Chapter 5 Transient Analysis

What is a Transient Analysis? When the response of a system over a time interval is required due to loads and boundary conditions which may (or may not) change over time, a Transient Analysis is performed. Time-varying loads Time-varying response Thermal energy storage effects are now included. Time has physical meaning. Analyses involving Phase Change are always transient. This special type of transient analysis is discussed in Chapter 9. Inventory #001445 March 15, 2001 5-2

Preprocessing Considerations for Transient Analysis In addition to thermal conductivity (k), the density (r) and specific heat (c ) material properties must be specified for entities which can conduct and store thermal energy*. Alternatively, enthalpy (H) may be defined (preferred form if performing phase change analysis). These material properties are used to calculate the heat storage characteristics of each element which are then combined in the Specific Heat Matrix [C]. If mass transport of heat effects are being modeled, then these properties are used to augment terms in the thermal conductivity matrix [K]. * the MASS71 Thermal Mass element is unique in that it can store thermal energy but cannot conduct it. Consequently, thermal conductivity is not required for this element. Inventory #001445 March 15, 2001 5-3

Preprocessing Considerations for Transient Analysis (Continued) Like steady-state analyses, transient analyses may be linear or nonlinear. If nonlinear, the same preprocessing considerations apply as for steady-state nonlinear analysis (described in Chapter 4). The most significant difference between steady-state and transient analyses lies in the Loading and Solution procedures. We will focus on these procedures after a brief presentation of the numerical methods employed during transient thermal analysis. Inventory #001445 March 15, 2001 5-4

Governing Equations Recall the governing equation for thermal analysis of a linear system written in matrix form. The inclusion of the heat storage term differentiates transient systems from steady-state systems: In a transient analysis, loads may vary with time . . . . . . or, in the case of a nonlinear transient analysis, time AND temperature: Heat Storage Term = (Specific Heat Matrix) x (time derivative of temperature) Inventory #001445 March 15, 2001 5-5

Time Integration The temperature of a linear thermal system changes continuously from instant to instant: When performing a thermal transient analysis, a time integration procedure is used to obtain solutions to the system equations at discrete points in time. The change in time between solutions is called the integration time step (ITS). Generally, the smaller the ITS, the more accurate the solution becomes. T t T t D t tn tn+1 tn+2 Inventory #001445 March 15, 2001 5-6

Time Step Size Guidelines Selection of a reasonable time step size is important because of its impact on solution accuracy and stability. If the time step size is too small, then spurious oscillations may be seen in midside-noded elements which could result in temperatures which are not physically meaningful. T t D t If the time step is too large, then temperature gradients will not be adequately captured. One approach is to specify a relatively conservative initial time step and allow Automatic Time Stepping to increase the time step as needed. The guidelines on the following slides are presented as a way to roughly approximate a reasonable initial time step size for use with Automatic Time Stepping. Inventory #001445 March 15, 2001 5-7

Time Step Size Guidelines (Continued) To roughly approximate a reasonable time step size for thermal transient analyses, one can make use of the Biot and Fourier numbers. The Biot Number is the dimensionless ratio of convective and conductive thermal resistances: Where D x is the mean element width, h is the average film coefficient, and K is an averaged conductivity. The Fourier Number is a dimensionless time (Dt/t ) which quantifies the relative rates of heat conduction vs. heat storage for an element of width D x : Where r and c are averaged density and specific heat, respectively. Inventory #001445 March 15, 2001 5-8

Time Step Size Guidelines (Continued) If Bi < 1: The time step size can be estimated by setting the Fourier number equal to a constant and solving for D t : The term a is known as the thermal diffusivity. A large a indicates that a material is better at conducting thermal energy than storing it. If Bi > 1: The time step size can be estimated from the product of the Fourier and Biot numbers: Solving for D t gives: (Again, where 0.1  b  0.5) Time step estimates will be more or less conservative depending on the choice of element width, property averaging method, and scaling factor b . Inventory #001445 March 15, 2001 5-9

Numerical Procedure A generalized trapezoidal rule is used for time integration. The current temperature vector, {Tn } is assumed to be known; either as an initial condition or from the previous solution. We define the temperature vector at the next time point as: Where q is called the Euler parameter and is equal to 1 by default. The temperature at the next time point is: We next solve for using equation (a) and substitute the result into equation (b): T Tn Tn+1 D t t tn Tn+1 If nonlinearities are present, the incremental form of this equation is iterated upon at every time point. Equivalent conductivity matrix Equivalent heat flow vector Inventory #001445 March 15, 2001 5-10

More on the Euler Parameter The value of the Euler Parameter, q, is restricted to be between 1/2 and 1. In this range, the time integration algorithm is implicit and unconditionally stable. Consequently, ANSYS will always compute a solution regardless of the magnitude of the ITS (assuming nonlinearities converge). However, there is no guarantee that the solution calculated will be accurate. Here are some guidelines for selection of time integration parameters: When q = 1/2, the time integration strategy is referred to as the “Crank-Nicolson” technique. This setting is typically the most accurate and efficient for the majority of thermal transient problems. When q = 1, the time integration strategy is referred to as the “Backward Euler” technique. This is the default and most numerically stable setting since it eliminates spurious oscillations that may arise when severe nonlinearities or higher order (i.e., midside-noded) elements are present. This technique will generally require a smaller ITS to achieve accuracy comparable to Crank-Nicolson. Inventory #001445 March 15, 2001 5-11

Evaluating Transient Solution Accuracy There are many potential sources of error in a transient thermal analysis. To help evaluate the accuracy of the time integration algorithm, ANSYS computes and reports some helpful quantities after every solution: The Response Eigenvalue represents the dominant system eigenvalue for the most recent time step solution: Where {DT} is the change in the temperature vector {T} across the last time step. It characterizes the relative rates of heat energy conduction and heat energy storage for a system. It is a dimensionless time and can be viewed as a kind of Fourier Number for the system matrices. Note that if nonlinearities are present [KT] replaces [K] in the above equation. Inventory #001445 March 15, 2001 5-12

Evaluating Transient Solution Accuracy (Continued) The Oscillation Limit is a dimensionless quantity that is simply the product of the Response Eigenvalue and the current time step size (i.e., ITS): It is typically desirable to maintain the oscillation limit below 0.5 to insure that the transient response of the system is being adequately characterized. Inventory #001445 March 15, 2001 5-13

Time Step Prediction By default, the Automatic Time Stepping (ATS) feature bases time step prediction on the Oscillation Limit. ATS seeks to maintain the Oscillation Limit below 0.5 within a tolerance, and will adjust the ITS to satisfy this criterion. Notice how ATS gradually reduces the ITS based on the Oscillation Limit. This sample was taken from the ANSYS Output Window during a nonlinear transient analysis. Inventory #001445 March 15, 2001 5-14

Loading and Solution Considerations for Transient Analysis All of the procedures described in Chapter 4 for nonlinear steady-state analysis apply to nonlinear transient analysis. Even if nonlinearities are not present, some of the same procedures still apply but for different reasons. For example, you may need to . . . divide loads into smaller increments to insure that the ITS is not too large and insure that the solution is sufficiently accurate manage the large volume of information that is typically generated during a transient analysis We will focus primarily on Loading and Solution procedures which are unique to transient thermal analysis in the following slides. Complete discussion of this material is beyond the scope of this seminar. Refer to the Thermal Analysis Guide available through on-line help for more details on transient thermal analysis. Inventory #001445 March 15, 2001 5-15

Load Steps and Substeps In a transient analysis, Load Steps and Substeps are defined in much the same manner as for nonlinear steady-state analysis. Loads are specified for the end of each Load Step, and are either ramped or stepped through time. Intermediate solutions are obtained at Substeps within each load step. The spacing of substeps is dependent upon the integration time step. Automatic Time Stepping (ATS) is also available for transient analysis to make the selection of the ITS easier. The selection (ITS) will affect the accuracy of the transient solution and the convergence behavior of nonlinearities (if present). Inventory #001445 March 15, 2001 5-16

Performing a Transient Analysis By default, ANSYS assumes that a steady-state analysis is desired. Use these Solution menus to indicate that a transient analysis is to be performed: “FULL” is the only option available for transient thermal analysis. 3 2 1 4 5 6 7. You will then be prompted to enter analysis options which are not unique to thermal analysis (e.g., equation solver, N-R options) Inventory #001445 March 15, 2001 5-17

Initial Conditions Initial Conditions must be specified for every temperature DOF in the model for the time integration procedure to commence. Initial conditions specified at nodes which have temperature constraints are ignored. Initial conditions may be specified one of several ways depending upon the characteristics of the initial temperature field: NOTE: If no initial temperature is specified, initial DOF values default to zero. Inventory #001445 March 15, 2001 5-18

Uniform Initial Temperature When the initial temperature of the entire model is uniform and non-zero, specify the value with these menus: 1 2 3 4 You may recall from Chapter 4 that this procedure was used to establish a starting point for nonlinear steady-state analyses. Inventory #001445 March 15, 2001 5-19

Non-Uniform Initial Temperatures 1 When the initial temperature distribution of a model is known but is not uniform, apply initial conditions to specific groups of nodes with these menus: 4. Graphically pick nodes or manually enter node numbers that you wish to establish initial temperatures for. 5. Press OK. NOTE: When the IC command is entered manually or via input file, a nodal component name may be used to identify nodes. 3 2 5 4 Inventory #001445 March 15, 2001 5-20

Non-Uniform Initial Temperatures (Continued) 6 7 8 NOTE: The initial temperature of DOF for which initial conditions have not been specifically defined defaults to the uniform temperature specified with the TUNIF command. When Solution Control is on, be sure to specify TUNIF before assigning initial conditions. 6. Select DOF label “TEMP”. 7. Specify initial temperature value. 8. Press OK if done. Press APPLY to repeat process and assign initial temperatures to other groups of nodes. Inventory #001445 March 15, 2001 5-21

Initial Temperatures from a Steady-State Analysis (Continued) When the initial temperature distribution in a model is non-uniform and unknown, a single load step steady-state thermal analysis can be performed to establish initial temperatures prior to the transient analysis. To do so, follow these steps: 1. Steady-state First Load Step: Enter Solution and request a Steady-state analysis type. Apply initial steady-state loads and boundary conditions for the end of the first load step. For convenience, specify a small end time (e.g., 1E-3 seconds). Avoid extremely small values of time (~ 1E-10) since these can cause numerical errors. Specify other controls or settings that might apply (e.g., nonlinear controls). Solve the current load step. NOTE: complete coverage of linear and nonlinear steady-state solution procedures are discussed in Chapters 3 and 4. Inventory #001445 March 15, 2001 5-22

Initial Temperatures from a Steady-State Analysis (Continued) 2. Subsequent Load Steps are Transient: In the second load step, apply loads and boundary conditions corresponding to the first transient load step. Remember to delete any temperature constraints imposed in Load Step 1 which are now free to vary during the transient. 1 Apply transient analysis controls and settings. Before solving, turn on time integration: Solve current transient load step. Solve subsequent load steps using standard procedures. Time integration effects will remain active unless turned off in a later load step. 2 3 4 Inventory #001445 March 15, 2001 5-23

Turning On/Off Time Integration Effects As just demonstrated, a steady-state analysis can quickly become a transient analysis simply by turning ON time integration effects in subsequent load steps. In the same manner, a transient analysis can become a steady-state analysis simply by turning OFF time integration effects in subsequent load steps. Conclusion: In terms of solution methodology, the only difference between a steady-state analysis type and a transient analysis type is time integration. ANTYPE,TRANS + TIMINT,OFF  ANTYPE,STATIC ANTYPE,STATIC + TIMINT,ON  ANTYPE,TRANS Inventory #001445 March 15, 2001 5-24

Another Time Integration Example In this example, rather than turning off time integration effects at the beginning of an analysis to establish initial conditions, they are turned off at the end of the analysis to “speed up” the transient. Notice the sharp change as the solution jumps to steady-state conditions. The end time of the final load step is arbitrary but must be higher than the end time of the preceding transient load step. Typically, the goal of an analysis is to characterize the most severe thermal gradients that occur during a transient thermal event. These gradients often occur in the initial stages of the event and decay with time as the system approaches steady-state. When system response has decayed to the extent that results are no longer of interest, the analysis could simply end or if the steady-state temperature distribution is also desired, time integration effects could be turned off for the final load step. Inventory #001445 March 15, 2001 5-25

Open Control Open Control is used in conjunction with Automatic Time Stepping to “open” (increase) the time step size when a transient thermal solution approaches steady-state. If the maximum temperature change between substeps is less than 0.1 temperature units (default) over 3 successive substeps, the time step size will be rapidly increased to improve efficiency. Available only with Solution Control ON. Settings may be changed using these menus: 1 3 4 5 6 2 3. Specify temperature. 4. Specify threshold value. 5. Specify number of substeps. 6. Press OK Inventory #001445 March 15, 2001 5-26

Stepping or Ramping? To accurately simulate the transient response of a structure it is important that loads be applied with the correct magnitudes, at the correct times, and at the correct rates. Recall that loads can be either stepped or ramped versus time within load steps: ANSYS ramps loads by default. Ramping is preferred to enhance transient solution stability and to avoid convergence problems if nonlinearities are present. Inventory #001445 March 15, 2001 5-27

Stepping or Ramping? (Continued) To approximate a stepped load, ramp the load to full value over a very short time step and then hold it constant for the following load step. QUESTION: You are performing a transient thermal analysis of a tea kettle. You apply a heat flux load to the bottom to simulate heating from an electric stove. Should the heat flux load be ramped or stepped if . . . 1. The kettle is on the stove when it first turned on 2. The kettle is placed on a stove that’s already hot Inventory #001445 March 15, 2001 5-28

Time Integration Controls Recall that the time integration parameter (q) and oscillation limit ( f ) influence the stability and accuracy of time integration. ANSYS permits the user to specify these parameters manually: 1 q f f tolerance 6 Time Integration must be ON Not applicable to thermal analysis 3 4 5 (-1) indicates that ANSYS will select value 2 Inventory #001445 March 15, 2001 5-29

Reviewing “Snapshots” of Results in Time The General Postprocessor may be used to list or plot transient results items at “snapshots” in time. There are a small number of elements used primarily for transient analysis (e.g., MASS71, COMBIN37) which may require special consideration when postprocessing. For most element types, however, there is no substantive difference between steady-state and transient analyses in terms of the kinds of results items that evaluated and the techniques used. When evaluating results from a transient analysis, remember that the energy balance now has a thermal energy storage term included. Consequently, nodal heat flow rates will not sum to the value of the applied loads until steady-state is reached. Inventory #001445 March 15, 2001 5-30

Animating Transient Results Typically, many “snapshots” in time must be viewed to adequately characterize the response of the system. The ANSYS Animation feature is available to automate this task. Use the following menus to animate results across time with constant time intervals: 2 3 1 NOTE: The “Animate Over Results” option is available if interpolation is not desired. 6 Specify sets to be animated Specify results items for animation 4 5 Inventory #001445 March 15, 2001 5-31

The Time-History Postprocessor The ANSYS Time-History Postprocessor (POST26) is used to view analysis results variables as functions of time or other results items. Variables can be displayed in tabular listings and in graphical format. The following table provides a summary of the kinds of variables that can be defined for thermal analysis with the Time-History Postprocessor: These variables can be plotted vs. time Inventory #001445 March 15, 2001 5-32

Defining Variables Use the following menus to define variables in the Time-History Postprocessor: These menus allow you to identify results data, set the number of variables which can be defined, and merge variables from other analyses. By default, up to 10 variables can be defined and ANSYS uses all data in the currently active results file. 1 2 Variables 1-5 have already been defined. Note that ANSYS always presets variable 1 to be Time automatically. Therefore, user defined variables must start with variable number 2. Inventory #001445 March 15, 2001 5-33

Defining Variables (Continued) 3 4 5 6 3. Specify the type of variable to be defined. We will specify a variable corresponding to the temperature at Node 234. 4. Press OK 5. Pick node 234 6. Press OK Inventory #001445 March 15, 2001 5-34

Defining Variables (Continued) 7. Specify the variable number. Notice how ANSYS has automatically specified the next available number 8. This is the number of the node that was picked in the previous step. This can be changed if desired. 9. This variable will be associated with a label that you specify here. 8 7 9 10. Temperature is the only DOF variable available for thermal analysis 10 11 Inventory #001445 March 15, 2001 5-35

Listing Variables Use these menus to list variables that have been defined: By default, variables are always listed and graphed versus the time variable 1 2 3 Inventory #001445 March 15, 2001 5-36

Listing Variables (Continued) This menu can be used to list the maximum and minimum values of a variable: 2 3 1 Inventory #001445 March 15, 2001 5-37

Graphing Variables Use these menus to graph variables: 2 1 3 Inventory #001445 March 15, 2001 5-38

Customizing Graph Features You can customize the style of graphs using these menus: 2 1 3 These menus allow you to modify a range of graph features including line widths, font sizes, markers, labels, colors, gridlines, axis ranges, etc. Inventory #001445 March 15, 2001 5-39

Using Variables in Math Operations A variety of mathematical operations can be performed with variables: The Table Operations menus allow you to transfer a variable in to an array parameter and vice-versa. Let’s do an example to see how these menus work. Inventory #001445 March 15, 2001 5-40

Example Time Derivative of Temperature In the preceding slides we defined a variable representing the temperature at Node 234 and plotted and graphed it versus time. Suppose that the rate of change of the variable with respect to time is desired . This can be done as follows: 1 2. Specify the variable number to be associated with the result of this operation. 3. The result can be scaled by a factor. We will use 1. 4. Variable to be differentiated. 5. Differentiation is done with respect this variable. 6. Label to associate with resulting variable. 2 3 4 5 6 7 Inventory #001445 March 15, 2001 5-41

Example Time Derivative of Temperature The resulting variable can now be listed or graphed: Inventory #001445 March 15, 2001 5-42