Radial Point Interpolation Research Articles

Inverse heat conduction problem with a nonlinear source term by a local strong form of meshless technique based on radial point interpolation method

Inverse heat conduction problem with a nonlinear source term by a local strong form of meshless technique based on radial point interpolation method

Simulation of mode II fracture propagation in adhesive joints using a meshless technique

Adhesive bonding is extensively used by commanding industries such as automotive and aircraft sectors. Nevertheless, due to the intricate mechanical behaviour of adhesively bonded joints, especially when crack propagation occurs at the adhesive layer, improvement of new numerical techniques to simulate this bonding approach is currently under development. In this work, a recent crack propagation algorithm based on a meshless technique is implemented to analyse mode II fracture propagation in adhesively bonded joints. After the problem domain is discretized with an independent set of field nodes, a background numerical integration mesh is constructed. The crack tip advancement is then simulated by iteratively rearranging the field nodes and integration cells at the crack tip. Radial point interpolation (RPI) functions are constructed using the concept of influence domains, allowing to obtain sparse and stable global matrices. To assess the suitability of the proposed method, End-notched flexure (ENF) specimens were manufactured and tested. The numerical simulation justly reproduces the experimental data provided and, thus, the numerical model can be applied to mode-II shear loadings and industrial applications for design purposes.

Buckling analysis of functionally graded plates in thermal environment by using a meshfree method

In material, the pre-stress state caused by temperature variation of the working environment is one of the main reasons which cause buckling in structures. This paper aims to analyse the critical temperature rise to cause buckling in functionally graded material (FGM) under different boundary conditions. Reissner-Mindlin plate theory is employed in this paper with the radial point interpolation method (RPIM). The RPIM has the advantage compared to other meshfree methods in the satisfaction of the Kronecker delta property which allows directly imposing boundary conditions on scattered nodes. Factors affecting to thermal buckling phenomenon are investigated. The accuracy of solutions will be shown by comparing them to reliable results in previous studies

A centroid-enriched strain-smoothed radial point interpolation method for nearly incompressible elastoplastic problems in solid mechanics

While edge-based (2D) and face-based (3D) smoothed radial point interpolation methods (ES-RPIM and FS-RPIM) have demonstrated excellent performance in solid mechanics, the volumetric locking problem limits their applications in nearly incompressible problems. This paper develops a centroid-enriched edge-based/face-based smoothed RPIM (CE-ES-RPIM and CE-FS-RPIM) to address the volumetric locking problem in nearly incompressible problems. The proposed method comprises the following elements: (1) a centroid-enriched scheme is creatively used and implemented into the traditional ES-RPIM and FS-RPIM framework, in which the centroids of triangular mesh are added as spatial discretization nodes; (2) the strain-smoothing technique is performed on triangle (2D) or tetrahedron (3D) mesh to maintain the linear exactness of the radial point interpolation method; and (3) the combination of the centroid-enriched scheme and strain smoothing technique makes the ratio of the number of displacement equations to the number of incompressible constraints reach the optimal value of 2, effectively eliminating volumetric locking. The accuracy, effectiveness and efficiency of the proposed method are validated on several 2D and 3D nearly incompressible benchmark examples. All results demonstrate that the proposed method can provide accurate numerical results under an incompressible limitation and provide better performance than conventional quadratic element methods.

Strength prediction of a single lap joint under impact using meshless methods

The use of adhesive joints in structures subjected to dynamic loads, such as wind turbines and cars, makes it important to study them under those conditions. Numerical models are an integral part of that. Commonly the Finite Element Method (FEM) is used, but meshless methods can be an interesting alternative. These models do not require elements, and as such they can model complex geometries more easily. The current work aims at performing a first study on adhesive joints under impact using a meshless method, the Radial Point Interpolation Method (RPIM). Since the strength prediction of adhesive joints is also an important field and because the commonly used Cohesive Zone Models (CZM) have some limitations, like the use of special cohesive elements, this work also aims to expand the use of the ISSF criterion to impact conditions. The results show that the RPIM can be used in this type of problem without numerical difficulties, and the ISSF gives acceptable strength predictions, with errors between 30.5% and 13.5%.Adhesive bonding is a joining technique that offers some advantages, when compared to other common joining techniques, like bolting or riveting. One of those advantages is that adhesive joints are generally lighter than the alternatives, which is very important in the search for more efficient modes of transportation since lighter vehicles consume less energy. Given the interest in the use of adhesive joints, it is important to study their behaviour under different conditions. Currently, the static behaviour of adhesive joints is very well documented, with many research works dedicated to it. However, the number of publications on their dynamic behaviour is still scarce, with only a few works dedicated to fatigue, impact and free vibrations. Additionally, the use of meshless methods to study adhesive joints is also currently mostly limited to static analysis, and even in that case it is still very incipient. Therefore, this work aims at extending the use of meshless methods to the dynamic analysis of adhesive joints, to help in the advancement of both fields.

A meshfree model of hard-magnetic soft materials

Hard-magnetic soft materials comprising a soft matrix embedded with hard-magnetic particles can exhibit large deformations under external magnetic fields, which have attracted much interest due to their non-contact activation, flexible programmability, and rapid response in various applications such as biomedical devices, soft robotics and flexible electronics. Precise predictions of large deformations of hard-magnetic soft materials would be a key for relevant applications. Given that the rational designs in the programmable shape morphing of ferromagnetic structures requires high precision, here we develop a meshfree model based on the radial point interpolation method to quantitatively predict large deformations of hard-magnetic soft materials. We apply the stabilized conforming nodal integration instead of direct nodal integration to eliminate the influence of zero-energy mode, and the central difference scheme is implemented for the resolution procedure. We explore the bending of a hard-magnetic beam and snap buckling of a bistable hard-magnetic beam under external magnetic stimuli. Uniform and non-uniform discretizations are compared to show the insensitivity of the radial point interpolation method to nodal mesh. To demonstrate the versatile programmability of our model in the design of magnetic robotics, we further simulate complex locomotion (crawling, walking and rolling) of hard-magnetic soft robots under external magnetic excitation. The results could be used to guide rational designs of ferromagnetic structures and soft continuum robots.

RPIM-RITSS Method for Large Deformation Analysis Using ABAQUS

In the standard Lagrangian finite-element framework, the “remeshing and interpolation technique with the small strain” (RITSS) method has been applied increasingly in geotechnical engineering. However, the mapping algorithm of the RITSS method suffers from either the onerous division of unique element, such as the modified unique element divisional method, or frequent mapping of the stress and other information between integration points and nodes, resulting in the loss of precision, such as superconvergent patch recovery (SPR) and recovery by equilibration of patches. In this study, within the original RITSS, the radial point interpolation method (RPIM) mapping algorithm is implemented to improve accuracy and convenience. The RPIM-RITSS method is integrated into the ABAQUS (6.14) commercial software package. With a master Python script, the whole analysis process is performed automatically and continuously without any intervention from the user. The efficiency and robustness of the RPIM-RITSS method are verified through three numerical examples: penetration of the strip footing, the pullout and the rotation of the anchor plate, and deep penetration of the T-Bar. Consequently, the RPIM-RITSS method is an alternative, powerful, and easily extensible way to tackle large deformation problems in geotechnical engineering.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

Variable RPIM and HOCM coupling for non-linear buckling and post-buckling analysis of transverse FG sandwich beams

This paper presents an analytical-based approach to analyze non-linear buckling and post-buckling of transverse functionally graded sandwich beams (TFGSB). The proposed approach combines a high order continuation method (HOCM) and a variable radial point interpolation method (RPIM). First of all, the strong form of the beam buckling governing equations are established in the framework of first order shear deformation theory (FSDT) by taking into account the non-linear tangential term. Then, the spatial approximation of the unknown variables and their partial derivatives using the RPIM with variable shape parameters is the first step of this approach. The RPIM shape parameter is considered to change according to Franke formula, modified Franke formula or reciprocal Franke formula. The non-linear resulting system of discrete equations is linearized by using the Taylor series expansion, whilst the continuation method is employed to obtain the whole solution. To demonstrate the effectiveness of the proposed approach, numerical examples on non-linear buckling and post-buckling analysis of TFGSB are presented. These numerical tests are carried out to define the optimal RPIM with variable shape parameters and to investigate the sensitivity of the accuracy regarding boundary conditions, effects of power-law index, Young’s modulus, skin-core-skin thickness and the span-depth ratios. The results obtained through the proposed approach are compared with those computed by finite element method (FEM) and the available data in the literature.

An inhomogeneous stabilized node-based smoothed radial point interpolation method for the multi-physics coupling responses of functionally graded magneto-electro-elastic structures

In this paper, an inhomogeneous stabilized node-based smoothed radial point interpolation method (ISNS-RPIM) is developed for analyzing functionally graded magneto-electro-elastic (FGMEE) structures in hygrothermal environment. The change of material gradient follows the exponential function distribution. Combining with the gradient smoothing technique (GST) and the coupling constitutive relationship, and considering the hygrothermo-magneto-electro-elastic (HMEE) coupling effects of FGMEE material, the equations for ISNS-RPIM are deduced. The generalized displacement of the FGMEE structures is obtained, and the error in total energy norm, CPU time and efficiency of ISNS-RPIM, NS-RPIM and FEM are researched. Numerical examples are provided to explore the influence of the following parameters in FGMEE structures including exponential factor, hygrothermal loading and empirical constants, and demonstrate the correctness, high effectivity and insensitivity to mesh distortion of ISNS-RPIM. The present method has shown great potential in solving practical complex problems.

Total knee arthroplasty coronal alignment and tibial base stress-a new numerical evaluation.

Total knee arthroplasty (TKA) is one of the most frequently performed orthopedic procedures. The correct positioning and alignment of the components significantly affects prosthesis survival. Considering the current controversy regarding the target of postoperative alignment of TKA, this study evaluated the tension at tibial component interface using two numerical methods. The stress of the prosthesis/bone interface of the proximal tibial component was evaluated using two numerical methods: the finite element method (FEM) and the new meshless method: natural neighbor radial point interpolation method (NNRPIM). The construction of the model was based on Zimmers NexGen LPS-Flex Mobile® prosthesis and simulated the forces by using a free-body diagram. Tibiofemoral mechanical axis (TFMA) for which a higher number of nodes are under optimal mechanical tension is between 1° valgus 2° varus. For values outside the interval, there are regions under the tibial plate at risk of bone absorption. At the extremities of the tibial plate of the prosthesis, both medial and lateral, independent of the alignment, are under a low stress. In all nodes evaluated for all TFMA, the values of the effective stresses were higher in the NNRPIM when compared with the FEM. Through this study, we can corroborate that the optimal postoperative alignment is within the values that are currently considered of 0 ± 3° varus. It was verified that the meshless methods obtain smoother and more conservative results, which may make them safer when transposed to the clinical practice.

Investigating nonlinear moving load responses of FG-GPLRC skew plates using meshfree radial point interpolation method

Investigating nonlinear moving load responses of FG-GPLRC skew plates using meshfree radial point interpolation method

Free vibration parametric study of a single lap joint using the Radial Point Interpolation Method

Free vibration parametric study of a single lap joint using the Radial Point Interpolation Method

Highly efficient mesh-free approach to simulate the non-linear bending analysis of FG porous beams and sandwich beams with FG face sheets

The main objective of the present work is to propose a coupling of the Radial Point Interpolation Method (RPIM) with variable shape parameter to High Order Continuation Method (HOCM) for analyzing the geometrically non-linear behavior of porous Functionally Graded (FG) isotropic and sandwich Timoshenko beams by taking into account the non-linear shear deformation. The FG and FG Sandwich Beams (FGSB) effective material properties are assumed to vary continuously in the thickness direction according to a power-law index. The porosity is modeled as the stiffness reduction criteria and included in the mixture rule. The strong form of the non-linear equations governing porous FG and FGSB beams are obtained using the principle of virtual displacements in the framework of First-Order Shear Deformation Theory (FOSDT). The HOCM is used to compute the solution of the non-linear problem by a transformation to recursive succession of linear problems using Taylor series expansion. The continuation technique is used to obtain the complete solution path in a step by step manner. The effectiveness and accuracy of the proposed coupling is brought out through numerical examples which concern porous FG isotropic and FG sandwich beams. The optimal parameters have been defined according to the stability analysis of the proposed solver, and then used to investigate the effect of boundary conditions, showing the effect of the power-law index, effect of geometric imperfections, and face sheet-core-face sheet thickness ratios on deflections. The validation is done by comparing the obtained results to those available in the literature and/or those computed using Finite Element Method (FEM).

Radial Point Interpolation-Based Error Recovery Estimates for Finite Element Solutions of Incompressible Elastic Problems

Error estimation and adaptive applications help to control the discretization errors in finite element analysis. The study implements the radial point interpolation (RPI)-based error-recovery approaches in finite element analysis. The displacement/pressure-based mixed approach is used in finite element formulation. The RPI approach considers the radial basis functions (RBF) and polynomials basis functions together to interpolate the finite element solutions, i.e., displacement over influence zones to recover the solution errors. The energy norm is used to represent global and local errors. The reliability and effectiveness of RPI-based error-recovery approaches are assessed by adaptive analysis of incompressibility elastic problems including the problem with singularity. The quadrilateral meshes are used for discretization of problem domains. For adaptive improvement of mesh, the square of error equally distributed technique is employed. The computational outcome for solution errors, i.e., error distribution and convergence rate, are obtained for RPI technique-based error-recovery approach employing different radial basis functions (multi quadratic, thin-plate splint), RBF shape parameters, different shapes of influence zones (circular, rectangular) and conventional patches. The error convergence in the original FEM solution, in FEM solution considering influence-zone-based RPI recovery with MQ RBF, conventional patch-based RPI recovery with MQ RBF and conventional patch LS-based error recovery are found as (0.97772, 2.03291, 1.97929 and 1.6740), respectively, for four-node quadrilateral discretization of problem, while for nine-node quadrilateral discretization, the error convergence is (1.99607, 3.53087, 4.26621 and 2.54955), respectively. The study concludes that the adaptive analysis, using error-recovery estimates-based RPI approach, provides results with excellent accuracy and reliability.

On Interpolative Meshless Analysis of Orthotropic Elasticity

As one possible alternative to the finite element method, the interpolation characteristic is a key property that meshless shape functions aspire to. Meanwhile, the interpolation meshless method can directly impose essential boundary conditions, which is undoubtedly an advantage over other meshless methods. In this paper, the establishment, implementation, and horizontal comparison of interpolative meshless analyses of orthotropic elasticity were studied. In addition, the radial point interpolation method, the improved interpolative element-free Galerkin method and the interpolative element-free Galerkin method based on the non-singular weight function were applied to solve orthotropic beams and ring problems. Meanwhile, the direct method is used to apply the displacement boundary conditions for orthotropic elastic problems. Finally, a detailed convergence study of the numerical parameters and horizontal comparison of numerical accuracy and efficiency were carried out. The results indicate that the three kinds of interpolative meshless methods showed good numerical accuracy in modelling orthotropic elastic problems, and the accuracy of the radial point interpolation method is the highest.

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.

The Meshfree Radial Point Interpolation Method (RPIM) for Wave Propagation Dynamics in Non-Homogeneous Media

This work presents a novel simulation approach to couple the meshfree radial point interpolation method (RPIM) with the implicit direct time integration method for the transient analysis of wave propagation dynamics in non-homogeneous media. In this approach, the RPIM is adopted for the discretization of the overall space domain, while the discretization of the time domain is completed by employing the efficient Bathe time stepping scheme. The dispersion analysis demonstrates that, in wave analysis, the amount of numerical dispersion error resulting from the discretization in the space domain can be suppressed at a very low level when the employed nodal support domain of the interpolation function is adequately large. Meanwhile, it is also mathematically shown that the amount of numerical error resulting from the time domain discretization is actually a monotonically decreasing function of the non-dimensional time domain discretization interval. Consequently, the present simulation approach is capable of effectively handling the transient analysis of wave propagation dynamics in non-homogeneous media, and the disparate waves with different speeds can be solved concurrently with very high computation accuracy. This numerical feature makes the present simulation approach more suitable for complicated wave analysis than the traditional finite element approach because the waves with disparate speeds always cannot be concurrently solved accurately. Several numerical tests are given to check the performance of the present simulation approach for the analysis of wave propagation dynamics in non-homogeneous media.

Crack Analysis by Using An Extended Meshfree Method and Cartesian Transformation Method without Creating Subdomains

This paper investigates the fracture behavior of plates with through-thickness crack by using the extended concept of the Radial Point Interpolation Method (RPIM). The attractiveness of the RPIM shape functions is the satisfaction of the Kronecker delta property providing direct imposition of essential boundary conditions. In the extended concept, the jump in deflection and rotation fields caused by crack, also the stress singularity near the crack tip are described by adding enriched functions to the interpolation equation. Particularly, Heaviside function and asymptotic enriched function. For numerical integration, the Cartesian Transformation Method (CTM) is employed. No integration background cell is required in CTM, this technique transforms a domain integral into a boundary integral and a 1D integral. For analysis of discontinuous problems, in this study, the distribution of integration points is manipulated to avoid the discontinuity caused by crack segmentation. Therefore, no subdomains are required, unlike other reference CTM studies. To achieve that, a virtual boundary is introduced that represents the discontinuity such as holes or cracks. This also matches the concept of the extended approach that no explicit discontinuity exists in the geometry, instead, the discontinuity is modelled by mathematics equation. The Stress Intensity Factors (SIFs) of the cracked plate are evaluated through the interaction integral technique. The efficiency of the proposed method is illustrated through various numerical examples. The accuracy of the obtained results are compared with other available numerical solutions and analytical solutions.

Analysis of cracked Reissner-Mindlin plate using an extended meshfree method

An extended meshfree method for analyzing cracked plates based on Reissner-Mindlin theory is presented in this paper. Among a variety of meshfree formulations, the radial point interpolation method (RPIM) is chosen in this study due to the satisfaction of the Kronecker delta property. The essential boundary conditions, therefore, are easily imposed in the RPIM. The shape function derived from RPIM is employed to interpolate the field variables. An extended RPIM formulation is used to model the crack segment without explicitly defining it in the discretized domain. The discontinuity due to the crack is defined by extrinsic enriched functions, particularly, the jump in the displacement field on two sides of the crack is modelled by the Heaviside function, and the stress singularity near the crack tip is described by the asymptotic enriched function. In this study, the stress resultant intensity factors (SRIFs) are evaluated through the interaction integral approach. The obtained SRIFs are shown in the paper through many numerical examples for comparison purposes. The trending variation of SRIFs is also observed from the numerical results. It can be remarked that the SRIFs depend on many factors: the number of cracks, crack orientation, load type and boundary conditions. The numerical examples show the accuracy of the present approach. The obtained results are compared with analytical solutions and other numerical methods.