FLOW4
A Finite Element Code for Fluid Flow

FLOW4 is a FORTRAN90 program, using double precision arithmetic, which uses the finite element method to solve a problem involving two dimensional steady incompressible Navier Stokes flow.

FLOW4 uses the finite element method to solve for the steady state velocity and pressure of an incompressible fluid flowing in a two dimensional region.

FLOW4 writes a graphics dump file that can be read in and displayed by DISPLAY3.

Related Data and Programs:

BUMP is a FORTRAN90 program which solves a fluid flow problem in a channel including a bump which obstructs and redirects the flow.

CHANNEL is a FORTRAN90 program which solves a fluid flow problem in a channel.

Reference:

1. Max Gunzburger,
   Finite Element Methods for Viscous Incompressible Flows,
   A Guide to Theory, Practice, and Algorithms,
   Academic Press, 1989,
   ISBN: 0-12-307350-2,
   LC: TA357.G86.

Source Code:

• flow4.f90, the source code;
• flow4.csh, commands to compile and load the source code;
• flow4_dictionary.txt, a list of internal variables.

Examples and Tests:

• flow4_input001.txt, input file for problem 1;
• flow4_input002.txt, input file for problem 2;
• flow4_input003.txt, input file for problem 3;
• flow4_input004.txt, input file for problem 4;
• flow4_input006.txt, input file for problem 6;
• flow4_input007.txt, input file for problem 7;
• flow4_input008.txt, input file for problem 8;
• flow4_input009.txt, input file for problem 9;

The list of routines includes:

• FLOWMAIN controls a package that solves a fluid flow problem.
• BMPBC computes the value of the boundary conditions for the
• BMPBC1 computes the value of the boundary conditions for the
• BMPBC2 computes the value of the boundary conditions for the
• BMPCST evaluates the cost of the bump control.
• BMPDER evaluates the derivative of the cost of the bump control
• BMPSEN sets up the right hand side F associated with the
• BMPSPL sets up or updates the spline data that describes the bump.
• BSP computes the value and spatial derivatives of the linear basis
• CH_CAP capitalizes a single character.
• CHKDAT performs some simple checks on the input data.
• CHRCTD accepts a string of characters, and tries to extract a
• CHRCTI accepts a STRING of characters and reads an integer
• CHRDB1 accepts a string of characters and removes all
• CHRUP2 copies STRING into STRNG2, up to, but not including, the
• CUBSPL is given data and boundary conditions for a cubic
• DISCST computes the discrepancy integrals for the pressure,
• DISDER computes the derivative of the discrepancy integrals for the
• FLODUV sets the derivative of the parabolic inflow in terms of
• FLOSEN sets up the right hand side F associated with the
• FLOSOL is given a set of flow parameters in PARA, and an
• FLOSPL sets up or updates the spline data that describes the inflow.
• FLOUV computes the specified boundary values of velocity for a
• FP computes the jacobian of the finite element Navier Stokes or
• FX computes the residual of the Navier Stokes or Stokes equations,
• GETCST is given the value of the solution, G, the target
• GETDER returns the derivatives of the cost with respect to the parameters.
• GETDU uses the finite element method to estimate the value of
• GETDU2 is damaged code.
• GETDU3 uses interpolating serendipity quadrilaterals in an attempt
• GETDU4 uses the Zienkiewicz-Zhou technique to attempt to improve the
• GETDU5 uses interpolating serendipity quadrilaterals in an attempt
• GETFIX computes GRADF, a correction for the finite difference
• GETGRD uses finite differences to compute GDIF, an estimate of the
• GETSEN computes the sensitivities of the state variables U, V and
• GQUAD1 returns the weights and abscissas for a 1 dimensional,
• HELLO prints the program name, date, and computer name.
• ILAENV is called from the LAPACK routines to choose problem-dependent
• INIT initializes the simple parameters which may be overridden
• INPUT takes symbolic commands of the form "name = value" and
• INTERV computes LEFT, the maximum value of I so that
• ISAMAX FINDS THE INDEX OF ELEMENT HAVING MAX. ABSOLUTE VALUE.
• LBASE evalualates the IVAL-th Lagrange polynomial based
• LMEMRY allows the user to define the name of a logical variable,
• LSAME returns .TRUE. if CA is the same letter as CB regardless of
• LSPOLY evaluates a polynomial in (1, x, y, x*x, xy, y*y) at
• MARCH carries out a one, two or three dimensional "march".
• MAROPN opens the marching file.
• NAMELS reads a line of user input which is similar in form
• NEWTON seeks a solution G of the nonlinear state equations.
• PLDX evaluates the derivative of a piecewise linear function with
• PLDX1 evaluates the X derivative of the piecewise linear
• PLTOPN opens the plotting file.
• PLTWRT writes information to a file which can be used to create
• PLVAL evaluates a piecewise linear function at a given point.
• PLVAL1 evaluates the piecewise linear polynomial which is 1
• PPVALU calculates the value at X of the JDERIV-th derivative of
• PQDX evaluates the derivative of a piecewise quadratic function with
• PQDX1 evaluates the X derivative of the piecewise quadratic
• PQVAL evaluates a piecewise quadratic function at a given point.
• PQVAL1 evaluates the piecewise quadratic polynomial which is 1
• PRCOST2 prints out the current cost function.
• PRCSEN prints out the cost sensitivities.
• PRDAT prints the user input file data.
• PRGS2 prints out the maximum entries in the finite difference and
• PROBAS accepts N vectors of length M, when N <= M, and
• PROJEC computes the projection of an M vector into an N
• PRPAR prints out the current parameters.
• PRPUV prints out the nodal values of the finite differences and
• PRSOL prints out information about a single solution.
• PRSPL prints the raw spline data for the simple cases ISHAPE = 1 or 2.
• PRSPLN prints out the value of a spline function or one of its
• QBF evaluates a particular quadratic basis function at a point
• QSOLVE carries out the optimization algorithm for a fixed grid,
• REFBSP evaluates one of the three linear basis functions,
• REFQBF evaluates one of the six quadratic basis functions,
• REYSEN sets up the right hand side RHS associated with the first
• RINT_TO_RINT maps a real interval to another real interval.
• S_EQI is a case insensitive comparison of two strings for equality.
• SCOPY copies a vector, x, to a vector, y.
• SDOT forms the dot product of two vectors.
• SERENE evaluates an interpolating function which is defined on
• SERTRN computes the value of dUdY at the central node of an
• SETBAN computes NLBAND, the lower band width of the Jacobian matrix,
• SETBAS computes the value of the basis functions at each
• SETNOD assigns numbers to the nodes.
• SETQ3 sets the abscissas and weights for a three point quadrature
• SETQ7 sets the abscissas and weights for a seven point quadrature
• SETXY sets the X and Y coordinates of the nodes.
• SGBTF2 computes an LU factorization of a real m-by-n band matrix A
• SGBTRF computes an LU factorization of a real m-by-n band matrix A
• SGBTRS solves a system of linear equations
• SGEMM performs one of the matrix-matrix operations
• SGEMV performs one of the matrix-vector operations
• SGER performs the rank 1 operation
• SGETF2 computes an LU factorization of a general m-by-n matrix A
• SGETRF computes an LU factorization of a general M-by-N matrix A
• SGETRS solves a system of linear equations
• SLASWP performs a series of row interchanges on the matrix A.
• SSCAL scales a vector by a constant.
• SSWAP interchanges two vectors.
• STBSV solves one of the systems of equations
• STRSM solves one of the matrix equations
• TRANS calculates the biquadratic transformation which maps the
• UPVALQ evaluates the sensitivities of the velocities and pressure,
• UVAL evaluates the velocities and pressure, and their X and Y
• UVALQ evaluates the velocities and pressure, and their X and Y
• WHODAT is damaged code.
• WRITES writes out scalar data for use by SAMPLE2 when comparing
• WRITEV writes out vector data for use by SAMPLE2 when comparing
• XERBLA is an error handler for the LAPACK routines.
• XOFXSI is given the XSI, ETA coordinates of a point in an

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