page_title: Stretched grid method
section_title: FEM and BEM mesh refinement
hierarchical_section_title: Stretched grid method / FEM and BEM mesh refinement
caption_attribution_description: English: A triangle grid inside plane polygon
context_page_description: The stretched grid method 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.
context_section_description: In recent decades the finite element and boundary element methods (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
   - symmetrical matrix in the banded form similar to global stiffness matrix of FEM assemblage,
   and - incremental vectors of co-ordinates of all nodes at axes 1, 2,
   and  - the right part vectors that are combined by co-ordinates of all nodes in axes 1, 2.
  The solution of both systems, keeping all boundary nodes conservative, obtains new interior node positions corresponding to a non-distorted mesh with pseudo-regular elements. For example, Fig. 2 presents the rectangular area covered by a triangular mesh. The initial auto mesh possesses some degenerative triangles (left mesh). The final mesh (right mesh) produced by the SGM procedure is pseudo-regular without any distorted elements.                               
  As above systems are linear, the procedure elapses very quickly to a one-step solution. Moreover, each final interior node position meets the requirement of co-ordinate arithmetic mean of nodes surrounding it and meets the Delaunay criteria too. Therefore, the SGM has all the positive values peculiar to Laplacian and other kinds of smoothing approaches but much easier and reliable because of integer-valued final matrices representation. Finally, the described above SGM is perfectly applicable not only to 2D meshes but to 3D meshes consisting of any uniform cells as well as to mixed or transient meshes.
caption: Fig. 1 A triangle grid bounded by plane polygonal single-coherent contour
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