Meshfree Point Interpolation Method Research Articles

Time-domain acoustic wave propagations in multi-fluids using a weak-form meshfree method

In this work, we focus on the derivation of the weak-form for meshfree methods in analyzing the time-domain acoustic wave propagation in multi-fluids. Due to the employment of meshfree shape functions not restricted to a local background cell, the continuity condition for acoustic particle velocity cannot be spontaneously satisfied, leading to inaccurate simulations of acoustic wave upon the interface. Thus, the penalty function method is utilized here in the weak-form for meshfree methods in order to cure this discontinuity issue without introducing any addition degrees of freedom. By conducting a number of numerical analyses, the numerical results obtained from a radial point interpolation meshfree method are in accord with the analytical solutions and the numerical results of bilinear and quadratic finite elements, validating the effectiveness of the weak-form derived for meshfree methods. Moreover, the numerical accuracy of the used meshfree method can surpass those of bilinear and quadratic finite elements when the same sets of nodes are employed, especially for relatively high frequencies.

Free vibration analysis of sandwich structure via point interpolation meshfree method based on polynomial basic function

The stability of circular and annular sector plate structures under aerodynamic pressure in supersonic conditions is crucial for aircraft components. Conventional uniform materials lead to heavy structures, which are undesirable for aircraft. This study explores the use of functionally graded materials to enhance flutter stability. It employs a refined logarithmic shear deformation theory to describe plate displacement, classical elasticity theory for constitutive equations, and Hamilton’s principle for equations of motion. Two methods, finite element, and Meshfree radial point interpolation, are used to solve these equations and verify the results. Parameter studies are conducted to assess the effects of grading index, Biot’s coefficient, span angle, and geometrical aspect ratio, revealing unexpected adverse effects of grading material on stability.

Application of point interpolation mesh-free method for magneto/electro rheological vibrations of sandwich conical panels

Magneto-rheological or electro-rheological materials, due to many possible control-based applications have acquired lots of attention. So, in the present work, as the first attempt, a point interpolation method is exploited to evaluate vibrations of the sandwich conical panel with magneto-rheological or electro-rheological core, and composite face sheets. Through the linear viscoelastic hypothesis, and in the pre/post-yield zones, the correlation of shear stress-strain for magneto-rheological fluid is applied. The equilibrium equations are governed by the principle of Hamilton in addition to the first-shear deformation (FSD) hypothesis. By satisfying the compatibility equations, the interfaces of the core along with composite face layers are simulated. Upon polynomial basic function, shape functions together with the approximate form of the displacement field are constructed. The correctness of the computational approach is certified through the analogical investigation. The results show that number of face layers, thickness, the pattern of composite layers, weight fraction, and electric/magnetic field, together with the geometry of the conical panel have an imperative impact on the system's natural frequency in addition to the loss factor index.

Airy-based static limit analysis of structures using stabilized radial point interpolation method

This paper presents a novel formulation for static limit analysis of structures, for which the Airy stress function is approximated using stabilized Radial Point Interpolation Mesh-free method (RPIM). The stress field is determined as second-order derivatives of the Airy function, and the equilibrium equations are automatically satisfied a priori. The so-called Stabilized Conforming Nodal Integration (SCNI) is employed to ensure a present method is truly a mesh-free approach, meaning that all constraints in problems are only enforced at nodes. With the use of the Airy function, SCNI, and Second-Order Cone Programming (SOCP), the size of the resulting problem is kept to be minimum. Several benchmark problems having arbitrary geometries and boundary conditions are investigated. The obtained numerical solutions are compared with those available in other studies to perform the computational aspect of the proposed method.

Dynamic response and optimization of functionally graded porous nanocomposite cylinders using a meshfree method

AbstractA new class of advanced ultralight composite materials has recently emerged through the use porous polymer matrix reinforced by carbon nanotubes. In this article, the dynamic response of functionally graded porous polymeric cylinders, reinforced by randomly oriented single‐walled carbon nanotubes, using a meshfree method is studied. Three different porosity distribution patterns are investigated: symmetric distribution (SYD), unsymmetric distribution, and uniform distribution. A thorough study on the effects of reinforcement volume fractions and porosity distribution patterns on the dynamic response of the structure has been carried out using the radial point interpolation meshfree method based on the 2D theory of elasticity. In addition, a Pareto front solution is obtained through a multiobjective optimization aimed at minimizing the weight and maximizing the natural frequency of the structure with porosity and reinforcement volume fraction as design variables. From a design perspective, the results indicate that the SYD porosity type is the best candidate for relatively thick cylinders because of its smaller mass and higher stiffness compared to the other distribution types. The current research presents a reliable computational framework to help provide an insight into the design of an optimum structure subject to dynamic loading.

Static flexural analysis of sandwich beam with functionally graded face sheets and porous core via point interpolation meshfree method based on polynomial basic function

A point interpolation meshfree method based on polynomial basic function is employed to analyze static behavior of sandwich beams with functionally graded face sheets and porous core whose mechanical properties vary continuously in the depth-direction. Transverse shear deformation is taken into account with the context of third-order beam theory which satisfies the vanishing of shear stress at the top and bottom surfaces. The equilibrium equations are derived from the principle of virtual work. Polynomial basic function is employed to construct shape functions and approximate the displacement field of computational domain. The accuracy of the computational method is confirmed by comparisons of computed results with those available in the literature. The convergence rate and effect of nodal distribution on the accuracy are examined in details. Numerical examples are performed to investigate the effects of span-to-height ratio, face sheet-core-face sheet thickness ratio, material volume fraction index, porosity coefficient, as well as different boundary conditions on transverse displacement, axial and shear stresses of the beams.

Modelling Steady Convection-Dominated Phenomena by Node-Adaptive Radial Point Interpolation Meshfree Method (RPIM) with Various RBFs

For numerically solving PDEs, node or grid adaptation schemes have been invented, applied, and improved mainly to improve solution accuracy while reducing the burden caused by CPU-limitation and/or users’ interference. This becomes crucial particularly for problems involving strong changes in physical phenomena. This work proposes a means of node-adaptation used in conjunction with a numerical meshfree method called ‘radial point interpolation (RPIM)’, carried out using the concept of interpolation via. radial basis functions and hence no grid or mesh required. The proposed method is developed to tackle one of the most challenging problems with convective force being overwhelming in the system. This phenomena is known not to be easily numerically simulated. The proposed node-adaptive scheme is carried out where locally-estimated Péclet number (Pe) being used to indicate the areas of interest, places where nodes are expected to be added or removed. The whole adaptation algorithm is dynamic requiring on user’s interference. When compared to solutions obtained from fixed-node cases and other benchmark work, the results obtained in this work strongly suggest benefits in terms of both solution accuracy improvement and CPU-storage reduction.

The variable node multiscale approach: Coupling the atomistic and continuum scales

Linking atomistic and continuum zones in multiscale methods is a necessity in order to overcome the deficiencies of the conventional molecular and continuum-based methods. In this paper, a novel approach is presented to bind the atomistic and continuum zones together based on the combined concepts of the finite element method and the radial point interpolation meshfree method. This variable node multiscale method (VNMM) introduces a new way to couple the finite element method with the molecular solutions with nodes coinciding with the atoms. In contrast to QC methods, VNMM does not need a mesh refinement in the vicinity of the molecular zone, nor it requires an overlapping zone in the continuum/atomistic interface which is necessary in bridging schemes. In VNMM, the total displacement of each atom is decomposed into coarse and fine displacements, in which the coarse displacement is computed using the finite element solution over the whole domain, while the fine displacement is calculated just in a limited atomistic zone with the use of molecular statics. Integration of the finite element method within the VNMM formulation potentially allows for enrichment of any field over the entire domain. Conventional interatomic potentials could be employed at the atomistic zone. The results are compared with the fully atomistic simulation, conducted with the molecular statics, and the multiscale quasicontinuum method. A good agreement is observed with substantially reduced degrees of freedom and computational costs.

Dispersion analysis for acoustic problems using the point interpolation method

A typical meshfree point interpolation method (PIM) is presented to investigate the dispersion error in the numerical solutions of acoustic problems which is governed by the Helmholtz equation. It is well-known that those results from several numerical approaches, such as the finite element method (FEM) and several meshfree techniques, will suffer from the pollution effect, leading to the incorrect acoustic wave propagation for high wave numbers. The reason for this phenomenon is that the numerical solutions of wave number do not accord with the exact wave number, which is the so-called dispersion issue. In addition, to overcome the possible singularity issue in constructing the shape functions for the PIM with the polynomial basis functions (PBFs), the Gauss–Jordan elimination (GJE) technique is employed here. Several numerical examples concerning dispersion analysis and acoustic wave propagation are performed to verify the accuracy of results from the PIM. It is found that the PIM can reduce the dispersion error effectively and hence generate more accurate results than the FEM with the same set of nodes.

Magnetotelluric three-dimensional forward modeling by meshfree point interpolation method

As an important supplement and development of mesh method numerical calculation, meshfree method is a kind of new numerical algorithm in last decade. Point interpolation method(PIM), as a simple and efficient meshfree method, avoids the mesh generation and has obtained many good results in the field of computational mechanics. This paper devotes PIM to magnetotelluric three-dimensional forward modeling, introduces the approximate principle of PIM in detail, and deduces the discrete system matrix expression which corresponding to the magnetotelluric three-dimensional variational problem by combining the Galerkin method and the gauss integral formula, then the boundary conditions of loading technique is briefly introduced. The effectiveness of three-dimensional PIM is verified by the numerical calculation of several models.

Nonlinear analyses of FGM plates in bending by using a modified radial point interpolation mesh-free method

A modified mesh-free radial point interpolation method (RPIM) which employs a new radial basis function capable of building the shape functions without any supporting fixing parameters is developed for the nonlinear bending analyses of functionally graded material (FGM) plates. The shape functions constructed by the RPIM possess Kronecker delta function property and thus special schemes for imposing boundary conditions are not needed. The nonlinear bending formulations of FGM plates are based on a higher-order shear deformation plate theory considering the von Kármán large deflection assumption. The nonlinear equations are solved by the modified Newton–Raphson iterative technique. The computed results by the modified RPIM are compared with the other numerical solutions reported in the literature to verify the effectiveness and accuracy of the proposed mesh-free method. Detailed parametric studies are then carried out to scrutinize the effects that the volume fraction, plate length-to-thickness ratio, boundary condition and initial deflection have on the nonlinear flexural responses of FGM plates. Results illustrate that the modified mesh-free RPIM can effectively predict the nonlinear bending behavior of FGM plates.

Thermal buckling analyses of FGM sandwich plates using the improved radial point interpolation mesh-free method

This study reports thermal buckling analyses of functionally graded material (FGM) sandwich plates using an improved mesh-free radial point interpolation method (RPIM). The buckling formulation of FGM sandwich plates is derived from the improved RPIM which employs a new radial basis function enabling the shape functions to be built without any supporting fixing parameters based on the higher-order shear deformation plate theory. Two types of FGM sandwich plates with different composition scheme, i.e. one with FGM skins and homogenous core and the other composed of homogenous skins and FGM core are considered in the analyses. The simulated results by the improved RPIM are compared with the analytical solutions found in the literature for the verification purpose. Detailed parametric studies are then carried out to scrutinize the effects of the volume fraction, plate length-to-thickness ratio, aspect ratio, boundary condition and FGM constituents on the critical buckling temperature changes of the FGM sandwich plates under various types of temperature variation through the thickness. Results demonstrate that the improved mesh-free RPIM can effectively predict the thermal buckling behavior of the FGM sandwich plates.

Factors affecting accuracy of radial point interpolation meshfree method for 3-D solid mechanics

Recently, the radial point interpolation meshfree method has gained popularity owing to its advantages in large deformation and discontinuity problems, however, the accuracy of this method depends on many factors and their influences are not fully investigated yet. In this work, three main factors, i.e., the shape parameters, the influence domain size, and the nodal distribution, on the accuracy of the radial point interpolation method (RPIM) are systematically studied and conclusive results are obtained. First, the effect of shape parameters (R, q) of the multi-quadric basis function on the accuracy of RPIM is examined via global search. A new interpolation error index, closely related to the accuracy of RPIM, is proposed. The distribution of various error indexes on the R-q plane shows that shape parameters q ∈ [1.2, 1.8] and R ∈ [0, 1.5] can give good results for general 3-D analysis. This recommended range of shape parameters is examined by multiple benchmark examples in 3D solid mechanics. Second, through numerical experiments, an average of 30–40 nodes in the influence domain of a Gauss point is recommended for 3-D solid mechanics. Third, it is observed that the distribution of nodes has significant effect on the accuracy of RPIM although it has little effect on the accuracy of interpolation. Nodal distributions with better uniformity give better results. Furthermore, how the influence domain size and nodal distribution affect the selection of shape parameters and how the nodal distribution affects the choice of influence domain size are also discussed.

Numerical Analysis of Quasistatic Contact between Rubber Roller and Rigid Roller

A quasistatic and steady state contact problems of rubber roller and rigid roller is modeled by using a stress function in the form of series. The model is derived from the paper feed unit of duplicating machine. The point interpolation meshfree method (PIM) is applied to obtain the solution of the dual series equations resulting from the boundary condition. Stress and deformation for the rubber roller are obtained based on elasticity theory. Effects of rubber thickness and normal load are observed through the numerical examples. The observation show that the contact zone is changing with the normal load F and the rubber thickness has effect on the stress distribution. In addition, the study shows that the PIM method is an efficient and promising method for simulating the problem.

Applications of point interpolation method for spatial general shells structures

The meshfree point interpolation method (PIM) based on polynomial function is presented for the static analysis of spatial general shell structures. The formulation of the discrete system equations is derived from stress resultant geometrically exact shell theory of shear flexible shells based on the Cosserat surface. The PIM technique constructs its interpolation functions through a set of arbitrarily distributed points in the problem domain and its shape function has the delta function property. Hence, the implementation of essential boundary conditions can be imposed with ease as in conventional finite element method. An improved matrix triangularization algorithm (MTA) is proposed to obtain proper node enclosure and basis selection in order to overcome the singularity problem in the point interpolation method using a polynomial basis. It is found that the shape functions are generally not compatible to ensure the compatibility of the solution and lead to non-conforming PIM (NPIM) when the energy principles are used to formulate PIM. To obtain compatible shape functions, the enforcement of compatibility is needed along the common edges of the background cells in the problem domain and a background cell-based nodal selection is utilized, which lead to conforming PIM (CPIM). Both NPIM and CPIM are evaluated in the paper. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the NPIM and CPIM. The phenomena of shear locking and membrane locking are illustrated by showing the membrane and shear energies as fractions of the total energy. The convergences and performances are investigated for both conforming and non-conforming PIMs. It is also shown that the NPIM and CPIM with the improved MTA are very easy to implement and very efficient in constructing shape functions in comparison with the Element-Free Galerkin method.

An experimental investigation of large-strain tensile behavior of a metal sheet

A method for determining the stress–strain curve of a larger strain is proposed when plastic instability occurs in standard tension tests. A thin tested sheet of clod-rolled steel is subjected to tension loading until a fracture occurs. The deformation process is captured with a digital camera and using a mesh-free point interpolation method (PIM) method, a displacement field and strain field of material deformation can be calculated. In order to verify that the local measuring stress–strain curve by the PIM method near the center of the specimen can describe a full stress–strain curve, a tensile experiment with different gauge lengths (25 mm, 35 mm and 50 mm) and numerical simulation at different locations along the specimen axial have been presented that different parts of specimen have a different deformation distribution in tensile and the center fracture part of the tensile specimen is the only region which can experience a full strain. The true stress–strain curves based on the estimated parameters are validated in all strain regions by comparison with curves from standard tension tests. The measured curves by the PIM method are very stable. Compared with several material constitutive equations, Ludwik’s and Swift’s equation is very close to the experimental curve at plastic deformation.

A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures

A mesh free method called point interpolation method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures. In the present method, the problem domain and its boundaries are represented by a set of properly scattered nodes. The displacements and the electric potential of a point are interpolated by the values of nodes in its local support domain using shape functions derived based on a point interpolation scheme. Techniques are discussed to surmount the singularity of the moment matrix. Variational principle together with linear constitutive piezoelectric equations is used to establish a set of system equations for arbitrary-shaped piezoelectric structures. These equations are assembled for all quadrature points and solved for displacements and electric potentials. A polynomial PIM program has been developed in MATLAB with matrix triangularization algorithm (MTA), which automatically performs a proper node enclosure and a proper basis selection. Examples are also presented to demonstrate the accuracy and stability of the present method and their results are compared with the conventional FEM results from ABAQUS as well as the analytical or experimental ones.