SERBA
Boundary Element Method for Elasticity

SERBA is an executable FORTRAN77 program, using single precision arithmetic, which applies the boundary element method to solve the elasticify equation in a 2D region.

The program uses linear continuous elements, and any kind of combination of boundary conditions in stresses and displacements can be considered.

Input File Format:

The input to the program involves a file containing seven sets of data:

1. The title line (a character string)
2. The parameter line, listing NP, NPI and NANU:
   • NP is the number of elements used to discretize the boundary; This is also the number of boundary points. This value must be positive.
   • NPI is the number of internal points where the displacement and stress will be computed; NPI can be zero if no information about interior points is desired.
   • IANUL is the number of components of displacements that are going to be fixed, regardless of the boundary conditions. This number should only be nonzero in cases where the specified boundary conditions do not eliminate a rigid body motion;
3. Elastic properties are specified on one line:
   • NTIP is 0 for plane strain, or else 1 for plane stress;
   • E is the value of Young's modulus;
   • POIS is the value of Poisson's ratio;
   • DM is the maximum (spatial) dimension of the problem to be solved; It is used for scaling.
4. The coordinates of internal points; For each of the NPI internal points, there must be a record specifying XINT(I) and YINT(I):
   • XINT(I) is the X coordinate of the I-th internal point;
   • YINT(I) is the Y coordinate of the I-th internal point;
5. The coordinates of boundary points; The boundary points are indexed from 1 to NP+1, and are assumed to be given in counterclockwise order; Logically, the boundary point information is given consecutively; however, if one or more points are omitted, their values will be determined by linear interpolation from the preceding and succeeding values that are given. Thus, the very least input would specify points 1 and NP+1; the fullest information would specify all NP+1 points. Most input would omit points that are interior to an equally spaced line.
   • I, the index of the boundary point.
   • X(I) is the X coordinate of the I-th boundary point;
   • Y(I) is the Y coordinate of the I-th boundary point;
6. The boundary conditions, specifying the boundary node index, the boundary code, and the values of the potential, and the "before" and "after" fluxes at the node. A typical boundary condition will only specify the potential or the flux; the values not specified can be given as 0. A boundary condition must be given for every boundary node. Logically, the boundary nodes are input in order; however boundary nodes may be omitted, in which case the data will be linearly interpolated; however, the preceding and succeeding boundary conditions must have the same value of NCOD for this to be done.
   • I, the index of the boundary point.
   • I1, the code for boundary condition 1.
   • V1 the value for boundary condition 1 (displacement or stress component)
   • I2, the code for boundary condition 2.
   • V2 the value for boundary condition 2 (displacement or stress component)
   • I3, the code for boundary condition 3.
   • V3 the value for boundary condition 3 (displacement or stress component)
   • I4, the code for boundary condition 4.
   • V4 the value for boundary condition 4 (displacement or stress component)
7. Support conditions on one line, if IANUL is nonzero. In this case, the positions of the displacements to be fixed must be given. Supposing the number of nodes is NP, then the horizontal displacement of node I is fixed by specifying I, while the vertical displacement is fixed by specifying I+NP.
   • NANU(1) is I to fix the horizontal displacement at node I, or I+NP to fix the vertical displacement at node I.
   • ... intermediate values;
   • NANU(IANUL) is I to fix the horizontal displacement at node I, or I+NP to fix the vertical displacement at node I.

The boundary condition codes are required to indicate which of 8 possible items are being specified. Displacements and stress components are given in local coordinates, based on the normal to the boundary, and the anticlockwise tangent vector. There are eight possible variables at a node, with only four of them being known. The paramters I1, I2, I3 and I4, which must be given in ascending order, take values between 1 and 8, identifying the boundary condition. The meanings of the eight possible codes for boundary conditions are:

1. normal displacement of the node corresponding to the element before the node.
2. tangential displacement at the node corresponding to the element before the node.
3. normal displacement of the node corresponding to the element after the node.
4. tangential displacement at the node corresponding to the element after the node.
5. normal stress component of the node corresponding to the element before the node.
6. tangential stress component of the node corresponding to the element before the node.
7. normal stress component of the node corresponding to the element after the node.
8. tangential stress component of the node corresponding to the element after the node.

Usage:

serba

• namedat, the name of the input file;
• namesal, the name of the output file; (note that the program will terminate if there is already a file named namesal). The program will not overwrite an output file, so it is the user's responsibility to delete an old, unwanted file of the same name.
• igauss, the number of points to use in the Gauss quadrature rule. a value of 4 is often sufficient. This value must be between 1 and 50.

Related Data and Programs:

BETIS is an executable FORTRAN77 program which solves Laplace's problem in a 2D region using the boundary element method.

FEM_50 is an executable MATLAB program which solves Laplace's equation in an arbitrary region using the finite element method.

FEM2D_POISSON is an executable FORTRAN77 program which solves Poisson's equation in a 2D rectangle using the finite element method.

FISHPACK is a FORTRAN77 library which solves Poisson, Laplace or Helmholtz equations in a variety of 2D and 3D geometries, using the finite difference method.

FREE_FEM_POISSON is an executable FORTRAN90 program which solves Poisson's equation in an arbitrary triangulated region using the finite element method.

PLTMG_SINGLE is a FORTRAN77 library which solves elliptic partial differential equations in an arbitrary triangulated region using the finite element method and the multigrid algorithm.

Reference:

1. Federico Paris, Jose Canas,
   Boundary Element Method: Fundamentals and Applications,
   Oxford, 1997,
   ISBN: 0-19-856543-7,
   LC: TA347.B69.P34.

Source Code:

• serba.f, the source code.
• serba.csh, commands to compile the source code.

Examples and Tests:

Example 1 models a square plate in traction, using stress conditions:

• input01.txt, the input file.
• output01.txt, the output file.

Example 2 models a square plate in traction, using displacement conditions:

• input02.txt, the input file.
• output02.txt, the output file.

Example 3 models a square plate in traction, using symmetry:

• input03.txt, the input file.
• output03.txt, the output file.

List of Routines:

• MAIN is the main program for the SERBA boundary element code.
• SERB1 reads input, generates geometry and boundary conditions.
• SERB2 builds and solves the system of equations.
• SERB3 calculates and prints the displacements and stresses.
• NUMER numerical calculates integration constants of the fundamental equation.
• ANA analytically calculates integration constants of the fundamental equation.
• CALA relates principal stresses to stress vectors in local coordinates.
• PIYAY calculates certain integration constants along boundary elements.
• PIVO applies Gauss elimination to solve the linear system.
• GAUSS_QN determines a Gauss quadrature rule.
• COEFIC sets the coefficients of a polynomial needed for the Gauss rule.
• EVALUA evaluates the polynomial defining the Gauss rule.
• ROOTS seeks roots of the polynomial defining the Gauss rule.

You can go up one level to the FORTRAN77 source codes.