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 structures.
FEM and BEM mesh refinement In recent decades, the finite element and boundary element methods (FEM and BEM) have become mainstays 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 that the reliability of the analysis 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 that 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 allows one to obtain pseudo-regular meshes very easily and quickly in a one-step solution.
Assume that there is an arbitrary triangle grid embedded into a 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 n {\displaystyle n} -dimensional vector with all network segments as its components. Thus, the potential energy takes the form
Π = D ∑ j = 1 n R j 2 , {\displaystyle \Pi =D\sum _{j=1}^{n}{R_{j}}^{2},}
where n {\displaystyle n} is total number of segments in the network, R j {\displaystyle R_{j}} is the length of segment number j {\displaystyle j} , and D {\displaystyle D} is an arbitrary constant. The length of segment number j {\displaystyle j} may be expressed by two nodal coordinates as
R = ( X 12 − X 11 ) 2 + ( X 22 − X 21 ) 2 {\displaystyle R={\sqrt {(X_{12}-X_{11})^{2}+(X_{22}-X_{21})^{2}}}}
It may also be supposed that coordinate vector { X } {\displaystyle \{X\}} of all nodes is associated with the non-distorted network, and the coordinate vector { X ′ } {\displaystyle \{X'\}} is associated with the distorted network. Vector { X } {\displaystyle \{X\}} may the be expressed as
{ X } = { X ′ } + { Δ X } {\displaystyle \{X\}=\{X'\}+\{\Delta X\}}
The vector { X } {\displaystyle \{X\}} determination is related to minimization of the quadratic form Π {\displaystyle \Pi } by incremental vector { Δ X } {\displaystyle \{\Delta \ X\}} ; that is,
∂ Π ∂ Δ X k l = 0 , {\displaystyle {\frac {\partial \Pi }{\partial \Delta X_{kl}}}=0,}
where l {\displaystyle l} is the number of interior node of the area, and k {\displaystyle k} is the number of coordinate After all transformations, we may write the following two independent systems of linear algebraic equations:
[ A ] { Δ X 1 } = { B 1 } {\displaystyle [A]\{\Delta X_{1}\}=\{B_{1}\}}
[ A ] { Δ X 2 } = { B 2 } , {\displaystyle [A]\{\Delta X_{2}\}=\{B_{2}\},}
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