Smoothed finite element method
Encyclopedia
Smoothed Finite Element methods (S-FEM) 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
. S-FEM are applicable to solid mechanics
as well as fluid dynamics
, but so far mainly applied for solid mechanics
problems.
theory were developed. The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. 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 NS-FEM, ES-FEM and CS-FEM. The major property of S-PIM can be found also in S-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 .
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
for the simulation of physical phenomena. It was developed by combining meshfree methods
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...
with the 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...
. S-FEM are applicable to 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...
, but so far mainly applied for 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...
problems.
Description
The essential idea in the S-FEM is to use a finite element mesh (in particular triangular mesh) to construct numerical models of good performance. This is achieved by modifying the compatible strain field, or construct a strain field using only the displacements, hoping a Galerkin model using the modified/constructed strain field can deliver some good properties. Such a modification/construction can be performed within elements but more often beyond the elements (meshfree concepts): bring in the information from the neighboring elements. Naturally, the strain field has to satisfy certain conditions, and the standard Galerkin weakform needs to be modified accordingly to ensure the stability and convergence.History
The development of S-FEM started from the works on meshfree methods , where the so-called weakened weak (W2) formulation based on the G spaceG 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...
theory were developed. The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. 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 NS-FEM, ES-FEM and CS-FEM. The major property of S-PIM can be found also in S-FEM.
List of S-FEM models
- Node-based Smoothed FEM (NS-FEM)
- Edge-based Smoothed FEM (NS-FEM)
- Face-based Smoothed FEM (NS-FEM)
- Cell-based Smoothed FEM (NS-FEM)
- Node/Edge-based Smoothed FEM (NS/ES-FEM)
- Alpha FEM method (Alpha FEM)
Applications
S-FEM has 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...
- 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....
- 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 point interpolate method
- G spaceG spaceG 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...