The stretched grid method (SGM) is a
numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior.
In particular, meteorologists use the stretched grid method for weather prediction and engineers use the stretched grid method to design tents and other
tensile structure
A tensile structure is a construction of elements carrying only tension and no compression or bending. The term ''tensile'' should not be confused with tensegrity, which is a structural form with both tension and compression elements. Tensile ...
s.
FEM and BEM mesh refinement
In recent decades the
finite element
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
and
boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, ele ...
s (FEM and BEM) have become a mainstay for industrial engineering design and analysis. Increasingly larger and more complex designs are being simulated using the FEM or BEM. However, some problems of FEM and BEM engineering analysis are still on the cutting edge. The first problem is a reliability of engineering analysis that strongly depends upon the quality of initial data generated at the pre-processing stage. It is known that automatic element
mesh generation techniques at this stage have become commonly used tools for the analysis of complex real-world models. With FEM and BEM increasing in popularity comes the incentive to improve automatic meshing algorithms. However, all of these algorithms can create distorted and even unusable grid elements. Several techniques exist which can take an existing mesh and improve its quality. For instance
smoothing (also referred to as
mesh refinement) is one such method, which repositions nodal locations, so as to minimize element distortion. The Stretched Grid Method (SGM) allows the obtaining of pseudo-regular meshes very easily and quickly in a one-step solution(see ).
Let one assume that there is an arbitrary triangle grid embedded into plane polygonal single-coherent contour and produced by an automeshing procedure (see fig. 1) It may be assumed further that the grid considered as a physical nodal system is distorted by a number of distortions. It is supposed that the total potential energy of this system is proportional to the length of some
-dimensional vector with all network segments as its components.
Thus, the potential energy takes the following form
:
where
*
- total number of segments in the network,
*
- The length of segment number
,
*
- an arbitrary constant.
The length of segment number
may be expressed by two nodal co-ordinates as
:
It may also be supposed that co-ordinate vector
of all nodes is associated with non-distorted network and co-ordinate vector
is associated with the distorted network. The expression for vector
may be written as
:
The vector
determination is related to minimization of the quadratic form
by incremental vector
, i.e.
:
where
*
- is the number of interior node of the area,
*
- the number of co-ordinate
After all transformations we may write the following two independent systems of linear algebraic equations
:
:
where
*