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<html>
<head>
<title>
ARBY4 - Reduced Basis Fluid Flow Code
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
ARBY4 <br> Reduced Basis Fluid Flow Code
</h1>
<hr>
<p>
<b>ARBY4</b>
is a FORTRAN90 program which
analyzes a 2D fluid flow using the reduced
basis method.
</p>
<p>
The reduced basis idea is related to the finite element method.
In the finite element method, a completely arbitrary basis is
set up. The reduced basis method seeks, instead, to use a much
smaller set of basis functions which somehow represent the most
typical behaviors of the solution.
</p>
<p>
Such a basis set might be determined by computing lots of solution
vectors, or from theoretical considerations. It can also be
determined by taking the state equations, repeatedly
differentiating them with respect to a parameter, and solving the
resulting systems. This then allows a sort of Taylor expansion
of the solution with the parameter acting as the independent
variable.
</p>
<p>
In a true Taylor expansion, the coefficients of the
basis vectors are strictly determined by the size of the increment
in the independent variable. But in this approach, it is assumed
that although the Taylor coefficients may quickly become inaccurate,
the corresponding Taylor basis vectors will still be useful
for representing the solution. It remains, then, to determine
the now unknown coefficients.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/bump/bump.html">
BUMP</a>,
a FORTRAN90 program which
seeks the appropriate channel inflow and bump shape which will cause the flow to most
closely match a given downflow profile.
</p>
<p>
<a href = "../../f_src/channel/channel.html">
CHANNEL</a>,
a FORTRAN90 program which
seeks the appropriate channel inflow which will cause the flow to most
closely match a given downflow profile.
</p>
<p>
<a href = "../../f_src/toms611/toms611.html">
TOMS611</a>,
a FORTRAN90 library which
minimizes a functional. It is used by ARBY4.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Janet Peterson,<br>
The Reduced Basis Method for Incompressible Viscous Flow Calculations,<br>
SIAM Journal of Scientific and Statistical Computing,<br>
Volume 10, Number 4, pages 777-786, July 1989.
</li>
</ol>
</p>
<h3 align = "center">
Source Code
</h3>
<p>
<ul>
<li>
<a href = "arby4.f90">arby4.f90</a>, the source code.
</li>
<li>
<a href = "arby4.sh">arby4.sh</a>,
commands to compile and load the source code.
</li>
<li>
<a href = "arby4_dict.txt">arby4_dict.txt</a>,
a "data dictionary" for the program.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests
</h3>
<p>
<ul>
<li>
<a href = "test08_input.txt">test08_input.txt</a>, sample input #8.
</li>
<li>
<a href = "test08_output.txt">test08_output.txt</a>, sample output #8.
</li>
<li>
<a href = "test09_input.txt">test09_input.txt</a>, sample input #9.
</li>
<li>
<a href = "test09_output.txt">test09_output.txt</a>, sample output #9.
</li>
<li>
<a href = "test10_input.txt">test10_input.txt</a>, sample input #10.
</li>
<li>
<a href = "test10_output.txt">test10_output.txt</a>, sample output #10.
</li>
<li>
<a href = "test11_input.txt">test11_input.txt</a>, sample input #11.
</li>
<li>
<a href = "test11_output.txt">test08_output.txt</a>, sample output #11.
</li>
<li>
<a href = "test12_input.txt">test12_input.txt</a>, sample input #12.
</li>
<li>
<a href = "test12_output.txt">test12_output.txt</a>, sample output #12.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines
</h3>
<p>
<ul>
<li>
<b>ARBY4</b> solves a fluid flow problem which has several parameters.
</li>
<li>
<b>DIFFPRB</b> estimates the jacobian of the reduced function, using finite differences.
</li>
<li>
<b>DIFSENFL</b> computes a central difference estimate for the first NCOFRB
</li>
<li>
<b>DIFSENRB</b> estimates the reduced sensitivities using finite differences.
</li>
<li>
<b>FLOWBC</b> computes the specified boundary values at a given position.
</li>
<li>
<b>FPBCRB</b> evaluates the jacobian of the reduced boundary conditions.
</li>
<li>
<b>FPFERB</b> evaluates the reduced basis jacobian directly.
</li>
<li>
<b>FPFL</b> computes the jacobian of the Navier Stokes residual function
</li>
<li>
<b>FPIRB</b> computes the jacobian of the reduced basis solution GRB
</li>
<li>
<b>FPRB</b> evaluates the reduced basis jacobian directly.
</li>
<li>
<b>FXBCRB</b> evaluates the reduced boundary conditions.
</li>
<li>
<b>FXFERB</b> evaluates the finite element portion of the reduced function.
</li>
<li>
<b>FXFL</b> computes the residual of the Navier Stokes equations,
</li>
<li>
<b>FXFL2RB</b> projects a full residual into a reduced residual.
</li>
<li>
<b>FXIRB</b> computes the residual of the reduced basis solution GRB
</li>
<li>
<b>FXRB</b> evaluates the boundary conditions and finite element equations
</li>
<li>
<b>GETGSEN</b> computes the coefficients of the sensitivity matrix S,
</li>
<li>
<b>GETBCRB</b> computes the vectors that will be placed into the set
</li>
<li>
<b>GETFERB</b> computes the finite element reduced basis vectors
</li>
<li>
<b>GETSENFL</b> computes the matrix SENFL of sensitivity vectors.
</li>
<li>
<b>GETSENRB</b> computes the value of the reduced sensitivities by
</li>
<li>
<b>GFL2RB</b> projects a full solution vector GFL into GRB, the
</li>
<li>
<b>GRB2FL</b> computes the full solution GFL represented by a set of
</li>
<li>
<b>HELLO</b> prints the program name, date of revision, time and date,
</li>
<li>
<b>HELP</b> prints out a list of the interactive commands which the
</li>
<li>
<b>INIT</b> sets problem data to default values.
</li>
<li>
<b>NEWTFL</b> is given an initial estimate of the solution of the full
</li>
<li>
<b>NEWTRB</b> is given an initial estimate of the solution of the reduced
</li>
<li>
<b>OPTDIFFL</b> optimizes the full problem, without gradient information.
</li>
<li>
<b>OPTDIFRB</b> optimizes the reduced problem, without gradient information.
</li>
<li>
<b>PICFL</b> carries out simple iteration on the full Navier Stokes equations.
</li>
<li>
<b>PICMFERB</b> evaluates the simple iteration matrix for a reduced problem.
</li>
<li>
<b>PICMFL</b> computes the Picard iteration matrix for the full Navier Stokes equations.
</li>
<li>
<b>PICRB</b> carries out simple iteration on the reduced Navier Stokes equations.
</li>
<li>
<b>PICVFERB</b> computes the finite element portion of the right hand
</li>
<li>
<b>PICVFL</b> computes the right hand side for Picard iteration on the
</li>
<li>
<b>REYSEN</b> sets up the right hand side RHS associated with the ISEN-th
</li>
<li>
<b>TEST2</b> compares U, V, and P for the full solution and the reduced
</li>
<li>
<b>TEST3</b> verifies that RB*RFACT = SENFL
</li>
<li>
<b>TEST4</b> ???
</li>
<li>
<b>TEST5</b> is given the QR factors of the reduced basis matrix, and
</li>
<li>
<b>PRUVPFL</b> prints the values of velocity and pressure for all nodes
</li>
<li>
<b>PRUVPRB</b> prints the values of the reduced velocity and pressure
</li>
<li>
<b>UVPRB</b> is given:
</li>
<li>
<b>BMPCST</b> evaluates the cost of the bump control.
</li>
<li>
<b>BMPSPL</b> sets up or updates the spline data that describes the bump.
</li>
<li>
<b>BSP</b> computes the value and spatial derivatives of the linear basis
</li>
<li>
<b>CAPCHR</b> accepts a STRING of characters and replaces any lowercase
</li>
<li>
<b>CAVITY</b> sets up the standard driven cavity problem.
</li>
<li>
<b>CAVITY2</b> sets up the H C Lee driven cavity problem.
</li>
<li>
<b>CHANNL</b> sets up the standard channel problem.
</li>
<li>
<b>CHRCTD</b> accepts a string of characters, and tries to extract a
</li>
<li>
<b>CHRCTI</b> accepts a STRING of characters and reads an integer
</li>
<li>
<b>CHRDB1</b> accepts a string of characters and removes all
</li>
<li>
<b>CHRUP2</b> copies STRING into STRNG2, up to, but not including, the
</li>
<li>
<b>DDETFL</b> computes the determinant of a double precision band matrix
</li>
<li>
<b>DDETRB</b> computes the determinant of a double precision matrix
</li>
<li>
<b>DELHMS</b> returns the number of seconds between TIME1 and TIME2.
</li>
<li>
<b>DFACFL</b> factors a double precision band matrix by elimination.
</li>
<li>
<b>DFACRB</b> factors a double precision dense matrix.
</li>
<li>
<b>DIFSET</b> computes the NCOF coefficients for a centered finite difference
</li>
<li>
<b>DISCST</b> computes the discrepancy integrals for the pressure,
</li>
<li>
<b>DSOLFL</b> solves the linear system
</li>
<li>
<b>DSOLRB</b> solves the linear system
</li>
<li>
<b>DVEQ</b> returns .TRUE. if the N elements of the double precision
</li>
<li>
<b>DVNEQ</b> returns .TRUE. if any of the N elements of the double precision
</li>
<li>
<b>FACT</b> computes the (real) factorial of a nonnegative integer.
</li>
<li>
<b>GETCST</b> is given the value of the solution, GFL, the target
</li>
<li>
<b>GQUAD1</b> returns the weights and abscissas for a 1 dimensional,
</li>
<li>
<b>GRID</b> computes the X or Y coordinate of the I-th gridpoint.
</li>
<li>
<b>INTPRS</b> interpolates the pressure at the midside nodes.
</li>
<li>
<b>L2NORM</b> computes the "big" L2 norm of the velocity over the flow region,
</li>
<li>
<b>LBASE</b> evalualates the IVAL-th Lagrange polynomial based
</li>
<li>
<b>LEQI</b> is a case insensitive comparison of two strings for
</li>
<li>
<b>LEQIDB</b> is a case insensitive comparison of two strings for
</li>
<li>
<b>NBINOM</b> calculates the number of combinations of M things taken N
</li>
<li>
<b>NRMFLO</b> returns norms of a flow solution or flow residual.
</li>
<li>
<b>PCVAL</b> evaluates a piecewise constant function at a given point.
</li>
<li>
<b>PLDX</b> evaluates the derivative of a piecewise linear function with
</li>
<li>
<b>PLDX1</b> evaluates the X derivative of the piecewise linear
</li>
<li>
<b>PLTOPN</b> opens the plotting file.
</li>
<li>
<b>PLVAL</b> evaluates a piecewise linear function at a given point.
</li>
<li>
<b>PLVAL1</b> evaluates the piecewise linear polynomial which is 1
</li>
<li>
<b>PQDX</b> evaluates the derivative of a piecewise quadratic function with
</li>
<li>
<b>PQDX1</b> evaluates the X derivative of the piecewise quadratic
</li>
<li>
<b>PQVAL</b> evaluates a piecewise quadratic function at a given point.
</li>
<li>
<b>PQVAL1</b> evaluates the piecewise quadratic polynomial which is 1
</li>
<li>
<b>PRBMAT</b> prints all nonzero entries of rows ILO to IHI, columns JLO to
</li>
<li>
<b>PRDAT</b> prints the problem information.
</li>
<li>
<b>PRDMAT</b> prints out a portion of a dense matrix.
</li>
<li>
<b>PRELEM</b> prints out data about one or more elements.
</li>
<li>
<b>PRFXFLN</b> prints out the norm of a full residual.
</li>
<li>
<b>PRGRB</b> prints out the reduced basis solution.
</li>
<li>
<b>PRINDX</b> prints out the integer variables that define the
</li>
<li>
<b>PRMATFL</b> prints entries from a matrix A associated with a full
</li>
<li>
<b>PRPAR</b> prints out the current parameters.
</li>
<li>
<b>PRSENN</b> prints out the norms of the sensitivities.
</li>
<li>
<b>PRVECFL</b> prints out some entries of a vector indexed by node number.
</li>
<li>
<b>PRVECRB</b> prints out entries ILO through IHI of a vector.
</li>
<li>
<b>PRXY</b> prints the X and Y coordinates of each node.
</li>
<li>
<b>QBF</b> evaluates a particular quadratic basis function at a point
</li>
<li>
<b>REFBSP</b> evaluates one of the three linear basis functions,
</li>
<li>
<b>REFQBF</b> evaluates one of the six quadratic basis functions,
</li>
<li>
<b>SETBAN</b> computes NLBAND, the lower band width of the Jacobian matrix
</li>
<li>
<b>SETGEO</b> is given a set of flow parameters in PAR, and an
</li>
<li>
<b>SETLOG</b> determines some data that depends on the user input.
</li>
<li>
<b>SETNOD</b> assigns numbers to the nodes and elements, decides which
</li>
<li>
<b>SETPFL</b> computes the value of the finite element basis functions at
</li>
<li>
<b>SETPRB</b> is given:
</li>
<li>
<b>SETQ3</b> sets the abscissas and weights for a three point quadrature
</li>
<li>
<b>SETXY</b> sets the X and Y coordinates of the nodes.
</li>
<li>
<b>STEP</b> sets up a forward facing step problem.
</li>
<li>
<b>TARGET</b> is called to save the current parameters and solution
</li>
<li>
<b>TRANS</b> calculates the biquadratic transformation which maps the
</li>
<li>
<b>UVPFL</b> evaluates the velocities and pressure, and their X and Y
</li>
<li>
<b>UVPNRM</b> returns the "norm" of the solution. Here, the norm of
</li>
<li>
<b>UVPQFL</b> evaluates the velocities and pressure, and their X and Y
</li>
<li>
<b>UVPQRB</b> is given:
</li>
<li>
<b>WRDIS</b> writes information to a file which can be used to create
</li>
<li>
<b>WRTEC</b> writes out information which can be used for with the
</li>
<li>
<b>XOFXSI</b> is given the XSI, ETA coordinates of a point in an
</li>
<li>
<b>DASUM</b> takes the sum of the absolute values of the entries of
</li>
<li>
<b>DAXPY</b> adds a multiple of one vector to another.
</li>
<li>
<b>DCOPY</b> copies a vector X to a vector Y.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DGBTF2</b> ???
</li>
<li>
<b>DGBTRF</b> ???
</li>
<li>
<b>DGBTRS</b> ???
</li>
<li>
<b>DGEMM</b> ???
</li>
<li>
<b>DGEMV</b> ???
</li>
<li>
<b>DGEQR2</b> computes a QR factorization of a real m by n matrix A:
</li>
<li>
<b>DGEQRF</b> computes a QR factorization of a real M-by-N matrix A:
</li>
<li>
<b>DGER</b> ???
</li>
<li>
<b>DGETF2</b> ???
</li>
<li>
<b>DGETRF</b> ???
</li>
<li>
<b>DGETRS</b> ???
</li>
<li>
<b>DLAPY2</b> returns sqrt(x**2+y**2), avoiding overflow.
</li>
<li>
<b>DLARF</b> ???
</li>
<li>
<b>DLARFB</b> ???
</li>
<li>
<b>DLARFT</b> ???
</li>
<li>
<b>DLASWP</b> ???
</li>
<li>
<b>DNRM2</b> returns the euclidean norm of a vector.
</li>
<li>
<b>DORG2R</b> ???
</li>
<li>
<b>DORGQR</b> ???
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>DSWAP</b> interchanges two vectors.
</li>
<li>
<b>DTBSV</b> ???
</li>
<li>
<b>DTRMM</b> ???
</li>
<li>
<b>DTRMV</b> ???
</li>
<li>
<b>DTRSM</b> ???
</li>
<li>
<b>IDAMAX</b> FINDS THE INDEX OF ELEMENT HAVING MAX. ABSOLUTE VALUE.
</li>
<li>
<b>ILAENV</b> ???
</li>
<li>
<b>LSAME</b> ???
</li>
<li>
<b>XERBLA</b> is an error handler for the LAPACK routines.
</li>
<li>
<b>DLAMCH</b> ???
</li>
<li>
<b>DLAMC1</b> ???
</li>
<li>
<b>DLAMC2</b> ???
</li>
<li>
<b>DLAMC3</b> is intended to force A and B to be stored prior to doing
</li>
<li>
<b>DLAMC4</b> ???
</li>
<li>
<b>DLAMC5</b> ???
</li>
<li>
<b>DLARFG</b> ???
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 13 December 2007.
</i>
<!-- John Burkardt -->
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