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NeoPZ
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NeoPZ
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As advanced finite element technologies we denominate finite element techniques which are generally not available in textbook finite element codes. NeoPZ is able to generate adaptive meshes, interpolation between meshes, nonlinear geometric maps, multigrid iterations, continuous and discontinuous approximation spaces, among others.
Commercial finite element libraries allow the user to choose between linear or quadratic elements. In most finite element textbooks, the map between the master element and its actual location in space is computed using the same shape functions that are used to approximate the solution. These elements are consequently called isoparametric elements.
The appproximation spaces implemented in NeoPZ use hierarchical shape functions that are not easily used for mapping the geometry of the elements. Therefore, in NeoPZ the definition of the geometric map is strongly separated from the definition of the approximation space. This approach allows us to implement any geometric map between a master element and a deformed element.
In his master's thesis Cesar Lucci implemented an extension of the concept of transfinite blending functions originally conceived by Gordon and Hall in 1973 to the complete family of known element topologies. This effort, developed under the sponsorship of ANP/Petrobras, allows to develop high order approximations of differential equations on curved domains.
Considering one the first motivations for developing NeoPZ was to develop a library which applies adaptive finite elements to different families of partial differential equations, it is natural that h-adaptivity is built in its structure.
In early versions each type of element (linear, quadrilateral, triangular, etc) was implemented using its proper class, derived from the abstract TPZGeoEl class. This approach was very intuitive, but very difficult to maintain. Each change in the interface of the geometric element had to be repeated in each element topology. Frequently a feature implemented in a topology that was frequently used did not work for other topologies. This was one of the main motivations for reimplementing the geometric maps within a template structure. In its current version, a single c++ class, specialized by a template implements the complete family of finite elements.
A geometric element is responsible for implementing/computing the mapping between the master element and deformed element. Within an adaptive context, the geometric element keeps track of its father and children.
One possibility of refining geometric elements is to divide the elements in a given pattern: a one dimensional element is divided in two elements, a quadrilateral element is divided into four elements, a hexahedral element is divided into eight elements, etc. Within the NeoPZ context this refinement pattern is called uniform, because each element is divided using elements.
The refinement of the individual elements is also specialized by a template class, allowing linear or nonlinear geometric maps to be divided alike.
In order to understand the concepts of the geometric map, the advanced user should consult the paper published by Calle, Devloo, Bravo and Gomes. There, the essential concepts which guide the development are explained.
One key feature of the implementation of the geometric refinement is that the elements division is performed without dynamic memory allocation.
A data structure item which distinguishes NeoPZ from traditional finite element codes is that within NeoPZ each element keeps track of its neighbours. The size of this data structure is also constant for each element type.
Adaptive geometric refinement using uniform refinement patterns is capable of improving the efficiency of the computations, but in most cases the singularity of the functions that are being approximated are one dimensional in nature. A typical example is a problem with boundary layers.
This was the motivation for developing a more advanced procedure for dividing geometric elements. In the concept of refinement patterns, each element can be divided in an arbitrary number of sub elements, as long as the sub elements form a partition of the original element. Although this concept is relatively simple, it creates issues with respect to the compatibility of the refinements of neighbouring elements.
This research effort was sponsored by FAPESP in a collaborative project with Embraer as a PICTA project.
Using refinement patterns very specialized meshes can be created starting from an essential geometric description of the computational domain.
On the downside is that creating refined meshes now becomes a programming issue instead of a decision of which element to refine. One technique which has worked well is to dicide on the most appropriate refinement pattern based on identifying the edges which need to be halfed.
The technology of refinement patterns was further extended by Cesar Lucci when applying refinement patterns to the numerical simulation of hydraulic fracturing. In this effort, refinement patterns are dynamically created to represent the edge of the fracture.
Within NeoPZ the approximation space is implemented in a separate class structure. One of the main reasons is that different approximation spaces can be generated based on a single geometry. At the geometry level the full refinement tree is kept. The approximation space at the other hand is a partition of the computational domain.
The definition of the approximation space, similarly to the geometry, is implemented as a specialization of computational element class. The geometry and approximation space class are derived from a single topology class. As such the pzgeom::TPZGeoQuad class and the pzshape::TPZShapeQuad classes are derived from pztopology::TPZQuadrilateral class
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