Radial Point Interpolation Research Articles

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.

Stable node-based smoothed radial point interpolation method for the dynamic analysis of the hygro-thermo-magneto-electro-elastic coupling problem

In the current work, the stable node-based smoothed radial point interpolation method (SNS-RPIM) was used to investigate the dynamic responses of magneto-electro-elastic (MEE) beams under hygrothermal environment with the generalized Newmark method. Based on the generalized smoothed Galerkin weak form and the gradient smoothing technique, the basic equations for hygro-thermo-magneto-electro-elastic (HTMEE) coupling problems were derived. The stable item of SNS-RPIM made the stiffness ‘close-to-exact’ and cured the ‘over-soft’ of the model. Through the numerical examples, the effect of empirical constants, temperature variation and moisture concentration on the elastic stiffness coefficients and generalized displacement was investigated. Besides, numerical results verified the accuracy, convergence and correctness of the present method compared with the traditional finite element method when analyzing the HTMEE coupling problems. Moreover, SNS-RPIM was very stable and accurate for extremely distributed meshes, which can have an excellent application for future studies of the coupled multi-physical problems.

Modeling fracture in viscoelastic materials using a modified incremental meshfree RPIM and DIC technique

In the present work, fracture mechanics of linear viscoelastic materials is investigated numerically and experimentally. Herein, an effective and accurate meshfree method is developed to analyze fracture mechanics in viscoelastic problems. To this end, an incremental approach with higher-order finite difference (FD) schemes for analysis of viscoelastic materials in the time domain, along with an improved meshfree method based on the global weak formulation, i.e. the radial point interpolation method (RPIM), is employed. An accurate and efficient integration approach with minimum computational cost, i.e. the background decomposition method (BDM), is also used to compute the domain integrals. To evaluate the accuracy and efficiency of the proposed numerical method, first, two example problems are provided and the robustness of the presented method is assessed. Then, a non-contact full-field optical method, i.e. the digital image correlation technique, is utilized to investigate the viscoelastic fields of a polymer, polymethyl-methacrylate (PMMA), during crack propagation in mode I. Also, in order to properly model the fracture behavior of this material by the meshfree RPIM, the mechanical properties of the PMMA, including the Prony viscoelastic parameters, are obtained by an ASTM test method. Moreover, the obtained results from the DIC test and also the load-displacement curve of the cracked specimen are compared with the results of the proposed meshfree RPIM and a very close agreement is observed.

Meshless analysis of the stress singularity in composite adhesive joints

Adhesives are an exceptionally well-suited method for joining composites. Unlike other methods, such as bolting or riveting, adhesives do not introduce holes in their joining material. This is a significant advantage in the case of composites because the holes required by bolting or riveting induce stress concentrations and can also lead to tears, burrs or delamination. A point of concern in adhesive joints is the adhesive/adherend interface corner where a stress singularity occurs, and failure usually initiates. Thus, it is crucial to study this stress singularity to better understand adhesive joints’ mechanical behaviour.The goal of this work is to validate the application of the Intensity of Singular Stress Fields (ISSF) criterion to meshless methods, in this case, the Radial Point Interpolation Method (RPIM). With this purpose, eight overlap lengths (LO) in single-lap joints (SLJ) composed of Carbon Fibre Reinforced Polymer (CFRP) and bonded with a brittle adhesive were experimentally and numerically tested. Furthermore, an extrapolation based method is implemented to determine the critical stress singularity components (Hc) necessary for the strength predictions. In the end, the experimental and numerical results are compared to assess the suitability of the method. It was found that the ISSF criterion can be accurately applied to meshless methods and composite materials successfully, given the simplicity of the method applied.

A novel centroid-enriched edge-based smoothed radial point interpolation method for upper bound limit analysis

This paper presents a novel centroid-enriched edge-based smoothed radial point interpolation method (CE-ESRPIM) to overcome the volumetric locking problem and increase the numerical accuracy for upper bound limit analysis. Based on the edge-based smoothed radial point interpolation method (ESRPIM), the centroid of the triangle element is used as a field node to discretize the problem domain. Simultaneously, the local support node selection scheme, which is established based on a modified centroid-enriched triangle mesh (CE-TScheme), is proposed to realize local radial point interpolation (RPIM). Using the Mohr–Coulomb yield criterion and associated flow rule, the upper bound theorem in the plane strain condition is discretized into a standard second-order cone programming (SOCP) problem through the CE-ESRPIM, which is solved using the primal–dual interior-point algorithm (PDIP). The proposed approach with the enriched centroid considered as the field node is straightforward and efficient and can directly inherit the formulation framework of the traditional ESRPIM. Moreover, this method has been proven to be effective in eliminating the volumetric locking problem under incompressible conditions.

Modelling adhesively-bonded T-joints by a meshless method

Bonding of non-parallel substrates has many applications in the transport industry. Adhesively-bonded T-joints are employed for that purpose. However, geometrical dimensions, substrates’ shape and material choices are broad. The effect that varying upper substrate thickness has on the joint strength (P max ) was investigated in this work through numerical analyses. The numerical analysis was performed using a meshless method, the natural neighbour radial point interpolation method (NNRPIM), which has been proven accurate and robust in another adhesive joint configuration. Materials were considered elastic-plastic. A yield criterion developed for rubber-like materials, the Exponent Drucker-Prager, was used for the adhesive layer, while the metallic substrates were analysed with the von Mises yield criterion. P max was determined numerically using a strain-based continuum mechanics failure criterion. Normalised peel and shear stress distributions along the bond-line are presented. Effective strains, both elastic and plastic, were also obtained. The estimated P max was compared with experimental data; a good agreement was found. The stress distribution along the bond line becomes asymmetric in joints with unbalanced substrate thicknesses. At P max , from 10 to 25% of the bond-line has entered into plastic regime. The results indicate that the proposed methodology is suitable to analyse adhesively-bonded joints under different load solicitations.

Task-oriented treadmill training to improve walking adaptability in children/adolescents with Cerebral Palsy

This paper presents a numerical analysis of bending plates considering Reissner’s hypothesis. A truly meshless method designated Meshless Local Petrov-Galerkin (MLPG) method is used to obtain a linear system of equation. A simplified Radial Point Interpolation Method (RPIM) approximation scheme, by centering both quadrature and local interpolation subdomains in the same field point, is proposed to increase the computational efficiency of the MLPG. Moreover, the shear locking effect is also analyzed. Results obtained by the application of the presented formulation are discussed in this work, considering plates with different geometries and boundary conditions, and compared, in terms of precision and efficiency, with solutions obtained via Finite Element Method.

An efficient reduced basis approach using enhanced meshfree and combined approximation for large deformation

This paper describes a new efficient approach based on the concept of reduced basis for large deformation analysis. The domain problem is discretized using the meshfree particle radial point interpolation method (RPIM), which inherently possesses the Kronecker’s delta property. Meshless numerical integration is evaluated by the Cartesian transformation method (CTM), which enhances the performance of the RPIM. In addition, we also introduce a new approach to further improve the capability of the current CTM in evaluation of numerical integration for problems with complex geometries, i.e., by incorporation of the non-uniform rational B-splines function (NURBS) into the CTM. The emphasis of the paper is on the nonlinear nature of the large deformation problems, which are often solved by an iterative scheme. Conventional Newton–Raphson technique usually requires high cost due to the fact that several load steps are usually performed, and multiple iterations are needed in each load step. This low computational efficiency can be overcome, as proposed in this work, by using the so-called combined approximation, which approximates the full-size solution by a set of reduced bases. In other words, reduction of the problem size can be obtained, leading to reduction of the computational time, while accuracy is almost preserved.

An effective crack identification method in viscoelastic media using an inverse meshfree method

In the present study, a numerical approach is proposed for identification of straight cracks in two-dimensional linear viscoelastic media. The identification procedure is based on the wave propagation in cracked viscoelastic media. The forward wave propagation problem in viscoelastic media is solved using the meshfree radial point interpolation method (RPIM), which allows the crack boundary to be easily modelled at any part of the domain. In the meshfree approach, the crack position can be changed freely during the iterations of the identification procedure. A modified higher-order incremental method is implemented for modelling dynamic behaviour of linear viscoelastic materials. Also, the integration points are determined using the background decomposition method (BDM). Firstly, the wave propagation behaviour in viscoelastic media is studied, and the effect of crack on wave propagation is investigated. Then, the inverse identification task is formulated as a nonlinear optimization problem to determine the unknown crack parameters. The optimization problem is solved by a gradient-based inverse method. Several numerical tests are simulated with different number of sensors, crack size, and various initial guesses, to demonstrate the capability of the proposed techniques in obtaining accurate results. Also, it is shown that the proposed method is robust to tackle reasonable levels of measurement error. In addition, the method is investigated for different placement patterns for several sensors and the effect of number of sensors on identification of the crack is studied.

An improved meshless artificial viscosity technology combined with local radial point interpolation method for 2D shallow water equations

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

A practical meshfree inverse method for identification of thermo-mechanical fracture load of a body by examining the crack path in the body

An inverse meshfree approach is presented for thermo-mechanical load identification of a fractured body. To this end, the crack propagation path is used as the input of the inverse analysis. This study can be used for finding the thermo-mechanical loads that have led to the fracture of a body, and is applicable to situations where the direct measurement of the boundary loadings on some part of a body is not possible. This highly ill-posed problem is solved using the meshfree radial point interpolation method (RPIM) with a global optimization approach. The inspected crack path is compared to the one obtained numerically and their difference is set as a cost function. By finding the minimum of this cost function, the actual loading conditions that have led to the fracture of the body is obtained. The density clustering method is used as the global optimization method in which the damped Gauss-Newton technique is used as the local search algorithm. To demonstrate the performance of the presented meshfree inverse approach, two examples are provided.

Elasto-plastic adhesive joint design approach by a radial point interpolation meshless method

ABSTRACT For efficient use of adhesive joints, reliable prediction techniques should be made available to the designer. Simulation of these joints’ behaviour is usually performed using the Finite Element Method (FEM). However, it is known that, in adhesive joints, the adhesive thickness (t A) is much smaller than the adherend thickness (t P), thus requiring a highly refined mesh to produce good results. Linked to this, the adhesive has to withstand high strains, causing mesh distortion and hindering the resolution. In these cases, meshless methods can be a good alternative. This work aims to implement the von Mises (vM) and Exponent Drucker-Prager (EDP) criteria combined with a meshless formulation based on the Radial Point Interpolation Method (RPIM), for the strength prediction of adhesively-bonded single-lap joints (SLJ). Validation with experiments is undertaken for joints with brittle to ductile adhesives, with varying overlap lengths (L O). Stress and strain distributions were plotted in the adhesive layer, and the failure load (P m) was assessed by strength of materials failure criteria. Significant adhesive and L O effects were found on P m. The RPIM proved to be a promising tool to predict the behaviour of bonded joints, although some limitations were found by using strength of materials criteria.

Abstract P345: A MRI Displacement-based Deep-learning Semantic Segmentation Tool For Left-ventricular Longitudinal Strain Analysis In Cardiotoxicity

Global longitudinal strain (GLS) computed in the left-ventricle (LV) is an established metric for detecting cardiotoxicity in breast cancer patients treated with antineoplastic agents. The purpose of this study was to develop a novel, MRI-based, deep-learning semantic segmentation tool that automates the phase-unwrapping for LV displacement computation in GLS. Strain analysis via phase-unwrapping was conducted on 30 breast cancer patients and 30 healthy females acquired with the Displacement Encoding with Stimulated Echoes (DENSE) sequence. A ResNet-50 deep convolutional neural network (DCNN) architecture for automated phase-unwrapping, a previously validated ResNet-50 DCNN for chamber quantification and the Radial Point Interpolation Method were used for GLS computation (Figure 1). The DCNN's performance was measured with F1 and Dice scores, and validated in comparison to the robust transport of intensity equation (RTIE) and quality guided phase-unwrapping (QGPU) conventional algorithms. The three techniques were compared by intraclass correlation coefficient with Cronbach’s alpha (C-alpha) index. Classification accuracy with the DCNN was F1 score of 0.92 and Dice score of 0.89. The GLS results from RTIE, QGPU and DCNN were -16.0 ± 2%, -16.1 ± 3% and -15.9 ± 3% (C-alpha = 0.89) for patients and -18.9 ± 3%, -19.0 ± 4% and -18.9 ± 3% (C-alpha = 0.92) for healthy subjects. Comparable validation results from the three techniques show the feasibility of a DCNN-based approach to LV displacement and GLS analysis. The dissimilarities between patients and healthy subjects demonstrate that DCNN-based GLS computation may detect LV abnormalities related to cardiotoxicity.

A Stochastic Radial Point Interpolation Method for Wideband Uncertainty Analysis

In this letter, a time-domain stochastic radial point interpolation method (SRPIM) is developed for uncertainty quantification of electromagnetic systems. Derivatives of field quantities in Maxwell's equations are obtained using radial basis function, and stochasticity in the dielectric constant of a part of the model space is incorporated into this formulation. This is validated using the numerical example of a parallel-plate waveguide with dielectrics inside, which is implemented using uniaxial perfectly matched boundary layers. The stochastic variations in the computed field in the time domain and the transmission coefficient in the frequency domain are evaluated. Accuracy of these simulation results is validated using Kolmogorov Smirnov test, with Monte Carlo (MC) simulation as the reference. The computation time of the proposed method is found to be significantly better than MC and superior to stochastic collocation. The proposed method performs better even for large variations.

Bending and free vibration and analysis of laminated plates on Winkler foundations based on meshless layerwise theory

Free vibration and bending of laminated plates on Winkler foundations are investigated based on a combination of generalized layerwise theory (LW) for higher-order shear deformation theory (HSDT) and the meshless radial point interpolation method (RPIM). The shape function of the meshless radial interpolation method is used to discretize the weak form control equations derived from the principle of virtual work. The shape functions have the properties of Kronecker functions. The slope components associated with the first-order derivatives of the transverse displacements are defined as approximate variables. The transverse shear stresses at each layer interface are continuous. A comparative study with literature solutions verified the numerical stability and applicability of the combined generalized LW-HSDT and meshless RPIM methods. The effect of Winkler modulus parameters on the vibration performance and bending performance of the laminated plate was investigated. The results of bending, vibration frequencies and modes of the laminated plates on different Winkler foundations are presented. It is found that initially, the Winker foundation had a significant effect on the frequency magnitude and bending deformation, but little influence on vibration mode. After the influence of the foundation on the frequency enters the stable period, the influence of foundation on the mode shape will be significant. With the relaxation of the boundary conditions, the vibration mode will still be affected when the foundation parameters are small. Highlights It is for the first time that the laminated plates on Winker foundations were investigated using a combination of higher-order layerwise theory and meshless radial interpolation method. In the proposed model, the slope component related to the first-order derivative of the transverse displacement is defined as an approximate variable and does not require the boundary to be treated by the isogeometric method. The transverse shear stresses at the interface of each layer are continuous for the bending analysis of laminated panels. The meshless results reveal the influence of the Winker foundation parameters on the vibration characteristics of the laminated plate

Predicting bone remodeling using a mechano-biological mathematical model combined with a natural neighbor meshless method

Bone tissue perceives certain mechanical stimuli, adapting its structure to a loading scenario through bone remodeling. Thanks to this ability, bone's functions of support and protection are assured. The mechanical response of bone is computationally reproduced in this work with a new bone remodeling model. Through a mechanobiological approach, the model includes two types of bone cells (i.e. osteoclasts and osteoblasts), describing its functioning and interaction with new sets of parameters. Mechanical stimulation induces a response by bone cells that ultimately is translated into variations of bone's apparent density and other mechanical properties by applying a material phenomenological law. The new algorithm here proposed is for the first time combined with an estabilished meshless method (Natural Neighbor Radial Point Interpolation Method), taking advantage of its features for biomedical applications. As a preliminary study, a two-dimensional bone patch is analyzed. A first study is performed to find the best parameters of NNRPIM for this type of simulation. Then, starting from a uniform bone distribution, a well-defined loading regime is imposed to the bone patch. Results revealed the expected bone adaption, attaining an optimized trabecular structure that reflects the orientation of the applied loads.

Mesh Free Radial Point Interpolation Based Displacement Recovery Techniques for Elastic Finite Element Analysis

The study develops the displacement error recovery method in a mesh free environment for the finite element solution employing the radial point interpolation (RPI) technique. The RPI technique uses the radial basis functions (RBF), along with polynomials basis functions to interpolate the displacement fields in a node patch and recovers the error in displacement field. The global and local errors are quantified in both energy and L2 norms from the post-processed displacement field. The RPI technique considers multi-quadrics/gaussian/thin plate splint RBF in combination with linear basis function for displacement error recovery analysis. The elastic plate examples are analyzed to demonstrate the error convergence and effectivity of the RPI displacement recovery procedures employing mesh free and mesh dependent patches. The performance of a RPI-based error estimators is also compared with the mesh dependent least square based error estimator. The triangular and quadrilateral elements are used for the discretization of plates domains. It is verified that RBF with their shape parameters, choice of elements, and errors norms influence considerably on the RPI-based displacement error recovery of finite element solution. The numerical results show that the mesh free RPI-based displacement recovery technique is more effective and achieve target accuracy in adaptive analysis with the smaller number of elements as compared to mesh dependent RPI and mesh dependent least square. It is also concluded that proposed mesh free recovery technique may prove to be most suitable for error recovery and adaptive analysis of problems dealing with large domain changes and domain discontinuities.

On applicability of MQ-RPIM and MLPG meshless methods with 3D extended-enriched base functions for estimation of mode I stress intensity factor and fatigue crack growth in cyclic tensile and bending load of an un-notched and notched shaft

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

Optimizing a meshless method for the simulation of the extrusion of non-Newtonian materials

In this work, the Radial Point Interpolation Method (RPIM), a meshless method, is used to simulate the extrusion of non-Newtonian materials. This is a key phase in the Fused Filament Fabrication (FFF), a 3D printing process based on the extrusion of semi-molten thermoplastic filaments. Using the flow formulation (following a Lagrangian approach) and considering the materials as non-Newtonian fluids, the velocity and pressure fields are computed. The developed numerical tool takes advantage of the easy remeshing procedure associated with meshless methods.The performance of the algorithm is optimized through several parametric studies by varying the nodal meshes, integration meshes, solution methods, and the number of nodes within the influence-domains of the meshless formulation. Thus, this work addresses the accuracy and stability of the present numerical approach and provides guidelines and optimal parameters to accurately simulate the process.Due to the similarity of its behaviour, the numerical examples obtained from the literature for the extrusion of aluminium (following a power-law constitutive relationship) are used to validate this new approach. In the end, this work applies, for the first time in the literature, an advanced discretization technique to the modelling and numerical simulation of the extrusion process, enhancing the state-of-the-art concerning meshless methods and the simulation of the extrusion process for the future improvement of forming processes and/or extrusion-based additive manufacturing processes like the FFF.

A meshless artificial viscosity method for wet-dry moving interfaces problems of shallow water flow

In this paper, an improved meshless artificial viscosity (AV) method is proposed and combined with the local radial point interpolation method (LRPIM) and the MacCormack method to establish a newly-developed meshless numerical model, which can efficiently and accurately solve wet-dry moving interfaces problems of shallow water flow. As an important model describing shallow water flow, the two-dimensional (2D) shallow water equations (SWEs) are a hyperbolic system of 1st-order nonlinear partial differential equations which have a characteristic of strong gradient. The LRPIM and MacCormack method are adopted to discretize the 2D SWEs spatially and temporally, respectively. Meanwhile, the method of selecting the different shape of support domain in the LRPIM is employed for properly cooperating with the MacCormack method to accurately capture the direction of wave propagation. For eliminating the non-physical oscillation at the discontinuity of shallow water flow, the improved meshless AV form is first presented. Four challenging numerical examples were selected to verify the proposed model by comparing with the other solutions and experimental observations. The results indicate that the proposed method with improved meshless AV can accurately capture the shock waves and efficiently simulate the wet-dry front changes.