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NESC0806 2DEPEP

2DEPEP, Partial Differencial Equation Solution and Eigenvalues for Potential and Diffusion Problems

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1. NAME OR DESIGNATION OF PROGRAM

2DEPEP

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2. COMPUTERS
To submit a request, click below on the link of the version you wish to order. Only liaison officers are authorised to submit online requests. Rules for requesters are available here.
Program name Package id Status Status date
2DEPEP NESC0806/01 Tested 23-DEC-1980

Machines used:

Package ID Orig. computer Test computer
NESC0806/01 CDC 7600 CDC 7600
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3. DESCRIPTION OF PROGRAM OR FUNCTION

2DEPEP solves the partial differential equation system:

   C1(X,Y,U,V,T)*DU/DT=D(OXX)/DX+D(OXY)/DY+F1(X,Y,U,V,T)
   C2(X,Y,U,V,T)*DV/DT=D(OYX)/DX+D(OYY)/DY+F2(X,Y,U,V,T) in a general two-dimensional region, R, with
   U=FB1(S)
   V=FB2(S) for S on BR1, and
   OXX*NX+OXY*NY=GB1(S,U,V,T)
   OYX*NX+OYY*NY=GB2(S,U,V,T) for S on BR2, where BR1 and BR2 are distinct parts of the boundary. (NX,NY)= unit outward normal.
   U=U0(X,Y)
   V=V0(X,Y) for T=T0, and
   OXX=OXX(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)
   OXY=OXY(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)
   OYX=OYX(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)
   OYY=OYY(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T).

 

The Jacobian matrices of derivatives of OXX,OXY,OYX,OYY with respect to DU/DX,DU/DY,DV/DX,DV/DY and of F1,F2 with respect to U,V and of GB1,GB2 with respect to U,V must be symmetric.

 

The related elliptic and eigenvalue problems are also solved by 2DEPEP and single equations can be handled efficiently. Examples of applications of the program are elasticity, one- or two- component diffusion, heat conduction, minimal surface, potential problems and the time-independent and time-dependent Schrodinger equations.

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4. METHODS

The user supplies  an initial  triangulation which defines the region R. This triangulation is refined as controlled by a user-supplied function. For singular elliptic problems, optimal order convergence is possible if this function approximates the function D3MAX(UV), which is the maximum of the third derivatives of U and V.

 

The problem is discretized by Galerkin's method, using a trial function space of piecewise quadratic polynomials with respect to the triangulation. An approximate solution is calculated, using either the implicit or the Crank-Nicolson method for each time- step. The non-linear equations which must be solved each time- step are solved by Newton's method. One iteration is sufficient since the solution on the previous time-step is used for starting values. In each application of Newton's method the symmetric, banded Jacobian matrix is inverted directly by Gaussian elimination, with the matrix stored out-of-core when necessary, according to the frontal method. For linear time-independent problems the Jacobian is the same each time-step, so the Cholesky factorization done in the first step is used on all following steps. The isoparametric method is used to handle curved boundaries.

 

When C1=C2=0 and all functions are independent of T, that is, when the problem is a steady-state (elliptic) problem, each time- step corresponds to one iteration of Newton's method. (One iteration is sufficient for linear problems.) An eigenvalue problem can be converted to a related parabolic problem where each time-step is equivalent to an iteration of the inverse power method for finding the smallest eigenvalue and the corresponding eigenfunction. A single equation can be handled with no loss of efficiency in storage or execution time.

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5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM

At most two simultaneous partial differential equations can be solved. The input data set is limited to 200 cards. This can be increased by changing the value of the variable MXCARD and increasing the dimensions of the arrays L, INDX, and LNAM in the preprocessor.

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6. TYPICAL RUNNING TIME

Execution time is problem dependent. NESC executed the sample problems in 30 seconds on a CDC7600.

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7. UNUSUAL FEATURES OF THE PROGRAM

Local mesh refinement capability makes 2DEPEP ideal for singular problems, such as elastic crack problems.

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9. STATUS
Package ID Status date Status
NESC0806/01 23-DEC-1980 Tested at NEADB
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10. REFERENCES
NESC0806/01, included references:
- Granville Sewell:
  2DEPEP, 2-D Elliptic, Parabolic and Eigenvalue Problems, User's
  Manual
  Purdue University (October 1978).
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11. HARDWARE REQUIREMENTS

120K (octal) words of memory are required together with an auxiliary storage device, such as disk or tape, for temporary use in large problems (unit 2), and another storage device for temporary use to store the preprocessor output (unit 4).

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12. PROGRAMMING LANGUAGE(S) USED
Package ID Computer language
NESC0806/01 FORTRAN-IV
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13. SOFTWARE REQUIREMENTS

SCOPE.

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14. OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS

The program is written to process input data for one problem only per execution.

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15. NAME AND ESTABLISHMENT OF AUTHORS

E. G. Sewell
Computer Science Department
Purdue University
West Lafayette, Indiana  47907, USA

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16. MATERIAL AVAILABLE
NESC0806/01
File name File description Records
NESC0806_01.001 INFORMATION FILE 4
NESC0806_01.002 PREPROCESSOR SOURCE 538
NESC0806_01.003 TDEPEP SOURCE 1061
NESC0806_01.004 PROBLEM 1 OUTPUT 56
NESC0806_01.005 PROBLEM 2 OUTPUT 174
NESC0806_01.006 PROBLEM 3 OUTPUT 120
NESC0806_01.007 PROBLEM 4 OUTPUT 160
NESC0806_01.008 PROBLEM 1 INPUT 59
NESC0806_01.009 PROBLEM 2 INPUT 31
NESC0806_01.010 PROBLEM 3 INPUT 43
NESC0806_01.011 PROBLEM 4 INPUT 38
NESC0806_01.012 JCL 70
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17. CATEGORIES
  • P. General Mathematical and Computing System Routines

Keywords: banded matrix, eigenvalues, finite element method, iterative methods, partial differential equations, two-dimensional.