Cell-based Smoothed Radial Point Interpolation Method Research Articles

Dispersion error control for underwater acoustic scattering problems using a coupled cell-based smoothed radial point interpolation method

A coupled cell-based smoothed radial point interpolation method is presented by combining the cell-based smoothed radial point interpolation method (CSRPIM) and the modified Dirichlet to Neumann boundary condition (MDtN-BC) to effectively control the dispersion error in predicting acoustic scattering. In the coupled CSRPIM, the MDtN-BC is used to satisfy the Sommerfeld radiation condition approaching infinity, and a cell-based smoothing operation together with the radial point interpolation method (RPIM) is adopted to soften the acoustic stiffness of the numerical model. The condensed RPIM shape functions are created by using real and virtual nodes, and virtual node selection schemes are studied to appropriately soften the stiffness. A major advantage of the coupled CSRPIM is that it can achieve very close-to-exact stiffness of the original model without introducing extra degrees of freedom. The properties of the proposed method are numerically studied. Simulations indicate that the six-virtual-node selection scheme performs the best. The results reveal that the developed method can significantly reduce the dispersion error and has excellent computational performance. In addition, the developed method shows good potential for practical acoustic-scattering prediction in engineering applications.

A hybrid CSRPIM/SEA method for the analysis of vibro-acoustic problems in mid-frequency range

Hybrid methods are commonly methods to solve the response of mid-frequency vibro-acoustic system. In this study, a novel hybrid CSRPIM/SEA method is proposed to calculate the coupling problem between enclosed cavity and flexible plate. Deterministic subsystem with low modal density is modeled by the Cell-based Smoothed Radial Point Interpolation Method (CSRPIM) and statistical subsystem with the high modal density is modeled by the Statistical Energy Analysis (SEA). Owing to the generalized gradient smoothing technique and the condensed shape function technique used, the stiffness of discrete model is softened and the dispersion error can be reduced obviously, which makes the hybrid CSRPIM/SEA method give satisfactory numerical results considering the examples studied. The proposed method has provided an effective way for analyzing mid-frequency virbo-acoustic problems.

Modeling, simulation and analysis of groundwater flow captured by the horizontal reactive media well using the cell-based smoothed radial point interpolation method

The Horizontal Reactive Media Treatment (HRX) well is a novel technology for in situ treatment of contaminated groundwater. The most significant technical performance risk associated with field-scale implementation of the HRX well is the potential for water to bypass the well untreated. Therefore, the groundwater capture simulation plays a very important role in the design of the HRX well. We build a model based on a cell-based smoothed radial point interpolation method (CS-RPIM) for simulating groundwater flow captured by the HRX well in both 3D test pit pilot scale and field scale. This method is compared with the finite element method and measured values. Capture width and groundwater velocity obtained by CS-RPIM are consistent with the measured values. We further develop MATLAB software modules to analyze the effects of the hydraulic conductivities, dimension of well and other parameters on the groundwater capture zone, velocity and residence time in the reactive media. The results show that increase of well diameter has a significant effect on enlargement of the capture zone and the extension of media treatment time. When well diameter increases by 100%, horizontal and vertical capture width increase by 57% and 50% respectively, and media treatment time can be extended by 68%.

Cell-Based Smoothed Radial Point Interpolation Method for Underwater Acoustic Scattering Problems

In this work, a novel cell-based smoothed radial point interpolation method (CSRPIM) is used to deal with underwater acoustic scattering problem. The nature of infinite domain has been changed by exact Dirichlet-to-Neumann (DtN) boundary condition. Owing to the use of gradient smoothing operation and consequent properly softened stiffness, the CSRPIM model is capable of reducing pollution error significantly. A theoretical analysis is carried out to elucidate the superiority in controlling pollution error of CSRPIM. By selecting virtual nodes properly in condensed shape functions, the interpolation error in the CSRPIM can also be reduced. Through several simple acoustic scattering numerical examples, advantages of CSRPIM have been verified. The results show that the CSRPIM can be applied directly to solve acoustic scattering problems and can obtain higher accuracy than traditional linear finite element method (FEM) even if in relatively high frequency range. Also, the use of linear triangular elements in the present model makes the analysis of practical underwater acoustic problems with complex shapes easier.

Analysis of acoustic radiation problems using the cell-based smoothed radial point interpolation method with Dirichlet-to-Neumann boundary condition

In this work, the cell-based smoothed radial point interpolation method (CSRPIM) with Dirichlet-to-Neumann (DtN) boundary condition is presented to resolve acoustic radiation problems. In the scheme of CSRPIM, additional virtual nodes are used in RPIM shape functions creation, which, together with the operation of cell-based gradient smoothing operation, enable the model provide proper stiffness for acoustic field without extra degrees of freedom. It is demonstrated that the present CSRPIM can reduce the dispersion error significantly and possess some superiorities, such as high accuracy and insensitivity to mesh distortion. For exterior acoustic problems, the well-known DtN artificial boundary has been employed to ensure that there are no spurious reflections from unbounded domain. Several numerical examples have been studied to verify the properties of the present CSRPIM with DtN condition. The results show that this CSRPIM is outstanding to tackle acoustic radiation problems and can generate more reasonable solutions than the standard FEM. Also, owing to the use of linear background mesh, the present method has a good potential to be applied to practical acoustic radiation engineering problems.

A cell-based smoothed radial point interpolation method using condensed shape functions for free and forced vibration analysis of solids

In this work, a novel cell-based smoothed radial point interpolation method (CS-RPIM) has been developed for free and forced vibration analysis of solids by using condensed radial point interpolation method (RPIM) shape functions with virtual points. In the scheme of the present method, the generalized gradient smoothing technique (GST) is used for constructing smoothed strains and the ingenious six virtual nodes are selected together with real nodes for creating condensed RPIM shape functions, which helps the proposed method avoid not only the overly-stiff problem existing in traditional finite element method (FEM) but also the overly-soft issue of original CS-RPIM models. Thanks to the well-constructed system stiffness, the proposed method has shown very good performance in terms of accuracy, efficiency and stability for the examples studied. As three-nodes triangular elements, that can be automatically generated for complex problem domains, are used as background mesh, the proposed method is supposed to possess good potential for solving practical engineering problems.

A Combination of Singular Cell-Based Smoothed Radial Point Interpolation Method and FEM in Solving Fracture Problem

The singular cell-based smoothed radial point interpolation method (CS-RPIM) has been previously proposed and shown good performance in solving fracture problems. Motivated from the fact that CS-RPIM performs over softly by providing an upper bound solution and the finite element method (FEM) is overly stiff by providing a lower bound solution, this work proposes a combination of singular CS-RPIM and FEM with a correlation coefficient [Formula: see text], and [Formula: see text] has been recommended through intensive numerical studies. Several numerical examples have been studied and the proposed method has been found perform quite well from both stress intensity factors and strain energy.

Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method

This study aims to improve the accuracy and efficiency when dealing with thermal and mechanical response in functionally graded cylinder, which is very important in modern aerospace industry. The cell-based smoothed radial point interpolation method (CS-RPIM) is formulated for such analysis. In CS-RPIM, triangular meshes are utilized to discretize problem domains, which can be easily generated. Each triangular element is then partitioned into several smoothing cells. Field functions are constructed by RPIM shape functions and system equations are obtained based on these smoothing cells. Finally, the performances of CS-RPIM are fully investigated through several numerical examples.

Analysis of Transient Thermo-Elastic Problems Using a Cell-Based Smoothed Radial Point Interpolation Method

The transient thermo-elastic problems are solved by a cell-based smoothed radial point interpolation method (CS-RPIM). For this method, the problem domain is first discretized using triangular cells, and each cell is further divided into smoothing cells. The field functions are approximated using RPIM shape functions which have Kronecker delta function property. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form. At first, the temperature field is acquired by solving the transient heat transfer problem and it is then employed as an input for the mechanical problem to calculate the displacement and stress fields. Several numerical examples with different kinds of boundary conditions are investigated to verify the accuracy, convergence rate and stability of the present method.

A Cell-Based Smoothed Radial Point Interpolation—Perfectly Matched Layer Method For Two-Dimensional Acoustic Radiation Problems

We present a cell-based smoothed radial point interpolation method (CS-RPIM) model for two-dimensional acoustic radiating problem by incorporating the perfectly matched layer method (PML). In this work, the computational region, truncated by PML, is discretized into triangular background cells. Each cell is further divided into several smoothing cells, and then the cell-based gradient smoothing operation is implemented throughout the smoothing cells. The pressure field function is approximated using the RPIM shape functions. The supporting node selection for shape function construction uses the T2L-scheme associated with edges of the background cells. The cell-based gradient smoothing operation provides proper softening effect, and makes the acoustic stiffness of the CS-RPIM model much softer than that of the FEM (finite element method)/PML model, which in turn significantly reduces the numerical dispersion error. Numerical results show that, compared with FEM–PML, the CS-RPIM achieves better absorbing effect in the PML, and higher accuracy in the computational region. This enables us to conclude that the CS-RPIM model with the PML can be well applied in solving acoustic radiation problems.

A cell-based smoothed radial point interpolation method (CS-RPIM) for three-dimensional solids

A cell-based smoothed radial point interpolation method (CS-RPIM) is formulated for three-dimensional elasticity problems. In present method, the problem domain is firstly discretized by tetrahedron background cells, and each tetrahedron cell is then further divided into several smoothing cells. The displacement field function is approximated using RPIM shape functions which have Kronecker delta function property. Supporting node selection for shape function construction uses the efficient T2L-scheme associated with the background cells. The smoothed Galerkin weak form is employed to create discretized system equations, and then the gradient smoothing operation is adopted to construct smoothed strain fields in every smoothing cell. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear interpolation and node-based smoothed finite element method, the CS-RPIM solutions can achieve better efficiency, higher accuracy, and greater stability in static and free vibration analysis in presented examples.

Dispersion and pollution of the cell-based smoothed radial point interpolation method (CS-RPIM) solution for the Helmholtz equation

Abstract The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion. This paper presents an application of the cell-based smoothed radial point interpolation method (CS-RPIM) and focuses on the dispersion analysis in two-dimensional (2D) Helmholtz problems. The dispersion error is mainly caused by the ‘overly stiff’ feature of the discrete model. Therefore a properly “softened” stiffness for the discrete model is much more essential to the root of the numerical dispersion error. Owing to the proper softening effects provided by the cell-based gradient smoothing operations, the CS-RPIM model, therefore, can significantly reduce the dispersion error in the numerical solution. Numerical results demonstrated that, the CS-RPIM yields considerably better results than the FEM and EFGM, because of the crucial effectiveness in handling dispersion.

A cell-based smoothed radial point interpolation method (CS-RPIM) for heat transfer analysis

Abstract A cell-based smoothed radial point interpolation method (CS-RPIM) is further extended to solve 2D and 3D heat transfer problems. For this method, the problem domain is first discretized using triangular elements or tetrahedral elements, and each element is further divided into several smoothing cells. Then, the field functions are approximated using RPIM shape functions. Finally, the CS-RPIM utilizes the smoothed Galerkin weak form to obtain the discretized system equations in these smoothing cells. Several numerical examples with different kinds of boundary conditions are investigated to verify the validity of the present method. It has been found that the CS-RPIM can achieve better accuracy and higher convergence rate, when dealing with the 2D and 3D heat transfer analysis.

Numerical treatment of 2D acoustic problems with the cell-based smoothed radial point interpolation method

This paper addresses the cell-based smoothed radial point interpolation method (CS-RPIM) for 2D acoustic problem. In present method, the acoustic domain is discretized using triangular background cells, and each cell is further divided into several smoothing cells and then the cell-based gradient smoothing operation is implemented through the smoothing cells. The pressure field function is approximated using RPIM shape functions. Supporting node selection for shape function construction uses the T2L-scheme associated with edges of the background cells. The system equations are derived using the smoothed Galerkin weak form, and the essential boundary conditions are imposed directly as in the finite element method (FEM). The cell-based gradient smoothing operation provides proper softening effect, makes the CS-RPIM model much softer than the “overly-stiff” FEM model and hence significantly reduces the numerical dispersion error. Numerical results show that the CS-RPIM achieves more accurate results and higher convergence rates as compared with the corresponding finite elements.

A singular cell-based smoothed radial point interpolation method for fracture problems

A cell-based smoothed radial point interpolation method (CS-RPIM) is developed for fracture problems. The strain smoothing is performed over background triangular cells. The stiffness matrix is calculated based on the weakened weak formulation, using shape functions obtained by radial point interpolation method. A layer of five-node singular elements are used to simulate the singularity around the crack tip. Different schemes are devised in the five-node elements to perform the strain smoothing. Several examples are presented to validate the newly developed method. The results are found in excellent agreement with the exact (or reference) solutions.

ANALYSIS OF MINDLIN–REISSNER PLATES USING CELL-BASED SMOOTHED RADIAL POINT INTERPOLATION METHOD

In this paper, a formulation for the static and free vibration analysis of Mindlin–Reissner plates is proposed using the cell-based smoothed radial point interpolation method (CS-RPIM) with sub-domain smoothing operations. The radial basis functions augmented with polynomial basis are employed to construct the shape functions that have the Delta function property. The generalized smoothed Galerkin (GS-Galerkin) weakform is adopted to discretize the governing differential equations, and the curvature smoothing is performed to relax the continuity requirement and to improve the accuracy and the rate of convergence of the solution. The present CS-RPIM formulation is based on the first-order shear deformation plate theory, with effective treatment for shear-locking and hence is applicable to both thin and relatively thick plates. To verify the accuracy and stability of the present formulation, intensive comparison studies are conducted with existing results available in the literature and good agreements are obtained. The numerical examples confirm that the present method is shear-locking free and very stable and accurate even using extremely distributed nodes.

A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids

A cell-based smoothed radial point interpolation method (CS-RPIM) based on the generalized gradient smoothing operation is proposed for static and free vibration analysis of solids. In present method, the problem domain is first discretized using triangular background cells, and each cell is further divided into several smoothing cells. The displacement field function is approximated using RPIM shape functions which have Kronecker delta function property. Supporting node selection for shape function construction uses the efficient T2L-scheme associated with edges of the background cells. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form, and the essential boundary conditions are imposed directly as in the finite element method (FEM). The effects of the number of divisions smoothing cells on the solution properties of the CS-RPIM are investigated in detail, and preferable numbers of smoothing cells is recommended. To verify the accuracy and stability of the present formulation, a number of numerical examples are studied to demonstrate numerically the efficiency of the present CS-RPIM.