Mech-HT 120 Lect-05 Nonlinear

Chapter 5

Nonlinear Thermal Analysis

5-1ANSYS, Inc. Proprietary© 2009 ANSYS, Inc. All rights reserved.

August 2009Inventory #002667

ANSYS Mechanical Heat Transfer


Training ManualChapter Contents

A. Nonlinear TheoryB. Nonlinear Solution SetupC. Solution FeedbackD. Workshop




Training ManualA. Nonlinear Theory

• The governing equation for thermal analysis of a linear system written in matrix form:

• If these quantities vary with temperature, the syst em is nonlinear and must be solved using an iterative process:

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



• Any of the following cause the analysis to be nonli near:– temperature-dependent material properties– temperature-dependent film coefficients– temperature-dependent heat sources

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


Training Manual. . . Nonlinear Theory

• Consider the case of steady-state nonlinear analysi s:

• This equation can be viewed equivalently as:

• Initially, internal nodal heat flows will NOT be equal to the applied

( )[ ]{ } ( ){ }TQTTK =

{ } { }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-b alance heat flow vector or “residual”:

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

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

{ } { } { }nra QQ −=Φ

{ } aQε<Φ


Training Manual. . . Nonlinear Theory

• To accomplish this the Newton-Raphson iterative sol ution technique is employed . . .

• Nodal temperatures are updated

• The convergence norm is computed

[ ]{ } { } { } .)..3,2,1( =−=∆ iQQTK nr

i

a

i

T

i

{ } { } { }iii TTT ∆+=+1

Single DOF System

Q

Q



• The convergence norm is computed

and compared against the criterion :

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

{ } aQε<Φ

Note: The default value for ε is 0.001

Q

TT

T

T

Tnr1iQ +

How Close is Close Enough? Convergence Criteria Says When.


Training Manual

• Temperature-Dependent Inputs:– The most common nonlinearities in a thermal model a re temperature-

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

– Radiation also contributes nonlinear terms to the c onductance matrix (K) which are functions of T 4

. . . Nonlinear Theory




Training Manual

• A nonlinear thermal solution often requires special loading controls to be employed in Mechanical:

– Divide loads into smaller increments to insure conv ergence.– Control convergence criteria.– Manage the large volume of information that is typi cally generated during

a nonlinear analysis .

• Steps: differentiate periods of loading. Steps are not un ique to nonlinear analysis.

B. Nonlinear Solution Setup



nonlinear analysis.

• Steps are typically used when loads change value.

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

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


Training Manual

• In a non-linear steady state analysis we might want to increment the solution to ramp a severely nonlinear load or prope rty 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 la ter.

– When complete, there will be 5 result points for po stprocessing.– Note each substep may also require multiple Newton -Raphson iterations

. . . Nonlinear Solution Setup



– Note each substep may also require multiple Newton -Raphson iterations in order to converge.


Training Manual

• Mechanical activates the nonlinear option when cond itions warrant (e.g. K(t), Q(t), etc.)

• Often one leaves Auto Time Stepping as Program Cont rolled and ANSYS will determine the initial, minimum, and maxi mum number of substeps to be used

• The user may set Auto Time Stepping to ON, and cont rol these values manually for more severe nonlinearities (see next s lide)

• In this example the solver will begin by attempting to solve 10




• In this example the solver will begin by attempting to solve 10 incremental solutions

– If convergence struggles, the solution may bisect t o take a smaller increment (up to 1000)

– If convergence is achieved quickly the solvermay increase the substeps and finish morequickly (down to 2)


Training Manual

• Automatic Time Stepping (ATS): – Time step size is adjusted automatically by ANSYS ba sed on the

response of the model – 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

converged substeps or the transient response of mod el (if not a steady state analysis)




state analysis)– It performs time step reduction when the solution is expected to require

more 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


Training Manual

• 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




– Analysis time increases

• A number of factors will affect the optimum time st ep size for a given problem such as:

– Types and severity of nonlinearities active– Location and types of loads– Mesh size– Previous convergence behavior– Transient effects (discussed in a later chapter)


Training Manual

• Nonlinear convergence criteria are used by Mechanic al 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)




(HEAT)– Temperature based convergence checking

is 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 removed and must be specified manually if desired.


Training Manual. . . Nonlinear Solution Setup

• With multiple solutions come increased result file sizes (multi-step linear analyses or nonlinear multiple load step/sub step 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



• 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)



• The reference “Value” can be specified, or it is ca lculated to be the norm of the applied heat flow rate (default)

• This reference value is multiplied by a “Tolerance” to establish the convergence criterion

• Recall from the nonlinear theory section that we wa nt to satisfy:

{ } aQε<Φ



• In Workbench terms we can think of this as:

• If then the minimum reference value is used

Residual < Tolerance × Value

Value < Minimum Reference



• Line Search is an augmentation to the Newton Raphso n nonlinear solution scheme

– Generally line search is applicable to unstable pro blems (e.g. phase change, radiation, etc.) and should not be activate d for general problems

– In situations where oscilliatory behavior is notice d while solving, line search can enhance convergence

– Line Search is program chosen by default




Training Manual

Heat Flow Convergence Norm is Decreasing.

Incremental Temperature Change is Decreasing.

• Much can be learned by examining the Solution Infor mation, “Solver Output” during a nonlinear thermal analysis

C. Solution Feedback



Conv.Value LessThan Criterion.

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

We are Done With LS 2, SS 3.


Training Manual

• In addition to solver output (previous page), a num ber of solution parameters can be plotted to track solution progres s

. . . Solution Feedback



Workshop 5

Fin Tube Heat Exchanger



ANSYS Mechanical Heat Transfer

Mech-HT 120 Lect-05 Nonlinear

Documents

Transcript of Mech-HT 120 Lect-05 Nonlinear