Thermal Analysis Using MSC SimXpert R3.2 Course Notes

Part Number: SIMX*R3.2*Z*Z*Z*SM-SMX124-NT Copyright2009 MSC.Software CorporationJanuary 2009

Thermal Analysis Using MSC SimXpertSMX124 Course Notes

MSC.Software Corporation

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2

Legal Information

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3

CONTENTS

Section Pages

1.0 Introduction to MSC SimXpert Thermal Analysis……………………………………………….. 1-12.0 Conduction………………………………………………………………………………………………. 2-1

3.0 Convection………………………………………………………………………………………………. 3-1

4.0 Transient Thermal Analysis……………………………………………..…………………………… 4-1

5.0 Radiation…………………………………………………………………………………………………. 5-16.0 SOL 400 Thermal Capability………………………………………………………………………….. 6-1

7.0 Thermal Analysis Theory……………………………………………………………………………… 7-1


4

S1-1SMX124, Section 1, January 2009Copyright2009 MSC.Software Corporation

SECTION 1

INTRODUCTION TOMSC SimXpert THERMAL

ANALYSIS



HEAT TRANSFER

● Should a Heat Transfer analysis be performed ?●When the solution for the temperature field in a solid (or fluid) is

desired, and the temperature is not influenced by other fields, a heattransfer analysis is appropriate.

vs

Coupled Fields

Thermal Field


MODES OF HEAT TRANSFER

● Mechanisms of Heat Transfer● Conduction (through a material)● Convection (to a fluid)

● Natural (free)● Forced

T1>T2

Convection

T2

Moving Fluid RemovesHeat From Solid

q

T1

ThAq

T2

Advection

T1>T2

T1

Heat Moves INMoving Fluid

q

)( 21 TTCmq p Radiation

T1>T2

21 q

12 q

T1 T2

)( 42

41 TTFAq

● Advection (energy carried in a fluid)● Radiation (energy transfer through

free space (no intervening medium)from one surface to another)● Ambient (to space)● Enclosure (enclosed radiation

system)


ANALYSIS TYPES

● Thermal problems can be categorized as● Steady-state

●Steady-state analyses are concerned with state point solutions withfixed boundary conditions

● Transient●Transient analyses are characterized by solution evolution over time,

and in addition to energy exchange with the environment, involvesthermal energy storage

● Linear●Properties; i.e. conductivity, convection coefficient.; do not change

with● Temperature

●The boundary conditions do not involve● Radiation


ANALYSIS TYPES (Cont.)

● Thermal problems can be categorized as (continued)● Nonlinear

●Temperature dependent material properties.● Conduction● Convection

● Natural● Forced

●Radiation boundary conditions.● Ambient● Enclosure

●Steady-state or transient solution.●All nonlinear analyses necessarily involve solution iteration, error

estimation, and convergence criteria.


THERMAL MODEL ELEMENTS

● MSC SimXpert thermal elements● 1D – conduction only in the direction of the centerline of the element● 2D – conduction only through the mid-plane of the element● 3D – conduction in all three directions●Axi-symmetry

●Non-zero thermal gradient in the radial and axial directions●Zero thermal gradient in the circumferential direction

●Special elements for conduction●Special element for lumped thermal capacitance


THERMAL MODEL MATERIAL PROPERTIES

● Thermal materials●Conduction

●Thermal conductivity● Can be a function of temperature

●Transient model● Specific heat (heat capacity per unit mass)

● Can be a function of temperature

● Density● Cannot be a function of temperature


THERMAL MODEL MATERIAL PROPERTIES

● Thermal materials● Convection

●Natural (free)● Heat transfer coefficient

● Can be a function of temperature● Can be a function of time

● h = function(Gr, Pr)

●Forced● Constant coefficient● h = function(Re, Pr)

● Radiation●Surface absorptivity

● Can be a function of temperature

●Surface emissivity● Can be a function of temperature


THERMAL MODEL BOUNDARY CONDITIONS

● Thermal boundary conditions● Convection

●Natural (free)●Forced (SimXpert R4.0)

● Radiation●Ambient (to space)●Enclosure

● Temperature●Temperature - constant or time varying temperature for set of nodes● “Initial conditions” for set of nodes

● Steady-state – initialization/starting temperature● Transient – initial temperature

●MPC – multipoint constraint (constraint for set of nodaltemperatures)


THERMAL MODEL LOADS

● Thermal loads●Heat flux

●Applied to surface, Normal Flux●From a distant radiation source, Radiant Flux

● Can specify direction not normal to the surface

●Nodal heat●Power input to a node

●Volumetric load● Internal (to element) heat generation


STEADY-STATE EQUATION

● The familiar conduction heat transfer equation is

● The MD Nastran Thermal steady-state equation derived from this equation is

● This is a nonlinear matrix equation. It is solved using the Newton-Raphsoniteration scheme. In concert with the Newton-Raphson method, the followingoptions are provided to improve the efficiency of the iteration process.● Tangential matrix update strategy● Line search method● Bisection of loads● Quasi-Newton (BFGS) updates

● This equation, and its solution, are described in Section 7.

tT

kq

T

12

}{}{}]{[}]{[ 4 NPTTTK abs R


TRANSIENT EQUATION


● The MD Nastran Thermal transient equation derived from this equation is

● This is a nonlinear matrix equation. It is solved using the Newton-Raphsoniteration scheme. In concert with the Newton-Raphson method, thefollowing options are provided to improve the efficiency of the iterationprocess.● Tangential matrix update strategy● Line search method● Bisection of loads● Quasi-Newton (BFGS) updates


tT

kq

T

12

}{}{}]{[}]{[}]{[ 4 NPTTTKTB abs R



SECTION 2

CONDUCTIONWITH

ELEMENTS, MATERIALS,CONSTRAINTS, LOADING



TOPICS ON CONDUCTION

● Laws affecting conduction● Supported element topologies

● 1D, 2D, 3D, Axisymmetric

● Assemble element conduction matrices to form system matrix● Create elements● Material for conduction

● Isotropic (K)● Orthotropic (Kxx, Kyy, Kzz)● Anisotropic (Kxx, Kxy, Kxz, Kyy, Kyz, Kzz)● Temperature independent or dependent

● Constraints and loads that can be applied to conduction elements● Temperature (constant, spatial dependent, or time dependent)● Heat flux to a surface: normal flux, radiant (vector) flux● Nodal heat input● Heat generation within conduction element


LAWS AFFECTING CONDUCTION

● Second law of thermodynamics●Heat only flows from regions of high temperature to regions of low

temperature.

● Fourier’s Law (empirical)●One dimension

●General

● First law of thermodynamics - conservation of energy

tT

kq

T

12

ix

Tkq}{

)/( dxdTkAq


SUPPORTED ELEMENT TOPOLOGIES

● Element types● The MD Nastran SOL 153 and SOL 159 elements are

isoparametric

CTUBE

CROD

CTRIA6CONROD

CTETRACTRIA3CBEND

CPENTACQUAD8CBEAM

CTRIAX6CHEXACQUAD4CBAR

Axisym3D2D1D


SUPPORTED ELEMENT TOPOLOGIES (Cont.)

● 1D element

● Isoparametric element with 1 dimension, (r)

● Interpolation within element

● Element conduction matrix

11

12

13

33

34

35

12

2

1

)(i

ii TrhT

,2/)1(1 rh 2/)1(2 rh

)(

)()()()()( ][][][][jV

jjjTjje dVBkBk

r1 2

0 r = 1r = -1



● 1D element● where

● = element j conduction matrix● = element j temperature gradient

interpolation matrix● = element j thermal conductivities●The integration is over the volume of element j

● An example is

11

12

13

33

34

35

12

)(][ jek

)(][ jk

)(][ jB

]0[]0[]0[

]0[1111

]0[

]0[]0[]0[

][)(

)(

j

je L

Akk



● 2D element

● Isoparametric element with 2 dimensions, (r,s)

● Interpolation within element

11

12

13

33

34

35

55

56

57

34

4

1

),(i

ii TsrhT

,4/)1)(1(1 srh 4/)1)(1(2 srh

,4/)1)(1(3 srh 4/)1)(1(4 srh

r

s

0

r = 1r = -1s = -1

s = 1 12

3 4



● 2D element

●Element conduction matrix11

12

13

33

34

35

55

56

57

34

)(

)()()()()( ][][][][jV

jjjTjje dVBkBk



● 3D element

● Isoparametric element with 3 dimensions, (r,s,t)● Interpolation within element

●Element conduction matrix

8

1

),,(i

ii TtsrhT

)(

)()()()()( ][][][][jV

jjjTjje dVBkBk



● Recommendations● Use linear elements, unless have substantial curvature and desire to

minimize the number of elements● If doing thermal analysis (calculate temperatures) to structural analysis

(stress analysis) mapping (loads are temperatures from thermalanalysis), it is best to use the same type of element that is to be usedfor the structural analysis, e.g. Tet10, and not Tet4.

● Loads and boundary conditions do not affect which type of elementshould be used

● Special Elements● This category is for elements that are not finite elements, but of course

they can be used in modeling a thermal process. There are two typesof special elements. They are● CELASi (scalar spring)

● Simple conduction element



● Special Elements (continued)● CELASi (scalar spring) (continued)

● Used to model a one-dimensional conduction element with a thermal resistance.● R= L/kA, where CELAS2 used inverse of R

● G=1/R = kA/L (Watt/0C, type of unit)

● DMI or DMIG (direct matrix input) -- complex component● DMI

● For defining matrix data blocks.

● where the elements Xij may be real or complex (real and imaginary part). Thematrix is defined by a single header entry and one or more column entries. Onlyone header is required. A column entry is required for each column with nonzeroelements.

● For more information see the MD Nastran Quick Reference Guide.

][][ ijXname


ASSEMBLE ELEMENT MATRICES

● Assemble all element conduction and other systemmatrices●Derive element conduction matrices

●Sum the element conduction matrices

●Create other necessary system matrices, and create theconduction system matrix equation

)(

)()()()()( ][][][][j

jjjTjj

eV

dVBkBk

j

jekK )(][][

}{}]{[ PTK


● Summary of the steady-state finite element method

ASSEMBLE ELEMENT MATRICES (Cont.)

Assemble loads into a global load vector {P}Represent continuous structure as a collection of

discrete elements connected by nodes

Derive element conduction matrices frommaterial properties, element properties, and geometry

Assemble all element conduction matrices into aglobal conduction matrix [K]

Apply boundary conditions to constrain themodel

Solve the matrix equation [K] {T} = {P} fornodal temperature

Compute thermal flux from temperature results


CREATING 1D ELEMENTS● 1D elements

● Create by meshing curve or edge● Create a material property set● Create an element property set● Mesh a geometric curve with 1D

elements● Select the curve for Curve to mesh,

CURVE/1● Specify the element size, 1● From the Model Browser tree select the

element property, PBAR_1


CREATING 1D ELEMENTS (Cont.)

● 1D elements (continued)● Create manually

● Select the Element Property under PID● Specify (X1,X2,X3) for the cross-section

orientation vector

1

2

● 1D elements● Create manually

● Specify the coordinates for thetwo ends of the 1D element


CREATING 2D ELEMENTS● 2D elements

● Create by meshing surface or face, method l● Create a material property set● Create an element property set● Mesh a geometric surface with 2D elements

● Screen select the surface for Surface to mesh● Specify the element size● Specify Mapped for Mesh method● From the Model Browser tree select the element

property PSHELL_1



● The four sets of coordinatesare (0,0,0), (1,0,0), (1,1,0),and (0,1,0)

● The image of the CQUAD4element created is



● Use the pick panel to define fourcorner nodes


CREATING 3D ELEMENTS

● 3D elements● Create by meshing solid, method I

● Create a material property set● Create an element property set● Mesh a geometric solid with 3D elements

● Select the solid for Solid To mesh● Specify the element size● From the Model Browser tree select the

element property PSOLID_1



● The eight sets of coordinatesare (0,0,0), (1,0,0), (1,1,0),(0,1,0), (0,0,1),…,(0,1,1)

● The image of the CHEXA8element created is



● Use the pick panel to defineeight corner nodes


MATERIAL FOR CONDUCTION

● Thermal conductivity of solid material● Isotropic

● K● Temperature independent● Temperature dependent

● Orthotropic● Kxx, Kyy, Kzz● Temperature independent● Temperature dependent

● Anisotropic● Kxx, Kxy, Kxz, Kyy, Kyz, Kzz● Temperature independent● Temperature dependent

● These material properties are for modeling conduction● Steady-state● Transient


MATERIAL FOR CONDUCTION (Cont.)

● In addition the following topics are covered for materials● Required material properties to define a model● Units● Method of creation of materials● Use of fields for material definition

● In the section on transient analysis the following topics are covered,in addition to those for steady-state analysis● Density● Specific heat at constant pressure● Lumped thermal capacitance



● Isotropic material● Temperature independent

Isotropic material property.● Access the material

property forms using theMaterials and Propertiestab, then the Material group.

● Select Isotropic.● Enter the name, ID, and

description.● Select Solid to define

thermal properties of solidmaterial.

● Enter the value for thethermal conductivity.

● Click OK.


MATERIAL FOR CONDUCTION(Cont.)

● Isotropic material (continued)● Temperature dependent Isotropic material property● For the Thermal Conductivity entry box click on the Pick… icon● Select a temperature dependent table in the Model Browser tree● Click OK



● Orthotropic and Anisotropic material● Temperature independent or dependent.● Access the material property forms using the Materials and Properties

tab, then the Material group.● Select an Orthotropic or Anisotropic material, similar to that for Isotropic

material.



● Units● The following table lists the units that correspond to the fields of the

Materials – Thermal (High Level Editor), or the fields of theMatIsotropic and RadMat entries (Single Card Image ).


TEMPERATURE CONSTRAINTS● Temperature constraints at node points

● Access the temperature constraint form using the LBCs tab, then theLBC group

● The form Defaults For Temperature BC appears● Its default setting is for a constant temperature at node points.

Enter the value for the temperature.● Click Store, then click Exit


TEMPERATURE CONSTRAINTS (Cont.)

● Temperature constraints at node points●Specify the application region using the Create Temperature pick

panel●Select an entity type to screen pick, e.g. Nodes●Screen pick the nodes to constrain their temperature●Click Done, then click Exit



● MPC temperature constraints

● Access the constraint formusing the LBCs tab, then the LBCgroup

● The form Defaults ForMPC BC appears

● Change to the formDefaults For MPC BC byusing an Entity Type pick

● DOF0 is for the DOF of thedependent node

● WT0 is for the weightingfactor for DOF0

k kk

TA 0



● MPC temperature constraints● DOFi is for the DOF of the i-th

independent node● WTi is for the weighting factor

for DOFi● Click Store, then click

Exit



● MPC temperature constraints● Specify the application region

using the Create MPC pickpanel

● Select an entity type to screenpick, e.g. Nodes

● Screen pick the nodes●First node picked is dependent

node (example, Node 25)●Subsequent nodes picked are

independent nodes (example,Node 50, 49, 24)

● Click Done● In Multi-Point Constraint form

that appears click Create


INITIALIZATION TEMPERATURE

● Temperature initialization● To perform nonlinear steady-state or transient analysis it is

necessary to specify initialization temperatures.●Steady-state, must specify starting temperature (as opposed to

“initial temperature” that is needed for transient analysis) at allthe nodes.

●Transient, must specify the initial temperature at all the nodes.


INITIALIZATION TEMPERATURE (Cont.)● Temperature initialization

● Access the initialization temperature form usingthe LBCs tab, then the LBC group.

● The form Defaults For TEMP_INIT BCappears.

● Enter the value for the initial temperature.● Click Store, then click Exit.



● Specify the application region using the CreateTEMP_INIT pick panel.

● Select an entity type to screen pick, e.g.Nodes.

● Screen pick the nodes to specify theirinitialization temperature.

● Click Done, then click Exit.


INITIALIZATION TEMPERATURE (Cont.)

● Temperature initialization● For the nodes not included in the

specified application regions, theirtemperatures are specified usingAnalysis: Solver Control / GeneralParameters / Properties / GeneralSolution Parameters / Default InitTemperature


CASE STUDY: CONDUCTION THRU PLATE



● Create Material property



● Create Property



● Create surface● Create two curves




● Create a surface using the curves



● Create surface● Create a surface using the curves



● Mesh the surface



● Apply temperature constraints






● Create an LBC Set



● Create SimXpert analysis file









● Run MD Nastran Thermal● Display thermal results

● Attach .xdb file



● Display thermal results● Display thermal fringe


LOADING

● Loads that can be applied to conduction elements● Heat flux to a surface

●Normal flux (Nastran QHBDY or QBDYi)● Heat Flux (power/area)● Time scaling function● Application region

● Heat into a node●Nodal heat

● Heat generation in conduction elements●Volumetric heat (Nastran QVOL)

● Power input per unit volume● Power produced by heat conduction elements

● Time scaling function● Application region


LOADING (Cont.)

● Heat flux to a surface● Normal Flux● Access the normal flux form using the LBCs tab, then the LBC group.

● The form Defaults For Normal Flux appears.● Enter the value for the heat flux (power/area).● Click Store, then click Exit.


LOADING (Cont.)

● Heat flux to a surface● Normal Flux

● Specify the application regionusing the Create Normal Fluxpick panel


● Screen pick the nodes to applythe heat flux to

● Click Done, then click Exit


LOADING (Cont.)

● Heat generation● Volumetric Heat● Access the volumetric heat form using the LBCs tab, then the LBC

group.● The form Defaults For Volumetric Heat BC appears.● Enter the value for the power input per unit volume (power/volume)● Click Store, then click Exit


LOADING (Cont.)

● Heat generation●Volumetric Heat

● Specify the application region using theCreate Volumetric Heat pick panel

● Select an entity type to screen pick, e.g.Elements

● Screen pick the elements to apply theinternal heat generation to



NOTE ON GEOMETRIC SURFACE ELEMENTS

● Heat flux or boundary condition surface elements● To apply heat flux, convection, or radiation to a

conducting surface (2D element, e.g. CQUAD4)or face (3D element, e.g. CHEXA) it is necessaryto apply them to a geometric surface, which inturn is connected to conducting elements. Thegeometric surface elements are named CHBDYE,CHBDYG, or CHBDYP, or simply CHBDYi. Thisis shown conceptually.

● The user does not explicitly create the CHBDYielements. They are created automatically by(from within) SimXpert as a result of creating theloads and boundary conditions.


LOADING (Cont.)

● The heat flow through the isoparametric elements is in the parametricdirections of the elements.

● 1D element● Heat flow is only along the centerline of the element, not normal to the centerline

of the element● 2D element

● Heat flow is only in the plane of the element, not normal to the plane of theelement

● 3D element● Heat flow is in all three directions of the element

● Axisymmetric element● Heat flow is only in the radial or centerline direction of the element, not in the

circumferential direction● So, do not use an isoparametric element that does not have a parametric

coordinate in a direction that heat flow must be modeled● The performance of linear finite elements, e.g. CQUAD4 element, is as good

as that of parabolic finite elements, e.g. CQUAD8 element, for 2D or 3D.


● Create an LBC set that combines the needed LBCs● Access the loads and boundary condition set form using the

LBCs tab, then the LBC Set group.

● The form LBC Set appears.● Select the two loads and boundary conditions● Click OK

LBC Set


ANALYSIS SETUP

● The Analysis forms in the Modeltree browser are used to specifythe parameters for an analysis● Model: Analysis

● Model: Analysis / Nastran Jobs

● Nastran Jobs : Create New Job●Job Properties form


ANALYSIS SETUP (Cont.)

● The Analysis forms in the Model● General Parameters, etc.

● General Parameters: Properties●General Solution Parameters

form



● The Analysis forms in the Model●Radiation Parameters: Properties

●Radiation Parameters form



● The Analysis forms in the Model●Cases: Add Common Case

●Common Case, Case Control● Specify titles to be

above all subcases



● The Analysis forms in the Model●Common Case, Add Output

Requests



● The Analysis forms in the Model●Cases: Add Subcase

●Select the desired LBC Set



● The Analysis forms in the Model● Subcase Parameters: Properties

●NINC = number of equalsubdivisions of the load changedefined for the subcase forNewton-Raphson methods.



● The Analysis forms in the Model●Matrix Update Method

● Automatic => the programautomatically selects the mostefficient strategy based onconvergence rates. At each stepthe number of iterations requiredfor convergence is estimated.The conduction matrix is updatedif 1) the estimated number ofiterations to converge exceedsMAXITER, 2) the estimated timerequired for convergence with thecurrent conduction matrixexceeds the estimated timerequired for convergence withupdated conduction, and 3) thesolution diverges.




● Semi-Automatic => the programfor each load increment 1)performs a single iteration basedon the new load, 2) updates theconduction matrix, and 3) followsthe normal Automatic approach.

● Controlled Iters => the programupdates the conduction matrix atevery KSTEPth iteration and onconvergence if KSTEP <=MAXITER. However, if KSTEP >MAXITER the conduction matrixis never updated.




● Controlled Iters => note that theNewton-Raphson iterationstrategy is used by selecting thisapproach (Controlled Iters) andKSTEP = 1, while the ModifiedNewton-Raphson iterationstrategy is used by selecting thisapproach (Controlled Iters) andKSTEP = MAXITER.

●KSTEP => for Automatic andSemi-Automatic methods theconduction matrix is updated atconvergence if KSTEP is < thenumber of iterations that wererequired for convergence with thecurrent conduction.



● The Analysis forms in the Model●MAXITER => the number of

iterations for a load increment islimited to MAXITER. If the solutionprocess does not converge inMAXITER iterations the loadincrement is bisected and theanalysis is repeated.

●Temperature Error, Yes or No●EPSU = error tolerance for

temperature criterion●Load Error, Yes or No●EPSP = error tolerance for load

criterion●Work Error, Yes or No●EPSW = error tolerance for work

criterion



● The Analysis forms in the Model● Output File: Properties

●Output File Properties form



● The Analysis forms in the Model● Submit an analysis file to the solver● ss_sample_problem: Run

●File name



● The Analysis forms in the Model● Choices for SOL type for steady-

state analysis.●SOL153●SOL600, 153


DISPLAY RESULTS

● Display thermal analysis results● Attach an .XDB result file


DISPLAY RESULTS (Cont.)

● Display thermal analysis results● Access the results form using the

Results tab, then the Results group.● Select the following

● Fringe● Result Cases -- SC1: Non-linear: 100 % of L● Result Type -- Temperatures● Click Update Plot



● Display thermal analysis results● Display the temperature results as a fringe plot


EXERCISE

Perform Workshop 1 “PCB With Heat Flux andConstant Temperature at Boundary” in your exerciseworkbook.


SECTION 3

CONVECTION



HEAT TRANSFER VIA CONVECTION

● The transfer of energy (heat) between a solid boundary and a fluidtakes place by a combination of conduction and mass transport

● The fluid motion can be induced by two different processes.● The fluid may be moved as a result of density differences due to a

temperature variation within the fluid. This type of heat transfer is calledfree (or natural) convection.

● The fluid motion may be caused by an external source, such as a pumpor blower. This type of heat transfer is called forced convection.

● A simple representation of convection heat transfer is given by

● where● qsurface to fluid = rate of heat transfer by convection● hc = convective heat transfer coefficient, or average unit thermal convective

conductance

)( 0TTAhq scfluidtosurface

.


HEAT TRANSFER VIA CONVECTION (Cont.)

● where●A = heat transfer area●Ts = surface temperature●T0 = fluid temperature

● The convective heat transfer coefficient is actually a complicatedfunction● Fluid flow● Thermal properties of the fluid medium● Geometry of the thermal system● Its numerical value is in general not uniform over a surface● Depends on the location where the fluid temperature T0 is specified

● Convective heat transfer is modeled with SimXpert usingempirically derived functions



● Free convection

● where●Nu = Nusselt number (ratio of the temperature gradient in the fluid

at the surface to a reference temperature gradient)●Gr = Grashof number (ratio of buoyant to viscous forces)●Pr = Prandtl number

● The Reynolds number (Re) is not needed for free convection

(Pr))( gGrfNu

f

c

kLh

Nu Lk

Nuh fc

)( 0TTLAkNuq sffluidtosurface



● Forced convection

● where●Nu = Nusselt number (ratio of the temperature gradient in the fluid

at the surface to a reference temperature gradient)●Re = Reynolds number (ratio of inertia to viscous forces)●Pr = Prandtl number

(Pr)(Re) gfNu

f

c

kLh

Nu Lk

Nuh fc

)( 0TTLAkNuq sffluidtosurface


FREE CONVECTION

● Free convection● Access the free convection form using the LBCs tab, then the LBC

group.● The Defaults For Free Convection form appears● Enter the value for the ambient temperature.


FREE CONVECTION (Cont.)

● Free convection●Convection coefficient – constant coefficient●ConvCoeff vs Time scaling function (for transient analysis click in

checkbox, and specify a time function using a table)●Formula type – type of formula for free convection (more on

subsequent page)●Reference temp location – location used to calculate the convection

film coefficient (more on subsequent page)●Conv coeff exponent – free convection exponent



● Free convection●ConvCoeff vs Temperature scaling function – free convection heat

transfer coefficient as a function of temperature

●Click Store, then click Exit



● Free convection●Formula type – choices for type of formula

for free convection:

● The reference temperature is the average of element node pointtemperatures (average) and the ambient point temperatures (average)

● The reference temperature is the surface temperature (average ofelement node point temperatures)

● The reference temperature is the ambient temperature (average ofambient point temperatures)

)()( AMBEXPF

AMBCNTRLND TTTTTHq

)( EXPFAMB

EXPFCNTRLND TTTHq



● Free convection●Specify the application region using

the Create Free Convection pickpanel

●Select an entity type to screen pick,e.g. Nodes

●Screen pick the nodes to apply thefree convection to

●Click Done, then click Exit


CASE STUDY: CONVECTION OFF PLATE






● Create Property



















● Apply free convection boundary condition





















EXERCISE

Perform Workshop 2 “Steady-state Analysis of HexMesh with Free Convection and Internal Heat” in yourexercise workbook.



SECTION 4

TRANSIENT THERMAL ANALYSIS



TRANSIENT THERMAL MODELS

● The preceding sections (conduction and convection)dealt with steady-state thermal models

● Before steady-state conditions can occur some timemust elapse after the heat transfer process is initiatedfor the transient conditions to decay to zero


TRANSIENT THERMAL MODELS (Cont.)

● For some models steady-stateconditions will never be reached, e.g.heating a brick


TRANSIENT EQUATION


● The MD Nastran Thermal transient equation derived from this equation is

● This is a nonlinear matrix equation. It is solved using the Newton-Raphsoniteration scheme. In concert with the Newton-Raphson method, thefollowing options are provided to improve the efficiency of the iterationprocess.● Tangential matrix update strategy● Line search method● Bisection of loads● Quasi-Newton (BFGS) updates


tT

kq

T

12

}{}{}]{[}]{[}]{[ 4 NPTTTKTB abs R


TRANSIENT TEMPERATURE CONSTRAINTS

● The Defaults For TemperatureBC form was shown for thesection on conduction

● The material given in the sectionon conduction is repeated here

● The Temperature entry was usedto specify a constanttemperature

● The Temp vs Time scalingfunction entry is discussed now



● Temperature at node pointconstraints● The constraints can be made

time dependent by clicking inthe Temp vs Time scalingfunction check box

● Click on the box with the “…” tothe right

● Can select Select or Create tospecify the time dependent table

● Select the entry TABLED1_1(this is a time dependent tablethat must have been previouslycreated)

● Click OK



● Temperature at node point constraints● Click Store, then click Exit● Specify the application region using the Create Temperature

pick panel● Select an entity type to screen pick, e.g. Nodes● Screen pick the nodes to constrain their temperature● Click Done, then click Exit


LOADING

● Loads that can be applied to conduction elements●Heat flux to a surface

●Normal Flux (Nastran QHBDY or QBDYi)● Heat Flux (power/area)● Time scaling function● Application region

●Heat generation in conduction elements●Volumetric Heat (Nastran QVOL)

● Power input per unit volume● Power produced by heat conduction elements

● Time scaling function● Application region


LOADING (Cont.)

● Normal Flux● Access the normal flux form

using the LBCs tab, then theLBC group.

● The form Defaults For NormalFlux appears

● Enter a value for the heat flux(power/area)

● The loading can be made timedependent by clicking in the Fluxvs Time scaling functioncheckbox



LOADING (Cont.)

● Normal Flux● Can select Select or Create● Select TABLED1_1● Click OK● Click Store, then click Exit


LOADING (Cont.)

● Normal Flux●Specify the application region

using the Create Normal Flux pickpanel

●Select an entity type to screenpick, e.g. Nodes

●Screen pick the nodes to apply theheat flux to



LOADING (Cont.)● Volumetric Heat

● Access the volumetric heat formusing the LBCs tab, then theLBC group.

● The form Defaults For VolumetricHeat BC appears

● Enter the value for power input perunit volume (power/volume)

● The loading can be made timedependent by clicking in theHeatGen vs Time scaling functioncheckbox



LOADING (Cont.)

● Volumetric Heat●Can select Select or Create●Select TABLED1_1●Click OK●Click Store, then click Exit


LOADING (Cont.)

● Volumetric Heat● Specify the application region using the

Create Volumetric Heat pick panel● Select an entity type to screen pick, e.g.

Elements● Screen pick the elements to apply the

internal heat generation to● Click Done, then click Exit


TRANSIENT CONVECTION CAPABILITY

● The Defaults For FreeConvection form wasshown for the section onfree convection

● There are two timedependent function entries● AmbTemp vs Time scaling

function – table used to definethe ambient temperature as afunction of time

● ConvCoeff vs Time scalingfunction – table used to definethe convection coefficient as afunction of time


TRANSIENT CONVECTION CAPABILITY (Cont.)

● To use either of thesefunctions do the following● Click in the check box for

one of the functions, e.g.AmbTemp vs Time

● Click in the box with the“…” to the right of thefunction entry box

● Can select Select or Create● Select Select● If a table corresponding to

the function has not beencreated prior to doing this amessage will be displayed.



● To use either of these functions do the following● It is necessary to create the time dependent table first● Access the time dependent table form using the Tables tab, then

the Tables group.



● To use either of these functionsdo the following● Now, AmbTemp vs Time has “1”

in its entry box● Click Store, then click Exit



● To use either of these functionsdo the following● Specify the application region

using the Create Free Convectionpick panel


● Screen pick the nodes to applythe free convection to



INITIALIZATION TEMPERATURE

● Temperature initialization● To perform nonlinear steady-state or transient analysis it is

necessary to specify initialization temperatures●Steady-state – must specify starting temperature (as opposed to

“initial temperature” that is needed for transient analysis) at allthe nodes

●Transient – must specify the initial temperature at all the nodes



● Access the initialization temperature form usingthe LBCs tab, then the LBC group.

● The form Defaults For TEMP_INIT BCappears.

● Enter the value for the initial temperature.● Click Store, then click Exit.



● Specify the application region using the CreateTEMP_INIT pick panel.

● Select an entity type to screen pick, e.g.Nodes.

● Screen pick the nodes to specify theirinitialization temperature.

● Click Done, then click Exit.


INITIALIZATION TEMPERATURE (Cont.)

● Temperature initialization● For the nodes not included in the

specified application regions, theirtemperatures are specified usingAnalysis: Solver Control / GeneralParameters / Properties / GeneralSolution Parameters / Default InitTemperature


CASE STUDY: TRAN CONV OFF PLATE






● Create Property



















● Apply free convectionboundary condition● See the next page for the

creation of the table



● Apply convection boundary condition























Time = 0.0 Time = 501.4 Time = 1000.6



● Display thermal results● Display thermal chart



● Display thermal results● Display thermal chart


ANALYSIS SETUP

● The Analysis forms in the ModelBrowser tree are used to specifythe parameters for an analysis● Model: Analysis

● Model: Analysis / Nastran Jobs

● Nastran Jobs : Create New Job●Job Properties form



● The Analysis forms in the Model● General Parameters, etc.

● General Parameters: Properties●General Solution Parameters

form



● The Analysis forms in the Model●Radiation Parameters: Properties

●Radiation Parameters form



● The Analysis forms in the Model●Cases: Add Common Case

●Common Case, Case Control● Specify titles to be

above all subcases



● The Analysis forms in the Model●Common Case, Add Output

Requests



● The Analysis forms in the Model●Cases: Add Subcase

●Select the desired LBC Set



● The Analysis forms in the Model● Subcase Parameters: Properties

●DT = initial time step size●NDT = approximate number of

time steps




● Automatic => (ADAPT) theprogram automatically adjuststhe incremental time and usesbisection. During the bisectionprocess the conduction matrixis updated every KSTEPconverged bisection solution.




● Semi-Automatic => same asAutomatic => (ADAPT)

● Controlled Iters => same asAutomatic => (ADAPT)

●KSTEP => the number ofconverged bisection solutions froma conduction matrix update to thenext conduction matrix update



● The Analysis forms in the Model●MAXITER => the number of

iterations for a time step is limitedto MAXITER. If MAXITER isnegative the analysis is terminatedwhen the divergence conditionoccurs twice during the same timestep or the solution processdiverges for five consecutive timesteps. If MAXITER is positive theprogram computes the bestsolution and continues the analysisuntil divergence occurs. If thesolution process does notconverge in MAXITER iterations itis treated as divergent.



● The Analysis forms in the Model●Temperature Error, Yes or No●EPSU = error tolerance for

temperature criterion●Load Error, Yes or No●EPSP = error tolerance for load

criterion●Work Error, Yes or No●EPSW = error tolerance for work

criterion



● The Analysis forms in the Model● Output File: Properties

●Output File Properties form



● The Analysis forms in the Model● Submit an analysis file to the solver● transient_sample_prob: Run

●File name



● The Analysis forms in the Model● Choices for SOL type for transient

analysis.●SOL159●SOL600, 159


DISPLAY RESULTS

● Display thermal analysis results● Attach an .XDB result file



● Display thermal analysis results● Access the results form using the

Results tab, then the Results group.● Set the following

● Fringe● Result Cases, SC1: Time = 5.7● Result Type, Temperatures● Click Update Plot



● Display thermal analysis results● Display the temperature results as a fringe plot


DISPLAY RESULTS (Cont.)● Display thermal analysis results● Access the results form using the Results tab, then

the Results group.● Set the following

● Chart● Result Cases, SC1: Time = 0. (thru) Time = 5.7● Result Type, Temperatures● Target Entities, Node 6431 (screen pick a node)● Click Add Curves



● Display thermal analysis results● Display the temperature results

for a node (e.g. Node 6431) asa function of time


EXERCISE

Perform Workshop 3 “Transient Analysis of Heating ofan Ice-Cream Block” in your exercise workbook.



SECTION 5

RADIATION



RADIATION CONCEPTS

● Thermal radiation● Radiation at wavelengths () in the domain 0.1 m < < 100 m● This domain includes all visible light (0.35 m < < 0.75 m),

portions of the ultraviolet light sub-domain (short wavelength),and infrared light sub-domain (long wavelength)

● Total emissive power, E● The total amount of radiation emitted by a body per unit area

and unit time

● Black body● Body absorbs all radiation for all wavelengths

● Grey body● Real bodies (grey bodies) emit radiation at a lower rate than

black bodies


RADIATION CONCEPTS (Cont.)

● Absorption, reflection, and transmission

● When radiation impinges on a body it is partially absorbed,partially reflected, and partially transmitted.

●= absorptivity●= reflectivity●= transmissivity

1



● Kirchhoff’s law● Body Bi

●Bi, Ai, i, Ei

●Emissivity (in general is a functionof T and )

●Grey body at thermal equilibrium

●Black body at thermal equilibrium

EEEG

2

2

1

1

1 bb

bEE

B1

B2

G

G



● Planck’s law

● where●h = Planck’s constant●k = Boltzmann’s constant

● Stefan-Boltzmann equation

● where●= Stefan-Boltzmann constant

)1/()0.8( )/()(5 kThcb ehcE

4

0TdEE bb



● Net rate of radiant heat transfer between a surface andits surroundings

● Net rate of radiant heat flow between two grey bodies(surfaces)

● where●FG is the geometric view factor

)( 44ambientsurface TTAq

)( 42

41 TTAFq G



● Net rate of heat flow byradiation between two greybodies (surfaces) A1 and A2

2121

221

4

2

4

121coscos)( dAdA

AA rTTq

r

A1

A2

dA1

dA2

1̂n 2̂n



● Viewfactor

● Reciprocity theorem

● In general

● Net radiation from Ai to othersurfaces in enclosure

● Sum of Fik for enclosure

jiji

ij

ji

iijdAdA

AA rAF 2

coscos1

jiijFF

jijijiFAFA

n

ikkbkbii EEFAq ikneti ,1

)(,

n

ikk ikF,1

1


RADIATION CAPABILITIES

● Ambient exchange● Emissivity,

●Constant, 0 <= <= 1●Temperature dependent

● Absorptivity, ●Constant, 0 <= <= 1●Temperature dependent

● Time dependent viewfactor


RADIATION CAPABILITIES (Cont.)

● Applied heat flux from a distant source● Heat Flux (Q0) (e.g. W/m2)

●Constant or time dependent

● Absorptivity●Constant, time and/or temperature dependent

● Emissivity●Constant or temperature dependent

● Radiation enclosures● Temperature dependent emissivity (= )● Diffuse surface view factor calculations with self and

3rd-body shadowing


RADIATION CAPABILITIES (Cont.)

● Radiation enclosures● Adaptive view factor calculations with error estimation● Net effective view factors● Radiation matrix control● Radiation enclosure control● Multiple radiation enclosures


RADIATION TO SPACE (AMBIENT)● Radiation to space (ambient)

● Access the rad to space form using the LBCs tab, then the LBC group.● The Defaults For Rad to Space form appears● Ambient temperature - constant ambient temperature● AmbTemp vs Time scaling function – function used to define the ambient

temperature as a function of time


RADIATION TO SPACE (AMBIENT) (Cont.)

● Radiation to space(ambient)● View factor – view factor

between face and ambient● ViewFact vs Time scaling

function – function used todefine the view factor as afunction of time

● Absorptivity – absorptivity,0 <= <= 1

● Emissivity – emissivity,0 <= <= 1

● Shell surface option●FRONT●BACK


RADIATION TO SPACE (AMBIENT) (Cont.)● Radiation to space

(ambient)● Detailed surface prop

definition – wavelengthand/or temperaturedependent surfaceproperties●Can select Select or

Create●First, create a

radiation materialproperty. Use theSolver Card option.

● In the form DefaultsFor Rad to Spaceclick on box with “…”.

●Select Select, thenselect RadMat_1



● Radiation to space (ambient)●Click Store, then click Exit●Specify the application region using

the Create Rad to Space pick panel●Select an entity type to screen pick,

e.g. Nodes●Screen pick the nodes to apply the

radiation to space them●Click Done, then click Exit



● Radiation to space (ambient)● Rate of heat flow by radiation to space (ambient)

)( 44ambee TTFAMBq


CASE STUDY: NONDIRECTION SOLAR LOAD






● Create Property
















● Create radiation to ambient



● Create uniform heat flux



● Create LBC Set



● Create Job Properties



● Specify General Solution Parameters values



● Specify Radiation Parameters values



● Create a subcase



● Specify desired output



● Specify Static Nonlinear Iterations parameter values



● Specify information and file type to be output



● Write SimXpert analysis file and run analysis



● Attach SimXpert analysis result file



● Plot temperature fringe



● Plot temperature fringe


EXERCISES

Perform Workshop 4 “Hex8 Mesh With Heat Flux,Ambient Convection, and Ambient Radiation” in yourexercise workbook.


RADIATION IN ENCLOSURES

● Radiation in enclosures● Access the enclosure radiation face

form using the LBCs tab, then the LBCgroup.

● The Defaults For Encl Rad Face formappears

● Shell surface option -- for specifyingwhat face the radiation is emitted from.●Front●Back


RADIATION IN ENCLOSURES (Cont.)

● Radiation in enclosures●Shading option -- for specifying

what is to be shaded or what isto shade.

● None● The face cannot shade or

be shaded by other faces

● Can shade● The face can shade other

faces

● Can be shaded● The face can be shaded by

other faces

● Both● The face can shade and be

shaded by other faces



● Radiation in enclosures●Subelem mesh size (b-dir (beta

direction)) and (g-dir (gammadirection)).

● Used for the calculation of viewfactors by a finite difference orcontour integration technique.They are used to specify howfine a “grid” (not finite elementgrids) for surface elements(CHBDYi) for integration thereare to be in the b-dir and g-dirdirection.

Subelem (b-dir) = 2

Subelem (g-dir) = 4


● Radiation in enclosures●Absorptivity

● Constant, 0 <= <= 1

●Emissivity● Constant, 0 <= <= 1

●Detailed surface prop definition● Create a temperature dependent

table for Absorptivity andEmissivity




● Radiation in enclosures●Click Modify●Click Store, then click Exit●Specify the application region

using the Create Encl Rad Facepick panel

●Select an entity type to screenpick, e.g. Elements

●Screen pick the elements todefine the face




● Radiation in enclosures● Now, that the face(s) have been

defined, it is necessary to definean enclosure(s) using the faces.

● Access the create radiationenclosure form using the LBCtab, then the LBC group.

● SHADOW -- control third-bodyshading calculation during viewfactor calculation.●Yes●No

● TAMB●Temperature of ambient

environment that radiation islost to



● Radiation in enclosures●SCALE

● Enclosure view factor sum isset equal to this if the sum >1

●PRTPCH● Control printing and punching

view factor to RADLST andRADMTX entries.

●NFECI. Controls whether afinite difference method orcontour integration is to beused for calculating viewfactors.

● Finite difference● Contour integration● Mixed method



● Radiation in enclosures●RMAX

● Subelement area factor.When NFECI is set to “Mixedmethod” the contourintegration method is usedonly ifwhere As = area ofsubelement s, and drs =distance between subelementr and subelement s.

●GITB● Gaussian integration order to

be used for calculating neteffective view factors withthird –body shadowing.

RMAXdA rss 2/



● Radiation in enclosures●GIPS

● Gaussian integration order tobe used for calculating neteffective view factors with selfshadowing.

●ETOL● Error estimate above which a

corrected view factor iscalculated using the semi-analytic contour integrationmethod.

●ZTOL● If calculated view factor is <

ZTOL it is set to zero.



● Radiation in enclosures●WTOL

● If amount of warpage ofsurface element, CHBDYi, is> WTOL the view factor Fii iscalculated using exactmethod.

●NUMBCS● Number of enclosure

radiation faces. After enteringthe value, and clicking in thecell just to the right of thevalue, the form updates sothere are NUMBCS ERFacecells.



● Radiation in enclosures●NUMBCS

● Double click in first ERFacecell.

● Entity Selection form appears.● Select an Encl Rad Face entity.● Click OK.● Repeat, by selecting the next

ERFace.● …● Click OK.● …● Click Create, then click Exit.


CASE STUDY: THREE RAD ENCLOSURES



● Import model of four 2Delements

● Select filemultiple_enclosures.bdf



● Display the four 2D elements showing their node numbers



● Display the element normals

● Create three enclosures

1 2 3



● Create first enclosure● Two faces for this enclosure









● First enclosure created and shown



● Create second enclosure● Two faces for this enclosure









● Second enclosure created and shown



● Create third enclosure● Two faces for this enclosure









● Third enclosure created and shown



● Fix the left element at 2000 0K



● Fix the left element at 2000 0K









● Create SimXpert analysisfile















● Execute MD Nastran Thermal analysis

● Attach the MD Nastran Thermal results file



● Display thermal results using a fringe plot



● Display thermal results using a fringe plot


EXERCISES

Perform Workshop 5 “Single Radiation Enclosure WithFive Faces” in your exercise workbook.


VECTOR FLUX FROM A DISTANT SOURCE

● Radiant Flux - heat fluxfrom a distant radiationsource● Access the radiant flux form

using the LBCs tab, then the LBCgroup

● The form Defaults For RadiantFlux appears

● Heat Flux – thermal heat flux intoelement face. It is positive forheat flow into a surface.

● Flux vs Time function – functionused to define the flux as afunction of time


VECTOR FLUX FROM A DISTANT SRCE (Cont.)

● Radiant Flux - heat flux from a distant radiation source● Temp of radiant source – temperature of source● Local coord system for specifying the thermal vector flux

direction● Local x-dir – direction cosine of flux relative to x-axis of local

coordinate system● Local y-dir -- direction cosine of flux relative to y-axis of local

coordinate system● Local z-dir -- direction cosine of flux relative to z-axis of local

coordinate system



● Radiant Flux - heat flux from a distant radiation source● Absorptivity

●Constant, 0 <= <= 1

● Absorp vs Time scaling function● function used to define absorptivity as a function of time

● Emissivity●Constant, 0 <= <= 1

● Shell surface option●FRONT or BACK

● Detailed surface prop definition●Wavelength and/or temperature dependent surface properties



● Radiant Flux - heat flux from adistant radiation source●Click Store, then click Exit●Specify the application region using

the Create Radiant Flux pick panel●Select an entity type to screen pick,

e.g. Nodes●Screen pick the nodes to apply the

vector heat flux to●Click Done, then click Exit


EXERCISES

Perform Workshop 6 “Variable Directional Solar Load toa Single Quad4 Element” in your exercise workbook.


SECTION 6

SOL 400 THERMAL CAPABILITYVIA STRUCTURES WORKSPACE



THERMAL PROBLEMS VIA SOL 400

● Currently SOL 400 is accessed via the StructuresWorkspace (SimXpert R3.2)

● It is similar to the Thermal Workspace● The following briefly covers several topics

● Material properties● Element properties● Loads and boundary conditions● SimXpert analysis file creation (Simulations)



● Materials

● Some of these apply to heat transfer● Isotropic● Orthotropic 2D, 2D Axi, and 3D● Anisotropic 2D and 3D● Radiation Band● Radiation Mat



● Properties (element)

● Some of these apply to heat transfer● 0D

●Spring (simple resistor) (PELAS)●Damper (simple lumped heat capacity) (PDAMP)

● 1D●Truss (PROD)●Beam (simple, library, or arbitrary cross-section) (PBAR)●Spring (simple resistor) (PELAS)●Damper (simple lumped heat capacity) (PDAMP)



● Properties (element)● 1D

●Tube (PTUBE)●Bend (PBEND)

● 2D●Shell (PSHELL)●Layered composite (PCOMP)●Plane strain composite

● 3D●Solid (PSOLID)●Solid composite (PCOMPLS)



● Loads and boundary conditions.

● Currently these are the available lbcs.● Future releases of SimXpert (>R3.2) will include forced

convection.



● Specify the parameter values for a SimXpert thermalanalysis.



● Under DefaultLoadCase specify the analysis type asnonlinear steady state heat transfer.



● Specify the parameter values forLoadcase Control.



● Observe what LBCs are associated to Loads/Boundaries;look at the set DefaultLbcSet.



● For a SimXpert chainedanalysis (e.g. thermal tostructural) this is what aModel Browser tree can looklike.


EXERCISE

Perform Workshop 7 “Concentric Spheres WithRadiation Between Them” in your exercise workbook.

Perform Workshop 8 “Transient Response of PowerElectronics” in your exercise workbook.

Perform Workshop 9 “Integrated Circuit Board ThermalStress Analysis” in your exercise workbook.



SECTION 7

THERMAL ANALYSIS THEORY OFMSC SimXpert THERMAL

ANALYSIS



FORMULATIONS

● Thermal problems can be categorized as● Steady-state

●Steady-state analyses are concerned with state point solutions withconstant/fixed boundary conditions

● Transient●Transient analyses are characterized by solution evolution over time,

and in addition to energy exchange with the environment, involvesthermal energy storage

● Linear●Properties; i.e. conductivity, convection coefficient; they do not

change with● Temperature

●The boundary conditions do not involve● Radiation


FORMULATIONS (Cont.)

● Thermal problems can be categorized as (continued)● Nonlinear

●Temperature dependent material properties.● Conduction● Convection

● Natural● Forced

●Radiation boundary conditions.● Ambient● Enclosure

●Steady-state or transient solution.●All nonlinear analyses necessarily involve solution iteration, error

estimation, and convergence criteria.


STEADY-STATE FORMULATION STEPS


● A transition is made from this equation (strong form) to a variationalformulation (weak form). The matrix equation corresponding to thevariational formulation is

● where, [B] and [K] are the heat capacity and conduction matrices

● {T} and {F} are the temperature and heat load vectors

● The MD Nastran Thermal steady-state equation derived from thisequation is

tT

kq

T

12

}{}]{[}]{[ FTKTB

}{}{}]{[}]{[ 4 NPTTTK abs R


STEADY-STATE FORMULATION STEPS (Cont.)

● This is a nonlinear matrix equation. It is solved using the Newton-Raphson iteration scheme. A residual load vector function is defined asthe difference between the applied thermal load vector and the thermalload vector due to element temperature.

● The residual load vector function is equal to zero for the solutiontemperature vector. The task is to determine the solution temperatures.The residual load vector function is approximated by its first-orderTaylor series expansion about the temperature vector from the i-thiteration.

)}]{[}]{([}){}({}{ 4absTTTKNPR R

)}{}({}{}{

)}}({{})}({{ i

i

i TTTR

TRTR



● The Taylor series expansion is evaluated at the temperatures for the(i + 1)-th iteration, and set equal to zero.

● From this the following equation is arrived at.

● where

● Note

}0{)}{}({}{}{

)}}({{)}}({{ 11

ii

i

ii TTTR

TRTR

iiiT RTK }{}{][

iii TTT }{}{}{ 1

i

absiiii

T TNTTKK

}{}{}{][4][][ 3R

iii TTT }{}{}{ 1



● This is an iterative process, not linear. Start with i = 0. Thus, the firstequation to be solved is

● From the solution to this equation the next temperature vector isfound from

● Next, use i = 1. The next equation to be solved is

● From the solution to this equation the next temperature vector isfound from

● This process is repeated until a converged solution is obtained, {T}m.

000 }{}{][ RTKT

001 }{}{}{ TTT

111 }{}{][ RTKT

112 }{}{}{ TTT



● Graphical representation of the solution process● Expression (F) for the thermal load vector due to element

temperature4}]{[}]{[ absTTTKF R

KT1

1

T0 T1 T2 Tm T

F

P + N

KT0

1

F0

F1

T0 T1

R0

R1



● Since matrix decomposition is time consuming, MD Nastran does notupdate the left-hand side matrix at each iteration. The tangential matrixis updated only when the solution fails to converge or the iterationefficiency can be improved. However, the residual vector is updated ateach iteration.

● In concert with the Newton-Raphson method, the following options areprovided to improve the efficiency of the iteration.● Tangential matrix update strategy● Line search method● Bisection of loads● Quasi-Newton (BFGS) updates


STEADY-STATE CONVERGENCE

● Convergence of nonlinearsteady-state solution process● The form, whos entries are used to

specify control of the nonlinearsteady-state solution process, isrepeated here for convenience. MDNastran parameter (alpha) namesare shown in the MSC SimXpertGUI


STEADY-STATE CONVERGENCE (Cont.)

● Convergence of nonlinear steady-state solution process● Iteration control

● The incremental and iterative solution processes are controlled by theparameters specified under Static Nonlinear Iterations.

● For each subcase, load, and SPC temperature, changes are processedincrementally with the number of subdivisions defined by the value of NINC.

T

F

P + N

f2(P + N)

f1(P + N)

OneIncrement

OneIncrement

OneIncrement

Total Load




● The incremental and iterative solution processes are controlled by theparameters specified under Static Nonlinear Iterations.

● The Matrix Update Method field is used to specify the tangential matrix updatestrategy. Three separate options may be selected.

● Automatic -- the program automatically selects the most efficient strategy based onconvergence rates. At each iteration, the number of steps (iterations) required to converge aswell as the computing time with and without matrix update are estimated. The tangentialmatrix is updated if (a) the estimated number of iterations to converge exceeds MAXITER, (b)the estimated time required for convergence with current matrix exceeds the estimated timeto converge with matrix update, or (c) the solution diverges. The tangential matrix is alsoupdated on convergence if KSTEP is less than the number of steps required for convergencewith the current matrix.

T

F

j

For iteration“j”

# iters toconvergeestimated

jTK ][ updated if

a) Est # iters to converge > MAXITER,

b) Est time to converge with current >Est time to converge with update, orc) Solution process diverges

Automatic




● The Matrix Update Method field is used to specify the tangential matrixupdate strategy. Three separate options may be selected.

● Semi-Automatic -- this option is identical to the AUTO option except that theprogram updates the tangential matrix after the first iteration.

● Controlled Iters -- the program updates the tangential matrix at every KSTEPthiteration and on convergence if KSTEP < MAXITER. However, the tangential matrixis never updated if KSTEP > MAXITER. Note that the Newton-Raphson method isobtained if KSTEP = 1, and the modified Newton-Raphson method is selected bysetting KSTEP = MAXITER.

jTK ][ updated every KSTEPth iteration or at

convergence if KSTEP <= MAXITER

T

F

j

For iteration“j”

Controlled Iters




● The number of iterations for a load increment is limited to MAXITER. If the solutiondoes not converge in MAXITER iterations, the current load increment is bisected andthe analysis for that increment is repeated.

T

F

P + N

f2(P + N)

f1(P + N)




● The divergence rate is defined as the ratio of energy error before and at the end ofthe current iteration, i+1

● The divergence parameter for the i-th iteration, NDIVi , is incremented as follows

● This computation is performed for all of the iterations of the current load increment

● The solution for the current load increment is taken as diverged when NDIVfinal >=MAXDIV

errorseverelessforNDIVNDIV ii ,11

errorseveremoreforNDIVNDIV ii ,21


STEADY-STATE CONVERGENCE (Cont.)● Convergence of nonlinear steady-state solution process

● Iteration output● For each iteration, the related output is printed under the following headings:

Number of occurrences of bisection conditions during the iterationMDV

Number of occurrences of probable divergence during the iterationNDV

Expected number of iterations for convergenceENIC

Number of line searches performed during the iterationNLS

Number of quasi-Newton vectors appendedNQNV

Error at the end of the line searchE-FINAL

Divergence rate, initial error before line searchE-FIRST

Final value for the line search parametersFACTOR

Absolute norm of the residual vectorDLMAG

Rate of convergenceLAMBDA

Relative error for energyEWI

Relative error for loadEPI

Relative error for temperatureEUI

Iteration iITERATIONS

DescriptionParameters


STEADY-STATE CONVERGENCE (Cont.)● Convergence of nonlinear steady-state solution process

● Convergence criteria● The convergence criteria are characterized by the dimensionless error

functions and the convergence tolerances. To ensure accuracy andefficiency, multiple criteria with errors measured about temperatures,loads, and energy are provided.

● Temperature error function● Since the error in temperatures is not known, a contraction factor q is

introduced to formulate the temperature error function, which is defined as

● The final form of the temperature error function is obtained by introducing aweighted normalization. The result is

● where the weighting function {is defined as the square root of the diagonalterms of the tangent matrix.

||||||||

||||||||

11

1

i

i

ii

ii

TT

TTTT

q

|}{||}{||}{||}{|

1 TT

qq

E T

T

U



● Convergence of nonlinear steady-state solution process● Convergence criteria

● Load error function● The load error function is defined as

● where

● where {Pld} is the applied thermal load at the previous load step, and {Pld} is theincremental load.

● Energy error function● The energy (or work) error function is defined as

|}{||}{||}{||}{|

TPTRE T

T

P

|}{||}{||}{| ldld PPP

|}{||}{||}{||}{|

TPTR

E T

T

W



● Convergence of nonlinear steady-state solution process● Convergence criteria

● Using the error functions● At every iteration, error functions are evaluated and the results printed in

the convergence table under the headings EUI, EPI, and EWI. Theconvergence test is performed by comparing the value of the errorfunctions with the convergence tolerances, e.g.

● EU < EPSU ?● EP < EPSP ?● EW < EPSW ?

● where the value of EPSU, EPSP, and EPSW are tolerances specified inthe MSC SimXpert Subcase Parameters form. The solution has convergedif these tests are satisfied. Note that only those criteria selected by the user(specified in the Convergence Criteria part of the form) are used to checkfor convergence. The tolerances should not be too restrictive so that manymore iterations are performed than need to be, or too un-restrictive so thatthere is poor accuracy. It is recommended that the default values be useduntil better values are found through iteration experience.


RECOMMENDATIONS FOR STEADY-STATE

● Recommendations● The following are recommendations, designed to aid the user.

● Highly nonlinear radiation problems● It is advised to use NLPARM,ID,NINC,DT,KMETHOD,KSTEP,…; where NINC = 1,

KMETHOD = ITER, and KSTEP = 1. This will cause the tangent conduction matrixto be calculated for each iteration.

● NLPARM settings for problems that do not involve a high-degree of nonlinearradiation

● NLPARM,ID,NINC,DT,KMETHOD,KSTEP,MAXITER,…; where NINC = 1,KMETHOD = AUTO, KSTEP = 5, and MAXITER = 25.

● Also, for some of these problems, the iterative solution is sensitive to the initialtemperature guess. It is recommended to overshoot (i.e., make a high initial guess)the estimated solution temperature vector.




● Incremental load● Incremental loading reduces the imbalance of the equilibrium equation caused by

applied loads. The single-point constraints (temperature specified by SPC in theBulk Data) and the applied loads (specified by QHBDY, QBDYi, VECFLUX, andQVOL) can be incremented. If the solution takes more iterations than the defaultvalues of the maximum number of iterations allowed for convergence (MAXITER),the increment size should be decreased. For linear problems, no incremental loadsteps are required.

● Convergence criteria● At the beginning stages of a new analysis, it is recommended that the defaults be

used for all options. However, all the Error options (Temperature, Load, and Work)may be selected to improve the efficiency of convergence. For problems with poorconvergence, the tolerances EPSU, EPSP, and EPSW can be increased within thelimits of reasonable accuracy.




● Convergence criteria● For radiation problems that do not converge, and have an error code SFM 4551,

use the full Newton Raphson method so the conduction matrix will be updatedevery iteration. This is done using NLPARM, ID, NINC, DT, KMETHOD, …, whereKMETHOD = FNT for SOL 400.

● SOL153● The best method determined through experience is to use the “full” Newton-

Raphson method. This involves updating the conduction tangent matrix everyiteration. To do this use Matrix Update Method: Controlled Iters, Number ofIterations per Update (KSTEP) = 1. Because the conduction tangent matrix iscalculated for each iteration it is necessary to have only one increment, NINC = 1.

● For problems that are not highly nonlinear it is acceptable to use Matrix UpdateMethod: Automatic.


TRANSIENT THERMAL ANALYSIS

● Transient analysis●Models are similar to those for steady-state analysis, except must

also have●Specific heat (heat capacity per unit mass)●Density●As needed

● Lumped thermal capacitance● Dynamic transfer function

● The equation for the variation of temperature with time, as shown onthe following page, will have to be solved as for steady-stateanalysis. Also, integrating the set of equations will require specifyingparameters, e.g. integration interval/step.


TRANSIENT FORMULATION STEPS


● A transition is made from this equation (strong form) to a variationalformulation (weak form). The matrix equation corresponding to thevariational formulation is

● where, [B] and [K] are the heat capacity and conductivity matrices

● {T} and {F} are the temperature and heat load vectors

● The MD NASTRAN Thermal transient equation derived from thisequation is

tT

kq

T

12

}{}]{[}]{[ FTKTB

}{}{}]{[}]{[}]{[ 4 NPTTTKTB abs R


TRANSIENT FORMULATION STEPS (Cont.)

● This is a nonlinear matrix equation. It is solved using the Newton-Raphson iteration scheme. A residual load vector function is definedas the difference between the applied thermal load vector and thethermal load vector due to element temperature and temperaturerate.

● The residual load vector function, at a given time, is equal to zero forthe solution temperature vector at that time. The task is to determinethe solution temperatures and their rates. The residual load vectorfunction is approximated by its first-order Taylor series expansionabout the temperature vector from the i-th iteration, for the n+1 timepoint (the solution is known for time point n).

)}]{[}]{[}]{([}){}({)}({ 4absTTTKTBNPtR R

)}{}({}{}{

}{}{ 111

11i

nn

i

n

inn TT

TR

RR



● The Taylor series expansion is evaluated at the temperatures for the(i + 1)-th iteration, and set equal to zero.

● From this the following equation is arrived at.

● where

● Note

}0{)}{}({}{}{

}{}{ 11

11

11

1

i

ni

n

i

n

in

in TT

TR

RR

in

in

in TTT }{}{}{ 1

111

in

in

inT RTK }{}{]~[ 111

in

in

in TTT }{}{}{ 11

11



● where (continued)

● and, where the approximation for the time derivative is given by

● Another approximation used is

nTninT

in

i

n

inT KB

tKB

tTR

K ][][1

][][1

}{}){(

]~[ 111

1

n

absnnnnT TNTTKK

}{}{}{][4][][ 3R

}){11(}){}({1}{ 11 nni

ni

n TTTt

T



● The residual load vector function for the i-th iteration is given by

● This process is the same as for the steady-state analysis, this is aniterative process. Start with i = 0. Thus, the first equation to be solvedis

● From the solution to this equation the next temperature vector is foundfrom

i

nin

in

in

inn

in TKTBNPR }{][}{]([)}{}({}{ 1111111

)}{][ 411

iabsn

in TT R

01

01

01 }{}{]~[ nnnT RTK

01

01

11 }{}{}{ nnn TTT



● Next, use i = 1. The next equation to be solved is

● From the solution to this equation the next temperature vector is foundfrom

● This process is repeated until a converged solution is obtained,{Tn+1}m.

● The process is repeated for the next time point, n+2. (tn+2 = tn+1 +tn+1). Set {Tn+2}0 = {Tn+1}m.

11

11

11 }{}{]~[ nnnT RTK

11

11

21 }{}{}{ nnn TTT



● Graphical representation of the solution process● Expression (F) for the thermal load vector due to element

temperature and temperature rate4}]{[}]{[}]{[ absTTTKTBF R

1

1

01nR

11nR

01nF

11nF

F

NP

T)(01

lnn TT

11nT 2

1nT mnT 1

01 nT 1

1 nT

01

~nTK

11

~nTK



● Since matrix decomposition is time consuming, MD NASTRAN doesnot update the left-hand side matrix at each iteration. The tangentialmatrix is updated only when the solution fails to converge or theiteration efficiency can be improved. However, the residual vector isupdated at each iteration.

● In concert with the Newton-Raphson method, the following options areprovided to improve the efficiency of the iteration process:

● Tangential matrix update strategy● Line search method● Bisection of loads● Quasi-Newton (BFGS) updates


FORMULATION STEP COMMENTS

● These options are specified under Analysis in SimXpert. In general, ifthe solution process diverges, a line search algorithm, a bisection ofloads method, or the quasi-Newton update method are implemented inan effort to improve the solution obtained. If the solution still fails toconverge using all the above methods, the tangential stiffness matrixis updated, and the iteration is resumed. The user may refer to the MDNASTRAN Handbook for Nonlinear Analysis for a detailed descriptionof the above mentioned algorithms.


TRANSIENT CONVERGENCE

● Convergence of nonlinear transientsolution process● The form, whos entries are used to

specify control of the nonlineartransient solution process, is repeatedhere for convenience. MD NASTRANparameter (alpha) names are shown inthe MSC SimXpert GUI


TRANSIENT CONVERGENCE (Cont.)

● Convergence of nonlinear transient solution process● Automatic time stepping

● An example of adjusting the time step● MD NASTRAN estimates an optimal time step size, and the step size evolves based

on the convergence condition. The time step is doubled (tn+1 = 2tn) as {Tn} (= {Tn} –{Tn-1}) becomes small.

● where is the maximum value of the norms computed from the previous timesteps, and UTOL is a tolerance on the temperature increment specified in MSCSimXpert.

● If the temperature increment exceeds the tolerance, a proper time step size canbe predicted from the following calculation, where n is the inverse of thecharacteristic time.

)1.0(||}{||||}{||

max

defaultUTOLTTn

max||}{|| T

}{}{

}]{[}{

nT

n

nTT

nn HT

TKTn



● Convergence of nonlinear transient solution process● Automatic time stepping

● The next time step is adjusted using

● where f(r) is a scaling factor defined as follows

● where

nnn trft )(1

RBrrfRBrrf

rRBrfRBrRBrf

RBrrf

nn

nn

nn

nn

nn

/0.3,0.4)(/0.30.2,0.2)(

0.2,0.1)(5.0,5.0)(

5.0,25.0)(

nnn tMSTEP

r121



● Convergence of nonlinear transient solution process● Integration and iteration control

●The initial time increment and the number of time steps are specified byDT and NDT, respectively. So, the duration of the solution of theproblem is approximately NDT * DT. This time is approximate becausethe time increment may be adjusted (increased or decreased) duringthe solution of the problem, so that the number of time steps may notbe equal to NDT. However, the total solution time is close to NDT * DT.

DTNDTtimeTotal




● During the bisection process (time step is bisected) the heat capacitance matrixand tangential conduction matrix are updated every KSTEPth convergedbisection solution.

T

F

P + N

f2(P + N)

f1(P + N)inTK 1][

inB 1][ update every KSTEPth

bisection solution




● The number of iterations for a time step is limited to MAXITER. If MAXITER isspecified as negative, the analysis is terminated on the second divergencecondition during the same time step or when the solution diverges for fiveconsecutive time steps. If MAXITER is specified as positive, the programcomputes the best solution and continues the analysis until divergence occursagain. If the solution does not converge in MAXITER iterations, the process isconsidered divergent. Either bisection or matrix update is activated when theprocess diverges.

Number of iterations fortime step <= MAXITER

Process diverges => Bisection of time step or matrix update




● The convergence criteria are defined under Convergence Criteria. Two types ofthings must be specified

● Error: Temperature, Load, Work

● Tolerance: Temperature (EPSU), Load (EPSP), Work (EPSW)

● The requested criteria (combination of Temperature Error (U), Load Error (P),and Work Error (W)) are satisfied upon convergence. Note that at least twoiterations are required to check the temperature convergence criterion.

● If the bisection option is used, the time step is bisected upon divergence.Otherwise, the left-hand side matrices are updated, and the computation for thecurrent time step is repeated.

● The BFGS update and the line search process are performed in the same wayas in steady state analysis.




● The number of bisections for a load increment is limited to |MAXBIS|. Differentactions are taken when the solution diverges, depending on the sign of MAXBIS.If MAXBIS is positive and the solution does not converge after MAXBISbisections, the best solution is computed and the analysis is continued to thenext time step. If MAXBIS is negative and the solution does not converge in|MAXBIS| bisections, the analysis is terminated.

● Iteration output● At each iteration or time step, the related output data are printed



● Convergence of nonlinear transient solution process● Integration and Iteration control

● The divergence rate is defined as the ratio of energy error before and at the end ofthe current iteration, i+1

● The divergence parameter for the i-th iteration, NDIVi , is incremented as follows

● This computation is performed for all of the iterations of the current time step

● The solution for the current time step is taken as diverged when NDIVfinal >=MAXDIV

errorseverelessforNDIVNDIV ii ,11

errorseveremoreforNDIVNDIV ii ,21



● Convergence of nonlinear transient solution process

Number of occurrences of bisection conditions during the iteration.MAT DIV

Number of bisections executed for the current time interval.NO. BIS

Ratio of time step adjustment relative to DT.ADJUST

Number of occurrences of divergence detected during the adaptive iteration.ITR DIV

Number of line searches performed during the iteration.NLS

Number of quasi-Newton vectors appended.NQNV

Error at the end of line search.E-FINAL

Divergence rate, initial error before line search.E-FIRST

Final value of the line search parameter.FACTOR

Absolute norm of the residual vectorDLMAG

Rate of convergenceLAMDBA(I)

Relative error in terms of workWORK

Relative error in terms of loadsLOAD

Relative error in terms of temperaturesDISP

Iteration count for each time step.ITER

Cumulative time for the duration of the analysis.TIME

DescriptionParameters


RECOMMENDATIONS FOR TRANSIENT

● Convergence of nonlinear transient solution process● Recommendations

● The following are recommendations designed to aid the user.

● Time step size● To avoid inaccurate results or results from an unstable analysis, a proper initial time step

associated with the spatial mesh size is suggested. The selection criterion is

● where t|initial is the initial time step, n is the modification number of the time scale, x is themesh size (dimension of the smallest element), is the material density, Cp is the specificheat at constant pressure, and k is the thermal conductivity. A suggested value for n is 10.For highly nonlinear problems a small initial time step size is recommended.

● Numerical stability● Numerical stability is controlled by the damping parameter . For linear problems = 0 (no

numerical damping) is adequate, but for nonlinear problems a larger value of may beadvisable. Increasing the value of improves numerical stability; however, the solutionaccuracy can be reduced. The recommended range of values is from 0.0 to 0.1 (the defaultvalue is 0.01). The numerical damping is specified using PARAM, NDAMP, (value of This is a Bulk Data PARAM entry.

kC

xn

t pinitial

2)(1|





● Initial temperatures and boundary temperatures

● The specification of initial temperatures and boundary condition temperatures shouldbe consistent. For a given point, the initial temperature should be equal to theboundary condition temperature at t = 0.

● Convergence criteria

● At the beginning stages of a new analysis, it is recommended that the defaults beused for all options. However, all three Error options (UPW) may be selected toimprove the efficiency of convergence. For nonlinear problems with time-varyingboundary conditions, the U Error option must be selected, because the largeconductance (internally generated) affects the calculations of the PW Error functions.For problems with poor convergence, the tolerances EPSU, EPSP, and EPSW canbe increased within the limits of reasonable accuracy.

● When there is a time varying temperature boundary condition, e.g. time varyingambient temperature, set the convergence criteria to just “U”. An example of this canbe seen for Example 7b of the exercises (Chapter 5) in the MD Nastran 2002 ThermalAnalysis User’s Guide.





● Fixed time step● If a fixed time step is desired, the adaptive time stepping can be deactivated by

clicking the checkbox for Fixed Time Steps.

Documents

Thermal Analysis Using MSC SimXpert R3.2 Course Notes