電子版一abaqus經(jīng)典聲學(xué)分析例題a1-appendix heat transfer eqs

1、Copyright 2006 ABAQUS, Inc.Heat Transfer TheoryAppendix 1Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.2OverviewSummary of Governing Equations for ConductionConstitutive RelationFouriers LawThermal Energy BalanceDifferential Form Thermal Energy BalanceEquivalent Variational Form Finite Ele

2、ment ApproximationTransient AnalysisEulerian Formulation for Convection Thermal Radiation FormulationAdiabatic Thermal-Stress AnalysisNonlinear Solution SchemeCopyright 2006 ABAQUS, Inc.Summary of Governing Equations for ConductionHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.4Summary of G

3、overning Equations for ConductionThe heat balance equation enforces the condition that the surface and volumetric heat flux into a body is balanced by the rate of internal energy generation in the body.Differential formVariational formThe variational form is used as the basis of the finite element d

4、iscretization.Surface SVolume VArbitrary volume controlrqnThe details of the derivations of these equations are discussed shortly.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.5Summary of Governing Equations for ConductionqHeat flux per unit of current area crossing surface S from environm

5、ent to body under consideration (scalar quantity); related to the flux vector q and outward surface normal n throughr Heat flux per unit volume generated within body, such as: 1. Ohmic heating (electrical resistance). 2. Heat caused by dissipation of mechanical energy (plastic work in metals, molecu

6、lar friction in rubbers or polymers, etc.).r Current material mass density.U Internal energy per unit mass assumed to be a function of temperature only. ( = material time rate of change of U.)dq Arbitrary variational temperature field.Copyright 2006 ABAQUS, Inc.Constitutive RelationFouriers LawHeat

7、Transfer and Thermal-Stress Analysis with ABAQUSA1.7Constitutive RelationFouriers LawExperimental observations have indicated that the heat flux q is proportional to the temperature gradient.The proportionality factor is the materials thermal conductivity k .In one dimension:The minus sign indicates

8、 heat flows in the direction of decreasing temperature.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.8Constitutive RelationFouriers LawGeneralizing to multi-dimensions,where k is the temperature-dependent anisotropic conductivity matrix.Introducing Fouriers Law into the weak statement of t

9、he energy balance givesCopyright 2006 ABAQUS, Inc.Thermal Energy BalanceDifferential FormHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.10Thermal Energy BalanceDifferential FormConstruction of the energy balance equation for an arbitrary control volume.We consider the control volume attache

10、d to the material (Lagrangian formulation) and assume that the material is not undergoing large strain.Surface SVolume VArbitrary volume controlHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.11Thermal Energy BalanceDifferential FormThermal energy balance:Stated mathematically,storage termsu

11、rface flux term q positive into the bodyvolumetric flux term (source)Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.12Thermal Energy BalanceDifferential FormTo convert the energy balance equation to differential form:First, introduce a heat flux vector, q, by the geometric relation, where n

12、 is the unit outward normal to the surface.-q = flux crossing surfaceHeat flux definitionHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.13Thermal Energy BalanceDifferential FormSubstituting the heat flux vectorand applying the divergence theoremyieldsHeat Transfer and Thermal-Stress Analysi

13、s with ABAQUSA1.14Thermal Energy BalanceDifferential FormSince the volume is arbitrary, we obtain the differential form of the heat balance equation:This “strong” form of thermal equilibrium holds pointwise in a body.Copyright 2006 ABAQUS, Inc.Thermal Energy BalanceEquivalent Variational FormHeat Tr

14、ansfer and Thermal-Stress Analysis with ABAQUSA1.16Thermal Energy BalanceEquivalent Variational FormWe cannot enforce the “strong” form of thermal equilibrium numerically.We need to relax the equation and enforce thermal equilibrium in an average sense over finite volumes (finite elements).To do thi

15、s, we multiply the differential thermal energy balance equation by an arbitrary variational temperature field dq and integrate over the volume:The chain rule allows us to rewrite the middle term:Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.17Thermal Energy BalanceEquivalent Variational Fo

16、rmWe again apply the divergence theorem, this time to convert a volume integral to a surface integral, then convert our heat flux vector back to its scalar form:This produces the “weak” form of the thermal energy balance equation, which is used as the basis of the finite element discretization:Copyr

17、ight 2006 ABAQUS, Inc.Finite Element ApproximationHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.19Finite Element ApproximationWe discretize the volume geometrically using finite elements and interpolate the temperatures within the elements from the element nodal temperatures, bywhere are t

18、he interpolation functions for the element (summation is implied over the repeated superscripts). Using the Galerkin approach, the variational temperature field dq is interpolated with the same functions: In ABAQUS the functions are linear or parabolic polynomial product forms for one-, two-, and th

19、ree-dimensional elements, as well as for axisymmetric elements.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.20Finite Element ApproximationIntroducing this interpolation into the heat balance equation we obtain the approximation Since the variational quantities are independent, this provid

20、esThis set of equations is the basis for uncoupled, conductive heat transfer in ABAQUS.Under steady-state conditions and so the first term on the left-hand side disappears.For transient problems the equation must be integrated in time. Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.21Finite

21、 Element ApproximationWe can write where is the specific heat of the material. The approximate heat balance can be written whereCopyright 2006 ABAQUS, Inc.Transient AnalysisHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.23Transient AnalysisRecall the finite element approximation to the ther

22、mal energy balance equation, The finite elements we introduced discretize the problem in space. For transient problems this equation must also be integrated through time. The time integration operator used in ABAQUS/Standard for transient solid body heat transfer is the backward difference algorithm

23、: Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.24Transient AnalysisIntroducing the time integration operator into the thermal energy balance equation results in the following system of equations:Copyright 2006 ABAQUS, Inc.Eulerian Formulation for ConvectionHeat Transfer and Thermal-Stress

24、 Analysis with ABAQUSA1.26Eulerian Formulation for ConvectionRecall the Lagrangian form of the energy balance for an arbitrary control volume:The divergence theorem has been applied to the heat flux term.An Eulerian formulation is used for forced convection problems, so expansion of the material tim

25、e rate of change of internal energy, is required:Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.27Eulerian Formulation for ConvectionNoting thatand using the chain rule we can writeand Applying these results to an arbitrarily small volume, the differential equation of the thermal energy bal

26、ance esHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.28Eulerian Formulation for ConvectionUpwindingPetrov-Galerkin finite elements are used to model systems with high Peclet numbers accurately. These elements use nonsymmetric weighting functions that are different from those used as shape

27、functions.That is, we interpolate the variational field asand the temperature field as The weighting functions are of the formHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.29Eulerian Formulation for ConvectionUpwinding formHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.30Eulerian

28、Formulation for ConvectionThese weighting functions are biased, or “upwinded,” with the extent of the upwinding defined by y on an element-by-element basis. The upwinding functions act to control numerical diffusion and dispersion and, thus, to stabilize results.The upwinding term is partly a functi

29、on of the element Peclet number, g. The full expression for the weighting functions is given in the ABAQUS Theory Manual.Copyright 2006 ABAQUS, Inc.Thermal Radiation FormulationHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.32Thermal Radiation FormulationSeveral assumptions are made to calc

30、ulate the matrix of geometric viewfactors, radiation fluxes, and resultant temperatures for a cavity.Geometric assumptionsCalculation of Fij requires that each cavity be divided into a series of individual surfaces. In ABAQUS/Standard these surfaces are further subdivided into a series of individual

31、 facets, which is easily plished in the context of finite elements. For the purpose of radiation calculations, each of these facets is considered to be at a uniform temperature (isothermal) and to possess a single value of emissivity over the facet (isoemissive).Heat Transfer and Thermal-Stress Anal

32、ysis with ABAQUSA1.33Thermal Radiation FormulationRadiative surface property assumptionsThe radiative characteristics of the surfaces are assumed to possess certain characteristics:The surfaces are assumed to be “gray” bodies.Thermal radiation is transported via electromagnetic waves. The thermal ra

33、diation emitted by a real surface passes a range of wavelengths. The amount of radiation can vary with the length of the propagating waves. This variance is best quantified by plotting emissivity as a function of wavelength for a typical engineering surface. In some materials variance of emissivity

34、with wavelength can be quite strong.A gray body has an emissivity that is independent of wavelength for all levels of radiation.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.34Thermal Radiation FormulationThus, by assuming that surfaces are gray, an average emissivity for the spectrum of e

35、lectromagnetic energy involved in thermal radiation is used.Emissivity 10111021031010.51Wavelength (m m)Typical emissivity vs. wavelengthgray bodyreal surface“range” of thermal radiationHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.35Thermal Radiation FormulationThe surfaces are assumed to

36、 be “opaque.”Thermal radiation is assumed to be a surface effect rather than a volumetric effect. At a surface incident radiation can be reflected and absorbed but not transmitted (see figure). Kirchhoffs Law states that the ability of the surface to absorb is equal to its ability to emit (relative

37、to a blackbody).incident radiationreflected radiationabsorbed radiationtransmitted radiation = 0surfaceOpaque surface behaviorHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.36Thermal Radiation FormulationThe surfaces are assumed to be “diffuse.”Incident radiation, reflected radiation, and e

38、mitted radiation at the surface are assumed to have no directional tendencies; they are diffuse. This assumption is reasonable for most surfaces. However, all surfaces exhibit some departure from diffuse behavior. Highly polished surfaces, for example, may tend to invalidate this assumption.Radiativ

39、e transfer occurs within cavities that are filled with a nonparticipating medium.Any gas or other medium within the confines of a cavity has no influence on the thermal radiation. The radiation exchange involves only the surfaces of the cavity. Energy exchange by radiation in an absorbing, emitting

40、medium (such as molten glass) is beyond the scope of these notes.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.37Thermal Radiation FormulationSurface emissivity is often a function of surface temperature (see figure) and other surface conditions (e.g., oxidation). Emissivity may depend on

41、temperature or predefined field variables.050010000.51.0aluminumstainless steelTemperature (C)EmissivityTypical emissivity vs. temperatureHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.38Thermal Radiation FormulationRadiation flux The ABAQUS/Standard implementation of cavity radiation is ba

42、sed on the definition of one or more cavities in the model.Each cavity is a collection of distinct surfaces.ABAQUS/Standard automatically creates the geometric viewfactor matrix for a cavity, including treatment of symmetry conditions (which are discussed later).Each surface is composed of facets.A

43、facet is a side of an element in axisymmetric and two-dimensional cases and is a face of a solid or shell element in three-dimensional cases.Each facet is assumed to be isothermal and to have a uniform emissivity.Radiative fluxes are imposed on each of these facets.Heat Transfer and Thermal-Stress A

44、nalysis with ABAQUSA1.39Thermal Radiation FormulationThe radiation flux per unit area into cavity facet i can be written as whereis the reflection matrix.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.40Thermal Radiation FormulationIn these expressions:ei, ej are the emissivities of facets

45、i, j;s is the Stefan-Boltzmann constant;Fij is the geometrical viewfactor matrix;Ti, Tj are the absolute temperatures of facets i, j; anddij is the Kronecker delta function.In the special case of blackbody radiation, e = 1, no reflection takes place: Ckj = dkj.Heat Transfer and Thermal-Stress Analys

46、is with ABAQUSA1.41Thermal Radiation FormulationNotation difference with the ABAQUS Theory ManualThe definition of the viewfactor given in these notes differs from that given in the ABAQUS Theory Manual.The expression given in the ABAQUS Theory Manual for the viewfactor does not include the term 1/A

47、i in front of the double integral. The two forms are related through:where is the form used in the Theory Manual. By substituting the above expression everywhere Fij appears in these notes, all equations are converted to a form that is appropriate for use with . Since the expression for the viewfact

48、or used in these notes is more commonly used in the literature, we adopt this form and acknowledge the difference with the Theory Manual. Copyright 2006 ABAQUS, Inc.Adiabatic Thermal-Stress AnalysisHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.43Adiabatic Thermal-Stress AnalysisRecall the

49、basic heat balance equation, where is the material time derivative of the thermal energy density;c is the specific heat; r is the mass density of the material; q is the temperature at a point;r is the heat source term: energy per unit time, per unit volume; and is the net rate of heat flux entering

50、the element by conduction.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.44The adiabatic assumption is that Thus, in adiabatic analysis the local heat balance is,where W pl is the rate of energy dissipation due to plastic work, which is a function of strain rate, and temperature, q ; and h

51、is the inelastic heat fraction, the percentage of inelastic work converted to heat. A typical value of h for plasticity in metals is 0.9.Adiabatic Thermal-Stress AnalysisHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.45Adiabatic Thermal-Stress AnalysisThe temperature-dependent mechanical co

52、nstitutive relationship can be written in the rate formwhere D is the tangent stiffness for instantaneously isothermal behavior.The last term on the right is the rate of change in stress due to temperature changes if the strains are kept fixed. It includes effects due to thermal expansion and temper

53、ature-dependent elastic and plastic properties.Heat Transfer and Thermal-Stress Analysis with ABAQUSA1.46Adiabatic Thermal-Stress AnalysisTo include the adiabatic effects, the heat balance equation can be combined with the constitutive equation to yield Therefore: Adiabatic analysis can be handled s

54、olely by modification of the mechanical constitutive equations.The extra term in the effective modulus usually has a softening effect, because is generally negative and other coefficients in that term are positive.Generally, the matrix will be nonsymmetric. In many cases, however, the nonsymmetric t

55、erms are relatively small, and good convergence can be obtained with the symmetric approximation.Copyright 2006 ABAQUS, Inc.Nonlinear Solution SchemeHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.48Nonlinear Solution SchemeABAQUS/Standard uses Newtons method to solve nonlinear equations, wh

56、ich may be described as follows: We write the thermal energy equilibrium equation (without convection) for a degree of freedom asRN is the “flux residual” at degree of freedom N: the out-of-balance flux at the node. Assume that at iteration i we have an approximate solution, If we assume is the corr

57、ection to needed to get the exact solution, thenHeat Transfer and Thermal-Stress Analysis with ABAQUSA1.49Nonlinear Solution SchemeNewtons method is based on expanding this equation in a Taylor series about the approximate solution, and ignoring higher-order terms to obtain a linear system of equations for the We can rewrite this as Here, is the Jacobian (tangent stiffne