerical solution of fluid-flow equations. The calculus problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005456
EISBN: 978-1-62708-196-2
... of friction do not change the direction of principal axes or distort the deformation field. This results in homogeneous deformation, with plane sections remaining plane. These assumptions converted the partial differential equations of plasticity to ordinary differential equations of first order that can...
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
... design Computational fluid dynamics has as its objective the numerical solution of fluid-flow equations. The calculus problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure, and temperature) is transformed...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005451
EISBN: 978-1-62708-196-2
... differential equations with variable coefficients, integral equations of the convolution, or Faltung, type, and partial differential equations. Several Laplace transforms are listed in Table 1 . Laplace transforms Table 1 Laplace transforms Entries f ( s ) give values of the Laplace transform ℒ...
Series: ASM Handbook
Volume: 21
Publisher: ASM International
Published: 01 January 2001
DOI: 10.31399/asm.hb.v21.a0003402
EISBN: 978-1-62708-195-5
.... Squeeze Flow Manufacturing Methods Squeeze flow manufacturing methods, or advanced thermoplastic composites manufacturing methods, usually involve continuous or long, aligned, discontinuous fibers either partially or fully preimpregnated with thermoplastic resin. In these processes, the fibers...
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
.... Such parameters are boundary conditions, material properties, and nonuniform initial temperature distribution. Underestimation of application specifics or overly simplified assumptions can lead to an incorrect mathematical model (including improperly chosen governing equations) that will not be able to provide...
Series: ASM Handbook
Volume: 18
Publisher: ASM International
Published: 31 December 2017
DOI: 10.31399/asm.hb.v18.a0006409
EISBN: 978-1-62708-192-4
... on solving a set of partial differential equations that express the conservation of mass, momentum, and energy, and can be solved either by mesh-based (e.g., finite-element modeling [FEM]) or mesh-free (e.g., smoothed particle hydrodynamics) methods ( Ref 26 ). The strength of macroscopic descriptions lies...
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
..., the aforementioned procedure becomes quite slow. To overcome this problem, the HTC is computed by an alternate procedure, which avoids explicit numerical differentiation of calculated temperatures. The heat-conduction equation ( Eq 10 ) is written in a space-integrated form, as a heat-conservation law...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005636
EISBN: 978-1-62708-174-0
.... Such understanding is crucial to the development of sound and robust mathematical models. Unlike the experimental approach that directly measures the temperature or molten metal flow velocity, the mathematical models describe the heat and mass transfer using a set of partial differential equations (PDEs). The models...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002482
EISBN: 978-1-62708-194-8
... to be defined in terms of mathematical equations, typically partial differential equations. These equations are dependent on time, space, field variables, and internal states. Ordinary differential equations can be used if the problem can be simplified so that the shape is not important and a lumped-parameter...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005425
EISBN: 978-1-62708-196-2
.../forming pressures during metal forming, for instance) and the solution of relatively simple partial-differential equations (for various heat flow, solidification, and diffusion problems) that evolved during the 1920s to 1980s. The development of mainframe, mini-, and then microcomputers and associated...