A truly meshless pre- and post-processor for meshless analysis methods
Introduction
The development of approximate methods for the numerical solutions of partial differential equations has attracted the attention of engineers, physicists and mathematicians for a long time. In recent years, meshless methods have been developed as an alternative numerical approach to eliminate known drawbacks in the well-known finite element method (FEM). Meshless methods do not require a mesh to discretize the problem domain, and the approximate solution is constructed entirely based on a set of scattered nodes. Meshless methods may also alleviate some other problems associated with FEM, such as element distortion, locking, and remeshing during large deformations. Moreover, nodes can be easily added or removed. As the deformation progresses, more accurate solutions can be obtained using meshless methods.
Several versions of meshless methods have been developed, which may be broadly divided into two categories: boundary type methods such as the boundary node method (BNM) [1], [2] and boundary point interpolation method (BPIM) [3]; and domain type methods such as diffuse element method (DEM) [4], element free Galerkin (EFG) method [5], reproducing kernel particle method (RKPM) [6], [7], [8], point interpolation method (NM) [9], and point assembly method (PAM) [10]. Most methods are “meshless” only in terms of the interpolation or approximation of the field or boundary variables, as compared to the usual boundary element method (BEM) or finite element method (FEM), but still have to use a background mesh to integrate a weak form over the problem domain or the boundary. The need for a background mesh for integration makes these methods not truly meshless.
Some truly meshless methods have been developed including the meshless local Petrov–Galerkin (MLPG) method [11], [12], [13], [14], the local boundary integral equation (LBIE) method [15], [16], [17], [18], [19] and the local point interpolation method (LPIM) [20]. These methods do not require a background mesh either for purposes of interpolation of the solution variables, or for the integration of the energy. To use a meshless method, the input file describing the problem of interest must be constructed; further the user may also need support for analyzing and visualizing the results obtained by the meshless method. There are commercial software packages that have these two functionalities, such as ANSYS [21], MSC/PATRAN [22] and CONPLOT [23], but these are element-based software packages designed for use with FEM solvers; for meshless methods, because there is no element at all, we cannot use these software packages. Liu [24] developed a post-processor for one meshless method, but this is element-based and cannot be used by other meshless methods.
In order to make use of the meshless method [19] more easily and for post-processing the output file of the meshless method more conveniently and efficiently, we have developed a generic meshless pre-processor and post-processor for use with meshless solvers. This pre-processor and post-processor can be used generally with any meshless method.
Section snippets
Regularized boundary integral equation
A meshless integral method based on the regularized local boundary integral equation approach has been developed by the authors [19]. The method is an improved version of the LBIE method proposed previously by Atluri and coworkers [15], [16]. The most critical improvement is the use of the subtraction technique to remove the strong singularity that results in a regularized governing integral equation. A special numerical integration is employed for the calculation of integrals with weak
The pre-processor
The pre-processor is used to define the data and discretization scheme for meshless analysis. This includes support for the geometric model, as well as definition of material, boundary conditions, and other parameters for the meshless method.
One salient feature of this pre-processor is that the troublesome and time-consuming node generation process is automated. For discretization of a problem domain, the pre-processor can construct nodal information for the entire domain with minimal user
Post-processor
The post-processor provides a convenient graphical user interface for visualization of numerical solutions obtained from the meshless solver. Generally, the post-processors that are designed for use with finite element packages use the idea of an element to produce contours of the desired field variable. The problem domain is usually traversed element by element, and a coloring scheme for the element under consideration is synthesized, based upon the values of the field variable of interest
Example
This section presents in detail an example to illustrate the capabilities of the pre-processor and the post-processor to produce graphical representations of the results obtained using the meshless method. The example is an infinite plate with a circular hole.
Fig. 5 shows the interface of the software package that was developed using JAVA. This software gives us the flexibility of changing the color degree and deformation factor and switching the color space between RGB and GRAY. Changing the
Concluding remarks
In recent years, meshless methods have been developed as an alternative numerical approach to eliminate the known drawbacks in the well-known finite element method, meshless methods do not require a mesh to discretize the problem domain, and the approximate solution is constructed entirely based on a set of scattered nodes. To use a meshless method, the user should generate an input file in which the nodes information is included. This paper presents a quadrilateral-based pre-processor to
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