mech-ht 13.0 l06 nonlinear

Customer Training Material

L t 6Lecture 6

Nonlinear ThermalAnalysis

ANSYS MechanicalANSYS MechanicalHeat Transfer

L6-1ANSYS, Inc. Proprietary© 2010 ANSYS, Inc. All rights reserved.

Release 13.0December 2010

ANSYS Mechanical Heat Transfer

Customer Training MaterialChapter ContentsA. Nonlinear TheoryB. Nonlinear Solution SetupC. Solution FeedbackC. Solution FeedbackD. Workshop 6, Fin/Tube Heat Exchanger




Customer Training MaterialA. Nonlinear Theory• The governing equation for thermal analysis of a linear system

written in matrix form:

[ ]{ } [ ]{ } { }&

• If these quantities vary with temperature, the system is nonlinear and

[ ]{ } [ ]{ } { }QTKTC =+

q y p , ymust be solved using an iterative process:

( )[ ]{ } ( )[ ]{ } ( ){ }TQTTKTTC +&

Any of the following cause the analysis to be nonlinear:

( )[ ]{ } ( )[ ]{ } ( ){ }TQTTKTTC =+

• Any of the following cause the analysis to be nonlinear:– temperature-dependent material properties– temperature-dependent film coefficients

t t d d t h t



– temperature-dependent heat sources


Customer Training Material. . . Nonlinear Theory• Consider the case of steady-state nonlinear analysis:

( )[ ]{ } ( ){ }TQTTK =• This equation can be viewed equivalently as:

{ } { }anr QQ =Vector of Internal Nodal Heat Flows

Vector of Nodal HeatFlows From Applied Loads

• Initially, internal nodal heat flows will NOT be equal to the applied nodal loads. The difference is called the out-of-balance heat flow

{ } { }

vector or “residual”:

Th l i t d i th it d f thi id l t

{ } { } { }nra QQ −=Φ

• The goal is to drive the magnitude of this residual to zero.

• In practice a convergence criteria ε is used to establish equilibrium.



{ } aQε<Φ


Customer Training Material. . . Nonlinear Theory• To accomplish this the Newton-Raphson iterative solution technique

is employed . . . [ ]{ } { } { } .)..3,2,1( =−=Δ iQQTK nri

ai

Ti

• Nodal temperatures are updated:

{ } { } { }iii TTT Δ+=+1

Single DOF System

Q

Q

• The convergence norm is computedand compared against the criterion:

{ } { } { }iii+1 Q

Tnr1iQ +and compared against the criterion:

{ } aQε<Φ

1i+

How Close is Close Enough? Convergence

• If equilibrium is not satisfied, [KT] is updated and another iteration is performed

QT

Convergence Criteria Says When.



performed.

Note: The default value for ε is 0.001

TT T


Customer Training Material. . . Nonlinear Theory• Temperature-Dependent Inputs:– The most common nonlinearities in a thermal model are temperature-

dependent boundary conditions and material properties:• Temperature dependent film coefficient h(t).• Temperature dependent thermal conductivity K(t).

– Radiation also contributes nonlinear terms to the conductance matrix (K) which are functions of T4which are functions of T4.




Customer Training MaterialB. Nonlinear Solution Setup• A nonlinear thermal solution often requires special loading controls

to be employed in Mechanical:– Divide loads into smaller increments to insure convergence.– Control convergence criteria.– Manage the large volume of information that is typically generated during

a nonlinear analysis .• Steps: differentiate periods of loading. Steps are not unique to

nonlinear analysis.

• Steps are typically used whenSteps are typically used when loads change value.

• Substeps are used to increment steps in a series of convergedsteps in a series of converged solutions.

• Each step and substep require a separate “solve”.



separate solve .


Customer Training Material. . . Nonlinear Solution Setup• In a non-linear steady state analysis we might want to increment the

solution to ramp a severely nonlinear load or property on gradually.• Multiple substeps are manually defined by turning “On” “Auto Time

Stepping” and choosing to define by Substeps.– Here we have forced 5 substeps by setting initial, minimum and maximum

all to 5. We will cover automatic time stepping later.– When complete, there will be 5 result points for postprocessing.– Note each substep may also require multiple Newton-Raphson iterations

in order to converge.




Customer Training Material. . . Nonlinear Solution Setup• Mechanical activates the nonlinear option when conditions warrant

(e.g. K(t), Q(t), etc.).• Users can leave Auto Time Stepping as “Program Controlled” and

ANSYS will determine the initial, minimum, and maximum number of substeps to be used.

• The user may set Auto Time Stepping to ON, and control these values manually for more severe nonlinearities (see next slide).

• In this example the solver will begin by attempting to solve 10 incremental solutions:– If convergence struggles, the solution may bisect to take a smaller

increment (up to 1000).– If convergence is achieved quickly the solver.

may increase the substeps and finish morequickly (down to 2).




Customer Training Material. . . Nonlinear Solution Setup• Automatic Time Stepping (ATS): – Time step size is adjusted automatically by ANSYS based on the

response of the model.Th li th I iti l i i d i ti t i– The user supplies the Initial, minimum and maximum time step sizes which “bound” the action of ATS.

• ATS has two important functions:– It performs time step prediction based on iterations for previously– It performs time step prediction based on iterations for previously

converged substeps or the transient response of model (if not a steady state analysis).

– It performs time step reduction when the solution is expected to require th th i ll d b f ilib i it ti tmore than the maximum allowed number of equilibrium iterations to

converge.• ATS is particularly valuable if the behavior of the model is expected

to vary widely from load step to load step.to vary widely from load step to load step.




Customer Training Material. . . Nonlinear Solution Setup• The incremental time step size is the most important solution

parameter relating to nonlinear solution robustness, accuracy, and efficiency.

• In general, when the time step size is reduced:+ The solution is less likely to diverge.+ Results become more accurate.+ Fewer equilibrium iterations are required for each solution.– Analysis time increases.

• A number of factors will affect the optimum time step size for a given p p gproblem such as:– Types and severity of nonlinearities active.– Location and types of loads.yp– Mesh size.– Previous convergence behavior.– Transient effects (discussed in a later chapter).



( p )


Customer Training Material. . . Nonlinear Solution Setup• Nonlinear convergence criteria are used by Mechanical to

determine if an iterative solution has “converged” or if more N-R iterations are needed.

• Convergence criteria may be chosen manually or program chosen (default):– By default, ANSYS selects a criterion

based on nodal heat flow rate imbalance (HEAT).

– Temperature based convergence checking is usually less conservative thanis usually less conservative than convergence based on heat flow rate.

– If more than one criterion is active, then the solution must meet all criteria before it is considered to be “converged”.

NOTE: If either criterion is chosen manually then ALL default criteria is



yremoved and must be specified manually if desired.


Customer Training Material. . . Nonlinear Solution Setup• With multiple solutions come increased result file sizes (multi-step

linear analyses or nonlinear multiple load step/substep analyses).• Output Controls:– Users may choose to omit heat flux results (default is to calculate heat

flux).– Users may control the frequency with which results are written to the

results file (default is to save all time points).• Choose to save only the last calculated time point (caution, no other results

will be available to review).• Choose to save results at equally spaced points (e g every 4th step/substep)Choose to save results at equally spaced points (e.g. every 4th step/substep).




Customer Training Material. . . Nonlinear Solution Setup• The reference “Value” can be specified, or it is calculated to be the

norm of the applied heat flow rate (default).• This reference value is multiplied by a “Tolerance” to establish the p y

convergence criterion.• Recall from the nonlinear theory section that we want to satisfy:

I W kb h t thi k f thi

{ } aQε<Φ

• In Workbench terms we can think of this as:

Residual < Tolerance × Value

• If then the minimum reference value isValue < Minimum Reference



If then the minimum reference value is used.

Value < Minimum Reference


Customer Training Material. . . Nonlinear Solution Setup• Line Search is an augmentation to the Newton Raphson nonlinear

solution scheme:– Generally line search is applicable to unstable problems (e.g. phase

change, radiation, etc.) and should not be activated for general problems.– In situations where oscillatory behavior is noticed while solving, line

search can enhance convergence.Li S h i h b d f lt– Line Search is program chosen by default.




Customer Training MaterialC. Solution Feedback• Much can be learned by examining the Solution Information, “Solver

Output” during a nonlinear thermal analysis:

Heat Flow Convergence Norm is Decreasing

Incremental Temperature Change is

Decreasing. Decreasing.

Conv Value Less

We are Done With LS 2, SS 3.

Conv.Value LessThan Criterion.

This Substep Took 7 Iterations



This Substep Took 7 Iterations to Converge. So Far, 28 Total Iterations Have Been Performed.


Customer Training Material. . . Solution Feedback• In addition to solver output (previous page), a number of solution

parameters can be plotted to track solution progress.



Customer Training Material

W k h 6Workshop 6

Fin/Tube Heat Exchanger

ANSYS MechanicalANSYS MechanicalHeat Transfer



mech-ht 13.0 l06 nonlinear

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