G space
Encyclopedia
G space is a functional space used in the formulation of general numerical methods based on meshfree methods
and/or finite element method
settings. These numerical methods are applicable to solve PDEs in particular solid mechanics
as well as fluid dynamics
problems.
), displacement functions needs to be in a proper Hilbert space, meaning that we have to need to make sure that the assumed displacement function is continuous over the entire problems domain. In a discrete setting (e.g., FEM), we construct the function using elements, but have to make sure that it is continuous a long all element interfaces. This is known also as compatibility conditions. To ensure the compatibility, however, care must be taken, and FEM techniques should apply.
The G space theory accommodates functions that may be discontinuous. This is done by using the so-called generalized gradient smoothing technique , with which one can approximate the gradient of displacement functions in a proper G space
. Since we do not have to actually perform even the 1st differentiation to the assumed displacement functions, the requirement on the consistence of the functions are further reduced, and hence the Weakened weak form
or W2 form can be used to create stable and convergent computational methods. The stability is ensured by the so-called positivity conditions, and the convergence to the exact solution is ensure the admissible conditions on the assumed gradient (strain) fields.
The S-FEM is largely the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler. It has also variations of Node-based Smoothed FEM (NS-FEM) Edge-based Smoothed FEM (NS-FEM) , Face-based Smoothed FEM (NS-FEM) , Cell-based Smoothed FEM (NS-FEM) , Edge/node-based Smoothed FEM (NS/ES-FEM) , as well as Alpha FEM method (Alpha FEM) .
1) Mechanics for solids, structures and piezoelectrics ;
2) Fracture mechanics and crack propagation ;
3) Heat transfer;
4) Structural acoustics ;
5) Nonlinear and contact problems;
6) Adaptive Analysis ;
7) Phase change problem ;
8) Limited analysis .
Meshfree methods
Meshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for calculations. Meshfree methods exist...
and/or finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
settings. These numerical methods are applicable to solve PDEs in particular solid mechanics
Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
as well as fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
problems.
Description
For simplicity we choose elasticity problems for our discussion. In a weak formulation (such as in the FEMFEM
FEM refers to a number of things, either as an acronym or otherwise:*Field emission microscopy*Finite element method*FEM *[Front End Module ]*Far East Movement*fem - alternative spelling of femme...
), displacement functions needs to be in a proper Hilbert space, meaning that we have to need to make sure that the assumed displacement function is continuous over the entire problems domain. In a discrete setting (e.g., FEM), we construct the function using elements, but have to make sure that it is continuous a long all element interfaces. This is known also as compatibility conditions. To ensure the compatibility, however, care must be taken, and FEM techniques should apply.
The G space theory accommodates functions that may be discontinuous. This is done by using the so-called generalized gradient smoothing technique , with which one can approximate the gradient of displacement functions in a proper G space
G space
G space is a functional space used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solve PDEs in particular solid mechanics as well as fluid dynamics problems.-Description:For simplicity we...
. Since we do not have to actually perform even the 1st differentiation to the assumed displacement functions, the requirement on the consistence of the functions are further reduced, and hence the Weakened weak form
Weakened weak form
Weakened weak form is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems....
or W2 form can be used to create stable and convergent computational methods. The stability is ensured by the so-called positivity conditions, and the convergence to the exact solution is ensure the admissible conditions on the assumed gradient (strain) fields.
History
The development of G Space theory started from the works on meshfree methods . The G Space theory forms the foundation for the W2 formulations, leading to various W2 models. The W2 models work well with triangular meshes and insensitive to mesh distortion. Because triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence automation in modeling and simulation. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully-compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM) . The S-PIM can be node-based (known as NS-PIM or LC-PIM) , edge-based (ES-PIM) , and cell-based (CS-PIM) . The NS-PIM was developed using the so-called SCNI technique . It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free . The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments.The S-FEM is largely the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler. It has also variations of Node-based Smoothed FEM (NS-FEM) Edge-based Smoothed FEM (NS-FEM) , Face-based Smoothed FEM (NS-FEM) , Cell-based Smoothed FEM (NS-FEM) , Edge/node-based Smoothed FEM (NS/ES-FEM) , as well as Alpha FEM method (Alpha FEM) .
Applications
Numerical methods built on G space theory have been applied to solve the following physical problems:1) Mechanics for solids, structures and piezoelectrics ;
2) Fracture mechanics and crack propagation ;
3) Heat transfer;
4) Structural acoustics ;
5) Nonlinear and contact problems;
6) Adaptive Analysis ;
7) Phase change problem ;
8) Limited analysis .
See also
- Meshfree methodsMeshfree methodsMeshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for calculations. Meshfree methods exist...
- Finite element methodFinite element methodThe finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
- Smoothed finite element methodSmoothed finite element methodSmoothed Finite Element methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method...
- Smoothed point interpolate method
- Weakened weak formWeakened weak formWeakened weak form is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems....