Radial Point Interpolation Research Articles

A meshfree-based topology optimization approach without calculation of sensitivity

This paper presents a novel topology optimization approach without calculation of sensitivity for the minimum compliance problems, based on the meshfree Radial Point Interpolation Method (RPIM). Relying on the algorithm of Proportional Topology Optimization (PTO), material is distributed using only information of the objective function (which is the elastic strain energy). Material properties are interpolated by the well-known Solid Isotropic Material with Penalization (SIMP) technique; however the pseudo density (design variables) are not defined on the element center as usually encountered in finite element-based approaches, but on integration points. Since no element exists in meshfree analysis, this would be a natural choice. More importantly, the number of integration points is in general larger than that of elements or that of nodes, resulting in higher resolution of the density field. The feasibility and efficiency of the proposed approach are demonstrated and discussed via several numerical examples.

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.

Improved finite element method based on radial point interpolation method (RPIM) for electro-thermal coupling

Efficient electro-thermal coupling calculation is of great significance to the monitoring and structural design of power equipment. Finite element method (FEM) is one of the powerful tools of electro-thermal analysis and is widely used in various fields. However, the finite element method itself cannot solve the first-order discontinuity, which causes the deterioration of accuracy. For example, in the electro-thermal coupling calculation, Joule heat is often used as the only heat source of the thermal field, and the calculation of Joule heat depends on the first-order variable field strength of the electric field. The calculation error of is further amplified in the thermal field solution, and the performance is more obvious in strong electromagnetic environment. Therefore, this paper proposes a finite element (RFEM) electrothermal coupling method based on the radial point interpolation method (RPIM). The idea of this method is to use the first-order continuous RPIM method to interpolate the variables in the overall solution domain to obtain continuous and smooth solution results. Examples show that RFEM has a higher solution efficiency than FEM, and it is better at capturing local high gradient changes in an electromagnetic environment.

Analysis of the interior acoustic wave propagation problems using the modified radial point interpolation method (M-RPIM)

Although the radial point interpolation method (RPIM), which is a typical meshless numerical technique, usually behaves much better than the conventional FEM in addressing the numerical dispersion error issue for acoustic computation and more accurate solutions can generally be yielded with the identical node distributions, the related numerical error still can not be completely removed and further improvements are still required. In this work, we proposed a modified RPIM (M-RPIM) to enhance the abilities of the original RPIM in suppressing the numerical dispersion error. In this M-RPIM, a simple and straightforward scheme is employed to ensure that the integrands in performing the numerical integration are continuously differentiable, while in the original RPIM the quadrature cells usually do not align with the shape function supports and then results in the discontinuously differentiable numerical approximation in the quadrature cells, hence considerable numerical integration error can be generated. Since the discontinuously differentiable entities in the system stiffness matrix can be completely avoided in the present M-RPIM, it is found that the numerical dispersion can be markedly suppressed and more accurate numerical solutions can be yielded than the original RPIM in solving acoustic problems. It should be pointed out that the numerical treatments and conclusions in this work are also applicable for most of other meshfree approximations which are similar to the RPIM.

Coupling of NURBS and Meshfree RPIM for plane stress of web with openings

This study presents a novel approach for the coupling of the non-uniform rational B-spline (NURBS) with the radial basis functions (RBF) under the framework of the meshfree radial point interpolation method (RPIM). In this method, denoted as N-RPIM, the NURBS basis functions are employed to model the geometry of the problem domain exactly, whilst the RPIM is used for field variables interpolation. The formulation is applied to the web of a cellular beam under plane stress condition to highlight the advantages. The results show that the method works efficiently, and provides favourable performance against FEM. The solution of the former converges faster, thus highlighting the potential of N-RPIM for future development.

Surrogate-based structural optimization design of large-scale rectangular pressure vessel using radial point interpolation method

Large-scale rectangular pressure vessel possesses wide application due to volume efficiency and the convenience of transporting and installing. Despite there being increasing emphasis on weight reduction, the optimization design method for large-scale rectangular pressure vessel is still under development due to complicated geometry and engineering constraints that make conventional optimization approaches incapable. Therefore, a surrogate-based optimization design method is proposed in this work to resolve the aforementioned difficulties. The proposed technique adopts radial point interpolation method to construct surrogate model, which is validated with classical numerical examples and a rectangular pressure vessel, i.e., the test section of a large-scale water tunnel. Fair convergence and precision have been revealed. Then, the surrogate-based optimization framework is established on the basis of the surrogate model, involving optimization scheme, sampling range and constraints, sampling method, standard optimization model and method. Subsequently, the investigated rectangular pressure vessel is optimized using the developed method. The optimized structure maintains good machinability and reduces weight by 10.06% in comparison to the results of preliminary routine structural design, proving that the developed technique possesses good capacity for the optimization design of large-scale rectangular pressure vessels.

Fracture Toughness Determination on an SCB Specimen by Meshless Methods

This work investigates fracture characteristics of a marble semi-circular bend (SCB) specimen with a pre-defined crack under a compressive loading condition. It aims at evaluating how the fracture toughness can be affected by the crack and span length variation. Numerically, the model is solved using meshless methods, extended to the linear elastic fracture mechanics (LEFM), resorting to radial point interpolation method (RPIM) and its natural neighbor versions (NNRPIMv1 and NNRPIMv2). Alternatively, to validate the meshless method results, the problem is resolved following the finite element method (FEM) model based on the standard 2D constant strain triangle elements. As a result, fracture toughness and the critical strain energy release rate are characterized following the testing method on the cracked straight through semi-circular bend specimen (CSTSCB). A comparison is drawn amongst the theoretical, meshless methods and FEM results to evaluate the capability of advanced numerical methods. Encouraging results have been accomplished leading to validate the supporting numerical methodologies.

Phase-field modelling of brittle fracture with Smoothed Radial Point Interpolation Methods

The phase-field modelling of brittle fracture is an efficient tool for the modelling of complex cracks, since it is capable to predict crack initiation and branching, as well as the merging of cracks. While there are a number of papers in the literature devoted to the application of different meshfree methods to the phase-field modelling of fracture, this paper is, to the authors knowledge, the first application of the methods of the Smoothed Point Interpolation Methods (SPIM) family. The smoothing operation, that is used to approximate the displacements gradient in standard SPIM strategies, is here extended to the additional variable of a phase-field model. A number of numerical simulations, performed considering different kind of smoothing domains and support nodes selection strategies, illustrates the characteristics of the proposed approach.

An evolutionary structural optimization algorithm for the analysis of light automobile parts using a meshless technique

PurposeThe purpose of the present work is to develop the combination of the radial point interpolation method (RPIM) with a bi-directional evolutionary structural optimization (BESO) algorithm and extend it to the analysis of benchmark examples and automotive industry applications.Design/methodology/approachA BESO algorithm capable of detecting variations in the stress level of the structure, and thus respond to those changes by reinforcing the solid material, is developed. A meshless method, the RPIM, is used to iteratively obtain the stress field. The obtained optimal topologies are then recreated and numerically analyzed to validate its proficiency.FindingsThe proposed algorithm is capable to achieve accurate benchmark material distributions. Implementation of the BESO algorithm combined with the RPIM allows developing innovative lightweight automotive structures with increased performance.Research limitations/implicationsComputational cost of the topology optimization analysis is constrained by the nodal density discretizing the problem domain. Topology optimization solutions are usually complex, whereby they must be fabricated by additive manufacturing techniques and experimentally validated.Practical implicationsIn automotive industry, fuel consumption, carbon emissions and vehicle performance is influenced by structure weight. Therefore, implementation of accurate topology optimization algorithms to design lightweight (cost-efficient) components will be an asset in industry.Originality/valueMeshless methods applications in topology optimization are not as widespread as the finite element method (FEM). Therefore, this work enhances the state-of-the-art of meshless methods and demonstrates the suitability of the RPIM to solve topology optimization problems. Innovative lightweight automotive structures are developed using the proposed methodology.

Errata: “Stabilized node‐based smoothed radial point interpolation method for micromechanical analysis of the magneto‐electro‐elastic structures in thermal environment”

Errata: “Stabilized node‐based smoothed radial point interpolation method for micromechanical analysis of the magneto‐electro‐elastic structures in thermal environment”

Analysis of stress singularity in adhesive joints using meshless methods

Recent years saw a rise in the application of bonding techniques in the engineering industry. This fact is due to the various advantages of this technique when compared to traditional joining methods, such as riveting or bolting. The growth of bonding methods demands faster and more powerful tools to analyze the behavior of products. For that reason, adhesive joints have been the subject of intensive investigation over the past few years. Recently, a fracture mechanics based approach emerged with great potential to evaluate joint behavior, called Intesity of Singular Stress Fields (ISSF), similar to the Stress Intensity Factor (SIF) concept. However, it allows the study of multi-material corners and does not require an initial crack. This approach was not yet tested with meshless methods. The present work intends to fill this gap, resorting to the Radial Point Interpolation Method (RPIM). With this purpose, adhesive joints with four different overlap lengths (LO) bonded with a brittle adhesive were studied. The interface corner's stresses were also evaluated. The predicted strengths were compared with the experimental data to assess the accuracy of the applied methods. In conclusion, the ISSF criterion proved to be applicable to meshless methods, namely the RPIM.

The Radial Point Interpolation Method combined with a bi-directional structural topology optimization algorithm

The Radial Point Interpolation Method combined with a bi-directional structural topology optimization algorithm

Topology optimization of light structures using the natural neighbour radial point interpolation method

Topology optimization of light structures using the natural neighbour radial point interpolation method

A Meshfree Approach for Simulating Fluid Flow in Fractured Porous Media

This paper presents a meshfree approach for simulating fluid flow in fractured porous media using a novel fracture (FM) mapping approach. Fracture mapping is a continuum-based approach which simulates the flow interaction between the porous matrix and existing fractures via a transfer function. The approach simulates fluid flow through both the matrix and the fractures and is well suited to models containing sparsely spaced, unconnected fractures. The presented approach determines the fluid flow using approximating functions constructed employing the radial point interpolation method (RPIM) meshfree formulation which uses radial basis functions (RBFs) augmented with polynomials. As part of this meshfree scheme a nodal integration procedure has been implemented thereby removing the need for background integration cells that are usually required for meshfree schemes that rely on Gaussian integration. Numerical test results illustrate the methods ability to adequately describe the fluid pressure fields within a fractured porous domain

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%.

FREE VIBRATION ANALYSIS OF CRACKED KIRCHHOFF-LOVE PLATE USING THE EXTENDED RADIAL POINT INTERPOLATION METHOD

The Kirchhoff-Love plate theory is appropriate for analysing thin plate structures. In a simple form, only one degree of freedom (per node) is needed to describe the behaviour of the plate, thus saving the computational cost. Besides, the analysis of cracked structures is important because it is related to the lifetime of the structures. Therefore, this paper uses the extended radial point interpolation method (XRPIM) to investigate the free vibration of the Kirchhoff-Love plate. The XRPIM is based on RPIM so the requirement for calculating the second-order derivative in Kirchhoff-Love theory is easily done. The numerical results from this study are compared with other researchers to verify the accuracy of the method.

Development of an adaptive CTM–RPIM method for modeling large deformation problems in geotechnical engineering

In this paper, a meshfree method called adaptive CTM–RPIM is developed to model geotechnical problems with large deformation. The developed adaptive CTM–RPIM is a combination of the Cartesian transformation method (CTM), the radial point interpolation method (RPIM) and the alpha shape method. To reduce the requirement for meshes, the CTM is adopted to transform domain integrals into line integrals, and the RPIM is applied to construct interpolation functions. The alpha shape method, which is capable of capturing severe boundary evolution due to large deformations, is then introduced into the CTM–RPIM to form the adaptive CTM–RPIM. The accuracy of CTM–RPIM is first verified by considering a cantilever beam under small deformation, where the influence of key parameters on the simulation results is explored. Afterward, the ability of the adaptive CTM–RPIM to handle large deformation problems is demonstrated by simulating cantilever beams with large deformations for which analytical solutions are available. Finally, the ability of the proposed method to model the geotechnical large deformations is illustrated from both quasi-static and dynamic aspects, where a slope failure problem and a footing bearing capacity problem are modeled to evaluate the stability of geotechnical structures; and a 2-D soil collapse experiment using small aluminum bars is simulated to show the method capability in describing the soil flows. These benchmark examples demonstrate that the adaptive CTM–RPIM is a numerical method with broad application prospects for modeling large deformation problems in geotechnical engineering.

Topology optimization using a natural neighbour meshless method combined with a bi-directional evolutionary algorithm

Due to recent developments in the additive manufacturing industry, topology optimization is nowadays a powerful computational tool that allows to design feasible lightweight components. Although the Finite Element Method (FEM) is the most applied discretization technique, meshless methods are currently established as accurate numerical methods with relevant advantages in several engineering fields. Nonetheless, the state-of-the-art of meshless methods in topology optimization is still scarce. This work develops the combination of a bi-direction structural optimization (BESO) algorithm with the Natural Neighbour Radial Point Interpolation Method (NNRPIM), a meshless method combining the natural neighbours geometric concept with the RPI shape functions. First, several benchmark examples are solved to evaluate the algorithm capability under several algorithm parameters. The proposed methodology is then implemented to design new automotive lightweight components. The results from the numerical applications demonstrate that the NNRPIM is a solid technique to be incorporated in optimization algorithms. Additionally, innovative automotive industry designs for additive manufacturing can be obtained using the presented approach.

A bio-inspired remodelling algorithm combined with a natural neighbour meshless method to obtain optimized functionally graded materials

Recent developments suggest the use of triply periodic minimal surfaces (such as the gyroid) as a possibility for bone tissue scaffold. Moreover, through functional gradients of cellular structures, the mechanical properties can be edited and enhanced to achieve the most efficient results. One of the main concerns when designing bone scaffold is avoiding stress shielding, which occurs when the Young’s modulus of the implant is higher than the Young’s modulus of the bone it is replacing. If so, bone decay occurs in the surrounding tissue. While the literature possesses some approaches exploring functional gradients of material density, there are no solutions based on bone tissue phenomenological laws. Thus, the gyroid infill obtained with PLA (E=3145 MPa) was characterized with mechanical tests, namely tensile and compression, and the obtained model was implemented in a bone remodelling algorithm. Using the natural neighbour radial point interpolation method (NNRPIM) it was found that similar bone density distributions were obtained for the gyroid infill and for bone tissue when subject to the same boundary conditions. Finally, the gyroid mechanical behaviour was extrapolated to other materials and it was concluded that similar properties can be obtained for bone tissue and titanium alloy (E=110GPa) scaffold.

Direct left-ventricular global longitudinal strain (GLS) computation with a fully convolutional network

This study's purpose was to develop a direct MRI-based, deep-learning semantic segmentation approach for computing global longitudinal strain (GLS), a known metric for detecting left-ventricular (LV) cardiotoxicity in breast cancer. Displacement Encoding with Stimulated Echoes cardiac image phases acquired from 30 breast cancer patients and 30 healthy females were unwrapped via a DeepLabV3 + fully convolutional network (FCN). Myocardial strains were directly computed from the unwrapped phases with the Radial Point Interpolation Method. FCN-unwrapped phases of a phantom’s rotating gel were validated against quality-guided phase-unwrapping (QGPU) and robust transport of intensity equation (RTIE) phase-unwrapping. FCN performance on unwrapping human LV data was measured with F1 and Dice scores versus QGPU ground-truth. The reliability of FCN-based strains was assessed against RTIE-based strains with Cronbach's alpha (C-α) intraclass correlation coefficient. Mean squared error (MSE) of unwrapping the phantom experiment data at 0 dB signal-to-noise ratio were 1.6, 2.7 and 6.1 with FCN, QGPU and RTIE techniques. Human data classification accuracies were F1 = 0.95 (Dice = 0.96) with FCN and F1 = 0.94 (Dice = 0.95) with RTIE. GLS results from FCN and RTIE were −16 ± 3% vs. −16 ± 3% (C-α = 0.9) for patients and −20 ± 3% vs. −20 ± 3% (C-α = 0.9) for healthy subjects. The low MSE from the phantom validation demonstrates accuracy of phase-unwrapping with the FCN and comparable human subject results versus RTIE demonstrate GLS analysis accuracy. A deep-learning methodology for phase-unwrapping in medical images and GLS computation was developed and validated in a heterogeneous cohort.