|
| Method | Element Formulation | Advantages | Disadvantages | Specialty of VPHE element compared to other techniques | Application fields and typology |
|
| VCFEM | Principle of minimum complementary energy. | 1. Computationally efficient compared to the conventional FEM. | 1. Perform poorly when the heterogeneity is in the form of voids.
2. Poorly defined stress functions within the interior of the element. | Stress functions within the element are well defined by monomials. | Simulation of microstructures (grains) and multiscale modelling. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| NCM-VCFEM | Hybrid of VCFEM with other method such as numerical conformal mapping. | 1. The variational principle is generalized.
2. Solution accuracy is competitive with that of conventional FEM package in ANSYS.
3. Reduces the computational cost when compared to FEM package in ANSYS. | 1. The NCM-based stress function construction is expensive in comparison with conformal mapping of regular shapes such as ellipses and circles.
2. NCM-based stress functions introduce singularity. Special technique (such as divergence theorem) is needed to reduce the order of these singularities. | Special techniques are not needed to handle singularities in the shape functions. | Real micrographs of heterogeneous materials with irregular shapes can be analyzed effectively. Applicable for 2D problems for time being. Applicable for both linear and nonlinear analyses. |
|
| HPE | Hybrid stress element method together with Muskhelishvili’s complex analysis. | 1. Stress functions within the interior of the element are defined by self-equilibrating stress field.
2. Better performance compared to conventional FEM for plane linear elasticity problems. | 1. Can contain only one irregular phase (void/inclusion) within the element. | - | Simulation of microstructures (grains) with heterogeneity. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| PFEM | Barycentric Coordinates. | 1. Able to take arbitrary form, with arbitrary number of sides and nodes. | 1. Evaluation of barycentric coordinates (complex rational functions) is neither simple nor efficient compared to the conventional displacement based FEM.
2. Not efficient for assembling the stiffness matrices associated with weak solutions of Poisson equations. | The shape functions consist of simple monomials irrespective on number of planes/sides. | Solid mechanics and heat transfer phenomena. Computer graphics, animation and geometric modelling. Quadtree/Octree mesh generation. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| nSFEM | Coupling of conventional FEM with meshless method. | 1. nSFEM is advantageous over the conventional FEM since it produces more accurate solutions, able to tolerate volumetric locking and do not require isoparametric mapping. | 1. nCS-FEM increases the computational cost for solid mechanics.
2. Computational time of nNS-FEM and nES-FEM is longer compared to conventional FEM for the same number of global nodes, due to larger bandwidth of stiffness matrices.
3. Disadvantage of nES-FEM is that there is tendency to overestimate or underestimate the strain energy of the model for some cases. | Formulation of the proposed/present element is similar to nSFEM. The current technique can be improved by carrying out smoothing technique, which will be then similar to nCS-FEM. | Solid mechanics and heat transfer phenomena. Fluid-solid interaction (FSI) problems. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| PSBFEM | Semi analytical method which combines boundary element method (BEM) and FEM. | 1. Analytical solutions are achieved inside the domain, discretization of free and fixed boundaries and interfaces between different materials are not required, and the calculation of stress concentrations and intensity factors based on their definition is straightforward.
2. Yields highly accurate solutions for problems involving stress singularities.
3. Superior to other techniques such as nSFEM and conforming PFEM within the context of linear elasticity and the linear elastic fracture mechanics. | 1. Not directly applicable for unbounded domains with strongly inclined interfaces.
2. PSBFEM cannot be directly used to process transient excitation as opposed to BEM.
3. Not as efficient as conventional FEM or BEM when solving problems involving smooth stress variations within bounded/enclosed domain. | - | Solid mechanics and polygonal mesh creation. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
MFD and
VEM | Surface representation of discrete unknowns (MFD) and unknown degrees of freedom are attached to trial functions within interior of the polygonal domain (VEM).
| 1. Efficient in solving problems involving polygonal meshes.
2. Difficulties faced in integration of complex functions resulting from barycentric coordinates in PFEM are entirely avoided.
3. Does not require extension of compatible interpolation functions to the interior of the element. | 1. Quite difficult to present MFD due to nonexistence of trial functions for the interior of the element.
2. Involve complex procedures and therefore require high computational effort. | Easier to execute due to simpler element formulation.
| Electromagnetic field problems, convection-diffusion problems, fluid flows problems, hydrodynamics problems, eigenvalue problems, solid mechanics, heat transfer, and topology optimization. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| VNM | The polygonal domain is divided into several triangular regions which use the conventional FEM shape functions. These triangular regions are then coupled together by using mean least square shape functions. | 1. Do not require formulation of complex stress functions for the element (which could be difficult for some cases, as reported for stress based FEMs such as HPE and HS-F
2. Numerical integration for the elements is simple and exact, as opposed to compatible PFEM.
3. Simpler and easier for computer applications compared to MFD and VEM. | 1. Integration within each tetrahedron can be simplified by mapping, but the mapping procedure imposes restriction to element geometry due to high aspect ratio (limited tolerance towards mesh distortion).
2. Prone to element locking.
| - | Adaptive computation, solid mechanics, and heat transfer phenomena. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| DGFEM | Problem domain is discretized into several polygonal cells which represent the polygonal elements. The interpolation for the elements is carried out based on a set of monomial functions which are totally independent of the element. | 1. Does not require any conforming shape function and the method is simple.
2. Adjacent elements can be of different order and the elements do not need to be conforming.
| 1. Interpolation functions do not comply with shape function requirements of conventional FEM (Not compatible).
2. Integrations are carried out on the boundaries of each cell. This step is an addition compared to the conventional FEM. | The element fulfills all the requirements of traditional FEM. | Solid mechanics, heat transfer, and eigenvalue problems on polygonal meshes. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| T-FEM or HT-FEM | Utilizes two different set of functions to approximate the solutions, one for the boundary and the other is for the interior domain. | 1. Able to produce more accurate results and higher convergence rates compare to the conforming PFEM with Laplace/Wachspress shape functions.
2. Elements with embedded cracks or voids can be constructed.
3. Able to handle geometry induced singularities and stress/force concentrations efficiently without mesh adjustment.
4. General polygonal elements with curved sides can be generated and the elements are tolerant to mesh distortions. | 1. T-complete sets for some problems are either complex or difficult to formulate. | - | Solid mechanics and heat transfer phenomena. Applicable for 2D problems for the time being. Applicable for both linear and nonlinear analyses. |
|
| BEM-based FEM | Trefftz-like
basis functions are defined implicitly and treated locally by means of Boundary
Element Methods (BEMs). | 1. Applicable to general polygonal meshes (immune to severe mesh distortion).
2. Computational effort is reduced, since only one subspace is needed to approximate the pricewise harmonic functions.
| Large linear systems of equations are generated | - | Adaptive mesh generation, time dependent problems, and boundary value problems. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |
|
| HS-F | Combination of principle of minimum complementary energy (similar principal used in VCFEM) with Airy stress function. | 1. Possess excellent performance compared to the conventional elements and especially independent of the element geometry. | 1. Not well desired for most of the engineering applications, due to the difficulties in obtaining suitable/compatible stress functions.
2. Difficult to acquire nodal displacements in stress based FEMs. | The element fulfills all the requirements of traditional FEM. | Solid mechanics phenomena. Applicable for 2D problems for the time being. Applicable for both linear and nonlinear analyses. |
|
| BFEM | Replaces the stress functions in the stress based FEMs with base forces which are easier to obtain. | 1. Able to approximate the strain and force fields accurately.
2. Found to be superior than conventional FEM when analyzing large strain contact problems and the nonlinear problems. | 1. The Lagrange multiplier method is used to deal with the equilibrium equation. So, the stiffness matrix is a full matrix. | - | Solid mechanics, bonding damage detection and damage mechanics. Applicable for 2D problems for the time being. Applicable for both linear and nonlinear analyses. |
|
