Computer implementation of the CBS algorithm
Contents
1. are used The reader is referred to the appropriate chapters of this volume Chapters 3 4 and 5 for different non dimensional parameters In Sec 9 2 we shall describe the essential features of data input to the program Here either structured or unstructured meshes can be used to divide the problem domain into finite elements Section 9 3 explains how the steps of the CBS algorithm are implemented In that section we briefly remark on the options available for shock capturing various methods of time stepping and different procedures for equation solving In Sec 9 4 the output generated by the program and postprocessing procedures are considered In the last section Sec 9 5 we shall consider the possibility of further extension of CBSflow to other problems such as mass transfer turbulent flow etc 9 2 The data input module This part of the program is the starting point of the calculation where the input data for the solution module are prepared Here an appropriate input file is opened and the data are read from it Unlike in Chapter 20 Volume 1 we have no mesh generator coupled with CBSflow However an advancing front unstructured mesh generator and some structured mesh generators are provided separately By suitable coupling the reader can implement various adaptive procedures as discussed in Chapters 4 and 5 Either structured or unstructured mesh data can be given as input to the program The general program structure and many more d2. the reader can refer to the theory discussed in Chapter 3 of this volume for a comprehensive review of the CBS algorithm Solution module 279 Preliminary routines Make changes Data check Passed Step 1 intermediate momentum Step 2 Density pressure Step 3 ae Time correction loop Energy coupling No Boundary conditions Steady No state S Ye Fig 9 2 Flow diagram for CBSflow Yes Energy temperature calculation 280 Computer implementation of the CBS algorithm 9 3 2 Time step In general three different ways of establishing the time steps are possible In problems where only the steady state is of importance so called local time stepping is used see Sec 3 3 4 Chapter 3 Here a local time step at each and every nodal points is calculated and used in the computation When we seek accurate transient solution of any problem the so called minimum step value is used Here the minimum of all local time step values is calculated and used in the computation Another and less frequently used option is that of giving a fixed user prescribed time step value Selection of such a quantity needs considerable experience from solving several flow problems The times loop starts with a subroutine where the above mentioned time step options are available In general the local time steps are calculated at every iteration for the initial few time steps and then they are calculated o
3. 1 TSTI shape function derivatives element length at node 1 external time step internal time step Solution module 283 ANX GEOME 3 IE ANY GEOME 6 IE ALEN 1 0 DSQRT ANX 2 ANY 2 TSTP ALEN VSUM TSTI ALEN VELN DELTP IP3 MIN DELTP IP3 TSTP DELTI IP1 MIN DELTI IP1i TSTI END DO IE DO IP 1 NPOIN DELTP IP SFACT DELTP IP SFACT safety factor END DO IP IF IALOT EQ 0 THEN DTP 1 0d 06 DO IP 1 NPOIN DTP MIN DTP DELTP IP END DO IP CALL TIMFIL MXPOI DELTP NPOIN DTP ENDIF END Fig 9 3 Continued The nodal quantities calculated in a manner explained above are averaged over elements and used in the relations of Eq 6 17 Chapter 6 Figure 9 5 shows the calculation of the nodal pressure switches for linear triangular elements In the next option available in the code the second derivative of pressure is calculated from the smoothed nodal pressure gradients see Sec 4 5 1 Chapter 4 by averaging Other approximations to the second derivative of pressure are described 5 a b Fig 9 4 Typical element patches a interior node b boundary node 284 Computer implementation of the CBS algorithm in Sec 4 5 1 Chapter 4 The user can employ those methods to approximate the second derivative of pressure if desired 9 3 4 CBS algorithm Steps Various steps involved in the CBS algorithm are described in detail in Chapter 3 There are three essential steps in the CBS a
4. 2 IW ANY1 IF ACH LT 0 2 THEN WNOR 1 IW 0 0D00 WNOR 2 IW 0 0D00 WRITE IWPOIN 1 IW is trailing edge e g aerofoil ENDIF END DO IW END Fig 9 1 Subroutine calculating surface normals on the walls mass matrix calculation and lumping and some allocation subroutines are necessary before starting the time loop The routine for establishing the surface normals is shown in Fig 9 1 On sharp narrow corners as at the trailing edge of an aerofoil the boundary contributions are made zero by assigning a zero value for the surface normal as shown 9 3 Solution module Figure 9 2 shows the general flow diagram of CBSflow As seen the data from the input module are passed to the time loop and here several subprograms are used to solve the steps of the CBS algorithm It should be noted that the semi implicit form is used here only for incompressible flows and at the second step we only calculate pressure as the density variation is here assumed negligible 9 3 1 Time loop The time iteration is carried out over the steps of the CBS algorithm and over many other subroutines such as the local time step and shock capture calculations As men tioned in the flow chart the energy can be calculated after the velocity correction However for a fully explicit form of solution the energy equation can be solved in step 1 along with the intermediate momentum variable Further details on different steps are given in Sec 9 3 4 and
5. 2 V 2 DO IP 1 NMAX I IP1i MAXCON IP I VNORM I VNORM I VMAG IP1 PRS I PRS I PRES IP1 RHO I RHOCI UNKNO 1 IP1 END DO IP Fig 9 3 Subroutine for time step calculation 282 Computer implementation of the CBS algorithm VNORM T PRS I RHO T SONIC I END DO I VNORM I FLOAT NMAX T PRS I FLOAT NMAX T RHO 1 FLOAT NMAX T DSQRT GAMMA PRS I RHO I DO IP 1 NPOIN DELTP IP 1 0d06 SONIC IP DSQRT GAMMA PRES IP UNKNO 1 IP speed of sound END DO IP c c loop for calculation of local time steps c DO IE 1 NELEM IP1 IP2 IP3 Ui Vi U2 V2 U3 V3 VN1 VN2 VN3 VELN CMAX VSUM ANX ANY ALEN TSTP TSTI DELTP IP1 DELTI IP1 ANX ANY ALEN TSTP TSTI DELTP IP2 DELTI IP1 Fig 9 3 Continued IX 1 IE IX 2 IE IX 3 IE UNKNO 2 IP1 UNKNO 1 IP1 UNKNO 3 IP1 UNKNO 1 IP1 UNKNO 2 IP2 UNKNO 1 IP2 UNKNO 3 IP2 UNKNO 1 IP2 UNKNO 2 IP3 UNKNO 1 IP3 UNKNO 3 IP3 UNKNO 1 IP3 DSQRT U1 2 U1 2 DSQRT U2 2 U2 2 DSQRT U3 2 U3 2 MAX VN1 VN2 VN3 connectivity ul velocity u2 velocity MAX SONIC IP1 SONIC IP2 SONIC IP3 VELN CMAX GEOME 1 IE GEOME 4 IE 1 0 DSQRT ANX 2 ANY 2 ALEN VSUM ALEN VELN MIN DELTP IP1 TSTP MIN DELTI IP1 TSTI GEOME 2 IE GEOME 5 IE 1 0 DSQRT ANX 2 ANY 2 ALEN VSUM ALEN VELN MIN DELTP IP2 TSTP MIN DELTI IP
6. A SSR NSA AAS SAG S The residuals difference between the current and previous time step values of parameters of all equations are checked at every few user prescribed number of itera tions If the required convergence steady state is achieved the program stops automatically The aimed residual value is prescribed by the user The program calculates the maximum residual of each variable over the domain The user can use them to fix the required accuracy We give the routine used for this purpose in Fig 9 7 287 288 Computer implementation of the CBS algorithm SUBROUTINE RESID MXPOI NPOIN ITIME UNKNO UNPRE PRES PRESN IFLOW a purpose IMPLICIT INTEGER REAL 8 REAL 8 REAL 8 REAL 8 EMAX1 EMAX2 EMAX3 EMAX4 D0 I i ERR1 ERR2 ERR3 ERR4 ERI ER2 ER3 ER4 IF ER1 EMAX1 ICON1 ENDIF IF ER2 EMAX2 ICON2 ENDIF IF ER3 EMAX3 ICON3 ENDIF IF ER4 EMAX4 ICON4 ENDIF END DO END calculations of residuals NONE I ICON1 ICON2 ICON3 ICON4 IFLOW ITIME MXPOI NPOIN EMAX1 EMAX2 EMAX3 EMAX4 ERR1 ERR2 ERR3 ERR4 ER1 ER2 ER3 ER4 PRES MXPOL PRESN MXPOI UNKNO 4 MXPOI UNPRE 4 MXPOI H NPOIN UNKNO 1 I UNKNO 2 1 UNKNO 3 1 UNKNO 4 1 DABS ERR1 DABS ERR2 DABS ERR3 DABS ERR4 GT EMAX1 ER1 GT EMAX2 ER2 I GT EMAX3 ER3 I GT EMAX4 ER4 I I 0 000d00 0 000d00 0 000d00 0 000d00 UNPRE 1
7. Chapter 3 of this volume We prefer to keep the compressible and incompressible flow codes separate to avoid any confusion However an experienced programmer can incorporate both parts into a single code without much memory loss Each program list ing is accompanied by some model problems which helps the reader to validate the codes In addition to the model inputs to programs a complete user manual is available to users explaining every part of the program in detail Any error reported by readers will be corrected and the program will be continuously updated by the authors Research Fellow Department of Civil Engineering University of Wales Swansea UK The data input module 275 The modules are constructed essentially as in Chapter 20 Volume 1 starting with 1 the data input module with preprocessing and continuing with 2 the solution module and 3 the output module However unlike the generalized program of Chapter 20 Volume 1 the program CBSflow only contains the listing for solving transient Navier Stokes or Euler Stokes equations iteratively Here there are many possibilities such as fully explicit forms semi implicit forms quasi implicit forms and fully implicit forms as discussed in Chapter 3 of this volume We concen trate mainly on the first two forms which require small memory and simple solution procedures compared to other forms In both the compressible and incompressible flow codes only non dimensional equations
8. Computer implementation of the CBS algorithm P Nithiarasu 9 1 Introduction In this chapter we shall consider some essential steps in the computer implementation of the CBS algorithm on structured or unstructured finite element grids Only linear triangular elements will be used and the notes given here are intended for a two dimensional version of the program The sample program listing and user manual along with several solved problems are available to down load from the publisher s web site http www bh com companions fem free of charge The program discussed can be used to solve the following different categories of fluid mechanics problems Compressible viscous and inviscid flow problems Incompressible viscous and inviscid flows Incompressible flows with heat transfer Porous media flows Shallow water problems BN With further simple modifications many other problems such as turbulent flows solidification mass transfer free surfaces etc can be solved The procedures presented here are largely based on the computer implementation discussed in Chapter 20 Volume of this book Many programming aspects will not be discussed here in detail and the reader is referred back to Chapter 20 Volume 1 Here it is assumed that the reader is familiar with FORTRAN and finite element procedures discussed in this volume as well as in Volume 1 We call the present program CBSflow since it is based on the CBS algorithm discussed in
9. ER INTEGER REAL 8 REAL 8 REAL 8 DO IELEM 1 IP1 IP2 IP3 PS1 PS2 PS3 PADD P11 P22 P33 PSWTH IP1 PSWTH IP2 PSWTH IP3 DELUN IP1 DELUN IP2 DELUN IP3 CSHOCK PSWTH IX DELUN ISIDE MODEL ITYPE calculates the pressure switch at each node and minimum value 0 NONE IB IELEM IP IP1 1P2 1P3 ITYPE MBC MODEL MXELE MXPOI NBS NELEM NPOIN ISIDE 4 MBC IX MODEL MKELE CSHOCK PADD P11 P22 P33 PS1 PS2 PS3 XPS XPD DELUN MXPOI PRES MXPOI PSWITH MXPOI NELEM IX 1 IELEM IX 2 TELEM IX 3 IELEM PRES IP1 PRES IP2 PRES IP3 PS 1 PS2 PS3 3 0d00 PSi PADD 3 0d00 PS2 PADD 3 0d00 PS3 PADD PSWTH IP1 Pil PSWIH IP2 P22 PSWTH IP3 P33 DELUN IPi DABS PS1 PS2 DABS PS1 PS3 DELUN IP2 DABS PS1 PS2 DABS PS2 PS3 DELUN IP3 DABS PS3 PS2 DABS PS1 PS3 END DO IELEM DO IB 1 NBS IPt IP2 PS1 PS2 XPS XPD PSWTH IP1 PSWTH IP2 DELUN IP1 DELUN IP2 ISIDE 1 IB ISIDE 2 IB PRES IP1 PRES IP2 PSi PS2 PS1 PS2 PSWTH IP1 XPD PSWTH IP2 XPD DELUN IP1 DABS XPD DELUN IP2 DABS XPD 286 Computer implementation of the CBS algorithm END DO IB DO IP i NPOIN IF DELUN IP LT 0 1 PRES IP DELUN IP PRES IP END DO IP DO IP 1 NPOIN PSWTH IP CSHOCK DABS PSWTH IP DELUN IP END DO IP END Fig 9 5 Calculation of n
10. I UNPRE 2 1 UNPRE 3 I UNPRE 4 1 THEN THEN THEN THEN Fig 9 7 Subroutine to check convergence rate density or pressure ul velocity or mass flux u2 velocity or mass flux energy or temperature References 289 9 4 Output module If the imposed convergence criteria are satisfied then the output is written into a separate file The user can modify the output according to the requirements of post processor employed Here we recommend the education software developed by CIMNE GiD for post and preprocessing of data The facilities in GiD include two and three dimensional mesh generation and visualization 9 4 1 Stream function calculation The stream function value is calculated from the following equation Oo Fb _ u dv 9 3 Ox x Ox Ox f This equation is derived from the definition of stream function in terms of the velocity components We again use the finite element method to solve the above equation 9 5 Possible extensions to CBSflow As mentioned earlier there are several possibilities for extending this code A simple subroutine similar to the temperature equation can be incorporated to solve mass transport Here another variable concentration needs to be solved Another subject which can be incorporated and studied is that of a free surface given in Chapter 5 of this volume Here another equation needs to be solved for the surface waves The phase change problems need
11. and banded solution are preferred by the user a flag activated by the user calculates the half bandwidth of the mesh and supplies it to the solution module Alternatively a diagonally preconditioned conjugate gradient solver can be used with an appropriate flag These solvers are necessary only when the semi implicit form of solution is used 9 2 2 Boundary data In general the procedure discussed in Chapter 20 Volume 1 uses the boundary nodes to prescribe boundary conditions However in CBSflow we mostly use the edges to store the information on boundary conditions Some situations require boundary nodes e g pressure specified in a single node and in such cases corresponding node numbers are supplied to the solution module 9 2 3 Other necessary data and flags In addition to the mesh data and boundary information the user needs to input a few more parameters used in flow calculations For example compressible flow computations need the values of non dimensional parameters such as the Mach number Reynolds number Prandtl number etc Here the reader may consult the non dimensional equations and parameters discussed in Sec 3 1 Chapter 3 and in Chapter 5 of this volume The necessary parameters for different problems are listed in Table 9 1 for completeness Several flags for boundary conditions shock capture etc need to be given as inputs For a complete list of such flags the reader is referred to the user manual and program
12. appropriate changes in the energy equation The liquid solid and mushy regions can be accounted for in the equations by simple modifications The latent heat also needs to be included in phase change problems The turbulent flow requires solution of another set or sets of equations similar to the momentum or energy equations as explained in Chapter 5 For the x e model the reader is referred to reference 13 The program CBSflow is an educational code which can be modified to suit the needs of the user For instance the modification of this program to incorporate a command language could make the code very efficient and compact 12 References 1 I Swith and D V Griffiths Programming the Finite Element Method Third Edition Wiley Chichester 1998 2 D R Will Advanced Scientific Fortran Wiley Chichester 1995 3 O C Zienkiewicz and R L Taylor The Finite Element Method Vol 1 The Basics Sth Edition Arnold London 2000 4 P Nithiarasu and O C Zienkiewicz On stabilization of the CBS algorithm Internal and external time steps Int J Num Meth Eng 48 875 80 2000 290 Computer implementation of the CBS algorithm 3 6 GiD International Center for Numerical Methods in Engineering Universidad Polit cnica de Catalufia 08034 Barcelona Spain P Nithiarasu K N Seetharamu and T Sundararajan Double diffusive natural convection in an enclosure filled with fluid saturated porous medium a gene
13. ents inside and on the boundaries For inside nodes Fig 9 4 a we calculate the nodal switch as S l4p1 P2 P3 Pa Psl 9 1 pi Pol pi p3l pi pal pi ps and for the boundary node Fig 9 4 b we calculate Sp 2p2 P3 2p4 9 2 2 pi P2 Pi p 2 pi pal 1 Solution module 281 SUBROUTINE TIMSTP MXPOI MXELE NELEM NPOIN IALOT IX SFACT amp amp DTFIX UNKNO DELTP DELTI SONIC PRES GAMMA GEOME X NMAX MAXCON MODEL NODEL c calculates the critical local time steps at nodes c calculates internal and external time steps c IMPLICIT IMPLICIT PARAMETER INTEGER INTEGER INTEGER REAL 8 REAL 8 REAL 8 REAL 8 REAL 8 NONE MPOI MPOI 9000 I IALOT IE IP IP1 IP2 IP3 MODEL MXELE MXPOI NELEM NODEL NPOIN IX MODEL MXELE MAXCON 20 MIXPOL NMAX MXPOT ALEN ANX ANY CMAX DTFIX DTP GAMMA SFACT TSTI TSTP U U1 U2 U3 V V1 V2 V3 VN1 VN2 VN3 VELN VSUM DELTI MXPOI DELTP MXPOI GEOME 7 MXELE PRES MXPOI SONIC MXPOI UNKNO 4 MXPOI X 2 MXPOI REAL 8 PRS MPOI RHO MPOI VMAG MPOI VNORM MPOI local arrays IF IALOT EQ 1 THEN CALL TIMFIL MXPOL DELTP NPOIN DTFIX CALL TIMFIL MXPOI DELTI NPOIN DTFIX RETURN ENDIF a smoothing the variables DO I 1 NPOIN VNORM I 0 00D 00 RHO I 0 00D 00 PRS I 0 00D 00 U UNKNO 2 1 UNKNO 1 1 V UNKNO 3 1 UNKNO 1 1 VMAG TI DSQRT U
14. essible flow computations the energy equation can be written in terms of the temperature variable and the dissipation terms can be neglected In general for compressible flows Eq 3 61 is used and Eq 4 6 is used for incom pressible flow problems 9 3 8 Thermal and porous media flows _ As mentioned earlier the heat transfer and porous medium flows are also included in the incompressible flow code Using the heat transfer part of the code the user can solve forced natural and mixed convection problems Appropriate flags and Solution module SUBROUTINE SYMMET MXPOI MBC NPOIN NBS UNKNO ISIDE RHOINF amp UINF VINF COSX COSY c c symmetric boundary conditions forced one component of velocity c forced to zero c IMPLICIT NONE INTEGER I IP J MBC MXPOI NBS NPOIN INTEGER ISIDE 4 MBC REAL 8 ANX ANY RHOINF UINF US VINF REAL 8 COSX MBC COSY MBC UNKNO 4 MXPOI DO I 1 NBS IF ISIDE 4 1I EQ 4 THEN symmetry flag 4 ANX COSX T ANY COSY I DO J 1 2 IP US UNKNO 2 IP UNKNO 3 IP END DO J ENDIF END DO I END ISIDE J I UNKNO 2 IP ANY UNKNO 3 IP ANX US ANY US ANX H Fig 9 6 Subroutine to impose symmetry conditions non dimensional parameters need to be given as input For the detailed discussion on these fiows the reader is referred to Chapter 5 of this volume 9 3 9 Convergence BUANN RRR AO SARRAR SEENE SE SAER DANAA EAEE SEARA EEAS EANA SA SAE S SEE ESAS SAE an AAEE ENESE SE
15. etails can be found in Chapter 20 Volume 9 2 1 Mesh data nodal coordinates and connectivity TONORA GASENIREN Aok itooo INGO HAND EO te SOD RENAN ao oN ISN RE SHERROD oaa ON OKANA SE ASENA AA ASENA Once the nodal coordinates and connectivity of a finite element mesh are available from a mesh generator they are allotted to appropriate arrays for a detailed descrip tion on the mesh numbering etc see Chapter 20 Volume 1 Essentially the same arrays as described in Chapter 20 Volume are used here The coordinates are allotted to X i j with i defining the appropriate cartesian coordinates x i 1 and x i 2 and j defining the global node number Similarly the connectivity is allotted to an array X k Here k is the local node number and is the global element number It should be noted that the material code normally used in heat conduction and stress analysis is not necessary 276 Computer implementation of the CBS algorithm Table 9 1 Non dimensional parameters Non dimensional number Symbol Flow types Conductivity ratio k Porous media flows Darcy number Da Porous media flows Mach number M Compressible flows Prandtl number Pr Compressible incompressible thermal and porous media flows Porosity E Porous media flows Rayleigh number Ra Natural convective flows Reynolds number Re Compressible incompressible thermal and porous media flows Viscosity ratio v Porous media flows If the structured meshes
16. lgorithm Fig 9 2 First an intermediate momentum variable is calculated and in the second step the density pressure field is determined The third step involves the introduction of density pressure fields to obtain the correct momentum variables In problems where the energy and other variables are coupled calculation of energy is necessary in addition to the above three steps In fully explicit form however the energy equation can be solved in the first step itself along with the intermediate momentum calculations In the subroutine stepl we calculate the temperature dependent viscosity at the beginning according to Sutherland s relation see Chapter 6 The averaged viscosity values over each element are used in the diffusion terms of the momentum equation and dissipation terms of the energy equation The diffusion convective and stabiliza tion terms are integrated over elements and assembled appropriately to the RHS vector The integration is carried out either directly or numerically Finally the RHS vector is divided by the lumped mass matrices and the values of intermediate momentum variables are established In step two in explicit form the density pressure values are calculated by the Eq 3 53 or Eq 3 54 The subroutine step2 is used for this purpose Here the option of using different values of 0 and 0 is available In explicit form 0 is identically equal to zero and 6 varies between 0 5 and 1 0 For compressible flow comp
17. listing at the publisher s web page 9 2 4 Preliminary subroutines and checks A few preliminary subroutines are called before the start of the time iteration loop Establishing the surface normals element area calculation for direct integration The data input module 277 SUBROUTINE GETNRW MXPOI MBC NPOIN NBS ISIDE IFLAG COSX COSY ALEN IWPOIN WNOR NWALL IMPLICIT NONE INTEGER I IB 1B2 IN IW J JJ MBC MXPOI NBS NN NPOIN NWALL INTEGER IFLAG MXPOI ISIDE 4 MBC IWPOIN 3 MBC REAL 8 ACH ANOR ANX1 ANY1 REAL 8 ALEN MBC COSX MBC COSY MBC WNOR 2 MBC DO I 1 NPOIN IFLAG I 0 END DO I DO I 1 NBS DO J 1 3 IWPOIN J I 0 END DO J END DO I NWALL i jo DO IN 1 2 DO I 1 NBS boundary sides flags on the wall points IF ISIDE 4 1 EQ 2 THEN flag 2 for solid walls NN ISIDECIN I JJ IFLAGCNN IF JJ EQ 0 THEN NWALL NWALL 1 IWPOIN I NWALL NN IWPOIN 2 NWALL I IFLAG NN NWALL ELSE IWPOIN 3 JJ I ENDIF ENDIF END DO I END DO IN DO IW 1 NWALL IB IWPOIN 2 IW IB2 IWPOIN 3 IW ANX1 ALEN IB COSX IB 278 Computer implementation of the CBS algorithm ANY1 ALEN IB COSY IB ACH 0 0D00 IF IB2 NE 0 THEN ANX1 ANX1 ALEN IB2 COSX IB2 ANY1 ANY1 ALEN CIB2 COSY IB2 ACH COSX IB COSX IB2 COSY IB COSY IB2 ENDIF ANOR DSQRT ANX1 ANX1 ANY1 ANY1 ANX1 ANX1 ANOR ANY1 ANY1 ANOR WNOR 1 IW ANX1 WNOR
18. nly after a certain number of iterations as prescribed by the user If the last option of the user specified fixed time step is used the local time steps are not calculated Figure 9 3 shows the subroutine used for calculating the local time steps for inviscid compressible flows with linear triangular elements As indicated in Sec 4 3 3 Chapter 4 two different time steps are often useful in getting better stabilization procedures Such internal DELTI and external DELTP time stepping options are available in the routine of Fig 9 3 The CBS algorithm introduces naturally some terms to stabilize the oscillations generated by the convective acceleration However for compressible high speed flows these terms are not sufficient to suppress the oscillations in the vicinity of shocks and some additional artificial viscosity terms need to be added see Sec 6 5 Chapter 6 We have given two different forms of artificial viscosities based on the second derivative of pressure in the program Another possibility is to use anisotropic shock capturing based on the residual of individual equations solved However we have not used the second alternative in the program as the second derivative based procedures give quite satisfactory results for all high speed flow problems In the first method implemented we need to calculate a pressure switch see Eq 6 16 Chapter 6 from the nodal pressure values Figure 9 4 gives a typical example of triangular elem
19. odal pressure switches for shock capturing solution module which type of boundary conditions are stored In this array i 1 2 correspond to the node numbers of any boundary side of an element i 3 indicates the element to which the particular edge belongs and i 4 is the flag which indicates the type of boundary condition a complete list is given in the user manual available at the publisher s web page Here j is the boundary edge number A typical routine for prescribing the symmetry conditions is shown in Fig 9 6 9 3 6 Solution of simultaneous equations semi implicit form The simultaneous equations need to be solved for the semi implicit form of the CBS algorithm Two types of solvers are provided The first one is a banded solver which is effective when structured meshes are used For this the half bandwidth is necessary in order to proceed further The second solver is a diagonal preconditioned conjugate gradient solver The latter can be used to solve both structured and unstructured meshes The details of procedures for solving simultaneous equations can be found in Chapter 20 of Volume 1 9 3 7 Different forms of energy equation aN ENRON ERR ENERO NIGER In compressible flow computations only the fully conservative form of all equations ensures correct position of shocks Thus in the compressible flow code the energy equation is solved in its conservative form with the variable being the energy However for incompr
20. ralised non Darcy approach Numerical Heat Transfer Part A Applications 30 413 26 1996 LR Idelsohn E Ofiate and C Sacco Finite element solution of free surface ship wave problems Int J Num Meth Eng 45 503 28 1999 K Morgan A numerical analysis of freezing and melting with convection Comp Meth Appl Mech Eng 28 275 84 1981 A S Usmani R W Lewis and K N Seetharamu Finite element modelling of natural convection controlled change of phase Int J Num Meth Fluids 14 1019 36 1992 S K Sinha T Sundararajan and V K Garg A variable property analysis of alloy solidi fication using the anisotropic porous medium approach Int J Heat Mass Transfer 35 2865 77 1992 R W Lewis K Morgan H R Thomas and K N Seetharamu The Finite Element Method for Heat Transfer Analysis Wiley Chichester 1996 P Nithiarasu An adaptive finite element procedure for solidification problems Heat and Mass Transfer to appear 2000 O C Zienkiewicz B V K S Sai K Morgan and R Codina Split characteristic based semi implicit algorithm for laminar turbulent incompressible flows Jat J Num Meth Fluids 23 1 23 1996
21. utations the semi implicit form with 0 greater than zero has little advantage over the fully explicit form For this reason we have not given the semi implicit form for compressible flow problems in the program For incompressible flow problems in general the semi implicit form is used In this 04 as before varies between 0 5 and 1 and 0 is also in the same range Now it is essential to solve the pressure equation in step2 of the algorithm Here in general we use a conjugate gradient solver as the coefficient matrix is not necessarily banded The third step is the one where the intermediate momentum variables are corrected to get the real values of the intermediate momentum In all three steps mass matrices are lumped if the fully explicit form of the algorithm is used As mentioned in earlier chapters this is the best way to accelerate the steady state solution along with local time stepping However in problems where transient solutions are of importance either a mass matrix correction as given in Sec 2 6 3 Chapter 2 or simultaneous solution using a consistent mass matrix is necessary 9 3 5 Boundary conditions As explained before the boundary edges are stored along with the elements to which they belong Also in the same array iside i j the flags necessary to inform the c Solution module 285 SUBROUTINE SWITCH MXPOI MXELE MBC NPOIN NELEM NBS PRES c this subroutine c maximum value i c IMPLICIT INTEGER INTEG
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