|Year : 2012 | Volume
| Issue : 2 | Page : 87-96
Intensification of the Subsisting Methodology to Enhance DFM without Discretization
Radhakrishnan Kumaravelan1, PSS. Srinivasan2, Palanisamy Tamilselvam3, Mohankumar Madhan4
1 Department of Mechanical Engineering, Velalar College of Engineering and Technology, Erode, India
2 Department of Mechanical Engineering, Knowledge Institute of Technology, Salem, India
3 Department of Mechanical Engineering, SNS College of Technology, Coimbatore, India
4 Department of Mechanical Engineering, Velammal Engineering College, Chennai, Tamilnadu, India
|Date of Web Publication||4-Aug-2012|
Department of Mechanical Engineering, Velalar College of Engineering and Technology, Erode
Source of Support: None, Conflict of Interest: None
| Abstract|| |
In the design for manufacturing, mesh generation is a time-consuming and expensive process in the assembly and solution of the finite element equations. The conversion of solid models to finite element data is also time-consuming and often introduces numerous ambiguities. In this study, a new mesh-less method has been proposed for analyzing the plane stress problems. The Moving Least Squares (MLS) approximation has been used in generating the shape function. The results obtained by the proposed method have been compared with the analytical solution and finite element method (FEM) results. The proposed method gives more accurate results than the finite element approximation, with less computational effort.
Keywords: Finite element method, Moving Least Square, plane stress
|How to cite this article:|
Kumaravelan R, Srinivasan P, Tamilselvam P, Madhan M. Intensification of the Subsisting Methodology to Enhance DFM without Discretization. J Eng Technol 2012;2:87-96
|How to cite this URL:|
Kumaravelan R, Srinivasan P, Tamilselvam P, Madhan M. Intensification of the Subsisting Methodology to Enhance DFM without Discretization. J Eng Technol [serial online] 2012 [cited 2021 Mar 2];2:87-96. Available from: http://www.onlinejet.net/text.asp?2012/2/2/87/99295
| 1. Introduction|| |
Computer-aided design (CAD) tools are necessary in the modeling and investigation of physical phenomena in complex engineering systems. The design of such systems necessitates either solving complex partial differential equations or discretizing the domain and using approximate methods. The most well-known approximation methods are the finite element method (FEM) and the finite difference method (FDM). FEM requires the discretization of the domain by a finite mesh. After meshing the problem domain in a decisive manner, complex partial differential equations are estimated by a set of algebraic equations. Then, by assembling the element equations, the system of algebraic equations for the problem domain can be obtained. Meanwhile, the mesh-free method (or mesh-less method) forms a system of algebraic equations for the problem domain without requiring a predefined mesh. Instead, mesh-free methods represent the problem domain and the boundaries, with sets of scattered nodes in the domain and on the boundaries. These sets of scattered nodes do not form meshes, unlike the other numerical methods. This removes the obligation of having a relationship between the nodes, at least for field variable interpolation. The mesh-free method is a new numerical analysis method. It has excellent accuracy and rapid convergence.
| 2. Literature Review|| |
In 1977, Lucy emerged with a method called the 'Smoothed Particle Hydrodynamics (SPH) Method'. It is a computational method used for the simulation of fluid flows. It has been used in several fields, including astrophysics, ballistics, volcanology, and oceanology. The SPH method is a mesh-free Lagrangian method, where the coordinates move with the fluid. In the SPH method, fluid is divided into a number of separate elements called particles and the distance between them is known as the'smoothing length'. The kernel function 'smoothed' the particles over the distance, which means the summation of the related properties of all the particles in the kernel range, gives the physical quantity of each particle. Lucy used the mesh-free method for modeling astrophysical phenomena without boundaries, such as, exploding stars and dust clouds. The SPH method displays tensile instability. This tension instability is solved by the Reproducing Kernel Particle Method (RKPM) under the scope of the Lagrangian Kernel. Swengle et al., have made a substantial contribution in the study of SPH method instabilities. Liu et al. have presented a correction function for Kernels, while they have presented a method to upgrade strain calculations. Belytschko et al. made notable modifications or corrections of the SPH method include the correction. Nayroles et al. moved the Least Square approximation,used in a mesh-free method (Galerkin Method), which was pioneered for solving partial differential equations, and they named that method,the Diffuse Element Method (DEM). DEM is advantageous over finite element methods in respect of not relying on a grid, and being more precise in the calculation of derivations of the reconstructed functions, Belytschko et al.. The method has been modified and refined by a mesh-less method. In this method, they used the moving least-squares interpolants to construct the trial and test functions for the variational principle (weak form) and weight functions. In contradistinction to DEM, they introduced certain key differences in the implementation, to improve the accuracy. Also in their article, they illustratethese modifications with examples, where no volumetric locking occurs and the rate of convergence highly exceeds that of the finite elements. It is evident that this type of method has considerable advantages, such as, consistency and stability, yet the SPH method is still cheaper. Moreover, these methods have an improvement toward the moving least squares and partition of unity. As the standard SPH method has a problem of getting accurate interpolation for the particles scattered arbitrarily, many developments have been made to improve the completeness of the SPH method, Nayroles et al. One of the most important ones is the normalization approximation and the other is the Moving Least Squares (MLS) approximation, first implicitly and then classified by Belytschko et al..Two approaches have been proposed to construct the EFG shape functions; one is the moving least squares approximation and the other is the partition of unity approximation. MLS is a method of reconstructing continuous functions from a set of scattered nodes. The reconstructed value is calculated for a node, around which a region is defined. A weighted least squares measure is tended to that region. Although MLS approximation has pioneered the development of many mesh-free methods, the shape functions generated by this method do not have the Kronecker delta function property. As mentioned earlier, there is another method for the construction of shape functions; the partition of unity method, Babuska et al.. The Generalized Finite Element Method (GFEM) has been introduced separately. They called the method by different names; Babuska et al. called it the finite element partition of unity method and they published several articles, while Duarte and Oden used the names hp clouds or cloud-based hp finite element method. The most common characteristic of these methods is the usage of a partition of unity (PoU). They described PoU as a set of function values, of which the sum to the unity at each node in a domain. Using partition of unity in the construction of the shape functions prevent from numerical integration problems related to the usage of moving least squares. Furthermore, it is claimed that the use of a finite element partition of unity helps to implement the essential boundary conditions. However, presently, the Moving Least Squares (MLS) approximation is the most popular method for generating the mesh-free shape functions, as it is much cheaper than the PoU method in the integration of the stiffness matrix, Lanson and Vila.Another mesh-free method has been introduced to this area, called the mesh-less method for conservation laws. It is aimed to obtain accurate approximation of derivatives under the scope of collocation approaches via this new mesh-free method. Another success has been achieved by Bouillard and Suleau. They have succeeded in introducing a mesh-free formulation to acoustic problems . Bonet and Lok  published an article,which presents a new variational framework for the various existing Smooth Particle Hydrodynamic (SPH) techniques and a new corrected SPH formulation. They claim that to preserve angular momentum, the gradient of a linear velocity field must be calculated correctly with the SPH equations. They have presented a corrected algorithm, which is a combination of the Kernel correction and gradient correction, and they have illustrated the theory with several examples related to fluid dynamics. Perez Pozo et al. introduced a stabilized Finite Point Method (FPM), where the stabilization is based on Finite Calculus (FIC), for solving nonlinear material problems. In thisarticle, it has beenproved, with several examples, that by using this method it hasa semi-implicit numerical solution for incompressible fluids. In contrast to SPH or some element-free methods, the Mesh-less Local Petrov-Galerkin (MLPG) method is a truly mesh-less method. This approach developed by Atluri and his colleagues, is based on writing the local weak form of partial differential equations over overlapping local sub-domains, and within these local sub-domains the integration of the weak form is also performed. This makes the method independent of any need for any kind of meshes or background cells, Atluri et al.(14 ,150) The MLPG method has been used in several problems in different areas, such as, fracture mechanics and Gu and Liu  forced vibration analysis, and so on, Yong Duan. One of the latest improvements in the mesh-free methods is the Mesh-less Galerkin method, using radial basis functions based on domain decomposition. The Mesh-less Galerkin method is developed with the help of the Interpolating Moving Least Squares technique. Thus, it is possible to have simplified the implementation of boundary conditions, because the kernel function enables the fulfillment of the Kronecker delta property. In hisarticle, Yong Duan has solved numerous examples to verify this method in different problems, such as, interpolation problems or PDEs. Wenjing Zhang et al. have published an article about a new mesh-free method, which is the point interpolation method based on the radial basis function or Radial Point Interpolation Method (RPIM). They asserted that this method has not only all the advantages of mesh-free methods, but also the Kronecker delta function property. The main difference of this new method is to have the shape function constructed by a combination of both radial and polynomial basis functions and this makes the implementation of the boundary conditions functional, like the traditional finite element methods. The article also has a solution for a two-dimensional static elastic problem, with success.
Moreover, in recent times, several authors have proposed to use mixed interpolations, which arecombinations of finite elements and mesh-free methods, in order to gain the advantages of both methods.
| 3. Materials and Methods|| |
3.1 The advantages of the mesh-freemethod
The finite element method is widely used in many fields of science to perform linear or nonlinear, static or dynamic, stress analysis, for solidstructures or fluid flows. In a structural simulation, the FEM enables visualization ofthe stiffness and strength of the structure parts. Although, at present, several modern and commercial FEM packages allow detailed visualization of where the structures bend or twist, and indicate the distribution of stresses and displacements, the following limitations of FEM appear to be obvious.
The very first step in FEM packages is to mesh the problem domain. This process is very expensive, asthe analysts spend most of their time in the generation of the mesh, and thistakes up the major share ofthe cost of a simulation project. In order to decrease the cost, the aim should be to use more computer power than manpower, but this is not the mannerin which FEM packages work. Thus, ideally the computer could accomplish the meshing without a man's contribution.
The FEM packages produce discontinuous and less accurate stress values. In problems having large deformations, the results are less accurate due to element distortions. Another disadvantage of FEM packages is the dependence on continuum mechanics. That is, the elements formed cannot be split or broken, which disables the simulation of the fracture of material into a number of pieces. Therefore, the elements should be in one piece or totally extinguished. Otherwise, it can cause significant errors, as the problem is essentially nonlinear and consequently the results are path-dependent.
To avoid errors from these types of problems, instead of splitting the elements, re-meshing of the problem domain is introduced. In re-meshing, the problem domain is re-meshed to prevent the distortion of the elements and make the nodal lines coincident. Thus, the meshes would be fine and smooth. Re-meshing can be performed by manpower or a mesh generation processor can be used. However, both solutions have considerable handicaps. Manpower is expensive and less accurate; while the mesh generation processors must be powerful, advanced, and adaptive. Furthermore, the cost of re-meshing is very high for large 3D problems.
3.2 Scope of the study
The main objective of this study is to implement a mesh-free method, which makes approximation based on nodes, not elements. In each plane stress problem solved, the capability and accuracy of the method is compared with the analytical and FEM results. For this purpose, the parameters of the proposed method are changed and the optimal parameters for the method are determined for each problem type.
The various steps involved in the suggested method is shown in [Figure 1]. In the proposed method, the moving least squares (MLS) approximation is used for construction of the shape functions and the Galerkin weak form is used to develop the discretized system equations.
3.4 Background mesh
A background mesh is confirmed,as MLS shape functions are consistent and compatible, and the constrained Galerkin approach is used to impose the essential boundary conditions. In this method, it is common to use high-order polynomials for shape functions, but even linear polynomial-based functions give quite accurate results for the curved boundaries, which are represented by nodes. The shape functions can interpolate the two nodes at any location on the boundary, as the shape functions are formed by nodes in a moving local domain. Essentially, there are numerous ways to perform geometric interpolation to simplify the geometry by different softwares in the computer. Moreover, it is very essential to simplify the model mathematically. The model of the plate with a background mesh is shown in [Figure 2].
3.5 Node generation
The problem domain is represented by a set of scattered nodes, as schematically illustrated in [Figure 3]. One of the advantages of using a mesh-free method is that it does not require meshes or elements. There is no need to use meshes or elements for field-variable interpolation. Instead, the nodes are scattered in the problem domain.
Moreover, node generation can be fully automated without any human intervention. These automated programs are based on the triangulation (as the most convenient mesh to use is a mesh of triangular cells) algorithm (e.g., Delaunay triangulation as shown in [Figure 4]), and are very simple, easy to find, and available for both 2D and 3D domains. For an analyst, using such algorithms certainly reduces the time of an analysis process. For some mesh-free methods, a background mesh is needed to be used in the integration of system matrices. However, the shape of the background mesh is not strict, provided theaccuracy in the integration is adequate.
[Figure 5] shows both the background mesh and nodes added to it. The nodes are created at the center and the three vertices of the triangular element. [Figure 6] shows the nodes and integration point distribution on a background mesh.
3.6 Shape functions
The construction of the shape function has been the main and the most important issue for the mesh-free methods, while in the finite element analysis methods the shape functions are based on elements, thus they can be computed directly and they satisfy the Kronecker delta function property.
First of all, in mesh-free methods, a shape function must satisfy a condition called 'partition of unity'. This condition is required in a shape function in order to be able to make any rigid motion of the problem domain. Second, there are other conditions that a shape function can satisfy, such as, 'linear field reproduction' or 'Kronecker delta function property'.
The linear field reproduction condition is required for a shape function to pass the standard patch test. The patch test is often used in testing finite elements. It is a simple indicator of the quality of a finite element. If the finite element solution is the same as the exact solution, it can be said that the elements have passed the patch test. Passing the patch test is not compulsory for a mesh-free shape function, because a shape function that does not pass this test can still be used if it provides a converged solution.
Nevertheless, many finite element methods that cannot pass the patch test are yet used in finite element packages. Next, the Kronecker delta function condition is also preferable for mesh-free method shape functions, as this condition simplifies putting the boundary conditions into effect.
In mesh-free methods, the shape functions are based on arbitrarily distributed nodes in a domain, without any relation between them. This makes it harder to construct the shape function. One of the tough issues in the area of mesh-free methods is to generate more effective methods for creating shape functions.
Considering a plane stress problem where,
Let there be MIP integration points (IP) and Mnodes in the domain. Note that(MIP ≥3or4M)must be satisfied. Each IPhasan influence domain of radiusρIP. Consider an IPatpoint (x,y). Let there be Nnodes in the influence domain of this IP. Let the distance of the ith node from IPbedi as shown in
[Figure 7]. The weight function and its derivatives are defined as follows:
Where u1 is the nodal displacement parameter vector in the u direction at node 1. Similarly,
Let us define the nodal stiffness matrix K ij as the basic component of assembling the global stiffness matrix of the system
This is the displacement parameter vector for the entire body. Then for each IP, the displacement parameter vectors u and v can be extracted from Δ.
Using the expressions u=pcu and v=pcv, the displacements of any point can be calculated by considering the nearest IP to that point.
The stress can be calculated as,
σ = Dε
3.8 Numerical examples
In this article, a new method is used for standard plane stress problems. The results are compared with analytical and FEM results. The parameters in the new method are the background mesh density, the number of nodes, and the number of Gauss integration points. Generally, a finer background mesh provides more accurate results. However, the density should be optimized considering the CPU time, modeling cost, and accuracy. On the other hand, increasing the number of Gauss integration points gives higher accuracy, but it is important not to have too coarse a background mesh when the number of integration points is very large. In this study, by using different densities of background mesh and different numbers of integration points, the displacement and stress distribution of the problem are found by using the proposed mesh-less method and the results are compared with the analytical and FEM results.
Two case studies are considered to validate the proposed method. One is a cantilever beam under parabolic end load and the other is a square plate with a central hole under a Uniform Distributed Load (UDL) at both sides.
3.8.1 Case study 1− Cantilever beam under parabolic end load
A cantilever beam subjected to a parabolic end load is shown in [Figure 8].
The exact solution is given by Timoshenko and Goodier . According to that the displacement in the x direction is,
The displacement in the y direction is,
The loading P is distributed in the form of a parabola at the right end of the beam:
In this example, the properties for this cantilever beam are taken as follows:
Young's modulus: E=200000 N/nm 2
Poisson's ratio: v = 0.3
Height of the beam: h = 120 mm
Length of the beam: L = 480 mm
3.8.2 Case study 2 Ϳ A square plate with a central hole under a uniform distributed load at both sides
A square plate with a central hole under uniform distributed load at both sides is shown in [Figure 9].
|Figure 9: A square plate with a hole subjected to uniform distributed side loading|
Click here to view
In this example, the properties for this cantilever beam are taken as follows:
Loading: w = 100 N/mm
Young's modulus: E=200000 N/nm 2
Poisson's ratio: v = 0.3
Height of the plate: a = 100 mm
Length of the plate: b = 600 mm
| 4. Results and Discussion|| |
In this study the results obtained from the analytical method, the finite element method, and the proposed method are compared
Initially the results of the proposed mesh-less method are compared with the analytical exact solutions, to validate the proposed one. This has been carried out in Case study 1. Then the results of proposed method are compared with the FEM results to prove its superiority. The same has been carried out in case study 2.
4.1 Cantilever beam under parabolic end load
[Table 1] shows the displacement results calculated by the proposed program and analytical method.
[Table 2] shows the stress results calculated by the proposed program and analytical method.
From [Table 1] it is seen that the error between the displacement (in the x and y directions) obtained from the mesh-free method and analytical exact solution is very less and its maximum value is 1.891%. It is highly within the acceptable range. Hence, the proposed methodis being validated for its correctness, for different sizes of the cantilever beam.
Similarly from [Table 2] it is concluded that the percentage error between the stresses (σx and σxy ) obtained from the mesh-free method and analytical exact solution is limited to 15. It is also lying in the acceptable range. Hence, the proposed method is again validated. The pattern of stress variation is also given in [Figure 10]. The magnitude of σxy calculated using the proposed method holds very less percentage of error, except for x=0 and y=-30, which has a variation of 27.81%, due to less influence of the integration points on the stress field.
|Figure 10: σx Distribution obtained by Patran/Nastran for cantilever beam|
Click here to view
4.2 A square plate with a central hole under uniform distributed load at both sides
[Table 3] shows the displacement results calculated by the proposed program and finite element method.
[Table 4] shows the stress results calculated by the proposed program and finite element method.
From [Table 3] it can be concluded that the percentage error between the displacement (in the x and y directions) obtained from the mesh-free method and FEM results are very less and its maximum value is 4.660. Hence, the correctness and accuracy of the proposed method is being checked against the FEM for different sizes of square plate with a central hole subjected to UDL on both sides.
Similarly [Table 4] shows the stresses induced is the same square plate using the proposed mesh-free method and FEM. The maximum value of percentage error between the stresses (σx and σy ) obtained from the mesh-free method and Fem are 13.325 and 42.155 for, σx and σy, respectively. The pattern of stress variation is presented in [Figure 11]. As FEM also yields approximate results, notable error percentages have been obtained.
|Figure 11: σx Distribution obtained by Patran/Nastran for square plate with hole|
Click here to view
| 5. Conclusion|| |
The development of the proposed method is a significant achievement in the improvement of mesh-free methods. It was observed that the proposed method performed satisfactorily and it compared favorably with the FEM results. In this study a cantilever beam with parabolic load was analyzed using the proposed method and compared with the analytical method and a cantilever beam with uniformly distributed load, and a plate with a circular hole may be analyzed using the proposed method and the results can be compared with FEM and the analytical method.
| 6. Nomenclature|| |
εx , εy , and γxy - Strain in X, Y, and XY directions, respectively
u, v, and w - Displacement in x, y, and z directions, respectively
dV - Change in strain energy vector
B - Stress strain relation vector
K ij - Nodal stiffness matrix
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| Authors|| |
Dr. R. Kumaravelan completed his M.E. Engineering Design at Bharathiar University, Coimbatore, India in 1999 with University First rank and Ph.D. in Mechanical Engineering at Anna University, Chennai, India in 2011 with a specialization in Functionally Graded materials and composite mechanics. He published 3 papers in international journals and 2 papers in Indian journals. He presented more than 15 papers in National and International conferences. He is member in professional bodies like ISTE, IIPE, IEI, SAE etc. His current research interest are composite mechanics and contact mechanics.
Dr. PSS. Srinivasan completed his M.Tech Thermal Engineering at IIT - Bombay, Mumbai, India in 1990 with Gold medal and Ph.D. in Mechanical Engineering at Bharathiar University, Coimbatore, India in 2002 with a specialization in Condensation over HIF tubes. He published 35 papers in international journals and 23 papers in Indian journals. He presented more than 134 papers in National and International conferences. He is member in professional bodies like ISTE, FMFP, ISHMT, ASME etc. His current research interest are Heat and Mass Transfer, Computational Fluid Dynamics (CFD), Refrigeration and Air-conditioning, Waste Heat Recovery, Renewable Energy, Solar Passive Design of Buildings, Design optimization, Taguchi method.
Dr. P. Tamilselvam completed his M.E. Engineering Design at Bharathiar University, Coimbatore, India in 1999 First Class with distinction and obtained Ph.D. in Mechanical Engineering from Anna University, Chennai, India in 2011 with a specialization in Condensation Heat Transfer. He published 2 papers in international journals. He presented more than 15 papers in National and International conferences. He is member in professional bodies like ISTE, IAENG etc. His current research interest are Heat and Mass Transfer and composite mechanics.
Mr. M. Madhan completed his M.E. Manufacturing Engineering from Anna University of Technology, Coimbatore, India in 2011. He is presently doing research in Self-healing ability of structural ceramics. He presented 3 papers in National and International conferences. He is Life member of ISTE.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11]
[Table 1], [Table 2], [Table 3], [Table 4]